J. theor. Biol. (1992) 158, 495-515
Temporal and Spatial Scales in Epidemiological Concepts DAVID W. ONSTAD
University of Illinois and Illinois Natural History Survey, 607 East Peabody Drive, Champaign, IL 61820, U.S.A. (Received on 4 December 1991, Accepted in revisedform on 25 March 1992) Traditional concepts in epidemiology are reviewed from ecological, cultural, and logical perspectives. In zoological epidemiology (including the study of human and livestock diseases caused by pathogens), temporal and spatial scales are typically not used in definitions, by hypotheses, and theories concerning epidemic and endemic diseases. The same is true for botanical and theoretical epidemiology, although these two subdisciplines use a different definition of an epidemic than does zoological epidemiology. If hypotheses are to be tested and implemented, more precise concepts that include general temporal and spatial scales are needed. Criteria proposed here for identifying temporal and spatial scales are based on the need for consistency of observation and ecological validity. Consistency of observation depends upon the relative life cycles of the hosts and pathogens and upon the environmental effects that lead to stable or unstable population structures. Pathogens are dassified as absent, sporadic, or persistent (endemic). Epidemics can occur in the latter two cases but require a separate evaluation. A definition of an epidemic based on temporal and spatial scales and statistics is proposed for use by all subdisciplines. An epidemic occurs when an indicator variable reaches a statistically unusually high value due to transmission of a pathogen in an ecologically proper space-time unit. Threshold theorems in botanical and theoretical epidemiology are also discussed. These proposals do not directly affect modeling, but changes to hypotheses may influence model analyses. 1. Introduction
Recently a number of ecologists have expressed concern about the lack of temporal and spatial scales in ecological hypotheses (Levandowsky & White, 1977; Allen & Starr, 1982; O'Neill et al., 1986; Peters, 1988; Roughgarden et al., 1989; Weins, 1989). The absence of scales in ecological concepts in general came to my attention when I evaluated the concept of the host-density threshold (herd immunity) in epizootiology (Onstad et al., 1990) and reviewed a number of epizootiological models (Onstad & Carruthers, 1990). For the past 60 years, the threshold concept has remained vaguely worded, and epidemics, which the thresholds predict, have never been precisely defined using temporal and spatial scales. Without scales we do not know, for example, whether a given concept pertains to a square meter and a day or to a million square kilometers and a year. Vagueness and the lack of operational definitions are indications of immature theories (Loehle, 1987) and prevent us from evaluating predictions about the dynamics of pathogens in time and space. The goal of this paper is to provide a framework for creating precise operational definitions and criteria for identifying temporal and spatial scales. For the purposes 495
0022-5193/92/200495 +21 $08.00/0
© 1992 AcademicPress Limited
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of this paper, epidemiology is defined as the study of two or more interacting populations of which at least one is an infectious pathogen that is contagious in or vectored by the others. The formulation of concepts (e.g. theories, hypotheses, definitions) is my focus of attention, not empirical studies or mathematical modeling and predictive techniques. As Loehle (I 987) states, "The mere attempt to define phenomena operationally can dramatically increase theory maturity". The major questions of epidemiology have been posed, now we must identify the scales and variables that will permit theorems to address these questions. Model computation or analysis may require smaller units of time and space to ensure proper calculation of functions and stable results, but these computational units are not the conceptual units of interest here. Empiricists may measure processes at different scales or over a range of scales to identify a pathogen and its rates of infection and virulence, but it is assumed here that the pathogen and its processes are known and that ecology, not etiology, is the subject. The sections of the paper are presented in the following manner. First, the definitions and theories described in well-known texts of zoological, botanical and theoretical epidemiology are reviewed. Many papers are not cited because they do not differ from those cited or because they do not provide definitions. Second, epidemics from ecological and cultural perspectives are considered to evaluate whether, in theory, epidemics should be measured as processes or states. Third, criteria are offered for identifying general temporal and spatial scales that can be used in many epidemiological concepts concerning the dynamics of human, animal, and plant pathogens. The scales cannot be species specific, but they must be measurable and useful in prediction. Because scales are defined by their minimum units, these scales are based on the minimum units required for the analysis of community dynamics. To emphasize how scales can be identified and to demonstrate the problems resulting from inadequate scales, examples are presented of modeling and empirical studies. Then definitions are proposed that are based on temporal and spatial scales and that can be used in all subdisciplines of epidemiology. Finally, these scales, the proposed definitions, and the threshold concepts are discussed. Tentative solutions are offered to important theoretical problems and it is hoped that subsequent discussion and improvements by others will provide more complete and comprehensive solutions.
