OPTIMAL PREVENTION OF HIV-AIDS WITH EMPHASIS ON UNPROTECTED AND UNNATURAL CANAL ACTIVITIES: A DETERMINISTIC MODELLING PERSPECTIVE

. HIV accounts for more than thirty three million death and approximately thirty eight million infected cases since it’s inception. The disease unfolds in three stages; Chronic, Acute and fully blown AIDS. Adhering to preventive protocols such as use of condoms, preventing oneself from unprotected sex and limiting one’s sexual partners could help minimize the disease spread. In this study, a deterministic model for HIV-AIDS is formulated. The equilibrium points, local and global stability of the equilibrium points, and HIV reproductive rate were determined and interpreted. The model was extended to optimal control by simulating the optimality system. This was done by incorporating the use of condoms and education of susceptible population as intervention strategies. It was established that the best and most effective control strategy was optimal education and sensitisation of susceptible population.


INTRODUCTION
The viral infection that thrive in the system of another living organism, especially humans, was detected in 1981 in the blood-streams of mostly gay men. The disease is identified with strains: HIV-1 and HIV-2. The HIV-1 is recognized as the lethal strain, causing pandemic in humans. The disease is transmissible when a contact of susceptible human is made with the infected person through unprotected and unnatural canal activity.
HIV accounts for more than 33 million death and 37.7 million infected cases since the disease's inception [1]. The disease unfolds in three stages; Chronic , Acute and full blown AIDS. The acute stage is characterized by rash, headache and fever. The acute stage is noted as the first two weeks to one month where the transmission of virus is very high, and one could easily get infected when in contact with the infected. The chronic stage is characterize by a decrease in the virus replication as the infected enters into clinical latency stage, but one can still get infected with the disease since it could be transferred by the infected. The last stage of the disease: AIDS happens when the person's immune system is substantially weakened and can no longer defend the body against foreign attacking pathogens.
Adhering to the preventive protocols such as use of condoms, preventing oneself from unprotected sex and limiting one's sexual partners could help minimize the number of the infected. However, the use of pharmaceutical drugs such as antiretroviral therapy (ART) may help reduce the multiplication of the virus and the swift progressing of the disease [2]. [3] proposed a model for the transition period of HIV/AIDS incorporated of weak uninfected CD4 + cells as T(t). For a weak uninfected CD4 + T cell, the authors estimate a very short transition period. As a result, when weak CD4 + T-cells engage with HIV, some of these weak CD4+ T-cells shift directly into the viral class, which is a key factor in the fast spread of HIV.
Another important finding was that the natural recovery of CD4 + T-cells cannot be overlooked because a large proportion of T cells have recovered. The results of the study through numerical technique using confirmed the analytical results of the model as several unmeasured parameter values were assumed and used.
[4] used seven-dimensional nonlinear ordinary differential equation to establish a mathematical model to analyze the spread of HIV epidemic within an antiretroviral therapy (ART) treatment as an alternative intervention. In absence of antiretroviral therapy (ART) treatment in the model showed transition rate among infected compartment reduced. However, model analysis showed sensitivity of the antiretroviral therapy treatment to the basic reproduction number along the numerical simulation. The findings of the study showed stationary in the number of susceptible humans, leading to a reduction in the number of infected individual who progressed to AIDS as a result of antiretroviral therapy treatment. [5] considered new deterministic mathematical model for the transmission dynamics of HIV/AIDS virus on the role of female sex workers in India. The study employed homotopy perturbation method to derive an analytical solution to each nonlinear deterministic system containing initial condition for those individual sub-groups. The analytical solution was compared with numerical solution obtained by MATLAB function; fourth order Runge-Kutta method. The analytical results obtained can run sensitivity analysis of the estimated parameters to better understand the spread mechanism of HIV/AIDS and suggest possible prevention strategies. Epidemiological models generally explain the transmission dynamics of diseases and can determine the status of infections with time. Models that are incorporated with some control can determine the best optimal control strategy in combating infections [7,8,9,10].

MODEL FORMULATION
Model divides the total population under study into five compartment of Susceptible, S, Exposed, E H , Fully blown HIV, A H , Infected HIV, I H and Treatment, T H . Recruitment into the susceptible population is denoted by the rate Λ. λ is the rate at which the exposed individuals leaves the exposed compartment to the infected compartment. Further, the model assumes that individuals die as a result of the disease at a rate δ . β is the transmission rate as a results of contact between susceptible, infected and fully blown HIV individuals. The model assumes that infected individuals seek medical attention at a rate γ, while a fraction of the infected individuals progressed to fully blown HIV status. Fully blown HIV individuals also seek treatment at a rate σ . µ is the natural death rate. Table 4 and Table 5 shows the parameters, variables, and their descriptions used in the model formulation.   Hence, the system differential equations describing the HIV model in Figure 1 is given by Proof.
Doing same for the entire compartments gives

Region of feasibility.
Theorem 3.2. The positive solution is a positively invariant set of the model and is given by Therefore, the positive solution set is an invariant set of the model and is given by 3.3. Disease-free equilibrium. The disease free equilibrium points of the HIV model is given

