Stochastic delay derivatives of Newcastle disease application in epidemic model: Stability analysis and approximation.

The illicit trade in wildlife for pet purposes poses a direct risk to animal populations through overharvesting, but it also serves as an indirect conduit for the spread of contagious diseases. The present study assessed the effects of the hypothetical release of captured, infected individuals of Newcastle disease into the natural population of the white-winged parakeet, the most trafficked psittacine species in Peru. This study analysed the computational dynamical analysis of the stochastic susceptible-exposed-infected-recovered model of Newcastle disease. We take two approaches to stochastic modelling: transition probabilities and the perturbation method. To investigate dynamical features such as positivity, boundedness, consistency, and stability, we introduced a stochastic non-standard finite-difference (SNSFD) scheme. Conventional numerical approaches such as Euler-Maruyama, stochastic Euler, and stochastic Runge-Kutta of order four were applied, but they failed to preserve the system's essential dynamical properties. Consequently, we formulated the SNSFD method to address this limitation. To support the proposed method, we presented several theorems that demonstrate it satisfies all dynamical properties of the model. In the end, we presented several simulations to compare the proposed method's efficiency to that of an existing method.