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AndronovHopf bifurcation and sensitivity analysis of a timedelay HIV model with logistic growth and antiretroviral treatment
Advances in Difference Equations volume 2017, Article number: 138 (2017)
Abstract
A mathematical model of the infection of CD4+ Tcells by HIV that includes the effects of treatment by a reverse transcriptase inhibitor (RTI) and a protease inhibitor (PI) is studied. The model includes three populations of CD4+ Tcells (healthy cells, latentlyinfected cells which cannot produce virus, and productivelyinfected cells which can produce virus) and two populations of free virus in the blood (infectious virus and noninfectious virus). The model includes a time delay between a Tcell becoming latently infected and productively infected. The model has a virusfree and a chronic infection equilibrium. It is shown that the model has AndronovHopf bifurcations leading to limit cycle behavior in the chronic infection region at critical values of the time delays. For three data sets obtained from the work of previous authors, numerical simulations have given critical delay values ranging from approximately 15 days to more than 200 days. This range includes the period of approximately 50 days for intermittent viral blips reported by Rong and Perelson (Plos Comp. Biol. 5(10), 118 (2009)). Simple formulas are derived for the sensitivity indices of the equilibrium populations and the basic reproductive number with respect to all parameters in the model. Numerical simulations are carried out to support the analytical results. The numerical results suggest that the most effective methods of reducing both the basic reproductive number and the chronic infection CD4+ Tcell and virus populations are the following: (1) to increase the efficacy of the antiretroviral treatments and (2) to increase virus clearance rate, decrease infection rate, or decrease viral reproduction rate.
Introduction
The development of antiretroviral therapy using reverse transcriptase inhibitors (RTI) and protease inhibitors (PI) has resulted in a big reduction in the disability associated with HIV and with the rate of progression to AIDS. Although there is evidence that antiretroviral therapy does not completely eliminate the virus (see, e.g., [2, 3]), there is recent evidence that antiretroviral therapy can reduce the level of virus in an HIV person below detectable levels (see, e.g., [1, 4–6]) and that it can depress the HIV level in an HIV+ person sufficiently to effectively stop transmission of HIV from an HIV+ person to an uninfected person (see, e.g., [7–9]). However, in many countries antiretroviral therapy is not available. Also, infection by HIV can be asymptomatic [3], and these asymptomatic infected people may interact normally with people and pass on the disease to uninfected people.
Many researchers have developed mathematical models in an attempt to develop an understanding of HIV transmission at either the cell level (see, e.g., [1, 5, 6, 10–15]) or the population level (see, e.g., [16, 17] ).
In this paper, we consider a model for HIV infection at the cell level recently discussed by Wang et al. [15]. The model includes three populations of CD4+ Tcells (healthy cells, latentlyinfected cells which cannot produce virus, and productivelyinfected cells which can produce virus) and two populations of free virus (infectious virus and noninfectious virus). The model also includes the effects of treatments with a reverse transcriptase inhibitor (RTI) and a protease inhibitor (PI). In their paper, Wang et al. showed that the model has virusfree and chronic infection equilibrium solutions and they proved local and global stability, boundedness and positivity of these solutions. They also used a latin hypercube sampling technique for sensitivity analysis of the parameters in their model.
The model of Wang et al. is based on a model discussed by Rong and Perelson [1, 5, 6] with the main difference being the addition of a logistic growth term for healthy CD4+ Tcells. One of the important questions discussed in the Rong and Perelson papers is the mechanism that produces an intermittent viral blip with a period of approximately 50 days when subjects are treated with highly active antiretroviral therapy (HAARV). It is well known (see, e.g., [18]) that time delays can produce bifurcations leading to limit cycle behavior in both discretetime and continuoustime dynamical systems. One of the aims of the present paper is to check if time delays for procession of latently infected Tcells to productively infected Tcells could result in viral bliptype behavior.
In the present paper, we develop simple analytical formulas for sensitivity indices for the basic reproductive number, and for the virusfree and chronic equilibrium populations of the timedelay model. We also show that the timedelay model can undergo AndronovHopf bifurcations in the chronic equilibrium solutions at critical time delays and that limit cycle behavior in all five populations occurs at time delays greater than the critical values.
