Topological Degree Methods for Age-Structured Epidemic Models

TL;DR

This paper applies topological degree methods to an age-structured SIRS epidemic model, establishing the existence, uniqueness, and continuous dependence of solutions within an abstract semigroup framework.

Contribution

It introduces the use of topological degree for condensing maps in analyzing age-structured epidemic models, replacing traditional fixed-point techniques.

Findings

Proves existence of a unique, global, nonnegative solution.

Demonstrates continuous dependence on initial data.

Models solutions in the $L^1$ space with respect to age.

Abstract

This paper is devoted to the study of an age-structured SIRS epidemic model, in which a population affected by a disease is divided into susceptible, infected, and removed individuals. We assume that the force of infection may be nonlinear and time-dependent. The model, originally introduced and studied by Iannelli and his co-authors, can be naturally formulated in an abstract setting and has traditionally been analyzed using fixed point techniques, most often the Banach contraction principle. Following the approaches of Inaba and Banasiak, our investigation is based on the semigroup theory, through which we study the existence of mild (integral) solutions. The main novelty of our work lies in the use of the topological degree for condensing maps instead of classical fixed-point arguments. We prove the existence of a unique, global, nonnegative solution to the model that satisfies the prescribed initial and nonlocal conditions and takes values in the space $L^1$ with respect to the age variable. Moreover, this solution depends continuously on the initial data.