Stability analysis of a delayed sir epidemic model with diffusion and saturated incidence rate
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ORIGINAL PAPER
Stability analysis of a delayed sir epidemic model with diffusion and saturated incidence rate Abdelhadi Abta1 • Salahaddine Boutayeb1 • Hassan Laarabi2 • Mostafa Rachik2 Hamad Talibi Alaoui3
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Received: 13 January 2020 / Accepted: 17 May 2020 Ó Springer Nature Switzerland AG 2020
Abstract In this paper, we investigate the effect of spatial diffusion and delay on the dynamical behavior of the SIR epidemic model. The introduction of the delay in this model makes it more realistic and modelizes the latency period. In addition, the consideration of an SIR model with diffusion aims to better understand the impact of the spatial heterogeneity of the environment and the movement of individuals on the persistence and extinction of disease. First, we determined a threshold value R0 of the delayed SIR model with diffusion. Next, By using the theory of partial functional differential equations, we have shown that if R0 \1, the unique disease-free equilibrium is asymptotically stable and there is no endemic equilibrium. Moreover, if R0 [ 1, the disease-free equilibrium is unstable and there is a unique, asymptotically stable endemic equilibrium. Next, by constructing an appropriate Lyapunov function and using upper–lower solution method, we determine the threshold parameters which ensure the the global asymptotic stability of equilibria. Finally, we presented some numerical simulations to illustrate the theoretical results. Keywords SIR epidemic model SEIR epidemic model Incidence rate Ordinary differential equations Delayed differential equations Partial differential equations Lyapunov function Global stability Mathematics Subject Classification 34K20 34K25 34K05 35B09 35B40 35B35
This article is part of the section ‘‘Theory of PDEs’’ edited by Eduardo Teixeira. & Abdelhadi Abta [email protected] 1
Department of Mathematics and Computer Science, Poly-disciplinary Faculty, Cadi Ayyad University, P.O. Box 4162, Safi, Morocco
2
Department of Mathematics and Computer Science, Faculty of Sciences Ben M’Sik, Hassan II University, P.O. Box 7955, Sidi Othmane, Casablanca, Morocco
3
Department of Mathematics, Faculty of Sciences El Jadida, Chouaib Doukkali University, P.O. Box 20, El Jadida, Morocco SN Partial Differential Equations and Applications
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SN Partial Differ. Equ. Appl. (2020)1:13
1 Introduction The Kermack–McKendrick model is the first one to provide a mathematical description of the kinetic transmission of an epidemic in an unstructured population [9]. In this model the total population is assumed to be constant and divided into three classes: susceptible, infected (and infective), and removed (recovered with permanent immunity) and assuming that the transfers between these classes are instantaneous. The spread of an infection governed by this simple model that integrates neither diseases that have a latency period nor the influence of space on the dynamics of this model, has allowed many scientists to parti
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