Demographic population cycles and ℛ0 in discrete-time epidemic models

Abstract

We use a general autonomous discrete-time infectious disease model to extend the next generation matrix approach for calculating the basic reproduction number, $ \mathcal {R}_{0} $ , to account for populations with locally asymptotically stable period k cycles in the disease-free systems, where $ k\geq 1 $ . When $ \mathcal {R}_{0} 1 $ , we prove that the disease-free period k population cycle is unstable and the disease persists. Using the Ricker recruitment function, we apply our results to discrete-time infectious disease models that are formulated for Susceptible-Infectious-Recovered (SIR) infections with and without vaccination, and Infectious Salmon Anemia Virus (ISA $ \mathit {v} $ ) infections in a salmon population. When $ \mathcal {R} _{0} > 1 $ , our simulations show that the disease-free period k cycle dynamics drives the SIR disease dynamics, but not the ISAv disease dynamics.

Keywords

• Next generation matrix method
• Period k cycles
• Ricker recruitment
• $ \mathcal {R}_{0} $