Asymptotic Analysis for a Nonlinear Reaction-Diffusion System Modeling an Infectious Disease

TL;DR

This paper analyzes a complex nonlinear reaction-diffusion model for infectious diseases like cholera, proving global existence, uniqueness, and detailed stability conditions of solutions, addressing open questions in the field.

Contribution

It provides the first rigorous proof of global solutions and stability criteria for a nonlinear, spatially and temporally variable reaction-diffusion system modeling infectious diseases.

Findings

Proved global existence and uniqueness of solutions.

Derived conditions for stability and instability of steady states.

Characterized long-term behavior of the disease spread model.

Abstract

In this paper we study a nonlinear reaction-diffusion system which models an infectious disease caused by bacteria such as those for cholera. One of the significant features in this model is that a certain portion of the recovered human hosts may lose a lifetime immunity and could be infected again. Another important feature in the model is that the mobility for each species is allowed to be dependent upon both the location and time. With the whole population assumed to be susceptible with the bacteria, the model is a strongly coupled nonlinear reaction-diffusion system. We prove that the nonlinear system has a unique solution globally in any space dimension under some natural conditions on the model parameters and the given data. Moreover, the long-time behavior and stability analysis for the solutions are carried out rigorously. In particular, we characterize the precise conditions on variable parameters about the stability or instability of all steady-state solutions. These new results provide the answers to several open questions raised in the literature.