Research Article  Open Access
Qilin Sun, Lequan Min, "Dynamics Analysis and Simulation of a Modified HIV Infection Model with a Saturated Infection Rate", Computational and Mathematical Methods in Medicine, vol. 2014, Article ID 145162, 14 pages, 2014. https://doi.org/10.1155/2014/145162
Dynamics Analysis and Simulation of a Modified HIV Infection Model with a Saturated Infection Rate
Abstract
This paper studies a modified human immunodeficiency virus (HIV) infection differential equation model with a saturated infection rate. It is proved that if the basic virus reproductive number of the model is less than one, then the infectionfree equilibrium point of the model is globally asymptotically stable; if of the model is more than one, then the endemic infection equilibrium point of the model is globally asymptotically stable. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of the two groups of patients’ antiHIV infection treatment. The numerical simulation results are in agreement with the evolutions of the patients’ HIV RNA levels. It can be assumed that if an HIV infected individual’s basic virus reproductive number then this person will recover automatically; if an antiretroviral therapy makes an HIV infected individual’s , this person will be cured eventually; if an antiretroviral therapy fails to suppress an HIV infected individual’s HIV RNA load to be of unpredictable level, the time that the patient’s HIV RNA level has achieved the minimum value may be the starting time that drug resistance has appeared.
1. Introduction
The human immunodeficiency virus (HIV) mainly targets a host’s T cells. Chronic HIV infection causes gradual depletion of the T cell pool and thus progressively compromises the hosts immune response to opportunistic infections, leading to Acquired Immunodeficiency Syndrome (AIDS) [1].
In recent years, there is much work done on HIV infection from different points of view, such as pathology [2], microbiology [3], and mathematics [4–7]. Mathematical models have become essential tools to make assumptions, suggest new experiments, or help easily explain complex processes [8]. The basic mathematical model widely used for studying the dynamics of HIV infection has the following form [4, 9]: where , and are the number of uninfected cells, infected cells, and free virus, respectively. Uninfected cells are produced at a constant rate , die at rate , and become infected at rate . Infected cells are produced at rate and die at rate . Free virus is produced from infected cells at rate and dies at rate .
Equation (1) has a basic virus reproductive number . According to Nowak and Bangham [4], is defined as the number of newly infected cells arising from any one infected cell; if is smaller than 1, then in the beginning of the infection, each virus infected cell produces on average less than one newly infected cell. Thus, the infection cannot spread, and the system returns to the uninfected state; if is larger than 1, then initially each virus infected cell produces on average more than one newly infected cell. The infected cell population will increase, whereas the uninfected cell population will decline and therefore provide less opportunity for the virus to infect new cells.
There is a discussion about the process of the HIV RNA transcribing into DNA: when an HIV enters a resting T cell, the HIV RNA may not be completely reverse transcribed into DNA [10]. A proportion of resting infected cells can revert to the uninfected state before the viral genome is integrated into the genome of the lymphocyte [11].
Recently, some mathematical models of HIV infection have been proposed based on the assumption that a fraction of infected T cells return to the uninfected class [12–14]. Srivastava and Chandra [13] have considered a model with three populations: uninfected T cells , infected T cells , and HIV population . The model has the following form: where the meanings of the variables , , and and the parameters , and are the same as those given in (1). The term is the rate of infected cells in the latent stage reverting to the uninfected class. Equation (2) also has a basic virus reproductive number . They have proved that if , the infectionfree steady state of (2) is globally asymptotically stable; if , the endemic steady state of (2) is globally asymptotically stable [13].
In (2), the mass action term used to model infection of T cells by free virions is biologically problematic for several reasons. Firstly, since represents the total number of T cells in the basic virus reproductive number , this causes to depend upon the total number of T cells in vivo. This implies the dubious prediction that individuals with more T cells will be more easily infected than individuals with less T cells. Secondly, the rate of HIV infection is assumed to be bilinear by the mass action term . However, the actual incidence rate is probably not linear over the entire range of virus and uninfected T cells [15–17].
On biological grounds, during primary HIV infection, the rate of virus infection should be approximately proportionate to the virus load because of a small amount of viral load with respect to a large number of T cells. However,since the total number of healthy T cells in vivo is limited, the HIV infection will approach saturation with more and more virus produced. In this case, it is more reasonable to assume that the rate of virus infection should be approximately proportionate to the number of healthy T cells .
