Statistical modelling of measles and influenza outbreaks
Identifieur interne : 000F37 ( Istex/Corpus ); précédent : 000F36; suivant : 000F38Statistical modelling of measles and influenza outbreaks
Auteurs : Ad Cliff ; P. HaggettSource :
- Statistical Methods in Medical Research [ 0962-2802 ] ; 1993-03.
English descriptors
- Teeft :
- Absolute deviation, Accounting identities, Adequate contact, American journal, Asian influenza, Baroyan, Basic model, Cambridge university press, Chain binomial, Chain structure, Cliff, Cliff etal, Conventional view, Deterministic form, Different families, Dynamic analysis, Early part, Effective contact, Elveback model, Epidemic, Epidemic cycles, Epidemic peak, Epidemic peaks, Epidemic recurrences, Epidemic size, Epidemiological, Epidemiological modelling, Epidemiological research laboratory, Epidemiological research unit, Epidemiology, Etal, Exponential family, General practitioners, Graphs show, Haggett, Herd immunity, High probability, Historical geography, Hong kong, Icelandic data, Independent events, Infection, Infection rate, Infectious disease, Infectious diseases, Infectious period, Infectiousness, Infective, Infective population, Infectives, Influenza, Influenza data, Influenza epidemics, Influenza morbidity, Influenza outbreaks, Influenza peaks, Influenza spread, Influenza virus, Influenzal pneumonia, Large numbers, Latent period, Linear predictor, Link function, Main drawback, Mathematical biosciences, Mathematical statistics, Mathematical theory, Measles, Measles cases, Measles contacts, Measles epidemics, Measles epidemiology, Measles outbreaks, Measles virus, Medical district, Modelling, Morbidity, Open circles, Other areas, Outbreak, Parameter, Past levels, Population size, Process models, Random shocks, Reasonable projections, Recurrence, Recurrent epidemics, Reference region, Royal college, Schematic model, Serial interval, Shortest chain, Simple form, Simulation, Small epidemic, Soviet union, Spatial structure, Spatial systems, State models, Statistical modelling, Susceptible, Susceptible population, Susceptible populations, Susceptibles, Time period, Time periods, Time series, Time series models, Total community, Total number, Total population, Vaccination, Viral diseases, Viral infections, Virus, Virus isolations, World health organization.
Abstract
This paper reviews the application of statistical models to outbreaks of two common respiratory viral diseases, measles and influenza. For each disease, we look first at its epidemiological characteristics and assess the extent to which these either aid or hinder modelling. We then turn to the models that have been developed to simulate geographical spread. For measles, a distinction is drawn between process-based and time series models; for influenza, it is the scale of the communities (from small groups to global populations) which primarily determines modelling style. Applications are provided from work by the authors, largely using Icelandic data. Finally we consider the forecasting potential of the models described.
Url:
DOI: 10.1177/096228029300200104
Links to Exploration step
ISTEX:3ADFCD29C14C33D2D4E1480C368E4FFE7E8C30A9Le document en format XML
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<contrib-group><contrib contrib-type="author" xlink:type="simple"><name name-style="western"><surname>Cliff</surname><given-names>AD</given-names></name><aff>Department of Geography, University of Cambridge</aff></contrib></contrib-group>
<contrib-group><contrib contrib-type="author" xlink:type="simple"><name name-style="western"><surname>Haggett</surname><given-names>P.</given-names></name><aff>Department of Geography, University of Bristol</aff></contrib></contrib-group>
<pub-date pub-type="ppub"><month>03</month><year>1993</year></pub-date><volume>2</volume><issue>1</issue><fpage>43</fpage><lpage>73</lpage><abstract><p>This paper reviews the application of statistical models to outbreaks of two common respiratory viral diseases, measles and influenza. For each disease, we look first at its epidemiological characteristics and assess the extent to which these either aid or hinder modelling. We then turn to the models that have been developed to simulate geographical spread. For measles, a distinction is drawn between process-based and time series models; for influenza, it is the scale of the communities (from small groups to global populations) which primarily determines modelling style. Applications are provided from work by the authors, largely using Icelandic data. Finally we consider the forecasting potential of the models described.</p></abstract><custom-meta-wrap><custom-meta xlink:type="simple"><meta-name>sagemeta-type</meta-name><meta-value>Journal Article</meta-value></custom-meta><custom-meta xlink:type="simple"><meta-name>search-text</meta-name><meta-value>43 Statistical modelling of measles and influenza outbreaks SAGE Publications, Inc.1993DOI: 10.1177/096228029300200104 AD Cliff Department of Geography, University of Cambridge P. Haggett Department of Geography, University of Bristol Address for correspondence: AD Cliff, Christ's College, Cambridge CB2 3BU, UK. This paper reviews the application of statistical models to outbreaks of two common respiratory viral diseases, measles and influenza. For each disease, we look first at its epidemiological characteristics and assess the extent to which these either aid or hinder modelling. We then turn to the models that have been developed to simulate geographical spread. For measles, a distinction is drawn between process-based and time series models; for influenza, it is the scale of the communities (from small groups to global populations) which primarily determines modelling style. Applications are provided from work by the authors, largely using Icelandic data. Finally we consider the forecasting potential of the models described. 1 Introduction Although the history of medical statistics extends earlier, it is useful to start any review of epidemiological modelling at 1840 with William Farr's second Registrar General's report for England and Wales. Farr fitted a Normal curve to smoothed quarterly data on deaths from smallpox and announced his hope that empirical laws could be discovered underlying the waxing and waning of epidemics. As the century proceeded, so the foundations were laid for more detailed work: statistics on notifiable diseases began to be collected centrally, bacteriological research by Pasteur and Koch allowed an understanding of disease processes, and new branches of applied mathematics (notably probability theory and mathematical statistics) were evolved. In this paper, we review how far that process of epidemiological modelling has now progressed with respect to two common respiratory viral diseases, measles and influenza. Roughly equal proportions of the paper are devoted to each disease, but we note that it is upon measles that most modelling work has focused and where greater success has been recorded. Model applications are illustrated with reference to our own research on measles outbreaks' and influenza outbreaks' for the Icelandic population. We have arranged our account in terms of the different families of models used, and our historical references are therefore incidental. A useful chronological view of the advances since Farr's time is given in the opening chapter of Bailey's Mathematical theory of infectious diseases4 and in Anderson and May's Infectious diseases of humans.5 5 2 Nature of measles epidemiology Among the human infectious diseases, measles has historically played a leading role as a test disease for statistical modelling. Seven factors make it of special interest. a) Simplicity of the spread process Measles has a straightforward transmission mechanism which allows the virus to be passed from person to person without the need for an intermediate host or vector. Black' 5044 Figure 1 The infection process as a chain structure. Schematic model for measles contacts between an infective (i) and a susceptible ( j). Open circles denote onset of infection. In the shortest chain (B), infective i makes his contact on day 1 of his infectious period, and the susceptible j is latent for as short a time as possible. (Source: Cliff et al., Figure 3.4, p 43.2) describes it having the simplest epidemiological behaviour of all the major human viral diseases, while the World Health Organization has observed for measles that `... its almost invariable direct transmission, the relatively fixed duration of infectivity, the lasting immunity which it generally confers, have made it possible to lay the foundations of a statistical theory of epidemics' .' 7 Figure 1 shows the measles infection process as a chain structure. This gives a schematic model for measles contacts between an infective, i, and a susceptible, j. Open circles denote the onset of the infection. In the shortest chain (B), infective i makes contact on day 1 of the infectious period, and the susceptible j as 'latent' for as short a time as possible. The longest chain arises when i transmits on the last possible day of infection and j is latent for as long as possible (C). In the epidemiological literature, the average chain length, shown in (D), (that is, the average time between the observation of symptoms in one case and the observation of symptoms in a second case directly infected from the first) is referred to as the serial interval of the disease. For measles, this average is 14 days. b) Very high rates of infection ' Measles is highly contagious with substantial attack rates in an unvaccinated population. It generates, therefore, large numbers of cases over short time periods to give distinct epidemic events. The high attack rate is supported by the many reliable estimates in the literature of the proportion of a population that has contracted measles. In a 1925 paper, Chapin cites figures for several cities - Aberdeen, Scotland, 90-95%; Willesden, London, 93%; and Providence, Rhode Island, 96% - and concluded that'... 