Andronov-Hopf bifurcation and sensitivity analysis of a time-delay HIV model with logistic growth and antiretroviral treatment
- Rachadawan Darlai†^{1}Email authorView ORCID ID profile and
- Elvin J Moore†^{2, 3}
https://doi.org/10.1186/s13662-017-1195-1
© The Author(s) 2017
Received: 31 January 2017
Accepted: 1 May 2017
Published: 16 May 2017
The Erratum to this article has been published in Advances in Difference Equations 2017 2017:265
Abstract
A mathematical model of the infection of CD4+ T-cells 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+ T-cells (healthy cells, latently-infected cells which cannot produce virus, and productively-infected cells which can produce virus) and two populations of free virus in the blood (infectious virus and non-infectious virus). The model includes a time delay between a T-cell becoming latently infected and productively infected. The model has a virus-free and a chronic infection equilibrium. It is shown that the model has Andronov-Hopf 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), 1-18 (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+ T-cell 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.
Keywords
1 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+ T-cells (healthy cells, latently-infected cells which cannot produce virus, and productively-infected cells which can produce virus) and two populations of free virus (infectious virus and non-infectious 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 virus-free 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+ T-cells. 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 discrete-time and continuous-time dynamical systems. One of the aims of the present paper is to check if time delays for procession of latently infected T-cells to productively infected T-cells could result in viral blip-type behavior.
In the present paper, we develop simple analytical formulas for sensitivity indices for the basic reproductive number, and for the virus-free and chronic equilibrium populations of the time-delay model. We also show that the time-delay model can undergo Andronov-Hopf 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.
2 Time-delay model
Population | Definition |
---|---|
T(t) | Healthy CD4+ T-cells at time t |
L(t) | Latently infected CD4+ T-cells at time t |
I(t) | Productively infected CD4+ T-cells at time t |
V(t) | Free infectious virus at time t |
W(t) | Free non-infectious (inhibited) virus at time t |
Parameter | Definition |
---|---|
Λ | Constant production rate of healthy CD4+ T-cells from precursors |
\(d_{T}\) | Natural death rate of healthy CD4+ T-cells |
r | Logistic growth rate of healthy CD4+ T-cells |
\(T_{\mathrm{max}}\) | Carrying capacity of healthy CD4+ T-cells |
k | Infection rate of healthy CD4+ T-cells by free infectious virus |
η | Fraction of infections leading to latently infected CD4+ T-cells |
\(d_{L}\) | Death rate of latently infected CD4+ T-cells |
\(d_{I}\) | Death rate of productively infected CD4+ T-cells |
a | Activation rate from latently to productively infected CD4+ T-cells |
N | Average number of free virus released by a productively infected CD4+ T-cell during its mean lifetime \(1/d_{I}\) |
c | Clearance rate of free virus |
\(n_{rt}\) | Drug efficacy of RTI \((0 \leq n_{rt} < 1)\) |
\(n_{p}\) | Drug efficacy of PI \((0 \leq n_{p} < 1)\) |
τ | Time delay for transformation from latently to productively infected CD4+ T-cells |
3 Equilibrium points
As shown by Wang et al. [15] the system (1)-(5) has a virus-free and a chronic equilibrium point. For completeness, we will briefly summarize the proof here.
Theorem 1
- 1.Virus-free:$$ \bigl(T_{1}^{*},L_{1}^{*},I_{1}^{*},V_{1}^{*},W_{1}^{*} \bigr) = \biggl(\frac{T_{\mathrm{max}}}{2r} \biggl(r-d_{T}+\sqrt{(r-d_{T})^{2}+4 \frac{r\Lambda}{T_{\mathrm{max}}}} \biggr),0,0,0,0 \biggr). $$(6)
- 2.Chronic infection:where \(\overline{k}=(1-n_{rt})k\), \(\overline{N}=(1-n_{p})N\).$$ \begin{aligned} &T_{2}^{*} = \frac{c(a+d_{L})}{\overline{k}\overline{N}(a+(1-\eta)d_{L})},\qquad L_{2}^{*} = \frac{c\eta}{\overline{N}(a+(1-\eta)d_{L})}V_{2}^{*} , \\ &I_{2}^{*} = \frac{c}{\overline{N}d_{I}}V_{2}^{*} ,\qquad V_{2}^{*} = \frac{1}{\overline{k}} \biggl(\frac{T_{1}^{*}}{T_{2}^{*}}-1 \biggr) \biggl(\frac{\Lambda}{T_{1}^{*}} +r\frac{T_{2}^{*}}{T_{\mathrm{max}}} \biggr), \\ &W_{2}^{*} = \frac{n_{p} N}{\overline{N}}V_{2}^{*} , \end{aligned} $$(7)
Proof
- 1.
