Basic Reproductive Numbers in Host-Parasite Systems with Demographic Bistability
The basic reproductive number, $\mathcal{R}_0$, is widely used to determine whether an infectious disease can persist in a population. However, its interpretation may become unclear when the underlying disease-free demographic dynamics exhibit multiple types of stable attractors. In this study, we investigate how demographic bistability between fixed points and periodic cycles affects the calculation and interpretation of $\mathcal{R}_0$. We consider two discrete-time host-parasite models with infection: a tick-host model motivated by Borrelia burgdorferi transmission and a Nicholson-Bailey host-parasitoid model with intraspecific competition and infection. For each model, we calculate reproductive numbers associated with different disease-free attractors using the next-generation operator method and compare their invasion thresholds. In the tick-host model, the reproduction number calculated on the stable demographic cycle differs from that calculated at the positive coexistence fixed point, resulting in parameter regions where infection can invade at one demographic attractor but not the other. In contrast, in the host-parasitoid model, the host-driven invasion threshold is necessarily the same at the positive coexistence fixed point and on the host-only periodic cycles. These contrasting results demonstrate that demographic bistability alone does not necessarily produce attractor-dependent invasion thresholds. Rather, the effect of demographic dynamics on invasion depends on the structure of pathogen transmission. In particular, a complete host--vector--host transmission pathway can allow differences between demographic attractors to alter the invasion threshold. Our results emphasize the importance of considering both demographic dynamics and transmission structure when interpreting $\mathcal{R}_0$ in populations with multiple stable disease-free states.