Analysis of global behavior in an age-structured epidemic model with nonlocal dispersal and distributed delay

The virus infections pose a significant threat to human health, necessitating a profound comprehension of these diseases to effectively manage epidemics and avert fatalities. This study delves into a comprehensive analysis of a mathematical model that incorporates the age of infection and distributed time delays in the context of pathogenic epidemics. The framework of this model accommodates the nonlocal diffusion of the pathogen within cells. Our investigation showcases the well-posed nature of the model and establishes the compactness of the solutions. We derive the basic reproduction number, denoted as R0$$ {R}_0 $$, and substantiate that when R0$$ {R}_0 $$ is less than 1, the disease-free equilibrium ( E0$$ {E} circumflex 0 $$) achieves global asymptotic stability. Furthermore, we construct a global attractor within a bounded set and affirm the existence of the complete trajectory ( psi$$ \psi $$). This research also delves into scenarios where R0$$ {R}_0 $$ surpasses 1, illustrating the presence of the endemic steady state ( E*$$ {E} circumflex {\ast } $$) through the utilization of super-subsolutions. We establish the globally asymptotic stability of E*$$ {E} circumflex {\ast } $$ for cases where R0$$ {R}_0 $$ is greater than 1 by employing a well-suited Lyapunov function. To validate the robustness of our mathematical computations, we conclude by presenting a set of numerical results.