5145 it is probable that in England, Canada and the United States [the countries he studied] over 90% of urban populations contract measles at some time during their lives'.' With vaccination, this proportion has fallen. c) High probability of clinical recognition The response of the human body to a virus may range from no apparent clinical symptoms to the maximum clinical severity, death. The ratio of sub-clinical to clinical responses varies greatly from one infectious disease to another, with poliomyelitis at one extreme ( 1000 :1 ) and measles at the other ( 1: 99).9 Thus, in clinical terms, measles is a readily-recognizable disease with a low proportion of both misdiagnosed cases and of sub-clinical cases. d) Large numbers of recorded cases The combination of (b) and (c) leads to large numbers of measles cases, and some proportion of those feed through as reported cases of morbidity and mortality. For morbidity, in 1965, the World Health Organization recorded some 4 million reported cases, although the real number of cases was at least ten times larger. As late as 1990, and despite massive vaccination campaigns, some 2 million deaths were attributable to measles, While the precise level of mortality and morbidity will vary both historically and regionally, the very high incidence rates in comparison with other infectious diseases give greater confidence in the reported data than with low-incidence diseases.9 e) Distinctive endemic and epidemic wave patterns Reports of measles cases indicate that outbreaks occur in distinct waves, often with a marked periodicity. Figure 2 shows the time series of reported cases between 1945 and 1970 for four countries, arranged in decreasing order of population size. In the United States, with a population then of 210 millions, epidemic peaks arrive every year, and in Britain (56 million), every two years. Denmark (5 million) has a more complex pattern, with a tendency to a three-year cycle in the latter half of the period. Iceland (0.2 million) stands in contrast to the other countries in that only eight waves occurred in the 25-year period, and several years are without cases. Since countries rarely represent distinct demographic or epidemiological units, research has tended to concentrate upon more 'natural' population clusters. In a classic 1957 paper, Bartlett11 investigated the relationship between the periodicity of measles epidemics and population size for urban centres on both sides of the Atlantic. He found that the largest cities have an endemic pattern with periodic eruptions (Type I), while cities below a certain population size threshold (c. 250000) have an epidemic pattern with disease fade-out between. A distinction can be drawn between urban areas above about 10000 people with a regular pattern of epidemics (Type II) and those where occasional epidemics may be missed altogether, giving a more irregular pattern (Type III). f ) Spatial and temporal stability of the mealses virus Although written records of probable measles outbreaks go back to the Roman period, the causative agent was not identified until 1954 when Enders and Peebles isolated the agent as one of the myxoviruses. In addition to measles, the myxovirus group contains influenza, mumps, canine distemper and rinderpest. As far as present knowledge extends, the measles virus is not thought to undergo significant changes in structure and, unlike the influenza virus which has produced major mutations in a few decades, may be assumed to have stable diffusion parameters over the modern period. 5246 Figure 2 Reported cases of measles per month, 1945 to 1970, for four countries arranged in descending order of population size. Note the characteristic cyclicity in all cases, the dramatic reduction in amplitude for the United States series after 1964 (because of vaccination programmes), and the fact that only Iceland has clear non-endemicity. (Source: Cliff et al., Figure 3.1, p 39.2) g) Potential for eradication Like smallpox, the measles virus is theoretically eradicable. Although the cost of global eradication is currently prohibitive, the campaign in the United States by the Centers for Disease Control for the eradication of endemic measles has met with 5347 significant success.lo,i2 Study of the spatial structure of this particular disease is therefore likely to be usable in planning future eradication campaigns. ' ~ ~~ ~ 3 Process models of measles outbreaks ; i I ' The classic work on models for measles epidemics was undertaken by Hamer13 and Soper14 for the Hamer-Soper model, and by Reed, Frost and Greenwood on the chain binomial.ls,16 An important distinction exists between the two models in the level of data aggregation they employ. The Hamer-Soper model is a so-called mass action model and operates with population aggregates. The chain binomial was developed to study chains of measles infection within household groups, school classrooms and so on. 3.1 The Hamer-Soper models We set out here the theoretical structure of the model for a single region and then for a multiple area system. Single-region .. : _ -. At any time t, we assume that the total population in the region can be divided into three classes: namely the population at risk or susceptible population of size St, the infected population of size h, and the removed population of size Rt . The removed population consists of people who have had the disease, but who can no longer pass it on to others because of recovery, isolation on the appearance of overt disease symptoms, or death. Four types of transition are allowed (see Table 1): 1 A susceptible being infected by contact with an infective. 2 An infective being removed. We assumed that infection confers life-long immunity to further attack after recovery. 3 A susceptible 'birth'. This can arise either through a child growing up into the critical age range (that is, reaching about six months of age and not undergoing vaccination), or else through a susceptible entering the population by migration into the region. 4 An infective entering the I population by migration into the region. For simplicity, we assume that there is no out-migration. Suppose that transition i occurs at the rate ri (i = 1, 2, 3, 4); that is, in a small time interval (t, t + 8t ) the probability of transition 1 occurring is r2 8t + o (8t ), where o (8t ) means a term of smaller order than 8t. All events are assumed to be independent and to depend only on the present state of the population. The probability density of the time between any pair of successive transitions is Table 1 The Hamer-Soper model. Transition types and rates (Source: Cliff etal., p 160.2) 5448 where and the probability that the transition is of type i is We assume, in transitions 1-4, that the infection rate is proportional to the product, SI, that the removal rate is proportional to I (homogeneous mixing), and that the birth- and immigration-rates are constant. We can thus prepare Table 1. The model was first put forward in its deterministic form by Hamer and was studied extensively by Soper. Haggett17 and Cliff and Murray18 have considered the utility of the model in a regional setting. The main drawback of this deterministic formulation is that it leads to damping of successive epidemic waves which is not observed in practice (Figure 3). Soper erroneously believed that this damping could be eliminated by allowing for the fact that individuals exposed to measles undergo a latent period before becoming infectious - instead of assuming, as in the deterministic model described, that they transmit infectious material for the entire time between initial exposure and removal. Even with this change and modifications to allow for seasonal variations in infection rates and for spatial factors, reduced damping still persists. We therefore consider the more realistic stochastic formulation. Even this relatively simple model is surprisingly intractable analytically and, except in special cases such as v = ~ = 0, it is best studied using the Monte Carlo techniques. It is, however, possible to Figure 3 Deterministic form of the Hamer-Soper model for an artificial data set. Note the characteristic damping of the S and I curves to the levels Sd and Id respectively. (Source: Cliff et al., Figure 7.1, p 161 .2) 5549 see intuitively how the model operates. An infective is isolated after an average period of 1/jjL days, and, while infectious, he or she causes new infections at the rate of r3S per day. If we ignore the changes in S during this period, one infective infects an average number of r3S/J.L (= x, say) susceptibles before the infective is removed. From the theory of the simple birth and death process, we would expect that, when x ---- 1, a small epidemic would die out. However, when x > 1, a small epidemic will spark off a major outbreak although, of course, as the epidemic spreads S will fall and r3S/J.L can become less than unity. Thus, the general pattern will be that the susceptible population will build up (transition type 3) to around the critical population size S = )JL/(3, when the epidemic will continue to spread until the susceptible population falls sufficiently for the epidemic phase to pass (cf. Kermack and McKendrick's Threshold Theorem discussed in Kenda1.19 The cycle will then repeat itself to yield the recurrent epidemics illustrated in Figure 2. In large communities where measles is endemic, the period between epidemic peaks is of approximate length ~,/~3v, the mean time for the birth of J.L/r3 susceptibles. In smaller communities, where there is fade-out, the period is longer because, once the critical susceptible population size is reached, there is a delay until infection is reintroduced into the region.. _ Multi-region To adapt the model to handle regional interactions, the transitions given in Table 1 must be redefined. Each region i, in an n-region system has its own susceptible (S ), infectious (I ), and removed (R ) populations; that is, Of the parameters in Table 1, jjb depends on the nature of the disease and for measles can be taken to be regionally invariant. Inter-regional interaction is then handled by defining the transition rate of infection for region i, r3i, as This sum includes the term, r3iiSiIi, as in the single-region model, and covers the ith intra-regional interaction. The remaining terms in (5) handle interactions between region 1 and other regions j. Again, for simplicity, we assume that there is no migration between regions although, as we shall see later in section 6.3, there is no difficulty in adding migration to the model. At the aggregate level, distance-decay ideas imply that the probability of contact between a person in I and a person in S will diminish with increasing distance between them. A plausible first step in modelling the {r3ij} is to use the gravity formulation where dij is the distance between the (demographic) centroids of regions i and j. In equation (6), the diagonal terms {r3ij}, the within-region probability interaction rates, represent the proportion of the total population of region i with which a susceptible in region i makes contact. r3ii can be taken to be inversely proportional to the 5650 total population of region 1 by the rough argument that any given susceptible might be expected to have approximately the same size of acquaintance and kinship circle (that is, contact group at home, school, etc.), and to have the same level of risk of infection as a result, whatever the population size of the region. The importance of allowing for non-homogeneous mixing via varying contact parameters {r3ij} across geographical space (and also between different ethnic and social groups) is discussed in detail in Fox.2° This is the distinctive feature of the multi-region version of the Hamer-Soper model compared with its single region equivalent. An application is provided in section 7 of this paper. 