On a stochastic epidemic SIR model with non homogenous population: a toy model for HIV

In this paper we generalise a simple discrete time stochastic SIR type model defined by Tuckwell and Williams. The SIR model by Tuckwell and Williams assumes a homogeneous population, a fixed infectious period, and a strict transition from susceptible to infected to recovered. In contrast, our model introduces two groups, $A$ and $B$, where group $B$ has a higher risk of infection due to increased contact rates. Additionally, the duration in the infected class follows a probability distribution rather than being fixed. Finally, individuals in group $B$ can transition directly to the recovered class R, allowing us to analyze the impact of this preventive measure on disease spread. Finally, we apply this model to the spread of HIV, analyzing how risk behaviors, rapid testing, and PrEP-like therapies influence the epidemic dynamics.