Fig. 2.

Dynamics of the disease in Patch 2. In the high-mobility case $p_{12}=p_{21}={\sigma }_{12}={\sigma }_{21}=1$ (and then $p_{11}=p_{22}={\sigma }_{11}={\sigma }_{22}=0$), the disease dies out (solid red curve) for $\mathbb{P}$ constant, with ${\stackrel{\sim }{R}}_0^2=\frac{{\beta }_1}{{\gamma }_2+d_2}=0.8571$. For the constant residence times matrix, the system is strangely decoupled because individuals of Patch 1 spend all their time in Patch 2, whereas individuals of Patch 2 spend all their time in Patch 1. Hence, Patch 2 individuals ($d_2$ and ${\mu }_2$) are subject exclusively to the environmental conditions that define Patch 1 (${\beta }_1$), and so the basic reproduction of the “isolated” Patch 1 is ${\stackrel{\sim }{R}}_0^2=\frac{{\beta }_1}{{\gamma }_2+d_2}$ and the disease dies out because ${\stackrel{\sim }{R}}_0^2=0.8571$. The disease persists if $\mathbb{P}$ state-dependent (dashed red curve) as $p_{12}(I_1,I_2)=\frac{1+I_1}{1+I_1+I_2},p_{21}(I_1,I_2)=\frac{1+I_2}{1+I_1+I_2},p_{11}(I_1,I_2)=\frac{I_2}{1+I_1+I_2}$ and $p_{22}(I_1,I_2)=\frac{I_1}{1+I_1+I_2}$. Figure courtesy of ref. 24.