3.2 Chain binomial models The chain binomial models, as originally developed, were designed to study intra- family transmission of infectious disease. Bailey' gives a full account, and Cliff and Ord21 have developed adaptations to model the spatial spread of measles. We suppose that it is possible to record the state of the system at times t = 0, 1, 2, ... Ideally, the time interval between recording points correspond in length to the serial interval of the disease. This ensures that we do not witness multiple epidemic cycles in a single time period, provided that each infected individual is isolated after the appearance of symptoms. The total population in a given medical district at the beginning of time period t, which we denote by Nt, will contain St susceptibles, h infectives who can transmit the disease to susceptibles, and Rt removals. During the rth time period, Nt may be modified by the addition of At arrivals (births and/or immigrants) and the loss of Dt departures (deaths and/or emigrants). Finally, let Xt denote the number of new cases which occur in time period t. The following accounting identities may then be written down for each district: If ottat denotes the number of individuals among the new arrivals who are infectives, while BtDt denotes the number of infectives among the departures, then As in the Hamer-Soper model, the arrivals term in (9) is required to resume spatial propagation of a disease into an area after local fade-out; these arrivals are the individuals who trigger a recurrence of the disease when the density of susceptibles is high. The model defined in equations (7)-(9) is shown schematically in Figure 4. In common with Bailey,22 we say that there has been adequate contact by time t between the jth susceptible, Sj, and the infectives, {Ii ~, if S~ contracts the disease during period t. The contact may have occurred one or more time periods earlier, depending upon the length of the serial interval and its relationship to the length of the data-recording interval. We now define the probability that Sj is identified as an infective during the time period t as 0'jt. One way of specifying 0'jt might be to assume the following: 1 the serial interval is relatively constant in length, and equal to the interval between 5751 Figure 4 Accounting identities for the typical medical district, i. For definitions, see text. (Source: Cliff et al., Figure 7.7, p 172.2) z observations, so that Sj is identified as being infected during period t if adequate contact was sustained during (t - 1); 2 the disease is transmitted only by adequate contact between a susceptible and an infective; 3 adequate contact with a single infective is sufficient to transmit the disease; 4 contacts between Sj and the different infectives are independent events; - 5852 5 there is homogeneous mixing between susceptibles and infectives within a district at some standard rate of contacts per unit time. Then O.~t = 1- p (no contact with any infectives) where 0~ = p (contact between h and S~ ). In equation (10) contacts between Sj and the different Ii are regarded as independent events [assumption (4)], and we need to consider only the infectives in circulation at time t [assumption ( 1 )] . The form of equation (10) is then given by assumption (3), with assumption (2) guaranteeing that the infection cannot arise from other courses. Finally, assumption (5) will allow simplification in particular cases. From (10), the probability generating function (pgf) for the number of new cases, Xt + 1, during period (t + 1) is given by Equation (11) represents a more general version of the Reed-Frost chain binomial model. The assumptions underlying the model will vary in plausibility from one application to the next, although assumption (1) is likely to be the most difficult to meet. We have shownlo how this can be relaxed by varying the length of the serial interval and by adopting a Poisson approximation. Multi-region extensions of the basic single region model outlined above are also considered by Evans.' An illustrative application of the chain binomial appears in section 6.2. .. 4 Time series models of measles outbreaks Given the rather regular incidence of measles epidemics over time shown in Figure 2, it is not surprising that epidemiologists have looked for time series regularities as a basis for modelling. In this section, we consider and illustrate the use of various kinds of time series of epidemic processes. 4.1 Box-Jenkins models . Model forms The basic philosophy underlying the so-called Box-Jenkins models is that the past behaviour of regional systems provides the key to how they will act in the future, and that this can be modelled using autoregressive and moving average components. Let xi, denote the value of a variate, X, in region i at time t. We index time periods earlier than t by t - k, and periods later than t by t + k. Let 1 denote the reference region and j other geographical units. The dependency of an area's behaviour upon what happened in that area in the past is referred to as an autoregressive component, and the dependency upon other areas as 5953 space-time covariances. In an extension of the Box-Jenkins notation proposed in Cliff et al. ,23 we may refer to such models as space-time autoregressive (STAR) models and write as a typical model. Here, (31 and r32 are parameters to be estimated, the fwijl are pre-specified, non-negative, structural weights indicating which areas, j, affect i, and the summation is over all areas j for which wij t- 0. We assume that there are n areas of interest in the space-time system. There is no reason why the temporal dependency should be restricted to a single past map, and equation (12) could be extended to earlier time periods (that is, time lags of order k = 2, 3, ...). As it stands, equation (12) is said to be of order one in time and n in space. It often happens that change over time cannot be accounted for solely in terms of systematic components such as those in equation (12), and we may envisage random shocks arriving in a region both from other areas from within itself on past maps. Thus, we might postulate a space-time random shocks, or moving average, model (STMA) as J where b and b2 are parameters, and the f e are random shocks with zero means, constant variances and zero intercorrelations. As before, the model could be extended to cover more than one time lag and any order of spatial interaction. Additionally, there is nothing to stop us combining equations (12) and (13) to define a general STARMA model. 24,2S The identification and estimation of STARMA models is described in Box and Jenkins26 and is illustrated in Cliff et al.2 2 '-. Model fitting To illustrate the fitting and performance of a STAR model, we applied the simple form to the number of reported cases of measles occurring in the capital city of Iceland, Reykjavik, in the epidemics of 1950-51 and 1958-59. The estimates of r31 were then used to generate the {Xt+ 1 }. Reported and modelled epidemic curves are illustrated in Figure 5. While reasonable correspondence is achieved between the fitted and actual sizes of the epidemics, the striking feature is that the estimated cases run one month in arrears. This is inherent in the model structure; model (14) can only respond to what happened in t at time t + 1, by which time the epidemic has either burgeoned further or begun to decline. Wide-ranging application of all the STARMA genre models to the Icelandic data showed that this limitation was persistent across the family of models. The reasons for this are to be found in the assumptions of the time series methods. They assume: (i) stationarity of the process generating the data series, and (ii) that the data come from a continuous distribution. One method of meeting (i) is by pre-whitening the data. However, as illustrated in Cliff et al.27 this approach generates other difficulties, notably loss of epidemiological `signal'.. 6054 Figure 5 Reported measles cases and one-month-ahead forecasts from AR(1 ) model for Reykjavik in the epidemics of 1950-51 and 1958-59. The graphs show reported cases against forecasts. Pecked line denotes coincidence of reported and forecast cases; letters indicate months to the wave. The graphs show vividly how the AR(1 ) model underestimates cases up to the epidemic peak and overestimates thereafter. (Source: Cliff etal., Figure 6.4, p 139.2 ) z . " .. Stationarity ' ~ .. " z Measles epidemic waves have been described by Bartlett2g as being quasi-stationary. They are stationary in the sense that the waves tend to repeat a basic shape at regular intervals - they have a cyclical wave form. Within any given epidemic, however, the process is clearly non-stationary; compare, for example, the build-up and fade-out phases in the number of cases reported in the epidemics graphed in Figures 2 and 5. To handle the changes between build-up and fade-out without producing the lags effects noted, models with time-varying parameters are required. It is also evident from Figure 2 that the underlying distribution is discrete in all but the largest geographical units, since long runs of months when zero cases are reported (inter-epidemic phases) frequently separate each epidemic. We now consider ways of adapting the time series models to match these data characteristics. 6155 4.2 Kalman filtering One obvious way of overcoming the lack of stationarity in the data is to formulate a STARMA model which is dynamic in its parameters; that is, the parameters of which can be changed and updated as fresh information arrives. The framework most commonly proposed for this updating uses the Kalman filter algorithm described by Harrison and Stevens,29 Martin3° and Bennett,25 and illustrated in Cliff et a1.2 A natural alternative to the continuous updating of model parameters is to develop mixed state models that handle the epidemic and non epidemic interludes of a time series separately. 