Based on the argument above, this paper describes an amended model. In this model, we use a saturated infection rate to replace the mass action term in (2). Under the formulation of this saturated infection rate, the basic virus reproductive number is independent of the total number of T cells. Meanwhile, the actual incidence rate is not linear over the entire range of virus and uninfected T cells any more. The global stabilities of the infectionfree state and the endemic infection state of the modified HIV infection model have been discussed. Based on the clinical data from HIV drug resistance database of Stanford University, using the proposed model simulates the dynamics of two groups of patients’ antiHIV infection treatment, and then make longterm predictions for the two groups’ antiHIV infection treatment, respectively.
The rest of this paper is organized as follows. Section 2 introduces a modified model and discusses the boundedness of the solutions of the model. Sections 3 and 4 discuss the global stability of the infectionfree state and the endemic infection state of the modified HIV infection model, respectively. Section 5 simulates the dynamics of two groups of patients’ antiHIV infection treatment. Section 6 summarizes this paper.
2. Modified HIV Infection Model
2.1. The Modified HIV Infection Model
Based on (2), our modified HIV infection model has the following form: where the meanings of the variables , , and and the parameters , and are the same as those given in (2). Equation (3) has two steady states: (1)The infectionfree steady state represents the virus infection free. is called infectionfree equilibrium point. Here, (2)The endemic infected steady state represents persistent virus infection. is called endemic infection equilibrium point. Here, Here, Since the total rate of disappearance of infected cells is , infected cells live on average for time . Each infected cell produces virus at rate . Thus, each infected cell produces on average a total of viruses. Since virus dies at rate per virion, each virus survives on average for time . During the time , each virus infects on average cells, where and are the preinfection target cells’ density and viruses’ density, respectively. Thus, the total number of cells infected by the viruses is . According to (4) and (5), and at the preinfection steady state. Then one can obtain that the total number of cells infected by each infected cell is . Hence, is the basic virus reproductive number of (3) which is independent of the total number of the uninfected T cells.
According to (4), (6), and (7), if , then is the unique infectionfree equilibrium point; if , then, in addition to the infectionfree equilibrium point, (3) has another equilibrium point .
2.2. Boundedness of Solutions
It is easy to show that the solutions of (3) with initial conditions , and have all positive components for . Hence, one begins the analysis of (3) by observing the nonnegative octant
According to the first two equations of (3), one can get and then So and are bounded. From the last equation of (3), it follows that and then So are bounded. Hence there is a bounded subset of : such that any solution trajectory of (3) with initial value in will keep in the subset .
According to (7), , , and . One can get that the endemic infection equilibrium point exists in the interior of : Therefore, the stability of the endemic infection equilibrium point only needs to be discussed in .
3. Stability of the InfectionFree Equilibrium Point
In this section, we discuss locally asymptotical stability and globally asymptotical stability of the infectionfree equilibrium point of (3).
3.1. Locally Asymptotical Stability of the InfectionFree Equilibrium Point
Theorem 1. If , then the infectionfree equilibrium point of (3) is locally asymptotically stable. If , then the infectionfree equilibrium point is unstable.
Proof. The Jacobi matrix of (3) at an arbitrary point is given by
where and .
Substituting the equilibrium point into matrix (16) gives
The corresponding eigenequation is
Solving
gives
Equation (21) can be written as
Solving equation (22) gives
If , then and . Hence the infectionfree equilibrium point is locally asymptotically stable. If , then such that the infectionfree equilibrium point is unstable.
3.2. Globally Asymptotical Stability of the InfectionFree Equilibrium Point
Theorem 2. If , then the infectionfree equilibrium point of (3) is globally asymptotically stable in .
Proof. Define a global Lyapunov function by
The derivative of along the positive solutions of (3) is
If , then holds in . Moreover, if and only if . Hence, the largest compact invariant set in is
According to the LaSalle’s invariance principle, . Then one can get limit equations:
Define a global Lyapunov function by
where
The derivative of along the positive solutions of (27) is
Since ,
Since the arithmetic mean is greater than or equal to the geometric mean, we obtain .
Therefore, holds in , and if and only if and . There is the largest compact invariant set in :
Hence if , all solution paths in approach the infectionfree equilibrium point .
4. Stability of the Endemic Infection Equilibrium Point
In this section, we analyze local asymptotical stability and global asymptotical stability of the endemic infection equilibrium point of (3).