Timedelay model
We consider the following model, which is generalized from the model of Wang et al. [15] by including a time delay for the procession of latently infected Tcells to productively infected Tcells. The variables in the model are defined in Table 1, and the parameters are defined in Table 2.
Equilibrium points
As shown by Wang et al. [15] the system (1)(5) has a virusfree and a chronic equilibrium point. For completeness, we will briefly summarize the proof here.
Theorem 1
The system (1)(5) has two equilibrium points:

1.
Virusfree:
$$ \bigl(T_{1}^{*},L_{1}^{*},I_{1}^{*},V_{1}^{*},W_{1}^{*} \bigr) = \biggl(\frac{T_{\mathrm{max}}}{2r} \biggl(rd_{T}+\sqrt{(rd_{T})^{2}+4 \frac{r\Lambda}{T_{\mathrm{max}}}} \biggr),0,0,0,0 \biggr). $$(6) 
2.
Chronic infection:
$$ \begin{aligned} &T_{2}^{*} = \frac{c(a+d_{L})}{\overline{k}\overline{N}(a+(1\eta)d_{L})},\qquad L_{2}^{*} = \frac{c\eta}{\overline{N}(a+(1\eta)d_{L})}V_{2}^{*} , \\ &I_{2}^{*} = \frac{c}{\overline{N}d_{I}}V_{2}^{*} ,\qquad V_{2}^{*} = \frac{1}{\overline{k}} \biggl(\frac{T_{1}^{*}}{T_{2}^{*}}1 \biggr) \biggl(\frac{\Lambda}{T_{1}^{*}} +r\frac{T_{2}^{*}}{T_{\mathrm{max}}} \biggr), \\ &W_{2}^{*} = \frac{n_{p} N}{\overline{N}}V_{2}^{*} , \end{aligned} $$(7)where \(\overline{k}=(1n_{rt})k\), \(\overline{N}=(1n_{p})N\).
Proof

1.
Virusfree equilibrium. Setting \(L_{1}^{*} = I_{1}^{*} = V_{1}^{*} = W_{1}^{*} = 0\), we have \(\frac{dL}{dt}=\frac{dI}{dt}=\frac{dV}{dt}=\frac{dW}{dt}= 0\). Then, setting \(\frac{dT}{dt}=0\) and choosing the positive solution for \(T_{1}^{*}\) gives the result in part 1 of the theorem.

2.
Chronic equilibrium. The equations for \(I_{2}^{*}\) and \(W_{2}^{*}\) follow immediately from the conditions \(\frac{dV}{dt}=\frac{dW}{dt} = 0\). Then, setting \(\frac{dL}{dt}=0\) gives
$$ L_{2}^{*} = \frac{\eta}{a+d_{L}}\overline{k}V_{2}^{*}T_{2}^{*}. $$(8)Then, adding equations (2) and (3), setting \(\frac{d(L+I)}{dt}=0\), and substituting in equation (8), we obtain the equation for \(T_{2}^{*}\) in (7). Then, substituting for \(T_{2}^{*}\) in (8), we obtain the equation for \(L_{2}^{*}\) in (7). Finally, setting \(\frac{dT}{dt}=0\), we obtain
$$\begin{aligned} V_{2}^{*} =& \frac{1}{\overline{k}T_{2}^{*}} \biggl(\Lambda +T_{2}^{*} \biggl(rd_{T} r\frac{T_{2}^{*}}{T_{\mathrm{max}}} \biggr) \biggr) \\ =&\frac{1}{\overline{k}} \biggl(\frac{T_{1}^{*}}{T_{2}^{*}}1 \biggr) \biggl( \frac{\Lambda}{T_{1}^{*}} +r\frac{T_{2}^{*}}{T_{\mathrm{max}}} \biggr). \end{aligned}$$(9)
The proof is complete. □
Note: The chronic infected virus population \(V_{2}^{*}\) is greater than 0, i.e., the chronic equilibrium exists, if and only if the parameter \(\displaystyle R_{0} = \frac{T_{1}^{*}}{T_{2}^{*}} > 1\). We shall show in the next section that \(R_{0}\) is the basic reproductive number of the model.