Virus-free 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 \(\frac{dL}{dt}=0\) givesThen, 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$$ L_{2}^{*} = \frac{\eta}{a+d_{L}}\overline{k}V_{2}^{*}T_{2}^{*}. $$(8)$$\begin{aligned} V_{2}^{*} =& \frac{1}{\overline{k}T_{2}^{*}} \biggl(\Lambda +T_{2}^{*} \biggl(r-d_{T} -r\frac{T_{2}^{*}}{T_{\mathrm{max}}} \biggr) \biggr) \\ =&\frac{1}{\overline{k}} \biggl(\frac{T_{1}^{*}}{T_{2}^{*}}-1 \biggr) \biggl( \frac{\Lambda}{T_{1}^{*}} +r\frac{T_{2}^{*}}{T_{\mathrm{max}}} \biggr). \end{aligned}$$(9)
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.
4 Basic reproductive number
For this model, as in most models with a virus-free 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 next-generation 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).
4.1 Next-generation method
4.2 Linearized equations and stability
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 virus-free and chronic equilibrium points for the case of zero time delay.
5 Sensitivity indices
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.
5.1 Sensitivity indices for virus-free healthy T-cell population \(T_{1}^{*}\)
Sensitivity indices for virus-free healthy T-cell population \(\pmb{T_{1}^{*}}\)
Parameter | Index | Parameter | Index |
---|---|---|---|
\(T_{\mathrm{max}}\) | \(1-\frac{2r\Lambda}{T_{\mathrm{max}}D}\) | Λ | \(\frac{2r}{T_{\mathrm{max}}D}\) |
r | \(-\frac{2r(\Lambda-d_{T}T_{1}^{*})}{T_{\mathrm{max}}D}\) | \(d_{T}\) | \(-\frac{d_{T}}{\sqrt{(r-d_{T})^{2}+\frac{4r\Lambda}{T_{\mathrm{max}}}}}\) |
5.2 Sensitivity indices for chronic healthy T-cell population \(T_{2}^{*}\)
Sensitivity indices for chronic healthy T-cell population \(\pmb{T_{2}^{*}}\)
Parameter | Index | Parameter | Index |
---|---|---|---|
\(d_{L}\) | \(\frac{a\eta d_{L}}{(a+d_{L})(a+(1-\eta)d_{L})}\) | a | \(-\frac{a\eta d_{L}}{(a+d_{L})(a+(1-\eta)d_{L})}\) |
c | 1 | k | −1 |
N | −1 | \(n_{rt}\) | \(\frac{n_{rt}}{1-n_{rt}}\) |
\(n_{p}\) | \(\frac{n_{p}}{1-n_{p}}\) | η | \(\frac{\eta d_{L}}{a+(1-\eta)d_{L}}\) |
5.3 Sensitivity indices for the basic reproductive number \(R_{0}\)
Sensitivity indices for the basic reproductive number \(\pmb{R_{0}}\)
Parameter | Index | Parameter | Index |
---|---|---|---|
\(T_{\mathrm{max}}\) | \(SI(T_{1}^{*}|T_{\mathrm{max}})\) | Λ | \(SI(T_{1}^{*}|\Lambda)\) |
r | \(SI(T_{1}^{*}|r)\) | \(d_{T}\) | \(\displaystyle SI(T_{1}^{*}|d_{T})\) |
\(d_{L}\) | \(-SI(T_{2}^{*}|d_{L})\) | a | \(\displaystyle -SI(T_{2}^{*}|a)\) |
