Complex dynamics of a delay multi-scale environmental disease transmission model with infection age and general incidence

A multi-scale model coupling within-host infection and between-host transmission with immune delay, infection age, multiple transmission routes and general incidence is developed based on the complexity of environmentally-driven infectious disease transmission. The model is composed of ordinary differential equations (ODEs), delay differential equations (DDEs), and a partial differential equation (PDE). Firstly, the dynamics of the within-host model are analyzed, including the existence and stability of infection-free equilibrium, immunity-inactivated equilibrium, immunity-activated infection equilibrium and Hopf bifurcation. Further, in the context of the coupled between-host model that disregards immune responses, the basic reproduction number ${\tilde{R}}_0^h$, the existence and stability of equilibria, the existence of backward bifurcation and the uniform persistence are obtained. And then, focusing on the within-host model with stable immunity-activated infection equilibrium, the results are achieved regarding the basic reproduction number $R_0^h$, the existence and stability of equilibria for the coupled between-host model. In addition, when stable periodic solutions exist for the within-host model, the existence and stability of the disease-free and positive periodic solutions for the coupled between-host model are determined by numerical simulations. Effective disease control is achieved when two crucial factors are met: a robust adaptive immune response in the host, coupled with an optimally shortened latency period for generating immune components following antigen exposure. Finally, numerical simulations are employed to substantiate these primary findings, illustrate the practical application of our model and propose control strategies for mitigating disease transmission.