Mathematical Epidemiological Model of the Propagation and Prevention of Infectious Diseases Considering COVID-19 Vaccination

The duration of the necessity of current restrictions and preventive countermeasures against the COVID-19 pandemic is of great concern. While several previous epidemiological studies have discussed controlling the course of the epidemic with regard to social distancing, vaccination, care capacities, and future scenarios, we adopt an alternative approach to provide insights into the time scale of epidemic propagation in human populations. We show that the reproduction number modified by preventive countermeasures R m implies a threshold R~ m to reach the herd immunity level at time T h . While T h varies moderately for large R m , it increases sharply around R m =R~ m , and T h is infinite below R m =R~ m . The transition region for this increase is minute, demonstrating that prevention of infectious diseases must consider the relatively asymptotic nature of their propagation, which varies unpredictably between steady transmission and explosive outbreaks. These results suggest the continuation of preventive countermeasures to suppress the transmission of COVID-19 for many years; if they were discontinued or reduced such that R m exceeded R~ m , the disease would be transmitted throughout the considered community, and the required herd immunity level would be reached within 250 days if R m > 1.6. However, the implementation of vaccination programs could drastically alter this dynamic.