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 6 n 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 ⇔ 2 ⇔ 3 ⇔ 1 , circular connection(CC), i.e., 1 ↔ 2 ↔ 3 ↔ 1 and bidirectional connection(BC), i.e., 1 ⇔ 2 ⇔ 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.