Repellors attract attention

NEWS AND VIEWS CHAOS AND EPIDEMIOLOGY-------------------------------------------------------- Repellors attract attention John J. Sidorowich RECURRENT epidemics of childhood diseases are an important public health issue and the subject of a great deal of epidemiological research. But although knowledge of the micro dynamics of infection and incubation has increased to the point where most occurrences can be cured, and in some cases even prevented, there is still much to learn about how the diseases spread through large populations. Despite speculation that chaos might playa role in the evolution of such epidemics, it has been extremely difficult to establish a coherent explanation that agrees with both theory and data. A report by D. R. Rand and H. B. Wilson in the Proceedings of the Royal Society (B246, 179-184; 1991) offers a fresh perspective on this issue by examining a new mechanism for chaotic behaviour in dynamical systems. The search for chaos in the real world has been going on for nearly a decade. The most spectacular successes have occurred in fluid dynamics, where the experimentalist can generate large quantities of data under well controlled conditions. Algorithms have been developed for calculating characteristic exponents and dimensions, and methods for constructing nonlinear predictive models have been refined. But the techniques that search for some signature of chaos frequently produce ambiguous results when the data sets are small or noisy. Such is the case for the data collected about epidemics. Instead of tens or hundreds of thousands of data points, only several hundred have been accumulated. Sampling errors can be very large, and clearly the epidemiologist can exercise little control over environmental effects. Consequently, the results from analysing the historical records are not conclusive. Several groups have tried to calculate Lyapunov exponents for the monthly records on the incidence of chickenpox and measles in major cities. The exponents characterize the sensitivity of a dynamical system to perturbations: a positive exponent indicates that small disturbances grow exponentially; a negative one that they exponentially decay. A positive Lyapunov exponent is the telltale sign of chaos, whereas if the system relaxes onto a periodic limit cycle, only negative exponents will be observed. The verdict for chickenpox is a split decision. The majority view is that there is no positive exponent, but there is contrary evidence that cannot be ignored. On the other hand, almost everyone agrees that a significant positive exponent exists for the measles data. Confusion about the estimated expo584 Attractors are invariant sets that trajectories relax onto as they evolve through the state space. The 'reduction of the state space' represented by attractors is the primary motivation behind nents is compounded by studies of the chaotic time series analysis - the hope epidemiologists' mathematical models. It that what seems like an outrageously is impossible to model an epidemic on a complicated dynamic could, after tranperson by person basis, so a mean field sients die out, result in simple behaviour approximation is adopted where the that can be described in terms of an state variables correspond to quantities invariant geometrical object. Repellors averaged over a large population. A are the unstable counterparts of the popular example is the SEIR model, a stable attractors. Points on the repellor set of coupled differential equations de- remain there as time progresses; but scribing the rates at which susceptible, points in the vicinity of the repellor, exposed, infected and recovered (SEIR) instead of being attracted onto the inportions of the host population grow and variant set, are repelled away from it. diminish. One of the parameters in the Because of their instability, repellors -2.2 +----'-----'-----'---,,,..---'----+ traditionally were not considered relevant for physical studies as they were -2.4 thought to be unobservable. Rand and Wilson investi-2.6 gated what happens when en the mean field approxima0)-2.8 .Q tion underlying the SEIR model fails. Normally, the -3 large number of individuals in the host population -3.2 prevents the occurrence of -3.4+----.-----.-----.----.-----1- significant fluctuations from -25 -20 15 10 -5 0 the average behaviour. But log ,the model equations allow A cross-sectional view of an approximation to the chaotic for near extinctions of some repel lor of the SEIR equations; S, individuals susceptible to subpopulations, requiring a infection, I, infectives, individuals capable of transmitting modification in the de scripthe disease. (Courtesy of D. R. Rand and H. Wilson.) tions of how these rare indiequations, the contact rate, is particular- viduals interact with the rest of the ly important because it is the coupling population. As their numbers dwindle, constant for the nonlinear interaction the stochastic nature of how particular that transforms susceptibles into exposed infected individuals happen to bump into individuals. Low values for the contact susceptible folks necessarily adds a ranrate generate periodic limit cycles, which dom element to the model. go through a period-doubling sequence Fluctuations introduced by the ranto chaotic behaviour as the contact rate dom interactions launch the system state increases. There is general agreement into what Rand and Wilson refer to as a that the value of the contact rate chaotic pinball machine. The system is appropriate for chickenpox lies within knocked off its periodic limit cycle into the periodic range. For measles the the surrounding region of state space, situation is much more contentious. The which is filled with a chaotic repellor. In SEIR model can produce a time series the absence of any further perturbations, with the same Lyapunov exponent as the the disturbed state would typically historical records, but many believe that display a short-lived chaotic transient that requires an unreasonably high value before relaxing back onto the periodic for the contact rate. However, if the attractor. However, the stochastic forvalue is reduced the model reverts to cing of the system does persist, and this periodic behaviour. 'noise' actually stabilizes the effects of An intriguing explanation for the dis- the chaotic repellor. The chaotic trancrepancy between the SEIR model and sients persist as they are jostled about the historical data is provided by a new the fractal repellor, producing long-term type of process named chaotic stochasti- trajectories with positive Lyapunov city. In their paper, Rand and Wilson exponents. Rand and Wilson's observations propoint out that in addition to the periodic attractor, an unstable chaotic repellor vide more than just an interesting interalso exists for the realistic values of the pretation of the SEIR model. The novel contact rate (see figure). The coexist-aspect of their proposal is the demonence of these two invariant (...truncated)