Seasonality is a driving force that has a major effect on the spatio-temporal dynamics of natural systems and their populations. This is especially true for the transmission of common infectious diseases (such as influenza, measles, chickenpox and pertussis), and is of great relevance for host-parasite relationships in general. Here we gain further insights into the nonlinear dynamics of recurrent diseases through the analysis of the classical seasonally forced SIR (susceptible, infectious or recovered) epidemic model. Our analysis differs from other modelling studies in that the focus is more on post-epidemic dynamics than the outbreak itself. Despite the mathematical intractability of the forced SIR model, we identify a new threshold effect and give clear analytical conditions for predicting the occurrence of either a future epidemic outbreak, or a 'skip'-a year in which an epidemic fails to initiate. The threshold is determined by the population's susceptibility measured after the last outbreak and the rate at which new susceptible individuals are recruited into the population. Moreover, the time of occurrence (that is, the phase) of an outbreak proves to be a useful parameter that carries important epidemiological information. In forced systems, seasonal changes can prevent late-peaking diseases (that is, those having high phase) from spreading widely, thereby increasing population susceptibility, and controlling the triggering and intensity of future epidemics. These principles yield forecasting tools that should have relevance for the study of newly emerging and re-emerging diseases controlled by seasonal vectors.

We reformulate several recent analyses of infection processes on highly heterogeneous networks (e.g., scale-free networks) which conclude that diseases will spread and persist even for vanishingly small transmission probabilities. The results of these latter studies contrast with conventional epidemiological models where there are clear threshold effects, namely, should the transmission probability fall below a critical threshold level the disease is expected to die out. Here we show that epidemic propagation depends equally on the infection scheme as well as the network structure. Connectivity-dependent infection schemes can yield threshold effects even in scale-free networks where they would otherwise be unexpected.

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