Toward the Impact of Non-pharmaceutical Interventions and Vaccination on the COVID-19 Pandemic With Time-Dependent SEIR Model

The outbreak of COVID-19, caused by the SARS-CoV-2 coronavirus, has been declared a pandemic by the World Health Organization (WHO) in March, 2020 and rapidly spread to over 210 countries and territories around the world. By December 24, there are over 77 M cumulative confirmed cases with more than 1.72 M deaths worldwide. To mathematically describe the dynamic of the COVID-19 pandemic, we propose a time-dependent SEIR model considering the incubation period. Furthermore, we take immunity, reinfection, and vaccination into account and propose the SEVIS model. Unlike the classic SIR based models with constant parameters, our dynamic models not only predicts the number of cases, but also monitors the trajectories of changing parameters, such as transmission rate, recovery rate, and the basic reproduction number. Tracking these parameters, we observe the significant decrease in the transmission rate in the U.S. after the authority announced a series of orders aiming to prevent the spread of the virus, such as closing non-essential businesses and lockdown restrictions. Months later, as restrictions being gradually lifted, we notice a new surge of infection emerges as the transmission rates show increasing trends in some states. Using our epidemiology models, people can track, timely monitor, and predict the COVID-19 pandemic with precision. To illustrate and validate our model, we use the national level data (the U.S.) and the state level data (New York and North Dakota), and the resulting relative prediction errors for the infected group and recovered group are mostly lower than 0.5%. We also simulate the long-term development of the pandemic based on our proposed models to explore when the crisis will end under certain conditions.