Figure 1

Mechanism for (a) $r$ synergy and (b) $d$ synergy. Dark (red online) and lighter (green online) circles correspond to infected and susceptible hosts, respectively. Arrows indicate possible events for transmission of infection. For $r$ synergy, the susceptibility of a recipient susceptible host $r$ is enhanced (constructive synergy) or diminished (interfering synergy) in response to multiple simultaneous challenges from two or more neighboring infected hosts (donors). The susceptibility of $r$ depends on the number $n_r$ of neighbors simultaneously challenging it ($n_r=3$ in this example). For $d$ synergy, the infectivity of a donor host $d$ depends upon the number $n_d$ of connections of this host to other infectious hosts that can share resources with the donor host. In this example, $n_d=1$ (connection indicated by an edge).Reuse & Permissions

Figure 2

Phase diagram for synergistic epidemics. The exact phase boundaries are marked by squares for $r$ synergy and by circles for $d$ synergy. Epidemics are invasive if the parameters ($\alpha$, $\beta$) are on the right of the phase boundary. The arrow indicates the limiting value of ${\alpha }_c\simeq 0.2$ reached asymptotically by the phase boundary for $r$ synergy in the limit of large $\beta$ (not shown on the scale of the graph). The approximate phase boundaries obtained analytically by neglecting correlations in transmission of infection (cf. 29) are shown for both $r$ (solid line) and $d$ synergy (dashed line).Reuse & Permissions

Figure 3

Patterns of infection. Illustration of the effect of (a) constructive and (b) interfering synergy on the patterns of infection spreading from the central host (marked by solid circle) in systems of linear size $L=31$. The synergy rates are $\beta =5$ and $\beta =-5$ for patterns with constructive and interfering synergy, respectively. All snapshots correspond to the final state of the epidemic with only $R$ and $S$ hosts remaining. Solid lines indicate those edges between $d\mathrm{\text{−}}r$ pairs that have transmitted the pathogen at some time during the course of the epidemic. The value of $\alpha$ has been chosen in each case such that the probability of invasion is $P_{\mathrm{inv}}=0.5$ for all snapshots: (a) $\alpha =0.47$ for $r$ synergy and $\alpha =0.18$ for $d$ synergy. (b) $\alpha =0.81$ for $r$ synergy and $\alpha =5.22$ for $d$ synergy.Reuse & Permissions

Figure 4

Branching and invasion transitions for $d$ synergy. (a) Diagram showing the branching transition line ${\alpha }_b(\beta )=-\beta$ together with the phase boundary for invasion and noninvasion transition, ${\alpha }_c(\beta )$. Branching is forbidden in the region under the continuous line where $\alpha <{\alpha }_c(\beta )$. Invasive epidemics can occur in the region above the dashed line corresponding to $\alpha >{\alpha }_c(\beta )$ where branching is present. The intermediate region between the continuous and dashed lines with ${\alpha }_b<\alpha <{\alpha }_c$ corresponds to epidemics that display branching but are not invasive. (b) Example of path for infection with $d$ synergy in the regime without branching ($\alpha \le {\alpha }_b$). Infection starts from the host marked by a solid circle and evolves along the path indicated by arrows. Arrow numbers define schematically a sequence of infection events. For the first infected host (solid circle), $n_d=0$ and infection can be transmitted to any of its neighbors at a rate ${\lambda }_{d\mathrm{\text{−}}r}=\alpha$. However, $n_d=1$ as soon as the pathogen is transmitted to one of the neighbors (arrow 1) and thus ${\lambda }_{d\mathrm{\text{−}}r}=0$ because $\alpha \le -\beta$. At this moment, transmission of infection is arrested until the initially infected host recovers. After this recovery, $n_d=0$ for the newly infected host and infection can be transmitted to one of its nearest neighbors at rate $\alpha$. Iteration of this process over time gives growing SAWs. The state after the event marked by arrow 10 illustrates the phenomenon of self-trapping.Reuse & Permissions