Stability analysis of HIV-1 model with multiple delays
- Nigar Ali^{1},
- Gul Zaman^{1}Email author and
- Obaid Algahtani^{2}
https://doi.org/10.1186/s13662-016-0808-4
© Ali et al. 2016
Received: 2 November 2015
Accepted: 10 March 2016
Published: 31 March 2016
Abstract
A mathematical model for HIV-1 infection with multiple delays is proposed. These delays account for (i) the delay in contact process between the uninfected cells virus, (ii) a latent period between the time target cells which are contacted by the virus particles and the time the virions enter the cells, and (iii) a virus production period for new virions to be produced within and released from the infected cells. For this model, the basic reproductive number is identified and its threshold property is discussed. The uninfected and infected steady states are shown to be locally as well as globally asymptotically stable. The value of the basic reproductive number shows that increasing any one of these delays will decrease this number. This may suggest a new direction for new drugs that can prolong the infection process and spreading of virus. The proved results have potential applications in HIV-1 therapy.
Keywords
1 Introduction
Mathematical modeling in epidemiology provides understanding of the mechanisms that influence the spread of a disease and suggests control strategies. Human immunodeficiency virus (HIV-1) is a lentivirus that causes acquired immunodeficiency syndrome (AIDS). The HIV infection is characterized by three different phases, namely, the primary infection, a clinically asymptomatic stage (chronic infection), and acquired immunodeficiency syndrome (AIDS). In recent years the population dynamics of infectious diseases have been extensively studied [1–19]. Clinical research combined with mathematical modeling has enhanced progress in the understanding of the HIV-1 infection [4]. This is because mathematical models can offer a way to study the dynamics of viral load in vivo and can be very useful in understanding the interaction between virus and host cell.
In the last decade, the HIV-infection models with time delays have been studied by many authors, and time delays of one type or another have been incorporated into biological models by many authors (e.g., [1–12, 14–19] and the references cited therein). The results presented in [17–19] have shown that larger intercellular delay may help eradicate the virus, while the activation of CTLs can only help reduce the virus load and increase the healthy \(\mathrm{CD}{+}4\) cells in the long term sense. Pawelek et al. [7] have studied that models of HIV-1 infection incorporating intracellular delays are more accurate representations of the biology, and that they can change the estimated values of the kinetic parameters when compared to models without delays. Therefore, we incorporate time delay terms in the model in the interaction of different cells.
We study the dynamical behavior of the proposed model and show how delays influence the stability. We discuss the well-posedness of the solution of equilibria and their stability. In order to properly define biologically meaningful equilibria, we find the basic reproduction number. It will be shown that an infection-free equilibrium, \(E_{0}\), is locally as well as globally asymptotically stable. We also show that the single-infection equilibrium, \(E_{1}\), is locally as well as globally asymptotically stable, and the double-infection equilibrium, \(E_{2}\), is also globally asymptotically stable.
The rest of this paper is organized as follows. The next section is devoted to the well-posedness and positivity of the solution. In Section 3, local and global stabilities of the infection-free equilibrium, \(E_{0}\), are discussed, and in Section 4 we discuss the infection-free equilibrium, \(E_{1}\), and the double-infection equilibrium, \(E_{2}\). A numerical simulation and conclusion are discussed in Section 5.
2 Positivity and well-posedness of the solution and basic reproductive number
In this section, first we discuss the positivity and well-posedness of the solution.
Theorem 2.1
All solutions of the system (1) remain non-negative, provided the given initial conditions are non-negative and bounded.
Proof
3 Stability of the disease-free equilibrium \(E_{0}\)
In this section, we show the dynamical behavior of the system (1) at \(E_{0}\).
Theorem 3.1
When \(R_{0} <1\), then the disease-free equilibrium \(E_{0}\) is locally asymptotically stable while for \(R_{0} >1\), \(E_{0}\) becomes unstable and the single-infection equilibrium \(E_{1}\) occurs.
Proof
Theorem 3.2
If \(R_{0} <1\), then the disease-free equilibrium \(E_{0}\) is globally asymptotically stable.
Proof
4 Stability of single- and double-infection equilibria
In this section, we discuss the single-infection-free equilibrium, \(E_{1}\).
