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Viral dynamics of an HIV model with latent infection incorporating antiretroviral therapy

Advances in Difference Equations20162016:225

https://doi.org/10.1186/s13662-016-0952-x

  • Received: 17 February 2016
  • Accepted: 23 August 2016
  • Published:

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/mm3, 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 [14]. 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, 46].

Mathematical models including the latent infection have been formulated to study HIV dynamics in-host in recent years [4, 712]. 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 [1520], 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

In this section, we will extend the basic HIV latent infection model with logistic growth for healthy CD4+ T-cells and drug therapy. The model includes healthy CD4+ T-cells (T), latently infected CD4+ T-cells (L), productively infected CD4+ T-cells (\(T^{*}\)), and free virus (V). Because of the use of protease inhibitors, we divide the free virus into infectious virus (\(V_{I}\)) and noninfectious virus (\(V_{NI}\)). The model can be described as
$$ \begin{aligned} &\frac{d}{dt}T(t)= \lambda-d_{T} T+rT \biggl(1-\frac{T}{T_{\max}} \biggr)-(1-n_{rt})kV_{I}T, \\ &\frac{d}{dt}L(t)=\eta(1-n_{rt})kV_{I}T-d_{L} L-aL, \\ &\frac{d}{dt}T^{*}(t)=(1-\eta) (1-n_{rt})kV_{I} T-\delta T^{*}+aL, \\ &\frac{d}{dt}V_{I}(t)=(1-n_{p}) N\delta T^{*}-cV_{I}, \\ &\frac{d}{dt}V_{NI}(t)=n_{p} N\delta T^{*}-cV_{NI}. \end{aligned} $$
(1)
Here, healthy CD4+ T-cells are produced from precursors at a constant rate λ, the parameter \(d_{T}\) is the natural death rate of CD4+ T-cells, r denotes the logistic growth rate, and \(T_{\max}\) represents the carrying capacity of the CD4+ T-cells population. The parameter k denotes the rate of infection between T-cells and infectious virus, and η is the fraction of infections leading to latency. \(d_{L}\) and δ are the death rates of latently infected T-cells and productively infected T-cells, respectively. The parameter a is the activated rate from latently infected cells to productively infected T-cells. We assume a productively infected T-cell can release on average N viral particles during its lifespan with the mean period \(1/\delta\), and the clearance rate of the HIV virus is c. \(n_{rt}\) and \(n_{p}\) represent the drug efficacy of RTI and PI (\(0 \leq n_{rt} <1\) and \(0 \leq n_{p} <1\)), respectively.
It should be noted that the fifth equation is independent from the first four equations in system (1). Therefore, the stability analysis of system (1) is equivalent to the following subsystem:
$$ \begin{aligned} &\frac{d}{dt}T(t)= \lambda-d_{T} T+rT \biggl(1-\frac{T}{T_{\max}} \biggr)-(1-n_{rt})kV_{I}T, \\ &\frac{d}{dt}L(t)=\eta(1-n_{rt})kV_{I}T-d_{L} L-aL, \\ &\frac{d}{dt}T^{*}(t)=(1-\eta) (1-n_{rt})kV_{I} T-\delta T^{*}+aL, \\ &\frac{d}{dt}V_{I}(t)=(1-n_{p}) N\delta T^{*}-cV_{I}. \end{aligned} $$
(2)
For simplicity, we denote
$$\bar{k}:=(1-n_{rt})k,\qquad \bar{N}:=(1-n_{p})N, \qquad \epsilon =1-(1-n_{rt}) (1-n_{p}). $$

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].

Now, we show the boundedness of solutions. From the first equation of (2), we have
$$\dot{T}\leq\lambda-d_{T} T+rT \biggl(1-\frac{T}{T_{\max}} \biggr), $$
so
$$ \lim_{t\rightarrow+\infty} \sup T(t)\leq T_{0}, $$
(3)
where
$$ T_{0}=\frac{T_{\max}}{2r} \biggl[r-d_{T}+ \sqrt{(r-d_{T})^{2}+\frac{4r\lambda }{T_{\max}}} \biggr]. $$
(4)
We define \(F=T+L+T^{*}\), and computing the derivative of F along the trajectories yields
$$\begin{aligned} \dot{F} =& \lambda-d_{T} T+rT \biggl(1-\frac{T}{T_{\max}} \biggr)- \delta T^{*}-d_{L}L \\ \leq& -d_{T} T-d_{L}L-\delta T^{*}+\lambda+rT \\ \leq& -hF+M_{0}, \end{aligned}$$
where \(M_{0}=\lambda+rT_{0}\), \(h=\min\{d_{T}, d_{L}, \delta\}\). Therefore, we see that F is ultimately bounded. From (3), we know that \(T(t)\) has an ultimate bound \(T_{0}\), therefore, \(L(t)\) and \(T^{*}(t)\) are ultimately bounded with some \(M_{1}\). From the fourth equation of system (2), we can easily see that \(V_{I}(t)\) is ultimately bounded with some \(M_{2}\). □

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.