4.1. Locally Asymptotical Stability of the Endemic Infection Equilibrium Point
Theorem 3. If , then the endemic infection equilibrium point of (3) is locally asymptotically stable.
Proof. Put the equilibrium point into matrix (16); then one obtains
Solving the eigenequation of the matrix above, here isIf , one obtains that
Hence all inequalities of the RouthHurwitz criterion are satisfied. Therefore, the endemic infection equilibrium point is locally asymptotically stable.
4.2. Globally Asymptotical Stability of the Endemic Infection Equilibrium Point
In this subsection, we firstly introduce a lemma outlined by Li and Wang [18], and then using this lemma discusses the globally asymptotical stability of the endemic infection equilibrium point of (3).
The lemma is briefly summarized as follows.
Let be a function for in an open set . Consider the differential system Denote by the solution to (36) such that . Let be an equilibrium point of (36). Li and Wang [18] made the following two basic assumptions:there exists a compact absorbing set ;equation (36) has a unique equilibrium in .
Li and Wang (see Theorem 2.5 in [18]) have given the following lemma.
Lemma 4 (see [18]). Assume that(1)assumptions () and () hold;(2)equation (36) satisfies the PoincaréBendixson Property;(3)for each periodic solution to (36) with , the linear system (the second additive compound system) is asymptotically stable, where is the second additive compound matrix of the Jacobian matrix ;(4). Then the unique equilibrium is globally asymptotically stable in .
Now one uses Lemma 4 to show the following.
Theorem 5. If , then the endemic infection equilibrium point of (3) is globally asymptotically stable in , where is defined by (15).
Proof. Based on Lemma 4, the proof of Theorem 5 has been implemented via the following four steps.
For epidemic models and many other biological models where the feasible region is a bounded cone, is equivalent to the uniform persistence of the system [19]. By (15), is bounded, so it only needs to show the uniform persistence of (3). According to Proposition 3.3 in [20], the necessary and sufficient condition for the uniform persistence of (3) is equivalent to the equilibrium point being unstable. Theorem 1 has shown that is unstable if . Therefore, (3) is uniformly persistent if so that holds if .
Meanwhile, by (4), so does not exist in . Hence, is the unique equilibrium point of (3) in so that holds.
The results above verify the condition of Lemma 4.
The Jacobian matrix of (3) is
where and .
If , then
and one can obtain that has nonpositive offdiagonal elements in . Therefore (3) is competitive in . It is known that 3dimensional competitive systems have the PoincaréBendixson Property [21]. Hence, (3) satisfies the PoincaréBendixson Property. This verifies condition of Lemma 4.
Let be a periodic solution in .
According to [22], if is a matrix, then the second additive compound of is
The Jacobian matrix of (3) is
where and .
And then the second additive compound matrix of the Jacobian matrix of (3) is given by and the second additive compound system of (3) along the periodic solution is
Define a global Lyapunov function by
where is the norm in set defined by
Suppose that the solution is periodic of least period and that . According to [23], (3) is uniformly persistent, if there exists a positive constant such that
Step has shown that (3) is uniformly persistent if . Hence, there always exists a positive constant which satisfies (46). The orbit of remains at a positive distance from the boundary of by the uniform persistence, and one can obtain that
Since by (15),
for all .
Along a solution of (43), becomes
The righthand derivative of along the positive solution of (43) is
Therefore
where
Denote , and then
By (52) and Gronwall’s inequality, one obtains
when , and then when by (48). The second additive compound system is asymptotically stable. This verifies the condition of Lemma 4.
The Jacobi matrix of (3) at the endemic infection equilibrium is
and then
According to (7), , and then one can obtain
Since is a matrix, one gets . Then
If , then holds in . This verifies condition of Lemma 4.
Hence, if , then the endemic infection equilibrium point is globally asymptotically stable in by Lemma 4.
5. Numerical Simulation
In the first subsection, we determine some parameter values of an antiHIV infection treatment model based on (3). In the second subsection, using the antiHIV infection treatment model simulates the dynamics of the Group I’s antiHIV infection treatment. In the third subsection, using the antiHIV infection treatment model simulates the dynamics of the Group II’s antiHIV infection treatment. In the fourth subsection, we make longterm predictions for the two groups’ antiHIV infection treatment, respectively.
