SIAM Journal on Mathematical Analysis, Volume 53, Issue 3, Page 2776-2810, January 2021.
In this paper, we are concerned with an epidemic reaction-diffusion system with nonlinear incidence mechanism of the form $S^qI^p\,(p,\,q>0)$. The coefficients of the system are spatially heterogeneous and time dependent (particularly time periodic). We first establish the $L^\infty$-bounds of the solutions of a class of systems that improve some previous results in [M. Pierre, Milan J. Math., 78 (2010), pp. 417--455]. Based on such estimates, we then study the long-time behavior of the solutions of the system. Our results reveal the delicate effect of the infection mechanism, transmission rate, recovery rate, and disease-induced mortality rate on the infection dynamics. Our analysis can be adapted to models with some other types of infection incidence mechanisms.

中文翻译：

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异构环境中扩散扩散系统的全局$ L ^ \ infty $界和长期行为

SIAM数学分析杂志，第53卷，第3期，第2776-2810页，2021年1月。
在本文中，我们关注的是一种具有非线性入射机制的流行病反应扩散系统，其形式为$ S ^ qI ^ p \，（p，\，q> 0）$。该系统的系数在空间上是异质的并且与时间有关（尤其是时间周期性的）。我们首先建立一类系统的解的L $ \ infty $-界，以改善[M. Pierre，Milan J. Math。，78（2010），第417--455页]。基于这样的估计，我们然后研究系统解的长期行为。我们的结果揭示了感染机制，传播率，恢复率和疾病致死率对感染动力学的微妙影响。我们的分析可以适合于具有其他一些类型的感染发生机制的模型。