Next, we will analyze the dynamics of system (2) in the following bounded feasible region:
$$\Gamma= \bigl\{ \bigl(T, L, T^{*}, V_{I}\bigr)\in\mathbf{R}_{+}^{4}: T\leq T_{0}, L, T^{*} \mbox{ and } V_{I} \leq M \bigr\} . $$
It can be observed that all solutions of (2) eventually enter and stay within Γ, and Γ is a positive invariant set.

3 Model analysis

We will find the possible equilibria of system (2) and will analyze their stabilities. First of all, we find the equilibria of system (2) by solving the following algebraic equations:
$$ \left \{ \textstyle\begin{array}{@{}l} 0=\lambda-d_{T} T+rT (1-\frac{T}{T_{\max}} )-(1-n_{rt})kV_{I}T,\\ 0=\eta(1-n_{rt})kV_{I}T-d_{L} L-aL,\\ 0=(1-\eta)(1-n_{rt})kV_{I} T-\delta T^{*}+aL,\\ 0=(1-n_{p}) N\delta T^{*}-cV_{I}, \end{array}\displaystyle \right . $$
(5)
for the unknown constants \(T, L, T^{*},V_{I}\). From equation (5), we can easily get an uninfected equilibrium \(E_{0}(T_{0}, 0, 0, 0)\), where \(T_{0}\) is given by the expression of (4). The other equilibrium is \(\bar{E}=(\bar{T}, \bar{L}, \bar{T^{*}}, \bar{V_{I}})\), where
$$\begin{aligned}& \bar{T} = \frac{c(a + d_{L})}{\bar{k}\bar{N}[a + (1-\eta)d_{L}]}, \\& \bar{L} = \frac{c\eta\bar{V}_{I}}{\bar{N}[a + (1-\eta)d_{L}]}, \\& \bar{T^{*}} = \frac{c\bar{V_{I}}}{\bar{N}\delta}, \\& \bar{V_{I}} = \frac{\lambda-{d_{T}}\bar{T} + r\bar{T}(1 -\frac{\bar {T}}{T_{\max}})}{\bar{k}\bar{T}}. \end{aligned}$$
There are some methods to determine the basic reproduction number \(\mathcal{R}_{0}\) [23, 24]. By the next generation method of Van den Driessche and Watmough [24], we obtain
$$ \mathcal{R}_{0}= \frac{a+(1-\eta)d_{L}}{c(a+d_{L})}\bar{k}\bar{N}T_{0}. $$
(6)
For the detailed calculation process refer to [7] with \(\tau_{1}=\tau_{2}=0\). If \(\eta=0\), then the basic reproduction number is \(\tilde{\mathcal{R}}_{0}= \bar{k}\bar{N}T_{0}/c\), which is the same as that of the HIV model without latent infection [1519]. Since \(0<\eta<1\), we see that \(\mathcal{R}_{0}<\tilde{\mathcal{R}}_{0}\). Biologically, the model without latent infection would overestimate the average number of infected T-cells.
It turns out that the value of \(\mathcal{R}_{0}\) determines the existence of the infected equilibrium, that is, Ē exists if and only if \(\mathcal{R}_{0}>1\). In fact,
$$\lambda=d_{T}T_{0}-rT_{0}\biggl(1- \frac{T_{0}}{T_{\max}}\biggr), $$
then
$$\begin{aligned} \bar{V_{I}} =& \frac{d_{T} T_{0}-rT_{0}(1-\frac{T_{0}}{T_{\max}})-{d_{T}}\bar{T} + r\bar{T}(1 -\frac{\bar{T}}{T_{\max}})}{\bar{k}\bar{T}} \\ =& \frac{(T_{0}-\bar{T})[d_{T} -r(1-\frac{T_{0}}{T_{\max}})+\frac{r\bar {T}}{T_{\max}}]}{\bar{k}\bar{T}} \\ =& \frac{(\frac{T_{0}}{\bar{T}} -1 )(\frac{\lambda}{T_{0}} + \frac{r\bar {T}}{T_{\max}})}{\bar{k}} \\ =& \frac{(\mathcal{R}_{0} -1 )(\frac{\lambda}{T_{0}} + \frac{r\bar {T}}{T_{\max}})}{\bar{k}} \\ >& 0. \end{aligned}$$
Through the expression of \(\mathcal{R}_{0}\), we can also rewrite \(\bar {T}=\frac{T_{0}}{\mathcal{R}_{0}}\).
To study the stability at the equilibrium \(\bar{E} (\bar{T}, \bar{L}, \bar{T^{*}}, \bar{V}_{I})\), we let \(y_{1}(t)=T(t)-\bar{T}, y_{2}(t)=L(t)-\bar{L}, y_{3}(t)=T^{*}(t)-\bar{T^{*}}, y_{4}(t)=V_{I}(t)-\bar{V}_{I}\). The linearized system of system (2) at \(\bar{E} (\bar{T}, \bar{L}, \bar{T^{*}}, \bar{V}_{I})\) is
$$\begin{aligned}& \frac{d}{dt}y_{1}(t) = \biggl(-d_{T}+r-\frac{2r\bar{T}}{T_{\max}}\biggr) y_{1} - \bar{k}\bar {T} y_{4}, \\& \frac{d}{dt}y_{2}(t) = \eta\bar{k}\bar{V_{I}} y_{1}-(d_{L}+a)y_{2} +\eta\bar{k} \bar{T} y_{4} , \\& \frac{d}{dt}y_{3}(t) = (1-\eta)\bar{k}\bar{V_{I}} y_{1} +a y_{2}-\delta y_{3}+(1-\eta)\bar{k}\bar{T}y_{4}, \\& \frac{d}{dt}y_{4}(t) = \bar{N}\delta y_{3}-cy_{4}. \end{aligned}$$

