diff --git "a/PdFAT4oBgHgl3EQfzh4f/content/tmp_files/load_file.txt" "b/PdFAT4oBgHgl3EQfzh4f/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/PdFAT4oBgHgl3EQfzh4f/content/tmp_files/load_file.txt" @@ -0,0 +1,1947 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf,len=1946 +page_content='Geometric approach for non pharmaceutical interventions in epidemiology Laurent Evain ∗, Jean-Jacques Loeb Abstract: Various non pharmaceutical interventions have been settled to minimise the burden of the COVID-19 outbreak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We build a framework to analyse the dynamics of non pharmaceutical interventions, to distinguish between mitigations measures leading to objective scientific improvements and mitigations based on both political and scientific considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We analyse two possible strategies within this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Namely, we consider mitigations driven by the limited resources of the health system and mitigations where a constant set of measures is applied at different moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We describe the optimal interventions for these scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Our approach involves sir differential systems, it is qualitative and geometrical rather than computational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Along with the analysis of these scenarios, we collect several results that may be useful on their own, in particular on the ground when the variables are not known in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' 1 Introduction In the pandemic situation, governments have settled policies, based on socio-economic appreciations, field studies, and modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The toolbox for the crisis management involved mitigations policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Numerical simu- lations suggest that these mitigations measures change the final share r∞ of infected people in the population, sometimes markedly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A possible roadmap for a scientific programme to select non pharmaceutical interventions could be as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Discuss the choice of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Which models are realistic ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Find the mathematical optimisations for the chosen model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Find the counterpart of the optimisation in real life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Determine possible concrete mitigations that ap- proach the targeted mathematical optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Social analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Analyse the public acceptance of the mitigation, the economic impact, and the indirect costs on the population health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the present article, we choose a variant of the sir-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We concentrate on the second item of this roadmap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Our target is to obtain theoretical results, and in particular proved qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Our results shed light and give an understanding of the the numerical simulations of the spread of a disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We have a particular interest in qualitative results independent of the input parameters, as these results could be more robust on field with little known parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The theoretical background, the constructions, and the mathematical results are exposed in full generality in the supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The main text is dedicated to a larger audience, it explains, contextualises and illustrates the results with simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Our work started with the article [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We were puzzled by some simulations exhibiting final fractions infected depending on the choice of the intensity of preventive measures, via a constant α in the next generation matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Several remarks were formulated, suggesting qualitative explanations of the phenomena observed on the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance, “preventive measures were not imposed from the start and were lifted before the epidemic was over” or “lifting restrictions gradually [ can prevent ] overshoot “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' So it was implicit but clear from the article [1] that an adequate scheduling was required to minimise the burden of the epidemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' However, we could not identify the adequate scheduling in precise mathematical terms, nor could we identify ∗laurent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='evain@univ-angers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='fr 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='08698v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='PE] 20 Jan 2023 the assumptions required to obtain the qualitative behaviour of the examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Thus our goal was to clarify and give a general picture of what could mean an “optimal scheduling” of the preventive measures for a pandemic outbreak described with a sir-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Our interest for qualitative results was reinforced by contradictory results in the literature with respect to the relevance of an early and strong set of mitigation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Whereas overshoot was pointed out as a risk in [1] and implicitly in [8], other other sources [7], [12] advocated for early or strong mitigations to save lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In contrast to papers where strategies are analysed at fixed dates [7], sometimes to wait for some new drugs or vaccines [8], we are concerned with the very long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We try to minimise the mortality after an infinitely long time using only non pharmaceutical interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In other words, this paper considers non pharmaceutical interventions as an active medical tool to minimise the burden of the epidemic in the long run rather than as a tool to postpone the mortality till some new drug comes on the market.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Here is a summary of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Even in the absence of medicine to wait for, finite time interventions may be considered to minimise the burden of an epidemic because of the dynamics involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The situation is analogous to a bike on a sloping road : it is not possible to stop before the low point using a finite time breaking, but breaking is nevertheless useful to avoid moving far beyond the low point due to inertia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In symbols, let r∞ be the ratio of finally infected people and let R0 be the classical reproduction number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A well scheduled finite time intervention can drive r∞ close to rherd = 1 − 1 R0 , whereas no intervention often leads to r∞ >> rherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The role of the dynamics is more important when R0 has a medium value (R0 ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='5) where nearly 30% of the population may avoid the disease thanks to a suitable finite time intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' There is a fundamental qualitative difference between finite time interventions and infinite time interven- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Infinite time interventions can lead to an arbitrarily small ratio r∞ of infected people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In contrast, finite time interventions result in situations where the inequality r∞ > rherd always holds, so that r∞ close to rherd is the best possible value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Planning is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Examples show that awkward planning lead to mitigations that are long, costly, with little effect on r∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The analogy with bikes on a sloping road makes sense again : breaking hard far from the low point hardly has an impact on the inertia and on the distance covered after the low point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In contrast, the analogy with a bike on a flat road is badly suggestive, as it encourages early intensive mitigations with poor results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This leads to a problem of control theory : what are the mitigations which minimise the effort on the population for a fixed result ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We build a scientific framework to distinguish between the political level and the scientific level in the decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Obviously, the mitigations have a social cost that require a personal subjective appreciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A rational decision process includes political considerations to aggregate the divergent wishes of the citizen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Nevertheless, some conclusions may be true independently of the subjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Considering two possible choices A and B, there is a scientific ground to prefer mitigation A to mitigation B if there are simultaneously fewer infected people and fewer restrictions on the population when A is chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In contrast, a political trade-off is necessary when a middle ground between infections and constraints has to be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We keep these notions informal in the main text, but the concepts are rigorously defined in the supplementary material in terms of cost functions and diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the following, when we say that a choice A is better than a choice B, we always refer to the scientific meaning : choice A leads both to less restrictions and to less infected people than choice B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We analyse the scenario where a same constant intervention is applied one or several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In this context, the two problems of minimising the duration of the constraint for a fixed burden or minimising the burden for a fixed duration are equivalent, and there is no compromise between the duration of the intervention and the number of infections to be found : both are minimised simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We thus approach the problem with a fixed predefined duration and we determine the adequate planning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This scenario includes for instance the comparison between two strategies, where the first strategy promotes a change every Monday for seven weeks, whereas the second strategy promotes the same change a whole week one month after the starting point of the epidemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' More