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and disease within the host, is the sum of these contributions. Acknowledgements

We thank Robert L. Sinsheimerfor criticallyreadingthe manuscript. This work was supportedby NIH grant AI36373,ACSJFRA554 and a BeckmanYoungInvestigatorAward (to M.J.M.) References

1 Falkow,S. (1997) Am. Soc. Microbiol. News 63, 359-365 2 Conner,C.P.,Heithoff,D.M. and Mahan, M.J. (1997) Current Topics in Microbiology and Immunology (Vol.225) (Vogt,ILK. and Mahan, M.J., eds), pp. 1-12, Springer-Verlag 3 Mahan,M.J., Slauch,J.M. and Mekalanos,J.J. (1993)Science 259, 666-668 4 Mahan,M.J. etal. (1995) Proc. Natl. Acad. Sci. U. S. A. 92, 669-673 5 Hensel,M. et al. (1995) Science 269, 400-403 6 Plum,G. and Clarke-Curtiss,J.E. (1994) Infect. Immun. 62, 476-483 7 Abu Kwaik,Y. and Pederson,L.L.(1996) Mol. Microbiol. 21,

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8 Valdivia,R.H. and Falkow,S. (1996) Mol. Microbiol. 22, 367-389 9 Burns-Keliher,L.L.,P0rtteus, A. and Curtiss,R., III (1997) J. Bacteriol. 11, !3604-3612 10 Hacker,J. et al. (1997) Mol. Microbiol. 23, 1089-1097 11 Shea,J.E. et al. (1996) Proc. Natl. Acad. Sci. U. S. A. 93, 2593-2597 12 Heithoff,D.M. et al. (1997) Proc. Natl. Acad. Sci. U. S. A. 94, 934-939 13 Lawrence,J.G. and Roth,J.R. (1996) Genetics 142, 11-24 14 Lawrence,J.G. and R0th,J.R. (1996) Genetics 143, 1843-1860 15 Valdivia,R.H. and Falkow,S. (1997)Science277, 2007-2011 16 Young,G.M.and Hiller, V.L. (1997) Mol. Microbiol. 25, 319-328 17 Wang,J. et al. (1996) Proc. Natl. Acad. Sci. U. S. A. 93, 10434-10439 18 Rainey,P., Heithoff,D.M. and Mahan, M.J. Mol. Gen. Genet. (in press) 19 Camilli,A. and Mekalanos,J.J. (1995) Mol. Microbiol. 18, 671-683

Modelling the persistence of measles Matthew J. Keeling n the developing world, For vaccination programs to be effective, it of the population is likely to be measles is a major killer is important to understand and predict the susceptible, so R, the effective R0 ( R = R o x s ) 2, must be conpersistence of the disease. By considering of malnourished children 1'2 sidered instead. The invasion the process at different scales (from the and, even in countries with high vaccine uptake (such as the USA individual to the population level), several threshold is concerned with the magnitude of R" below the models allow the persistence of diseases, and UK), measles still persists. threshold, R is less than one, so, such as measles, to be captured. To clarify the impact of vaccion average, each infectious innation, we first need to underM.J. Keeling is in the Zoology Dept, dividual causes less than one stand the spatial persistence Downing Street, Cambridge, UK CB2 3EJ. secondary case. For the majority of the disease. The simplest tel: +44 1223 336644, fax: +44 1223 334466, of vaccination schemes, we atscenario (and best data) is proe-mail: [email protected] tempt to force R below orle by vided by the persistence of reducing the contact rate with measles in developed countries infected individuals: when this is achieved, the disease before vaccination: this has been the subject of much recent work. Here, we will discuss the concept of disease rapidly dies out and cannot re-invade. This was essenpersistence and go on to review the performances of tially the process that led to the global eradication of smallpox 2. However, for measles and many other inseveral different models at capturing this phenomenon. When considering the local and global eradication of fections, vaccination levels have not yet reached the invasion threshold, even in regions where the uptake infection, two distinct processes are important. First, is very high. to eradicate disease from a given region the herd imThis leads us to the second important concept in the munity needs to be raised sufficiently, by vaccination, to prevent re-invasion of infection from outside. This is maintenance of infection: its stochastic persistence in usually measured by an invasion threshold, below which the troughs between epidemics. Even when infection is, on average, above the invasion threshold, there is some infection will inevitably decrease and the disease canprobability that the chain of transmission may be. bronot re-enter a population once it has been eradicated. ken in the troughs. This is also a threshold phenomenon, This concept has been successfully represented by deterbut a much more subtle one. The persistence threshministic models 2,3, in which the fundamental quantity is the basic reproductive ratio of infection (Ro), defined old measures the lowest population size and susceptible density that can maintain a disease in the long term once as the average number of secondary cases produced by it has invaded, and it can only be assessed using stochasan infectious individual in a totally susceptible poputic concepts because it depends on the population size as lation 2'3. For endemic infections, only a proportion (s)