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

The characteristic equation for system (2) at the uninfected equilibrium \(E_{0}\) is
$$ \biggl(\xi+d_{T}-r+\frac{2r T_{0}}{T_{\max}}\biggr) \bigl( \xi^{3}+a_{1}\xi^{2}+a_{2} \xi+a_{3}\bigr)=0, $$
(7)
where
$$\begin{aligned}& a_{1} = a+d_{L}+c+\delta, \\& \begin{aligned}[b] a_{2} &= (a+d_{L}) (c+\delta)+c \delta-(1-\eta)\delta\bar{k}\bar{N}T_{0} \\ &= (a+d_{L}) (c+\delta)+c\delta \biggl( 1-(1-\eta)\frac{\mathcal{R}_{0} (a+d_{L})}{a+(1-\eta)d_{L}} \biggr) \\ &= (a+d_{L}) (c+\delta)+c\delta \biggl( 1-\mathcal{R}_{0}+ \frac{a\eta \mathcal{R}_{0}}{a+(1-\eta)d_{L}} \biggr), \end{aligned} \\& \begin{aligned}[b] a_{3} &= (a+d_{L})c\delta-a\eta\delta \bar{k}\bar{N}T_{0}-(a+d_{L}) (1-\eta )\delta\bar{k} \bar{N}T_{0} \\ &= (a+d_{L})c\delta-a \delta\bar{k}\bar{N}T_{0}-(1-\eta) \delta d_{L} \bar {k}\bar{N}T_{0} \\ &= (a+d_{L})c\delta-\mathcal{R}_{0} c\delta(a+d_{L}) \\ &= (a+d_{L}) (1-\mathcal{R}_{0})c\delta. \end{aligned} \end{aligned}$$
Clearly, \(\xi=-d_{T}+r-\frac{2r T_{0}}{T_{\max}}=-\frac{\lambda}{T_{0}}-\frac {rT_{0}}{T_{\max}}<0\) is a negative root of equation (7). The remaining roots of equation (7) are determined by the following equation:
$$ \xi^{3}+a_{1}\xi^{2}+a_{2} \xi+a_{3}=0. $$
(8)
It is obvious that \(a_{1}=a+d_{L}+c+\delta>0\), and \(a_{3}=(a+d_{L})(1-\mathcal {R}_{0})c\delta>0\) if \(\mathcal{R}_{0}<1\) is satisfied. Furthermore,
$$\begin{aligned} a_{1}a_{2}-a_{3} =& (a+d_{L}+c+ \delta) \biggl[ (a+d_{L}) (c+\delta)+c\delta \biggl( 1- \mathcal{R}_{0}+\frac{a\eta\mathcal{R}_{0}}{a+(1-\eta)d_{L}} \biggr) \biggr] \\ &{}-(a+d_{L}) (1-\mathcal{R}_{0})c\delta \\ =& (a+d_{L}) (c+\delta) (a+d_{L}+c+\delta)+(a+d_{L}) \frac{c\delta a\eta \mathcal{R}_{0}}{a+(1-\eta)d_{L}} \\ &{} +(c+\delta)c\delta \biggl( 1-\mathcal{R}_{0}+\frac{a\eta\mathcal {R}_{0}}{a+(1-\eta)d_{L}} \biggr), \end{aligned}$$
and it follows that \(a_{1}a_{2}-a_{3}>0\) if \(\mathcal{R}_{0}<1\) is satisfied. By the Routh-Hurwitz criterion, we know that equation (8) has negative real roots. Therefore, the uninfected equilibrium \(E_{0}\) is locally asymptotically stable if \(\mathcal{R}_{0}<1\).

