Abstract. HIV treatment is much more difficult and complicated than other virus disease
treatments because of the unclear HIV starting moment and due to the up and down
behavior of the body resistance system. In this work we created a simple dynamic model for
HIV infection with three parameters: the concentration of HIV virus, the concentration of
health and the infected T-cells. Using a ‘therapy starting moment’ and this dynamic model,
we investigated the effectiveness of HIV drug treatment.

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JOURNAL OF SCIENCE OF HNUE DOI: 10.18173/2354-1059.2016-0048
Mathematical and Physical Sci., 2016, Vol. 61, No. 7, pp. 182-188
This paper is available online at
MODELING THE DYNAMIC MECHANISMS OF HIV INFECTION
AND THE EFFECTIVENESS OF DRUG TREATMENT
Nguyen Minh Hoa1 and Nguyen Ai Viet2
1Hue University of Medicine and Pharmacy
2Hanoi Institute of Physics
Abstract. HIV treatment is much more difficult and complicated than other virus disease
treatments because of the unclear HIV starting moment and due to the up and down
behavior of the body resistance system. In this work we created a simple dynamicmodel for
HIV infection with three parameters: the concentration of HIV virus, the concentration of
health and the infected T-cells. Using a ‘therapy starting moment’ and this dynamic model,
we investigated the effectiveness of HIV drug treatment.
Keywords: Dynamic model, genetic material, starting time, satisfied, immune system,
reverse transcriptase inhibitors, protease inhibitors.
1. Introduction
The first identification of the HIV-1 virus was in the Congo in 1959 and 1960 when genetic
studies indicated that it passed into the human population from chimpanzees some fifty years
earlier [1]. Since then, according to the 2009 UNAIDS report, some 60 million people have become
infected worldwide, with some 25 million deaths, and 14 million orphaned children in southern
Africa since the epidemic began [2]. The development of potent antiviral drugs began in the
mid-1990’s. Current treatment for HIV infection consists of HAART (highly active antiretroviral
therapy) [3]. It improves patients quality of life, reduces complications, and reduces HIV viremia
below the limit of detection.
The main target of the HIV virus is CD4T cells. It attack and deposit its genetic material
into the cell. Once inside, it uses host cell machinery to make copies of its viral DNA in the same
manner as other retrovirus [4]. When the viral attack leads the level of the T-cells below a critical
level. Reflecting on the fact that most HIV patients develop AIDS several years after infection, the
assumption is that there is a slow dynamic time scale of infection [5].
In the last few years, there has been considerable interest in developing drugs models. In
one of the most recent models, drug effectiveness is considered to be a dynamical variable [5].
The efficacy of therapy is thought to evolve dynamically. The virus population fitness has been
Received February 26, 2016. Accepted September 20, 2016.
Contact Nguyen Minh Hoa, e-mail address: minhhoa2806@gmail.com
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Modeling the dynamic mechanisms of HIV infection and the effectiveness of drug treatment
introduced, and in presence of drug treatment, and this fitness is related to the effectiveness of the
drug. This fitness of the virus population depends on an adaptive mechanism.
In this paper, we investigate drug effectiveness considering the starting moment of the
therapy. It is delicate to introduce a ‘therapy starting moment’ notion for HIV treatment therapy
because it’s not easy to determine the infection time. The HIV damages the immune system so the
disease schema is complicated. A treatment schema for any other disease could not be used in this
case. This is why the dynamic model is needed for HIV treatment.
2. Content
2.1. Subjects and methods
We have looked at the long-term dynamics of HIV infection and analyzed the effectiveness
of drugs at different starting times on 20 people who were infected with the HIV virus.
In mono-therapy, there are two specific methods for using drugs: using a drug once in
time-treatment or using continually in time-treatment. Successful therapy would give a high
number of uninfected cells, a low number of infected cells and a long latency period.
A combination of drugs might result in active anti-retroviral therapies and prevent virus
drug resistance The simplest therapy is a combination of two drugs. Our model combines RTI
therapy (Reverse Transcriptase Inhibitors) and PI therapy (Protease Inhibitors).
The schematic diagram of the experimental setup used in this experiment is shown in
Figures 1-7.
2.2. Results and discussion
2.2.1. The basic hiv dynamic model
The HIV virus attacks the immune system by penetrating T-cells. It uses the genetic material
of the T-cells to produce copies of itself and eventually it kills the T-cells and releases the newly
produced viruses into the blood stream to infect other T-cells.
With plasma concentration of T-cells T, concentration of the infected T-cells t, and
concentration of the HIV viruses V, the evolution of the disease could be described [6]:
dT
dt
= s− µTT + rT (1− T + T ∗ {Tmax})− kV T (2.1)
in which s, r, µT , and Tmax are the source terms for CD4+T cells, the natural death rate of
CD4+T cells, the growth rate of CD4+T cells and the maximal population level of CD4+T cells,
respectively. The concentration of infected T-cell T and HIV virus V are defined:
dT∗
dt
= kV T − µT∗T∗ (2.2)
dV
dt
= NµT∗T ∗ − cV (2.3)
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Nguyen Minh Hoa and Nguyen Ai Viet
where k is rate of infection for CD4+T cells, µT ∗ is natural death rate of infected CD4+T
cells, N is number of viruses produced by infected T-cells and c is natural death rate of the virus.
Table 1. Parameters of mean values
s µT r Tmax k µ(T ∗) N c
0.02 10 0.03 1500 2.4 ×10−5 0.26 500 2.4
day−1 mm−3 day−1 day−1 mm3 day−1mm3 day−1
In the following sections , we use the parameters in Table 1.
2.2.2. Mono-therapy
Due to the envelopment of the HIV virus, there are two anti-viral therapies.