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cases of the disease - three weeks was chosen to allow for under-reporting of cases. Figure I shows the average number of fade-outs of measles per year for 60 communities in England and Wales before vaccination (1946-1968). The CCS is clearly -300 000-500 000: data from American cities and isolated islands demonstrate a similar value for the CCS. However, despite the accuracy of deterministic models at predicting the large-scale dynamics of epidemics s, these models cannot directly address the stochastic question of persistence 6,v. The CCS provides a natural measure of stochastic persistence for any community.

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One of the simplest models commonly used for simulating measles dynamics is the SEIR model. This is a compartmental model and splits the population according to whether individuals are susceptible, exposed, infectious or have recovered from the infection2,~,8,9 (Fig. 2a). In practice, a more com1945 1947 1949 plex formulation incorporating age structure and allowing for the Year seasonal aggregation of children (d) ~ . . . . . . . . . . . . . . . . in school is used to model the detailed patterns s,~°. This RAS (realistic age-structured) model captures the overall biennial pattern of epidemics and the input of vaccination extremely well. However, when this model is made stochastic - so that each event (e.g. birth, 1945 1947 1949 death or transmission of infection) Year occurs randomly - frequent fadeFig. 1. (a) The average number of fade-outs (local extinctions) of measles per year against population outs in the inter-epidemic troughs size for 60 communities in England and Wales. Parts (b)-(d} demonstrate the three levels of persistare seen and the CCS for the model ence and dynamics as defined by Bartlett4. (b) Type I dynamics (e.g. London, population 3.4 million) is found to be significantly larger are regular: measles is endemic and does not fade-out. (¢) With type II dynamics (e.g. Nottingham, population 300 000) there are fade-outs in the troughs - represented by black dots - but the epithan that observed 7. Thus, despite demics are regular. (d) Type III dynamics (e.g. Teignmouth, population 11000) display irregular epithe overall realism of the model, demics with long fade-outs between them. it still overestimates the CCS by an order of magnitude 7'1°. Three main processes have recently been proposed well as the contact and infection rates. For small poputo deal with this problem. Each acts :as an extension to lations, there is a high probability that the disease will the standard models, with the ultimate goal of predictbe eradicated by chance, and as the population increases ing the remarkably high persistence of measles. These so does the persistence. Both the persistence and invaideas all rely on incorporating greater biological realsion thresholds are important if we are to fully underism, in terms of heterogeneity at different scales. This stand vaccination. need to examine processes at a variety of scales paralThe persistence of measles is usually addressed in terms of the critical community size (CCS), which is lels recent work in theoretical ecology T M (Fig. 2). the smallest population without any extinctions (a temLarge-scale heterogeneities and metapopulations porary absence of the disease). In his seminal work on In large communities, there is never complete mixing the CCS, Bartlett 4 defined a fade-out (local extinction) between all individuals. This leads to variations in the as three or more consecutive weeks with no reported

