Numerical Simulation of SIA Model in HIV/AIDS Using Euler and 4th OrderRunge-Kutta Method
Abstract
The HIV/AIDS epidemic remains a major public health issue, particularly in regions such as West Nusa Tenggara, Indonesia, where the number of reported cases continues to rise. To better understand and predict the spread of the disease, mathematical modeling provides a valuable analytical tool. This study employs the SIA (Susceptible-Infected-AIDS) compartmental model to describe the dynamics of HIV/AIDS transmission within the population. The model incorporates a latent phase, distinguishing it from simpler models and making it more suitable for diseases with long incubation periods such as HIV/AIDS. Two numerical methods, that is Euler and the 4th Order Runge–Kutta (RK4), are applied to solve the system of nonlinear differential equations derived from the model. Using official epidemiological data from 2023 in West Nusa Tenggara, the simulation tracks the evolution of the population across the three compartments over a 12-month period. The results indicate a consistent decline in the number of susceptible individuals and an increase in both infected and AIDS-diagnosed individuals. The basic reproduction number, , suggesting that the disease is endemic in the region. Stability analysis further confirms that the disease-free equilibrium is unstable, while the endemic equilibrium is locally asymptotically stable. These findings highlight the urgency of targeted interventions and demonstrate the importance of mathematical models in guiding public health strategies for disease control.
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