Dynamics of a Fractal-Fractional and Stochastic COVID-19 Model with Levy Noise and its Computational Modelling

This paper investigates the dynamics of Coronavirus Disease 2019 (COVID-19) by incorporating vaccination, partial immunity, and stochastic perturbations into a comprehensive mathematical framework. Three versions of the model are considered: a classical ordinary differential equation model, a fractal-fractional order model, and a stochastic differential equation model driven by Levy noise. The fractal-fractional order COVID- 19 (FFoCOVID-19) model is numerically solved using a fourth-order RungeKutta method with a power-law kernel to capture memory and hereditary effects in disease transmission. For the stochastic model, sufficient conditions for the existence and uniqueness of positive solutions, disease extinction, and persistence in the mean are established using Lyapunov functions and stochastic analysis techniques. The basic reproduction number, [Formula: see text] is derived and shown to serve as a threshold parameter governing the spread and control of the disease. Furthermore, a LevenbergMarquardt backpropagation neural network (L-MBNN) is employed to obtain numerical solutions of the COVID-19 model under three different initial-condition scenarios. The accuracy and efficiency of the proposed neural-network approach are evaluated through mean squared error (MSE) analysis, with Case 1 achieving training, testing, and validation MSE values of [Formula: see text], and [Formula: see text] respectively. Numerical simulations confirm the reliability of the proposed computational techniques and demonstrate that Levy noise can significantly influence disease dynamics, while the neural-network framework provides an effective and accurate tool for solving complex epidemiological models.