Stochastic SEIR dynamics via graph Laplacian diffusion under transmission uncertainty

Epidemic spatial spread across interconnected populations depends on both mobility between locations and random fluctuations in transmission. We extend the Susceptible-Exposed-Infected-Removed (SEIR) epidemic model to a network setting by introducing graph Laplacian diffusion to capture population mobility, and further incorporate stochastic perturbations in the transmission rate to capture environmental noise. Using Lyapunov function analysis, we first establish that the stochastic system admits a unique global solution which remains non-negative for all time, so that the model is mathematically well-posed and its state variables retain their epidemiological meaning as population counts. Building on this well-posedness, we then apply Lyapunov techniques a second time to analyse the long-term behaviour of the solution. The disease-free equilibrium is shown to be globally asymptotically stable when the basic reproduction number does not exceed one, indicating disease extinction. When the basic reproduction number exceeds one, the disease instead persists, in the sense that the stochastic solution exhibits a bounded time-averaged deviation from the endemic equilibrium of the corresponding deterministic system. Sequential Monte Carlo (SMC) methods are further employed for parameter estimation and uncertainty quantification, applied to COVID-19 incidence data across all 47 prefectures in Japan. The proposed framework brings together network-driven mobility, stochastic transmission dynamics, and Bayesian uncertainty quantification within a SEIR model. Finally, computational experiments on a small-world Watts-Strogatz network are used to validate and visually illustrate the theoretical results.