# Global $L^\infty$-bounds and long-time behavior of a diffusive epidemic system in heterogeneous environment

## TL;DR

This paper investigates the long-term behavior of a spatially heterogeneous epidemic reaction-diffusion system with nonlinear infection mechanisms, establishing bounds and analyzing how various parameters influence disease dynamics.

## Contribution

It introduces improved $L^
abla$-bounds for solutions and analyzes the impact of infection, transmission, recovery, and mortality rates on long-term epidemic behavior.

## Key findings

- Established $L^
abla$-bounds for the system solutions.
- Analyzed the influence of parameters on infection persistence or extinction.
- Results applicable to other nonlinear infection mechanisms.

## Abstract

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, which improve some previous results in [58]. 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 some other types of infection incidence mechanisms.