Mathematical analysis of a new nonlinear stochastic hepatitis B epidemic model with vaccination effect and a case study

Abstract

This work present a detailed analysis of a stochastic delayed model which governs the transmission mechanism of the Hepatitis B virus (HBV) while considering the white noises and the effect of vaccinations. It is assumed that the perturbations are nonlinear and an individual may lose his/her immunity after the vaccination, that is, the vaccination can produce temporal immunity. Based on the characteristics of the disease and the underlying assumptions, we formulated the associated deterministic model for which the threshold parameter $R_0^D$ is calculated. The model was further extended to a stochastic model and it is well-justified that the model is both mathematically and biologically feasible by showing that the model solution exists globally, bounded stochastically and is positive. By utilizing the concepts of stochastic theory and by constructing appropriate Lyapunov functions, we developed the theory for the extinction and persistence of the disease. Further, it is shown that the model is ergodic and has a unique stationary distribution. The stochastic bifurcation theory is utilized and a detailed bifurcation analysis of the model is presented. By using the standard curve fitting tools, we fitted the model against the available HBV data in Pakistan from March 2018 to February 2019 and accordingly the parameters of the model were estimated. These estimated values were used in simulating the model, theoretical findings of the study are validated through simulations and predictions were drawn. Simulations suggest that for a complete understanding of HBV dynamics, one must include time delay into such studies, and improvements in every vaccination program are unavoidable.