Basic reproductive number
For this model, as in most models with a virusfree and a chronic equilibrium state, there are three methods of determining the basic reproductive number. They are: (1) Lyapunov’s first method of checking the eigenvalues of the Jacobian of the linearized system at an equilibrium point (see, e.g., [19]), (2) the nextgeneration method of van den Driessche and Watmough [20], or (3) by finding the condition for the existence of the chronic equilibrium as in equation (9).
Nextgeneration method
In using the nextgeneration method, it is necessary to identify a suitable infected population in the model to choose as an initial infected population. Possible populations are \(L,I,V\). In this model L and I cannot be considered as separate initially infected populations since both infections come from infection of the susceptible population T by contact with the V population. The population V comes only from one population (the I population), and therefore we choose it as the initial infected population for the nextgeneration method and order the variables as \([V,T,L,I,W]^{T}\). As stated in Wang et al. [15], the nextgeneration method gives
This formula for \(R_{0}\) is in agreement with the result derived in (9) as the condition for existence of the chronic equilibrium.
Linearized equations and stability
For the timedelay model (1)(5), the linearized equations about an equilibrium point are (we let \(T(t)=T^{*}+x_{1}(t)\), \(L(t) = L^{*}+x_{2}(t)\), \(I(t)=I^{*}+x_{3}(t)\), \(V(t) = V^{*}+x_{4}(t)\), and \(W(t) = W^{*}+x_{5}(t)\), where \(x_{j}(T)\) are small perturbations)
Equation (11) can be written in matrix form as \(\frac{dx}{dt}= Jx(t)\), where \(x=(x_{1},x_{2},x_{3},x_{4},x_{5})^{T}\) and J is a Jacobian. Then, assuming a trial solution of the standard form \(x(t) = e^{\lambda t}v\), where v is a constant vector, we obtain the Jacobian
Using the RouthHurwitz conditions (see, e.g., [19]), Wang et al. [15] showed for the case of zero time delay that the eigenvalues of the Jacobian for the virusfree equilibrium have negative real parts for \(R_{0} =\frac{T_{1}^{*}}{T_{2}^{*}} < 1\) and the eigenvalues of the Jacobian for the chronic equilibrium have negative real parts for \(R_{0} > 1\). In a later section, we will use the Jacobian to find the critical time delays for AndronovHopf bifurcations.
Note: With modern mathematical software such as Matlab, Maple or Mathematica, it is, of course, very easy to numerically compute the eigenvalues of the Jacobian for both the virusfree and chronic equilibrium points for the case of zero time delay.
Sensitivity indices
We define normalized sensitivity indices for a quantity Q with respect to a parameter h as
There are at least three possible methods of computing sensitivity indices: (1) direct computation by differentiation of formulas for the quantity Q, (2) the method of Chitnis et al. [21] of linearizing the original nonlinear model equations to set up a system of linear algebraic equations for the sensitivity indices and then numerically solve these equations, and (3) the method used in Wang et al. [15] based on a Latin hypercube sampling technique.
In this paper, we use the first method of direct differentiation as it gives explicit formulas for the indices. We will first compute the sensitivity indices for \(T_{1}^{*}\), \(T_{2}^{*}\) with respect to the parameters by using the formulas in Theorem 1. We will then compute sensitivity indices for \(R_{0}\) using the formula in (10). Finally, we will compute the sensitivity indices for \(V_{2}^{*}\), \(L_{2}^{*}\), \(I_{2}^{*}\) and \(W_{2}^{*}\) using the formulas in Theorem 1.
Sensitivity indices for virusfree healthy Tcell population \(T_{1}^{*}\)
From (6), we have
The virusfree equilibrium Tcell population \(T_{1}^{*}\) is a function of the four parameters \(T_{\mathrm{max}}\), Λ, r and \(d_{T}\). By differentiation of (14) with respect to these four parameters, we obtain the sensitivity indices shown in Table 3.
