Semianalytical solution of SEIR-like models of epidemic spreading

We consider a generalization of the well-known SEIR model of epidemic spreading with two susceptible compartments and without permanent immunity. It is shown that the set of ordinary differential equations defining its dynamics can be cast in a single one in the form of an Abel's differential equation of the second kind. While a general solution of this equation is not known, we obtain exact analytical expressions for the important short-term growth and long-term decay regimes of the epidemic outbreak. In particular, exact expressions for the exponential growth and decay rates and the exact relation between the growth rate and the basic reproduction number are obtained. Finally, we discuss the relevance of the initial conditions in these type of models and show an application to the Omicron wave of the Covid-19 pandemic in different countries.