3.4.
HIV reproductive rate, R 0 . Using the approach in [11,12,13,14,15,16], the infection compartments are as follows The matrices F and V are generated from the infection compartments 7 as The Jacobian of matrix F is given by Similarly, finding the Jacobian of matrix v gives The Local stability of the disease free equilibrium. The section presents the stability analysis of model equation 1 at the disease-free equilibrium. The linearization method is adopted in studying the asymptomatic stability of model equation 1 at the disease-free equilibrium [17,18].
The Jacobian matrix at disease free equilibrium becomes When J B is evaluated at the disease-free equilibrium point Matrix A is the remaining matrix of (J B 1 − l), given by Referring to 19, the model system 1 is locally unstable since according to Gershgorin circle, 3.6. Global Stability of the Disease-free equilibrium. We investigate the global asymptotic stability of the model system 1 by using the Castillo-Chavez's method. This is presented as follows; Consider with p 1 and p 2 denote number of uninfected and infected individuals respectively. Thus, we denote p 1 = S ∈ R 2 and p 2 = E H , I H , T H , A H ∈ R 4 . The disease-free equilibrium f 0 for the model system (1) is given by f 0 = (p 0 1 , 0). Thus, the global stability at D 0 exists based on these conditions where D = Wy 2 C(z 0 1 , 0) is an M-matrix, with a positive off-diagonal entries and τ is the feasible biological region of model (1). When the above conditions are satisfied by model system (1), then the underlying theorem holds. Proof. From model (1), we can deduce Hence Y (p 1 , 0) becomes, The Jacobian of X(p 1 , p 2 ) is given by Hence using the expression we deduce the following Applying the equation 23, and solving for the expressionw(p 1 , p 2 ) gives  [19,20].
When 24 is evaluated at the endemic equilibrium The remaining matrix of J y becomes Hence, the HIV model system 1 is not lacally stable.

Global
The time derivative of L becomes individuals u 2 to the non-control model.
We consider a quadratic function for the objective functional as in other literature [25]. Here, we seek to minimize the exposed and infected. The control of personal protection: condom use u 1 would be employed to achieve the above mentioned purpose of minimizing the exposed and infected population. Hence, the objective functional J is given by The quantities of objective functional (29) G 1 and G 2 are the weight coefficients of the exposed, infected, treatment and asymptomatic population. In addition, the expressions C 1 u 2 1 2 and C 2 u 2 2 2 are the cost that comes with minimizing the the exposed and infected population. Hence, we seek an optimal control u * 1 such that Theorem 4.1. There exists an optimal control U * = (u * 1 , u * 2 ) ∈ U such that subject to the control system (28) with the initial conditions.
Proof. By the work of [26], the existence of optimal control is proved. We observe that the state and control variables are non-negative. We also observe that in minimizing the control problem, the necessary and convexity of the objective functional in u 1 are satisfied. The control space is also convex and closed by definition. The optimal system is bounded which verifies the compactness needed for the existence of the optimal control. Also, the integrand in functional 29, is convex on the control u. Therefore, we see that there exist a constant k > 1, positive numbers u 1 , u 2 such that, Hence, there exist an optimal control. In the quest to find the optimal solution, the Pontryagin's maximum principle [27,21] is applied to the Hamiltonain 32 such that if (w, u) is an optimal solution of the optimal control problem, then there exist a non-trivial vector function λ = (λ 1 . . . λ 6 ) satisfying the below equation Hence, the necessary condition associated to the Hamitonian (32) is applied.
Theorem 4.2. Given that S, E H , I H , T H and A H are optimal state solutions with associated control variables (u * 1 , u * 2 ) for the optimal control problem 28 and ??, then there exist adjoint variables λ i for i = 1, . . . , 6, satisfying The optimal control u * 1 are given by Proof. = 0 are determined on the interior of the control set and using the optimal conditions and the property of the control space u 1 and u 2 , and we derive 28. From (28), The control is characterize by solving the optimal system. Thus, the transversality and the charcterisation of the optimal control (u 1 ) are use in solving the optimal system [28,29,30].
The controls u * 1 and u 2 when substituted into the control system (28) gives

NUMERICAL SIMULATIONS
In determining the best control strategy that would help combat the spread of infection, an iterative scheme that uses a fourth-order Runge-Kutta method to run the optimal system is designed. This approach runs state equation forward and the adjoint system backwards in time.
Iteration runs until a stopping criterion is met, and it stops. Effectiveness of the considered controls on the model are assessed, these controls are paired, and a numerical simulation carried out.
Output plots generated for each considered strategy are carefully assessed for consideration. Table 6 shows some of the parameter values used in the numerical simulations that generated these outputs. Following are the observations from various plots as indicated in Figures 2, 3

CONCLUSION
In this study, a deterministic model for HIV-AIDS is formulated. The equilibrium points, local and global stability of the equilibrium points, and HIV reproductive rate were determined and interpreted. The model was extended to optimal control and it was established that the best and most effective control strategy was optimal education and sensitisation of susceptible population.
We simulated the optimality system by incorporating the use of condoms as the only intervention. It can be observed that there have been an exponential decrease in the number of susceptible and infected populations. Then the optimality system was simulated by incorporating education as the only intervention. It was observed that there have substantial change in population of susceptible individuals. Moreover, there have been a reduction in the of individuals getting infected with infection. An indication of the possibility of this intervention.
In combating the infection, more resources should placed on sensitisation and education of the susceptible population.

ACKNOWLEDGMENT
Authors expressed their appreciation to other researchers for their numerous review comments and suggestions.

SOURCE OF FUNDING
No sources of funding for this study.