Abstract
In the framework of homogeneous susceptible-infected-recovered (SIR) models, we use a control theory approach to identify optimal pandemic mitigation strategies. We derive rather general conditions for reaching herd immunity while minimizing the costs incurred by the introduction of societal control measures (such as closing schools, social distancing, lockdowns, etc.), under the constraint that the infected fraction of the population does never exceed a certain maximum corresponding to public health system capacity. Optimality is derived and verified by variational and numerical methods for a number of model cost functions. The effects of immune response decay after recovery are taken into account and discussed in terms of the feasibility of strategies based on herd immunity.
Highlights
The recent outbreak of the illness COVID-19, caused by the SARS-CoV-2 virus, has resulted in a pandemic with unprecedented impact on societies all over the globe
We have shown that for a wide class of cost functions, in order to reach herd immunity without vaccination, it provides an optimal control strategy to keep the effective reproduction number, R, at unity during the majority of the duration of the pandemic
Deviations which depend upon the specific form of the cost function are limited to a narrow time window and can be considered negligible for practical purposes
Summary
The recent outbreak of the illness COVID-19, caused by the SARS-CoV-2 virus, has resulted in a pandemic with unprecedented impact on societies all over the globe. The task we address in this study is to limit, during the whole period of the pandemic, the current number of infected individuals such as to prevent the number of those needing intensive care from exceeding the capacity of the deployed HSS Such control may be described by a control parameter α(t), which quantifies the effect of mitigation strategies upon the infection rate. The (more or less) accurate determination of these subsets is the task of careful social (e.g., infection history) data analysis among citizens This is illustrated, where mitigation strategies, followed by the public authorities, are indicated by the dashed and dotted curves, within a space spanned by the effect of the measures upon the infection rate, α, and the cost incurred for economy and society as a whole, f(α). We do not consider potential vaccination scenarios here (as is done elsewhere [9, 10]), so immunization can only be achieved via infection with the virus in the present study
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