c | \(-SI(T_{2}^{*}|c)\) | k | \(\displaystyle -SI(T_{2}^{*}|k)\) |
N | \(-SI(T_{2}^{*}|N)\) | \(n_{rt}\) | \(\displaystyle -SI(T_{2}^{*}|n_{rt})\) |
\(n_{p}\) | \(-SI(T_{2}^{*}|n_{p})\) | η | \(\displaystyle -SI(T_{2}^{*}|\eta)\) |
5.4 Sensitivity indices for the chronic productive virus population \(V_{2}^{*}\)
Sensitivity indices for chronic productive virus population \(\pmb{V_{2}^{*}}\)
Parameter | Index |
---|---|
\(T_{\mathrm{max}}\) | \(\frac{R_{0} SI(T_{1}^{*}|T_{\mathrm{max}})}{R_{0}-1}-\frac{\Lambda T_{\mathrm{max}}SI(T_{1}^{*}|T_{\mathrm{max}})+rT_{1}^{*}T_{2}^{*}}{\Lambda T_{\mathrm{max}}+rT_{1}^{*}T_{2}^{*}}\) |
Λ | \(\frac{R_{0} SI(T_{1}^{*}|\Lambda)}{R_{0}-1}+\frac{\Lambda T_{\mathrm{max}}(1-SI(T_{1}^{*}|\Lambda))}{\Lambda T_{\mathrm{max}}+rT_{1}^{*}T_{2}^{*}}\) |
r | \(\frac{R_{0} SI(T_{1}^{*}|r)}{R_{0}-1}+\frac{r T_{1}^{*} T_{2}^{*}-\Lambda T_{\mathrm{max}}SI(T_{1}^{*}|r)}{\Lambda T_{\mathrm{max}}+rT_{1}^{*}T_{2}^{*}}\) |
\(d_{T}\) | \(\frac{R_{0} SI(T_{1}^{*}|d_{T})}{R_{0}-1}-\frac{\Lambda T_{\mathrm{max}}SI(T_{1}^{*}|d_{T})}{\Lambda T_{\mathrm{max}}+rT_{1}^{*}T_{2}^{*}}\) |
\(d_{L}\) | \(-\frac{R_{0} SI(T_{2}^{*}|d_{L})}{R_{0}-1}+\frac{r T_{1}^{*} T_{2}^{*}SI(T_{2}^{*}|d_{L})}{\Lambda T_{\mathrm{max}}+rT_{1}^{*}T_{2}^{*}}\) |
a | \(-\frac{R_{0} SI(T_{2}^{*}|a)}{R_{0}-1}+\frac{r T_{1}^{*} T_{2}^{*}SI(T_{2}^{*}|a)}{\Lambda T_{\mathrm{max}}+rT_{1}^{*}T_{2}^{*}}\) |
c | \(-\frac{R_{0} SI(T_{2}^{*}|c)}{R_{0}-1}+\frac{r T_{1}^{*} T_{2}^{*}SI(T_{2}^{*}|c)}{\Lambda T_{\mathrm{max}}+rT_{1}^{*}T_{2}^{*}}\) |
k | \(- \frac{R_{0} SI(T_{2}^{*}|k)}{R_{0}-1}+\frac{r T_{1}^{*} T_{2}^{*}SI(T_{2}^{*}|k)}{\Lambda T_{\mathrm{max}}+rT_{1}^{*}T_{2}^{*}}-1\) |
N | \(-\frac{R_{0} SI(T_{2}^{*}|N)}{R_{0}-1} +\frac{r T_{1}^{*} T_{2}^{*}SI(T_{2}^{*}|N)}{\Lambda T_{\mathrm{max}}+rT_{1}^{*}T_{2}^{*}}\) |
\(n_{rt}\) | \(-\frac{R_{0} SI(T_{2}^{*}|n_{rt})}{R_{0}-1} +\frac{r T_{1}^{*} T_{2}^{*}SI(T_{2}^{*}|n_{rt})}{\Lambda T_{\mathrm{max}}+rT_{1}^{*}T_{2}^{*}}+\frac{n_{rt}}{1-n_{rt}}\) |
\(n_{p}\) | \(-\frac{R_{0} SI(T_{2}^{*}|n_{p})}{R_{0}-1}+\frac{r T_{1}^{*} T_{2}^{*}SI(T_{2}^{*}|n_{p})}{\Lambda T_{\mathrm{max}}+rT_{1}^{*}T_{2}^{*}}\) |
η | \(-\frac{R_{0} SI(T_{2}^{*}|\eta)}{R_{0}-1}+\frac{r T_{1}^{*} T_{2}^{*}SI(T_{2}^{*}|\eta)}{\Lambda T_{\mathrm{max}}+rT_{1}^{*}T_{2}^{*}}\) |
5.5 Sensitivity indices for the chronic infected T-cell populations \(L_{2}^{*}\), \(I_{2}^{*}\) and nonproductive virus population \(W_{2}^{*}\)
