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Viral dynamics of an HIV model with latent infection incorporating antiretroviral therapy
- Yan Wang^{1}Email authorView ORCID ID profile,
- Jun Liu^{1} and
- Luju Liu^{2}
https://doi.org/10.1186/s13662-016-0952-x
© Wang et al. 2016
- Received: 17 February 2016
- Accepted: 23 August 2016
- Published: 31 August 2016
Abstract
In this paper, we construct an HIV infection model which includes latent infection, logistic growth for healthy CD4^{+} T-cells, and antiretroviral therapy. We obtain the global asymptotic stability of the uninfected equilibrium by constructing a Lyapunov function, and we give a sufficient condition for the local asymptotic stability of the infected equilibrium. We also use the latin hypercube sampling technique to identify the key parameters in determining the stability of the infected equilibrium. By numerical simulations, we observe that the model without logistic growth would underestimate the number of infectious virions, while the model without latent infection would overestimate the number of infectious virions.
Keywords
- HIV
- CD4^{+} T-cells
- latent infected T-cells
- antiretroviral therapy
- asymptotic stability
1 Introduction
Acquired immune deficiency syndrome (AIDS) is caused by human immunodeficiency virus (HIV), and it has become a very serious threat to the health of the people all over the world since it was first found in 1981. HIV-1 infects CD4 lymphocytes with CD4 molecules in the human body selectively, especially CD4^{+} T-cells. When individuals are infected with HIV-1 over a long period of time (5-10 years), the body’s CD4^{+} T-cell count will gradually decline to 200 cells/mm^{3}, and the viral load will increase sharply. Finally, the body’s immune system will be severely damaged and the human body will be much more vulnerable to a series of opportunistic infections. Fortunately, the virus replication can be suppressed through the use of highly active antiretroviral therapy (HAART). HAART is commonly composed of reverse transcriptase inhibitors (RTI) and protease inhibitors (PI). RTI can prevent the formation of HIV RNA and DNA in the CD4^{+} T-cell host, so that the virus infection could not form provirus, and PI can restrain the virus protease hydrolysis and inhibit infected T-cells to produce mature infectious virions. However, people living with HIV-1 cannot recover in the process of long-time use of antiretroviral therapy, and the HIV-1 virus cannot be eradicated thoroughly [1–4]. Due to the existence of the latent reservoir, some HIV-1 virus particles can escape the immune clearance by hiding in the static memory CD4 T-cells [1]. Consequently, the latent infection is a major barrier to the elimination of HIV-1 virus [1, 4–6].
Mathematical models including the latent infection have been formulated to study HIV dynamics in-host in recent years [4, 7–12]. Banks et al. investigated that the model with HIV latent infection was in accordance with the actual measured patient data especially when the viral load was lower than the detectable level, and their model could better predict the trend of the virus [8]. Rong and Perelson [4, 11, 12] developed a kind of mathematical models with latent infection and antiretroviral therapy, examined the relationships among combination drug therapy, viral blips, and the number of latent infected T-cells, and showed that the viral replication from the latent reservoir might result in low-level persistence of viraemia during combination drug therapy. However, in each of these studies, a linear growth rate for healthy CD4^{+} T-cells with latent infection models was performed. In fact, Ho et al. in [13] and Sachsenberg et al. in [14] indicated that the mitosis for CD4^{+} T-cells was density-dependent on the number of T-cells, and the mitosis would decrease if the number of T-cells increased to a certain value. Recently, mitosis in healthy CD4^{+} T-cells was represented by a logistic growth term in within-host HIV-1 models [15–20], but these studies ignored the effect of latent infection. Also, Chomont et al. showed that T-cell survival and homeostatic mitosis could drive the number of latently infected T-cells [6]. Therefore, the inclusion of both a logistic growth term and latent infection in-host model is more reasonable, and the model would have further influence on the model progression.
The paper is organized as follows. In the next section, we will formulate an HIV infection model including latent infection, drug therapy, and logistic growth for healthy CD4^{+} T-cells, and we will address the positivity and boundedness of the model solutions. In Section 3, the basic reproduction number is derived by the next generation method. Stability analyses of the uninfected equilibrium and the infected equilibrium are given. In Section 4, sensitivity analyses with the latin hypercube sampling method are conducted. Numerical simulations with realistic parameter values are illustrated to demonstrate model behaviors in Section 5. Finally, we conclude our work and mention future work.
2 Model and well-posedness
The following theorem illustrates that the solutions of system (2) are positive and bounded.
Theorem 2.1
Let \((T(t), L(t), T^{*}(t), V_{I}(t))\) be the solution of system (2) with the initial values \((T(0), L(0), T^{*}(0), V_{I}(0))\in\mathbf{R}_{+}^{4} \), where \(\mathbf{R}_{+}^{4}=\{ (x_{1}, x_{2}, x_{3}, x_{4})|x_{j}\geq 0, j=1, 2, 3, 4\}\), Then \(T(t), L(t), T^{*}(t)\), and \(V_{I}(t)\) are all unique non-negative and ultimately bounded.