4.3 Mixed state models Generalized linear model (GLM) Generalized linear models were proposed by Nelder and Wedderburn,31 where statistical details are given. We envisage a set of observations tyt 1, namely the number of cases of measles reported in a given medical district at time t, whose variability we wish to model. The yt } are assumed to have come from the exponential family of distributions. Secondly, we have a set of known variables (such as the size of the susceptible population or population at risk; the number of people who contracted measles in earlier time periods f xjt 1, which we are going to use to model the variability in the {Yt}. These known variables are combined to form a linear predictor, with the r3j either known or to be estimated. Then the model is that the expected value, ~,t , of y, is related to the linear predictor by a link function. The link function must be a specified monotonic differentiable function, so that It is this link function which can be modified between epidemic and non-epidemic phases to enable model switching to occur. Thus, a generalized linear model is where F is some exponential family of distributions, indexed by the values of the mean. The model is specified by stating the family of distributions, F, the link function, ho and the composition of the linear predictor. Models are fitted to data by iterative solution of the maximum likelihood equations. Since, in any epidemic, new infectives are created by contact between existing infectives, a simple random mixing argument as in the basic Hamer-Soper model of section 3.1 suggests the use of a Poisson model with a linear link function. In existing notation, we might use to model the number of new cases in a geographical area as a function of (i) past levels of infection in the area (term in a), (ii) mixing between infectives and susceptibles within the area (term in r3) and (iii) contact between susceptibles within the area and infectives 6256 in other areas (inter-regional transmission, term in y). In inter-epidemic phases, the model is reasonable, since this will track low levels of virtually constant activity. The models for the epidemic/inter-epidemic phases given in (18) and (19) may then be combined by the simple specification where K is the combined model, E denotes model (18) and NE model (19). Set 8 = 0 when an epidemic starts and 8 = 1 in inter-epidemic phases. Bayesian-entropy formulations The combined model (20) considers two different states of the generating process: state I: no epidemic; state II: epidemic, and we now put the approach within a Bayesian framework. Assume that, for a particular study area affected by an epidemic, we observe a Poisson process, In for the number of infectives at time t, with niean level Ot which depends upon past levels of the process. We suppose that, at any time t, the generating model is a random choice between two models (that is, two states) where: M11~: Model 1: (no epidemic); Ol = 0,; 0, is a small positive constant. (21) A,ft2): Model 2: (epidemic); Ot is a gamma-distributed random variable. (22) Equation (21) states that when there is no epidemic, the observations come from a Poisson process with a constant, low-valued rate (Ol = 0~)- This implies the assignment of a high probability to the occurrence of small-valued observations (depending on the selected value for 0~), and almost zero probability to the occurrence of high-values observations. On the other hand, with M12~ in (22), e, is a gamma-distributed random variable, and the model itself corresponds to the single-state Poisson-gamma BEF (Bayesian-entropy forecasting) model described in Souza.32 The various components of the model are as follows: ~ ~ = 1 initial (probably subjective) estimates for the parameters of the start time (time t = 0); 2 a probabilistic description of the state of the model at time (t -1 ) given Ht_ [the information in the system at time (t -1 )]; 3 an updating procedure for the description in (2) which take into account the new information on infectives, It; this implies that H,=(H,-i, It, plus any new subjective information). Given a complete specification for time (t - 1), we can evaluate the one-step- (generally, m-step-) ahead entropy-maximizing predictors for Ot and It . An application of the model to reported measles cases in Reykjavik in the 1950-51 1 epidemic, based on Cliff et al., pages 150-562, appears in Table 2. The results may be 6357 Table 2 Bayesian entropy forecasts for measles in Reykjavik, September 1951-November 1952. State probabilities and one step ahead forecasts _________ 1 E(lt+,1 Ht) __ 1 2 Pr~~ = 1 -Pt2~ - 3 Pt (k, j) = prob(M§i and Mi ¡) I Ht), j, k = 1,2; all omitted entries are zero to two decimal places (Source: Cliff etal., p 155 .2) compared with those in Figure 5 for the AR model. Table 2 shows that the start of the epidemic is correctly identified, but the slow fade-out over July-October leads to some uncertainty which is reflected in the state and joint probabilities. Nevertheless, the end of the epidemic is forecast for September, which is reasonable. The numerical forecasts are unconvincing, showing the 'one month behind' effect of the AR models (cf. Figure 5). 4.4 Summary Time series models are generally less effective than process-based models in modelling measles epidemics. The self-regulation built into the process models via the threshold effect is absent from time series approaches. Hence, unlike the Hamer-Soper/chain- binomial models, they follow events rather than swinging naturally through epidemic cycles in response to changes in the density of the susceptible population and the presence/absence of infection at any moment in time. 5 Nature of influenza epidemiology ° ' ' ' In the case of measles (see Section 2 of this paper) we proposed seven characteristics which made that disease well suited to statistical modelling. Influenza is a more complex disease which poses greater modelling problems. It differs from measles on four of the seven factors. 6458 5.1 Some contrasts with measles epidemiology a) Contrasts in virology While measles is caused by a single virus, influenza is caused by at least three: the influenza A virus was isolated in 1932, the B virus in 1940 and the C virus in 1949. Types A and B are both associated with major epidemics with type A generally more prevalent. The C virus appears to be endemic. In the case of the A virus, the virus antigens are unstable over time. Stuart-Harris and Schild33 describe infrequent but major changes termed shifts and more frequent but minor changes called drifts. When the A virus is involved, a change in the antigenic structure will permit it to attack both individuals previously immune by virture of an earlier attack to some other form of that virus, as well as individuals recently born and not exposed to any A virus. Epidemics may thus be more frequent than with the B virus. In modelling 'influenza', therefore, it is important whenever possible to be sue of the particular virus that is responsible. the different strains are categorized by a conventional WHO code based on (a) the antigenic type of nucleoprotein core, A, B, or C; (b) the host of origin if nonhuman; (c) the geographical place of first isolation; (d) the strain number; and (e) the year of isolation. b) Complexity of the influenza spread process While there is a uniform view about the way the measles virus is spread, there remains some controversy over the influenza contagion process. The conventional view is that influenza passes from person to person as droplets of respiratory secretions exhaled by an infected individual and inhaled by others. The potential for transmission will vary with three factors. First, infected individuals may differ in the amount of virus that they shed into the environment. Second, the proximity and degree of crowding of the susceptibles in the population will affect the transmission rate. Third, ambient conditions in the environment outside the host will bear upon virus viability and hence transmission. As Fine" observes, the second and third factors are often compounded because the winter period may bring together both crowding and low temperatures. As in the case of measles, the process of transfer is chainlike (cf. Figure 1). But influenza has a shorter (four days as against fourteen days) and more variable serial interval than measles. This, plus the higher proportion of subclinical cases, makes it more difficult to establish the existence of secondary waves of influenza outbreaks. The conventional view outlined above is that influenza is an infectious virus disease spread by person-to-person contact in basically the same way as measles. The high variability of the influenza virus and the many puzzling features in its epidemiology have led to variants on this view. The explosive simultaneous occurrence of influenza A in different locations, the relatively rare documentation of secondary cases within households, and the sudden appearance of influenza in isolated locations, has led Hope- Simpson35 to propose a latent virus hypothesis. The hypothesis assumes that the influenza virus persists in some form in the human host and is reactivated, possibly in a genetically changed form, by seasonal triggers in a subsequent influenza winter. There is also the possibility of transfer of the virus between animal and human populations (see Stuart-Harris and Schild, pp. 78-91).33 c) Problems in influenza records Measurement of the prevalence of influenza in the community has proved much more difficult than for those infectious diseases such as measles with more clear-cut clinical 6559 Figure 6 Surveillance of influenza from July 1967 to June 1975 by multiple sources. (a) Virus isolations for England, Wales, and Scotland reported to the Epidemiological Research Laboratory, Colindale. (b) Death certificates for England and Wales for influenza and influenzal pneumonia; those from all causes are given by the inset axis (70-270). (c) Family doctors' reports of influenza in England and Wales to the Epidemiological Research Unit, Royal College of General Practitioners. (d) New weekly sickness benefit claims for England, Wales and Scotland. (Source: Cliff etal., Figure 25, p 19.3) signs.37 Influenza may lead to illnesses that are too slight to warrant medical attention and thus escape the medical record, or the illness may present symptoms which, though severe enough to call for medical attention, can be confused with those produced by other agents. Many countries make no attempt to record the incidence of influenza. Figure 6 illustrates the range of sources from which the presence of influenza in a community is reconstructed. Four different indicators are plotted for England and Wales over a seven-year period 1967-1975.36 Virus isolations (Figure 6a) are relatively expensive and are performed for only a minutely small fraction of the infected population. Isolations of influenza A and B viruses reported to the Epidemiological Research Laboratory, Colindale, England are shown. Although the maximum number of isolates rarely exceeds 500 in any one week (about one for every 100000 people in the resident population of England, Wales and Scotland), the pattern of influenza outbreaks over time is very clearly shown. Figure 6b shows the total number of deaths in England and Wales from influenza and influenzal pneumonia over a seven-year period; the association with peaks in virus isolations in 1969-70, and to a lesser extent in 1967-68 and 1972-73, is clear. In the majority of countries, including the United Kingdom and the United States, where notification is not required, information on influenza morbidity has to be derived from other sources. In the former, two indirect measures of influenza morbidity have been developed. First, a panel of general practitioners in different geographical regions of England and Wales notify the occurrence of cases of acute respiratory illness in their practice to the Epidemiological Research Unit of the Royal College of General Practitioners. The weekly totals of such returns are plotted for a seven-year period in Figure 6c. They give a sensitive index of influenza prevalence and can be refined further 6660 ---.. by tests on throat swabs from cases diagnosed as influenza. Second, use is made of the weekly data on new claims for sickness benefit in the working population recorded in National Insurance returns. As Figure 6d shows, although the sickness benefit claims are for all forms of sickness, they show a consistent seasonal increase in the winter half of the year. Large epidemics such as that of the Hong Kong virus of 1969-70 show up as major peaks in the sickness returns. d) Potential for eradication Unlike measles, which has seen a substantial reduction in incidence since the start of mass vaccination in 1965, eradication of influenza is not under consideration. Because of its shifting virological status, vaccination against influenza is largely a protective measure to reduce risks to certain vulnerable groups (notably the old and young) within a population. 