Characterization of the Basic Reproduction Number R0 of Pde Epidemic Models

The basic reproduction number R0 has been used in epidemic models as an important threshold for controlling the spread of infections.In this paper we give a theoretical characterization of R0 for compartmental models that are based on partial differential equations.For any of such models we show that R0 is the spectral radius of the basic operator of the translation semigroup of operators that is solution of the model.The stability of steady states and the asymptotic behavior of the solutions are also established even if the important irreducibility property is satisfied only for a projection of the basic operator.For illustration, an age structured Susceptible-Infected-Recovered epidemic model is considered and R0 is explicitly computed in terms of the model's parameters using the established characterization.