| Abstract: |
| We investigate a novel epidemic model known as the SIRV$^{3}$S model, which incorporates three types of vaccinations and introduces stochastic perturbations in one of our key parameters, considering the presence of a long-memory effect. This formulation enables us to express our model as a stochastic differential equation driven by fractional Brownian motion with a Hurst parameter ($H > \frac{1}{2}$), which we denote as FSDE. By employing the Wick-It^{o}-Skorohod (WIS) integral framework, we establish the existence and uniqueness of a global positive solution using the random Lyapunov function method in conjunction with It^{o}`s formula.
In our numerical modelling, we examine an example based on the COVID-19 epidemic. Our objective is to ascertain the most appropriate Hurst parameter for our specific context. To achieve this objective, we first generate fractional Brownian motion utilising the fast Fourier transform method. Following this step, we apply an Euler-type discretisation tailored to the increments of fractional Brownian motion. Throughout this simulation, we compare three distinct Hurst parameters. Through these comparisons, we identify the parameter that best corresponds to our particular scenario. |
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