5.2 Parallels with measles epidemics Despite contrasts between the two diseases, there are features which encourage similar epidemiological modelling. Both produce very large numbers of recorded cases, especially under epidemic conditions, and both show a well-marked tendency to occur in strongly-peaked epidemic waves separated by periods of very low or nil recorded cases. Figure 7 shows a 75-year run of records for Iceland. Figure 7 Time series of reported cases of influenza, Iceland 1901-1974. (Source: Cliff et al., Figure 5.5, p 139.3) 6761 Influenza follows the typical seasonal pattern of many respiratory virus diseases in having a low summer incidence and a high winter incidence. This tendency to peak in colder months leads to a characteristic oscillation of influenza activity between the northern and southern hemispheres at approximately six-monthly intervals. The precise cause of the cold-season concentration is unknown but almost certainly involves host as well as environmental factors: that is, crowding as well as low temperatures. Examples of coincidence in the start of the 'influenza season' with the start of the school term given by Carey et al. 37 and Dunn et al.3g and of the coincidence of influenza peaks with exceptionally severe weather, illustrated by Semple,39 lend support to both factors. Over a longer time scale, the regular annual cycle of influenza peaks is interrupted by major pandemics at intervals of ten to forty years associated with shifts in the H and N antigens in the A virus. The conventional argument is that the antigenic shifts produce a virus which has not been met before by most members of a population (and certainly not by most children) and to which they have little or no antibody protection. Such pandemics involve a large proportion of the population, perhaps between one third or two thirds, and are followed by smaller annual epidemics involving drift variants of the current pandemic strain. A schematic model of this relationship developed by Kilbourne4° appears in Figure 8. , .j± 6 Models of influenza spread z As we noted earlier in this paper, attempts to model the progress of an infectious disease through the human community date back more than a century. For influenza, we look here at models in ascending spheres of operation from the level of the family, through the small community, to that of the large population. 6.1 Influenza spread within families Models of the spread of influenza within the family describe how successive cases occur in a chain of transmission. Families are useful as a focus of study because they comprise a well-defined group of individuals, share in part a common environment, and Figure 8 Kilbourne's model of the decline of severity of influenza epidemics in a postpandemic period. The broken line shows the rise in specific A2 antibody levels in the exposed population. Mutants unlike A2 will have survival advantages and eventually lead to a new A3 pandemic. (Source: redrawn from Kilbourne, Figure 10, p 480.40) 6862 Table 3 Example of the chain binomial model for a family of n = 4 ~~~ ~~ 1 Either a member of S, the susceptible population, or I, the infective population, or R, the recovered population 2 p, = prob(infection in week t) (Source: Cliff etal., p 30.3) have a high rate of internal contact which increases the probability of contagious infection. The models most often applied to infectious spread within the family are the chain binomials outlined in section 3.2 for measles. The detailed mathematical structure of the models as applied to influenza is given in Cliff et a1.,3 but the simple example in Table 3 shows their basic structure. Here, we assume, as before, that the population can be exhaustively partitioned into three groups, St, h and R, at time t. Suppose the family consists of n = 4 individuals and that at the start of the process all are members of S. In week 1, infection is introduced into the family, so that S1 = 3, 1, = 1. In weeks 2 and 3, infection spreads through the family. For convenience, we assume that an infective recovers after one week and can no longer pass on the disease to others. Let pt denote the probability that any given member of the family is infected in week t. Common forms for pt are ( 1 ) pt = p for all t, and (2) pt, 1 = 1- ( 1- p)Xt, where X denotes the number of new infectives in week t. By using (1) in Table 3, the row corresponding to week 2 gives the probabilities that the initial infective passes on the disease to one, two, or all remaining members of the family. For example, if p2 is the probability that an individual is infected in week 2, then ( 1- p2 ) is the probability he or she is not infected. Hence the probability that no new infectives occur in week 2 will be ( 1- p2 ) x ( 1- p2 ) x ( 1- p2 ) _ ( 1- p2 )3. Similar arguments lead to the other probabilities in Table 3, if we remember to allow for all the permutations of infectives and susceptibles that are possible in the family. At each stage, an individual can be in one of two possible states as far as the active transmission process is concerned, namely I with probability pt or S with probability (1- Pt) = q, say. Such a simple two-state process can be described by the binomial distribution and, at each step in Table 3, binomial probabilities are being multiplied together to define the overall state of the system; hence the name chain binomials used here and in Section 3.2. 6.2 Influenza spread within small communities Although models with a chain binomial structure can be successfully applied to communities much larger than the family,16 the equations become unwieldy and the probability of collecting accurate information on chain histories becomes low. As the 6963 population grows, the assumption of random mixing can be modified as before to take account of different contact rates within different segments of the population (for example, subdivided by age, location, occupation). One approach to this problem has been pioneered by a group of workers in the United States who have successfully adapted the chain binomial by using simulation techniques to describe the spread of influenza in a highly structured community. Their papers stretch back over a fifteen-year period and we describe below one of the later versions of their models; it will be termed for simplicity the Elveback-Fox mode1.41,42 The model examines a suburban American community of 1000 imhabitants. The population of the community is taken to be structured on an age basis into 1 140 preschool age children; 2 320 school age and adolescent children; _ 3 316 young adults; and 4 224 older adults. The individuals in the population are allowed to mix in particular ways. Thus the preschool children are assumed to be divided among thirty playgroups, each with two to six members. Neighbourhood and social mixing groups (called 'clusters') are also specified for the population subdivisions. Fifty such clusters are assumed to exist, each containing three to six different families. Figure 9 shows the structure of the community and the probability that an individual in one of the age groups ( 1 ~(4) above will have sufficient contact with an individual in one of their mixing groups (family, playgroup, cluster, school or total community) in the course of one day for transmission of the influenza virus to occur; these probabilities are given for Asian influenza (unbracketed numbers) and Hong Kong influenza (bracketed numbers). Thus Figure 9a shows that the total community consists of patterns of mixing involving the family, playgroups, school and clusters. If we take Asian influenza as an example, for a preschool child (Figure 9b) the probability of effective contact with an infected individual within the family setting is 0.02, within the playgroup is 0.10, and within the cluster is 0.005. By definition, preschool age children do not move within the school group and this empty set is denoted by the stippled box. Finally, the probability that a preschool age child will have adequate contact with someone not in one of the specified groups but in the community at large is 0.00025; this number is given below the four individual boxes in Figure 9b. Figures 9c-9e provide the same information for the three other population subdivisions. The probability values are based upon empirical data collected by the authors. The model keeps an account of the history of each individual in the community on a daily basis. For example, let us assume a young male adult, i, is part of a family that has no Asian influenza cases in it on the day in question, and that he belongs to a cluster that has two infected cases; finally suppose that there are fifty cases in the community at large. Then by making use of Figure 9d and assuming independence of infection from different sources, the probability that individual i is infected on that day may be written as ( 1- probability not infected), where p (not infected) = p (not infected in family) x p (not infected in cluster) x p (not infected in community) ,. ,L,. .,.._ .,_ 7064 Figure 9 Contact probabilities used in simulating the spread of influenza in a community. (a), structure of the community. (b) to (e), probability of effective contact per day between a member of one of the age groups and an infected individual within the mixing group indicated by comparison with diagram (a). Unbracketed values refer to Asian (H2N2) influenza and bracketed to Hong Kong (H3N2) influenza (given only if different from Asian influenza estimate). Stippled boxes are