Mathematical Epidemic Model of HIV / AIDS in Pakistan

In Pakistan the effect population mobility, specifically labor migration and refugees is also thought to have been important in explaining the rapid spread of HIV/AIDS. One of the effects labor migration is likely to have had increased the prevalence of the overlap of sexual partnership. A nonlinear fractional differential equation model is discussed for transmission and control of HIV/AIDS in Pakistan. We shall also discuss the disease free equilibrium and stability behavior of the model.


Introduction
AIDS is the disease which is caused by the human immunodeficiency virus (HIV).HIV primarily targets CD4+ cells and without treatment this leads to the collapse of the host immune system and ultimately death.The clinical syndrome was called acquired immune deficiency syndrome ( AIDS) in 1982 and four years later the causative virus was named HIV-I.A large number of modeling studies has been focused on HIV since its discovery.Presently there is no remedy for HIV so that once the AIDS stage of HIV infection is attained then ultimately death follows.Usually HIV is much less infectious than the short duration bacterial STI because the value of is possibly minimum as compared to short duration bacterial STI.However, HIV is infectious for far longer than the short-duration STIs, increasing its relative to short-duration bacterial STIs.Depending on the research questions, we may also need to consider that the infectiousness of HIVinfected individuals varies obviously with time since infection.

Preliminaries
In recent advanced research, fractional calculus field is developed as it has much application in engineering and medical sciences.Application of fractional derivative will be discussed to understand some more related definitions.These are given below: Definition 1 Gamma function is given by We can also define function by using gamma function which will be more suitable for offering alternative form of the fractional integral is given by Definition 2 In terms of the gamma function, Beta integral and its solution can be shown as (1) (2) The general formula of the Grunwald-Letnikov fractional derivative Definition 6 Caputo defined the fractional derivative of a function( ) as In our research, we will generalized HIV/AIDS model to a fractional order system of order  in the sense of caputo definition because it is equivalent to ordinary differential equation when  =

𝑛𝑛 Description of the Model
To construct the model, all parameters are supposed to be non-negative.We divide the population into four subclasses, the susceptible class s(t), the infective class I(t) that do not know that they have HIV and the known infective class J(t) and the AIDS class A(t).Thus we formulated the fractional order system model as: Leading Eigen values are denoted by the equation given below: Where If  � > 1, a positive endemic equilibrium Λ * = ( * ,  * ,  * ,  * ) is given by the following:

