Qualitative Analysis of Unpredictability: A Case Study from Childhood Epidemics

The unpredictability of the recurrent outbreaks of childhood epidemics has been a matter of scientific dispute. At first sight the deterministic SEIR model, which divides the host population into four classes (Susceptible, Exposed, Infectious, Recovered), fails to explain the unpredictability, because realistic parameter values lead to periodic attractors. We show that these periodic attractors coexist with chaotic transients. The detailed geometrical analysis of this phenomenon suggests that chaotic transients have been underestimated in their importance for the dynamics on observable time scales. A second problem of the SEIR model is the high extinction probability of epidemics in finite populations. We argue that the immigration of infectives from outside is an essential parameter in this context. Immigration and the process of infection itself are sources of demographic fluctuations, which undergo subtle interaction with chaotic transients. The stochastic simulation of the SEIR model shows that the chaotic transients are permanently revisited. The demographic noise integrates chaotic transients and intermittent periodic episodes.