Evaluating the effectiveness of stochastic CTMC and deterministic models in correlating rabies persistence in human and dog populations

Rabies continues to pose a significant zoonotic threat, particularly in areas with high populations of domestic dogs that serve as viral reservoirs. This study conducts a comparative analysis of Stochastic Continuous-Time Markov Chain (CTMC) and deterministic models to gain insights into rabies persistence within human and canine populations. By employing a multitype branching process, the stochastic threshold for rabies persistence was determined, revealing important insights into how stochasticity influences extinction probabilities. The stochastic model utilized 10,000 sample paths to estimate the probabilities of rabies outbreaks, offering a rigorous assessment of the variability in disease occurrences. Additionally, the study introduces a novel mathematical formulation of rabies transmission dynamics, which includes environmental reservoirs, free-ranging dogs, and domestic dogs as essential transmission factors. The basic reproduction number ($\mathcal{R}_0$) was derived and analyzed within stochastic frameworks, effectively bridging the gap between these two modeling approaches. Numerical simulations confirmed that the results from the stochastic model closely aligned with those from the deterministic model, while also highlighting the importance of stochasticity in scenarios with low infection rates. Ultimately, the study advocates for a comprehensive approach to rabies control that integrates both the predictable trends identified through deterministic models and the impact of random events emphasized by stochastic models.