Solution of epidemic models with quenched transients.

We consider a model for single-season disease epidemics, with a delay (latent period) in the onset of infectivity and a decay ("quenching") in host susceptibility described by time-varying rates of primary and secondary infections. The classical susceptible-exposed-infected (SEI) model of epidemiology is a special case with constant rates. The decaying rates force the epidemics to slow down, and eventually stop in a "quenched transient" state that depends on the full history of the epidemic including its initial state. This equilibrium state is neutrally stable (i.e., has zero-value eigenvalues), and cannot be studied using standard equilibrium analysis. We introduce a method that gives an approximate analytical solution for the quenched state. The method uses an interpolation between two exactly solvable limits and applies to the whole, five-dimensional parameter space of the model. Some applications of the solutions for analysis of epidemics are given.