2. Review of Definitions and Theories 2.1. T H E O R E T I C A L
EP1DEMIOLOGY
For the past 60 years, theorists have attempted to describe the mechanisms leading to epidemics or to the persistence of infectious diseases in animal, particularly human, populations (Kermack & McKendrick, 1927; Bailey, 1975; Hethcote, 1976; Anderson & May, 1981; Black & Singer, 1987). Most theorists assume that the host population is homogeneous and can be considered constant except for losses due to the disease (i.e. removals). Kermack & McKendrick (1927) were the first to develop
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the concept of a host-density threshold and to show with a model that this threshold is a function of transmission, recovery, and virulence. They state that an epidemic can occur and continue as long as the host population remains above this threshold; below this threshold, the epidemic either cannot begin or ultimately dies out. The simplest formula for the host-density threshold is ( v / b ) < S or ( b S / v ) > 1, where S is the density of susceptible hosts, b is the disease transmission parameter (proportion of susceptibles becoming infected per day per infected host), and v is the proportion being removed (e.g. dying or recovering with immunity) per day (Bailey, 1975; Nokes & Anderson, 1988). When S = (v/b), the differential equation(s) for infected hosts equals zero (Kermack & McKendrick, 1927; Bailey, 1975). However, as Onstad et al. (1990) point out, theorists have defined the threshold in at least two other ways (e.g. persistence or epidemic occurrence) and have often used more than one defintion in a single report (e.g. May & Anderson, 1982; Becker, 1989). For some vector-borne or sexually transmitted diseases, density thresholds may not exist (Getz &Pickering, 1983). In general, theoretical epidemiologists imply that an epidemic is any growth or spread of disease. They also vaguely discuss recurrent epidemics and persistence of disease. Although some models include spatial heterogeneity or movement of organisms (Hethcote, 1976; Post et al., 1983; Moilison, 1986; Mollison & Kuulasmaa, 1985; Hethcote & van Ark, 1987; Travis & Lenhart, 1987), when a concept is created, it does not include either a spatial or a temporal scale. The failure to include a temporal scale may be due to the interest of theorists in equilibria and in the behavior of the differential equations. One approach, chain-binomial models, uses infectious period as an implicit temporal unit in the chain of infection and ignores latent period (Becker, 1989). An example of a theoretical paper that fails to include scales and precise definitions is the mathematical study of Post et al. (1983). This paper epitomizes the results of modeling studies both because it is similar to most and because the authors clearly state that their theorems are not model-dependent. Post et al. (1983) urge readers, as they should, to interpret and use their theorems for all situations identified in the theorems. (The same applies to any theorem developed from an empirical study.) Post et aL (1983) postulate that "A disease will become established in a spatially heterogeneous population if and only if -A22()~7/) is not an M-matrix". This theorem or hypothesis has at least two problems in interpretation. The first is the absence of scales. Without the temporal and spatial scales, we cannot use or test the predictive capabilities of this hypothesis. Because the hypothesis transcends or is independent of the model from which it was induced (as Post et al. suggest), the scales or units cannot be found in their mathematical model. The second problem is the lack of definition for the word established. The rest of their paper does not help with the interpretation, and the authors use the following four terms in similar contexts: established, maintained, endemic, and supported. Although these words may have the same meaning in the paper by Post et al., other authors have used the terms differently. Therefore, even though the modeling by Post et al. (1983) was good, the
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epidemiological community was left with a useless theorem because they followed the same vague course as almost all other theorists. 2.2. Z O O L O G I C A L
EPIDEMIOLOGY
Most of the literature in zoological epidemiology deals with diseases of humans and their domesticated animals. Almost all commonly used texts define an epidemic as the occurrence of a disease in a community or region at unusually high levels or in excess of normal expectancy based on past experience (Fox et al., 1970; MacMahon & Pugh, 1970; Benenson, 1975; Schwabe et al., 1977; Last, 1983). None of these definitions includes temporal scales, although Last (1983) states that seasons should be compared only across years and not in the same year. Sinnecker (1976) defines an epidemic as the accumulated occurrence of a disease with limitations in time and space; he also provides a tentative time scale in terms of the pathogen's latent period. An endemic disease, on the other hand, is restricted by space but not time. Sinnecker's third scenario, a pandemic, is limited only by time. Last (1983) and Benenson (1975) define the endemic state as the constant presence or usual prevalence of a disease in a given area. No author cited above provides a spatial scale for any concept. The literature concerning the epidemiology of wild animals is equally vague (Onstad & Carruthers, 1990). Fuxa & Tanada (1987) describe an epidemic as an unusually large number of cases of a disease at a given place and time period. Endemic disease is constantly present in a population. Fuxa & Tanada (1987), unlike the zoological epidemiologists cited above, briefly mention a threshold concept. 2.3. B O T A N I C A L E P I D E M I O L O G Y
Van der Plank (1963, 1975) was the first botanical epidemiologist to describe epidemic and endemic diseases according to the value of iR. This term is the length of the infectious period, i, multiplied by the number of germinating propagules produced by each infectious unit per time unit, R. Van der Plank's (1963) threshold theorem states that when iR > 1 a plant disease will increase and that such an increase in disease is an epidemic. Van der Plank (1975) also states that an endemic disease is one that is always present with iR = 1 on average. No scales were explicitly incorporated into any of the concepts created by Van der Plank (1963, 1975). Zadoks & Schein (1979) state that operational definitions should be simple, specific, consistent, and applicable to realistic situations. Butt & Royle (1980) also state that operational definitions are important but do not define the terms epidemic and endemic. According to Burdon & Chilvers (1982), the time span for prediction of disease dynamics must be included in epidemiological concepts and expressed in terms of a series of time units related to the generation times of the host and pathogen. Zadoks & Schein (1979) define an epidemic as an increase in disease in a population of plants in time and space. Hence, a small increase is a small epidemic. Zadoks & Schein (1979) also argue that an endemic disease is limited to a region with an average value of i R = 1 over the long term. Endemic diseases, therefore, can be epidemic within a portion of a region or epidemic over a season in an entire region.