Proof
The right hand side functions of system (2) are continuous and satisfy the Lipschitz condition; by the existence and uniqueness of solutions for ordinary differential equations [21], we see that system (2) has a unique solution \((T(t), L(t), T^{*}(t), V_{I}(t))\in\mathbf{C}([0,+\infty), \mathbf {R}_{+}^{4})\) with non-negative initial values.
By the first equation of system (2), we obtain \(\dot{T}|_{T=0}=\lambda>0\). Then we see that \(T(t)\geq0\) for every \(t\geq0\) [22].
By the second equation of system (2), we get \(\dot{L}|_{L=0}=\eta\bar{k} V_{I}T \geq0\). Thus, we derive that \(L(t)\geq0\) is established. In the following, we will use reduction ad absurdum to prove the correctness of this statement.
Assume there is a \(t_{1}>0\) with \(t_{1}=\operatorname{inf}\{t|L(t)=0, t>0\}\), such that \(\dot{L}(t_{1})|_{L(t_{1})=0}=\eta\bar{k} V_{I}(t_{1}) T(t_{1})< 0\). That is to say, \(L(t_{1})=0\), \(L(t)>0\) with \(t\in[0, t_{1})\) and \(V_{I}(t_{1})<0\). As \(V_{I}(0)\geq0\), there exists a \(t_{2}>0\) with \(t_{2}=\operatorname{inf}\{t|V_{I}(t)=0, t\in[0, t_{1})\}\), and thus \(\dot {V}_{I}(t_{2})\leq0\). Moreover, we get \(\dot{V}_{I}(t_{2})=\bar{N}\delta T^{*}(t_{2})\leq0\) from the fourth equation of system (2). Therefore, we can deduce that \(T^{*}(t_{2})\leq0\). Since \(T^{*}(0) \geq0\), there exists a \(t_{3}>0\) with \(t_{3}=\operatorname{inf}\{t|T^{*}(t)=0, t\in[0, t_{2})\}\), and thus \(\dot {T}^{*}(t_{3})\leq0\). On the other hand, from the third equation of system (2), \(\dot{T}^{*}(t_{3})=(1-\eta)\bar{k}V_{I}(t_{3})T(t_{3})+a L(t_{3})>0\) (\(0< t_{3}< t_{2}< t_{1}\)), which is a contradictory to the hypothesis. Similarly, we can verify that \(T^{*}(t)\geq0\) for every \(t\geq0\).
From the last equation of system (2), we get \(\dot {V}_{I}|_{V_{I}=0}=\bar{N} \delta T^{*}\geq0\), so we have \(V_{I}(t)\geq0, t\geq0\) [22].
Denote \(M=\max\{M_{1}, M_{2}\}\). It follows that \(T(t)\leq T_{0}, L(t)\leq M, T^{*}(t)\leq M\) and \(V_{I}(t)\leq M\), for sufficiently large time t.
3 Model analysis
Theorem 3.1
The uninfected equilibrium \(E_{0}\) for system (2) is locally asymptotically stable if \(\mathcal{R}_{0}<1\), and it is unstable if \(\mathcal{R}_{0}>1\).
Proof
If \(\mathcal{R}_{0}>1\), it is easy to see that \(a_{3}<0\). So, equation (8) has at least one positive root. Therefore, the uninfected equilibrium \(E_{0}\) is unstable when \(\mathcal{R}_{0}>1\). □
Theorem 3.2
If \(\mathcal{R}_{0}<1\), the uninfected equilibrium \(E_{0}\) for system (2) is globally asymptotically stable.
Proof
Theorem 3.3
When \(\mathcal{R}_{0}>1\), the infected equilibrium Ē for system (2) is locally asymptotically stable if the condition \(d_{T}-r+\frac{2r\bar {T}}{T_{\max}}>0\) is satisfied.