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

Define a Lyapunov function \(W_{1}: R\times R\times R\times R\rightarrow R\)
$$\begin{aligned} W_{1} = \bigl[a+(1-\eta)d_{L}\bigr] \biggl( T-T_{0}-T_{0} \ln\frac{T}{T_{0}} \biggr) +aL+(a+d_{L})T^{*}+\frac{a+d_{L}}{\bar{N}}V_{I} \end{aligned}$$
for \(T>0, L\geq0, T^{*}\geq0, V_{I}\geq0\), and it is obvious that \(W_{1}\geq 0\). By the calculus knowledge of a multi-variable function, we can derive that \(W_{1}\) has a global minimum value attained at \(E_{0}\) and thus, \(W_{1}=0\) if and only if \((T, L, T^{*}, V_{I})=(T_{0}, 0, 0, 0)\). Computing the derivatives of \(W_{1}\) along the trajectories of system (2), and using the equalities \(\lambda=d_{T} T_{0}-rT_{0}(1-\frac{T_{0}}{T_{\max}})\) and \(\bar{k}\bar{N}T_{0}[a+(1-\eta )d_{L}]=c\mathcal{R}_{0}(a+d_{L})\), we derive that
$$\begin{aligned} \dot{W}_{1} =& \bigl[a+(1-\eta)d_{L}\bigr] \biggl[ \lambda-d_{T} T+rT \biggl(1-\frac {T}{T_{\max}} \biggr)- \bar{k}V_{I}T \biggr]\biggl(1-\frac{T_{0}}{T}\biggr) \\ &{} +a(\eta\bar{k}V_{I}T-d_{L} L-aL)+(a+d_{L}) \bigl((1-\eta)\bar{k}V_{I} T-\delta T^{*}+aL\bigr) \\ & +\frac{a+d_{L}}{\bar{N}}\bigl(\bar{N}\delta T^{*}-cV_{I} \bigr) \\ =& \bigl[a+(1-\eta)d_{L}\bigr] \biggl[(T_{0}-T) \biggl(d_{T}-r+\frac{rT_{0}}{T_{\max}}+\frac {rT}{T_{\max}}\biggr)- \bar{k}V_{I}T \biggr]\biggl(1-\frac{T_{0}}{T}\biggr) \\ &{} +\bigl[a+(1-\eta)d_{L}\bigr]\bar{k}V_{I}T- \frac{c(a+d_{L})}{\bar{N}}V_{I} \\ =& \bigl[a+(1-\eta)d_{L}\bigr] \biggl[(T_{0}-T) \biggl( \frac{\lambda}{T_{0}}+\frac {rT}{T_{\max}}\biggr)-\bar{k}V_{I}T \biggr] \biggl(1-\frac{T_{0}}{T}\biggr) \\ &{} +\bigl[a+(1-\eta)d_{L}\bigr]\bar{k}V_{I}T- \frac{c(a+d_{L})}{\bar{N}}V_{I} \\ =& \bigl[a+(1-\eta)d_{L}\bigr] \biggl[-\frac{(T-T_{0})^{2}}{T}\biggl( \frac{\lambda}{T_{0}}+\frac {rT}{T_{\max}}\biggr)-\bar{k}V_{I}(T-T_{0}) \biggr] \\ &{} +\bigl[a+(1-\eta)d_{L}\bigr]\bar{k}V_{I}T- \frac{c(a+d_{L})}{\bar{N}}V_{I} \\ =& -\bigl[a+(1-\eta)d_{L}\bigr]\frac{(T-T_{0})^{2}}{T}\biggl( \frac{\lambda}{T_{0}}+\frac {rT}{T_{\max}}\biggr)-\frac{c(a+d_{L})}{\bar{N}}(1- \mathcal{R}_{0})V_{I}. \end{aligned}$$
It is clear to see that \(\dot{W}_{1}<0\) if \(\mathcal{R}_{0}<1\). \(\dot {W}_{1}=0\) if and only if
$$\begin{aligned} T= T_{0}, \qquad L=T^{*}=V_{I}=0. \end{aligned}$$
So, the maximum invariant set in \(\{\psi\in\Gamma|\dot{W}_{1}=0\}\) is only the set \(\{E_{0}\}\). By the LaSalle invariance principle [25], we get the global attraction of \(E_{0}\). This and Theorem 3.1 indicate the global asymptotic stability of \(E_{0}\). □
To prove the stability of the infected equilibrium Ē, we denote
$$\begin{aligned} B_{1} :=& d_{T}-r+\frac{2r\bar{T}}{T_{\max}}+\bar{k} \bar{V}_{I}=\frac{\lambda}{\bar{T}}+\frac{r\bar{T}}{T_{\max}}>0, \\ B_{2} :=& c\delta-(1-\eta)\delta\bar{k}\bar{N}\bar{T}=c\delta \biggl(1-\frac{(a+d_{L})(1-\eta)}{a+(1-\eta)d_{L}} \biggr) \\ =&\frac{c\delta\eta a}{a+(1-\eta)d_{L}}>0. \end{aligned}$$

