Optimal immunity control by social distancing for the SIR epidemic model.
Main result
The authors show that if the total containment strategy does not reduce the infection rate of the epidemic below a certain value, or if the time allowed to control the epidemic is too short, then there is no control strategy to avoid an "overshoot". They also show that the control strategy to minimize this overshoot is to avoid containment until a certain date when the number of infected people exceeds a critical value, and then apply the maximum containment policy until the end of the control period allowed for the epidemic. Indeed, by choosing the right date for containment, the number of persons infected during containment decreases to a value close to 0, while the number of susceptible persons barely exceeds the group immunity value. When containment is lifted, the epidemic can no longer spread and disappears.
Takeaways
The aim of this paper is to demonstrate the existence, uniqueness and value of a strategy of control by containment (decreasing in the rate of infection) over a defined time period for an SIR-type epidemic, with the aim of minimizing the "overshoot" (the number of contaminated individuals above the threshold of group immunity).
Strength of evidence Undetermined
The proposed model is deliberately simplistic, and the optimal control strategy corresponds to intuition. However, it represents a first step in the construction of epidemic control strategies. It may be unfortunate that the proposed optimization is carried out with a fixed time horizon, and not with a fixed containment period. Many extensions to this optimization problem can be proposed, such as control with penalization of the number of infected beyond a threshold value, or of the time spent in confinement, etc.
Objectives
The aim of this study is to determine an optimal control for a SIR-type epidemic by containment, i.e. by controlling the contamination rate between two minimum (total containment) and maximum (no containment) values for a defined period of time.
Method
The objective of the control is to minimize the "overshoot" of the process i.e. the number of people who have been contaminated above the group immunity threshold. Indeed, above this threshold, an epidemic cannot start from a small number of infected people, and the population becomes immune to the disease. However, if the number of infected people is too large when this threshold is exceeded, many more people will become ill before the epidemic "naturally" dies out.
Methods to show the existence, uniqueness and shape of the optimal containment strategy are based on the analysis of the system of differential equations. Thanks to the existence of integrable quantities (preserved by the dynamics of the epidemic), it is possible to transform this optimization problem on the set of control functions into a real optimization problem, whose resolution is then simplified.
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