MODELING THE DYNAMICAL MECHANISMS OF HIV INFECTION AND EFFECTIVENESS OF DRUG TREATMENT

Abstract. The HIV treatment is much more difficult and complicate than other virus disease treatments because unclear the HIV starting affection moment and due to the up and down behavior of body resistance system. In this work we develop a simple dynamical model for the HIV infection with three parameters: the concentration of HIV virus and the concentrations of the health and infected T-cells. By introducing the “therapy starting moment”, using this dynamical model we investigate the effectiveness of the HIV drug treatment.

Key word: dynamic model, genetic material, starting time, satified, immune system, RTI, PI.

I. INTRODUCTION

The earliest known positive identification of the HIV-1 virus comes from the Congo in 1959 and 1960 though genetic studies indicate that it passed into the human population from chimpanzees around fifty years earlier[1]. Since then, according to UNAIDS 2009 report, worldwide some 60 million people have been infected, with some 25 million deaths, and 14 million orphaned children in southern Africa alone since the epidemic began[2]. The development of potent antiviral drugs began in the mid 1990’s. Current treatment for HIV infection consists of highly active antiretroviral therapy, or HAART[3]. Its goals include improvement the patients quality of life, reduction in complications, and reduction of HIV viremia below the limit of detection.

The main target of HIV virus is CD4+ T cell (referred as T-cells in this paper). The virus attack and deposit its genetic material into the cell. Once inside, it uses the 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 after several years with the infection, lead to the assumption that there has a slow dynamical time scale of the infection[5].

In the last few years, there has been considerable interests in developing drugs models due to their effectiveness. One of the most recent models is to consider the drug effectiveness as a dynamical variable[5]. The efficacy of therapy is considered as evolve dynamically. The virus population fitness has been introduced, and in presence of drug treatment, 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 the drug effectiveness in considering starting moment of the therapy. It is delicate to introduce the ”therapy starting moment” notion for HIV treatment therapy, car it’s not easy to determine infection time. Thus, the HIV damages the immune system, so the disease schema is complicated. The treatment schema for other disease could not be used in this case. That’s why the dynamical model is needed to be introduced for HIV treatment.

II. SUBJECTS AND METHODS

We research the long-term dynamics of HIV infection and analyze the effectiveness of drug at different starting time on 20 patients of the infected HIV viruts.

In the mono-therapy, there are two specific methods using drug: using drug once in time-treatment and using continuance in time-treatment. The good therapy gives high uninfected cells number, low infected cells and viruses number and long latency period.

The combination of drugs might result in high active anti-retroviral therapies and prevent to resist drug of virus. The simplest combination of therapies is the combination of two drugs. The model combination of one PIT therapy and one PI therapy.

The schematic diagram of the experimental setup used in this experiment is shown in Fig 1. to Fig 7.

III. RESULTS AND DISCUSSION

III.1. THE BASIC HIV DYNAMIC MODEL

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The HIV attack the immune system by sending virus to penetrate T-cell, use the genetic material from the T-cell to produce copies of itself, and eventually kill the T-cell to release the newly produced viruses into the blood stream to infect other T-cells.

With T the plasma concentration of T-cells, T the concentration of the infected T-cells, and V the concentration of the HIV viruses, the evolution of the disease could be described[6]:

in which is source term for CD4+T cells, natural death rate of CD4+T cells, growth rate of CD4+T cells and maximal population level of CD4+T cells respectively.

The concentration of infected T- cell T and HIV virus V are defined:

where k is rate of infection for CD4+T cells, is natural death rate of infection CD4+T cells, N is number of viruses produced by infected T-cells and c is natural death rate of viruses.

Table 1. Table 1: Parameters mean values

s μT r Tmax k μT ¿ N c

0.02 10 0.03 1500 2.4 10−5 0.26 500 2.4∗

day−1mm−3 day−1 day−1 mm3 day−1mm3 day−1

In the following sections , we use the parameter in table 1.

III. 2. MONO-THERAPY

According to the envelopment of virus HIV, there are two anti-viral therapies.

The first one based on Reverse Transcriptase Inhibitors (RTI). These drugs are able to inhibit the Reverse Transcriptase enzyme, which is necessary to the virus in order 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 will not successfully infect it.

In this case, Eq. (2.2) could be written [5]:

in which the effectiveness of the drug on infected T-cells η is chosen in the interval [0, 1], particularly when η =1 the drug is 100% effective.