Sensitivity indices for chronic healthy Tcell population \(T_{2}^{*}\)
From Theorem 1, we have
The chronic equilibrium Tcell population is a function of eight parameters, \(d_{L}\), a, c, k, N, \(n_{rt}\), \(n_{p}\) and η. The sensitivity indices of \(T_{2}^{*}\) with respect to these parameters are shown in Table 4.
Sensitivity indices for the basic reproductive number \(R_{0}\)
From equation (10), we have \(\ln(R_{0}) = \ln(T_{1}^{*})\ln(T_{2}^{*})\). We note that \(T_{1}^{*}\) and \(T_{2}^{*}\) are functions of different sets of parameters. The sensitivity indices are shown in Table 5.
Sensitivity indices for the chronic productive virus population \(V_{2}^{*}\)
Using the formula for \(V_{2}^{*}\) from Theorem 1, we have
The formulas for the sensitivity indices can then be written in the form given in Table 6.
Sensitivity indices for the chronic infected Tcell populations \(L_{2}^{*}\), \(I_{2}^{*}\) and nonproductive virus population \(W_{2}^{*}\)
Using the formulas for \(L_{2}^{*}\), \(I_{2}^{*},W_{2}^{*}\) from Theorem 1, we have
The sensitivity indices for \(L_{2}^{*}\), \(I_{2}^{*},W_{2}^{*}\) are shown in Tables 7, 8, 9.
Numerical results
We give examples of numerical results using the parameter values listed in Table 10 selected from the work of previous authors.
Dependence of \(R_{0}\) on antiretroviral therapy
Figures 1(a), (b) and (c) show critical values of the antiretroviral parameters \(n_{rt}\) and \(n_{p}\) separating the virusfree and chronic infection equilibrium regions (\(R_{0} =1\)) for data sets 1, 2 and 3 of Table 10, respectively. For the three sets it can be seen that the model (1)(5) predicts that for sufficiently high antiretroviral therapy the HIV infection levels can be reduced to zero. However, as there is evidence (see, e.g., [2, 3]) that the virus cannot be completely eliminated from an HIV+ person, these results suggest that the model requires adjusting in the region of high levels of antiretroviral therapy.
Sensitivity analysis
For the sensitivity analysis, we consider three cases: (1) Virusfree equilibrium with \(R_{0} < 1\), (2) Chronic equilibrium with \(R_{0} \approx 1\), and (3) Chronic equilibrium with \(R_{0} > 1\).
Virusfree case. For this case, we use data set 1 in Table 10 and values of \(n_{rt} = 0.8\), \(n_{p}=0.6\). For these parameter values, \(R_{0} = 0.47087 <1\) and the equilibrium population values and eigenvalues are as follows:
As predicted for \(R_{0} < 1\), the virusfree equilibrium is locally asymptotically stable as the real parts of all eigenvalues are negative. Also, the chronic equilibrium does not exist because the infected Tcell and free virus populations are negative. It is also unstable because at least one eigenvalue has a positive real part.
The sensitivity indices for the virusfree healthy Tcell population (\(T_{1}^{*}\)) are shown in Table 11 and for the basic reproductive number \(R_{0}\) in Table 12.
From the numerical results in Table 12, it can be seen that the most effective methods of reducing \(R_{0}\) are the following: (1) to try to increase the efficacies \(n_{rt}\) and \(n_{p}\) of the antiretroviral therapy and (2) to increase virus clearance rate c, decrease infection rate k, or decrease viral reproduction rate N.
Chronic case ( \(R_{0} \approx 1\) ). For this case, we use data set 1 in Table 10 and values of \(n_{rt} = 0.575\), \(n_{p}=0.6\). For these parameter values, \(R_{0} = 1.0006 \approx 1\) and the equilibrium populations and eigenvalues are as follows:
As predicted for \(R_{0} > 1\), the virusfree equilibrium is unstable as the real part of at least one eigenvalue is positive. Also, the chronic equilibrium exists because the infected Tcell and free virus populations are positive. It is also locally asymptotically stable because the real parts of all eigenvalues are negative. Also, since \(R_{0} \approx 1\), the infected Tcell and free virus levels are close to zero and could easily be undetectable. The sensitivity indices for the chronic equilibrium with \(R_{0} \approx 1\) are shown in Table 13 for \(R_{0}\) and for the infected Tcell and virus populations \(T_{2}^{*}\), \(L_{2}^{*}\), \(I_{2}^{*}\) and \(V_{2}^{*}\).