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The smallest time unit for endemicity seems to be the growing season or the span of a single host generation. To identify an endemic disease using Zadoks & Schein's (1979) definition, one must consider history and experience; apparently that stipulation is not required to evaluate an epidemic. Heesterbeek & Zadoks (1987) define three types of epidemics: a zero-order epidemic, the spread of a disease within the boundaries of a field of host plants (i.e. focus expansion) during a single growing season; a first-order epidemic, the spread of a disease across many fields during one growing season ; and a second-order epidemic, the spread of a disease over many fields and several growing seasons. A field in these definitions refers to a set of homogeneous hosts distributed uniformly throughout space. The spatial scales for first- and second-order epidemics are unknown; for the zero-order epidemics, they are imprecise or not applicable. An historical context for evaluating epidemics is again missing, and an epidemic, therefore, is any increase, small or large. According to Burdon & Chilvers (1982), the time span for prediction of disease dynamics must be included in epidemiological concepts and expressed in terms of a series of time units related to the generation times of the host and pathogen. Several plant epidemiologists have discussed pathogen generation times. Gaumann (1950: 81, 156) stated that each link in an infection chain is a generation, which equals the length of a pathogen's life cycle. Van der Plank (1963: 41) defined the latent period as the generation time. Later, he discussed various operational definitions for the latent period (Van der Plank, 1975: 109). Zadoks & Schein (1979: 17) equated their infection cycle with Gaumann's infection chain and with a pathogen's generation time. They also provided an operational definition for infection cycle as the sum of the latent and infectious periods (Zadoks & Schein, 1979: 79). Gaumann (1950) and Zadoks & Schein (1979) believed that the population dynamics of a pathogen is a series of infection chains or cycles. 3. Epidemic as Both Process and State
Although special terms are used to discuss population growth and spread in time and space in various ecosystems (e.g. pest outbreaks, algal blooms), this practice is not universal even within the subject of parasitism. The term epidemic was originally used to describe situations that were unusually threatening to humans, their crops, and their domesticated animals (i.e. cultural concerns). Because high levels of disease were harmful, emphasis was placed on identifying those levels or states. In the study of parasitism of insects, particularly pest species, peak levels of parasitism may be recorded, but they are rarely given special attention. Two other characteristics, the mean level and the stability of the fluctuations about that mean, are the main interests of scientists (Hassell & Waage, 1984) for several reasons. First, peaks of parasitism are of no human value, as in the ease of natural systems, or are considered propitious, as in the case of pests. Second, if the host can be regulated on average far below the typical unregulated density, then peaks in the cycles of parasitism will be considered insignificant relative to the overall effect. Third, extensive analyses performed by theorists have focused on the stability of long-term equilibria under regulation or on
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the extinction of the parasitoid. Thus, with parasitism by parasitoids, two outcomes are typically discussed: stable coexistence and regulation by the parasite or extinction of the parasite. With parasitism by pathogens, three states are evaluated: endemic pathogen levels, epidemic pathogen levels, and pathogen extinction. Yet, the fluctuations of recurring epidemics and the long-term cycling of parasitism are indistinguishable (Fig. 1). From an ecological point of view, this similarity suggests that as states of the system epidemics p e r se are not important except as a part of the overall community dynamics. 100"/* ~ (a) Pathogen Epidemic ?
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100% II (b) Parasitoid
Stablecycles
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FIG. I. Comparison of temporal dynamics of communities consisting of (a) host and pathogen and (b) host and parasitoid. Although the dynamics are similar, the peaks and states are described differently.
In an ecological analysis of the relationships between a host and its pathogen, the pathogen should be viewed as an organism with interesting population dynamics. For the purposes of this paper, disease is the infection and resulting physiological effects on the host. Infection means that a host is inhabited by a viable pathogen that is in a latent or infectious stage. When a host is infected, it has an infection. When a host is diseased, it may be infected, it may be affected by post-infectious (removed) states of the pathogen, or it may be affected by residual by-products of the infection. Pathogen levels in a host population are often measured as the proportion or density of infected hosts. Many epidemiologists think of the overall spatial and temporal dynamics of a pathogen as an epidemic. This appreciation for the dynamics of pathogens does not
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require the identification of epidemics as processes nor does it eliminate the vagueness and imprecision mentioned in the introduction. In fact, this perspective is similar to my description of epidemiology as a subject. If we are to test hypotheses and implement theory, we cannot define an epidemic as a process that leads to high levels of pathogens because we are often unable to determine the starting point, even when levels are monotonically increasing. How should we reconcile (1) an ecological perspective which suggests that epidemics as high levels of pathogens are unimportant, (2) cultural concerns for these states, (3) the ecological interest in process, and (4) the logical requirement for a measurable, testable, and applicable variable? Although I am not an advocate of the excessive or unrestricted use of the term epidemic, I believe that defining it as a state of the ecological system in which pathogen levels are unusually high is the best choice for epidemiology. Certainly, ecology and process are part of the epidemiological perspective, but by themselves they do not address tradition, logic, and cultural problems. More will be said later about a definition of epidemics. 3.1. I N D I C A T O R (STATE) VARIABLES
To measure the state of an ecological system, one or more indicator variables are needed. This variable represents all infected hosts, including infectives (Kermack & McKendrick, 1927), or it represents the pathogen density. It must be measured consistently for an age class, plant organ, or age structure. If, for example, adults were measured in the past, then adults must be measured in the identification of an epidemic. Such consistency is especially important when age structure is unstable. For communities in which pathogen density can be directly measured (i.e. those with external plant pathogens), pathogen density may be a reliable indicator variable. Again, the measurement must be consistent for life stage of the pathogen. For most communities, however, an epidemic can be identified only by measuring prevalence (proportion of hosts infected) or density of infected hosts.