5.1. Modeling
Baxter et al. [24] have reported a randomized study of antiretroviral management based on plasma genotypic antiretroviral resistance testing in HIV patients failing therapy, which was enrolled from 14 units of the Terry Beirn Community Programs for Clinical Research on AIDS and the Walter Reed Army Medical Center (see the HIV drug resistance database of Stanford University [25]). These patients were failing virologically on a combination antiretroviral regimen containing protease inhibitors (PI) and nucleoside reverse transcriptase inhibitors (NRTI) [24]. The patients were seen at 4, 8, and 12 weeks. At each followup visit, changes in antiretroviral treatment were recorded and the tested items included patients’ plasma T cells counts and plasma HIV1 RNA levels by the Chiron bDNA assay [24].
In the following subsections, we select, from [24, 25], two group patients’ mean uninfected T cells counts and mean HIV RNA levels to simulate and make longterm predictions for the patients’ treatment outcomes. Group I consists of 15 patients. Group II consists of 13 patients. The two groups of patients received the same PI: ritonavir (RTV) and saquinavir (SQV). Additionally, Group I received NRTI: strvudine (D4T). Group II received NRTI: strvudine (D4T) and dideoxyinosine (DDI) [24, 25].
Based on (3), the antiHIV infection treatment model has the form where are the efficacy variables of the treatment.
The infectionfree equilibrium point of (60) is the same as that defined by (4):
The endemic infection equilibrium point of (60) is given by where, and is the basic virus reproductive number of (60):
Determine the parameter value ranges of (60).(1)Naive T cells decayed with an average halflife of 50 days [26]. Therefore, one obtains Hence . Because the apoptosis of T cells is raised by HIV infection [27, 28], should be more than 0.0139 during the simulation.(2)Since healthy individuals have an average of 830/μL T cells [29], (3)Since the cells that produce the virus are also shortlived, with a halflife of approximately 1.2 days [30], one obtains (4)Since the halflife of HIV1 in the plasma appears to be only 1 to 2 days [30], one selects (5)Because only a small fraction of cells in the eclipse phase will revert to the uninfected state, it assumes that [12]. Hence one obtains (6)According to reference [31], one can determine the other parameter value ranges as follows:
In each group, there was one patient whose clinical data was not complete. Therefore we do not conclude the two patients’ clinical data in the following simulations.
5.2. The Mean Dynamics Simulation of Group I’s AntiHIV Infection Treatment
Using the other 14 patients’ clinical data determines the equation parameter values as follows:
The value changes of the parameters , , and and the basic virus reproductive number are listed in Table 1.

The simulation results of the mean dynamics of antiHIV infection treatment of Group I are shown in Figure 1. During the first 4 weeks, the treatment reduced the basic virus reproductive number from to . Hence the patients’ mean HIV RNA levels decreased rapidly to approach infectionfree steady state defined by (4) as Theorem 2 predicts.
(a)
(b)
However, after the 4th week, the resistance to antiretroviral drugs appeared. It made the therapy efficacy parameters and decrease from 0.53 to 0.45 and 0.84 to 0.76, respectively. Meanwhile the apoptosis of T cells was raised by HIV more strongly. Hence rose from to . The suboptimal treatment increased value of Group I from to . As a result, the patients’ mean HIV RNA levels increased slowly to converge to a new infected steady state defined by (62) and (63) as Theorem 5 predicts. On the other hand, observe that the mean uninfected T cell counts of Group I increased rapidly in the first 4 weeks and decreased slowly in the following 8 weeks.
5.3. The Mean Dynamics Simulation of Group II’s AntiHIV Infection Treatment
Using 12 patients’ clinical data determines the equation parameter values as follows:
The value changes of the parameters , , and and the basic virus reproductive number are listed are listed in Table 2.

The simulation results of the mean dynamics of antiHIV infection treatment of Group II are shown in Figure 2. During the first 4 weeks, the treatment reduced the basic virus reproductive number from to . Hence the patients’ mean HIV RNA levels decreased rapidly to approach to infectionfree steady state defined by (4) as Theorem 2 predicts.
(a)
(b)
However, after the 4th week, the resistance to antiretroviral drugs appeared. It made the therapy efficacy parameters and decrease from 0.65 to 0.55 and 0.68 to 0.59, respectively. Meanwhile the apoptosis of T cells was raised by HIV more strongly. Hence rose from 0.041 to 0.0428. The suboptimal treatment made value of Group II increase from to . As a result, the patients’ mean HIV RNA levels increased slowly to converge to a new infected steady state defined by (62) and (63) as Theorem 5 predicts. On the other hand, observe that the mean uninfected T cell counts of Group II increased rapidly in the first 4 weeks but rose slowly in the following 8 weeks.