Chronic case ( \(R_{0} > 1\) ). For this case, we use data set 1 in Table 10 and values of \(n_{rt} = 0.4\), \(n_{p}=0.6\). For these parameter values, \(R_{0} = 1.4126 > 1\) and the equilibrium populations and eigenvalues are as follows:
As predicted for \(R_{0} > 1\), the virusfree equilibrium is unstable as the real part of at least one eigenvalue is positive. Also, the chronic equilibrium exists because the infected Tcell and free virus populations are positive. It is also locally asymptotically stable because the real parts of all eigenvalues are negative. The complex conjugate eigenvalue with the small negative real part indicates that the infected populations will oscillate with a slowly decreasing amplitude to the chronic equilibrium solution.
The sensitivity indices for the chronic equilibrium with \(R_{0} > 1\) are shown in Table 14 for \(R_{0}\) and for the infected Tcell and virus populations \(T_{2}^{*}\), \(L_{2}^{*}\), \(I_{2}^{*}\) and \(V_{2}^{*}\).
From the numerical results in Tables 13 and 14, it can be seen that in the chronic infection region the most effective methods of reducing the free virus population \(V_{2}^{*}\) are the following: (1) to try to increase the efficacies \(n_{rt}\) and \(n_{p}\) of the antiretroviral therapy and (2) to increase virus clearance rate c, decrease infection rate k, or decrease viral reproduction rate N.
Dynamic behavior of solutions
We used Matlab to integrate the system (1)(5) for the parameter values in data set 1 in Table 10 and the \(n_{rt}\) and \(n_{p}\) values for the virusfree case (\(n_{rt}=0.8\), \(n_{p}= 0.6\)) and for the chronic case with \(R_{0} > 1\) (\(n_{rt}=0.4\), \(n_{p}= 0.6\)). These parameter values correspond to cases 1 and 3, respectively, discussed in the sensitivity analysis section.
Examples of the timedependence of the solutions for zero time delay for infected Tcells and free virus are shown in Figure 2(a) for the virusfree case and (b) for the chronic case. The populations converge to the virusfree equilibrium \((1177.4,0,0,0,0)\) in Figure 2(a) and to the chronic equilibrium populations \((833.5, 4.3624, 22.026, 440.52, 660.78)\) in Figure 2(b). As noted for the chronic case with \(R_{0}>1\) in the previous section, the solutions oscillate with decreasing amplitude due to the dominant complex conjugate pair of eigenvalues of the chronic equilibrium Jacobian.
AndronovHopf bifurcation for chronic solutions
One of the main conditions for the existence of an AndronovHopf bifurcation (see, e.g., [18]) is the existence of a purely imaginary pair of eigenvalues of the Jacobian of the chronic equilibrium at a critical value of a delay time with all other eigenvalues having negative real parts. In this paper, we find purely imaginary eigenvalues and the critical delay time by direct numerical solution of the characteristic equation \(\det(\lambda I  J) = 0\), where J is the Jacobian in equation (12).
We have used Matlab to obtain numerical solutions of the characteristic equation \(\det(\lambda I  J) = 0\) for parameter values given in data sets 1, 2 and 3 of Table 10 for a selection of \(n_{rt}\) and \(n_{p}\) values corresponding to \(R_{0} \approx 1\) and \(R_{0} > 1\).
Examples of these critical values for data sets 1, 2 and 3 of Table 10 are shown in Table 15. The critical delay values for a bifurcation are clearly very different for the three data sets. A noticeable difference between the data sets is also that the critical values for data sets 1 and 2 appear to change slowly as the level of the antiretroviral therapy is increased, whereas the critical values for data set 3 change rapidly as the level of the antiretroviral therapy is increased. The reason for this difference is at present unknown to the authors and requires a sensitivity analysis of the critical delay times.
