Stability and structure-preserving numerical analysis of a stochastic delayed pine wilt disease model.

Pine wilt disease is a destructive forest disease involving pine trees, pinewood nematodes, and insect vectors. This study develops deterministic and stochastic delayed host-vector models to examine waiting periods and variability in transmission. For the deterministic system, positivity, boundedness, equilibria, and the basic reproduction number R0 are established. The disease-free equilibrium is locally and globally asymptotically stable when R0<1, while the endemic equilibrium is assessed through delay-dependent local stability. The stochastic model includes multiplicative environmental noise and is shown to possess positive solutions with bounded first moments. Exact Itô logarithmic identities demonstrate that long-term infection behaviour depends on current and delayed host-vector states. A delay-aware stochastic nonstandard finite difference scheme is constructed using delayed indices, nonstandard denominator functions, and positive stochastic exponential factors. Numerical experiments illustrate the effects of delay, noise, and step size, with the proposed scheme showing controlled behaviour at larger steps.