Stability analysis of a novel Delay Differential Equation of HIV Infection of CD4$^+$ T-cells

TL;DR

This paper analyzes the stability of a new HIV infection model incorporating delay differential equations, revealing how time delays can destabilize the system and induce periodic solutions, supported by numerical simulations.

Contribution

It introduces a novel delay differential equation model for HIV infection and studies its stability, highlighting the impact of incubation time on disease dynamics.

Findings

Delay can destabilize the disease-free equilibrium

Hopf bifurcation leads to periodic solutions

Numerical simulations confirm theoretical results

Abstract

In this paper, we investigate a novel 3-compartment model of HIV infection of CD4$^+$ T-cells with a mass action term by including two versions: one baseline ODE model and one delay-differential equation (DDE) model with a constant discrete time delay. Similar to various endemic models, the dynamics within the ODE model is fully determined by the basic reproduction term $R_0$. If $R_0<1$, the disease-free (zero) equilibrium will be asymptotically stable and the disease gradually dies out. On the other hand, if $R_0>1$, there exists a positive equilibrium that is globally/orbitally asymptotically stable within the interior of a predefined region. To present the incubation time of the virus, a constant delay term $\tau$ is added, forming a DDE model. In this model, this time delay (of the transmission between virus and healthy cells) can destabilize the system, arising periodic solutions through Hopf bifurcation. Finally, numerical simulations are conducted to illustrate and verify the results.