22.07.2020

Modeling Infectiology

Elie R et al

Pre-prints

The authors calculate optimal strategies of quarantine (individual and centralized) and show that an individual strategy greatly increases the total cost of the epidemic. Essentially, in contrast to a centralized strategy, individuals have a tendency to minimize the personal rather than societal costs of the epidemic by self-isolating earlier, less stringently, and for less time than the situation would require.

The authors study an SIR model where the prevalence of infection can be controlled using different policies. They compare a “centralized” and “individual” strategy of isolation, where the goal is to minimize the time passed in quarantine as well as the probability of being infected, individually or for the group.

This article presents in a pedagogical fashion the theory of mean field games and its applications for controlling an SIR epidemic. The principal result is an illustration of a “tragedy of the commons,” where each individual is motivated to break quarantine earlier than would be advised for the group’s interest. It would be interesting to consider extensions of these results, for example by considering a system with multiple populations with different costs of infection, or taking into account the effect of saturation of intensive care unites, greatly increasing the cost of infection once the number of cases passes a certain threshold.

The goal of this article is to study the impact of individual considerations on organization of quarantine measures to control an epidemic. The model considered here is that of an SIR-type epidemic; that is, a dynamic epidemic in a large population. The goal of the centralized quarantine strategy is to minimize the total number of people infected and the time spent in quarantine, while the goal of the individuals is not to be infected, and to be in quarantine for as little time as possible. The article compares the results of the centralized strategy and the individual strategy, and thus measures the “price of anarchy,” the number of secondary infections due to the absence of centralized planning.

The article uses classic game theory methods, particularly mean field games, and analysis of differential equations to determine optimal individual and centralized strategies. Numerical approximations are used to determine an explicit solution as well as the behavior of the epidemic in an individual or centralized method of quarantine.

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