An Advection and Age-Structured Approach to Modeling Bird Migration and Indirect Transmission of Avian Influenza

Abstract

We model indirect transmission, via contact with viruses, of avian influenza in migratory and nonmigratory birds, taking into account age structure. Migration is modeled via a reaction-advection equation on a closed loop parametrized by arc length (the migration flyway) that starts and ends at the location where birds breed in summer. Our modeling keeps the birds together as a flock, the position of which is implicitly determined and known for all future time. Births occur when the flock passes the breeding location and are modeled using ideas of impulsive differential equations. For a migratory species the model derivation starts from age-structured reaction-advection equations with location-dependent parameters that describe local conditions. In the derivation of delay equations for the time-dependent variables representing numbers of juvenile and adult birds, these location-dependent parameters are evaluated at the flock's position, so that seasonal effects are captured indirectly but through rigorous modeling whereby we keep track of the flock's exact position and local conditions there. Sufficient conditions are obtained for the local stability of the disease-free equilibrium (for a nonmigratory species) and for the disease-free periodic solution (for a migratory species).

A stage-structured delayed advection reaction-diffusion model for single species

“…Then they present results on positivity and boundedness, existence and local stability of equilibria, global stability of the extinction state under certain conditions, and existence of periodic solutions. Gourley et al, [13] presented demonstrate circuitous transmission, through contact with infections, of avian influenza in transitory and nonmigratory fowls, taking under consideration age structure. Relocation is modeled through a reaction-advection condition on a closed circle parametrized by bend length (the migration flyway) that begins and closes at the area where fowls breed in summer.…”