Propagation dynamics in epidemic models with two latent classes

Abstract

This article is concerned with the propagation dynamics in diffusive epidemic models that involve two classes of latent individuals. We formulate the spatial expansion process of latent and infected classes in terms of spreading speeds of initial value problems and minimal wave speed of traveling wave solutions. With several kinds of decaying initial conditions, different leftward and rightward spreading speeds are obtained by constructing proper auxiliary systems. To prove the existence of traveling wave solutions, we use the recipes of generalized upper and lower solutions, the theory of asymptotic spreading as well as a limit process. Our conclusions imply that when the basic reproduction ratio of the corresponding ODEs is larger than the unit, the disease has a minimal spatial expansion speed that equals to the minimal wave speed. When the ratio is not larger than the unit, the disease vanishes and there is not a nontrivial traveling wave solution.