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Impact of Vaccination on the Spatial Correlation and Persistence Of Proc. Natl. Acad. Sci. USA Vol. 93, pp. 12648-12653, October 1996 Population Biology Impact of vaccination on the spatial correlation and persistence of measles dynamics (spatial heterogeneity/disease eradication/critical community size/simulation models) B. M. BOLKER* AND B. T. GRENFELL Department of Zoology, Cambridge University, Downing Street, Cambridge CB2 3EJ, United Kingdom Communicated by Burton H. Singer, Princeton University, Princeton, NJ, August 14, 1996 (received for review April 21, 1996) ABSTRACT The onset of measles vaccination in England local extinction is fragile; reintroduction of the disease sparks and Wales in 1968 coincided with a marked drop in the a new epidemic, which cannot occur below the invasibility temporal correlation of epidemic patterns between major threshold. cities. We analyze a variety of hypotheses for the mechanisms Persistence is more difficult to analyze than invasibility driving this change. Straightforward stochastic models sug- because invasibility requires only a linear calculation (around gest that the interaction between a lowered susceptible pop- the disease-free equilibrium), while persistence demands con- ulation (and hence increased demographic noise) and non- sideration of the full stochastic system. In addition, local linear dynamics is sufficient to cause the observed drop in extinctions ("fade-outs") interact with nonlinear dynamics and correlation. The decorrelation of epidemics could potentially seasonality to change the dynamical behavior of measles lessen the chance of global extinction and so inhibit attempts epidemics qualitatively. at measles eradication. The detailed interaction of spatial structure with persis- tence, nonlinear dynamics, and seasonality is just beginning to The combination of practical importance and theoretical receive attention (28-30). Vaccination, and in particular the interest has made measles in urban populations one of the most mass vaccination programs begun in developed countries in studied epidemiological systems, but spatial and stochastic the late 1960s, serve as a natural experiment for exploring effects remain less well explored (1). Practically, the continu- these topics. At the simplest level, vaccination of young ing failure of mass vaccination campaigns to eradicate measles children reduces the recruitment rate of new susceptibles and in developed countries and the importance of measles mor- therefore the reproductive rate of the infection (13). A number tality among children in developing countries has led to of theoretical studies have shown that vaccination should continuing interest. Theoretically, recent work showing that significantly reduce the amplitude of epidemics (and hence nonlinear dynamics can reduce the extinction rate of spatially disease incidence) as well as changing their phase (4, 13). subdivided populations (2, 3) may be testable with the body of Various authors have explored the effects of spatial hetero- theory and data gathered to evaluate the persistence of geneity on the necessary effort and optimum strategy for measles and other childhood infections under mass vaccination vaccination (31) and the nonlinear interactions of seasonality (4-7). and vaccination (32). What has not been done, however, is to The basic dynamics of measles epidemics have been exten- consider simultaneously the important factors of spatial het- sively studied (8-13). Large measles epidemics exhaust a pool erogeneity, nonlinear dynamics, and vaccination along with of susceptible children, who gain lifelong immunity after data from the onset of mass vaccination. having the disease; the interepidemic period is determined by In this paper, we explore the impact of mass measles the severity of the epidemic and the length of time taken to vaccination on the spatial dynamics of measles. Specifically, we build up the pool of susceptibles again from new births. The show that the onset of measles vaccination in England and seasonal pattern of aggregation of children in schools is also Wales corresponded to a marked reduction in the temporal important in determining the size and pattern of epidemics correlation of epidemics between cities. Explicitly spatial (14-16). Furthermore, seasonality strongly influences the re- models indicate that this reduction is probably a nonlinear current epidemic behavior of measles, where there is strong dynamic effect: vaccination eliminated large epidemics, which evidence for nonlinear (and possibly chaotic) effects (17-23). had acted to correlate measles dynamics in different cities In small isolated communities, large epidemics are often before vaccination. Finally, we. demonstrate that this decorre- followed by extinction of disease as the chain of