Exact moment equivalence and structural nonidentifiability in nonlinear epidemics
Transmission through temporary groups does not necessarily preserve complete information about the group-size distribution in aggregate epidemic data. We show that, in finite-population SIS and SIR models, the group-size distribution enters the dynamics only through a finite set of moments selected by the order of the nonlinear transmission kernel. Consequently, markedly different distributions can generate identical stochastic dynamics when their relevant moments coincide. This equivalence extends to transient evolution, fluctuations, extinction-time statistics, and final outbreak sizes. In the deterministic limit, the first moment sets the invasion threshold, whereas the second controls the nature of the transition and the emergence of bistability and hysteresis. The two distributions become distinguishable only when a higher-order transmission mechanism activates their first unmatched moment; even a weak additional channel can shift the phase boundary and place the systems in different dynamical regimes. Solutions of the master equation and stochastic simulations support these analytical predictions. These results establish an intrinsic limit on epidemic inference: a single aggregate dynamical protocol can identify only an equivalence class of group-size distributions, rather than uniquely reconstructing the full distribution.