empty sets for the population sections concerned. (Source: Cliff et al., Figure 2.11, p 32.3 ) so that the probability of infection is 1- 0.9777 = 0.0223. The notation p f is used to denote the probability of infection in the family, I f the number of infectives in the family, and so on. The independence assumption underpinning the model will only be approximately true. Thus the probability that individual i is infected on the day is about 1 in 45. The simulation proceeds by first generating a random number to determine whether the individual is infected. If the individual is infected, other random numbers are used to determine (a) the length of the latent period, (b) the duration of infectiousness, (c) whether the illness is subclinical or clinical, and, if the latter, then (d) whether he is withdrawn from the population for bedrest at home. Should the initial result indicate no infection, then individual i remains in the population on the next day as a susceptible. Clearly since the generating process is stochastic, the pattern and spread of any single epidemic will reflect the particular random numbers in that run. But by repeating the process many times the general shape of any epidemic can be determined with confidence. Figure 10 illustrates the averaged results of 500 simulations of Asian influenza 7165 Figure 10 Frequency distribution of the total number of cases in epidemics in a structured population of 1000. The parameter values are those shown in Figure 9 and the distribution is based on 500 simulated Asian (H2N2) influenza and 200 Hong Kong (H3N2) influenza epidemics. (Source: Cliff et al., Figure 2.12, p 34.42) epidemics and 200 of Hong Kong influenza epidemics. It shows that the characteristics of the two virus types are distinctive. The Asian epidemic simulation displays a sharp peak at 500 cases but a narrow range; the Hong Kong epidemic simulation has a similar peak value but a larger degree of variability. The Elveback model allows a wide range of epidemiological situations to be explored. Part of a population can be protected by vaccination, playgroups or schools can be closed, 'superspreaders' can be introduced into the population, varying infectiousness assumed, and so on. Additional factors can readily be programmed into the model, and adaptations can be made to fit it to communities with different demographic or social characteristics. The simplicity of the model and its inherent flexibility give it great potential as an exploratory tool. The main drawback with the Elveback model is that its simulation structure means that a large number of runs (say 100 or more) have to be made before one can be confident about the frequency distributions of the predicted patterns. Computational problems are being overcome both by the steady increase in computer capacity and by the translation of the model into a deterministic form by Longini et <2/.~ 6.3 Influenza spread within large populations Modifications and restatements of the Hamer-Soper model to make it usable for influenza have come from a number of sources. These are summarized in Fine.34 Geographically one of the most interesting is the study of the spread of influenza between 100 large cities and 20 non-urban areas in the Soviet Union by Baroyan, Rvachev, and their associates. This work has been ongoing for some fifteen years; many of their publications are elusive and in Russian. English summaries appear in Baroyan et al.44,4s Spicer46 and Fine 34 give a full list of sources. To specify the model, we define the following notation for city i in an n city system: Mji the rate of migration to city i from some other city j ; Sit the number of susceptibles in city i at time t; Iit the number of infectives in city I at time t; Pi the population of city i ; 7266 (3i as in the aspatial Hamer-Soper model, denotes the infection parameter. Thus r3JPi is the infection rate, implying average contact proportional to Pi 1. In addition, T denotes time and g (T) is a function which describes the duration of infectiousness of cases. It is assumed that all groups in the population of a given city migrate at the same rate. These assumptions lead to a differential equation for the change in the number of susceptibles as a result of infection and net migration as where The other equations describe the change in the number of infectives due to net migration (superscript M ) and due to infection (superscript S ). The migration effect is given by and the conversion of susceptibles to infectives by Thus the change in infectives is Measurement and calibration of each of the elements in the Baroyan-Rvachev model did not prove simple. The transmission or diffusion parameter (3 was assumed to be specific to each city and nonurban area. It was made up of two components, namely: (1) a part which reflected the population structure of each area and therefore varied geographically, and (2) a second part which reflected the infectiousness of the particular influenza virus and therefore varied from outbreak to outbreak. Although different values of r3 are discussed in the Baroyan-Rvachev papers, it appears that in the operational runs of the model the value of r3 was estimated for the first city or n cities in which influenza appeared and, despite the above remarks, that this value was then used as a constant for all other (128 - n) cities and nonurban areas in the system. The duration of the infectiousness of influenza, g (T), was based on clinical information. Table 4 shows the distribution used by Baroyan et al. 4' The population values Pi 1, were directly determined from census data, but estimating the rate of migration, Mij) and {M~Z }, between pairs of cities posed greater problems. Early versions of the 7367 Table 4 Duration of infectiousness, g(T), as used in the Baroyan model (Source: Cliff etal., p 38.3) model used three values from a crude type of spatial interaction model as in equation (6). More recent versions have estimated daily migration rates from the official statistics of bus, rail and air travel and include a 128 x 128 matrix. Given the geographical size of Russia, it is not surprising to find that many cells in the matrix are zero or very small. Operation of the model has produced some interesting results. The early work of Rvachev concentrated on forecasting the timing, shape and duration of the Moscow 1965 epidemic from a model calibrated on the slightly earlier Leningrad epidemic. The work was later extended to include 19 other cities, and Figure 11 shows the results for a sample of six cities including Leningrad and Moscow. In each case, the graph shows the predicted morbidity (one-day-ahead forecasts) as a broken line compared with the observed prevalence (solid line). Prevalence is based on daily sickness absence claims. Since the six cities shown are only part of a large urban system, too much should not be deduced from Figure 11. Nevertheless, it does indicate a general geographical progression from Leningrad (onset 2 January) through Moscow (onset 12 January) to Prokopjevsk (onset 10 February). A general characteristic of this model is a tendency to overpredict cases in the early part of the epidemic and at the epidemic peak and to Figure 11 I Observed and predicted influenza morbidity in six cities in the Soviet Union, January to March 1965. The model was calibrated with the use of data from the early part of the Leningrad outbreak. The vectors indicate probable spread directions. (Source: redrawn from graphs in Baroyan et a/45; Cliff et al., Figure 2.14, p 40.3) 7468 underpredict in the later stages. How far this relates to the underreporting and overreporting believed to occur during the course of an epidemic is not clear. A further application by Rvachev and Logini47 extends the geographical range of the Soviet model to the global scale. They apply the model to the Hong Kong pandemic of 1968-69 with the use of parameters estimated from data for Hong Kong, the initial city to report the new strain of influenza A. Influenza data on the spread of the epidemic at the world scale were drawn from WHO records, and the spatial interaction matrix between world cities used the mean daily numbers of airline passengers recorded in official air-transport sources. Results show the comparison between the 32 cities for which dates of estimated and observed peaks are available. The mean absolute deviation between the two peaks for all cities is 16 days, with a maximum value of 53 days for the most remote city from Hong Kong (Santiago, Chile). The accuracy of the estimate is partly dependent on the latitude of the cities; the mean absolute deviation is 13 days for northern mid-latitude cities, 16 days for the tropical cities, and 25 days for the southern mid-latitude cities. From the forecasting viewpoint the model has a disappointing tendency to be tardy in its estimates of peaking. Some 25 of the cities were missed in the sense that the forecast peak occurred after the actual peak. In only seven of the cities were epidemics anticipated. The Baroyan-Rvachev model has attracted great interest among influenza modellers for its inclusion of a spatial structure, its use of migration data, and its apparent success in modelling the progress of the epidemic. Yet its basic equations retain a very simple form, little changed, from the original Hamer-Soper format. Much of the success of the Russian team may lie in their access to daily (rather than weekly or monthly) data, thus allowing more precise estimation of the crucial parameter, r3, than has been possible in other studies. It remains puzzling why the estimates based on a single city outbreak should apparently prove so stable when applied to all cities, of vastly different population sizes, in the system. 7 Forecasting epidemic waves , ~. ~.:. ~ Converting a general understanding of an epidemic spread process into a model with forecasting capability has long been a central epidemiological goal. We illustrate the general issues through an application of the Hamer-Soper model to Iceland." The basic approach is encapsulated in the inset of Figure 12, in which the future number of new infectives in a population is generated by the mixing of susceptibles with existing infectives. When the Hamer-Soper mixing parameter, r3, is modified to allow for the clinical history of the disease, vaccination rates and recovery rates, reasonable approximations to epidemic wave trains can be generated. In the case of Iceland, we were able to use our geographical knowledge of the contact system between areas implied by the movements of index cases to augment the basic model in two ways. First, a single, country-wide version could be replaced by one disaggregated into a series of local-area models. Each of these was able to incorporate the mixing parameter appropriate to the social structure of the local community being characterized. Second, these local models could then be linked to allow an exchange of susceptibles and infectives between adjacent areas. Clearly, what we mean by 'adjacent' will depend critically on the degree of spatial contact between places. 7569 a 0 V) -Ne co 2 0 CIJ r- C,4 Q 11 O~ ra q> *cD 0 u- E </meta-value> </custom-meta></custom-meta-wrap></article-meta></front><back><ref-list>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Farr W.</surname></name>