generally, we compare strategies with a same type of intervention, and the same total duration, and we analyse the optimal planning of the mitigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We show that in this scenario, splitting the mitigations through several short periods is never optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The mitigation minimising the number of finally infected people has always exactly one unique long lasting mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Our model does not support the idea sometimes expressed on the media to plan 2 a strong mitigation as soon as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' It is quite the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Intensity and timing have to be tuned in a consistent balanced manner : an earlier mitigation must be lighter than an intervention that starts later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Early and strong mitigations are not balanced and yield to poor results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The timing of an optimal planning is understood : it boils down to “as soon as possible” if the herd immunity threshold has been crossed, and around the herd immunity threshold otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Among the possible consistent choices of timing and intensity, the case of a late intensive mitigation, starting when the herd immunity threshold is crossed, has a special interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The corresponding strategy can be implemented on the ground using measurements in wastewater as in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' It may be more easily planned than alternative optimal strategies since estimating R0 is not necessary, thus bypassing the difficulty of its estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In a second scenario, we analyse the case where the health system would be saturated in the absence of interventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Non pharmaceutical interventions are used to maintain the health system below its maximal load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We compare several mitigation strategies with different loads, possibly different from 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance, the mitigation may start when the health system is filled at 90%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This second scenario is divided in two sub-scenarios : – 2a) : The mitigation is shaped so that the health system stays filled at 90% and it is relaxed when the herd immunity threshold is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The relaxation occurs when the epidemic naturally decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' – 2b) : This scenario starts like scenario 2a), but the mitigation is relaxed sooner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A rebound of the epidemic occurs and the limit of 90% is exceeded after relaxing, but the total load remains below 100% forever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In other words, the relaxation is launched as soon as returning to normal does not overload the health system in the future despite of a rebound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Instead of considering arbitrarily a load of 90% as in the above example, we address the problem of determining the optimal load between 0% and 100% for the health system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' What are the optimal loads for scenarios 2a) and 2b) ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' First, we show that in scenario 2a), there is no scientific answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Political trade-offs are unavoidable : a higher load abuts to fewer infected people at the price of more constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In scenario 2a), the duration of the mitigation tends quickly to infinity when the considered load goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Simulations show that the time of mitigation is often very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' It is thus natural to consider scenario 2b) which comes with fewer constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We show that, maybe surprisingly, all the strategies considered in the scenario 2b) have the same number of finally infected people, independently of the chosen load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Consequently, a load A is preferable than a load B if and only if the corresponding strategy leads to fewer constraints on the population for the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Small loads are ine��cient in scenario 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A minimal load is necessary, otherwise the policy is surpassed by other better planned strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the simulations considered, this minimal load of the health system to reach before launching the intervention is large : more than 80% of the maximal load of the health system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This may be viewed as an other incarnation in the context of possibly overloaded health systems of the slogan “A strong and early mitigation is inefficient”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' When this minimal load is reached, the question of still enlarging the launching load becomes a political one, it is not a scientific question any more : for the same number of finally infected people, a higher launching load requires an effort for the population which lasts longer, but the maximal effort is lower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The above scenarios are built upon a more general analysis which carries several results useful on their own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We give a focus on a function h which plays a role similar to energy in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In mechanics, a falling object undergoes important damages on the ground if it was thrown with a high kinetic or potential energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Similarly in our model, if a mitigation is relaxed with a high value of h, this will lead to many infected people and fatal cases because of the implied dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Since h is an indirect measure of the finally infected people, a mitigation that lowers h has a positive impact on the final burden.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In contrast, if h is only slightly changed by a mitigation, the mitigation hardly has an impact on the final burden : The mitigation is a temporal shift rather than an amelioration, the infections will happen later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A sensible objective for the public policy is thus to lower h using interventions of short duration, hence the importance of the derivative dh dt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The computation shows that dh dt is proportional to the ratio i(t) of infected people for a fixed intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This leads to the very important qualitative result that for a fixed level of constraint, the interventions are more efficient if they occur when many people are infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The same phenomena has been observed using numerical simulations in [8], however for a different model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This suggests that our qualitative result for the sir model could be extended and could provide an explanation of the numerical observations in other contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The variation of the energy function h shows that an early intensive intervention acts on mortality similarly to a free loan, with a positive effect on the short 3 term but not on the long term after the loan is repaid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This explains the apparent contradiction between the authors studying the aftermath of the intervention at a fixed date with a positive result [7], whereas [1],[8] have an opposite conclusion with a further time horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Context and limitations of the findings Mitigation strategies may be promoted through coercive laws, or they may be exposed to the public as mere recommendations with no obligation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Citizen may have more choice when the intervention occurs ( new option for remote work or for a day off for instance) or fewer choices due to restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Our paper is agnostic about the implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We consider a model which modifies the coefficients of the sir systems when people modulate their interactions, but we make no assumption on the social tools used to modify these coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Several questions have not been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We do not discuss the economical or global health impact nor the social acceptance in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Field testing, studies with animal models and experiments to support the numerical and theoretical results would be welcome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Insect pathogens have been used to test equations because of their tractability [2, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Minimising the spread of the epidemic in plants while minimising the intervention is a natural question, and we have not explored how our results could be enlightening for agriculture or other epidemiological contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' These questions are out of the scope of the article, and they belong to a field of work where we have no expertise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' However, we believe that these are important questions to be discussed by qualified researchers in these fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We discuss now the choice of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Some modellings rely on simple models, whereas other modellings require many interacting parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Both have their pros and cons, depending on the objectives, quantitative or qualitative, and on the quality of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' As a rule of thumb, simple models with few variables allow qualitative explanations, they have a lower sensitivity to the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' When high quality data and model is available, complex models with more variables lead to more precise predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Qualitative interpretation of the changes implied by the modifications of the input constants is difficult for complex models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Both approaches are complementary rather than opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Since our goal was to identify the qualitative phenomena that drive and circumscribe the computations, a variant of the simple and robust sir-model was a sound choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the variant considered, a coefficient that was constant in the original sir model varies with the mitigation policies implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Although the sir model is suitable for qualitative