Demographic population cycles and ℛ₀ in discrete-time epidemic models

We use a general autonomous discrete-time infectious disease model to extend the next generation matrix approach for calculating the basic reproduction number, [Formula: see text], to account for populations with locally asymptotically stable period k cycles in the disease-free systems, where [Formula: see text]. When [Formula: see text] and the demographic equation (in the absence of the disease) has a locally asymptotically stable period k population cycle, we prove the local asymptotic stability of the disease-free period k cycle. That is, the disease goes extinct whenever [Formula: see text]. Under the same period k demographic assumption but with [Formula: see text], 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[Formula: see text]) infections in a salmon population. When [Formula: see text], our simulations show that the disease-free period k cycle dynamics drives the SIR disease dynamics, but not the ISAv disease dynamics.