Transmission dynamics of mumps epidemic model through stochastic analysis with delay effect

Stochastic delayed modeling (stochastic differential equations (SDEs) with delay parameters) has a significant non-pharmaceutical intervention to control transmission dynamics of infectious diseases and its results are close to the reality of nature. Mumps is a viral disease specified with swollen jaws and inflated cheeks. Direct contact with saliva or respiratory drop less from the mouth is the major causes of its outbreak. According to the World Health Organization (WHO), still, 20% of young adult males develop mumps worldwide. No doubt, the vaccination of Mumps exists. The main cause is to study the transmission dynamics of Mumps through stochastic with delay approaches. How is the stochastic delay the best strategy to study the dynamics of disease in a population? For this, we consider the existing deterministic model in literature, with the whole population, divided as susceptible human population S(t), exposed human population E(t), symptomatic infectious I(t), asymptomatic infectious A(t), isolated and treated symptomatic Q(t), recovered humans R(t). After that, we extend the deterministic model into a stochastic delay model (Stochastic delay differential equations (SDDEs) by using the transition probabilities and non-parametric perturbation ways. The positivity, boundedness, extinction, and persistence of disease study with essential properties of reproduction number rigorously. The mump-free equilibrium (MFE) and mumps existing equilibrium (MEE) are two states, local, and global stability of second order and sensitivity analysis of parameters analyzed to verify the model validations. Due to the highly nonlinear stochastic delay differential equations of the model, we used both standard and nonstandard methods such as Euler Maryama, stochastic Euler, stochastic Runge-Kutta, and stochastic nonstandard finite difference with a delayed sense to visualization of results. In the end, the comparison of the methods is presented to support the efficiency of non-standard methods in the sense of stochastic with delay parameters.