analysis, our modelling carries the simplifications and limitations attached to this model : people are infected only once, deaths are not considered, the population is supposed to be geographically homogeneous, all individuals are equally susceptible, viruses undergo no mutations, to cite a few limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' There are slight variations of the sir-model for which we can carry our results or follow an approach along the same lines ( remark 19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' However, there are other models to experiment to approach reality, and they may be quite different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance, “on the experimental side, Dwyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' (1997) measured nonlinear relationships between transmission and densities of susceptible hosts, implying that the bilinear term in the classical susceptible-infected-recovered (SIR) model may not be appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' “ [3, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Many variations are possible, and probably many are necessary depending on the problem under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Closed formulas for a differential system are an exception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Most variations from the SIR-system will lead to models which are not computable with closed formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' While a large part of our analysis is geometrical and clarifies the involved phenomenons, other arguments still depend on the closed formulas of the sir system and will not apply to non computable models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Generalisations based on a better understanding of the geometrical architecture could be explored, to prove our results for a larger class of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Many parameters are not constant, their value evolves with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance, we do not know how many people will be vaccinated, how often, and the efficiency of the vaccines on the variants to come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In this rapidly changing environment, we hope that our qualitative results may be useful, in particular those independent of the numerical data in input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' 2 Main text Mitigation as an active tool In a pandemic context, mitigation policies are usually understood as a tool to give deciders and physicians time for dealing with the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Delaying the epidemic gives time for researchers to find new remedies and gives time to setup the logistics to vaccinate people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Our approach in this paper is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We consider 4 mitigations as a tool to minimise the number of finally infected people, even in the absence of change in the remedies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The goal of this article is to develop this point of view of mitigation strategies as an active tool for driving the epidemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In this section, we exhibit simulations to expose the problematic involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A mitigation that improves the final situation without any new drug is illustrated in the next figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The red drawing is the trajectory of the pandemic without mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The green drawing is the same pandemic, where a mitigation is applied from week 22 to week 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' During the four weeks of mitigation, the reproduction number R0 has been reduced from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 in the absence of mitigation to R1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Thanks to the mitigation, the share r∞ of finally infected people dropped from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='88 to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The evolution of the pandemic is shown in the (s, r)-plane, more precisely in the triangle s > 0, r ≥ 0, s+r ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Here s and r are the share of susceptible people and the share of removed people in the population, with the standard notations of the sir model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The share i of infected people is implicit since i = 1 − r − s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The point M(t) = (s(t), r(t)) that represents the epidemic at time t starts at t = 0 with M(0) at the bottom right (M(0) ≃ (1, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' As t increases, M(t) moves to the left (s decreases) and the limit point M∞ = (s∞, r∞) is on the diagonal r + s = 1 (i∞ = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Note that by construction r∞ is the share of people that have been infected at some time t during the epidemic, with 0 < t < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The limit point M∞ is lower for the green curve than for the red curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This means means that the mitigation has been efficient, it has lowered r∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Figure 1: A simple mitigation What is a good mitigation strategy ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The problem can be thought in analogy with the braking of a bike on a slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' All braking strategies are not equivalent : a rider does not apply a constant braking force on downslopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' There are moments where braking is useless, and other moments where braking is necessary to take the turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Technically, this is a question of control theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The riders on the Tour de France unconsciously apply some control theory to find the optimal timing and intensity for the braking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The same phenomenon appears in an epidemiological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A mitigation measure is the analogue of a braking action to slow down the epidemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Since mitigations limit the possibilities for citizen, one wants to minimise the duration and intensity of these mitigations for the same braking performance, or to minimise the number of infected people for a fixed level of braking effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A na¨ıve approach would suppose that control theory is straightforward, that no planning is required, and that the value of r∞ depends only on how strict and how long the mitigations are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This is not the case : adequate scheduling is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A lighter and shorter mitigation may outperform a harsher mitigation thanks to a better scheduling, as illustrated with the following example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The mitigations for the green trajectory are shorter-lived than those for the red trajectory ( 4 weeks vs 6 weeks ), are less intensive ( reproduction during the mitigation R1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 vs R1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='7), yet the share r∞ of finally infected people is lower for the green curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A public policy should recommend the green trajectory over the red trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' 3 Orders of magnitude What is the difference between a perfect mitigation and no mitigation ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' How many saved lives and how many people may avoid infection using an efficient mitigation ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We give some estimates in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In a free environment without mitigation, the share r∞,free of finally infected people in a sir model is the unique positive solution of the equation ln(1 − r∞,free) + R0r∞,free = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' 5 Onepointperday,6oweeks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 R0=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4, mu=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 Mitigation 2 weeks (w22-w26)with R1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 Ratio ofremovedpeople 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 RatioofsusceptiblepeopleFigure 2: Two mitigations with different plannings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' ( Theorem 16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' With an optimal mitigation, the ratio of finally infected people is about r∞,opt = 1 − 1 R0 ( Theorem 12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The share ∆r of the population avoiding an infection with an optimal mitigation is thus ∆r = r∞,free − r∞,opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The share ∆deaths of lives saved is ∆death = (r∞,free − r∞,opt)IFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' where IFR denotes the infection fatality rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' R0 IFR in % r∞,free r∞,opt ∆r in % ∆death in % 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='05 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='7 14.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='035 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='115 Figure 3: Orders of magnitude Some estimates of these quantities are given in the table of figure 3 for different values of R0 and IFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' When R0 is slightly above 2 : up to 30% of the general population avoids an infection with an optimal mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This high figure means that, from a mathematical point of view, mitigation as a tool to reduce the burden of an epidemic makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' There is no guarantee however that this strategy is possible in real life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We remarked quite surprisingly that the importance of mitigation increases for medium R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The natural guess that mitigation should play a more important role for R0 large is wrong : for large R0, the difference ∆r = (r∞,free − r∞,opt) is small because 0 < r∞,opt < r∞,free < 1 and r∞,opt = 1 − 1 R0 is close to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This phenomenon is illustrated in figure 4 showing ∆r as a function of R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' 6 64 weeksfrom the starting point, R0=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4,mu=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 Mitigation 6 weeks (w20-w26)with R1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='7 Mitigation 4 weeks (w24-w28) with R1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 people Ratio of removed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 Ratio ofsusceptiblepeopleFigure 4: ∆r as a function of R0 Note that the figures in table 3 for ∆r are only upper-bounds for the objectives of the public policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' People often react naturally when a brother or a friend is ill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In a growing epidemic, the population limits its interactions by itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance, it was remarked in [10] that “most of the decline in mobility in [the] sample happened before the introduction of lockdowns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Failing to account for voluntary changes in behaviour leads to substantially over-estimated effects of non pharmaceutical interventions ”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We call ∆nat the share of the population avoiding an infection due to this reaction of the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The share ∆public policy of the population protected against infection by the public policies is in addition to ∆nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The population that avoids an infection thanks to mitigation is ∆nat + ∆public policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If the combined effect of natural reaction and public policies is optimal, ∆nat + ∆public policy = ∆r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Without the optimality hypothesis, ∆public policy ≤ ∆r − ∆nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In our views, the role of the political institutions is to coordinate and amplify if necessary the natural movement of the public to maximise ∆public policy, in accordance to the history and social context of the country.