Modeling and NSFD discretization of HIV dynamics under Crowley–Martin incidence

In this paper, we consider an in-host HIV infection model which takes into account logistic growth for CD4 ⁺ T cells, the recovery rate of infected cells, and Crowley–Martin form of two-sided saturated incidence function. The main novelty of the proposed model lies in extending a classical logistic HIV framework by introducing the Crowley–Martin incidence mechanism, which captures mutual interference and saturation effects between healthy cells and free virions, thereby providing a more realistic description of infection dynamics at high population densities. In addition, a nonstandard finite difference NSFD scheme is developed to preserve the essential qualitative properties of the continuous model. First, we show that all solutions of the model are positive and bounded and calculate the basic reproduction number R₀ using the next-generation matrix technique. It follows from our analysis that the disease-free equilibrium is locally asymptotically stable if R₀ < 1 and globally asymptotically stable under an additional sufficient condition, while the existence of at least one biologically interpretable endemic equilibrium and its local stability occurs if R₀> 1 . The existence of an endemic equilibrium is described by a quartic polynomial whose constant term changes sign at R₀=1 . In order to preserve biologically relevant properties of the continuous problem in the discrete-time case, we propose a semi-implicit NSFD scheme that is designed to preserve the qualitative behavior of the continuous model.