Dynamics of SIRDS epidemic model with fractional Brownian motion

In this study, we explore a stochastic SIRDS epidemic model that is influenced by fractional Brownian motion (fBm) with a Hurst index H ∈ (1/6, 1/2). By integrating stochastic fluctuations into the transmission rate, we develop a four-compartment framework, which is represented through two ordinary differential equations and two stochastic differential equations driven by fBm. Utilising the symmetric stochastic integral, along with a version of Itô’s formula tailored for fBm and a suitably constructed random Lyapunov function, we establish the existence and uniqueness of a global, positive solution. We subsequently derive sufficient conditions that guarantee the extinction of the disease. For the numerical simulations, we generate sample paths of fBm using the Fast Fourier Transform method (FFT) and apply a modified Euler scheme to manage the resulting fBm increments. The model parameters are calibrated based on empirical epidemiological data, enabling us to evaluate how well the simulated infection paths align with the observed dynamics by identifying the parameters that best fit our context. We specifically compared two simulated infection trajectories corresponding to H = 0.38 and H = 0.5. This comparison indicated that the Hurst index has a significant impact on the model's dynamics. By employing error metrics to support our conclusions, we found that H = 0.38 corresponds more closely with the data than H = 0.5, thereby making it the most appropriate choice for our scenario.