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Planning and scheduling the information to the public is an ingredient of this maximisation, as media campaigns can increase the level of compliance to safe attitudes at the appropriate time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Prophylactic measures are dependent on the mode of transmission of the disease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Promoting the correct prophylactic attitudes at the key moments could also be an objective of the public policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Studying scenarios Our next goal is to give a qualitative analysis of the involved phenomena during the mitigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' To this aim, we tried to go beyond numerical simulations, because their qualitative interpretation is often difficult, and because extracting a general behaviour from examples depending on the input data is in our opinion a slippery methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Rather, we consider two scenarios, and we prove qualitative results that apply independently of the numerical data in input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The first scenario has fixed mitigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The second scenario considers situations where the health system is overwhelmed in the absence of mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the first scenario, we consider a fixed action : for instance a larger part of the population works remotely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This leads to a mitigation with reproduction number R1 which is lower than the initial reproduction number R0 > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Both cases R1 > 1 and R1 < 1 make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We fix a total duration d for the mitigation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This whole mitigation is split in k + 1 shorter uninterrupted mitigations of duration d0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' , dk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The total duration of the mitigation is the sum of the duration of the uninterrupted mitigations, hence � di = d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In this context, the question is : what is the optimal value for the number k and when should these k + 1 mitigations occur ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Our answer is that the optimal strategy satisfies k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The optimal strategy is an uninterrupted mitigation, which is not split in several shorter mitigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This is a general fact independent of the values of R0 and R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' It is illustrated in the left part of figure 5 : a 60 days mitigation lowers the reproduction from R0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 to R1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 ( µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This 60 days mitigation is not split (blue curve, partially hidden by the orange curve), split in two shorter mitigations of 30 days (orange curve), or five shorter mitigations of 12 days (green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The pause between two successive mitigations is three times the duration of the mitigations, namely 90 days ( orange curve) and 36 days ( green curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For each of these three scenarios, the start time for the first mitigation has been chosen to give the smallest possible r∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If the start time of the first mitigation is changed, the value of r∞ depending on the start time (in weeks ) is given in the right part of the figure for each scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance, for two mitigations of 30 days distant of 90 days, the orange curve on the right part of figure 5 says that in our simulations, the optimal start time for the first mitigation is a little more than 8 weeks after the first few imported cases, and yields to r∞ between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='775 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The corresponding epidemic is drawn on the left side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Moreover, in the unsplit case k = 0, we have an estimate for the optimal moment for the intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' 7 30 infection 25 th voiding ofpeople 20 15 Delta re Shar 10 1 2 1 4 1 1 7 8 1 3 6 ReproductionnumberRoFigure 5: Splitting mitigations Optimality requires that the equality s(t) = sherd = 1 R0 occurs at a moment t during the mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This is also a general result independent of the numerical values of R0 and R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If the mitigation ends at a time t with s(t) < sherd, it is too early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If the mitigation starts with s(t) > sherd, it is too late ( Theorem 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the example of the figure, with R0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6, we have sherd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The optimal strategy illustrated by the blue curve of the figure has numerical results consistent with the general theorem : the vertical line s = sherd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='38 is crossed during the mitigation period, represented by the most vertical part of the left blue curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Using the analogy with a bike on a slope, and considering that the low point on the road is the analogue of the herd ratio, our theorem says that the optimal breaking occurs in one step, and we should be breaking in a zone which encompasses the bottom of the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Ending the breaking before the lowest point of the valley would be inefficient, starting the breaking after the low point would be inefficient too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the scenario considered so far, the constraint was the same for all strategies ( same restrictions, same total duration for the mitigation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the next scenario, it will be necessary to compare heterogeneous constraints, which are not constant in time nor have the same duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The comparison is easy in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance, it is less constraining to have a soft mitigation lasting two days than a harsh mitigation lasting five days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For some other cases, the comparison is not possible : There is no natural choice between a long soft constraint, and a short harsh constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Finally, there are comparisons which are possible but may require a moment to reflect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' As an example, a strategy S1 imposing a partial set of constraints for 2 days and a total set of constraints for 3 days is less constrained than an alternative strategy S2 imposing the same partial mitigation for 1 day, and the same total mitigation for 5 days ( reason: S2 is obtained from S1 by replacing one day of partial constraints with two days of complete constraints).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This approach to order the constraints on the above examples can be formalised and written rigorously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the appendix, we formalise and extend these ideas to compare the constraints of two different strategies S1 and S2 to the case of mitigations whose constraints vary continuously with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' To keep things simple yet intuitive, there are fewer constraints for the strategy S1 than for the strategy S2 if every person prefers S1 than S2, whatever her personal cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This occurs in particular when S2 is obtained from S1 by replacing mitigations of duration d with harsher mitigations of duration d′ > d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the second scenario, we consider a risk of overwhelmed health systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We fix an upper bound itrig for the maximal ratio of infected people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance, itrig = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='1 means that when 10% of people are infected simultaneously, a mitigation is triggered to prevent the increase of the number of hospitalised patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The level of mitigation is then settled so that the ratio of infected people stays exactly at i = itrig = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Equivalently, there are as many people getting sick as people being cured by unit of time during the mitigation period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The process is illustrated in the next figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The epidemic starts with very few infected persons, and the initial propagation is drawn in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Then, the level of infection which triggers the mitigation measures is reached and the mitigation period with constant i corresponds to the blue segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' After some time (13 weeks in the example of the figure, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='33 ), the level of infected people naturally decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The mitigations are relaxed as they are not necessary any more ( in orange on the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A similar strategy can be set up with an other value for