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

The characteristic equation for system (2) at the infected equilibrium Ē is
$$ \xi^{4}+b_{1}\xi^{3}+b_{2} \xi^{2}+b_{3}\xi+b_{4}=0, $$
(9)
where
$$\begin{aligned}& b_{1} = B_{1}+a+d_{L}+c+\delta>0, \\& b_{2} = B_{1}(a+d_{L}+c+\delta)+(a+d_{L}) (c+\delta)+B_{2}>0, \\& b_{3} = B_{1}\bigl[(a+d_{L}) (c+ \delta)+B_{2}\bigr]+(1-\eta)\delta\bar{k}^{2}\bar{N}\bar {V}_{I}\bar{T}>0, \\& b_{4} = c\delta\bar{k}\bar{V}_{I}(a+d_{L})>0. \end{aligned}$$
In the following, we will use the Routh-Hurwitz criterion to verify the stability of the infected equilibrium. We have
$$\begin{aligned} H_{1} =& b_{1}>0, \\ H_{2} =& b_{1}b_{2}-b_{3} \\ =& (B_{1}+a+d_{L}+c+\delta) \bigl[B_{1}(a+d_{L}+c+ \delta)+(a+d_{L}) (c+\delta)+B_{2}\bigr] \\ &{} - B_{1}\bigl[(a+d_{L}) (c+\delta)+B_{2} \bigr]-(1-\eta)\delta\bar{k}^{2}\bar{N}\bar {V}_{I}\bar{T} \\ =& B_{1}^{2}(a+d_{L}+c+\delta) + (a+d_{L}+c+\delta)\bigl[(a+d_{L}) (c+\delta)+B_{2} \bigr] \\ &{} +B_{1}(a+d_{L}) (a+d_{L}+c+\delta) + B_{1}(a+d_{L}) (c+\delta) + B_{1}(c+\delta )^{2} \\ &{} -(1-\eta)\delta\bar{k}^{2}\bar{N}\bar{V}_{I}\bar{T} \\ =& (a+d_{L}+c+\delta)\bigl[B_{1}^{2}+(a+d_{L}) (c+\delta )+B_{2}\bigr]+B_{1}(a+d_{L}) (a+d_{L}+2c+2\delta) \\ &{} +B_{1}\bigl(c^{2}+\delta^{2}\bigr)+\bigl(2c \delta B_{1}-(1-\eta)\delta\bar{k}^{2}\bar{N}\bar {V}_{I}\bar{T}\bigr) \\ =& (a+d_{L}+c+\delta)\bigl[B_{1}^{2}+(a+d_{L}) (c+\delta )+B_{2}\bigr]+B_{1}(a+d_{L}) (a+d_{L}+2c+2\delta) \\ &{} +B_{1}\bigl(c^{2}+\delta^{2}\bigr)+c\delta \biggl(2B_{1}-\frac{(1-\eta )(a+d_{L})}{a+(1-\eta)d_{L}}\bar{k}\bar{V}_{I} \biggr) \\ >& (a+d_{L}+c+\delta)\bigl[B_{1}^{2}+B_{2}+(a+d_{L}) (c+\delta )\bigr]+B_{1}(a+d_{L}) (a+d_{L}+2c+2 \delta) \\ &{} +B_{1}\bigl(c^{2}+\delta^{2}\bigr)+c \delta(2B_{1}-\bar{k}\bar{V}_{I}), \\ H_{3} =& b_{3}H_{2}-b_{4}b_{1}^{2} \\ >& \bigl[B_{1}\bigl[(a+d_{L}) (c+\delta)+B_{2} \bigr]+(1-\eta)\delta\bar{k}^{2}\bar{N}\bar {V}_{I}\bar{T} \bigr] \bigl\{ (a+d_{L}+c+\delta)\bigl[B_{1}^{2}+B_{2} \\ &{}+(a+d_{L}) (c+\delta)\bigr]+B_{1}(a+d_{L}) (a+d_{L}+2c+2\delta)+B_{1}\bigl(c^{2}+\delta ^{2}\bigr)+c\delta(2B_{1}-\bar{k}\bar{V}_{I}) \bigr\} \\ &{}-c\delta\bar{k}\bar{V}_{I}(a+d_{L}) (B_{1}+a+d_{L}+c+\delta)^{2} \\ =& D_{1}+D_{2}+D_{3}+D_{4}, \end{aligned}$$