<source>Progress of epidemics. Second report of the registrar general of England and Wales</source>. <publisher-loc>London</publisher-loc>: <publisher-name>His Majesty's Stationery Office</publisher-name>, <year>1840</year>: <fpage>91</fpage>-<lpage>98</lpage>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Cliff AD</surname></name>, <name name-style="western"><surname>Haggett P.</surname></name>, <name name-style="western"><surname>Ord JK</surname></name>, <name name-style="western"><surname>Versey R.</surname></name>
<source>Spatial diffusion: an historical geography of epidemics in an island community</source> . <publisher-loc>Cambridge</publisher-loc>: <publisher-name>Cambridge University Press</publisher-name>, <year>1981</year>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Cliff AD</surname></name>, <name name-style="western"><surname>Haggett P.</surname></name>, <name name-style="western"><surname>Ord JK</surname></name>
<source>Spatial aspects of influenza epidemics</source>. <publisher-loc>London</publisher-loc>: <publisher-name>Pion</publisher-name>, <year>1986</year>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Bailey Ntj.</surname></name>
<source>The mathematical theory of infectious diseases</source>. <publisher-loc>London</publisher-loc> : <publisher-name>Griffin</publisher-name>, <year>1975</year>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Anderson RM</surname></name>, <name name-style="western"><surname>May RM</surname></name>
<source>Infectious diseases of humans</source>. <publisher-loc>Oxford</publisher-loc>: <publisher-name>Oxford University Press</publisher-name>, <year>1991</year>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Black FL</surname></name>
<source>Measles</source>. In: <name name-style="western"><surname>Evans AS</surname></name> ed. <source>Viral infections of humans: epidemiology and control</source>. <publisher-loc>New York</publisher-loc>: <publisher-name>Plenum Medical</publisher-name>, 1984: <fpage>397</fpage>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>World Health Organization.</surname></name>
<source>Epidemiological and vital statistics report</source>. <year>1952</year>; 5: <fpage>332</fpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Chapin CV</surname></name>
<article-title>Measles in Providence, R.I</article-title>. <source>American Journal of Hygiene</source>
<year>1925</year>; <volume>5</volume>: <fpage>635</fpage>-<lpage>55</lpage>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Evans AS</surname></name>
<source>Epidemiological concepts and methods</source>. In: <name name-style="western"><surname>Evans AS</surname></name> ed. <source>Viral infections of humans: epidemiology and control</source>. <publisher-loc>New York</publisher-loc>: <publisher-name>Plenum Medical</publisher-name> , <year>1984</year>: <fpage>20</fpage>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Cliff AD</surname></name>, <name name-style="western"><surname>Haggett P.</surname></name>, <name name-style="western"><surname>Smallman-Raynor M.</surname></name>
<source>Measles: the historical geography of a human viral disease, 1840-1990</source> . <publisher-loc>Oxford</publisher-loc>: <publisher-name>Blackwell</publisher-name>, <year>1993</year>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Bartlett MS</surname></name>
<article-title>Measles periodicity and community size</article-title>. <source>Journal of the Royal Statistical Society A</source>
<year>1957</year>; <volume>120</volume>: <fpage>48</fpage>-<lpage>70</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Hinman AR</surname></name>
<article-title>World eradication of measles</article-title>. <source>Review of Infectious Diseases</source>
<year>1982</year>; <volume>4</volume>: <fpage>933</fpage>-<lpage>39</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Hamer WH</surname></name>
<article-title>Epidemic diseases in England</article-title>. <source>Lancet</source>
<year>1906</year>; <volume>i</volume>: <fpage>733</fpage>-<lpage>39</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Soper HE</surname></name>
<article-title>Interpretation of periodicity in disease prevalence</article-title>. <source>Journal of the Royal Statistical Society A</source>
<year>1929</year>; <volume>92</volume>: <fpage>34</fpage>-<lpage>73</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Greenwood M.</surname></name>
<article-title>On the statistical measure of infectiousness</article-title> . <source>Journal of Hygiene</source>
<year>1931</year>; <volume>31</volume>: <fpage>336</fpage>-<lpage>51</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Abbey J.</surname></name>
<article-title>An examination of the Reed-Frost theory of epidemics</article-title>. <source>Human Biology</source>
<year>1952</year>; <volume>24</volume>: <fpage>201</fpage>-<lpage>203</lpage>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Haggett P.</surname></name>
<source>Regional and local components in elementary space-time models of contagious processes</source>. In: <name name-style="western"><surname>Carlstein T</surname></name>, <name name-style="western"><surname>Parkes DN</surname></name>, <name name-style="western"><surname>Thrift NJ</surname></name> eds. <source>Timing space and spacing time</source>, volume 3. <publisher-loc>London</publisher-loc>: <publisher-name>Arnold</publisher-name>, <year>1978</year>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Cliff AD</surname></name>, <name name-style="western"><surname>Murray GD</surname></name>
<article-title>A stochastic model for measles epidemics in a multi-region setting. Institute of British Geographers</article-title>, <source>Transactions New Series</source>
<year>1977</year>; <volume>2</volume>: <fpage>158</fpage>-<lpage>74</lpage>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Kendall DG</surname></name>
<source>La propagation d'une épidémie au d'un bruit dans une population limitée. Publications of</source>
<publisher-name>the Institute of Statistics</publisher-name>
<publisher-name>University of Paris</publisher-name>
<year>1957</year>; <fpage>6</fpage>: <fpage>307</fpage>-<lpage>11</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Fox JP</surname></name>
<article-title>Herd immunity and measles</article-title>. <source>Reviews of Infectious Diseases</source>
<year>1983</year>; <volume>5</volume>: <fpage>463</fpage>-<lpage>66</lpage>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Cliff AD</surname></name>, <name name-style="western"><surname>Ord JK</surname></name>
<source>Forecasting the progress of an epidemic</source>. In: <name name-style="western"><surname>Martin RL</surname></name>, <name name-style="western"><surname>Thrift NJ</surname></name>, <name name-style="western"><surname>Bennett RJ</surname></name> eds. <source>Towards the dynamic analysis of spatial systems</source>. <publisher-loc>London</publisher-loc>: <publisher-name>Pion</publisher-name> , <year>1978</year>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Bailey Njt.</surname></name>
<source>The elements of stochastic processes</source>. <publisher-loc>New York</publisher-loc> : <publisher-name>Wiley</publisher-name>, 1964: <fpage>183</fpage>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Cliff AD</surname></name>, <name name-style="western"><surname>Haggett P.</surname></name>, <name name-style="western"><surname>Ord JK</surname></name>, <name name-style="western"><surname>Bassett K.</surname></name>, <name name-style="western"><surname>Davies RB</surname></name>
<source>Elements of spatial structure: a quantitative approach</source>. <publisher-loc>London</publisher-loc>: <publisher-name>Cambridge University Press</publisher-name>, <year>1975</year>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Haggett P.</surname></name>, <name name-style="western"><surname>Cliff AD</surname></name>, <name name-style="western"><surname>Frey A.</surname></name>
<source>Locational analysis in human geography</source>, <edition>second</edition> edition. <publisher-loc>London</publisher-loc>: <publisher-name>Arnold</publisher-name>, <year>1977</year>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Bennett RJ</surname></name>
<source>Spatial time series</source>. <publisher-loc>London</publisher-loc>: <publisher-name>Pion</publisher-name> , <year>1979</year>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Box Gep</surname></name>, <name name-style="western"><surname>Jenkins GM</surname></name>
<source>Time series analysis, forecasting and control</source>. <publisher-loc>San Francisco</publisher-loc> : <publisher-name>Holden-Day</publisher-name>, <year>1970</year> .</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Cliff AD</surname></name>, <name name-style="western"><surname>Haggett P.</surname></name>, <name name-style="western"><surname>Stroup D.</surname></name>, <name name-style="western"><surname>Cheney E.</surname></name>
<article-title>The changing geographical coherence of measles morbidity in the United States, 1962-88</article-title>. <source>Statistics in Medicine</source>
<year>1992</year> ; <volume>11</volume>: <fpage>1409</fpage>-<lpage>24</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Bartlett MS</surname></name>
<article-title>Deterministic and stochastic models for recurrent epidemics</article-title>. <source>Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability</source>, <year>1956</year>; <volume>4</volume>: <fpage>81</fpage>-<lpage>108</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Harrison PJ</surname></name> , <name name-style="western"><surname>Stevens CF</surname></name>
<article-title>Bayesian forecasting</article-title>. <source>Journal of the Royal Statistical Society B</source>
<year>1976</year>; <volume>38</volume>: <fpage>205</fpage>-<lpage>47</lpage>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Martin RL</surname></name>
<source>Kalman filter modelling of time-varying processes in urban and regional analysis</source> . In <name name-style="western"><surname>Martin RL</surname></name>, <name name-style="western"><surname>Thrift NJ</surname></name>, <name name-style="western"><surname>Bennett RJ</surname></name> eds. <source>Towards the dynamic analysis of spatial systems</source>. <publisher-loc>London</publisher-loc> : <publisher-name>Pion</publisher-name>, <year>1978</year>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Nelder JA</surname></name>, <name name-style="western"><surname>Wedderburn Rwm.</surname></name>