itrig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The question in this context is the determination of the optimal itrig ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We show that there is no scientific answer to this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A political trade-off is necessary, as minimising constraints and minimising the ratio r∞ of finally infected people are opposite objectives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A small itrig corre- sponds to a smaller r∞ at the price of more constraints (Theorem 40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Figure 7 illustrates this fact with two 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='900 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 Ratio of Finally Infected People removedpeople 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='800 Ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 Ratio of susceptible people Start TimeFigure 6: Fixing the maximum level of infected people Figure 7: Comparing two different values of itrig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' mitigation levels itrig = 5% and itrig = 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The red curve represents an epidemic without mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The two possible mitigations are drawn in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The ratio r∞ is smaller for itrig = 5%, but the mitigation lasts far longer than for itrig = 15% ( 35 weeks vs 8 weeks with R0 = 3, µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Moreover, the initial reproduction number of the mitigation ( defined as the reproduction number when the mitigation has just started) is smaller for itrig = 5% ( 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='08 vs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='32 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This means that harsher and longer constraints are necessary for a small itrig value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' As itrig tends to 0, the constraint duration tends rapidly to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This is illustrated in figure 8 where the duration of the mitigation in weeks is plotted in red, and the initial reproduction number defined above is plotted in green ( scaled by a factor 100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Both are plotted as functions of itrig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Figure 8: Duration and intensity of mitigation as functions of itrig when R0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' To minimise this long constraint when itrig is small, we consider a variant of this scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For this variant, when the ratio i of infected people reaches i = itrig, a mitigation is set up as above to preserve the health system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' But the mitigation is relaxed sooner in comparison to the previous scenario : as soon as it is possible to stop the mitigation without overwhelming the health system in the future, the mitigation is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance, suppose that the health system is totally full when i = ihosp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A mitigation is triggered when i = itrig = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='05, and it is maintained for a moment so that i(t) stays blocked at the constant value i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' When the mitigation is relaxed, the ratio i of infected people increases again from i = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='05, but it never exceeds ihosp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In other words, a rebound of the epidemic occurs but the health system remains not full and viable after the mitigation is over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This strategy Sitrig,ihosp is illustrated in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' It depends on the two constants itrig and ihosp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The constant ihosp depends on the health system of the country and is not changeable in the short term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In contrast, different values of itrig are possible and lead to different public policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 Ratioofremovedpeople 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 Ratio ofsusceptiblepeople1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 Ratio ofremoved people 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 Ratio ofsusceptiblepeopleDuration in Weeks,mu=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 1oo*InitialReproductionNumber 250 200 150 100 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='30 Value of i_(trig}Figure 9: A mitigation is settled, then relaxed with a possible rebound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The duration of the mitigation in the scenario Sitrig,ihosp is shown as a function of itrig in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We see Figure 10: Duration of the mitigation in weeks as a function of itrig for ihosp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='15, R0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' that the duration in weeks (represented by the blue curve) is shorter when a rebound is allowed, in comparison to the previous scenario without rebounds ( orange curve).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The difference is significant, thus the objective of lowering the time of mitigation by allowing a rebound is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Besides a shorter mitigation time, there are several differences that make the scenario with rebound quite different from the scenario without rebound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Figure 11: Duration of the mitigation in weeks as a function of itrig for ihosp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='15, R0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' First, when a final rebound is allowed, the duration is not a decreasing function of itrig any more, as illustrated by a zoom on the previous blue curve ( Figure 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The minimal duration of around 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 weeks for the mitigation is obtained for itrig = imin := 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Second, the variation of r∞ is different too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the scenario without rebound, an early intervention lowered the ratio r∞ at the price of a longer and harsher mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the scenario with rebound, a harsher or longer mitigation is not rewarded by a smaller r∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' All strategies Sitrig,ihosp have the same ratio r∞ of finally infected people independently of itrig for a fixed hospital capacity ihosp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In other words, the level itrig that triggers mitigations does not influence how many people will be finally ill or dead (Theorem 44 in the appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This surprising phenomenon is illustrated in figure 12 with ihosp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='15 and two different values of itrig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The mitigations in blue are triggered when itrig = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='05 and itrig = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='10 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' They are relaxed as soon as ihosp is never exceeded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The value of r∞ which is common to the two mitigations is the r-coordinate of the point at the end of the yellow curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' As a consequence, the mitigations with itrig < imin must be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Indeed, they are longer and harsher 10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='85 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='80 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='70 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='65 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='60 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='1401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 people Ratio of removed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 Ratio of susceptible people100 80 60 40 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='14Figure 12: Mitigations with ihosp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='15, itrig = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='05 or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' than the mitigation with itrig = imin and the supplementary constraint is not compensated by an amelioration of r∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We have excluded the cases itrig ∈]0, imin[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Let us now consider the remaining range itrig ≥ imin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In this range, all values of itrig may be considered and lead to non comparable mitigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' More precisely, if mitigations are triggered above the minimal load imin, then a higher load itrig gives a longer but less intensive mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Thus a political trade-off between intensity and duration of the mitigation is required to make the choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Some people may prefer a short intensive mitigation (itrig close to imin) while other people may prefer a long cool mitigation (itrig close to ihosp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The remarks formulated on the example are illustrations of general qualitative results proved in Theorem 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The theorem can be summarised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the variant with a rebound allowed, the choice of the level itrig which triggers the mitigation has no impact on the share r∞ of finally people infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The choice of itrig impacts only the subjective human or economical cost of the mitigation, but the direct burden of the disease remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' There exists a minimal load imin characterised by the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If itrig < imin, the strategy is to be rejected as the mitigation is unnecessarily long and harsh for the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' All the choices with itrig ∈ [imin, ihosp] are possible and correspond to different trade-offs between length and intensity : itrig closer to imin corresponds to a shorter and harsher mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the example with ihosp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='15, we have imin = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Since imin is close to ihosp, this means that the mitigation must be triggered when the health system is nearly full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Other simulations also give imin close to ihosp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' These simulations express the idea that, for the scenario