5.4. The LongTerm Predictions for the Two Groups’ AntiHIV Infection Treatment
According to 2013 HIV therapy guidelines published by World Health Organization (WHO) [32], viral load is recommended as the preferred monitoring approach to diagnose and confirm antiretroviral treatment failure; treatment failure is defined by a persistently detectable viral load exceeding copies/mL after at least six months of using antiretroviral drugs. However, HIV RNA levels of the two groups of patients were only tested at 4, 8, and 12 weeks in the study [24]. Therefore, it is necessary to make longterm predictions to detect whether the treatments for the two groups are failure.
Assume that after ending the 12 weeks’ antiretroviral treatment testing, the two groups keep receiving the antiHIV infection treatment for 2 years. During the 2 years, the drug resistance does not become worse and all parameter values do not change. Using the numerical simulation of Equation (60) makes the longterm predictions for the two groups’ antiHIV infection treatment. The longterm prediction for Group I’ antiHIV infection treatment is shown in Figure 3. Two years’ outcomes of the therapy for Group II are shown in Figure 4. Observe that after finishing the 12 weeks’ antiretroviral treatment testing, the mean uninfected T cells counts of the two groups both decline a little and finally keep at a level larger than the mean baseline values; the mean HIV RNA levels of the two groups both rise a lot to a level less than the mean baseline values but keep exceeding copies/mL all the time. The longterm predictions suggest that the treatments for the two groups are failure and better antiHIV infection therapies should be considered.
(a)
(b)
(a)
(b)
6. Conclusion
This paper introduces a modified HIV infection differential equation model with a saturated infection rate and the proportion of infected cells reverting to the uninfected state.
The basic virus reproductive number of the model is independent of a patient’s plasma total T cell counts , and the actual incidence rate is not linear over the entire range of virus and uninfected T cells any more. This suggests that our model is more reasonable than the model proposed by [13].
The modified model has two equilibrium points: infectionfree equilibrium point and endemic infection equilibrium point . This paper discusses the locally asymptotical stabilities and globally asymptotical stabilities of the two equilibrium points, simulates the dynamics of two group patients’ antiHIV infection treatment, and makes longterm predictions for the two groups’ antiHIV infection treatment.
The theoretical results suggest the following.(1)If the basic virus reproductive number , then the infectionfree equilibrium point of (3) is globally asymptotically stable in . This means that if a person with , the person can recover automatically even if infected with a large amount of HIV; if a treatment makes a patient’s , the patient will be cured eventually even if infected with a large amount of HIV. The recent reports on three HIV infected patients have shown that some HIV infected patients may be cured via bone marrow transplants. The Berlin Patient was the first person cured of HIV [33]. After the Berlin Man, two cases reported cured of HIV in Kenya [34]. These reports can make one postulate that most individuals who connect HIV virus will not be infected by it and are not infecting other people. They will recover automatically without any treatment. The fact has not been well recognized since AIDS has been discovered in 1983. Mathematically, such phenomena can be described also via (3) where , and thus .(2)If the basic virus reproductive number , then the endemic infection equilibrium point of (3) is globally asymptotically stable in the interior of . This means that a person with will have endemic infection even if infected with only one HIV; if a treatment cannot make a patient’s , the patient’s HIV RNA in vivo cannot be cleared up eventually.
Based on the simulation results, one can propose the following hypotheses.(1)For a poor HIV treatment response patient, the drug resistance appears when the patient’s HIV RNA level reduces to the first lowest level. This hypothesis may interpret why the two group patients’ mean T cells counts rose and mean HIV RNA levels declined rapidly in the first 4 weeks but contrary in the following weeks (see Tables 1 and 2 and Figures 1 and 2).(2)Once a patient’s drug resistance appears, the patient’s HIV in vivo promotes the apoptosis of T cells more strongly. This hypothesis may interpret why between 4th week and 8th week the mean HIV RNA levels of the two groups kept a lower level than the first 4 weeks, but Group I’s mean T cells counts started to decrease slowly and Group II’s mean T cells counts rose more slowly than before (see Tables 1 and 2 and Figures 1 and 2).(3)According to the 2013 HIV therapy guidelines published by WHO [32], treatment failure is defined by a persistently detectable viral load exceeding copies/mL after at least six months of using antiretroviral drugs. Our longterm numerical simulation predictions suggest that after ending the 12 weeks’ antiretroviral treatment [24], the two group patients’ mean HIV RNA levels keep exceeding copies/mL during the additional 2 years’ antiHIV infection treatment. The treatments for the two groups are failure and better antiHIV infection therapies should be considered. This means that the additional 2 years’ treatments are not able to make patients obtain better outcomes. This may interpret why WHO defines that half a year’s treatment cannot suppress a patient’s HIV level below copies/mL to be treatment failure.