Sensitivity indices for chronic latently infected T-cell population \(\pmb{L_{2}^{*}}\)
Parameter | Index | Parameter | Index |
---|---|---|---|
\(T_{\mathrm{max}}\) | \(SI(V_{2}^{*}|T_{\mathrm{max}})\) | Λ | \(SI(V_{2}^{*}|\Lambda)\) |
r | \(SI(V_{2}^{*}|r)\) | \(d_{T}\) | \(SI(V_{2}^{*}|d_{T})\) |
\(d_{L}\) | \(SI(V_{2}^{*}|d_{L})-\frac{(1-\eta)d_{L}}{a+(1-\eta)d_{L}}\) | a | \(SI(V_{2}^{*}|a)-\frac{a}{a+(1-\eta)d_{L}}\) |
c | \(1+SI(V_{2}^{*}|c)\) | k | \(SI(V_{2}^{*}|k)\) |
N | \(SI(V_{2}^{*}|N)-1\) | \(n_{rt}\) | \(SI(V_{2}^{*}|n_{rt})\) |
\(n_{p}\) | \(SI(V_{2}^{*}|n_{p})+\frac{n_{p}}{1-n_{p}}\) | η | \(1+SI(V_{2}^{*}|\eta)+\frac{\eta d_{L}}{a+(1-\eta)d_{L}}\) |
Sensitivity indices for chronic productively infected T-cell population \(\pmb{I_{2}^{*}}\)
Parameter | Index | Parameter | Index |
---|---|---|---|
\(T_{\mathrm{max}}\) | \(SI(V_{2}^{*}|T_{\mathrm{max}})\) | Λ | \(SI(V_{2}^{*}|\Lambda)\) |
r | \(SI(V_{2}^{*}|r)\) | \(d_{T}\) | \(SI(V_{2}^{*}|d_{T})\) |
\(d_{L}\) | \(SI(V_{2}^{*}|d_{L})\) | \(d_{I}\) | −1 |
a | \(SI(V_{2}^{*}|a)\) | c | \(1+SI(V_{2}^{*}|c)\) |
k | \(SI(V_{2}^{*}|k)\) | N | \(SI(V_{2}^{*}|N)-1\) |
\(n_{rt}\) | \(SI(V_{2}^{*}|n_{rt})\) | \(n_{p}\) | \(SI(V_{2}^{*}|n_{p})+\frac{n_{p}}{1-n_{p}}\) |
η | \(1+SI(V_{2}^{*}|\eta)\) |
Sensitivity indices for chronic non-infected free virus population \(\pmb{W_{2}^{*}}\)
Parameter | Index | Parameter | Index |
---|---|---|---|
\(T_{\mathrm{max}}\) | \(SI(V_{2}^{*}|T_{\mathrm{max}})\) | Λ | \(SI(V_{2}^{*}|\Lambda)\) |
r | \(SI(V_{2}^{*}|r)\) | \(d_{T}\) | \(SI(V_{2}^{*}|d_{T})\) |
\(d_{L}\) | \(SI(V_{2}^{*}|d_{L})\) | \(d_{I}\) | 0 |
a | \(SI(V_{2}^{*}|a)\) | c | \(1+SI(V_{2}^{*}|c)\) |
k | \(SI(V_{2}^{*}|k)\) | N | \(SI(V_{2}^{*}|N)\) |
\(n_{rt}\) | \(SI(V_{2}^{*}|n_{rt})\) | \(n_{p}\) | \(SI(V_{2}^{*}|n_{p})+\frac{1}{1-n_{p}}\) |
η | \(SI(V_{2}^{*}|\eta)\) |
6 Numerical results
Parameter values (adapted from Wang et al. [ 15 ])
Parameter | Set 1 | Set 2 | Set 3 | Unit | Source |
---|---|---|---|---|---|
\(T_{\mathrm{max}}\) | 1500 | 1500 | 1500 | μl^{−1} | |
Λ | 10 | 10 | 10 | μl^{−1} day^{−1} | |
r | 0.1 | 0.03 | 0.1 | day^{−1} | |
\(d_{T}\) | 0.03 | 0.01 | 0.01 | day^{−1} | |
\(d_{L}\) | 0.001 | 0.004 | 0.2 | day^{−1} | |
a | 0.1 | 0.01 | 0.3 | day^{−1} | |
c | 20 | 3 | 15 | day^{−1} | |
k | 0.0001 | 0.0001 | 0.0001 | μl^{−1} day^{−1} | |
N | 1000 | 200 | 500 | virions/cell | |
η | 0.02 | 0.001 | 0.5 | ||
