Chaos in Childhood Epidemics

Abstract

Whereas childhood infections such as chickenpox evidence effectively periodic dynamics, case reports for diseases such as measles and rubella fluctuate more erratically [1]. Elsewhere [2,3], it has been suggested that the observed fluctuations correspond to low dimensional chaos of the sort which arises in epidemiological models [5] of the SEIR variety, i.e., differential equations, subject to periodic forcing.¹ The basis for this assertion is as follows:

• 1. Reconstructed phase portraits, Poincaré sections, and return maps constructed from actual data are remarkably similar to those obtained from the models (Fig. 1).

• 2. For measles, both the models and the data yield correlation dimensions of about 25. A comparable value is obtained for the Lyapunov dimension as computed directly from the differential equations.

• 3. Both the models and the data yield estimated positive Lyapunov exponents of between 0.4 and 0.5 bpy.