Special Issue in honor of Alan C. Lazer. Electron. J. Diff. Eqns., Special Issue 01 (2021), pp. 315-325.

SIR epidemic models with spatial spread in bounded domains

Helmut Knolle, Jairo Santanilla

Abstract:
The spread of an infectious disease which confers immunity after recovery from infection, can be described by a SIR model, i.e. a system of three differential equations for the dependent variables $S$, $I$, and $R$, which are the numbers (densities) of susceptible, infectious and recovered (immune) individuals, respectively. The equations for $S$ and $I$ are typically nonlinear. In this article, we consider two spatio-temporal SIR models. The first model is similar to reaction-diffusion systems in chemistry. A simple birth-and-death process is incorporated, and it is assumed, that a fraction f of the newborns is vaccinated and is then immune for life. We show how the smallest eigenvalue of the eigenvalue problem associated with the linearized equation for I is related to the basic reproduction number $\mathcal{R}_0$, a key concept in the mathematical theory of infectious diseases. Here it is defined by a variational principle. We show that the disease-free equilibrium is asymptotically stable if $\mathcal{R}_0<1$, or if $\mathcal{R}_0\ge 1$ and $f>1-1/{\mathcal{R}_0}$, and unstable if $\mathcal{R}_0>1$ and $f<1-1/\mathcal{R}_0$. In the other model we assume that the population consists of sedentary individuals who leave their home only temporarily. Both models suggest that restriction of mobility may be counterproductive for the control of an epidemic outbreak.

Published March 18, 2022.
Math Subject Classifications: 92D30.
Key Words: SIR epidemic model; spatial spread; basic reproduction number; mass action; disease-free equilibrium; stability; nonlinear parabolic differential equations.
DOI: https://doi.org/10.58997/ejde.sp.01.k2

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Helmut Knolle
Scheyenholzstrasse 9, 3075 Rüfenacht, Switzerland.
Institute Of Social And Preventive Medicine
University Of Berne
email: helmut.knolle@bluewin.ch
Jairo Santanilla
Department of Mathematics
University of New Orleans
2000 Lakeshore Dr.
New Orleans, LA 70148, USA
email: jsantani@uno.edu

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