Risk propagation in heterogeneous supply chain networks with saturated contagion pressure and environmental uncertainty

This paper investigates risk contagion within a heterogeneous supply chain network by formulating a degree-indexed susceptible-infected-recovered-susceptible (SIRS) stochastic dynamical system. The proposed system incorporates nonlinear saturated incidence rates and environmental perturbations to capture the complex interplay between network topology and external uncertainty. We first perform a deterministic analysis, derive the basic reproduction number $\mathcal{R}_0$ using the next-generation matrix approach, and establish stability criteria for both risk-free and endemic risk equilibria. Then, we extend the proposed framework to a stochastic environment by incorporating proportional (multiplicative) noise. Mathematically, we verify the existence and uniqueness of a global positive solution to ensure the system's well-posedness. By constructing Lyapunov functions, we derive stochastic thresholds $ \mathcal{R}_0^s $ and $ \mathcal{R}_0^p $ that dictates the system's asymptotic behavior. Our results show that if $\mathcal{R}_0^s < 1$, the risk may disappear, while when $\mathcal{R}_0^p > 1$, the system converges to a unique ergodic stationary distribution, which means that the risk will continue to exist under a stable probability state. Numerical simulations utilizing the Euler-Maruyama scheme validate these theoretical thresholds and further elucidate the degree-temporal evolution of contagion across the network. The findings suggest that risk managers should transition from traditional deterministic point forecasting toward distribution-based buffering strategies, prioritizing hub-centric resilience interventions to effectively mitigate cascading failures within highly coupled systems.