Highlights
(1)This paper introduces a modified HIV infection (antiHIV infection therapy) differential equation model with a saturated infection rate. The basic virus reproductive number of the model is independent of a patient’s plasma total T cell counts . This suggests that our model is more reasonable than the model proposed by Srivastava and Chandra in 2010 [13] whose basic virus reproductive number is dependent on which follows that the more T cell counts an individual has, the more easily the individual is infected by HIV.(2)This paper proposes and proves two theorems (Theorems 2 and 5) on the globally asymptotical stabilities of the infectionfree equilibrium point and the endemic infection equilibrium point of the modified model.(3)This paper points out the implications of the two theorems which are ignored by other similar researches on modelling HIV infection (antiHIV infection therapy):(a)A person with the basic virus reproductive number will recover automatically even if the person is infected with a large amount of HIV. If a treatment makes a patient’s , this patient will be cured eventually even if infected with a large amount of HIV.(b)A person with will have endemic infection even if the person is infected with only one HIV. If a treatment cannot make a patient’s , the patient’s HIV RNA in vivo cannot be cleared up eventually.
The recent reports on three HIV infected patients show that HIV infected patients may be cured via bone marrow transplants (e.g., see: Berlin Patient, first person cured of HIV, may soon have company, Los Angeles Times, July 27, 2012; after Berlin Man, two reported cured of HIV in Kenya, Africa Review, May 6, 2013).
These reports can make one postulate that most individuals who connect HIV virus will not be infected by it and are not infecting other people, who will recover automatically without any treatment. In this case, where . The fact has not been recognized since AIDS has been discovered in 1983.
In the report [35], a small proportion of human immunodeficiency virus type 1 (HIV1) infected individuals, called elite and viremic controllers, spontaneously control plasma HIV RNA levels to undetectable (elite controller) or 2000 copies/mL (viremic controller) in the absence of antiretroviral therapy.
These phenomena can be interpreted by our Theorem 2: HIV infected people’s basic virus reproductive number .(4)Based on the simulation results, we can propose the following hypotheses:(a)for a poor HIV treatment response patient, the drug resistance appears when the patient’s HIV RNA level reduces to the first lowest level;(b)Once a patient’s drug resistance appears, the patient’s HIV promotes the apoptosis of T cells more strongly;(c)According to the 2013 HIV therapy guidelines published by WHO [32], treatment failure is defined by a persistently detectable viral load exceeding copies/mL after at least six months of using antiretroviral drugs. Our longterm numerical simulation predictions suggest that after ending the 12 weeks’ antiretroviral treatment [24], the two group patients’ mean HIV RNA levels keep exceeding copies/mL during the additional 2 years’ antiHIV infection treatment. The treatments for the two groups are failure and better antiHIV infection therapies should be considered. This means that the additional 2 years’ treatments are not able to make patients obtain better outcomes. This may interpret why WHO defines that half a year’s treatment cannot suppress a patient’s HIV level below copies/mL to be treatment failure.
Conflict of Interests
All authors have no financial or other conflict of interests pertaining to this paper.
Acknowledgments
The authors gratefully acknowledge anonymous reviewers for valuable suggestions and Dr. Xiao Chen for her help in this study. The authors would like to thank the HIV drug resistance database of Stanford University which provides the clinical data on antiHIV infection treatment. This work is jointly supported by the National Natural Science of China (no. 61074192) and Doctoral Research Funds of University of Science and Technology Beijing (no. 06108126).
