An endemic model with variable re-infection rate and applications to influenza

An epidemic model is considered, where immunity is not absolute, but individuals that have recovered from the disease can be re-infected at a rate which depends on the time that has passed since their recovery (recovery age). Such a model, e.g., can account for the genetic drift in the influenza virus. In the special case that the model has no vital dynamics, there is no obvious disease-free equilibrium and so the model lacks the usual interplay between the basic replacement ratio being >1 and the disease-free equilibrium being unstable. In fact, this relatively simple model which combines ordinary differential equations with a transport equation shares with general structured population models the feature that the appropriate state space of the solution semiflow is a space of measures, here on the compactified right real half line, with the weak* topology. The disease-free equilibrium, in terms of recovered individuals, is then represented as a Dirac measure concentrated at infinity. Still it is difficult to linearize about it. This makes the concept of persistence very important, for one can show the following: if the basic replacement ratio is >1, the disease is uniformly strongly persistent, i.e., the number of infectives is ultimately bounded away from 0 with the bound not depending on the initial data. We also derive various conditions for the local and global stability of the endemic equilibrium in terms of the re-infection rate. For instance, the endemic equilibrium is likely to be locally asymptotically stable if the re-infection rate is a highly sub-homogeneous function of recovery age. Conversely, if the re-infection rate is a step function which is zero at small recovery age, the endemic equilibrium can be unstable.