Analysis of an HIV/AIDS Model with sexual transmission and travel in a patchy environment.

In this paper, a multi-patch HIV/AIDS epidemic model with heterosexual transmission is formulated to investigate the impact of travel among patches. It is a system of $6n^2$ ordinary differential equations describes HIV/AIDS spread in an environment divided into n patches. We derive the basic reproduction number $R_0$ . Lower and upper bounds on $R_0$ are given. We prove that if $R_0<1$ , the disease-free equilibrium is locally asymptotically stable, and if $R_0>1$ , there is at least an endemic equilibrium. We apply the model to three patches in which the disease spreads in a patch and dies out in the other two patches when there is no travel between them. We considered three types of connection between three patches: full connection(FC), i.e., $1\iff 2\iff 3\iff 1$ , circular connection(CC), i.e., $1\leftrightarrow 2\leftrightarrow 3\leftrightarrow 1$ and bidirectional connection(BC), i.e., $1\iff 2\iff 3$ . We set different travel and return rates to study the impact of travel on the spread of HIV/AIDS. Numerical simulations confirm the theoretical results and indicate that travel may increase or decrease the spread of HIV/AIDS, i.e. HIV/AIDS may become endemic or die out in three patches when travel occurs, and three types of connection may have different impacts on disease transmission.