An example of the dynamical behavior of the solutions of the model for data set 1 of Table 10 for the antiretroviral therapy levels \(n_{rt}=0.5\) and \(n_{p}=0.6\) are shown in Figure 3 for delay times just less than and just greater than the critical delay time. The plot for delay time just less than the critical time shows convergence to the chronic latent Tcell equilibrium population, whereas the plot for the delay time just greater than the critical time shows convergence to an oscillating population value. An example of the limit cycle behavior of the infectious free virus population for data set 1 in Table 10 for a time delay just greater than the critical delay is shown in Figure 4. The phase plane plot in Figure 4(b) shows a clear limit cycle behavior as predicted by AndronovHopf bifurcation theory (see, e.g., [18]).
In numerical results, not shown in this paper, we have found that in the limit cycle region the model (1)(5) predicts that the latently infected CD4+ Tcell population L can become negative. Since this is physiologically impossible, it is necessary to put a lower bound on the L population. The evidence (see, e.g., [2, 3]) that the virus cannot be completely eliminated suggests that placing a positive lower bound on L would give a more realistic model.
Conclusion
We have obtained simple analytical formulas for the sensitivity indices of this timedelay HIV model and used them to compute numerical values. We have found the following:

(1)
For the virusfree equilibrium, the virusfree healthy Tcell population depends on parameters that cannot be changed easily and the chronic healthy Tcell population is only useful because it is used to compute the sensitivity indices for \(R_{0}\). A reduction in \(R_{0}\) is important because it corresponds to a faster convergence of the infected populations to zero. From the numerical results, it can be seen that the most effective methods of reducing \(R_{0}\) are the following: (1) to try to increase the efficacies \(n_{rt}\) and \(n_{p}\) of the antiretroviral therapy and then (2) to increase virus clearance rate c, decrease infection rate k, or decrease viral reproduction rate N.

(2)
For the chronic equilibrium, reduction of the productively infected viral population \(V_{2}^{*}\) is the most important method of reducing the HIV infection. From the numerical results, the most effective methods of reducing \(V_{2}^{*}\) are the same as for the virusfree case.

(3)
The numerical results show that AndronovHopf bifurcations occur in the time delay model and that the critical delay times can vary over a wide range. For three data sets published by [15] and selected from the work of previous authors (Table 10), we have found delay times ranging from approximately 1520 days to more than 200 days.
As stated in the introduction, one aim of examining the effect of introducing a time delay for procession of latently infected CD4+ Tcells to productively infected Tcells was to check if AndronovHopf bifurcations could produce limit cycle behavior in the free virus populations that might be associated with the intermittent viral blips with period of approximately 50 days observed by Rong and Perelson [1, 5, 6]. Our results show that AndronovHopf bifurcations associated with this time delay in procession can produce limit cycle behavior with periods similar to the viral blip period. However, the present authors are not able to claim that this behavior actually causes the viral blips.
In numerical results, not shown in this paper, we have found that for a range of antiretroviral levels and delay times the model (1)(5) predicts that the latently infected CD4+ Tcell population L can become negative. However, in these cases, all other populations remain positive. The evidence (see, e.g., [2, 3]) that the virus cannot be completely eliminated suggests that placing a positive lower bound on L is necessary to obtain a more realistic timedelay model.
Change history
01 September 2017
An erratum to this article has been published.
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An erratum to this article is available at https://doi.org/10.1186/s1366201713277.
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Darlai, R., Moore, E.J. AndronovHopf bifurcation and sensitivity analysis of a timedelay HIV model with logistic growth and antiretroviral treatment. Adv Differ Equ 2017, 138 (2017). https://doi.org/10.1186/s1366201711951
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DOI: https://doi.org/10.1186/s1366201711951
Keywords
 HIV model
 RTI and PI treatment
 limit cycles
 viral blips
 sensitivity analysis