旨在建立一个含有预防接种的霍乱最优控制模型, 并对无病平衡点和地方病平衡点进行稳定性分析, 当$R_0<1$时, 无病平衡点是局部渐近稳定以及全局渐近稳定的; 当$R_0>1$时, 地方病平衡点是局部渐近稳定和全局渐近稳定的; 其次再使用最优控制理论和Pontryagin原理分析最优控制策略. 数值模拟的结果验证了最优控制率的有效性, 并表明在传染病爆发后接种疫苗具有重要的现实意义. 在预算有限的情况下, 可以只采用单一最优控制$u_1$作为最佳控制策略.

In this paper, we aim to construct and analyze an optimal control model with vaccination to study cholera epidemic. Firstly, we show if the basic reproduction number $R_0<1$, the disease-free equilibrium is locally asymptotically stable and globally asymptotically stable; if $R_0>1$, the disease-free equilibrium becomes unstable, while the endemic equilibrium is locally asymptotically stable and globally asymptotically stable as well. In addition, we analyze optimal control strategy by using optimal control theory and the Pontrjagin maximum principle. Furthermore, we present numerical simulation results to verify the analytical predictions. In the case of a limited budget, we can only use optimal control $u_1$ as the best control strategy.