Stability Analysis of HIV/AIDS Model with Educated Subpopulation

We had constructed mathematical model of HIV/AIDS with seven compartments. There were two different stages of infection and susceptible subpopulations. Two stages in infection subpopulation were an HIV-positive with consuming ARV such that this subpopulation can survive longer and an HIV-positive not consuming ARV. The susceptible subpopulation was divided into two, uneducated and educated susceptible subpopulations. In this paper, we consider the transmission only from uneducated to infection subpopulations. In other hand, educated subpopulation did not have contact with the infected subpopulations. We investigated local stability of the equilibrium points according to the basic reproduction number (R0) as a threshold of disease transmission. The disease-free and endemic equilibrium points were locally asymptotically stable when R0 < 1 and R0 > 1 respectively. To support the analytical results, numerical simulation was conducted.


INTRODUCTION
AIDS (Acquired Immune Deficiency Syndrome) is a disease of the immune system caused by HIV (human immunodeficiency virus) (HIV). AIDS is a threat in the world because people infected with HIV can cause death. WHO seeks to campaign for the dangers of this disease and provide various controls including the use of condoms or consume ARV (Antiretroviral Treatment).
Mathematical models have made a significant contribution to understanding the spread of HIV infection. Mathematical model HIV/AIDS have been studied by [1]- [4] where they formulated the mathematical model of HIV/AIDS with the treatment stated in the SIATR. In [4] and [5], they constructed and conducted a dynamic analysis of the HIV / AIDS epidemic model with different stages of infection and different stages of susceptible subpopulations respectively. [6] studied dynamical analysis the model 2 locally and globally. The results, the disease-free and endemic equilibrium points were locally and globally asymptotically stable.
In this research, we propose mathematical model of HIV/AIDS with educated subpopulation. The proposed model is more realistic. We determined the disease-free and endemic equilibrium points as the solution of the model, the basic reproduction number (R0), and analyzed the stability of equilibrium points locally following [7]- [14]. The disease-free equilibrium point is locally asymptotically stable when R0 < 1 and the endemic equilibrium point is locally asymptotically stable when R0 > 1. Numerical simulations were performed using values of selected parameters to support the results of the analysis.

METHODS
To this end the research, we started by literature review. We modify the mathematical model of HIV/AIDS from [6] by adding educated subpopulation. Next, we analyzed the constructed model dynamically. Firstly, we should find the equilibrium points (diseasefree and endemic equilibrium points). Then, we find the basic reproduction number used as a threshold endemic-occurred. Furthermore, we analyze local stability using the Routh-Hurwitz criteria. Numerical simulation is performed to see behavior of the model solution using the Runge-Kutta 4th order method.

RESULTS AND DISCUSSION
The model of HIV/AIDS with educated subpopulation consist of seven compartments ( Figure 1  We establish an HIV/AIDS model with educated subpopulation in the form of a system of non-linear differential equations as follows. = − 1 1 − 2 2 − , = − , 1 = 1 1 + 1 − 1 , 2 = 2 2 − 2 , where = + + , = 1 + , = 2 + 3 + , = 1 + 2 + 2 + , dan = 1 + . Parameter is recruitment rate of the population, β1 and β2 are transmission coefficient from uneducated subpopulation to infection stage I1 and I2 respectively, β3 and β4 are transmission coefficient from educated subpopulation to infection stage I1 and I2 respectively, p is natural mortality rate, α1 is the proportion of successful treatment, α2 is the proportion of treatment failure, K1 is progression rate from I1 to T, K2 is progression rate from I2 to A, K3 is progression rate from I2 to T, δ1 is the disease-related death rate of the AIDS, δ2 is the disease-related death rate of being treated, µ is the rate of susceptible individuals who changed their habits, and is the rate of educated individuals who received information of HIV/AIDS. The transmission coefficients from educated and uneducated subpopulations to infection stages were β i where i = 1,2,3,4 ((β 1 and β 2 ) > (β 3 and β 4 )) where in this research we consider the case of β 3 and β 4 were zero. It means that E(t) subpopulation is free of infection.