4. Identification of Temporal and Spatial Scales The analysis of certain theoretical models in ecology requires the implicit use of time scales, which may be inconsistent with the assumptions of other analysts (Allen & Starr, 1982: chapter 14). For example, in the analysis of the population dynamics of a single species, the temporal unit is typically the generation time (May, 1976). For hypotheses concerning the interactions of predator and prey (parasitoid and host), three major assumptions are usually made (Hassell, 1976; May, 1976): the temporal unit is assumed to be the generation time, both species are assumed to have the same generation time, and both species are assumed to be in synchrony and to occupy the same space. In more detailed studies of the functional response of predators to prey density (Hassell, 1976), time is identified as total time for predation, handling time, or searching time. The spatial unit is considered to be the area of discovery or search. Most hypotheses concerning interspecific competition ignore scales and assume that the system is in equilibrium without temporal or spatial
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variability (Pianka, 1976). Thus, almost all hypotheses regarding the dynamics of populations and communities lack spatial scales. Furthermore, ecologists are able to use a simple temporal scale in these cases because of their underlying assumptions about synchronies and equivalent generation times among species.
4.1. C__K)ALSAND REQUIREMENTS The identification of appropriate temporal and spatial scales is based on several goals. First, a general time unit that permits an analysis of community dynamics of the host and pathogen (epidemiology) is needed. Second, this time unit should account for changes due to the host-pathogen interaction, not changes due to migration, maturation, or normal births and deaths. Third, a general spatial unit is needed to permit analysis of community dynamics without migration interfering in analysis. Fourth, emphasis will be on minimum units, not maximum units, useful for analysis. Many of these goals and the following requirements will be discussed in more detail in later sections. Temporal and spatial units must correspond for logical reasons. Both units must be measurable. Units must account for behavior and longevity of both healthy and diseased hosts as well as the pathogen. Units must be ecologically relevant and correct or proper. Units must provide consistency in measurement of indicator variables across plant organs, age classes, or age structures. Units must be effective for both discrete and overlapping generations.
4.2. G E N E R A T I O N T I M E
Generation time (GT) is often used to conceptualize temporal dynamics of organisms. Three measures of GT are time to first reproduction or immature stage, life span, and mean or median age at reproduction. The length of the immature (prereproductive) stage is an adequate estimate of the fastest time between generations, but this time unit would not permit the inclusion of adult movement in the calculation of spatial unit. A host's life span certainly is a time unit that accounts for all movement, but life span (or pathogen infection cycle) is often too long for proper analysis of overlapping generations. A compromise between these two alternatives is the mean or median age at reproduction calculated from a cohort's age-specific life table. The advantage of this time unit is that it is related to the temporal distribution of offspring. It accounts for reproduction that is not accounted for in the immature stage and is useful for the analysis of overlapping generations. The disadvantage of mean or median age at reproduction is that it may not account for all movement by adults. If we assume that most movement occurs by the mean or median age, then this compromise is fine. This assumption is valid if survival of adults decreases greatly after this age or if most movement occurs at or before the beginning of the reproductive period. Of course, if the time and spatial units are based on the pathogen population, then movement is not an issue because it all occurs at the start of the immature (e.g. spore) stage.
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For discrete generations, mean or median GT is easily measured. For many hosts with discrete generations the time units will be a year. For overlapping generation, the mean age at reproduction is the mean calculated from the lxm~ distribution in the age-specific life table (Birch, 1948; Laughlin, 1965). This is the product of survival rate times reproductive rate distributed over the organism's ages x. The median time is the age at which the sum of (~mx equals half of its total value. The total is R0, the net reproductive rate calculated from a life table or study of fecundity. For symmetrical distributions of lxmx, the mean and median are the same. The median may be more appropriate than the mean because it may be easier to calculate and it is not sensitive to extreme values. The pathogen's GT is calculated from the distribution of infective units released from the host over time. It is the period between release from the mother pathogen or host and the mean or median time of the dispersal of infective units (offspring) from the host. This period is longer than the latent period, which is the minimum time to reproduction or release of infective units. Often the pathogen's generation time can be calculated from knowledge about the latent period, infectious period, and the reproductive rate per pathogen per day (Onstad, 1992).
4.3. T E M P O R A L SCALES
In most discussions of epidemiology, three main components of the ecological system are identified: host, pathogen, and environment. When host and pathogen interact, each is part of the other's environment. Here, the environment is considered to be the set of components that directly affect the pathogen or host. Often the environment, particularly climate, influences population structure. Changes in population structure (i.e. set of age classes or plant organs) that depend on seasonal change are called here the host's phenological cycle. Phenological cycle and GT may differ in such species as perennial plants that require several years to complete a generation but have seasonal changes in plant organs that repeat every year. It is assumed that host population structure and community dynamics are unaffected by environments that do not change over a very long period (>ten host generations) or that change quickly over short time intervals, such as diurnally. The identification of temporal scale depends on the influence of the environment on the organisms and on the relative durations of the life cycles of the pathogen and host (Table 1). Either hosts have unstable population structures because seasonal or other environmental changes and disturbances significantly affect the host, or they have stable population structures that are unaffected by their environments and pathogens. Humans are the best example of the latter group. Most hosts have unstable population structures; plants and poikilothermic animals are included in this group. These hosts can be further subdivided into three categories (column 1, Table 1): those with phenological cycles that occur more than once per generation, those with cycles that match the GT, and those host populations that are unstable but are not affected phenologically by climate. Many birds and mammals fit this third category (see fox example).
