Partial Differential Equations of an Epidemic Model with Spatial Diffusion

El Mehdi Lotfi,Mehdi Maziane,Noura Yousfi,Khalid Hattaf

doi:10.1155/2014/186437

Abstract

The aim of this paper is to study the dynamics of a reaction-diffusion SIR epidemic model with specific nonlinear incidence rate. The global existence, positivity, and boundedness of solutions for a reaction-diffusion system with homogeneous Neumann boundary conditions are proved. The local stability of the disease-free equilibrium and endemic equilibrium is obtained via characteristic equations. By means of Lyapunov functional, the global stability of both equilibria is investigated. More precisely, our results show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than or equal to unity, which leads to the eradication of disease from population. When the basic reproduction number is greater than unity, then disease-free equilibrium becomes unstable and the endemic equilibrium is globally asymptotically stable; in this case the disease persists in the population. Numerical simulations are presented to illustrate our theoretical results.

Highlights

In this paper, we consider the following SIR epidemic model with a specific nonlinear incidence rate described by dS dt = Λ− μS −1 + α1S βSI + α2I + α3SI dI dt βSI α1S + α2I + α3SI −
Using the results presented by Hattaf et al in [7], it is easy to get that the basic reproduction number of disease in the absence of spatial dependence is given by βΛ α1Λ)
We investigated the dynamics of a reactiondiffusion epidemic model with specific nonlinear incidence rate

Summary

Introduction

We consider the following SIR epidemic model with a specific nonlinear incidence rate described by dS dt. In the classical epidemic models, this rate was assumed to be linear with respect to the numbers of susceptible and infected individuals. We consider the following SIR epidemic model with specific nonlinear incidence rate and spatial diffusion:. We establish the global existence, positivity, and boundedness of solutions of problem (3)–(5) because this model describes the population. For any given initial data satisfying the condition (5), there exists a unique solution of problem (3)–(5) defined on [0, +∞) and this solution remains nonnegative and bounded for all t ≥ 0.

Qualitative Analysis of the Spatial Model

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Conclusion

Conflict of Interests