Modelling and optimal control strategy for HIV/AIDS disease in the presence of infective immigrants and non-linear incidence

A non-linear mathematical model is proposed to study the optimal control strategy for HIV/AIDS disease in the presence of infective immigrants and non-linear incidence. In modeling the dynamics of the system, the population is divided into three classes namely; HIV negatives but susceptible, HIV infective and full blown AIDS patients. Positivity of solutions were analyzed quantitatively. Sensitivity indices of the effective reproductive number “ ” to the parameters in the model was calculated. We estimate the model state initial conditions and parameters values from HIV/AIDS data of other literature. We use Pontryagins maximum principle to derive the optimality system and solve the system numerically. The existence and stability of the disease free equilibrium point were also analyzed. It is found that using linearization method by next generation matrix approach, as well as eigenvalues of Jacobian matrix, the disease free equilibrium point is locally asymptotically stable when the reproduction number is less than unity is significantly reduce the infective and AIDS population compared to the case when reproduction number is greater than unity which is unstable. In the absence of control the endemicity of the infection increases results in the increase of infective and AIDS population. Further, disease eradication associated with β is only feasible when the transmission rate is low. We also discuss the characterization of the optimal control via adjoint variables. Finally, we present numerical results obtained by simulating the optimality system using ODE solvers in matlab programming language. Numerical results show that application of control leads to decrease in transmission dynamics hence reduction of the disease. The fraction of the population averted from HIV infection due to the control measures starts to decrease with time gradually after its maximum value that is after six and half years of implementation.