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.

In this paper, a dynamic HIV model with cell-to-cell transmission, two immune responses, and induced apoptosis is proposed and studied. First, the non-negativity and boundedness of the solutions of the model are given, and then the exact expression of the basic reproduction number $R_0$ is obtained by using the next generation matrix method. Second, criteria are obtained for the local stability of the disease-free equilibrium, immune response-free equilibrium, and the infected equilibrium with both humoral and cellular immune responses. Furthermore, the threshold conditions are also derived for the global asymptotic stability of the disease-free equilibrium, immune response-free equilibrium, and the infected equilibrium with both humoral and cellular immune responses by constructing the suitable Lyapunov function. Finally, some numerical simulations are conducted to verify the theoretical results; the numerical simulation results show that the increase of apoptosis rate had a positive role in the control of viral infection.

Considering the influence of media reports and environmental transmission on disease transmission, a SEVIQBR infectious disease model is established. Firstly, the basic reproduction number ${\mathcal{R}}_0$ of the model is obtained by the next generation matrix method. Based on the basic reproduction number, the existence of disease-free equilibrium and endemic equilibrium of the model is discussed. Then, the local stability of disease-free equilibrium is studied by means of Hurwitz criterion, and the global asymptotic stability of disease-free equilibrium is studied by constructing an appropriate Lyapunov function. The results show that when the basic reproduction number ${\mathcal{R}}_0$ < 1, the disease-free equilibrium of the model is globally asymptotically stable. When ${\mathcal{R}}_0$ satisfies certain conditions, the endemic equilibrium of the model exists. Finally, we give some numerical simulations to explain the theoretical results.