Noise corrected thresholds for a stochastic chikungunya host-vector model with mosquito incubation, multiplicative diffusion, and Lévy jumps

We develop and analyse a stochastic host-vector model for chikungunya virus (CHIKV) transmission that explicitly incorporates the extrinsic incubation period in mosquitoes. The model couples a human SIR structure with a mosquito SEI structure and is driven by both multiplicative Brownian perturbations and compensated jump noise, capturing gradual environmental variability and abrupt shocks. Under minimal regularity and integrability assumptions on the diffusion coefficients and jump kernels, we derive a deterministic basic reproduction number R₀ and a noise-corrected effective threshold R˜₀^⋄, showing that stochastic perturbations always decrease the transmission potential in the sense that 0<R˜₀^⋄≤R₀². On this basis, we obtain explicit sufficient conditions for three contrasting regimes: almost sure exponential extinction of the infection, persistence in the mean, and the existence of an endemic stationary distribution for the coupled human and mosquito infectious classes. Our results reveal that the classical condition R₀>1 is no longer sufficient to guarantee persistence in the stochastic setting; it must be complemented by a moderate-noise requirement R˜₀^⋄>1, whereas strong noise and jump activity can enforce extinction even when R₀>1. The analytical findings are illustrated by numerical simulations implemented in Python. An extinction scenario is calibrated to the 2025 Foshan outbreak data, while a persistence scenario is constructed from theoretically motivated parameters. In both cases, sample paths and joint stationary densities exhibit qualitative behaviour in excellent agreement with the thresholds R₀ and R˜₀^⋄, providing a coherent picture of how environmental noise can either suppress or sustain CHIKV transmission.