Positivity of solutions
We proof positivity of solutions of the model follows [6].
Proof. From the first equation of system (1), we have where 1 ( ) = 1 1 + 2 2 + . We multiply equation (2) with ∫ 1 ( ) 0 to give which implies Next, we write the left hand side of equation (4) as derivative of ∫ 1 ( ) 0 wtith respect to t, to yield then we integrate with respect to v from 0 to t, we get We multiply equation (6) which is said that the solution of the first equation of system (1) is positive.
Furthermore, the solutions of system (1) can be written as

Equilibrium points
We will find two equilibrium points, disease-free and endemic equilibrium points. The equilibrium points are obtained by solving the equations system (1) when = 0, = 0, The basic reproduction number ( 0 ) is obtained by using the next generation matrix method [15]. The constituent components of the next generation matrix method only consist of infected population groups, namely Before we find the endemic equilibrium point, we will find the basic reproduction number. First, define ′ = ( 1 ′ , 2 ′ ) , the system of equations (9) can be stated as  ).
The partial derivative of V is We substitute the point 0 to DF matrix to get ).
We substitute point 0 into DV to yield ). Then, the eigenvalues of R matrix are obtained as follows.

Stability Analysis of HIV/AIDS Model with Educated Subpopulation
Ummu Habibah 193 The basic reproduction number ( 0 ) is obtained from the spectral radius of the R or the largest modulus of the eigenvalues of the matrix R and the value of 1 < 2 , then we get Furthermore, the endemic equilibrium point * = ( * , * , 1 * , 2 * , * , * , * ) can be written as follows

Local stability analysis
The local stability of the equilibrium point is obtained by linearizing the system around the equilibrium point. We linearize the system (1) to get the Jacobi matrix Determinant of equation (12) is We can write 5 = 2 − such that we get 5 = ( 0 − 1), where 5 < 0 when 0 < 1. 6,7 is obtained when satisfies | 1 − − 1 1 − − | = 0.

Stability Analysis of HIV/AIDS Model with Educated Subpopulation
Ummu Habibah 195 Theorem 2. The endemic equilibrium * is locally asymptotically stable when R0 > 1 and unstable otherwise.
Proof. The Jacobian matrix at the * equilibrium point is  (14) is 1 2 * , Sometime, it is difficult to find roots of the characteristic polynomial, therefore the Routh-Hurwitz criteria can be used to find stability characteristic of * equilibrium. * equilbrium point is asyptotically stable if and only if it meets the following conditions. i.

NUMERICAL SIMULATION
We give numerical simulation to illustrate the main result. Numerical simulations are solved by using the 4th order Runge-Kutta method. Numerical simulation is conducted in order to understand the behavior of the proposed HIV/AIDS model and to confirm the stability analysis of the equilibrium points (disease-free and endemic equilibrium points) in the previous section. We will show that the disease-free equilibrium point is asymptotically stable when 0 < 1 and the endemic equilibrium point is asymptotically stable when 0 > 1. We use the parameter values for numerical simulation in Table 1.   Table 1, the condition of the Routh-Hurwitz criterias are satisfied The simulation shows the endemic equilibrium point is asymtotically stable when 0 > 1, and the numerical results support the analysis results. Based on the numerical simulation results above, it can be seen that over time the number of individuals infected with HIV with symptoms will go to 3.1928 and individuals infected with HIV without symptoms will go to 4.4043. Thus, it can be interpreted that individuals infected with HIV / AIDS will always exist, so that there will be the spread of HIV / AIDS infection in that environment.
Next, we simulate the stability of model solutions for the disease-free equilibrium point numerically. We choose the parameter values in order to satisfy the basic reproduction number 0 > 1 as shown in Table 1 except the values 1 = 0.0023, 2 = 0.0033, and = 0.3. We obtain 0 = 0.0069 < 1. Figure 3 shows the solutions of HIV/AIDS model with initial values = (30, 10, 25, 35, 20, 16, 50) lead to desease-free equilibrium point 0 = (1.5732 , 24.0799 , 0 , 0 , 0 , 0 , 2.4107) as in Figure 3. The numerical simulation results obtained support the results of the analysis in the previous section that if ℛ 0 < 1, then the HIV / AIDS disease-free equilibrium point 0 , is asymptotically stable, which means that after quite a long time, the infected individual will vanish.

CONCLUSIONS
The mathematical model of HIV/AIDS with educated subpopulation have been established. The model consists of seven compartments (susceptible, educated, infected with and without treatment, AIDS, treatment, and recovered subpopulations). The infected subpopulation are in HIV-positive with consuming ARV I1 such that this subpopulation can survive longer and an HIV-positive without consuming ARV I2. The susceptible subpopulation was divided into two, uneducated and educated susceptible subpopulations.
The stability analysis of HIV/AIDS model is determined according the basic reproduction number. The disease-free equilibrium is locally asymptotically stable when R0<1 and unstable when R0>1. The endemic equilibrium is locally asymtotically stable when R0<1 and unstable otherwise. The endemic equilibrium is globally asyptotically stable when R0>1 and unstable otherwise. Numerical simulation are performed using values of selected parameters to support the analytical results.