Theorem 4.1
\(E_{1}\) is locally asymptotically stable if \(1< R_{0}<1+\frac{b\beta k e^{-m\tau_{1}}}{cdu}\), provided that \(\tau_{1}, \tau_{2}\geq0\), otherwise \(E_{1}\) is unstable.
Proof
Thus, we conclude that all the roots of equation (16) have a negative real part when \(\tau_{2}\geq0\). Therefore, the equilibrium \(E_{1}\) is locally asymptotically stable, when \(1< R_{0}<1+\frac{b\beta k e^{-m\tau_{1}}}{cdu}\) and \(\tau_{1},\tau_{2}\geq0\). □
Theorem 4.2
The single-infection-free equilibrium, \(E_{1}\), is globally asymptotically stable, if \(1< R_{0}<1+\frac{b\beta k e^{-m\tau_{1}}}{cdu}\), while for \(R_{0}>1+\frac{b\beta k e^{-m\tau_{1}}}{cdu}\), \(E_{1}\) is unstable.
Proof
Theorem 4.3
If \(\tau_{1}\neq0\) and \(\tau_{2}\neq0\) and \(R_{0}> 1+\frac{b\beta k e^{-m\tau_{1}}}{cdu}\), then the double-infection equilibrium, \(E_{2}\), is globally asymptotically stable.
Proof
5 Numerical simulation and discussion
Parameters used for numerical simulation
Notation | Parameter definition | Value | Source |
---|---|---|---|
N | recruitment rate | 160 | [20] |
d | death rate of uninfected target cells | 0.16 | assumed |
P | infection rate of uninfected cells by virus | 0.002 | [21] |
a | death rate of productively infected cells | 1.85 | [20] |
P | killing rate of infected cells by CTL response cells | 0.2 | assumed |
k | rate of the virus particles produced by infected cells | 1,200 | [20] |
u | viral clearance rate constant | 8 | assumed |
c | rate at which the CTL response is produced | 0.2 | [21] |
b | death rate of the CTL response | 0.4 | assumed |
T1 | intracellular delay | 0.2 | [21] |
T2 | delay in antigenic stimulation | 2.4 | [21] |
In this paper we discuss a HIV-infection model by introducing delay terms in different interaction terms. Dynamical analysis of the system (1) shows that delays play an important role in the stability of the equilibrium. The detailed analytic study has shown that the extended model with delays, like the model with no delay, also has three equilibrium solutions. The disease-free equilibrium \(E_{0}\), the single-infection equilibrium, \(E_{1}\), and the double-infection equilibrium, \(E_{2}\), and a series of bifurcations occur as the basic reproduction number is increased. One has shown that \(E_{0}\) is globally asymptotically stable for \(R_{0}\in(0,1)\) and becomes unstable at the transcritical bifurcation point \(R_{0}=1\) and bifurcates into \(E_{1}\), which is globally asymptotically stable for \(R_{0}>1\). However, it loses it stability at another bifurcation point \(R_{0}>1+\frac{b\beta k e^{-m\tau_{1}}}{cdu}\) and \(E_{2}\) occurs. Also, it has been shown that \(E_{2}\) is globally asymptotically stable.
From the reproductive number \(R_{0}(\tau_{1})= \frac{k\beta N e^{-m\tau_{1}}}{adp} \), which is a function of \(\tau_{1}\), we see that it is decreasing in delay \(\tau_{1}\) with \(R_{0} (\infty)= 0 \), which means that the intracellular delay \(\tau_{1}\) plays a positive role in preventing the virus. Because the larger \(\tau_{1}\) can bring \(R_{0} \) to a level lower than one. When the delay is chosen as the bifurcation parameter, it is shown that the delay plays an important role in determining the dynamical behavior of the system. This indeed suggests that delay is a very important fact which should not be missed in HIV-1 modeling. Finally, through numerical simulations, it can be concluded that delays in the infection process and virus production period play an important role in the disease control.
Declarations
Acknowledgements
We would like to thank the anonymous referees for their careful reading of the original manuscript and their many valuable comments and suggestions that greatly improved the presentation of this study. This work has been partially supported by King Saud University, Saudi Arabia.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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