On the repeated epidemic waves

Based on the discrete model of infection spread in a closed population, appearance of epidemic waves associated with strengthening and weakening quarantine measures in the present paper is shown. The effect of quarantine measures is considered in the model under consideration via time dependence in the infection transmission rate. It was shown that the epidemic development is controlled by four key parameters: the number of infected persons, the average virus carriage time (as applied to COVID-19-type epidemics caused by the virus SARS-CoV-2 spread), the average number of dangerous contacts (which can cause virus transfer from an infected person to healthy men) of one infectious person (virus carrier) per day and the probability of infection due to a hazardous contact. The two latter parameters enter the model only as a product called the indicator of infection growth (IG) in population. The found solutions depend also on the population size and on the initial number of infected persons. The IG in the model under consideration is similar in meaning to the reproductive number in continuous SIR and SEIR models. At the same time, due to the prolonged virus carriage characteristic of COVID-19, the solutions proposed here are based on the equations with delay, and even without temporal variations of the IG, differ significantly from the SIR and SEIR models. The effect of the feedback between the epidemic spread rate and variations in the IG, caused by strengthening or weakening quarantine measures is studied. It results in a principal change in the epidemic behavior, which not reaching the saturation mode, transforms to its wavy flow mode. The dependence of the onset of epidemic waves on characteristic times of quarantine restriction weakening was revealed. In the model under consideration, the possibility of complete epidemic end in the case of long-term restricted quarantine measures was shown. The possible existence of the quasi-steady mode of low-intensity epidemic was detected. In this mode, the number of virus carriers remains unchanged for a long time due to the balance of the number of infections and recoveries per day.