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Flg. 2. (a) A schematic diagram showing the movement between the four classes of individual, together with the transition rates. Individuals are born susceptible (S); once infected, they move into the exposed (E) class; later (after about 8 days), they enter the infectious (I) class when they begin to spread the disease; finally, they reach the recovered (R) class and are assumed to possess life-long immunity. Vaccination acts to move individuals from the susceptible class directly to the recovered class. Parts (b)-(d) represent the three scales of heterogeneity that are important to the persistence of measles. (b) Large-scale heterogeneity, as represented by metapopulation models: a town or city can be separated into several patches, each with its own dynamics but experiencing weak coupling with the other patches. In the figure, the height of a patch is proportional to the number of cases within it. (¢) Local-scale heterogeneity, as modelled by cellular automata or correlation equations: at this scale, the system is composed of individuals, each of which is in one of several states (S, E, I or R). (dJ Heterogeneity between individuals is achieved when the times spent exposed or infectious are explicitly modelled. The graph shows, for two very different assumptions, the probability that an individual has not yet recovered. For the usual assumption of constant transition rates (red), there is an unrealistic exponential decay, with a few individuals remaining infectious for a very long time. The assumption of constant periods (blue) leads to greater persistence of the disease.

density of both infectious and susceptible individuals across the community. The simplest way of incorporating heterogeneities is to separate the population into smaller subpopulations, each with its own independent dynamics but including some transmission or coupling between them 6,13-1s (Fig. 2b). This is termed a metapopulation or patch model. As the subpopulation dynamics can become desynchronized, it is likely that when some patches have faded out, measles will remain in others and these will continue to spread the disease, preventing global fade-outs of infection (Fig. 3). Metapopulations have proved an invaluable tool for introducing large-scale heterogeneities into epidemiological models. They have found favour as a simple modification of the standard models; however, there are difficulties with estimating the necessary parameters, such as the size of the subpopulations and the degree of coupling between them. Although heterogeneities exist at all scales, there is sufficient coupling within the school population (where most infections occur) that each school can be assumed to behave homogeneously.

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This still leaw.'s the transmission between subpopulations to be estimated without any obvious method of calculation from the data. If the coupling is too high, the epidemics become synchronized and heterogeneity is lost, whereas if the coupling is too weak, transmission of infection between subpopulations is rare and the disease rapidly dies out. Without detailed data at a variety of spatial scales, striking the correct balance is very difficult. One area where the use of metapopulations seems imperative is predicting the CCS after vaccination. If, for example, 60% of the population is vaccinated, so that the number of cases is reduced, a greater number of fade-outs and a larger CCS would be expected. However, this is not the case: there was little or no change in the CCS for England and Wales until vaccination levels reached 90 % in the late 1980s. This phenomenon is attributable to the greater heterogeneity of infection between and within communities after vaccination 16. This scale of heterogeneity is best captured by the metapopulation approach, which is a simple and informative

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Time (years) Rg. 3. (a) Twenty subpopulations, each of 25 000 individuals, are weakly coupled together to form a metapopulation version of the RAS (realistic age-structured) model. Individually, the subpopulations experience a high level of fade-outs (represented by black dots), displaying type I behaviour. (b) However, as the epidemics in the subpopulations become desynchronized, the aggregated population of 500 000 individuals persists, demonstrating that the critical community size can be reduced to the observed level using metapopulation models.

means of incorporating spatial structure. However, heterogeneities are also present at much smaller scales, such as at the family level, and these are best captured by modelling the behaviour of individuals. Local heterogeneities and spatial correlations

Although large-scale heterogeneities and metapopulation models have long been considered in epidemiology, it is only the recent advent of powerful computers that has allowed the study of local-scale heterogeneity. Work in this field has been driven by results and phen-