with rebound within a sir-system, it is a wrong idea to anticipate much and to launch mitigation measures far before the saturation of the health system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Temporary versus definitive mitigations, and temporal shifts versus improvements The idea “the sooner the restrictions, the better” is often implicit or explicit in the debate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance in [6], several epidemiologists called for early mitigation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' It is not supported by the above simulations and the scenarios we studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' As this may be surprising for many readers, we precise in this section where the misunderstandings come from and the underlying phenomenons that explain this apparent paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In this article, we consider temporary mitigation policies : the time of intervention is finite and then people return to their normal life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The duration of the intervention may be long, but it is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Considering instead infinite time intervention can alter the assessment of a strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance, suppose that a starting epidemic is annihilated with a drastic mitigation launched at the very beginning when the first people are infected ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' then the epidemic never starts again if the mitigations go on forever and if normal life never returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' However, for finite time interventions, the mitigation measures eventually stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' When normal life starts again, the situation is similar to the situation before the mitigation measures, with a na¨ıve population and no immunisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The epidemic will rise again from the few remaining viruses or from the viruses imported from abroad ( see for instance the simulations by Ferguson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' [8, fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='3] where infections rise after the mitigations are relaxed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The problem has been postponed, rather than solved, by the finite time mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We consider the mortality in the long run whereas other papers in the literature consider the mortality at a precise date [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This may lead to conclusions which are apparently in contradiction, but the results of the two approaches turn out to be compatible once the paradox is understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Suppose that a first strategy with a rebound of cases after relaxation is settled more early than a second more efficient strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The first strategy appears to be preferable if the evaluation occurs before the rebound of cases or when the second strategy has not yet been launched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' But if the burden of the epidemic is looked at later, when the efficient late strategy has produced its effects, then the conclusion becomes opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This phenomenon explains why Sofonea et 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 people Ratio of removed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 Ratio ofsusceptiblepeopleal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' [7] find good estimates for early mitigations whereas our conclusions are inverse for the long term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' As an illustration, we revisit the example of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We draw the epidemic with the ratio of removed people r(t) as a function of t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' After 26 weeks, the red mitigation seems preferable, but in the long run, the opposite conclusion holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Figure 13: Two mitigations with different plannings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If only finite time interventions are allowed, and if assessment is done in the long term, shifting the epidemic with no improvement of the situation is not neutral, it is a waste of resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If some measures are politically sustainable for 3 months, and if one month is spent in a inefficient set of measures which postpones the problem, then only two months of mitigation policies are left to improve the situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This explains why in many simulations, early interventions are inefficient in the context of finite time interventions : They are temporal shifts rather than improvements, time is wasted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Energy h of the system To make more rigorous the distinction between temporal shift and improvements induced by an intervention, we introduce the energy h of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A mitigation that lowers h ameliorates the situation while a mitigation that lets h roughly unchanged acts as a temporal shift .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In formula, the energy of a point (s, r) is h(s, r) = R0 − 1 − ln(R0s) − R0r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The function h is positive for every possible (s, r) and minimum at point Pherd = (1/R0, 1 − 1/R0) = (sherd, rherd), where its value is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We denote by Ch the equienergy curve containing the points (s, r) with energy level h, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' h(s, r) = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' They are the red curves on figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the absence of mitigation, the energy of the point M(t) = (s(t), r(t)) is unchanged, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' the world M(t) moves along these red curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The herd point (sherd, rherd) is drawn in green on the figure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The blue curve on the figure is the trajectory of an epidemic Figure 14: Equienergy curves and a pandemic with 2 mitigations 12 64 weeksfrom the starting point, R0=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4,mu=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 Ratio of removed people 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 Mitigation 6 weeks (w20-w26)with R1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 Mitigation 4 weeks (w24-w28)with R1=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0 10 20 30 40 50 60 Time in weeks1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 Ratioof removedpeople 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 Ratio ofsusceptiblepeoplewhere two mitigations have been launched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' During the 2 mitigation periods, the epidemic crosses these level lines, dissipates energy, and the energy h in the final situation is smaller than the initial energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Thus h is analogous to energy in physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' It is constant when the system evolves freely (no mitigation), and it diminishes when breaking/mitigation dissipates energy from the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The geometry of the equienergy curves show that h is an indirect measurement of the total burden ( past and future, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' including the mortality to come) in the absence of further intervention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Two points on the same equienergy curve go to the same point at infinity if no mitigation measures are set any more : two situations S1 = S(s1, i1, r1) and S2 = S(s2, i2, r2) with the same value of h will give the same number r∞ of finally infected people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Moreover, in the absence of further mitigation, the higher the value of h, the more people will be infected : r∞ is an increasing function of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' It follows that the goal of the policy maker is to propose measures that lower h as much as possible before the mitigations are definitively relaxed, with the minimal level of constraint on the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Variations of h during mitigations How much h decreases during a mitigation is governed by the following differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Suppose the epidemic is modelled by a sir system with propagation number R0 > 1 in the absence of mitigation, and by a sir system with propagation number R1 < R0 when some restrictions are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' During the mitigation, h is submitted to the differential equation dh dt = (R1 − R0)i(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This equation is of particular qualitative importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Indeed, the goal is to lower h rapidly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For a short time dt, the variation of h is dh = (R1 − R0)i(t)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This means that for a fixed mitigation with number R1 and a fixed small duration dt, the decline of h is more important when i(t) is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' A short time intervention is more efficient when it is applied when the number of infected people i(t) is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This result is consistent and may be an explanation for the numerical observation of [8] for an other model :”the majority of the effect of such a [mitigation] strategy can be achieved by targeting interventions [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='] around the peak of the epidemic.” At the other extreme, if i(t) = 0, dh = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We recover the qualitative fact that mitigations applied when i(t) is very small postpone the problem with no improvement since h does not decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In particular, an intensive mitigation at the beginning of the epidemic is inefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' On the minimal r∞ Since the ratio of finally infected people r∞ is an increasing function of h after the last mitigation, and since h is minimal at the herd point, the inequality r∞ > rherd holds whatever the finite time mitigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In other words, finite time mitigation measures cannot be used to maintain the epidemic at level zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the long term, the minimal share of people that have been infected is at least rherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Graphically, the limit of the red curves in figure 14 is always located above the herd point Pherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This impossibility of a zero case strategy by mitigations, or more generally of strategies to reach r∞ ≤ rherd, is valid only for the finite time strategies considered in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Figure 15 compares a finite time and an infinite time strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' On the left part, an infinite time strategy is settled and the limit point is below the herd point, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' r∞ < rherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' On the middle part of the figure, the same mitigation is relaxed after 20 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' There is a rebound when relaxing occurs, and r∞ > rherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' On the right part, the mitigation is relaxed after 60 weeks, which makes nearly no difference with 20 weeks, apart from the delay in the 60 weeks case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The sporadic cases that remain active yield quite the same rebound of the epidemic in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This theoretical result is consistent with on the ground situations for COVID-19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Several countries first tried to develop a zero case strategy, and most of them finally desisted from this strategy [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Our analysis suggests a little more : For a given epidemic, it will be difficult if not impossible to maintain r < rherd in the long run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The inequality r∞ > rherd for any finite time strategy is an epidemiological analogue of the mechanical situation with a bike on a sloppy road.