A free boundary problem-in time-for the spread of Covid-19

In this paper we deal with two aspects of the Covid epidemic. The first is a phase change during the epidemic. The empirical observation is that once a certain threshold of active infections is reached, the rate of infection is increasing significantly. This threshold depends, among others, also on the season. We model this phenomenon as a jump in the coefficient of the virus exposition, giving the force of infection. In a chemical mass action law this coefficient corresponds to the reaction rate. We get a free boundary problem in time, which exhibits deterministic 'metastability'. In a population which is in a state of herd immunity, still, if the number of imported infections is large enough, an epidemic wave can start. The second aspect is the two scale nature of the infection network. On one hand side, there is always a finite number of reoccuring-deterministic-contacts, and on the other hand there is a large number of possible random contacts. We present a simple example, where the group size of deterministic contacts is two, and the graph of random contacts is complete.