Seasonality is a driving force that has major impact on the spatio-temporal dynamics of natural systems and their populations. This is especially true for the transmission of common infectious diseases (eg., influenza, measles, chickenpox, pertussis), and of great relevance for host-parasite relationships and ecological systems in general. Here we gain new insights into the nonlinear dynamics of recurrent diseases through the analysis of the classical seasonally forced SIR epidemic model. (Note that the results are of equal relevance for standard laser models in physics which are based on almost identical equations.) Unfortunately, given the model's intrinsic mathematical intractability, it has been difficult to come by general quantitative or analytical insights, despite many attempts over the past decades. These difficulties are an outcome of the complex synchronization effects that arise between the external forcing and the natural oscillations of the nonlinear model. The analysis advanced here attempts to make progress in this direction by focusing on what we term ``skipping dynamics. In many epidemiological time-series, there are a significant number of years in which major epidemics do not appear to trigger at all, i.e. they `"skip" in an unpredictable fashion. Skipping events are also intrinsic to the forced SIR model when parameterised in the chaotic regime. Curiously, it is difficult if not impossible to locate realistic chaotic parameter regimes in which outbreaks occur regularly each year. This contrasts with the well known Rossler oscillator, and several ecological foodweb models whose outbreaks recur regularly but whose amplitude vary chaotically in time (i.e. the Uniform Phase Chaotic Amplitude oscillations). The goal of the present study is to develop a ``language of skips that makes it possible to predict under what conditions the next outbreak is likely to occur, and how many ``skips'' might be expected after any given outbreak (Stone et al. 2007, Olinky et al. 2008).

Our work begins by numerically exploring the seasonally forced model's bifurcations and routes to chaos. We then proceed to make approximations that facilitate model analysis. Firstly we develop mathematical approximations of the model's trajectory in phase-space during each of two separate seasons of the year. The two different reconstructions are "glued" together alternately shadowing the seasonal changes. In this way the trajectory of the nonlinear model may be calculated as time evolves. As the seasons change, the trajectory is attracted to the quasi-equilibrium associated with each season. Non-equilibrium behaviour arises as the trajectory is attracted to each equilibrium but kicked away before ever reaching it. The trajectory is thus kicked away from one equilibrium to the next. Biennial behaviour, for example, is characterized by a large outbreak in the upper portion of the phase plane in the first year, followed by a 'skip' and extended stay in the lower portion of phase plane in the second year. For realistic parameters, each regime "outbreak" or "skip" is characterized by its own specific time scale.

The calculations lead to an inductive formulation and finally result in the identification of a new threshold effect. The threshold gives 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. The prediction proves to be a more reliable indication than existing standard techniques based on the classical epidemiological reproductive number Ro.

In addition, the time of occurrence (i.e., phase) of an outbreak proves to be a useful new parameter that carries important epidemiological information. In forced systems, seasonal changes can prevent late-peaking diseases (i.e., 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 reemerging diseases. The implications for epidemics triggered on structured population networks (clustered, small world, scale-free, - see Berchemko et al. 2008) will be discussed.

References: Stone, L., Olinky, R. and Huppert, A. Seasonal dynamics of recurrent epidemics. Nature 446, 533-536 (2007).

 Olinky R, Huppert A. and Stone L. Seasonal dynamics and threshold governing recurrent epidemics. J. Math Biol.56: 827-839. 2008.

Berchenko, Y., Artzy-Randrup Y., Teicher M., Stone L. The emergence of the giant component in clustered networks. 2008