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Topological Degree Methods for Age-Structured Epidemic Models

Luisa Malaguti, Stefania Perrotta

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.

Topological Degree Methods for Age-Structured Epidemic Models

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 with respect to the age variable. Moreover, this solution depends continuously on the initial data.
Paper Structure (5 sections, 18 theorems, 146 equations)

This paper contains 5 sections, 18 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminary results
  3. The relative topological degree for condensing multimaps
  4. Existence of a unique nonnegative solution for the SIRS model
  5. A nonlinear, time-dependent, force of infection

Key Result

Theorem 2.2

Let $A$ be the infinitesimal generator of a $C_0$-semigroup $\{ S(t) \}_{t \geq 0}$. If $f \colon [0,T]\times X \to X$ is continuously differentiable, then the Cauchy problem eq:2:CP admits locally one and only one mild solution. Moreover, if $x_0\in D(A)$, then the mild solution is a classical sol

Theorems & Definitions (40)

  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Theorem 3.1
  • ...and 30 more