Global dynamics for an SIR patchy model with susceptibles dispersal.

Abstract

An $SIR$ epidemiological model with suscptibles dispersal between two patches is addressed and discussed. The basic reproduction numbers $R_{01}$ and $R_{02}$ are defined as the threshold parameters. It shows that if both $R_{01}$ and $R_{02}$ are below unity, the disease-free equilibrium is shown to be globally asymptotically stable by using the comparison principle of the cooperative systems. If $R_{01}$ is above unity and $R_{02}$ is below unity, the disease persists in the first patch provided $S_2^{1\ast }<S_2^{2\ast }$ . If $R_{02}$ is above unity, $R_{01}$ is below unity, and $S_1^{2\ast }<S_1^{1\ast }$ , the disease persists in the second patch. And if $R_{01}$ and $R_{02}$ are above unity, and further $S_2^{1\ast }>S_2^{2\ast }$ and $S_1^{2\ast }>S_1^{1\ast }$ are satisfied, the unique endemic equilibrium is globally asymptotically stable by constructing the Lyapunov function. Furthermore, it follows that the susceptibles dispersal in the population does not alter the qualitative behavior of the epidemiological model.