Prediction of the COVID-19 outbreak based on a realistic stochastic model

Yuan Zhang; Chong You; Zhenghao Cai; Jiarui Sun; Wenjie Hu; Xiao-Hua Zhou

(2.15) Intuitively, the mean-field ODE system above serves as a degenerated case of our stochastic model, where all randomness has been averaged out. When the differential equations are linear, which is not true in our case, the ODE also describes the evolution of the expectation of the stochastic system (Dynkin's Formula, see Da Prato and Zabczyk (2014) Section 4.4 for example). Besides, according to and , we may see that this deterministic model can also serve, under a much weaker condition, as a scaling approximation of the stochastic one after recaled by the renormalizing factor N . To be more specific, the renormalized random path ξ N (t)/N as a stochastic process is "almost" deterministic, and fluctuates closely around the deterministic trajectoryξ N (t)/N if the total population N is large and the renormalized initial values ξ N (0)/N andξ N (0)/N are close to each other,