HIV stands for human immunodeficiency virus. The virus attacks the immune system, and weakens our ability to fight infections and disease. HIV/AIDS pro- gresses in body slowly and its symptoms are shown after 6 - 8 years sometimes even later. At present, the most burning issue at the same time, the most dangerous phenomena is Human Immunodeficiency Virus (HIV) [1] . Since the beginning of the epidemic, more than 70 million people have been infected with the HIV virus and about 35 million people have died of HIV. Globally, 36.7 million [34.0 - 39.8 million] people were living with HIV at the end of 2015 [2] . An estimated 0.8% [0.7% - 0.9%] of adults aged 15 - 49 years worldwide are living with HIV, although the burden of the epidemic continues to vary considerably between countries and regions. Sub-Saharan Africa remains the most severely affected, with nearly 1 in every 25 adults (4.4%) living with HIV and accounting for nearly 70% of the people living with HIV worldwide [2] . Acquired Immunodeficiency Syndrome (AIDS) was first discovered in 1981, since then it has been considered as the most leading cause of mortality [3] . A detailed background and survey on HIV/AIDS is described in [4] [5] [6] [7] . HIV mainly targets CD⁴⁺ T cells. The continuous attack HIV causes the depletion of CD⁴⁺ T cells and this leads people to gradually become a victim of Acquired Immunodeficiency Syndrome (AIDS). For this reason, the count of CD⁴⁺ T cells is considered as the primary indicator of progression of HIV. In recent times, mathematical modeling has become the most powerful tool to incorporate the dynamic behaviors of infectious diseases. Mathematical modeling is basically referred to as a method of simulating real-life situations with mathematical equations to forecast their future behavior [8] . Numerous mathematical models have been developed to identify the characteristics of human immunodeficiency virus [9] [10] [11] . HIV dynamic model, a set of ordinary differential equations (ODE) that describe the interaction between HIV virus and human body cells, has been proven useful for understanding the pathogenesis of HIV infection and developing treatment strategies [12] . In this paper, we have shown the present scenario of HIV/AIDS in Bangladesh. Also we have studied a three-compartmental HIV model and investigated their stability at disease free and endemic equilibrium points.

HIV is a worldwide curse. There is no such country where this pandemic disease does not exist. Although Bangladesh is still considered to be a low responded HIV infected country in world, the present situation indicate that the influence of this pandemic disease is gradually increasing. The main reason for this low prevalence could be the early and sustained HIV prevention programs targeting high risk groups backed by a state-of-the-art surveillance system. Another contributing protective factor could be the high rates of male circumcision. There is, however, a concentrated HIV epidemic among injecting drug users (IDU), primarily due to sharing of unclean syringes and needles. As a result, the rate of new infections is still on the rise and Bangladesh is the only country in the South Asia Region where new infections are rising [13] .

In Bangladesh, the first case of HIV was detected in 1989 [3] . Since then, it has been enhanced considerably. In 2015 (December 2014 to November 2015), the number of newly HIV infected people is 469 and the number of HIV/AIDS related death is 95. Till December 2015, there were 4143 reported cases of HIV and among them 658 died [6] . Here we show a graphical representation of HIV surveillance of Bangladesh (see Figure 1) from 1989 to 2015 (except 2008) [14] .

To generate a realistic model of T cell infection by HIV, we first need to consider the population dynamics of T cells in the absence of HIV. Our interest is to present a mathematical model of HIV infection and analyze the model. In this paper, we present a three compartmental model of HIV which has been taken from [15] . We have modified this model and added a drug efficacy parameter whose value is in the range between 0 and 1 [16] . The total population size is divided into three stages of HIV/AIDS progression; the susceptible population, HIV infected individuals and HIV virus The total population is given by The population CD⁴⁺ T cells starts with a source or production rate and dead cells with rate are reduced from the

susceptible class. It has a logistic growth with where is the

proliferation rate. Parameters and are natural turnover rate of uninfected CD⁴⁺ T cells, infected CD⁴⁺ T cells and virus. Whereas is the maximum level of CD⁴⁺ T cell concentration in the body [17] . Infected CD⁴⁺ T cells it has an infection rate which is concentrated as. The transfer diagram of the model is shown in Figure 2.

The model is positively invariant and bounded in the region

We have determined the basic reproduction number which was first introduced by Ross (1909), which is defined in epidemiological modeling as the average number of infected individuals produced by one infected immigrant in a population which is completely susceptible [18] . Finding the basic reproduction number, we can determine the endemic result of disease in populations. If, the disease vanishes and if, the disease spreads and goes to the endemic level.

If one wishes to use a mathematical model to make predictions about a particular individual or population, estimation of model parameters from data is crucial. All the parameters and their values used for model (1) are taken from [15] [16] and presented in Table 1.

Here we investigate the positivity of the model, find out different equilibrium points, formulate the basic reproduction number and check the stability at disease free and endemic equilibrium points.

We have discussed the locally asymptotically stability of both infection free equilibrium and chronic infection equilibrium above. When, the endemic equilibrium may only be stable for small or large. Our numerical solutions consistently show the existence of periodic solutions when is unstable. For the numerical result we use the parametric values used in Table 1 taken from [15] and [16] but with the variation of. Considering, we have shown local stability of both the healthy CD⁴⁺ T cells and HIV virus at and (see in Figure 3 and Figure 4). Whereas at, is unstable and a periodic solution exists (see in Figure 5).

We observe is unstable within the range of between 0.093453 and 1.9118. From Figure 4, we observe viral load 600 mm⁻³ persists when while it is below 100 mm⁻³ at. Again when the initial oscillation disappears after 145+ days whereas at the damped oscillation are clearly visible after 2000 days. We also note that, the values of in these three cases are 0.5081, 0.8227 and 0.48431 respectively.

Bangladesh government and several NGO’s have played a magnificent role in keeping the HIV prevalence low by enhancing awareness to people. But this low prevalence rate is increasing day by day and becoming a great threat to us. In this paper, we have shown a brief report of HIV/AIDS of Bangladesh from 1989 to 2014 (except 2008). Again we have discussed the mathematical presentation of HIV infection in a three-compartmental model. In the model, we added a probability term with the infected T cells. Then we have calculated the basic re-

production number, where is considered as equilibrium of

CD⁴⁺ T cells in the absence of HIV infection. At disease free equilibrium point, the model is assumed to be stable and later we conclude the stable and unstable condition for the chronic infected equilibrium points. With the proliferation term and reproduction number, we find the solution of it. We find the numerical solution at different equilibrium points and have observed the curve in periodic and damped oscillation.

Sahani, S.K. and Biswas, M.H.A. (2017) Mathematical Modeling Applied to Understand the Dynamical Behavior of HIV Infection. Open Journal of Modelling and Simulation, 5, 145-157. https://doi.org/10.4236/ojmsi.2017.52010