Public health risk assessment of rodent-driven hantavirus spillover in Europe using a stochastic delay model.

Introduction: Hantavirus is a rodent-borne zoonotic disease in which rodents serve as the primary reservoir, while humans acquire infection through spillover exposure. To better understand the effects of environmental variability and delayed disease dynamics on transmission, this study develops a stochastic delay differential model that incorporates biologically relevant features of hantavirus spread. Methods: An SIR-SIR transmission framework is formulated for interacting rodent and human populations. The deterministic model is extended by incorporating Brownian stochastic perturbations, discrete time delays, and exponential delay-survival effects to capture incubation, survival, and environmental uncertainty. The qualitative properties of the model, including positivity, boundedness, feasibility, and well-posedness, are established. Disease-free and endemic equilibria are derived, and both deterministic and stochastic reproduction numbers are obtained. Extinction and persistence conditions are investigated using stochastic threshold analysis. Numerical simulations are carried out using the Euler-Maruyama, stochastic Runge-Kutta, and stochastic nonstandard finite difference (SNSFD) methods. Results: The analysis shows that hantavirus persistence is primarily governed by rodent-to-rodent transmission, whereas human infections arise through spillover from infected rodents. Stochastic perturbations and delay-survival effects reduce the effective reproduction threshold under certain conditions and can promote disease extinction. Theoretical findings are supported by numerical simulations, which demonstrate that all three numerical methods produce consistent solutions for small time steps. However, the SNSFD scheme preserves positivity and exhibits superior numerical stability under larger time steps and stronger stochastic perturbations. Reported hantavirus updates from 2026 are incorporated to motivate the model and emphasize the continued importance of reservoir-driven transmission dynamics. Discussion: The proposed stochastic delay model provides a mathematically rigorous and biologically realistic framework for studying hantavirus transmission. The results highlight that controlling infection within the rodent reservoir is the most effective strategy for reducing disease persistence and limiting human spillover. The improved stability of the SNSFD method also makes it a suitable computational approach for simulating stochastic epidemic systems with delays.