Fig. 5.

Endemicity boundaries in the presence of periodically forced transmission (model (3)). Long-term pathogen establishment is possible (${\mathcal{R}}_0^F>1$) on the right of the endemicity curves, which correspond to ${\mathcal{R}}_0^F=1$. A) Periodic veriations of exposure/transmission risk (black: ${\alpha }_{\varphi }=\alpha$ and ${\alpha }_N={\alpha }_H=0$) or of the population abundance of either host species (gray: ${\alpha }_{\varphi }=0$ and either ${\alpha }_N=\alpha$ or ${\alpha }_H=\alpha$). B) Coupled fluctuations of exposure/transmission risk and of the population abundance of either host species (black: ${\alpha }_{\varphi }=\alpha ,$ and either ${\alpha }_N=\alpha$ and ${\alpha }_H=0$ or ${\alpha }_H=\alpha$ and ${\alpha }_N=0$), or of the population abundances of the two host species (gray: ${\alpha }_{\varphi }=0,$${\alpha }_N={\alpha }_H=\alpha$ and ${\psi }_N=0$), for different phase shifts between the relevant periodic signals (labels). C) As in panel B, for synchronous coupled fluctuations with different amplitudes (black: ${\alpha }_{\varphi }={\alpha }_1,$ and either ${\alpha }_N={\alpha }_2,$${\alpha }_H=0$ and ${\psi }_N=0$ or ${\alpha }_H={\alpha }_2,$${\alpha }_N=0$ and ${\psi }_H=0$; gray: ${\alpha }_{\varphi }=0,$${\alpha }_N={\alpha }_1,$${\alpha }_H={\alpha }_2$ and ${\psi }_N={\psi }_H=0$). Results are shown for different values of the basic homogeneous reproduction number r₀ (labels). In all panels, $\beta N=\chi H=\sqrt{r_0\gamma \mu }$; other parameters as in Fig. 1.