Solving the HIV-AIDS Transmission Mathematical Model using Second-Order Runge-Kutta Methods

Abstract

This study focuses on the mathematical modelling of human immunodeficiency virus - acquired immunodeficiency syndrome (HIV-AIDS) dynamics using the Susceptible-Infected-Chronic-AIDS (SICA) model. The objective is to approximate the solution of HIV-AIDS transmission in a population by estimating the proportion of individuals in the subpopulations using second-order Runge-Kutta (RK2) methods specifically the midpoint and Ralston’s methods. The results are then compared to the existing solution using the fourth-order Runge-Kutta (RK4) method obtained by [1], with all computations performed in MATLAB R2023a. The results show that the susceptible subpopulation decreases, HIV infected subpopulation peaks then decline, the chronic subpopulation increases, and the AIDS subpopulation decreases, with all stabilizing at equilibrium. The comparison shows that the midpoint and Ralston methods produce results nearly identical to the RK4 method, with minor discrepancies in regions of rapid transitions. Norm analysis shows Ralston’s method outperforms the midpoint method, but both proved to be reliable for solving the SICA model.