\(d_{I}\) | 1 | 1 | 0.8 | day^{−1} |
6.1 Dependence of \(R_{0}\) on antiretroviral therapy
6.2 Sensitivity analysis
For the sensitivity analysis, we consider three cases: (1) Virus-free equilibrium with \(R_{0} < 1\), (2) Chronic equilibrium with \(R_{0} \approx 1\), and (3) Chronic equilibrium with \(R_{0} > 1\).
As predicted for \(R_{0} < 1\), the virus-free equilibrium is locally asymptotically stable as the real parts of all eigenvalues are negative. Also, the chronic equilibrium does not exist because the infected T-cell and free virus populations are negative. It is also unstable because at least one eigenvalue has a positive real part.
Sensitivity indices for \(\pmb{T_{1}^{*}}\)
Parameter | Sensitivity index | Parameter | Sensitivity index |
---|---|---|---|
\(T_{\mathrm{max}}\) | 0.9024 | Λ | 0.00976 |
r | 0.2472 | \(d_{T}\) | −0.3449 |
Sensitivity indices for virus-free basic reproductive number \(\pmb{R_{0}}\)
Parameter | Sensitivity index | Parameter | Sensitivity index |
---|---|---|---|
\(T_{\mathrm{max}}\) | 0.9024 | Λ | 0.00976 |
r | 0.2472 | \(d_{T}\) | −0.3449 |
\(d_{L}\) | −0.000196 | a | 0.00196 |
c | −1 | k | 1 |
N | 1 | \(n_{rt}\) | −4 |
\(n_{p}\) | −1.5 | η | −0.000198 |
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.
Sensitivity indices for \(\pmb{R_{0}}\) and for chronic infected T-cells and virus populations for \(\pmb{R_{0} \approx 1}\)
Parameter | \(\boldsymbol{R_{0}}\) | \(\boldsymbol{T_{2}^{*}}\) | \(\boldsymbol{L_{2}^{*}}\) | \(\boldsymbol{I_{2}^{*}}\) | \(\boldsymbol{V_{2}^{*}}\) |
---|---|---|---|---|---|
\(T_{\mathrm{max}}\) | 0.9024 | 0 | 1525.9 | 1525.9 | 1525.9 |
Λ | 0.00976 | 0 | 16.618 | 16.618 | 16.618 |
r | 0.2472 | 0 | 419.24 | 419.24 | 419.24 |
\(d_{T}\) | −0.3449 | 0 | −583.54 | −583.54 | −583.54 |
\(d_{L}\) | −0.000196 | 0.000196 | −0.3414 | −0.332 | −0.33164 |
a | 0.00196 | −0.00196 | 2.3261 | 3.3164 | 3.3164 |
c | −1 | 1 | −1690.2 | −1690.2 | −1691.2 |
k | 1 | −1 | 1690.2 | 1690.2 | 1690.2 |
N | 1 | −1 | 1690.2 | 1690.2 | 1691.2 |
\(n_{rt}\) | −1.3529 | 1.3529 | −2286.7 | −2286.7 | −2286.7 |
\(n_{p}\) | −1.5 | 1.5 | −2535.3 | −2535.3 | −2536.8 |
η | −0.000198 | 0.000198 | −0.6652 | 0.6650 | −0.3350 |
\(d_{I}\) | 0 | 0 | 0 | −1 | 0 |
As predicted for \(R_{0} > 1\), the virus-free equilibrium is unstable as the real part of at least one eigenvalue is positive. Also, the chronic equilibrium exists because the infected T-cell 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.
