The study of viral dynamics of HIV/AIDS has resulted in a deep understanding of host-pathogenesis of HIV infection from which numerous mathematical modeling have been derived. Most of these models are based on nonlinear ordinary differential equations. In Bangladesh, the rate of increase of HIV infection comparing with the other countries of the world is not so high. Bangladesh is still considered to be a low prevalent country in the region with prevalence < 1% among MARP (Most at risk populations). In this paper, we have presented the current situation of HIV infection in Bangladesh and also have discussed the mathematical representation of a three-compartmental HIV model with their stability analysis. We have determined the basic reproduction number and shown the local and global stability at disease free and chronic infected equilibrium points. Also we have shown that if the basic reproduction number , then HIV infection is cleared from T cell population and it converges to disease free equilibrium point. Whereas if , then HIV infection persists.
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) [
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 [
In Bangladesh, the first case of HIV was detected in 1989 [
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 [
susceptible class. It has a logistic growth with
proliferation rate. Parameters
The model is positively invariant and bounded in the region
We have determined the basic reproduction number
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 [
Description | Symbols | Values |
---|---|---|
CD4+ T cell source rate | 0.1 mm−3∙day−1 | |
Natural turnover rate of uninfected CD4+ T cell | 0.02 day−1 | |
Natural turnover rate of infected CD4+ T cell | 0.3 day−1 | |
Natural turnover rate of virus | 2.4 day−1 | |
Drug efficacy | 0.5 | |
CD4+ T cell infection rate | 0.0027 mm−3∙day−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.
Here we check the positivity of each compartments such as susceptible
Lemma 1. Let
Proof: To prove the Lemma 1, we have used the system of equations of the model (1).
in order to find the positivity we have,
Multiplying both sides of (2) by
Now Integrating (3)
where
Putting the value of
Hence
Therefore, it is true that,
The disease free equilibrium of the above HIV model (1) can be obtained by setting
thus we have,
Since we have considered the disease free equilibrium, hence
Thus, the disease free equilibrium is
Again for the endemic equilibrium point
Now we calculate the basic reproduction number
Basic reproduction number represents the average number of secondary infection caused by a single infected T cell in an entirely susceptible T cell population, throughout its period. In order to find the basic reproduction number of the model (1), we need to identify the classes which are relevant to each other. Form the model (1), we observe that the classes
Gains to
Since basic reproduction number is to be calculated at disease free equilibrium point
Matrix for the loss terms;
Inverse of
Now we have to evaluate a matrix
Hence the largest eigen value of the matrix
Firstly, we investigate the local stability at disease free equilibrium point
Theorem 1: If
Proof: To prove the above theorem, the following variation matrix is computed corresponding to equilibrium point
then the system (1) reduces to,
The Jacobian Matrix of the system (1) is
at
Now we have to find out the characteristic equation. To do that, first we have to calculate
To find out the characteristic equation we need to perform
Thus, the characteristic equation is
where
We observe that, first root of the characteristic equation is
If
Now we investigate the local stability of chronic infection equilibrium point
Lemma 2: Let
Before we apply the Lemma 2, we need the following definition of second additive compound matrix.
Definition 1: Let
Theorem 2: The chronic infection equilibrium point
Proof: From Equation (5), we have
at chronic infection equilibrium point
where
Now the second additive compound matrix
Now we compute
Hence by Lemma 2,
We have discussed the locally asymptotically stability of both infection free equilibrium
We observe
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
production number
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
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