Epidemic transmission with quarantine measures: application to COVID-19

Equations for infection spread in a closed population are found in discrete approximation, corresponding to the published statistical data, and in continuous time in the form of delay differential equations. We consider the epidemic as dependent upon four key parameters: the size of population involved, the mean number of dangerous contacts of one infected person per day, the probability to transmit infection due to such contact and the mean duration of disease. In the simplest case of free-running epidemic in an infinite population, the number of infected rises exponentially day by day. Here we show the model for epidemic process in a closed population, constrained by isolation, treatment and so on. The four parameters introduced here have the clear sense and are in association with the well-known concept of reproduction number in the continuous susceptible--infectious--removed, susceptible--exposed--infectious--removed (SIR, SEIR) models. We derive the initial rate of infection spread from the published statistical data for the initial stage of epidemic, when the quarantine measures were absent. On this basis, we can found the corresponding basic reproduction number mentioned above. Our approach allows evaluating the influence of quarantine measures on free pandemic process that leads to the time-dependent rate of infection and suppression of infection. We found a good correspondence of the theory and reliable statistical data. The initially formulated discrete model, describing epidemic course day by day is transferred to differential form. The conditions for saturation of epidemic are found by solving the delay differential equations. They differ essentially from ones in SIR model due to finite delay, typical for COVID-19. The proposed model opens up the possibility to predict the optimal level of social quarantine measures. The model is quite flexible and it can be extended to more complex cases.