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Temporal scalesfor nine ecological scenarios consisting of a variety of host phenological cycles (pc) and generation times of the host (hgt) and pathogen (pgt) Host population structure and phenologyt
Minimum temporal unit based on relative duration of generation times:l:
A. Unstable population structure (1) pc
Temporal and Spatial Scales in Epidemiological Concepts DAVID W. ONSTAD
University of Illinois and Illinois Natural History Survey, 607 East Peabody Drive, Champaign, IL 61820, U.S.A. (Received on 4 December 1991, Accepted in revisedform on 25 March 1992) Traditional concepts in epidemiology are reviewed from ecological, cultural, and logical perspectives. In zoological epidemiology (including the study of human and livestock diseases caused by pathogens), temporal and spatial scales are typically not used in definitions, by hypotheses, and theories concerning epidemic and endemic diseases. The same is true for botanical and theoretical epidemiology, although these two subdisciplines use a different definition of an epidemic than does zoological epidemiology. If hypotheses are to be tested and implemented, more precise concepts that include general temporal and spatial scales are needed. Criteria proposed here for identifying temporal and spatial scales are based on the need for consistency of observation and ecological validity. Consistency of observation depends upon the relative life cycles of the hosts and pathogens and upon the environmental effects that lead to stable or unstable population structures. Pathogens are dassified as absent, sporadic, or persistent (endemic). Epidemics can occur in the latter two cases but require a separate evaluation. A definition of an epidemic based on temporal and spatial scales and statistics is proposed for use by all subdisciplines. An epidemic occurs when an indicator variable reaches a statistically unusually high value due to transmission of a pathogen in an ecologically proper space-time unit. Threshold theorems in botanical and theoretical epidemiology are also discussed. These proposals do not directly affect modeling, but changes to hypotheses may influence model analyses. 1. Introduction
Recently a number of ecologists have expressed concern about the lack of temporal and spatial scales in ecological hypotheses (Levandowsky & White, 1977; Allen & Starr, 1982; O'Neill et al., 1986; Peters, 1988; Roughgarden et al., 1989; Weins, 1989). The absence of scales in ecological concepts in general came to my attention when I evaluated the concept of the host-density threshold (herd immunity) in epizootiology (Onstad et al., 1990) and reviewed a number of epizootiological models (Onstad & Carruthers, 1990). For the past 60 years, the threshold concept has remained vaguely worded, and epidemics, which the thresholds predict, have never been precisely defined using temporal and spatial scales. Without scales we do not know, for example, whether a given concept pertains to a square meter and a day or to a million square kilometers and a year. Vagueness and the lack of operational definitions are indications of immature theories (Loehle, 1987) and prevent us from evaluating predictions about the dynamics of pathogens in time and space. The goal of this paper is to provide a framework for creating precise operational definitions and criteria for identifying temporal and spatial scales. For the purposes 495
0022-5193/92/200495 +21 $08.00/0
© 1992 AcademicPress Limited
496
D . W . ONSTAD
of this paper, epidemiology is defined as the study of two or more interacting populations of which at least one is an infectious pathogen that is contagious in or vectored by the others. The formulation of concepts (e.g. theories, hypotheses, definitions) is my focus of attention, not empirical studies or mathematical modeling and predictive techniques. As Loehle (I 987) states, "The mere attempt to define phenomena operationally can dramatically increase theory maturity". The major questions of epidemiology have been posed, now we must identify the scales and variables that will permit theorems to address these questions. Model computation or analysis may require smaller units of time and space to ensure proper calculation of functions and stable results, but these computational units are not the conceptual units of interest here. Empiricists may measure processes at different scales or over a range of scales to identify a pathogen and its rates of infection and virulence, but it is assumed here that the pathogen and its processes are known and that ecology, not etiology, is the subject. The sections of the paper are presented in the following manner. First, the definitions and theories described in well-known texts of zoological, botanical and theoretical epidemiology are reviewed. Many papers are not cited because they do not differ from those cited or because they do not provide definitions. Second, epidemics from ecological and cultural perspectives are considered to evaluate whether, in theory, epidemics should be measured as processes or states. Third, criteria are offered for identifying general temporal and spatial scales that can be used in many epidemiological concepts concerning the dynamics of human, animal, and plant pathogens. The scales cannot be species specific, but they must be measurable and useful in prediction. Because scales are defined by their minimum units, these scales are based on the minimum units required for the analysis of community dynamics. To emphasize how scales can be identified and to demonstrate the problems resulting from inadequate scales, examples are presented of modeling and empirical studies. Then definitions are proposed that are based on temporal and spatial scales and that can be used in all subdisciplines of epidemiology. Finally, these scales, the proposed definitions, and the threshold concepts are discussed. Tentative solutions are offered to important theoretical problems and it is hoped that subsequent discussion and improvements by others will provide more complete and comprehensive solutions.
