Age structured discrete-time disease models with demographic population cycles

Abstract

We use juvenile-adult discrete-time infectious disease models with intrinsically generated demographic population cycles to study the effects of age structure on the persistence or extinction of disease and the basic reproduction number, $\mathcal {R}_{0} $. Our juvenile-adult Susceptible-Infectious-Recovered (SIR) and Infectious-Salmon Anemia-Virus (ISA $v) $ models share a common disease-free system that exhibits equilibrium dynamics for the Beverton-Holt recruitment function. However, when the recruitment function is the Ricker model, a juvenile-adult disease-free system exhibits a range of dynamic behaviours from stable equilibria to deterministic period k population cycles to Neimark-Sacker bifurcations and deterministic chaos. For these two models, we use an extension of the next generation matrix approach for calculating $\mathcal {R}_{0} $ to account for populations with locally asymptotically stable period k cycles in the juvenile-adult disease-free system. When $\mathcal {R}_{0} \lt 1 $ and the juvenile-adult demographic system (in the absence of the disease) has a locally asymptotically stable period k population cycle, we prove that the juvenile-adult disease goes extinct whenever $\mathcal {R}_{0} \lt 1 $. Under the same period k juvenile-adult demographic assumption but with $\mathcal {R}_{0} \gt 1 $, we prove that the juvenile-adult disease-free period k population cycle is unstable and the disease persists. When $\mathcal {R}_{0} \gt 1 $, our simulations show that the juvenile-adult disease-free period k cycle dynamics drives the juvenile-adult SIR disease dynamics, but not the juvenile-adult ISAv disease dynamics.

Keywords

• adults
• beverton-holt model
• juveniles
• population cycles
• ricker model