Computational Analysis of Hepatitis B Epidemic Model With Incorporating a Delay Effect Into Stochastic Differential Equations

Stochastic delayed modeling (stochastic differential equations [SDEs] with delay parameters) has a significant nonpharmaceutical intervention to control transmission dynamics of infectious diseases, and its results are close to the reality of nature. According to the report of the World Health Organization (WHO), 296 million people are infected, and approximately 1.5 million people report positive tests for Hepatitis B each year. Hepatitis B is still a threat in developing and developed countries and appears in different variants globally. The present study is an extension of the deterministic hepatitis B model into a stochastic delayed model. The incorporation of stochasticity with delay in each compartment of the population like the S(t) susceptible population, P(t) acute infection, Q(t) chronic carriers' cases, R(t) immunization cases, and H(t) vaccinated class. The essential properties like positivity, boundedness, existence, uniqueness, equilibria (hepatitis-free equilibrium [HFE] and hepatitis-existing equilibrium [HEE]), reproduction number, sensitivity analysis, and stability result in the sense of local and second-order stability result in the sense of global are studied rigorously before an extension of the model. The nonparametric perturbation and transition probabilities ways used to extend analysis. Also, the positivity, boundedness, extinction, and persistence of disease are studied rigorously. Due to the high complexity of nonlinear stochastic delay differential equations, computational methods are used to visualize the results with data from the model. Unfortunately, the existing methods (Euler-Maruyama, stochastic Euler, and stochastic Runge-Kutta) in the literature fail to restore the dynamical properties of the model and make a comparison with the newly proposed construction of nonstandard finite difference method in the sense of stochastic. In the proposed method, all dynamic properties of the model are restored with a free choice of time steps, and it is efficient. For convenience, a computational code of the nonstandard discrete model may be provided to the readers at their request.