transmission lation of epidemics could inhibit attempts at measles control by breaks down (8, 24, 25). The critical community size (8, 24, 26), lessening the chance of simultaneous extinction of disease in the threshold population size above which measles can persist all subpopulations. through interepidemic troughs, may depend on the spatial structure and connectedness of the regional population. The persistence behavior of measles is of particular interest in DATA SETS AND MODELS connection within the eradication of a disease by mass vacci- Data Sets. Spatially disaggregated weekly case reports are nation. Most theoretical estimates focus on the invasibility available in published form for England and Wales for the threshold, the level of control at which a disease will not only period 1948-1988, giving 20 years of data before and after the go extinct but also be unable to reinvade the population (13). onset of mass vaccination. The analysis is based on data from One can also locally eradicate infection simply by driving it seven large cities (London, Birmingham, Liverpool, Manches- below its persistence threshold, the point at which the disease ter, Sheffield, Bristol, and Newcastle), with a population size is likely to go extinct during the troughs between epidemics of to 10 million (out of a total population of (27). The persistence threshold is greater than the invasibility range -300,000 threshold and hence easier to reach, but eradication through Abbreviations: SEIR, susceptible/exposed/infective/recovered; RAS, realistic age-structured. The publication costs of this article were defrayed in part by page charge *To whom reprint requests should be sent at the present address: payment. This article must therefore be hereby marked "advertisement" in Department of Ecology and Evolution, Princeton University, Prince- accordance with 18 U.S.C. §1734 solely to indicate this fact. ton, NJ 08544-1003. 12648 Downloaded by guest on September 26, 2021 Population Biology: Bolker and Grenfell Proc. Natl. Acad. Sci. USA 93 (1996) 12649 50 million), to obtain a preliminary idea of the geographic annual blocks rather than continuously. [For more details, see coherence of measles epidemics in England; since we are Schenzle (16) or Bolker and Grenfell (29, 35).] interested in patterns of relative incidence through time, The spatial models we use mimic a population of 1 million incompleteness of case reports is not a significant difficulty. individuals, subdivided into 10 identical subpopulations ("cit- These cities are generally widely separated in space, although ies"). Contact between susceptibles and infectives in different Manchester and London are within 50 miles of each other. For cities occurs at a rate equal to 1% of the within-city contact comparative purposes, we also analyze the change in correla- rate, adding to the infections from within-city contacts. Indi- tion on a smaller spatial scale, among the boroughs of London. viduals do not move explicitly, but Sattenspiel and Dietz (44) Data Analysis. We apply a variety of standard classical and have shown that individual movement can be expressed in time-series statistics to the data to estimate the geographic terms of between-city contact rates. We use a between-city coherence of epidemics. contact rate of 1% of within-city contact rate, which is in the Measuring overall coherence. The primary estimate of coher- center of the range of previous estimates (45) [estimating the ence is the mean of the correlation [Pearson's r of log(1 + n); actual contact rate from data is a difficult and largely unex- n = number of cases] for all pairwise combinations of cities in plored problem (44), which we will not tackle here]. [Trial the data set. Short-term correlations (4-year blocks) identify simulations with different between-city contact rates gave rapid changes in the correlation structure but may magnify the average correlations before vaccination that were unrealisti- effects of transient differences in epidemic cycles, while long- cally low (with a relative cross-contact rate of 0.01%) or high term correlations (20-year blocks) give an accurate measure- (with a cross-contact of 10%); otherwise, the simulations gave ment of correlation but blur rapid changes in correlation. A results qualitatively similar to those reported below for a good summary statistic for changes in correlation is the size of cross-contact rate of 1%.] All age classes mix equally between the drop in mean pairwise correlation between the periods cities, which is unrealistic, but in the absence of data, we aim immediately before and after the start of mass vaccination; for parsimony. As in most stochastic measles models (10, 11), correlations for 4-, 10-, and 20-year blocks indicate the short-, we also allow for a low level of infective
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