2. Review of Definitions and Theories 2.1. T H E O R E T I C A L
EP1DEMIOLOGY
For the past 60 years, theorists have attempted to describe the mechanisms leading to epidemics or to the persistence of infectious diseases in animal, particularly human, populations (Kermack & McKendrick, 1927; Bailey, 1975; Hethcote, 1976; Anderson & May, 1981; Black & Singer, 1987). Most theorists assume that the host population is homogeneous and can be considered constant except for losses due to the disease (i.e. removals). Kermack & McKendrick (1927) were the first to develop
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the concept of a host-density threshold and to show with a model that this threshold is a function of transmission, recovery, and virulence. They state that an epidemic can occur and continue as long as the host population remains above this threshold; below this threshold, the epidemic either cannot begin or ultimately dies out. The simplest formula for the host-density threshold is ( v / b ) < S or ( b S / v ) > 1, where S is the density of susceptible hosts, b is the disease transmission parameter (proportion of susceptibles becoming infected per day per infected host), and v is the proportion being removed (e.g. dying or recovering with immunity) per day (Bailey, 1975; Nokes & Anderson, 1988). When S = (v/b), the differential equation(s) for infected hosts equals zero (Kermack & McKendrick, 1927; Bailey, 1975). However, as Onstad et al. (1990) point out, theorists have defined the threshold in at least two other ways (e.g. persistence or epidemic occurrence) and have often used more than one defintion in a single report (e.g. May & Anderson, 1982; Becker, 1989). For some vector-borne or sexually transmitted diseases, density thresholds may not exist (Getz &Pickering, 1983). In general, theoretical epidemiologists imply that an epidemic is any growth or spread of disease. They also vaguely discuss recurrent epidemics and persistence of disease. Although some models include spatial heterogeneity or movement of organisms (Hethcote, 1976; Post et al., 1983; Moilison, 1986; Mollison & Kuulasmaa, 1985; Hethcote & van Ark, 1987; Travis & Lenhart, 1987), when a concept is created, it does not include either a spatial or a temporal scale. The failure to include a temporal scale may be due to the interest of theorists in equilibria and in the behavior of the differential equations. One approach, chain-binomial models, uses infectious period as an implicit temporal unit in the chain of infection and ignores latent period (Becker, 1989). An example of a theoretical paper that fails to include scales and precise definitions is the mathematical study of Post et al. (1983). This paper epitomizes the results of modeling studies both because it is similar to most and because the authors clearly state that their theorems are not model-dependent. Post et al. (1983) urge readers, as they should, to interpret and use their theorems for all situations identified in the theorems. (The same applies to any theorem developed from an empirical study.) Post et aL (1983) postulate that "A disease will become established in a spatially heterogeneous population if and only if -A22()~7/) is not an M-matrix". This theorem or hypothesis has at least two problems in interpretation. The first is the absence of scales. Without the temporal and spatial scales, we cannot use or test the predictive capabilities of this hypothesis. Because the hypothesis transcends or is independent of the model from which it was induced (as Post et al. suggest), the scales or units cannot be found in their mathematical model. The second problem is the lack of definition for the word established. The rest of their paper does not help with the interpretation, and the authors use the following four terms in similar contexts: established, maintained, endemic, and supported. Although these words may have the same meaning in the paper by Post et al., other authors have used the terms differently. Therefore, even though the modeling by Post et al. (1983) was good, the
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epidemiological community was left with a useless theorem because they followed the same vague course as almost all other theorists. 2.2. Z O O L O G I C A L
EPIDEMIOLOGY
Most of the literature in zoological epidemiology deals with diseases of humans and their domesticated animals. Almost all commonly used texts define an epidemic as the occurrence of a disease in a community or region at unusually high levels or in excess of normal expectancy based on past experience (Fox et al., 1970; MacMahon & Pugh, 1970; Benenson, 1975; Schwabe et al., 1977; Last, 1983). None of these definitions includes temporal scales, although Last (1983) states that seasons should be compared only across years and not in the same year. Sinnecker (1976) defines an epidemic as the accumulated occurrence of a disease with limitations in time and space; he also provides a tentative time scale in terms of the pathogen's latent period. An endemic disease, on the other hand, is restricted by space but not time. Sinnecker's third scenario, a pandemic, is limited only by time. Last (1983) and Benenson (1975) define the endemic state as the constant presence or usual prevalence of a disease in a given area. No author cited above provides a spatial scale for any concept. The literature concerning the epidemiology of wild animals is equally vague (Onstad & Carruthers, 1990). Fuxa & Tanada (1987) describe an epidemic as an unusually large number of cases of a disease at a given place and time period. Endemic disease is constantly present in a population. Fuxa & Tanada (1987), unlike the zoological epidemiologists cited above, briefly mention a threshold concept. 2.3. B O T A N I C A L E P I D E M I O L O G Y
Van der Plank (1963, 1975) was the first botanical epidemiologist to describe epidemic and endemic diseases according to the value of iR. This term is the length of the infectious period, i, multiplied by the number of germinating propagules produced by each infectious unit per time unit, R. Van der Plank's (1963) threshold theorem states that when iR > 1 a plant disease will increase and that such an increase in disease is an epidemic. Van der Plank (1975) also states that an endemic disease is one that is always present with iR = 1 on average. No scales were explicitly incorporated into any of the concepts created by Van der Plank (1963, 1975). Zadoks & Schein (1979) state that operational definitions should be simple, specific, consistent, and applicable to realistic situations. Butt & Royle (1980) also state that operational definitions are important but do not define the terms epidemic and endemic. According to Burdon & Chilvers (1982), the time span for prediction of disease dynamics must be included in epidemiological concepts and expressed in terms of a series of time units related to the generation times of the host and pathogen. Zadoks & Schein (1979) define an epidemic as an increase in disease in a population of plants in time and space. Hence, a small increase is a small epidemic. Zadoks & Schein (1979) also argue that an endemic disease is limited to a region with an average value of i R = 1 over the long term. Endemic diseases, therefore, can be epidemic within a portion of a region or epidemic over a season in an entire region.