<article-title>Generalised linear models</article-title>. <source>Journal of the Royal Statistical Society A</source>
<year>1972</year>; <volume>135</volume>: <fpage>370</fpage>-<lpage>84</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Souza RC</surname></name>
<article-title>The forecasting of epidemics' progress using a Bayesian-entropy framework</article-title> . <source>Environment and Planning</source>
<year>1982</year>; <volume>14</volume>: <fpage>49</fpage>-<lpage>60</lpage>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Stuart-Harris CH</surname></name>, <name name-style="western"><surname>Schild GC</surname></name>
<source>Influenza: the viruses and the disease</source>. <publisher-loc>London</publisher-loc>: <publisher-name>Arnold</publisher-name>, <year>1976</year>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Fine Pem.</surname></name>
<source>Applications of mathematical models to the epidemiology of influenza: a critique</source> . In: <source>Influenza models: prospects for development and use</source>. <publisher-loc>Lancaster</publisher-loc>: <publisher-name>MTP Press</publisher-name>, <year>1982</year>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Hope-Simpson RE</surname></name>
<article-title>Epidemic mechanisms of type A influenza</article-title>. <source>Journal of Hygiene</source> (<publisher-loc>Cambridge</publisher-loc>) <year>1979</year> ; <volume>83</volume>: <fpage>11</fpage>-<lpage>26</lpage>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Smith Jwg.</surname></name>
<source>Surveillance of influenza</source>. <publisher-loc>Colondale</publisher-loc>: <publisher-name>Public Health Laboratory Centre</publisher-name>, <year>1976</year>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Carey DE</surname></name>, <name name-style="western"><surname>Dunn FL</surname></name>, <name name-style="western"><surname>Robinson RQ</surname></name>, <name name-style="western"><surname>Jensen KE</surname></name>, <name name-style="western"><surname>Martin JD</surname></name>
<article-title>Community-wide epidemic of Asian strain influenza: clinical and subclinical illnesses among school children. Journal of the</article-title>
<source>American Medical Association</source>
<year>1958</year>; <volume>167</volume>: <fpage>1459</fpage>-<lpage>63</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Dunn FL</surname></name>, <name name-style="western"><surname>Carey DE</surname></name>, <name name-style="western"><surname>Cohen A.</surname></name>, <name name-style="western"><surname>Martin JD</surname></name>
<article-title>Epidemiological studies of Asian influenza in a Louisiana parish</article-title>. <source>American Journal of Hygiene</source>
<year>1959</year>; <volume>70</volume>: <fpage>351</fpage>-<lpage>71</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Semple AB</surname></name>
<article-title>Epidemiology of the influenza epidemic in Liverpool in 1950-51</article-title>. <source>Proceedings of the Royal Society of Medicine</source>
<year>1951</year> ; <volume>44</volume>: <fpage>794</fpage>-<lpage>96</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Kilbourne ED</surname></name>
<article-title>The molecular epidemiology of influenza</article-title>. <source>Journal of Infectious Diseases</source>
<year>1973</year>; <volume>127</volume>: <fpage>478</fpage>-<lpage>87</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Fox JP</surname></name>, <name name-style="western"><surname>Elveback I.</surname></name>, <name name-style="western"><surname>Scott W.</surname></name>, <name name-style="western"><surname>Gatewood I.</surname></name>, <name name-style="western"><surname>Ackerman E.</surname></name>
<article-title>Herd immunity: basic concept and relevance to public health immunization practices</article-title> . <source>American Journal of Epidemiology</source>
<year>1971</year>; <volume>94</volume>: <fpage>179</fpage>-<lpage>89</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Elveback LR</surname></name> , <name name-style="western"><surname>Fox JP</surname></name>, <name name-style="western"><surname>Ackerman E.</surname></name>, <name name-style="western"><surname>Langworthy A.</surname></name>, <name name-style="western"><surname>Boyd M.</surname></name>, <name name-style="western"><surname>Gatewood L.</surname></name>
<article-title>An influenza simulation model for immunization studies</article-title>. <source>American Journal of Epidemiology</source>
<year>1976</year>; <volume>103</volume>: <fpage>152</fpage>-<lpage>65</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Longini I.</surname></name>, <name name-style="western"><surname>Ackerman G.</surname></name>, <name name-style="western"><surname>Elveback LR</surname></name>
<article-title>An optimization model for influenza A epidemics</article-title>. <source>Mathematical Biosciences</source>
<year>1978</year>; <volume>38</volume>: <fpage>141</fpage>-<lpage>57</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Baroyan OV</surname></name>, <name name-style="western"><surname>Genchikov LA</surname></name>, <name name-style="western"><surname>Rvachev LA</surname></name>, <name name-style="western"><surname>Shashkov VA</surname></name>
<article-title>An attempt at large-scale influenza epidemic modelling by means of a computer</article-title> . <source>Bulletin of the International Epidemiological Association</source>
<year>1969</year>; <volume>18</volume>: <fpage>22</fpage>-<lpage>31</lpage>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Baroyan O.</surname></name>, <name name-style="western"><surname>Rvachev LA</surname></name>, <name name-style="western"><surname>Ivannikov YG</surname></name>
<source>Modelling and prediction of influenza epidemics in the USSR [Russian]</source> . <publisher-loc>Moscow</publisher-loc>: <publisher-name>Gamaleia Institute for Epidemiology and Microbiology</publisher-name> , <year>1977</year>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Spicer CC</surname></name>
<article-title>Mathematical modelling of influenza epidemics. British Medical</article-title>
<source>Bulletin</source>
<year>1979</year>; <volume>35</volume>: <fpage>23</fpage>-<lpage>28</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Rvachev LA</surname></name>, <name name-style="western"><surname>Longini IM</surname></name>
<article-title>A mathematical model for the global spread of influenza</article-title>. <source>Mathematical Biosciences</source>
<year>1985</year>; <volume>75</volume>: <fpage>3</fpage>-<lpage>22</lpage>.</citation>
</ref>
<ref>
<citation citation-type="book" xlink:type="simple">
<name name-style="western"><surname>Cliff AD</surname></name>, <name name-style="western"><surname>Haggett P.</surname></name>
<source>Atlas of disease distributions</source>. <publisher-loc>Oxford</publisher-loc>: <publisher-name>Blackwell Reference</publisher-name>, <year>1988</year>: <fpage>271</fpage>-<lpage>73</lpage>.</citation>
</ref>
<ref>
<citation citation-type="journal" xlink:type="simple">
<name name-style="western"><surname>Sugihara G.</surname></name> , <name name-style="western"><surname>May RM</surname></name>
<article-title>Nonlinear forecasting as a way of distinguishing chaos from measurement error in time</article-title> series. <source>Nature</source>
<year>1990</year>; <volume>344</volume>: <fpage>734</fpage>-<lpage>41</lpage>.</citation>
</ref>
</ref-list></back></article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="en"><title>Statistical modelling of measles and influenza outbreaks</title></titleInfo><titleInfo type="alternative" lang="en" contentType="CDATA"><title>Statistical modelling of measles and influenza outbreaks</title></titleInfo><name type="personal"><namePart type="given">AD</namePart><namePart type="family">Cliff</namePart><affiliation>Department of Geography, University of Cambridge</affiliation></name><name type="personal"><namePart type="given">P.</namePart><namePart type="family">Haggett</namePart><affiliation>Department of Geography, University of Bristol</affiliation></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="research-article" authority="ISTEX" authorityURI="https://content-type.data.istex.fr" valueURI="https://content-type.data.istex.fr/ark:/67375/XTP-1JC4F85T-7">research-article</genre><originInfo><publisher>Sage Publications</publisher><place><placeTerm type="text">Sage CA: Thousand Oaks, CA</placeTerm></place><dateIssued encoding="w3cdtf">1993-03</dateIssued><copyrightDate encoding="w3cdtf">1993</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><abstract lang="en">This paper reviews the application of statistical models to outbreaks of two common respiratory viral diseases, measles and influenza. For each disease, we look first at its epidemiological characteristics and assess the extent to which these either aid or hinder modelling. We then turn to the models that have been developed to simulate geographical spread. For measles, a distinction is drawn between process-based and time series models; for influenza, it is the scale of the communities (from small groups to global populations) which primarily determines modelling style. Applications are provided from work by the authors, largely using Icelandic data. Finally we consider the forecasting potential of the models described.</abstract><relatedItem type="host"><titleInfo><title>Statistical Methods in Medical Research</title></titleInfo><genre type="journal" authority="ISTEX" authorityURI="https://publication-type.data.istex.fr" valueURI="https://publication-type.data.istex.fr/ark:/67375/JMC-0GLKJH51-B">journal</genre><identifier type="ISSN">0962-2802</identifier><identifier type="eISSN">1477-0334</identifier><identifier type="PublisherID">SMM</identifier><identifier type="PublisherID-hwp">spsmm</identifier><part><date>1993</date><detail type="volume"><caption>vol.</caption><number>2</number></detail><detail type="issue"><caption>no.</caption><number>1</number></detail><extent unit="pages"><start>43</start><end>73</end></extent></part></relatedItem><identifier type="istex">3ADFCD29C14C33D2D4E1480C368E4FFE7E8C30A9</identifier><identifier type="ark">ark:/67375/M70-J7JWK6T0-M</identifier><identifier type="DOI">10.1177/096228029300200104</identifier><identifier type="ArticleID">10.1177_096228029300200104</identifier><recordInfo><recordContentSource authority="ISTEX" authorityURI="https://loaded-corpus.data.istex.fr" valueURI="https://loaded-corpus.data.istex.fr/ark:/67375/XBH-0J1N7DQT-B">sage</recordContentSource></recordInfo></mods><json:item><extension>json</extension><original>false</original><mimetype>application/json</mimetype><uri>https://api.istex.fr/ark:/67375/M70-J7JWK6T0-M/record.json</uri></json:item></metadata><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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