## Abstract

Measles epidemics in UK cities, which were regular and highly synchronous before vaccination, are known to have become irregular and spatially uncorrelated in the vaccine era. Whooping cough shows the reverse pattern, namely a shift from spatial incoherence and irregularity before vaccination to regular, synchronous epidemics afterward. Models show that these patterns can arise from disease-specific responses to dynamical noise. This analysis has implications for vaccination strategies and illustrates the power of comparative dynamical studies of sympatric metapopulations.

A central debate in ecology concerns the relative importance of deterministic and stochastic processes in shaping the dynamics of populations (1). This issue has become more complex since the recognition that patterns of population fluctuation are substantially influenced by spatial structure and synchrony (2). Progress has been made in studies showing close dynamical agreement between models and data (3,4), the exploration of spatial structure (5–7), and com parative analyses of population dynamics across species (8). However, models and long-term observed spatiotemporal data have not been combined to analyze the comparative contemporaneous dynamics of sympatric metapopulations. This can be achieved in the study of childhood bacterial and viral infections (for example, measles, whooping cough, rubella, and chicken pox), for which there are unusually extensive spatiotemporal data and realistic models (6,9, 10).

We present an analysis of weekly incidence data from 60 cities in England and Wales from 1944 to 1994 for measles and whooping cough (11, 12). The time series encompass dramatic changes in effective demographic parameters, particularly as a result of nationwide vaccination schemes (starting in 1968 for measles and 1957 for whooping cough). This permits the study of the comparative dynamical effects of vaccination as a “natural experiment.” Apart from their broad ecological interest, understanding the mechanisms that drive the dynamics of disease metapopulations is also fundamental in the design of successful vaccination programs.

The two infections have a similar reproductive ratio (*R*
_{0}) of 16 to 18 (9, 12). The measles virus is transmitted by means of aerosol particles and is highly infectious. It conforms very closely to the assumptions of the SEIR (susceptible, exposed, infectious, recovered) class of models, with a latent period of about 8 days, an effective infectious period of about 5 days, and life-long immunity after infection (9). Before mass immunization, the dynamics of measles were highly synchronized across England and Wales with a pronounced 2-year period from 1950 (Fig. 1, A and B) (6, 13, 14). In contrast, the dynamics of measles in the vaccine era show little temporal or spatial pattern, consisting of annual and 2- to 3-year fluctuations (15) that are spatially uncorrelated (13) (Fig. 1, A and B).

Whooping cough is a respiratory disease caused by the highly infectious bacterium *Bordetella pertussis*, with an average latent period of 8 days and an infectious period of 14 to 21 days (16–18). The immunological response to pertussis varies, with known instances of immunity waning after infection (18). Despite this potential complication, whooping cough dynamics are relatively well described by the SEIR formalism (9, 17, 19, 20).

The spatiotemporal dynamics of whooping cough are dramatically different from those of measles. Figure 1, C and D, shows that, compared with measles, pertussis exhibits the reverse pattern of spatial synchrony, before and during the vaccine era. In the prevaccination era (1944–57), there is little spatial correlation in epidemics of whooping cough; each city exhibits bursts of about 2- to 2.5-year fluctuations, punctuated with periods of annual outbreaks. Mass vaccination in 1957, however, stimulated highly synchronized 3.5-year pertussis outbreaks across England and Wales. These spatial correlations are analyzed statistically in Fig. 2; compared with measles, the spatial dynamics of whooping cough do indeed show the opposite response to vaccination, with significantly higher correlations among communities in the vaccine era.

For measles, the predominantly biennial prevaccination pattern in large cities after 1950 is accurately reproduced by simple deterministic SEIR models with seasonally varying contact rates, as dictated by the pattern of school terms (21, 22). Similar deterministic models of whooping cough have been unable to capture the observed outbreaks, both before and after the start of mass vaccination. We investigated whether the missing ingredient might be stochasticity and found that, in the absence of vaccination, an event-driven, seasonally forced Monte Carlo SEIR model (12,22) generated a mixture of 2- to 2.5-year and annual cycles, consistent with the observed time series (22). The multiannual cycles are stimulated by stochastic resonance around the annual deterministic attractor. The model also generates epidemics in the vaccine era that are very similar to the data, with a pronounced 3.5-year period (Fig. 3, C and D).

These findings are in agreement with Hethcote's study of pertussis case reports in the United States (17). Using a complex age-structured model that incorporated different levels of vaccine-induced and infection-induced immunity, Hethcote also concluded that stochasticity is needed if the observed multiannual oscillations are to be accurately captured.

A detailed analysis reveals that the deterministic whooping cough attractor is extremely sensitive to stochastic effects (23). Small perturbations can cause relatively long transient behavior that is qualitatively different from the asymptotic dynamics. This is in contrast with measles, where trajectories that are pushed away from the biennial attractor by noise return to it rapidly and these brief transients are themselves strongly biennial.

