In this paper, a reaction–diffusion SEI epidemic model with nonlinear incidence rate is proposed. The well-posedness of solutions is studied, including the existence of positive and unique classical solution and the existence and the ultimate boundedness of global solutions. The basic reproduction numbers are given in both heterogeneous and homogeneous environments. For spatially heterogeneous environment, by the comparison principle of the diffusion system, the infection-free steady state is proved to be globally asymptotically stable if R0 < 1, if R0 > 1, the system will be persistent and admit at least one positive steady state. For spatially homogenous environment, by constructing a Lyapunov function, the infection-free steady state is proved to be globally asymptotically stable if R0 < 1, and then the unique positive steady state is achieved and is proved to be globally asymptotically stable if R0 > 1. Finally, two examples are given via numerical simulations, and then some control strategies are also presented by the sensitive analysis.

中文翻译：

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具有非线性发病率的反应-扩散 SEI 流行病模型的动力学分析

在本文中，提出了一种具有非线性发病率的反应-扩散 SEI 流行病模型。研究了解的适定性，包括正唯一经典解的存在性和全局解的存在性和最终有界性。在异构和同质环境中都给出了基本的再生数。对于空间异质环境，通过扩散系统的比较原理，证明无感染稳态是全局渐近稳定的，如果R0 < 1,如果R0 > 1，系统将是持久的并承认至少一个正稳态。对于空间同质环境，通过构造 Lyapunov 函数，证明无感染稳态是全局渐近稳定的，如果R0 < 1,然后达到唯一的正稳态，并证明它是全局渐近稳定的，如果R0 > 1. 最后，通过数值模拟给出了两个例子，然后通过敏感分析给出了一些控制策略。