Estimating Effective Reproduction Number for SIR Compartmental Model: A Stochastic Evolutionary Approach

Compartmental pandemic models have become a significant tool in the battle against disease outbreaks. Despite this, pandemic models sometimes require extensive modification to accurately reflect the actual epidemic condition. The Susceptible-Infectious-Removed (SIR) model, in particular, contains two primary parameters: the infectious rate parameter $\beta$ and the removal rate parameter $\gamma$, in addition to additional unknowns such as the initial infectious population. Adding to the complexity, there is an obvious challenge to track the evolution of these parameters, especially $\beta$ and $\gamma$, over time which leads to the estimation of the reproduction number for the particular time window, $R_T$. This reproduction number may provide better understanding on the effectiveness of isolation or control measures. The changing $R_T$ values (evolving over time window) will lead to even more possible parameter scenarios. Given the present Coronavirus Disease 2019 (COVID-19) pandemic, a stochastic optimization strategy is proposed to fit the model on the basis of parameter changes over time. Solutions are encoded to reflect the changing parameters of ${\beta }_T$ and ${\gamma }_T$, allowing the changing $R_T$ to be estimated. In our approach, an Adaptive Differential Evolution (ADE) and Particle Swarm Optimization (PSO) are used to fit the curves into previously recorded data. ADE eliminates the need to tune the parameters of the Differential Evolution (DE) to balance the exploitation and exploration in the solution space. Results show that the proposed optimized model can generally fit the curves well albeit high variance in the solutions.