The Contagion Landscape

Time-varying group membership — people joining and leaving social groups — should intuitively blur contagion dynamics, diluting the contacts that drive spreading. Static network models predict a single epidemic threshold: below it the disease dies out, above it a single endemic state exists.

Dynamic group turnover creates a richer attractor landscape. Instead of one endemic state, the system exhibits multiple coexisting endemic states — multistable active phases entirely absent in the static case. The contagion onset requires stronger nonlinear reinforcement than static models predict, meaning turnover raises the spreading threshold. But simultaneously, once spreading does occur, the system can settle into any of several distinct endemic equilibria depending on initial conditions.

The mechanism involves collective reinforcement within transient groups. When group membership changes, the reinforcement history is partially preserved — individuals carry their infection status between groups — but the group-level reinforcement is disrupted and must rebuild. This creates a landscape where multiple levels of endemic prevalence are locally stable, separated by unstable thresholds that depend on the rate of group turnover.

The structural observation: dynamics that weaken individual transmission strengthen the system's capacity for complex equilibria. Group turnover suppresses simple spreading while enabling multistability — making it harder for contagion to start but giving it more distinct modes of persistence once it does. The same mechanism that raises the threshold creates the landscape.