We analyze a mean field game model of SIR dynamics (Susceptible, Infected, and Recovered) where players choose when to vaccinate. We show that this game admits a unique mean field equilibrium (MFE) that consists in vaccinating at a maximal rate until a given time and then not vaccinating. The vaccination strategy that minimizes the total cost has the same structure as the MFE. We prove that the vaccination period of the MFE is always smaller than the one minimizing the total cost. This implies that, to encourage optimal vaccination behavior, vaccination should always be subsidized. Finally, we provide numerical experiments to study the convergence of the equilibrium when the system is composed by a finite number of agents ( $N$ ) to the MFE. These experiments show that the convergence rate of the cost is $1/N$ and the convergence of the switching curve is monotone.

中文翻译：

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接种疫苗后 SIR 动力学的平均场博弈分析

我们分析了 SIR 动态（易感、感染和恢复）的平均场博弈模型，其中玩家选择何时接种疫苗。我们表明，这个游戏承认一个独特的平均场平衡 (MFE)，它包括在给定时间之前以最大速率接种疫苗，然后不接种疫苗。使总成本最小化的疫苗接种策略与 MFE 具有相同的结构。我们证明了 MFE 的疫苗接种周期总是小于总成本最小化的周期。这意味着，为了鼓励最佳的疫苗接种行为，疫苗接种应始终得到补贴。最后，我们提供数值实验来研究当系统由有限数量的代理（ $N$ ) 到 MFE。这些实验表明，成本的收敛速度是 $1/N$ 并且切换曲线的收敛是单调的。