Threshold behaviour of a SI epidemiological model with two structuring variables

In this article, we study a SI epidemic model describing the spread of a disease in a perfectly mixed managed population, representing an animal herd in a fattening farm. The epidemic process is characterized by a non-neglectable and variable incubation period, during which individuals are infectious but cannot be easily detected. The susceptible and infected populations are structured according to age and, for infected, to time remaining before the end of the incubation, where they show detectable clinical signs. We study the well posedness and the asymptotic behaviour of the problem and show that in some cases, even if the farm is fed with healthy animals, disease persistence can occur. We give an explicit formula for the basic reproduction number and the biological interpretation of this threshold on a specific example. We finally illustrate the asymptotic behaviour of the model by numerical simulations.