We would like to determine the biological mechanisms that underlie this contrast in the dynamics of measles and whooping cough. The most important differences between the two diseases lie in their epidemiological parameters: Whooping cough has an infectious period about three times as long as that of measles. The dynamics of measles, with its shorter infectious period, are strongly dictated by the pattern of seasonality, wheareas the relatively brief periods of low contact rates between children when schools are out of term are likely to have a much less profound effect on a disease with a long duration of infection. Pertussis epidemics do appear much less locked into phase by seasonal forcing, although it is an important feature of the dynamics (17).

These differences in the interaction between nonlinearity, dynamical noise, and external forcing also account for the observed spatial patterns in the prevaccination and vaccine eras. The large-scale spatial coherence in prevaccination measles epidemics may be due to the presence of a unique (typically biennial) attractor at the individual city scale and epidemiological coupling, which transfers the infection to other centers and tends to push epidemics into phase. The vaccine era is more complicated. Elsewhere, we have demonstrated that vaccination moves model measles dynamics into a parameter region that may contain multiple stable solutions, with finely intertwined basins of attraction (21). In the presence of stochasticity, trajectories in different cities might switch between attractors in a complex manner, resulting in spatial asynchrony of epidemics. A fully stochastic SEIR model captures the qualitative dynamics of measles in the vaccine era (Fig. 3, A and B).

The absence of spatial synchrony in the prevaccine pertussis data is not surprising, given the more crucial roles that stochasticity and transients play in determining its dynamics and the weaker phase-locking with seasonality (Fig. 1D). We would like to determine why whooping cough outbreaks become so highly synchronized with a period of 3.5 years in the vaccine era. There are probably several factors responsible for this transition. First, vaccination increases the mean age at infection, which in turn increases the natural period of the system, giving rise to coherent 3.5-year cycles (9,24). Second, in the presence of this unique dynamical attractor, a concerted nationwide vaccination effort starting in 1957 is likely to have acted as a strong synchronized “shock” to the system, forcing different communities into phase. We explored these two factors using a stochastic SEIR model for 10 coupled populations, with vaccination starting at year 100. Despite its simplicity, the model accurately captures the qualitative effects of vaccination on spatial synchrony (Fig. 3, C and D). The spatially synchronized initiation of vaccination drives the model dynamics to 3.5-year cycles, with a lower effective *R*
_{0}(24).

The increase in the mean age at infection in the vaccine era might also result in greater amounts of movement in the infected population, thus increasing epidemiological coupling. We would expect this to enhance epidemic synchronization (when there is a unique attractor).

Our findings have implications for vaccination strategies and metapopulation dynamics. Since the influential work of Fine and Clarkson (25), it has been thought that although pertussis vaccination reduced the total number of new cases, it did not lengthen the period between whooping cough outbreaks. The general consensus, therefore, has been that moderate levels of immunization cannot disrupt the transmission of pertussis, although it prevents disease (18, 20). The post-1957 region of Fig. 1D shows, however, that vaccination had two major effects on pertussis epidemics: (i) Their period markedly increased from at most 2.5 years to 3.5 years, and (ii) they became highly synchronized, with very deep troughs between outbreaks. These dynamical transitions as a result of pertussis vaccination suggest that immunization is having a pronounced effect on the circulation of the infection.

Another implication of this work concerns the use of vaccination pulsed across all ages. This alternative immunization strategy has been proposed in an important body of theoretical work (26) and has gained much popularity, mainly for economic and logistic reasons. Superimposing pulses on continuous vaccination may also provide a spatial benefit by synchronizing local fade-outs, thus increasing the likelihood of global eradication (6, 14). This is potentially a useful effect for measles, for which vaccination had a spatially decorrelating effect (13). However, our findings indicate that the spatial consequences of pulsed vaccination are more difficult to predict for pertussis, for which continuous vaccination alone has synchronized outbreaks. In particular, our results suggest that dynamical factors need to be taken into account: The timing of any pulses should avoid disturbing both the deep troughs and the long interepidemic period currently observed.

This work also demonstrates the ecological benefits of studying comparative spatiotemporal dynamics in sympatric disease metapopulations. An intricate balance between stochasticity, seasonality, and nonlinearity makes a profound difference for the dynamics of two childhood infections, in the same host metapopulation, with largely the same dispersal patterns and the same pattern of seasonal forcing. The prevaccine dynamics of measles in large UK cities are relatively well explained with a deterministic model; in contrast, prevaccine pertussis dynamics exhibit a subtle interaction between noise and determinism (4, 27) and cannot be modeled adequately without stochasticity. The scenario is reversed in the vaccine era, during which measles appears to be highly influenced by stochasticity and pertussis outbreaks can be explained with simple, deterministic arguments (24). Childhood diseases are likely to prove excellent systems for further investigation of these issues.