## Abstract

A basic issue in ecology is the relation between extinction and population size. One of the clearest manifestations of a population threshold for extinction is the critical community size below which infections like measles do not persist. The current generation of stochastic models overestimates the observed critical community size for measles, generating much less persistence of infection than is observed. The inclusion of a more biologically realistic model for the duration of infection produced a much closer fit to the actual critical community size and explains previously undescribed high-frequency oscillations in measles incidence.

The relation between disease persistence and community size can be explored through the pattern of fadeouts of infection [three or more weeks without reported cases (1, 2, 3, 4, 5)]. The observed critical community size (CCS) for measles is about 250,000 to 400,000 (Fig. 1A). These figures are based on a large prevaccination data set for 60 towns in England and Wales for the years 1944 to 1968, but they are also typical of the pattern observed for U.S. cities (2) and islands (3). Below the CCS, the infection often becomes extinct in the troughs between epidemics and must be reintroduced from an external source.

Fadeout pattern predictions from the best current nonspatial stochastic model (6, 7, 8, 9, 10, 11, 12, 13) significantly overestimate the CCS (Fig. 1A), generating many more fadeouts than observed for towns with populations over 250,000. This discrepancy is even more marked in a comparison of the observed and expected total weeks of fadeout per year (Fig. 1B). Although this standard model [the realistic age-structured (RAS) model (10, 11, 12, 13, 14)] captures the deterministic dynamics of measles epidemics very well (5, 10, 13), its stochastic dynamics are unstable in populations below about 1 million, generating many more fadeouts than observed (4, 5, 6). A number of authors have sought an explanation for this failure of current models in terms of spatial heterogeneities in transmission, on both large spatial scales (4, 15, 16, 17) and at the individual (family and school) level (16). Although inclusion of spatial heterogeneities reduces the predicted degree of fadeout, even quite complicated spatial models cannot currently capture the low observed level of the CCS (4, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25).

The fact that the CCS applies to a wide range of communities, from cities to islands, indicates that we should seek a more generic explanation, one rooted in the biology of transmission. We propose that the problem arises because current models are too sensitive to stochastic fluctuations, which arise from the use of long-tailed exponential distributions for the incubation and infectious periods. This exponential formulation arises from the standard assumption that movement from the exposed to the infectious class and then into the recovered class occurs at constant rates *a* and *g*, respectively (6).

We can modify the standard model to allow for these effects by assuming normal distributions for the incubation and infectious periods (19). More complex distributions could be used (6), but evidence from the detailed study of transmission in families (20) indicates that the periods show limited variation, and the data are well described by infectious periods normally distributed about their means. The use of more discrete periods has been considered previously (21) but seldom in this context of seasonally forced stochastic models. The revised stochastic model tends to produce more concentrated pulses of infection, so we will call it the pulsed realistic age-structured (PRAS) model.

Support for the new model is provided by a comparison of Fourier spectra for simulated epidemic time series from the standard (RAS) and modified (PRAS) models with the observed pattern for England and Wales (Fig. 2). The observed spectrum (Fig. 2A) shows the well-known peaks associated with biennial and annual (seasonal) epidemics (4, 6, 8, 9). In addition, the analysis reveals a previously unreported high-frequency peak, reflecting low-amplitude epidemics with periods of 2 to 3 weeks. Both models capture the annual and biennial peaks accurately (Fig. 2B) (6, 10, 13); however, the standard RAS model cannot reproduce the high-frequency pulses, whereas the PRAS model does reflect variation at this time scale. Essentially, the more uniform infectious period of the PRAS model produces “generation pulses” of infection (22), which are apparent in the real data. These pulses are especially apparent in the data for small communities.

The impact of this uniformity of infection on the predicted CCS is shown in Fig. 1. The match to the observed number and total duration of fadeouts is much closer than with the standard RAS model. This correspondence occurs essentially because the normally distributed infection periods of the PRAS model cause infectiousness to be more evenly distributed among individuals, and the model is therefore much less vulnerable to stochastic fadeout. The mechanism can be illustrated with a simple analytical model (Fig. 3). Figure 3A shows the difference between the assumptions about the duration of infection made for the two models: The PRAS model, with its more discrete infectious period, has the majority of infections lasting between 4 and 6 days, whereas the standard RAS model assumes a lower proportion of people infected at the early stages, with a long exponential tail generating an appreciable probability of infection after 20 days. The associated amount of variation in the infectious periods is reflected in the variance of the basic reproduction ratio of infection *R*_{0} (6). The parameter Var(*R*_{0}) represents the stochasticity in the spread of infection and is therefore related to the probability of zero transmission (Fig. 3B), which is much higher with the exponential distribution produced by the standard model (23). This difference occurs because the RAS model relies on a few infectious individuals who retain the disease for a long time to spread the infection. Therefore, the RAS model has far more individuals who do not produce any secondary cases, as compared to the PRAS model (Fig. 3B). The greater persistence of the pulsed model is unaffected if we equate the average generation gap of the infection [serial interval between cases (24)] rather than the infectious period.

All of these patterns of infection were generated using isolated populations with a low stochastic influx of infectives to reintroduce infection (5, 9, 13); however, changing this influx level within realistic limits does not qualitatively alter the improvement made by the PRAS model. Of course, explicit spatial heterogeneity will be required to explain fully the persistence of the disease in a metapopulation where there is no influx from an external source (5).

In summary, inclusion of a more realistic infection period in childhood disease models generates the high-frequency pulsing seen in the real data and produces more realistic levels of persistence, as reflected in the lower CCS. This improved fit is likely to be a generic result for infections that occur as self-extinguishing epidemics. More generally, this well-documented example underlines (25) the idea that the assumption of constant transition rates (and therefore exponentially distributed times), which is often made in ecology, may need to be reevaluated if we are to fully understand patterns of stochastic fluctuations and extinctions.