where
$$\begin{aligned} D_{1} =& \bigl[B_{1}B_{2}+(1-\eta)\delta \bar{k}^{2}\bar{N}\bar{V}_{I}\bar {T}-c\delta\bar{k} \bar{V}_{I}\bigr](a+d_{L}) (c+\delta+B_{1}) (a+d_{L}+c+\delta+B_{1}) \\ =& \bigl\{ B_{1}B_{2}+\bar{k}\bar{V}_{I} \bigl[(1-\eta)\delta\bar{k}\bar{N}\bar {T}-c\delta\bigr] \bigr\} (a+d_{L}) (c+\delta+B_{1}) (a+d_{L}+c+ \delta+B_{1}) \\ =& B_{2}(B_{1}-\bar{k}\bar{V}_{I}) (a+d_{L}) (c+\delta+B_{1}) (a+d_{L}+c+\delta +B_{1}), \\ D_{2} =& \bigl(B_{1}B_{2}+(1-\eta)\delta \bar{k}^{2}\bar{N}\bar{V}_{I}\bar{T}\bigr) \biggl[B_{2}(a+d_{L})+(c+\delta) \bigl(B_{1}^{2}+B_{2} \bigr)+B_{1}\bigl(c^{2}+\delta^{2}\bigr) \\ &{}+c\delta\biggl(2B_{1}-\frac{(1-\eta)(a+d_{L})}{a+(1-\eta)d_{L}}\bar{k}\bar {V}_{I}\biggr)\biggr] \\ >& \bigl(B_{1}B_{2}+(1-\eta)\delta\bar{k}^{2} \bar{N}\bar{V}_{I}\bar{T}\bigr) \bigl[ B_{2}(a+d_{L})+(c+ \delta) \bigl(B_{1}^{2}+B_{2}\bigr)+B_{1} \bigl(c^{2}+\delta^{2}\bigr) \\ &{}+c\delta(2B_{1}-\bar{k}\bar{V}_{I})\bigr], \\ D_{3} =& \bigl[B_{1}(c+\delta)^{2}-c\delta \bar{k}\bar {V}_{I}\bigr](a+d_{L})^{2}(a+d_{L}+c+ \delta+B_{1})+B_{1}^{2}(c+\delta)^{2}(a+d_{L})^{2}, \\ D_{4} =& B_{1}(c+\delta) (a+d_{L})\biggl[ \bigl(B_{1}^{2}+B_{2}\bigr) (a+d_{L}+c+ \delta)+B_{1}(a+d_{L})^{2} \\ &{} +B_{1}(c+\delta)^{2}-\frac{(1-\eta)(a+d_{L})}{a+(1-\eta)d_{L}}c\delta\bar {k} \bar{V}_{I}\biggr] \\ >& B_{1}(c+\delta) (a+d_{L}) \bigl\{ \bigl(B_{1}^{2}+B_{2} \bigr) (a+d_{L}+c+\delta)+B_{1}(a+d_{L})^{2} \\ &{} +\bigl[B_{1}(c+\delta)^{2}-c\delta\bar{k} \bar{V}_{I}\bigr]\bigr\} . \end{aligned}$$
With the condition \(d_{T}-r+\frac{2r\bar{T}}{T_{\max}}>0\), we have \(B_{1}>\bar{k}\bar{V}_{I}\), then
$$\begin{aligned}& B_{1}(c+\delta)^{2}-c\delta\bar{k}\bar{V}_{I} = B_{1}\bigl(c^{2}+\delta^{2}\bigr)+c\delta (2B_{1}-\bar{k}\bar{V}_{I})>0, \\& D_{1}>0, \qquad D_{2}>0, \qquad D_{3}>0, \qquad D_{4}>0. \end{aligned}$$
Thus,
$$H_{2}>0 \quad\mbox{and}\quad H_{3}>0. $$
It is clear to see that
$$\begin{aligned} H_{4} = b_{4}H_{3}>0. \end{aligned}$$
The Routh-Hurwitz criterion is satisfied, thus we see that equation (9) has negative real roots. Therefore, the infected equilibrium Ē is locally asymptotically stable with the condition \(d_{T}-r+\frac{2r\bar {T}}{T_{\max}}>0\). □

