Dynamics of a nonlocal dispersal multi-group epidemic model with media-related nonlinear incidence in heterogeneous environment

In order to investigate the combined impact of multi-group structure and media coverage on disease transmission dynamics in heterogeneous environments, this paper develops a nonlocal diffusion multi-group SVIR epidemic model with nonlinear incidence and nonlinear contact rates driven by media coverage. Initially, we establish the well-posedness of the solution, define the threshold parameter $R_0^m$, and confirm the existence problem of the principal eigenvalues of the model. Subsequently, we obtain the threshold dynamics of the system, indicating that when $R_0^m<1$, the disease becomes extinct, whereas when $R_0^m>1$, the disease persists. Specifically, we examine a single-group degenerate diffusion model, addressing the basic reproduction number $R_0^s$ and the presence of compact global attractors for the semi-flow. Furthermore, for $R_0^s>1$, we demonstrate the existence of a unique positive steady state in the system and establish the global asymptotic stability of the endemic steady state by constructing a suitable Lyapunov functional. In the numerical simulation section, we validate the states of disease extinction and persistence by approximating the estimated basic reproduction number $R_{0,m}$. Our findings indicate that the disease dynamics are heavily influenced by the nonlocal diffusion kernel function. Additionally, the treatment rate plays a pivotal role in determining the ultimate outcome of the disease. While media reports may not alter the final outcome of the disease, they can effectively impede its spread and reduce the infection density.