Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells

doi:10.1016/j.mbs.2005.12.026

Mathematical Biosciences

Volume 200, Issue 1, March 2006, Pages 44-57

https://doi.org/10.1016/j.mbs.2005.12.026 Get rights and content

Abstract

A mathematical model that describes HIV infection of CD4⁺ T cells is analyzed. Global dynamics of the model is rigorously established. We prove that, if the basic reproduction number R₀ ⩽ 1, the HIV infection is cleared from the T-cell population; if R₀ > 1, the HIV infection persists. For an open set of parameter values, the chronic-infection equilibrium P^∗ can be unstable and periodic solutions may exist. We establish parameter regions for which P^∗ is globally stable.

Introduction

The Human Immunodeficiency Virus (HIV) mainly targets a host’s CD4⁺ T cells. Chronic HIV infection causes gradual depletion of the CD4⁺ T cell pool, and thus progressively compromises the host’s immune response to opportunistic infections, leading to Acquired Immunodeficiency Syndrome (AIDS). For this reason, the count of CD4⁺ T cells is a primary indicator used to measure progression of HIV infection. In a normal person, the level of CD4⁺ T cells in the peripheral blood is regulated at a level between 800 and 1200 mm⁻³. The body is believed to produce CD4⁺ T cells from precursors in the bone marrow and thymus at a constant rate s, and T cells have a natural turn-over rate α. When stimulated by antigen or mitogen, T cells multiply through mitosis with a rate r. Thus the CD4⁺ T cell dynamics can be modelled by the following logistic equation: $\frac{d T}{d t} = s - α T + rT (1 - \frac{T}{T_{\max}}),$ where T is the concentration of CD4⁺ T cells, and T_max is the maximum level of CD4⁺ T-cell concentration in the body [1], [2]. HIV infection will interrupt the normal CD4⁺ T-cell dynamics. The total concentration of CD4⁺ T cells is now T + T^∗, where T is the concentration of susceptible CD4⁺ T cells and T^∗ the concentration of infected CD4⁺ T cells by the HIV viruses. The T-cell dynamics will be determined by the interactions among susceptible CD4⁺ T cells, infected CD4⁺ T cells, and free HIV viruses. Several mathematical models have been proposed to describe the in vivo dynamics of T cell and HIV interaction, see [1], [2], [3], [4], [5], [6] for review and references. Of particular interest to us is a model in [1], which is given by the following system of differential equations: $\begin{matrix} \frac{d T}{d t} = s - α T + rT (1 - \frac{T + T^{*}}{T_{\max}}) - kVT, \\ \frac{d T^{*}}{d t} = kVT - β T^{*}, \\ \frac{d V}{d t} = N β T^{*} - γ V . \end{matrix}$ In this model, T, T^∗ and V denote the concentration of susceptible CD4⁺ T cells, infected CD4⁺ T cells, and free HIV virus particles in the blood, respectively. Parameters α, β, and γ are natural turn-over rates of uninfected T cells, infected T cells, and virus particles, respectively. Because of the viral burden on the HIV infected T cells, we assume that α ⩽ β. The logistic growth of the healthy CD4⁺ T cells is now described by $rT (1 - \frac{T + T^{*}}{T_{\max}})$ , and proliferation of infected CD4⁺ T cells is neglected. The term kVT describes the incidence of HIV infection of health CD4⁺ T cells, where k > 0 is the infection rate. Each infected CD4⁺ T cell is assumed to produce N virus particles during its life time, including any of its daughter cells.

A model for HIV infection similar to (2) but using a simplified logistic growth rT(1 − T/T_max) for susceptible CD4⁺ T cells has been proposed in Perelson and Nelson [2], its global dynamics are analyzed in De Leenheer and Smith [6]. The global dynamics of model (2), however, have not been rigorously established in the literature. The difference in the proliferation term does not change the basic reproduction number. It will not change the CD4 count at equilibrium level, as we will show, but it changes the equilibrium level of viral load during chronic infection. It is of interest to investigate if the difference in logistic terms will cause qualitative changes in the dynamics. The main difficulty of the mathematical analysis lies in the determination of the basin of attraction of the chronic-infection equilibrium P^∗. This is done by identifying the range of parameters for which P^∗ is globally asymptotically stable in the entire feasible region. The global-stability analysis is significant since models of this type are known to possess periodic solutions for an open set of parameter values.

Models considered in [2], [6] with a simplified logistic term are competitive systems. The global analysis in [6] relies in an essential way on properties of competitive systems. With a full logistic term, model (2) is no longer competitive, and the global stability of P^∗ needs to be established using a different approach. In the present paper, we adopt the approach developed in Li and Muldowney [7], which has been successfully applied to many epidemic and in-host models that are not competitive or monotone (see [8], [9]).

For system (2), we show that, if the basic reproduction number R₀ ⩽ 1, the infection-free equilibrium P₀ is globally asymptotically stable, the virus is cleared and no HIV infection persists. If R₀ > 1, P₀ becomes unstable and the HIV infection persists in the T-cell population. In this case, a unique chronic-infection equilibrium P^∗ exists. The local stability of P^∗ is described in term of the proliferation rate r of healthy T cells. We show that P^∗ can be unstable for a range of r. Numerical simulations show periodic solutions may exist. It is therefore important to investigate the basin of attraction of P^∗ when it is locally stable. For an open set of r values that are biologically reasonable, we show that the basin of attraction of P^∗ includes the whole feasible region.

