Bifurcation and Stabillity Analysis of HIV Transmission Model with Optimal Control

Abstract

A mathematical model of HIV transmission is built and studied in this paper. The system’s equilibrium is calculated. A next-generation matrix is used to calculate the reproduction number. The novel method is used to examine the developed model’s bifurcation and equilibrium stability. The stability analysis result shows that the disease-free equilibrium is locally asymptotically stable if $0<R_0<1$ but unstable if $R_0>1$ . However, the endemic equilibrium is locally and globally asymptotically stable if $R_0>1$ and unstable otherwise. The sensitivity analysis shows that the most sensitive parameter that contributes to increasing of the reproduction number is the transmission rate $\left({\beta }_2\right)$ of HIV transmission from HIV individuals to susceptible individuals and the parameter that contributes to the decreasing of the reproduction number is identified as progression rate $\left(\eta \right)$ of HIV-infected individuals to AIDS individuals. Furthermore, it is observed that as we change $\eta$ from $0.1$ to $1$ , the reproduction number value decreases from 1.205 to 1.189, where the constant value of ${\beta }_2=0.1$ . On the other hand, as we change the value of ${\beta }_2$ from $0.1$ to $1$ , the value of the reproduction number increases from 0.205 to 1.347, where the constant value of $\eta =0.1$ . Further, the developed model is extended to the optimal control model of HIV/AIDS transmission, and the cost-effectiveness of the control strategy is analyzed. Pontraygin’s Maximum Principle (PMP) is applied in the construction of the Hamiltonian function. Moreover, the optimal system is solved using forward and backward Runge–Kutta fourth-order methods. The numerical simulation depicts the number of newly infected HIV individuals and the number of individuals at the AIDS stage reduced as a result of taking control measures. The cost-effectiveness study demonstrates that when combined and used, the preventative and treatment control measures are effective. MATLAB is used to run numerical simulations.