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The smallest time unit for endemicity seems to be the growing season or the span of a single host generation. To identify an endemic disease using Zadoks & Schein's (1979) definition, one must consider history and experience; apparently that stipulation is not required to evaluate an epidemic. Heesterbeek & Zadoks (1987) define three types of epidemics: a zero-order epidemic, the spread of a disease within the boundaries of a field of host plants (i.e. focus expansion) during a single growing season; a first-order epidemic, the spread of a disease across many fields during one growing season ; and a second-order epidemic, the spread of a disease over many fields and several growing seasons. A field in these definitions refers to a set of homogeneous hosts distributed uniformly throughout space. The spatial scales for first- and second-order epidemics are unknown; for the zero-order epidemics, they are imprecise or not applicable. An historical context for evaluating epidemics is again missing, and an epidemic, therefore, is any increase, small or large. According to Burdon & Chilvers (1982), the time span for prediction of disease dynamics must be included in epidemiological concepts and expressed in terms of a series of time units related to the generation times of the host and pathogen. Several plant epidemiologists have discussed pathogen generation times. Gaumann (1950: 81, 156) stated that each link in an infection chain is a generation, which equals the length of a pathogen's life cycle. Van der Plank (1963: 41) defined the latent period as the generation time. Later, he discussed various operational definitions for the latent period (Van der Plank, 1975: 109). Zadoks & Schein (1979: 17) equated their infection cycle with Gaumann's infection chain and with a pathogen's generation time. They also provided an operational definition for infection cycle as the sum of the latent and infectious periods (Zadoks & Schein, 1979: 79). Gaumann (1950) and Zadoks & Schein (1979) believed that the population dynamics of a pathogen is a series of infection chains or cycles. 3. Epidemic as Both Process and State
Although special terms are used to discuss population growth and spread in time and space in various ecosystems (e.g. pest outbreaks, algal blooms), this practice is not universal even within the subject of parasitism. The term epidemic was originally used to describe situations that were unusually threatening to humans, their crops, and their domesticated animals (i.e. cultural concerns). Because high levels of disease were harmful, emphasis was placed on identifying those levels or states. In the study of parasitism of insects, particularly pest species, peak levels of parasitism may be recorded, but they are rarely given special attention. Two other characteristics, the mean level and the stability of the fluctuations about that mean, are the main interests of scientists (Hassell & Waage, 1984) for several reasons. First, peaks of parasitism are of no human value, as in the ease of natural systems, or are considered propitious, as in the case of pests. Second, if the host can be regulated on average far below the typical unregulated density, then peaks in the cycles of parasitism will be considered insignificant relative to the overall effect. Third, extensive analyses performed by theorists have focused on the stability of long-term equilibria under regulation or on
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the extinction of the parasitoid. Thus, with parasitism by parasitoids, two outcomes are typically discussed: stable coexistence and regulation by the parasite or extinction of the parasite. With parasitism by pathogens, three states are evaluated: endemic pathogen levels, epidemic pathogen levels, and pathogen extinction. Yet, the fluctuations of recurring epidemics and the long-term cycling of parasitism are indistinguishable (Fig. 1). From an ecological point of view, this similarity suggests that as states of the system epidemics p e r se are not important except as a part of the overall community dynamics. 100"/* ~ (a) Pathogen Epidemic ?
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FIG. I. Comparison of temporal dynamics of communities consisting of (a) host and pathogen and (b) host and parasitoid. Although the dynamics are similar, the peaks and states are described differently.
In an ecological analysis of the relationships between a host and its pathogen, the pathogen should be viewed as an organism with interesting population dynamics. For the purposes of this paper, disease is the infection and resulting physiological effects on the host. Infection means that a host is inhabited by a viable pathogen that is in a latent or infectious stage. When a host is infected, it has an infection. When a host is diseased, it may be infected, it may be affected by post-infectious (removed) states of the pathogen, or it may be affected by residual by-products of the infection. Pathogen levels in a host population are often measured as the proportion or density of infected hosts. Many epidemiologists think of the overall spatial and temporal dynamics of a pathogen as an epidemic. This appreciation for the dynamics of pathogens does not
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require the identification of epidemics as processes nor does it eliminate the vagueness and imprecision mentioned in the introduction. In fact, this perspective is similar to my description of epidemiology as a subject. If we are to test hypotheses and implement theory, we cannot define an epidemic as a process that leads to high levels of pathogens because we are often unable to determine the starting point, even when levels are monotonically increasing. How should we reconcile (1) an ecological perspective which suggests that epidemics as high levels of pathogens are unimportant, (2) cultural concerns for these states, (3) the ecological interest in process, and (4) the logical requirement for a measurable, testable, and applicable variable? Although I am not an advocate of the excessive or unrestricted use of the term epidemic, I believe that defining it as a state of the ecological system in which pathogen levels are unusually high is the best choice for epidemiology. Certainly, ecology and process are part of the epidemiological perspective, but by themselves they do not address tradition, logic, and cultural problems. More will be said later about a definition of epidemics. 3.1. I N D I C A T O R (STATE) VARIABLES
To measure the state of an ecological system, one or more indicator variables are needed. This variable represents all infected hosts, including infectives (Kermack & McKendrick, 1927), or it represents the pathogen density. It must be measured consistently for an age class, plant organ, or age structure. If, for example, adults were measured in the past, then adults must be measured in the identification of an epidemic. Such consistency is especially important when age structure is unstable. For communities in which pathogen density can be directly measured (i.e. those with external plant pathogens), pathogen density may be a reliable indicator variable. Again, the measurement must be consistent for life stage of the pathogen. For most communities, however, an epidemic can be identified only by measuring prevalence (proportion of hosts infected) or density of infected hosts.
