A Stochastic Differential Equation Model for HIV/AIDS Transmission Dynamics in Heterosexual Populations

A mathematical HIV/AIDS transmission model for heterosexual populations integrates environmental noise to analyze population variations. The model uses three distinct compartments for the population: susceptible S(t), infected I(t), and AIDS A(t). The mathematical well-posedness of the model was confirmed through positivity and boundedness tests which led to the determination of both deterministic R0 and stochastic \(R^S_0\) reproduction numbers using the method of next generation matrix approach and the results showed how environmental noise influences transmission. The system demonstrates local asymptotic stability at the disease-free equilibrium when \(R^S_0\) < 1 with noise intensities significantly influencing the stability conditions and an endemic equilibrium occurs when \(R^S_0\) > 1characterized by fluctuations around deterministic predictions. The simulation results demonstrate substantial fluctuations in disease prevalence when noise levels reach σ1 = 0.1, σ2 = 0.15 and σ3 = 0.1. The model extended to a discrete-state continuous-time Markov chain framework, enabling analysis of transition probabilities and stationary distributions under noise effects. Numerical simulations using the Euler-Maruyama method demonstrated that noise introduces substantial variability in compartmental trajectories, with the infected population exhibiting particularly pronounced fluctuations due to its higher noise sensitivity. Multiple stochastic realizations revealed significant outcome diversity despite identical parameters, emphasizing the importance of incorporating environmental variability in HIV/AIDS disease forecasting. The Markov chain extension facilitated analysis of transition probabilities between susceptible, infected, and AIDS compartments, enabling policymakers to track disease progression dynamics. This approach informs long-term treatment strategies, particularly in ensuring sustainable antiretroviral therapy resource allocation as infected individuals advance to AIDS.