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In which, red : therapy starting 1st infection HIV virus; blue: therapy starting 34 th days infection HIV virus; Green: therapy starting 50th days infection HIV virus; Yellow: therapy starting 97 th days infection HIV virus; Brown: therapy starting 128 th

days infection HIV virus; Black: therapy starting 200th days infection HIV virus.

When the process of protease is inhibited, the virion produced using the genetic material of T cells are unable to fully mature and thus unable to reproduce. Eventually, these virions diet out without contributing to the infection. This therapy is the Protease Inhibitors (PI). Eq. (2.3) for this therapy could be written[5]:

Where is the effectiveness of the drug on virus HIV, which has the same meaning of parameter η in equation (2.4), thus, the drug is 100% effective when .

In the literature both η and have been considered as constant due to the unchanged effectiveness of drug in a very short time period. In the long- term dynamics, the effectiveness of drug was considered as time-varying coeffcients[7], η = η(t),

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Fig.2. Using continuance in time-treatment η=η(t ) ;δ=δ( t )

In this paper, we research the long-term dynamics of HIV infection and analyze the effectiveness of drug at different starting time. Assuming the combination of the initial conditions satisfied the basic set of equations with no drug treatment. In the mono-therapy, there are two specific methods using drug: using drug once in time-treatment and using continuance in time-treatment. The good therapy gives high uninfected cells number, low infected cells and viruses number and long latency period.

Comparing the cells number of patient without treatment in Fig 1 to the cell number of patients using one drug during the time-treatment in Fig 3 . The life time of HIV patients using mono-therapy treatment, using once time drug is not longer than the life time of HIV patients without treatment.

III.3. THE COMBINATION OF THERAPIES

When the patients use a drug during the time-treatment, virus is able to resist drug. The combination of drugs might result in high active anti-retroviral therapies and prevent to resist drug of virus.

The simplest combination of therapies is the combination of two drugs. The model combination of one PIT therapy and one PI therapy was introduced in[5]:

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Fig.3.Using drug once in time-treatment α=0,8 ; β=0,05

Fig.4. Using continuance in time-treatment η=η(t ) ;δ=δ(t )

The effectiveness of drug using once during time-treatment is are constants.

The schematic diagram of the experimental setup used in this therapy is shown in following:

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Fig.5.Using drug once in time-treatment combined tow drugsη( t )=γ e( t )=α e

−βt ;α=0,8 ; β=0,05

Fig.6.Using drug once in time-treatment with RTI method η( t )=γ e( t )=α e− βt ;

α=0,8 ; β=0,05 using continuance in time-treatment with PI method

γ (t ) ¿(1−cos2πt) /2❑

Fig.7. using continuance in time-treatment combined tow drugsγ (t ) ¿(1−cos2πt) /2❑

IV. CONCLUSION

The relation between the fitness of the viruses and the effectiveness of the drug has been investigated. Based on that results, the dynamics of the drug effectiveness has been described. It is important to introduce the dynamical model into the HIV treatment, as the HIV schema is much more complicated than the other disease one. The HIV effects varies in function of time, and the drug must be used in the right moment, when the HIV has the less effects on the immune system.

We have shown mathematically by model that the earlier the treatment start, the better status the patient would stay. This ”seemed to be trivial” result has been shown in the increasing healthy cells number, decreasing infected cells and viruses number, also match to the medical statistical data. The decreasing of viruses number also confirm mathematically a statistical result which has been published recently that early HIV treatment minimize the viruses spread[8].

REFERENCES

[1] Worobey M, Gemmel M, Teuwen DE, et al., Nature, 455 (2008) (7213): 6614.

[2] data.unaids.org.

[3] A Pocket Guide to Adult HIV/AIDS Treatment, Department of Health and Human Services. February 2006.

[4] C. Graziozi, G. Pantaleo and A. S. Fauci, New Engl J. Med, 328, 327 (1993). J. M. Coffin, Science, 267, 483 (1996).

[5] Giulio Della Rocca, Marco Sammartino and Luciano Seta, Riceche di matematica, 54(1), 313-327 (2005).

[6] Perelson A.S. and Nelson, P.W. SIAM Rev., 41, 3-44, (1999).

[7] Yangxin Huang and Taolu, Ann. Appl. Stat, 2, 1384-1408 ( 2008).

[8] Washington Post (2011)

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