We consider an SIR model with vaccination strategy on a sparse configuration model random graph. We show the convergence of the system when the number of nodes grows and characterize the scaling limits. Then, we prove the existence of optimal controls for the limiting equations formulated in the framework of game theory, both in the centralized and decentralized setting. We show how the characteristics of the graph (degree distribution) influence the vaccination efficiency for optimal strategies, and we compute the limiting final size of the epidemic depending on the degree distribution of the graph and the parameters of infection, recovery and vaccination. We also present several simulations for two types of vaccination, showing how the optimal controls allow to decrease the number of infections and underlining the crucial role of the network characteristics in the propagation of the disease and the vaccination program.

中文翻译：

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大型配置模型中带有疫苗接种的 SIR 动力学。

我们在稀疏配置模型随机图上考虑具有疫苗接种策略的 SIR 模型。我们展示了当节点数量增加时系统的收敛性并描述了缩放限制。然后，我们证明了在博弈论框架中制定的限制方程的最优控制的存在，无论是在集中式还是分散式设置中。我们展示了图表的特征（度数分布）如何影响最佳策略的疫苗接种效率，并且我们根据图表的度数分布以及感染、恢复和疫苗接种的参数计算了流行病的最终限制规模。我们还提供了两种疫苗接种的几种模拟，