Periodic systems of delay differential equations and avian influenza dynamics

Xiang-Sheng Wang ; Jianhong Wu

Contemporary Mathematics. Fundamental Directions, Proceedings of the Sixth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2011). Part 1, Tome 45 (2012), pp. 32-42

Résumé

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.