This paper studies the dynamic behavior of a stochastic SEIRM model of COVID-19 with a standard incidence rate. The existence of global solutions for dynamic system models is proven by integrating stochastic process theory and the concept of stopping times, together with the contradiction method. Moreover, we construct appropriate Lyapunov functions to analyze system stability and apply Dynkin’s formula and Fatou’s lemma to handle stopping times and expectations of stochastic processes. Notably, the extinction study provides mathematical proof that under the given system dynamics, the total population does not grow indefinitely but tends to stabilize over time. The properties of the diffusion matrix are harnessed to guarantee the system’s stationary distribution. Conclusively, numerical simulations confirm the model’s extinction outcomes.

For a connected graph, the Fermat difference of three vertices is defined as the minimum sum of their distances to a common vertex minus half the sum of their pairwise distances. In this paper, we investigate the average Fermat difference of unicyclic graphs with prescribed order and cycle length. By establishing a projection identity, we reduce the problem to a weighted cubic form on the unique cycle. We obtain an exact formula for cycles and establish a sharp lower bound together with a characterization of all extremal graphs attaining equality. We further derive a general upper bound for unicyclic graphs with prescribed order and cycle length. Finally, we determine the maximum average Fermat difference among all unicyclic graphs of fixed order and characterize the corresponding extremal graphs.