Modeling the dynamic mechanisms of HIV infection and the effectiveness of drug treatment

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 182 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) 183 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. 184 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: 186 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)2 187 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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