References
 L. Wang and M. Y. Li, “Mathematical analysis of the global dynamics of a model for HIV infection of CD4^{+} T cells,” Mathematical Biosciences, vol. 200, no. 1, pp. 44–57, 2006. View at: Publisher Site  Google Scholar
 C. J. Miller and R. J. Shattock, “Target cells in vaginal HIV transmission,” Microbes and Infection, vol. 5, no. 1, pp. 59–67, 2003. View at: Publisher Site  Google Scholar
 T. Bronshtein, N. Toledano, D. Danino, S. Pollack, and M. MacHluf, “Cell derived liposomes expressing CCR5 as a new targeted drugdelivery system for HIV infected cells,” Journal of Controlled Release, vol. 151, no. 2, pp. 139–148, 2011. View at: Publisher Site  Google Scholar
 M. A. Nowak and C. R. M. Bangham, “Population dynamics of immune responses to persistent viruses,” Science, vol. 272, no. 5258, pp. 74–79, 1996. View at: Google Scholar
 M. A. Nowak, S. Bonhoeffer, G. M. Shaw, and R. M. May, “Antiviral drug treatment: dynamics of resistance in free virus and infected cell populations,” Journal of Theoretical Biology, vol. 184, no. 2, pp. 203–217, 1997. View at: Publisher Site  Google Scholar
 J. E. Mittler, B. Sulzer, A. U. Neumann, and A. S. Perelson, “Influence of delayed viral production on viral dynamics in HIV1 infected patients,” Mathematical Biosciences, vol. 152, no. 2, pp. 143–163, 1998. View at: Publisher Site  Google Scholar
 L. Rong and A. S. Perelson, “Modeling HIV persistence, the latent reservoir, and viral blips,” Journal of Theoretical Biology, vol. 260, no. 2, pp. 308–331, 2009. View at: Publisher Site  Google Scholar
 J. M. Orellana, “Optimal drug scheduling for HIV therapy efficiency improvement,” Biomedical Signal Processing and Control, vol. 6, no. 4, pp. 379–386, 2011. View at: Publisher Site  Google Scholar
 P. W. Nelson and A. S. Perelson, “Mathematical analysis of delay differential equation models of HIV1 infection,” Mathematical Biosciences, vol. 179, no. 1, pp. 73–94, 2002. View at: Publisher Site  Google Scholar
 J. A. Zack, S. J. Arrigo, S. R. Weitsman, A. S. Go, A. Haislip, and I. S. Y. Chen, “HIV1 entry into quiescent primary lymphocytes: molecular analysis reveals a labile, latent viral structure,” Cell, vol. 61, no. 2, pp. 213–222, 1990. View at: Publisher Site  Google Scholar
 P. Essunger and A. S. Perelson, “Modeling HIV infection of CD4^{+} tcell subpopulations,” Journal of Theoretical Biology, vol. 170, no. 4, pp. 367–391, 1994. View at: Publisher Site  Google Scholar
 L. Rong, M. A. Gilchrist, Z. Feng, and A. S. Perelson, “Modeling withinhost HIV1 dynamics and the evolution of drug resistance: tradeoffs between viral enzyme function and drug susceptibility,” Journal of Theoretical Biology, vol. 247, no. 4, pp. 804–818, 2007. View at: Publisher Site  Google Scholar
 P. K. Srivastava and P. Chandra, “Modeling the dynamics of HIV and CD4^{+} T cells during primary infection,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 612–618, 2010. View at: Publisher Site  Google Scholar
 B. Buonomo and C. VargasDeLeón, “Global stability for an HIV1 infection model including an eclipse stage of infected cells,” Journal of Mathematical Analysis and Applications, vol. 385, no. 2, pp. 709–720, 2012. View at: Publisher Site  Google Scholar
 X. Wang, Y. Tao, and X. Song, “Global stability of a virus dynamics model with BeddingtonDeAngelis incidence rate and CTL immune response,” Nonlinear Dynamics, vol. 66, no. 4, pp. 825–830, 2011. View at: Publisher Site  Google Scholar
 X. Song and A. U. Neumann, “Global stability and periodic solution of the viral dynamics,” Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 281–297, 2007. View at: Publisher Site  Google Scholar
 G. Huang, W. Ma, and Y. Takeuchi, “Global properties for virus dynamics model with BeddingtonDeAngelis functional response,” Applied Mathematics Letters, vol. 22, no. 11, pp. 1690–1693, 2009. View at: Publisher Site  Google Scholar
 M. Y. Li and L. Wang, “Global stability in some SEIR epidemic models,” in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory, pp. 295–311, Springer Science+Business Media, New York, NY, USA, 2002. View at: Google Scholar
