Fig. 2: The inadequacy of local reproduction numbers in spatially explicit systems.

A catalog of dynamical behaviors for the two-community system of Fig. 1 (red: large outbreaks and endemic transmission are possible because \({{\mathcal{R}}}_{0}\,> \, 1\); gray: transient epidemic waves may be possible because e₀ > 0, but transmission fades away eventually because \({{\mathcal{R}}}_{0}\, <\, 1\); blue: no epidemic waves because e₀ < 0). See text for an explanation of the specific dynamical outcomes in different parameter subregions (① to ⑧). Note that points ①, ②, and ⑧ correspond to the parameter combination used in Fig. 1 for (a–c), (d–f), and (g–i), respectively. The two communities are characterized by different values of the local basic reproduction number \({{R}_{0}}_{i}=\frac{{\delta }^{E}}{\mu +{\delta }^{E}}\frac{{\delta }^{P}}{\mu +{\delta }^{P}}\left[\frac{{\beta }_{i}^{P}}{{\delta }^{P}}+\sigma \frac{{\beta }_{i}^{I}}{\mu +\alpha +\eta +{\gamma }^{I}}+(1-\sigma )\frac{{\beta }_{i}^{A}}{\mu +{\gamma }^{A}}\right]\) because of differences in the local values of the transmission rates. Parameters and other details as in Fig. 1.