4 Sensitivity analysis

In Section 3, we have discussed the dynamics for system (2) at the uninfected and infected equilibria, respectively. It has been proved that the conditions \(H_{2}>0\) and \(H_{3}>0\) (they are more easily to satisfy than the condition \(d_{T}-r+\frac{2r\bar {T}}{T_{\max}}>0\) numerically) are sufficient to guarantee the local asymptotic stability of the infected equilibrium Ē if \(\mathcal {R}_{0}>1\). From Table 1, we observe that some parameters have large variations in our model, which may affect the outcomes greatly. Therefore, it is necessary to do an uncertainty analysis and a sensitivity analysis, so that the key parameters which can impact the stability of the infected equilibrium can be discovered.
Table 1

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

[4, 11, 12, 20]

\(d_{T}\)

Death rate of healthy T-cells

day−1

0.03

0.01

0.01

0.01-0.1

[4, 11, 12, 20]

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

[4, 11, 12, 16, 20]

η

Fraction of infections that result in latency

 

0.02

0.001

0.5

0.001-0.5

[4, 11, 12]

\(d_{L}\)

Death rate of latently infected T-cells

day−1

0.001

0.004

0.2

0.001-0.2

[4, 11, 12]

a

Transition rate

day−1

0.1

0.01

0.3

0.01-0.3

[4, 11, 12]

δ

Death rate of infected T-cells

day−1

1

1

0.8

0.5-1.4

[4, 11, 12, 20]

N

Burst term

virions/cell

1,000

200

500

200-3,000

[4, 11, 12, 16, 20]

c

Clearance rate of virus

day−1

20

3

15

3-36

[4, 11, 12, 20]

\(n_{rt}\)

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 [2628]. 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.

To detect the key parameters to impact the local stability of the infected equilibrium, we compute the partial rank correlation coefficients (PRCCs) between each input parameter and three corresponding outputs. Figure 1 shows the PRCCs results for each input parameters. The sign of the PRCCs indicates that the input parameter has a positive or negative effect on the corresponding output. From Figure 1, we observe that parameters \(k, N\), and λ have a positive effect on the size of \(H_{2}\) and \(H_{3}\) (\(|\mathrm{PRCC}|>0.5\)), while the parameters \(n_{p}\) and \(n_{rt}\) have a negative effect, and we also find that the parameters \(k, N\), and r have a positive effect on the size of \(\mathcal{R}_{0}-1\), while the parameters \(n_{p}, n_{rt}, d_{T}\), and c have a negative effect. As we have seen, the local asymptotic stability of the infected equilibrium depends on the sign of these three outputs (\(H_{2}>0, H_{3}>0\), and \(\mathcal{R}_{0}-1>0\)) from Section 3, Figure 1 further indicates that the parameters k and N have a positive effect on the three outputs, while \(n_{p}\) and \(n_{rt}\) have a negative effect.
Figure 1
Figure 1

The PRCCs between input parameters and three outputs ( \(\pmb{H_{2}, H_{3}}\) and \(\pmb{\mathcal{R}_{0}-1}\) ). The parameters k and N have a positive effect on the three outputs, while \(n_{p}\) and \(n_{rt}\) have a negative effect.