 P. Waltman, “A brief survey of persistence,” in Proceedings of a Conference in Honor of Kenneth Cooke, pp. 31–40, 1991. View at: Google Scholar
 M. Y. Li, J. R. Graef, L. Wang, and J. Karsai, “Global dynamics of a SEIR model with varying total population size,” Mathematical Biosciences, vol. 160, no. 2, pp. 191–213, 1999. View at: Publisher Site  Google Scholar
 M. W. Hirsch, “Systems of differential equations which are competitive or cooperative, IV: structural stabilities in three dimensional systems,” The SIAM Journal on Mathematical Analysis, vol. 21, pp. 1225–1234, 1990. View at: Google Scholar
 X. Wang and X. Song, “Global stability and periodic solution of a model for HIV infection of CD4^{+} T cells,” Applied Mathematics and Computation, vol. 189, no. 2, pp. 1331–1340, 2007. View at: Publisher Site  Google Scholar
 P. Waltman, “A brief survey of persistence in dynamical systems,” in Delay Differential Equations and Dynamical Systems, vol. 1475 of Lecture Notes in Mathematics, pp. 31–40, 1991. View at: Google Scholar
 J. D. Baxter, D. L. Mayers, D. N. Wentworth et al., “A randomized study of antiretroviral management based on plasma genotypic antiretroviral resistance testing in patients failing therapy,” AIDS, vol. 14, no. 9, pp. F83–F93, 2000. View at: Publisher Site  Google Scholar
 “Clinical Trial Datasets GART Study,” http://hivdb.stanford.edu/pages/clinicalStudyData/GART.html. View at: Google Scholar
 C. Bourgeois, Z. Hao, K. Rajewsky, A. J. Potocnik, and B. Stockinger, “Ablation of thymic export causes accelerated decay of naive CD4^{+} T cells in the periphery because of activation by environmental antigen,” Proceedings of the National Academy of Sciences of the United States of America, vol. 105, no. 25, pp. 8691–8696, 2008. View at: Publisher Site  Google Scholar
 P. U. Cameron, M. Pope, S. Gezelter, and R. M. Steinman, “Infection and apoptotic cell death of CD4^{+} T cells during an immune response to HIV1pulsed dendritic cells,” AIDS Research and Human Retroviruses, vol. 10, no. 1, pp. 61–71, 1994. View at: Google Scholar
 G. Herbein, C. van Lint, J. L. Lovett, and E. Verdin, “Distinct mechanisms trigger apoptosis in human immunodeficiency virus type 1infected and in uninfected bystander T lymphocytes,” Journal of Virology, vol. 72, no. 1, pp. 660–670, 1998. View at: Google Scholar
 M. Bofill, G. Janossy, C. A. Lee et al., “Laboratory control values for CD4 and CD8 T lymphocytes. Implications for HIV1 diagnosis,” Clinical and Experimental Immunology, vol. 88, no. 2, pp. 243–252, 1992. View at: Google Scholar
 D. R. Kuritzkes, “HIV Pathogenesis and Viral Markers,” April 2000, http://www.medscape.org/viewarticle/421048. View at: Google Scholar
 Y. Wang, Y. Zhou, J. Wu, and J. Heffernan, “Oscillatory viral dynamics in a delayed HIV pathogenesis model,” Mathematical Biosciences, vol. 219, no. 2, pp. 104–112, 2009. View at: Publisher Site  Google Scholar
 World Health Organization, Consolidated Guidelines on General HIV Care and the Use of Antiretroviral Drugs for Treating and Preventing HIV Infection: Recommendations for a Public Health Approach, WHO Press, Geneva, Switzerland, 2013.
 “Berlin Patient, first person cured of HIV, may soon have company,” July 2012, http://articles.latimes.com/2012/jul/27/news/lahebhivcureaidsmeeting20120727. View at: Google Scholar
 “After Berlin Man, two reported cured of HIV in Kenya,” May 2013, http://www.africareview.com/SpecialReports/AfterBerlinMan–tworeportedcuredofHIVinKenya//979182/1843388//3saq6o//index.html. View at: Google Scholar
 B. Julg, F. Pereyra, M. J. Bouzón et al., “Infrequent recovery of HIV from but robust exogenous infection of activated CD_{4}^{+}T cells in HIV elite controllers,” Clinical Infectious Diseases, vol. 51, no. 2, pp. 233–238, 2010. View at: Google Scholar
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Copyright © 2014 Qilin Sun and Lequan Min. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.