Nonlocal diffusion model of HIV/AIDS transmission dynamics among MSM populations with ART in heterogeneous environments

In this paper, we develop a spatial heterogeneous compartment model to investigate the combined impacts of nonlocal diffusion and antiretroviral therapy (ART) on the transmission dynamics of human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS) within men who have sex with men (MSM). We have overcome the difficulty caused by the lack of compactness of the nonlocal operator. With the updated equation, we obtain the function formulation of the next generation operator $\mathcal{R}$. Then, the basic reproduction number $R_0$ is obtained, which is the spectral radius of $\mathcal{R}$. Concerning analytical investigation, according to Lipschitz continuity assumption of the parameters, the linearized model under the assumption of no disease at steady state has a dominant eigenvalue associated with a non-negative eigenfunction with positive values. By specifying the eigenfunction to be the integral kernel of the Lyapunov function, we demonstrate that, for $R_0<1$, as time progresses, all solutions approach the disease-free steady state globally. Based on the uniform persistence theory applicable to dissipative systems, it is proved that when $R_0>1$, uniform persistence is maintained in the model, which implies the presence of at least one positive equilibrium. In the numerical simulation part, the theoretical results are verified, and these findings indicate that the MSM population experiences significantly reduced HIV transmission rates through effective ART implementation.