5 Numerical simulations

In this section, we carry out numerical simulations to explore the cell and viral dynamics of system (2). Using Data1 values in Table 1 and computing, \(\mathcal{R}_{0}=1.7657>1\) with \(n_{rt}=0.4, n_{p}=0.5\) (combination drug efficacy \(\epsilon=0.70\)), \(\mathcal{R}_{0}=1.1772>1\) with \(n_{rt}=0.5, n_{p}=0.6\) (\(\epsilon=0.80\)), and \(\mathcal{R}_{0}=0.3531<1\) with \(n_{rt}=0.7, n_{p}=0.8\) (\(\epsilon=0.94\)), respectively. To show how antiretroviral therapy impacts the T-cell count and viral load, we choose the infected equilibrium without drug therapy \((T(0), L(0), T^{*}(0), V_{I}(0))=(200.0396, 4.2248, 21.3308, 1{,}066.5412)\) as the initial value. Figure 2 shows that the infected equilibrium is locally asymptotically stable when the combination drug efficacy \(\epsilon=0.70\) and \(\epsilon=0.80\), while the infected equilibrium does not exist and the uninfected equilibrium is globally asymptotically stable when the combination drug efficacy \(\epsilon=0.94\). With the increase of combination drug efficacy, the number of healthy CD4+ T-cells becomes larger, while the number of latently infected CD4+ T-cells, productively infected CD4+ T-cells and infectious virions become smaller. Also, the influence of drug therapy on T-cell count and viral load changes dramatically. Figure 2 also demonstrates that the stability of the equilibrium changes as combination drug efficacy increases.
Figure 2
Figure 2

With the increase of the combination drug therapy ϵ , the number of the healthy CD4 \(\pmb{^{+}}\) T-cells becomes larger (a), while the number of the latently infected CD4 \(\pmb{^{+}}\) T-cells, productively infected CD4 \(\pmb{^{+}}\) T-cells and infectious virions (b), (c), (d) become smaller. The virus can be eliminated when \(\epsilon=0.94\). For the parameters see the Data1 values of Table 1.

In order to examine the effect of the logistic term \(rT(1-\frac {T}{T_{\max}})\) in system (2), we choose the infected equilibrium without drug therapy when \(r=0\) (that is, the model in [10, 11]) \((T(0), L(0), T^{*}(0), V_{I}(0))=(150.0429, 0.6071,8.4971,566.4762)\) as the initial value, and we use Data2 values in Table 1. We observe that the logistic term can cause transient oscillations at the beginning of the viral progression, and it has no effect on the stability of the infected equilibrium (see Figure 3). Figure 3 also demonstrates that the model without logistic term would underestimate the number of the latently infected CD4+ T-cells, productively infected CD4+ T-cells, and infectious virions, while it does not change the number of healthy CD4+ T-cells obviously.
Figure 3
Figure 3

When \(\pmb{\mathcal{R}_{0}>1}\) , compared system ( 2 ) to the model without logistic term, the logistic term can cause transient oscillations to all populations at the beginning, and finally the model without logistic term may underestimate the number of the latently infected CD4 \(\pmb{^{+}}\) T-cells, productively infected CD4 \(\pmb{^{+}}\) T-cells, and infectious virions (b), (c), (d), while there is no too much difference with the number of healthy CD4 \(\pmb{^{+}}\) T-cells (a). For the parameters see the Data2 values of Table 1.

Researchers have done a lot of work about HIV model, and most of them did not consider latent infection. Here, we consider the model without latent infection which has been studied in [1519]:
$$ \begin{aligned} &\frac{d}{dt}T(t)= \lambda-d_{T} T+rT \biggl(1-\frac{T}{T_{\max}} \biggr)-(1-n_{rt})kV_{I}T, \\ &\frac{d}{dt}T^{*}(t)=(1-n_{rt})kV_{I} T-\delta T^{*}, \\ &\frac{d}{dt}V_{I}(t)=(1-n_{p}) N\delta T^{*}-cV_{I}. \end{aligned} $$
(10)
It is a special case of system (2) with \(\eta=d_{L}=a=0\). In order to observe the effect of latent infection of system (2), we choose the infected equilibrium of system (10) without drug therapy (\((T(0), T^{*}(0), V_{I}(0))=(300,38.7500,1{,}033.3333)\) with tiny latently infected T-cells 10−3) as the initial value, and use Data3 values in Table 1. The dynamics of system (2) and system (10) is shown in Figure 4, which indicates that system (10) would underestimate the total number of CD4+ T-cells and overestimate the number of infectious virions.
Figure 4
Figure 4

When \(\pmb{\mathcal{R}_{0}>1}\) , compared to system ( 2 ), system ( 10 ) may underestimate the total number of T-cells (a), and overestimate the number of infectious virions (b). For the parameters see the Data3 values of Table 1.

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 [1519], 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

(1)
College of Science, China University of Petroleum (East China), Qingdao, Shandong, 266580, China
(2)
School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, Henan, 471023, China

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