Numerical analysis of an age-structured model for HIV viral dynamics with latently infected T cells based on collocation methods

In this paper, we consider the numerical threshold for an age-structured HIV model with latently infected T cells. Based on the continuous collocation methods, a semi-discrete scheme is constructed by discretizing the age variable and a numerical basic reproduction number $R_h$ is provided. With the study of higher-order convergence to the real basic reproduction number $R_0$, the relations between $R_h$ and local stability of disease-free are presented. From the viewpoint of full discretization, an equivalent block-Leslie matrix expression is obtained by embedding into a piecewise-discontinuous polynomial space rather than the piecewise-continuous polynomial space. An implicit full-discrete scheme is considered based on a linearly implicit Euler (IMEX) method, of which the computational cost is almost the same as an explicit scheme. It is more important that the dynamical behavior of the age-semi-discretization system is also preserved for any time step whenever $R_h$ is the threshold for the numerical dynamical system of the age-semi-discretization. Finally, numerical applications are shown to HIV models to illustrate our analysis.