Interepidemic intervals in forced and unforced SEIR models

Many infectious diseases give rise to recurrent epidemics. The time interval between epidemics is consequently an important property that epidemiologists and public health officials would like to be able to predict. Accurate estimates have been made for certain diseases by associating the observed interepidemic interval with the natural period of damped oscillations near the stable equilibrium solution of the standard (unforced) SEIR model. For childhood infections, this successful prediction is surprising because seasonal variation in contact rates (due to school terms) is known to have significant effects on patterns of disease incidence. Here, we show that the natural damping period of transients near the annual attractor of the seasonally forced SEIR model is usually well-approximated by the damping period obtained without forcing. This explains why naive calculations of interepidemic intervals have yielded accurate results in certain cases. However, the unforced approximation cannot be justified if the forced model has a non-annual attractor with a non-negligible basin of attraction, as is typically the case for measles; consequently, agreement between the interepidemic interval predicted by the unforced model for measles and real measles time series, appears to be coincidental.