(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Understanding and reducing the spread of epidemics depends much on the modelling and study of infectious disease dynamics. Among several compartmental models, the Susceptible-Infected-Quarantined-Recovered framework has become more popular as it can include the influence of quarantine, a significant intervention in many actual epidemics. Fundamental epidemic processes are naturally vulnerable to random variations brought on by environmental variability, population stochasticity, and other unknown elements. Hence, including stochastic influences in such models is crucial. Furthermore, spatial dispersion and diffusion effects are essential, particularly in significant populations and varied settings, which call for stochastic partial differential equations (SPDEs). A two-stage mixture of exponential integrator and Runge-Kutta scheme is proposed for solving stochastic epidemic disease models. The scheme is more accurate than the existing Euler Maruyama method. The stability and consistency of the scheme in the mean square sense are provided. The scheme only discretizes time-dependent terms in given stochastic partial differential equations. Moreover, a stochastic SIQR diffusive model is presented with the effect of incidence rate. The deterministic and stochastic models are solved using the Euler-Maruyama method, the proposed scheme, and the nonstandard finite difference method. The comparison shows that the proposed scheme provides less error than the existing nonstandard finite difference method. The results indicate that the proposed scheme attains superior accuracy and diminished errors relative to current methods. This underscores its capability as an effective instrument for simulating complex stochastic epidemic models incorporating spatial effects and non-linear dynamics.