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omena from cellular automata models 17-19,where space is divided into a grid of sites (Fig. 4a). In these caricature models, each site corresponds to one individual, who has only a few surrounding neighbours (usually four or eight) through whom the infection can spread. This limits the growth of the infection and means that an infectious individual can significantly affect its environment by rapidly infecting all neighbouring susceptible individuals 2°. Un:Fortunately, cellular automata models are notoriously complex to analyse and many of their underlying assumptions are somewhat artificial. A recent, more tractable, development, which captures some of the spatial structure, is the use of pair-wise models21,22.These represent pairs of individuals in a network. There is assumed to be contact between the two individuals comprising a pair and, hence, the possibility of transmitting infection between them (Fig. 4b). This approach calculates and incorporates the local correlations, whereas the standard models, which just consider the number of individuals, assume that correlations are absent. This formulation can be compared with work on sexually transmitted diseases, where transmission through sexua]L partnerships is modelled 2. Pair-.wise models have successfully described the departure of toy spatial models from their non-spatial (mean field) counterparts; however, it is only recently that they have been used in their own right for disease

systems23, 24.

Both cellular automata and pairwise models consider interactions at an individual level: hence, they require parameters derived from detailed, fine-scale data rather than from the global dynamics. Models with local-scale heterogeneities have effectively modified the standard mass-action form for transmission rates by including strong local correlations. This limits the rapid growth phase of an epidemic and hence controls the subsequent inevitable cr~Lsh, increasing the level of persistence (Fig. 4c). Local-scale models are of most use in the study of small communities 24,2s and in examining the effect of family size and structure, as they do not easily capture the large-scale heterogeneities of larger communities. The impact of family structure can be seen in the

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T i m e (years) Rg. 4. (a) A probabilistic cellular automata model of the spread of a disease, showing the early stages in the spread of measles, based on the SEIR (susceptible, exposed, infectious or recovered) model. The sites are colour coded green for susceptible (S), red for exposed (E), yellow for infectious (I) and grey for recovered (R) individuals. There is a high degree of aggregation within classes as the disease advances as a wave. (b) An idealized network, as used in the pair-wise models. The spread of infection from a site at the bottom right clearly forms strong local correlations between connected pairs. (c) A typical example of dynamics from an SEIR cellular automaton model on a lattice with a population of 500 000. This example also shows persistence at the observed population level.

England and Wales data, where cities with larger than average family sizes and higher birth rates demonstrate a tendency for annual, as opposed to biennial, measles epidemics 7. In developing countries, where birth rates are even higher and families are even larger, the need to consider local-scale dynamics is even more important. Another level of biological realism can be achieved by considering the heterogeneities within individuals in the infectious classes. Distribution of latent and infectious periods One of the most recent advances in predicting the correct level of fade-out is obtained by adding more biological realism to the assumptions underlying the standard SEIR and RAS models. These models assume that individuals move from the exposed to the infectious class and then from the infectious to the recovered class at a constant rate 2 (Fig. 2a). Mathematically, this is the simplest assumption and is a reasonable approximation in deterministic models for large populations. However, if we track the history of individuals, this simple assumption generates an exponential distribution for the time spent being infectious 2 (Fig. 2d). This in turn means that there is large variation in the number of sec-

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ondary cases produced, which is not biologically realistic. In fact, the classical work of Hope Simpson 26and Bailey27has shown that the times spent in the exposed and infectious classes have small variances and are approximately normally distributed. A recent development 2s, which allows for the norreal distribution of exposed and infectious periods, has been shown to reduce the CCS to -500 000. Inclucling these more discrete periods is another form of heterogeneity, as hosts behave differently depending on the amount of time that has elapsed since being infected. The use of more-discrete periods, rather than constant rates, has significant implications to many other fields in which stochasticity is important. At low population levels, using more discrete periods will often promote the stability of the stochastic system in question, as the variance at the level of the individual has been reduced. Conclusions To date, the use of more realistic distributions for incubation and infectious periods has provided the simplest method of obtaining an accurate CCS for measles. However, the use of both metapopulation models and