Proof
4 Sensitivity analysis
List of parameters
Paras | Definition | Unit | Data1 | Data2 | Data3 | Range | Source |
---|---|---|---|---|---|---|---|
λ | T-cells source term | \(\mu l^{-1}~\mbox{day}^{-1}\) | 10 | 10 | 10 | 1-10 | |
\(d_{T}\) | Death rate of healthy T-cells | day^{−1} | 0.03 | 0.01 | 0.01 | 0.01-0.1 | |
r | Growth rate of T-cells | day^{−1} | 0.1 | 0.03 | 0.1 | 0.03-0.1 | [20] |
\(T_{\max}\) | Carrying capacity of T-cells | \(\mu l^{-1}\) | 1,500 | 1,500 | 1,500 | 1,500 | [20] |
k | Infection rate | \(\mu l~\mbox{day}^{-1}\) | 10^{−4} | 10^{−4} | 10^{−4} | 10^{−5}-10^{−2} | |
η | Fraction of infections that result in latency | 0.02 | 0.001 | 0.5 | 0.001-0.5 | ||
\(d_{L}\) | Death rate of latently infected T-cells | day^{−1} | 0.001 | 0.004 | 0.2 | 0.001-0.2 | |
a | Transition rate | day^{−1} | 0.1 | 0.01 | 0.3 | 0.01-0.3 | |
δ | Death rate of infected T-cells | day^{−1} | 1 | 1 | 0.8 | 0.5-1.4 | |
N | Burst term | virions/cell | 1,000 | 200 | 500 | 200-3,000 | |
c | Clearance rate of virus | day^{−1} | 20 | 3 | 15 | 3-36 | |
\(n_{rt}\) | RTI efficacy | 0.4, 0.5, 0.7 | 0.4 | 0.3 | 0-1 | - | |
\(n_{p}\) | PI efficacy | 0.5, 0.6, 0.8 | 0.5 | 0.4 | 0-1 | - |
To examine the uncertainty analysis of the local stability of the infected equilibrium, we use the latin hypercube sampling (LHS) method to sample parameter ranges [26–28]. We choose the sample size \(n=1{,}000\), and we treat each input variable (\(\lambda, d_{T}, r, k, \eta, d_{L}, a, \delta, N, c\), \(n_{rt}\), and \(n_{p}\)) as a uniform distribution and treat each set of value of \(H_{2}, H_{3}\), and \(\mathcal {R}_{0}-1\) as the output variables. All the parameter ranges can be found in Table 1 in detail. A thousand data sets are generated from the 12 input variables distributions and 1,000 output data sets of the three output variables are obtained. Repeating the above procedure ten times, we investigate that there are a minimum of 855 and a maximum of 881 in 1,000 values satisfying \(\mathcal{R}_{0}>1\) for determining the existence of the infected equilibrium, and we also observe that there are a minimum of 855 and a maximum of 881 in 1,000 values satisfying \(H_{2}>0, H_{3}>0\), and \(\mathcal{R}_{0}>1\) for identifying the local stability of the infected equilibrium. Therefore, we conclude that the probability of the local stability for the infected equilibrium is between 0.855 and 0.881, and thus the local stability for the infected equilibrium is likely to occur. This phenomenon is consistent with the actual situation. Moreover, we observe that the sufficient conditions \(H_{2}>0\) and \(H_{3}>0\) are always satisfied when the basic reproduction number \(\mathcal{R}_{0}>1\). That is to say, there is no stability changes at the infected equilibrium if \(\mathcal{R}_{0}>1\), numerically.
5 Numerical simulations
6 Conclusions
We have studied an HIV model including latent infection, antiretroviral therapy and a logistic growth for healthy CD4^{+} T-cells. If the basic reproduction number is less than one, we obtained the global asymptotic stability of the uninfected equilibrium. If the basic reproduction number is greater than one, we proved local asymptotic stability of the infected equilibrium. Through latin hypercube sampling method, we investigate that, as long as the infected equilibrium exists, it should be locally asymptotically stable ultimately. Furthermore, we derive that the infection rate k and the burst size N have a positive effect on the stability of the infected equilibrium, while the drug efficacy \(n_{p}\) and \(n_{rt}\) have a negative effect.
Comparing our model behaviors with those established in Perelson et al. [10] and Rong and Perelson [11] without logistic growth, we found that their models would underestimate the number of the latently infected CD4^{+} T-cells, productively infected CD4^{+} T-cells and infectious virions. Also, comparing our model with [15–19], which do not include latently infected T-cells, we observed that those models would underestimate the total number of T-cells, and overestimate the number of infectious virions. Our conclusion can be regarded as an extension of the work of Perelson et al. [10] and Rong and Perelson [11] when \(r=0\).
In fact, cytotoxic T lymphocyte (CTL) is closely related to the suppression of viral replication and disease progression, and it plays a major part in the control of viral infection [29]. A mathematical model including latent infection and the influence of CTLs will be our future research work.
Declarations
Acknowledgements
The authors are grateful to thank the three anonymous referees for their careful reading and valuable suggestions. This research was supported by National Natural Science Foundation of China (Grant No. 11301543 (YW), No. 11401589 (JL) and No. 11301314), by the Natural Science Foundation of Shandong Province, China (Grant No. ZR2013AL018) (JL), by the Scientific Research Foundation for Doctoral Scholars of Haust (09001535) (LL), and by the Educational Commission of Henan Province of China (14B110021) (LL).
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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