Dynamics of a susceptible–infected–susceptible epidemic reaction–diffusion model

Keng Deng; Yixiang Wu

doi:10.1017/S0308210515000864

Dynamics of a susceptible–infected–susceptible epidemic reaction–diffusion model

Part of: Partial differential equations Parabolic equations and systems Applications - Dynamical systems and ergodic theory Mathematical sociology

Published online by Cambridge University Press: 19 July 2016

Keng Deng and

Yixiang Wu

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Keng Deng: Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, USA (deng@louisiana.edu)
Yixiang Wu: Affiliation:
Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504, USA (deng@louisiana.edu)

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We study a susceptible–infected–susceptible reaction–diffusion model with spatially heterogeneous disease transmission and recovery rates. A basic reproduction number is defined for the model. We first prove that there exists a unique endemic equilibrium if . We then consider the global attractivity of the disease-free equilibrium and the endemic equilibrium for two cases. If the disease transmission and recovery rates are constants or the diffusion rate of the susceptible individuals is equal to the diffusion rate of the infected individuals, we show that the disease-free equilibrium is globally attractive if , while the endemic equilibrium is globally attractive if .