Sensitivity indices for chronic infected T-cells and virus populations for \(\pmb{R_{0} > 1}\)
Parameter | \(\boldsymbol{R_{0}}\) | \(\boldsymbol{T_{2}^{*}}\) | \(\boldsymbol{L_{2}^{*}}\) | \(\boldsymbol{I_{2}^{*}}\) | \(\boldsymbol{V_{2}^{*}}\) |
---|---|---|---|---|---|
\(T_{\mathrm{max}}\) | 0.9024 | 0 | 2.1023 | 2.1023 | 2.1023 |
Λ | 0.00976 | 0 | 0.1647 | 0.1647 | 0.1647 |
r | 0.2472 | 0 | 1.6811 | 1.6811 | 1.6811 |
\(d_{T}\) | −0.3449 | 0 | −1.1350 | −1.1350 | −1.1350 |
\(d_{L}\) | −0.000196 | 0.000196 | −0.01021 | −0.000501 | −0.000501 |
a | 0.00196 | −0.00196 | −0.9852 | 0.00501 | 0.00501 |
c | −1 | 1 | −1.5562 | −1.5562 | −2.5562 |
k | 1 | −1 | 1.5562 | 1.5562 | 1.5562 |
N | 1 | −1 | 1.5562 | 1.5562 | 2.5562 |
\(n_{rt}\) | −0.6667 | 0.6667 | −1.0375 | −1.0375 | −1.0375 |
\(n_{p}\) | −1.5 | 1.5 | −2.3344 | −2.3344 | −3.8344 |
η | −0.000198 | 0.000198 | 0.9997 | 0.9995 | −0.000506 |
\(d_{I}\) | 0 | 0 | 0 | −1 | 0 |
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.
6.3 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 virus-free 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.
6.4 Andronov-Hopf bifurcation for chronic solutions
One of the main conditions for the existence of an Andronov-Hopf 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 critical delay times \(\pmb{\tau_{c}}\) (days) for Andronov-Hopf bifurcation
Set 1 | Set 2 | Set 3 | ||||
---|---|---|---|---|---|---|
\(\boldsymbol{(n_{rt},n_{p})}\) | \(\boldsymbol{\tau_{c}}\) | \(\boldsymbol{(n_{rt},n_{p})}\) | \(\boldsymbol{\tau_{c}}\) | \(\boldsymbol{(n_{rt},n_{p})}\) | \(\boldsymbol{\tau_{c}}\) | |
\(R_{0} > 1\) | (0.2, 0.3) | 15.429 | (0.2, 0.3) | 216.215 | (0.2, 0.3) | 14.171 |
(0.5, 0.5) | 16.077 | (0.5, 0.5) | 216.299 | (0.2, 0.4) | 26.563 | |
(0.4, 0.6) | 16.292 | (0.4, 0.6) | 216.313 | (0.25, 0.4) | 44.064 | |
(0.5, 0.6) | 19.175 | (0.5, 0.6) | 216.405 | (0.25, 0.43) | 132.673 | |
\(R_{0} \approx 1\) | (0.575, 0.6) | 18.052 | (0.633, 0.7) | 231.147 |
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+ T-cell 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.
7 Conclusion
- (1)
For the virus-free equilibrium, the virus-free healthy T-cell population depends on parameters that cannot be changed easily and the chronic healthy T-cell 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 virus-free case.
- (3)
The numerical results show that Andronov-Hopf 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 15-20 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+ T-cells to productively infected T-cells was to check if Andronov-Hopf 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 Andronov-Hopf 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+ T-cell 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 time-delay model.
Notes
Declarations
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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