Epidemic outbreaks in a network of cities with distributed sizes

Recurrent epidemics of infectious childhood diseases such as measles are a
major health problem and have been subject to extensive theoretical
research. Here we develop a theory for the dynamics of epidemic outbreaks
and their synchronization in a network of coupled cities with distributed
sizes. Each city is described by a seasonally forced SEIR model. The model
generates chaotic dynamics with annual and biennial dynamics in excellent
agreement with long-term data sets. A new qualitative criterion based on
the attractor topology is developed to distinguish between major outbreaks
and epidemic fade-outs. This information is coded into a symbolic
dynamics. We are able to deduce a one dimensional first return map of the
chaotic SEIR equations, which upon iteration is able to generate the
symbolic sequence of major outbreaks. The synchronization of epidemic
outbreaks in a network of cities is defined as measure based on the
symbolic dynamics. When applied to real data sets we find an excellent
agreement between these spatio-temporal patterns and the results from the
numerical simulation, when in the model the distribution of city sizes is
taken into account. For this, we assume that the coupling strength between
two interacting cities depends on the size ratio of the two cities. This,
effectively gives rise to a hierarchical network topology, which translates
to a hierarchic dynamics of the outbreaks.