Numerical dynamics of infection-age models with logistic growth and general nonlinear incidence

We consider an age-structured viral dynamics model with Logistic growth and a general nonlinear incidence rate. We present the basic reproduction number of the continuous model and conduct a theoretical analysis of the model. For such a hybrid infinite-dimensional system with abstract nonlinear terms, the comprehensive numerical analysis is still pending. We address this problem by establishing a fully discrete linearly implicit scheme, and the non-negativity of the numerical scheme is confirmed by utilizing the theory of M-matrix. With a solvability analysis, the finite time convergence is proved for strong solutions. For long-time dynamics, by utilizing the exponential decay characteristic of the fundamental solution matrix, we established a 1-order convergence analysis for the numerical reproduction number $R_0^{\Delta t}$, and further proved the 1-order convergence property of numerical equilibria. By applying linearization techniques and comparison principles, we demonstrate that the disease-free equilibrium is globally asymptotically stable when $R_0^{\Delta t}<1$, and the endemic equilibrium is locally asymptotically stable when $R_0^{\Delta t}>1$. Hence, numerical processes almost completely replicate the dynamic properties of continuous system. At last, some numerical experiments demonstrate the obtained results.