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 stochasti incorporate the distributed nature of ecological interactions. In between, moment approx imations have been proposed that represent the effect of correlations on the dynamics o mean quantities. As an alternative closer to the typical temporalmodels used in ecology, we present here results on ‘‘modifiedmean-field equations’’ for infectious disease dynamics, in which only mean quantities are followed and the effect of heterogeneous mixing i incorporated implicitly. We specifically investigate the previously proposed empirica parameterization of heterogeneousmixing inwhich the bilinear incidence rate SI is replaced by a nonlinear term kSI, for the case of stochastic SIRS dynamics on different contac 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 modifiedmean-field model approximates successfully the steady state dynamics as well as the respective short and long transients of decaying cycles. Thi 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 net work. The approach fails in the more complex case of persistent cycles observed within the narrow range of small-world configurations. # 2006 Elsevier B.V. All rights reserved avai lable at www.sc iencedi rec t .com journal homepage: ht tp : / /www.e lsev ier .com/ locate /ecocom