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London Ser. B 348, 309-320 7 Grenfell, B.T. (1992)J. R. Stat. Soc. B 54, 383-398 8 Grenfell, B.T. and Dobson, A.P. (1995) Ecology of Infectious Diseases in Natural Populations, Cambridge UniversityPress 90lsen, L.F. and Schaffer,W.M. (1990) Science249, 499-504 10 Bolker, B.M. (1993) IMA J. Math. Appl. Med. Biol. 10, 83-95 11 Kareiva, P. (1990) Philos. Trans. R. Soc. London Ser. B 330, 175-190 12 Grenfell, B.T. and Harwood, J. (1997) Trends Ecol. Evol. (in press) 13 Grenfell,B.T., Bolker, B.M. and Kleczkowski,A. (1995) Proc. R. Soc. London Ser. B 259, 97-103 14 Lloyd, A.L. and May, R.M. (1996)J. Theor. Biol. 179, 1-11 15 Ferguson, N.M., Anderson, R.M. and May, R.M. (1997) in

Questions for future research • Why do moderate vaccination levels (60%) have little effect on the critical community size? • What role does family structure play? • How can these models be extended to developing countries? • What can we determine about optimal vaccination strategies? • Can we apply these techniques to other diseases?

local-scale heterogeneities will be important if we are to understand the full range of dynamics. It is only once we have successfully modelled the persistence of measles before vaccination - when there are large data sets available - that we can begin to consider the more complex problem of persistence during vaccination. The study of disease by realistic spatial, stochastic models is still in its infancy: combining the.se techniques with reliable data is likely to be the only way that epidemiologists can answer many of the biologically important questions in this area.

Spatial Ecology: The Role of Space in Population Dynamics and Interspecific Interactions (Tilman, D. and Kareiva, P., eds),

Acknowledgements This research was supported by the WellcomeTrust. I am indebted to Bryan Grenfell for his comments and advice. References 1 McLean, A.R. and Anderson, R.M. (1988) Epidemiol. Infect. 100, 111-133 2 Anderson, R.M. and May, R.M. (1992) Infectious Diseases of Humans, Oxford UniversityPress 3 Roberts, M.G. (1995)Parasitol. Today 11,172-177 4 Bartlett, M.S. (1957)J. R. Star. Soc. A 120, 48-70 5 Schenzle,D. (1984) IMA J. Math. Appl. Med. Biol. 1,169-191 6 Bolker, B.M. and Grenfell, B.T. (1995) Philos. Trans. R. Soc.

Princeton UniversityPress 16 Bolker, B.M. and Grenfell, B.T. (1996) Proc. Natl. Acad. Sci. U. S. A. 93, 12648-12653 17 Durrett, R. (1988) Math. Intell. 10, 37-4. 7 18 Johansen, A. (1996)J. Theor. Biol. 178, 45-51 19 Rhodes, C.J. and Anderson, R.M. (1996)or. Theor. Biol. 180, 125-133 20 Rand, D.A., Keeling,M.J. and Wilson, H.B. (1995) Proc. R. Soc. London Ser. B 259, 55-63 21 Levin, S.A. and Durrett, R. (1996) Proc. R. Soc. London Ser. B 351, 1615-1621 22 Sato, K., Matsuda, H. and Sasaki, A. (1994)J. Math. Biol. 32, 251-268 23 Altmann, M. (19951J. Math. Biol. 33, 661.-675 2,4 Keeling, M.J., Rand, D.A. and Morris, A.J. (1997) Proc. R. Soc. London Ser. B 264, 1149-1156 25 Rhodes, C.J. and Anderson, R.M. (1996) Philos. Trans. R. Soc. London Ser. B 351, 1679-1688 26 Hope Simpson, R.E. (1952) Lancet 2, 549--554 27 Bailey, N.J.T. (1956) Biomatrika 43, 15-22 28 Keeling, M.J. and Grenfell, B.T. (1997) Science275, 65-67

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lower eukaryotes, by S.A. Osmani and X.S. Ye

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