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If an infinite time breaking is possible, the bike can be stopped in the middle of the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In contrast, if the breaking time is finite, the breaks are eventually released, the bike will move to the low point of the road and beyond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In this sense, the herd line s = sherd is the analogue of the bottom of the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' No finite time breaking can stop the epidemic before it crosses the herd line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' 13 Figure 15: Finite and infinite mitigations Dynamics and Inertia of the system after the herd ratio The analogy with the bike makes it easier to understand the role of the herd immunity threshold, measured equivalently by one of two herd ratios sherd = 1 R0 or rherd = 1− 1 R0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the vaccine pre-epidemic context, there are no dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The herd ratio rherd is the proportion of people that need to receive a vaccine to nip in the bud any propagation of the epidemic, preventing its launching from imported cases or remanent cases in the population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In a situation with dynamics, when the epidemic has already started, the situation is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The epidemic will not stop instantly when the herd ratio is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In this dynamical context, rherd and sherd still make sense, but have a different interpretation : i(t) will decrease with time if s(t) < sherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In other words, sherd is the threshold that guarantees the fall of the number of infected people with no mitigation measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Since i(t) is proportional to the derivative dr dt , i(t) is a measure of speed when one tries to minimise the total quantity r∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The decrease of i(t) without mitigation after the rational ratio sherd is the epidemiological counterpart to the fact that a bike that reaches the bottom of the slope will slow down without breaking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' How far the epidemic will go beyond the herd line when mitigations are released is the analogue of the question of how far goes a bike after the bottom of the slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' It depends on the energy h of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If all the mitigations are released at a point (s, r) with energy h(s, r) = h0, the share r = r∞ of finally infected people at infinity satisfies the equation h(1 − r∞, r∞) = h0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Some comments in the media suggest that once the herd ratio s = sserd is reached, the situation is under control and needs no more supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Our models suggest quite the opposite !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Because of the inertia and of the dynamics, it may be necessary to control and slow down the epidemic after the herd ratio is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Moreover, in many examples, i(t) is maximal at the herd ratio s(t) = sherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This is true for instance when no mitigation is launched before the herd ratio is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In such situations, the equation dh dt = (R1 − R0)i(t) tells that the moment the herd threshold is reached co¨ıncides with maximum effectiveness of the mitigation measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Thus, not only are the interventions often still necessary when the herd ratio s = sherd is reached, but also, launching the mitigations measures around the herd immunity ratio is good practice according to the sir-model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Building mitigations with small r∞ What is the infimum possible r∞,opt for the number r∞ of finally infected people and what are the strategies to lead to this infimum ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We have seen in the previous section that r∞ > rherd for any finite time strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In this section, we explain that r∞,opt = rherd, which means that the difference between r∞ and rherd may be arbitrarily small for a well-designed strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We discuss the strategies that lead to this infimum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' the strategies with r∞ close to rherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The na¨ıve and falsely convincing argument that the more restrictive mitigations give the better results on r∞ is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If the mitigation is too intensive, there will be a large rebound in the epidemic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If the mitigation is too loose, the epidemic is not stopped enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' An adequate calibration of the mitigation is thus necessary 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 people people f removed people removed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 - removed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 Ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 atio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 Ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 Ratio of susceptible people Ratio of susceptible people Ratio of susceptible peopleto get an optimal result, not too intensive to avoid a large rebound, nor too loose to sufficiently dampen the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This is illustrated in figure 16 with 3 mitigations starting at the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The black curve is the epidemic when no mitigation occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The mitigation in blue is too loose and the trajectory goes beyond the herd point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The mitigation in green is suitable towards the herd point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The mitigation in red is too strict : a large rebound occurs after the mitigation is stopped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The optimal constant mitigation is understood : its Figure 16: three mitigations : too harsh, correct, and too loose reproduction number is R1 = ln(sR0)/(1 − 1 R0 − r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' (1) where we suppose that the world is in situation (s, r) with r < rherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Such a mitigation applied an infinite time would lead to the herd point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In practice, the strategy is applied a finite long enough time, the herd point is approached, and r∞ is close to rherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This is illustrated by the green curve in figure 16 which uses the above formula for R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' An other question is the timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Is it necessary to launch a strategy long before the herd immunity threshold to settle a strategy with r∞ ≃ rherd ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The answer is no from a theoretical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' As long as s(t) > sherd, it is not too to late to launch a strategy that leads to r∞ close to rherd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance a mitigation strategy with R1 as above works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If s(t) < sherd, the formula implies R1 < 0 which is an impossible physically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In the formula, R1 tends to zero when s tends to sherd, which is a rephrasing of the high intensity required for a late mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This means that on the ground, there are limitations besides theoretical limitations since R1 cannot be arbitrarily low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The two questions, intensity and timing, are correlated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' There is a large range of possibilities for the start date of an optimal mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The formula for R1 shows that the later the start date, the more vigorous the intervention must be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' But on the other hand, the later the intervention, the shorter the duration of the mitigation for the same result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We can use this correlation between timing and intensity to revisit and explain the initial example of figure 2 or the red mitigation in figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The early mitigation of figure 2 was inefficient because the intensity was not consistent with the starting point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The mitigation was too intensive for its earliness, leading to poor results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' To get order of magnitudes, we illustrate in table 17 possible consistent values for the intensity and duration of the mitigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We fix R0 = 3 and µ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' An epidemic rises, and a mitigation is launched when s = sstart with the above optimal formula for R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The mitigation is relaxed so that r∞ = l∗rherd, with l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='1 or l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Thus by construction, all scenarios have a burden dependent on l but not on the choice of sstart for a fixed l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The