Periodic systems of delay differential equations and avian influenza dynamics

Modelling the spread of avian influenza by migratory birds between the winter refuge ground and the summer breeding site gives rise to a periodic system of delay differential equations exhibiting both the cooperative dynamics (transition between patches) and the predator-prey interaction (disease transmission within a patch). Such a system has two important basic reproductive ratios, each of which being the spectral radius of a monodromy operator associated with the linearized subsystem (at a certain trivial equilibrium): the (ecological) reproduction ratio $R_0^c$ for the birds to survive in the competition between birth and natural death, and the (epidemiological) reproduction ratio $R_0^p$ for the disease to persist. We calculate these two ratios by our recently developed finite-dimensional reduction and asymptotic techniques, and we show how these two ratios characterize the nonlinear dynamics of the full system.