By increasing τ
We try to find about it if  �  �,� change sign to be positive.If we get pure imaginary Eigen values (  �,� = ± ).Then Putting the value of  = ± in Eq. ( 18), then we get: Removing τ from the above equations ( 19), we get 20) Where y =  � , hence there is no positive roots for Eq. ( 18) In this case the values of Re �,� will remain positive.Since By applying the condition (22) we have: Since we can write: 16) proof is completed.Numerical Analysis, Simulation and Discussion.The local stability of the model for fractional order time derivative is evaluated using fractional Routh-Hurwitz stability criterion.The fractional derivative is described in Caputo sense.The results obtained through numerical procedure show that the technique is effective and reliable.To study the behavior of system numerically a fourth order Range-Kutta method is needed for this we used a computer simulation software Berkeley Madonna and data was collected from Pakistan Demographic Health Survey for this purpose.We intend the numerical simulations that shows the stability and equilibrium state of the disease and effectiveness of the model.In this study, we have proposed reported data of HIV/AIDS and Sentinel surveillance centers' reports on HIV/AIDS.These parameter values is to be included into the entire theme for development of appropriate models to predict the spread of HIV/AIDS in Pakistan.Treatment rate from symptomatic phase to asymptomatic phase D disease related death We examined the prediction, incident rate and intensity of HIV and secondary infection through this model .It has been seen that (Figure 1 shows recruitment rate of new higher risk individuals.It will be increasing with raised in susceptible individuals.While earlier prevalence rate is increased then attained constant trend.Th the higher risk individual exterminates through the population and new higher risk partners are not inducted at the similar rate.Therefore, partner rate change will be dropped with the passage of time.If nothing else changes, HIV prevalence infections and per year number of HIV deaths, (Figure 2). the system (Figure 4) are analyzed and obtain some significant results.These shows that HIV infection control efficiently if we increase the incubation time period and minimize the intensity of secondary infections with the proper treatment.However, i because variation in HIV prevalence and incidence are also due t and result of intervention.073 We examined the prediction, incident rate and intensity of HIV and secondary infection through this model .It has been seen that (Figure 1), the incidence of HIV/AIDS is predicted to rise steadily that shows recruitment rate of new higher risk individuals.It will be increasing with raised in susceptible individuals.While earlier prevalence rate is increased then attained constant trend.Th the higher risk individual exterminates through the population and new higher risk partners are not inducted at the similar rate.Therefore, partner rate change will be dropped with the passage of time.If nothing else changes, HIV prevalence will become steady with the stability in the new HIV infections and per year number of HIV deaths, (Figure 2).Equilibria and corresponding stability of the system (Figure 4) are analyzed and obtain some significant results.These shows that HIV control efficiently if we increase the incubation time period and minimize the intensity of secondary infections with the proper treatment.However, it is very difficult to explain HIV trend because variation in HIV prevalence and incidence are also due to the natural dynamics of infection 1 Prediction of people living with HIV/AIDS.We examined the prediction, incident rate and intensity of HIV and secondary infection through this ), the incidence of HIV/AIDS is predicted to rise steadily that shows recruitment rate of new higher risk individuals.It will be increasing with raised in susceptible individuals.While earlier prevalence rate is increased then attained constant trend.This is because the higher risk individual exterminates through the population and new higher risk partners are not inducted at the similar rate.Therefore, partner rate change will be dropped with the passage of time.
will become steady with the stability in the new HIV Equilibria and corresponding stability of the system (Figure 4) are analyzed and obtain some significant results.These shows that HIV control efficiently if we increase the incubation time period and minimize the intensity of is very difficult to explain HIV trend o the natural dynamics of infection We examined the prediction, incident rate and intensity of HIV and secondary infection through this model .It has been seen that (Figure 1), the incidence of HIV/AIDS is predicted to rise steadily that shows recruitment rate of new higher risk individuals.It will be increasing with raised in susceptible individuals.While earlier prevalence rate is increased then attained constant trend.This is because the higher risk individual exterminates through the population and new higher risk partners are not inducted at the similar rate.Therefore, partner rate change will be dropped with the passage of time.If nothing else changes, HIV prevalence will become steady with the stability in the new HIV infections and per year number of HIV deaths, (Figure 2).Equilibria and corresponding stability of the system (Figure 4) are analyzed and obtain some significant results.These shows that HIV infection control efficiently if we increase the incubation time period and minimize the intensity of secondary infections with the proper treatment.However, it is very difficult to explain HIV trend because variation in HIV prevalence and incidence are also due to the natural dynamics of infection and result of intervention.

Conclusion
In our study, a nonlinear mathematical model having equilibrium and corresponding stability of the system are analyzed and obtain some significant results i.e.HIV infection control efficiently, if we increase the incubation time period and minimize the intensity of secondary infections with the proper regimen fractional order is presented.Otherwise leads to the collapse of the host immune system and ultimately death.Ethical clearance: this article was approved by the ethics committee of Navy Engineering College -National University of Sciences and Technology, Islamabad (Pakistan) before submission

Table 2 :
Values of parameters of the model Parameters Description ì Death rate constant k Average number of contacts of an individual per unit time â � In the first stage, Probability of disease transmission per contact by an infective person â � Probability of disease transmission per contact by an infective person in the second stage I Transfer rate constant from asymptomatic phase to symptomatic phase  � Transfer rate constant from symptomatic phase to asymptomatic phase A Individual with AIDS ã

Figure. 1
Figure. 1 Prediction of people living with HIV/AIDS.