Section snippets

Equilibria

The non-negative octant $R_{+}^{3}$ is positively invariant with respect to (2). In the absence of infection, the dynamics of healthy T cells are governed by Eq. (1). It can be shown that the T-cell concentration stabilizes at a level T₀ given by $T_{0} = \frac{T_{\max}}{2 r} [(r - α) + \sqrt{(r - α)^{2} + \frac{4 sr}{T_{\max}}}] .$ From the first equation of (2), we know T(t) ⩽ T₀ if T(0) ⩽ T₀. Adding the first two equations of (2) gives T′ + T^∗′ ⩽ s + rT₀ − αT − βT^∗ ⩽ s + rT₀ − α(T + T^∗), since α ⩽ β. Therefore T + T^∗ is bounded, and thus T^∗ is bounded, say by M. Clearly, V is

Stability of the infection-free equilibrium P₀

The Jacobian matrix of (2) at P₀ is $J (P_{0}) = [\begin{matrix} - α + r (1 - \frac{T_{0}}{T_{\max}}) - \frac{{rT}_{0}}{T_{\max}} & - \frac{{rT}_{0}}{T_{\max}} & - {kT}_{0} \\ 0 & - β & {kT}_{0} \\ 0 & N β & - γ \end{matrix}] .$ An eigenvalue of J(P₀) is $- α + r (1 - \frac{T_{0}}{T_{\max}}) - \frac{{rT}_{0}}{T_{\max}} = - \frac{s}{T_{0}} - \frac{{rT}_{0}}{T_{\max}} < 0$ from the first equation of (2). The other two eigenvalues have negative real parts if and only if βγ − NβkT₀ > 0, i.e., R₀ < 1. If R₀ = 1, one eigenvalue is 0 and it is simple. If R₀ > 1, J(P₀) has a positive eigenvalue. P₀ is thus unstable with a two-dimensional stable manifold and a one-dimensional unstable manifold. We arrive at the following local stability

Local stability of the chronic-infection equilibrium P^∗

To investigate the fashion in which the HIV infection persists when R₀ > 1, we examine the local stability of P^∗. The Jacobian matrix of (2) at P^∗ is $J (P^{*}) = [\begin{matrix} - \bar{a} & - \frac{r \bar{T}}{T_{\max}} & - k \bar{T} \\ k \bar{V} & - β & k \bar{T} \\ 0 & N β & - γ \end{matrix}],$ where $\bar{a} = α - r (1 - \frac{\bar{T} + {\bar{T}}^{*}}{T_{\max}}) + \frac{r \bar{T}}{T_{\max}} + k \bar{V} = \frac{s}{\bar{T}} + \frac{r \bar{T}}{T_{\max}} > 0$ . The characteristic polynomial of J(P^∗) is $P (λ) = λ^{3} + (\bar{a} + β + γ) λ^{2} + [\bar{a} (β + γ) + \frac{kr}{T_{\max}} \bar{V} \bar{T}] λ + \frac{kr γ}{T_{\max}} \bar{V} \bar{T} + k β γ \bar{V} .$ Note that all coefficients of P(λ) are positive. Thus by the Routh–Hurwitz criteria, all zeros of P(λ) have negative real parts if and only if $Δ = (\bar{a} + β + γ) [\bar{a} (β + γ) + \frac{kr}{T_{\max}} \bar{V}]$

Global stability of P^∗

When r belongs to the range for which P^∗ is locally asymptotically stable, it is of interest to know the basin of attraction of P^∗. In particular, we would like to know if the basin of attraction includes all points in the feasible region Γ, namely, if P^∗ is globally asymptotically stable. Establishing the range of r values for which P^∗ is globally stable is especially important given that model (2) is capable of having periodic solutions.

Global stability of P^∗ for model (2) when the logistic

Numerical simulations

We have shown in Theorem 5(b) that, when R₀ > 1, the chronic-infection equilibrium P^∗ may only be stable for r small or large. In these parameter regions, the stability of P^∗ can be lost as r increases and then regained as r further increases. We carried out numerical simulations using Mathematica to illustrate such a change in stability. Furthermore, our numerical simulations consistently show the existence of periodic solutions when P^∗ is unstable. For the simulations, we use a similar set of

Discussion

In-host models for the HIV infection of CD4⁺ T cells are considered in [2], [6], where the growth of susceptible T cells is assumed to be unaffected by the HIV infection, and follows a simplified logistic growth: $rT (1 - \frac{T}{T_{\max}})$ during the course of infection. In model (2) considered in the present paper, a full logistic growth term $rT (1 - \frac{T + T^{*}}{T_{\max}})$ is used for the growth of healthy T cells, where T^∗ is the concentration of infected CD4⁺ T cells. This assumes that the growth of healthy T cells slows down

Acknowledgments

The research of L. Wang was supported in part by Georgia Southern University Summer Stipend Grant 2003. The research of M.Y. Li is supported in part by grants from Natural Sciences and Engineering Research Council of Canada (NSERC) and Canada Foundation for Innovation (CFI). The authors would like to thank an anonymous referee whose suggestions helped to improve the presentation of the paper.

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Mathematical analysis of the global dynamics of a model for HIV infection of CD4+ T cells

Abstract

Introduction

Section snippets

Equilibria

Stability of the infection-free equilibrium P0

Local stability of the chronic-infection equilibrium P∗

Global stability of P∗

Numerical simulations

Discussion

Acknowledgments

Math. Biosci.

Math. Biosci.

Math. Biosci.

Math. Biosci.

J. Math. Anal. Appl.

Mathematical analysis of HIV-I dynamics in vivo

SIAM Rev.

Modeling the interaction of the immune system with HIV

Using mathematics to understand HIV immune dynamics

Notices AMS

Virus dynamics: a global analysis

SIAM J. Appl. Math.

Mathematical analysis of the global dynamics of a model for HIV infection of CD4⁺ T cells

Stability of the infection-free equilibrium P₀

Local stability of the chronic-infection equilibrium P^∗

Global stability of P^∗