Parameter estimation in the stochastic SIR model via scaled geometric Brownian motion

The stochastic SIR epidemiological model offers a comprehensive understanding of infectious diseases dynamics by taking into account the effect of random fluctuations. However, because of the nonlinear nature of the stochastic SIR model, accurately estimating its parameters presents a significant challenge, crucial for unraveling the intricacies of disease propagation and developing effective control strategies. In this study, we introduce a novel approach for the estimation of the parameters within the stochastic SIR model, including the often-neglected noise in the transmission rate (volatility). We employ a quasi-deterministic approximation, where the number of infected (susceptible) individuals evolves deterministically, whereas the number of susceptible (infected) individuals evolves stochastically. The solutions of the resulting stochastic equations are scaled geometric Brownian motions (SGBM). Based on the maximum likelihood method applied to the log-returns of susceptible (infected) individuals, we propose algorithms that yield numerical evidence of unbiased estimates of transmission and recovery rates. Our approach maintains robustness even in the presence of increasing volatility, ensuring reliable estimations within reasonable limits. In more realistic scenarios where the model parameters vary with time, we demonstrate the adaptability of our algorithms for successful parameter estimation in sliding time windows. Notably, this approach is not only accurate but also straightforward to implement and computationally efficient.