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Long-time behavior of the heterogeneous SIRS epidemiological model

Romain Ducasse, Maxime Laborde

TL;DR

Under some hypotheses on the coefficients, it is proved that the solutions of the SIRS model converge to an equilibrium that is identified and some estimates on the speed of propagation are established.

Abstract

We study the long-time behavior of solutions of the SIRS model, a reaction-diffusion system that appears in epidemiology to describe the spread of epidemics. We allow the system to be heterogeneous periodic. Under some hypotheses on the coefficients, we prove that the solutions converge to an equilibrium that we identify and establish some estimates on the speed of propagation.

Long-time behavior of the heterogeneous SIRS epidemiological model

TL;DR

Under some hypotheses on the coefficients, it is proved that the solutions of the SIRS model converge to an equilibrium that is identified and some estimates on the speed of propagation are established.

Abstract

We study the long-time behavior of solutions of the SIRS model, a reaction-diffusion system that appears in epidemiology to describe the spread of epidemics. We allow the system to be heterogeneous periodic. Under some hypotheses on the coefficients, we prove that the solutions converge to an equilibrium that we identify and establish some estimates on the speed of propagation.
Paper Structure (22 sections, 13 theorems, 129 equations, 1 figure)

This paper contains 22 sections, 13 theorems, 129 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Presentation of the problem
  3. Related results
  4. Reaction-diffusion equations
  5. Periodic principal eigenvalues.
  6. Reaction-diffusion systems and epidemiology
  7. Main result
  8. Existence of stationary solutions
  9. Step 1: Lipschitz estimate on $T$.
  10. Step 2: Estimate for $Z$.
  11. Step 3: Conclusion.
  12. Long-time behavior of the evolution system.
  13. Definitions
  14. Preliminary results.
  15. Step 1. Proof of \ref{['lem int']}: reduction to a problem on entire solutions.
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $\gamma, \alpha \in C^\delta_{per}$Here and in the sequel, for $\delta \in (0,1)$, $C^\delta_{per}$ denotes the set of functions which are $\delta$-Hölder continuous and $1$-periodic with respect to the $x$ variable, that is, $f(x+k)=f(x)$ for all $k\in \mathbb{Z}^N$. We only consider $1$-period with initial datum $u(0,\cdot)$ continuous, compactly supported, non-negative and non-zero and let

Figures (1)

  • Figure 1: Evolution of the SIRS model. Starting from a small density of infectious and a constant density of susceptible, the disease spreads by forming front-like solutions.

Theorems & Definitions (26)

  • Theorem 1.1: Freidlin-Gartner
  • Theorem 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • proof : Proof of Theorem \ref{['th existence']}.
  • Proposition 2.4
  • ...and 16 more