On representing network heterogeneities in the incidence rate of simple epidemic models

Mean-field ecological models ignore space and other forms of contact structure. At the opposite extreme, high-dimensional models that are both individual-based and stochastic incorporate the distributed nature of ecological interactions. In between, moment approximations have been proposed that represent the effect of correlations on the dynamics of mean quantities. As an alternative closer to the typical temporal models used in ecology, we present here results on “modified mean-field equations” for infectious disease dynamics, in which only mean quantities are followed and the effect of heterogeneous mixing is incorporated implicitly. We specifically investigate the previously proposed empirical parameterization of heterogeneous mixing in which the bilinear incidence rate SI is replaced by a nonlinear term kSpIq, for the case of stochastic SIRS dynamics on different contact networks, from a regular lattice to a random structure via small-world configurations. We show that, for two distinct dynamical cases involving a stable equilibrium and a noisy endemic steady state, the modified mean-field model approximates successfully the steady state dynamics as well as the respective short and long transients of decaying cycles. This result demonstrates that early on in the transients an approximate power-law relationship is established between global (mean) quantities and the covariance structure in the network. The approach fails in the more complex case of persistent cycles observed within the narrow range of small-world configurations.