The first is based on Reverse Transcriptase Inhibitors (RTI). These drugs are able to inhibit
the Reverse Transcriptase enzyme, which is necessary for the virus to use the DNA of the T-cell
to replicate itself. Therefore, when the enzyme is inhibited, a virus can penetrate a T-cell but not
successfully infect it.
In this case, equation (2.2) could be written [5]:
dT
dt
= s− µTT + rT (1− T + T ∗ {Tmax})− kV T
dT∗
dt
=
(
1− η(t)
)
kV T − µT∗T∗
dV
dt
= NµT∗T ∗ −cV (2.4)
in which the effectiveness of the drug on infected T-cells η is chosen in the interval [0, 1]. When η
=1, the drug is 100% effective.
Figure 1. Using a drug once in time-treatment α = 0.8; β = 0.05
In Figure 1, red line: therapy starting at the time of initial HIV infection; blue line: therapy
starting 34 days after HIV virus infection; Green line: therapy starting 50 days after HIV virus
infection; Yellow line: therapy starting 97 days after HIV virus infection; Brown line: therapy
starting 128 days after HIV virus infection; Black line: therapy starting 200 days after HIV virus
infection.
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Modeling the dynamic mechanisms of HIV infection and the effectiveness of drug treatment
Figure 2. Using continual time-treatment
When the protease process is inhibited, the virion produced using the genetic material of
T cells is unable to fully mature and thus unable to reproduce. Eventually, these virions die out
without contributing to the infection. This therapy is the Protease Inhibitors (PI). Equation (2.3)
for this therapy could be written [5]:
dT
dt
= s− µTT + rT (1− T + T ∗ {Tmax})− kV T
dT∗
dt
= kV T − µT∗T∗
dV
dt
= (1− α)NµT∗T ∗ −cV (2.5)
where α is the effectiveness of the drug on the HIV virus which has the same meaning of
parameter η in equation (2.4), thus, the drug is 100% effective when α = 1 .
In the literature both η and α have been considered as constant due to the unchanged
effectiveness of drug during a very short time period. In long-term dynamics, the effectiveness
of drug was considered to be a time-varying coefficient[7], η = η(t), δ = δ(t).
Figure 3. Using a drug once in time-treatment α = 0.8; β = 0.05
185
Nguyen Minh Hoa and Nguyen Ai Viet
Figure 4. Using a drug continually in time-treatment η = η(t) = (1− cos2πt)/2
In this paper, we looked at the long-term dynamics of HIV infection and analyzed the
effectiveness of drugs at different starting times. We assumed that the combination of the initial
conditions satisfied the basic set of equations with no drug treatment. In the mono-therapy,
there were two methods for using a drug: using a drug once in time-treatment and using a drug
continually in time-treatment. Good therapy gives a high uninfected cell number, a low infected
cell and virus number, and a long latency period.
We compared the cells numbers in people not being treated in Figure 1 to the cell numbers
in people receiving one drug during time-treatment in Figure 3. The lifetime of the HIV infected
people receiving mono-therapy treatment, using a drug once, was not longer than the lifetime of
the HIV infected people who received no treatment.
2.2.3. The combination of therapies
When the patients used a drug during time-treatment, the virus was able to resist the drug.
A combination of drugs might result in improved anti-retroviral therapies and prevent resistance
to the drugs.
The simplest combination of therapies is the combination of two drugs. RTI therapy and PI
therapy were combined in [5]:
dT
dt
= s− µTT + rT (1− T + T ∗ {Tmax})− kV T
dT∗
dt
= kV T − µT∗T∗
dV
dt
= NµT∗T ∗ − cV
(2.6)
The effectiveness of using a drug once during time-treatment is α. e−γt, α, γ, which are
constants.
The schematic diagram of the experimental setup used in this therapy is shown in the
following:
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Modeling the dynamic mechanisms of HIV infection and the effectiveness of drug treatment
Figure 5. Using a drug once in a time-treatment that combined two drugs
η(t) = γe(t) = αe
−βt;α = 0.8;β = 0.05
Figure 6. Using drug once in time-treatment
with RTI method η(t) = γe(t) = αe
−βt;α = 0.8;β = 0.05 using continuance
in time-treatment with PI method γ(t) = (1− cos2πt)2
Figure 7. using continuance in time-treatment combined tow drugs γ(t) = (1− cos2πt)2
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Nguyen Minh Hoa and Nguyen Ai Viet
3. Conclusion
The relation between the fitness of the viruses and the effectiveness of the drug was
investigated. Based on results obtained, the dynamics of the drug effectiveness was described.
It is important to make use of the dynamical model in HIV treatment because the HIV schema is
much more complicated than that of other diseases. HIV effects vary as a function of time and the
drug must be used at the right moment, when the HIV has minimum effect on the immune system.
We have shown by mathematical model that the earlier treatment is started, the better the
status of the patient. This “seemed to be trivial” those results can be seen in the increased number
of healthy cells, a decreased number of infected cells and virus, and this matches the medical
statistical data. The decrease in virus number also confirms mathematically a recently published
statistical result which shows that early HIV treatment minimizes virus spread.
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[2] data.unaids.org.
[3] Department of Health and Human Services, 2006. A Pocket Guide to Adult HIV/AIDS
Treatment.
[4] C. Graziozi, G. Pantaleo and A. S. Fauci, 1993. New Engl., 328, 327. J. Med, 1996. J. M.
Coffin, Science, 267, 483.
[5] Giulio Della Rocca, 2005. Marco Sammartino and Luciano Seta. Riceche di matematica,
54(1), 313-327.
[6] Perelson A.S. and Nelson, 1999. P.W. SIAM Rev., 41, 3-44.
[7] Yangxin Huang and Taolu, 2008. Ann. Appl. Stat, 2, 1384-1408.
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