The impact of a delay on the evolution of epidemics

Based on a discrete model of the spread of infection in a closed population, the corresponding form of differential equations with delay is found. It is shown that the development of the epidemic is determined by four key parameters: the number of infectious persons, the average number of dangerous contacts of one infectious person per day, the probability of infection as a result of such contact, and the average time interval during which the sick person is able to infect others. The decision also depends on the size of the population and on the initial number of infected persons. The four named parameters have a clear meaning and are related to the well-known concept of reproductive number in the continuous Susceptible–Infectious–Recovered (SIR) and Susceptible–Infected–Infectious–Recovered (SEIR) models. The epidemic saturation conditions are established by solving the obtained differential equations. It is shown that, due to the long virus carrying characteristic of COVID-$19$, the solutions proposed here differ significantly from the SIR model.