table reports the values of R1 and the duration of the mitigation for different values of sstart ∈]sherd = 1 3, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Remark that there is some flexibility in the choice of R1 for a fixed r∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' When R1 is chosen small in the set of possible values, there will be a rebound in the epidemic whereas a large R1 will yield a strategy with no rebound after the mitigation is relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' However, both strategies have the same r∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Thus the existence of a rebound is not an indicator of an overshoot and of a badly calibrated mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Indeterminacy of R0 and timing implications Many computations above rely on the estimate of the reproduction number R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance, the computation of rherd, the energy h, the reproduction number R1 of the optimal mitigation depend on R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' However the value 15 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 Ratio of removed people 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='0 Ratio of susceptiblepeoplel sstart r1 duration in weeks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='5 Figure 17: R1 and duration of the mitigation of R0 is not well known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Determining R0 is an important and difficult question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' See [10], for the methodology implemented in Our world in data, and the many references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In this section, we discuss the implications of this indeterminacy on the choice of a strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In particular, we exhibit a strategy implementable on the ground without knowing R0, thus bypassing the difficulty to estimate R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' There are many reasons that make it difficult to estimate R0 = β0 µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For a virus, β0, hence R0, depend on the variants that propagate, they evolve with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The number R0 to be used in the modelling can be different from the R0 that has been computed at the beginning of the epidemic because of the change in the variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This problem is amplified by the interaction between vaccines and variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If a vaccine has efficacy e with 0 < e < 1, and the share of vaccinated people is r, then the fraction of the population protected by the vaccine is re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' These people are to be placed in the removed compartment of the sir system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Thus some continuous exchanges between the s-compartment and the r-compartment occur along with the evolution of the variants and their interactions with vaccinated people, making the calibration of R0 difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' An other difficulty is that the sir model is valid only locally because of heterogeneity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For a large territory, one can choose a constant R0 suitable to aggregate the constants of each local area where homogeneity makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' However, the local areas evolve independently and the choice of R0 may lead to a coherent modelling only for a short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' An other heterogeneity was pointed out in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The more the individuals have contacts, the more they spread the virus and the sooner they are infected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Heavy spreaders are in average removed sooner, and R0 decreases with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Using the basic reproduction number R0 computed at the beginning of the epidemic is thus expected to yield overestimation of r∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We may formalise these remarks as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' There is a propagation number R0(t) depending on time, on the (unknown) distribution of heterogeneity, on the immunity evolving with vaccination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For a short enough period of time, all parameters can be seen as constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The time-varying R0(t) can thus be approximated by a constant and the sir model makes sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' There exists an optimal intervention which is as late and as strict as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' If s = sherd = 1 R0 in formula 1, we get R1 = 0 independently of the value of R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This corresponds to a harsh mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Moreover, in the absence of mitigation, s(t) = sherd is reached at the moment the number of infected people starts to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Thus there exists an optimal mitigation that starts when the ratio i(t) of people infected naturally decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In contrast to the estimation of R0 which is very difficult, the moment when i(t) starts to decrease is easier to monitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' It can be estimated with a detection of the virus in the sewage system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This has already done in France through the “R´eseau Obepine” [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' To sum up, the start time and the intensity of the mitigation should be consistent, and this consistency is difficult to achieve because of the indeterminacy of R0(t) which is a constant only locally in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' This suggests that the following policy independent of R0 could be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The public authorities organise the monitoring of the virus in the sewage system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' When i(t) decreases, it may be interpreted as “the herd immunity ratio is reached”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Then it is a good moment to launch the communication to the public to slow down the spreading and the dynamics with a vigorous mitigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We think that the feasibility of this proposal needs to be considered as it has several advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The mitigation time is shorter than for other intervention times, making the acceptance of the strategy easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The public confidence in the political decisions is preserved because there are fewer risks of contradictory decisions induced by bad estimates of R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' 16 4 Supplementary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='1 Description of the model In this section, we describe the sir-controlled model that we use throughout the article, which involves some non continuity considerations that need to be clearly formulated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We consider an epidemic starting at time t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Three functions s, i, r of the time t are considered : s(t) is the share of people not infected up to t, i(t) is the share of people infected at t and r(t) is the share of people who were infected before t, but are cured at t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' By construction, for every non negative t, s(t) + i(t) + r(t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We do not consider births in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The classical sir equations are: � � � � � s′ = −β0si i′ = −µi + β0si r′ = µi where µ is a strictly positive constant and depends on the epidemiological context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' It governs the speed at which infected people are removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' β0 is a strictly positive constant and β0 µ is the initial propagation number, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' the average number of infections originated from the first infected persons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We use the classical notation R0 = β0 µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The classical SIR-system is often presented using absolute numbers, whereas the above version considers ratios rather than sizes of populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' For instance, in [11], the Kermack-McKendrick SIR epidemic model is presented with the number I of infected people, and N the size of population, whereas we use the share i = I N instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We use the lowercase notation sir-system rather than the uppercase notation SIR-system as a reminder of this choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' When mitigation policies apply, β0 is not a constant any more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We consider a model where we replace the constant β0 with a function β = β(t) depending on the time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' When no mitigation strategy is set up, β(t) = β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' When mitigation strategies occur, 0 ≤ β(t) < β0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' The condition β(t) > β0 is mathematically possible, but corresponds to people gathering and transmitting the virus more than expected, thus is hardly realistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' On the other hand, µ is constant as before and is independent of the mitigation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Summing up, our model to include mitigation strategies is a derivation of the classical sir model where β0 is replaced by a non negative function β = β(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In mathematical terms : (∗) � � � � � s′ = −βsi i′ = −µi + βsi r′ = µi In the following, we will use the expression ”sir-controlled model” for this model, or sometimes only ”sir-model” for simplicity, assuming it is implicit and clear that β is not a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' When a mitigation is launched at time t, a discontinuity occurs for the function β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Thus we need to consider non continuous functions β(t) and we suppose only that β(t) is piecewise continuous on [0, +∞[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' In this non continuous context, we define a solution of a controlled sir-system as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' We suppose that there exists a subdivision a0 = 0 < a1 < · · · < ak−1 < ak = +∞ such that β is continuous on each ]aj, aj+1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' Moreover, we suppose that for j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' , k − 1}, the right limit β(a+ j ) := limt→aj,t>aj β(t) and for j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFAT4oBgHgl3EQfzh4f/content/2301.08698v1.pdf'} +page_content=' , k − 2}, the left limit β(a− j+1) := limt→aj+1,t