Continuous and discrete dynamics of a deterministic model of HIV infection

TL;DR

This paper analyzes a mathematical model of HIV infection dynamics, examining stability of disease states, bifurcations, and comparing continuous and discrete models to understand long-term behavior and control thresholds.

Contribution

It provides analytic solutions for the endemic steady state and demonstrates the robustness of the discrete estimation method, advancing understanding of HIV infection modeling.

Findings

Stable endemic equilibrium appears when the reproduction number exceeds one.

Disease-free equilibrium is globally asymptotically stable under certain conditions.

Discrete model estimation is numerically stable regardless of time step size.

Abstract

We will study a mathematical model of the human immunodeficiency virus (HIV) infection in the presence of combination therapy that includes within-host infectious dynamics. The deterministic model requires us to analyze asymptotic stability of two distinct steady states, disease-free and endemic equilibria. Previous results have focused on investigating the global asymptotic stability of the trivial steady state using an implicit finite-difference method which generates a system of difference equations. We, instead, provide analytic solutions and long term attractive behavior for the endemic steady state using the theory of difference equations. The dynamics of estimated model is appropriately determined by a certain quantity threshold maintaining the immune response to a sufficient level. The result also indicates that a forward bifurcation in the model happens when the disease-free equilibrium loses its stability and a stable endemic equilibrium appears as the basic reproduction number exceeds unity. In this scenario, the classical requirement of the reproduction number being less than unity becomes a necessary and sufficient condition for disease mitigation. When the associated reproduction number is in excess of unity, a stable endemic equilibrium emerges with an unstable disease-free equilibrium (leading to the persistence and existence of HIV within the infected individuals). The attractivity of the model reveals that the disease-free equilibrium is globally asymptotically stable under certain assumptions. A comparison between the continuous and estimated discrete models is also provided to have a clear perception in understanding the behavioral dynamics of disease modelling. Finally, we show that the associated estimation method is very robust in the sense of numerical stability since the equilibria and the stability conditions are independent of the time step.