Modeling HIV-HBV Co-infection Dynamics: Stochastic Differential Equations and Matlab Simulation with Euler-Maruyama Numerical Method

HIV/AIDS and Hepatitis B co-infection complicates population dynamics and brings forth a wide range of clinical outcomes which makes it a difficult situation for public health. In particular designing treatment plans for the co-infection. A Stochastic Differential Equation (SDE) model is a special class of a stochastic model with continuous parameter space and continuous state space. Deterministic model lacks randomness while an SDE model accounts for randomness and uncertainties. In this study, an SDE model was formulated from an existing deterministic model to examine the variability of dynamic behavior. The analysis and numerical schemes were derived based on Euler-Maruyama SDE algorithms. The model utilized epidemiological insights with current developments in mathematical modeling approaches to represent the interaction between these two viruses. Matlab software was used to obtain SDE numerical results alongside the deterministic solution. Descriptive statistics of the sample paths indicated that the variability of infection outcomes oscillates around the deterministic trajectory. None of the sample paths are absorbed during the time steps. This shows the persistence of the co-infection in the population, in particular  The variability of the infections ranges between 1.972 and 202.4, being lowest in AIDS infectives and highest in acute Hepatitis B infectives. An indication that variability cannot be ignored in designing control interventions of co-infections. These results provide new insights into the dynamics of co-infection through in-depth research and simulation, which helps to understand the inherent nature of deterministic model by incorporating the stochastic effects. These understanding will further help the policy makers in health sector to take care of the variability and uncertainty in designing treatment and management strategies.