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

In contrast to the deterministic models (ODE or DE) summarized above, the transmission of a real world disease is inevitably random in nature. As a result, numerous stochastic dynamics models (see Section 3 for references) have been developed since the pioneering randomization of SIR model in . In fact, a deterministic ODE model can often be seen as the mean-field equation of the corresponding stochastic counterpart. Under certain conditions, the mean-field equation may represent the evolution of the expectation of the corresponding stochastic model. In some more generalized cases, the mean-field equation is a large scale approximation of the corresponding stochastic model, which can be seen as a process version of Law of Large Numbers. However, when the size of outbreak is not comparable to that of the total population, the approximation aforementioned may not be interpreted that the stochastic and deterministic models are close to each other by themselves, since they have both been rescaled to approximately a constant in these scenarios. See Section 3 for more details. Thus compared with deterministic models, the stochastic model may be a better choice to account for the non-negligible random nature of the COVID-19 outbreak. Furthermore, the stochastic dynamic model is also known for its expandability to incorporate individual variations , or even spatial structures Durrett (1988) , which may not be fully captured by its mean-field equations. To our knowledge, the stochastic dynamic modeling for COVID-19 is yet relatively rare comparing to its deterministic counterparts, though preliminary approaches such as statistic exponential growth models was considered in recent studies of Zhao et al. (2020) . Very recently in , an existing discrete time, stochastic model ) was employed to estimate the "effect of travel restrictions on the spread" of COVID-19. However, unique features of SARS-CoV-2, such as the infectious incubation and asymptomatic carriers, as well as control measures such as medical tracking, are still yet to be captured in .