4. Identification of Temporal and Spatial Scales The analysis of certain theoretical models in ecology requires the implicit use of time scales, which may be inconsistent with the assumptions of other analysts (Allen & Starr, 1982: chapter 14). For example, in the analysis of the population dynamics of a single species, the temporal unit is typically the generation time (May, 1976). For hypotheses concerning the interactions of predator and prey (parasitoid and host), three major assumptions are usually made (Hassell, 1976; May, 1976): the temporal unit is assumed to be the generation time, both species are assumed to have the same generation time, and both species are assumed to be in synchrony and to occupy the same space. In more detailed studies of the functional response of predators to prey density (Hassell, 1976), time is identified as total time for predation, handling time, or searching time. The spatial unit is considered to be the area of discovery or search. Most hypotheses concerning interspecific competition ignore scales and assume that the system is in equilibrium without temporal or spatial
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variability (Pianka, 1976). Thus, almost all hypotheses regarding the dynamics of populations and communities lack spatial scales. Furthermore, ecologists are able to use a simple temporal scale in these cases because of their underlying assumptions about synchronies and equivalent generation times among species.
4.1. C__K)ALSAND REQUIREMENTS The identification of appropriate temporal and spatial scales is based on several goals. First, a general time unit that permits an analysis of community dynamics of the host and pathogen (epidemiology) is needed. Second, this time unit should account for changes due to the host-pathogen interaction, not changes due to migration, maturation, or normal births and deaths. Third, a general spatial unit is needed to permit analysis of community dynamics without migration interfering in analysis. Fourth, emphasis will be on minimum units, not maximum units, useful for analysis. Many of these goals and the following requirements will be discussed in more detail in later sections. Temporal and spatial units must correspond for logical reasons. Both units must be measurable. Units must account for behavior and longevity of both healthy and diseased hosts as well as the pathogen. Units must be ecologically relevant and correct or proper. Units must provide consistency in measurement of indicator variables across plant organs, age classes, or age structures. Units must be effective for both discrete and overlapping generations.
4.2. G E N E R A T I O N T I M E
Generation time (GT) is often used to conceptualize temporal dynamics of organisms. Three measures of GT are time to first reproduction or immature stage, life span, and mean or median age at reproduction. The length of the immature (prereproductive) stage is an adequate estimate of the fastest time between generations, but this time unit would not permit the inclusion of adult movement in the calculation of spatial unit. A host's life span certainly is a time unit that accounts for all movement, but life span (or pathogen infection cycle) is often too long for proper analysis of overlapping generations. A compromise between these two alternatives is the mean or median age at reproduction calculated from a cohort's age-specific life table. The advantage of this time unit is that it is related to the temporal distribution of offspring. It accounts for reproduction that is not accounted for in the immature stage and is useful for the analysis of overlapping generations. The disadvantage of mean or median age at reproduction is that it may not account for all movement by adults. If we assume that most movement occurs by the mean or median age, then this compromise is fine. This assumption is valid if survival of adults decreases greatly after this age or if most movement occurs at or before the beginning of the reproductive period. Of course, if the time and spatial units are based on the pathogen population, then movement is not an issue because it all occurs at the start of the immature (e.g. spore) stage.
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For discrete generations, mean or median GT is easily measured. For many hosts with discrete generations the time units will be a year. For overlapping generation, the mean age at reproduction is the mean calculated from the lxm~ distribution in the age-specific life table (Birch, 1948; Laughlin, 1965). This is the product of survival rate times reproductive rate distributed over the organism's ages x. The median time is the age at which the sum of (~mx equals half of its total value. The total is R0, the net reproductive rate calculated from a life table or study of fecundity. For symmetrical distributions of lxmx, the mean and median are the same. The median may be more appropriate than the mean because it may be easier to calculate and it is not sensitive to extreme values. The pathogen's GT is calculated from the distribution of infective units released from the host over time. It is the period between release from the mother pathogen or host and the mean or median time of the dispersal of infective units (offspring) from the host. This period is longer than the latent period, which is the minimum time to reproduction or release of infective units. Often the pathogen's generation time can be calculated from knowledge about the latent period, infectious period, and the reproductive rate per pathogen per day (Onstad, 1992).
4.3. T E M P O R A L SCALES
In most discussions of epidemiology, three main components of the ecological system are identified: host, pathogen, and environment. When host and pathogen interact, each is part of the other's environment. Here, the environment is considered to be the set of components that directly affect the pathogen or host. Often the environment, particularly climate, influences population structure. Changes in population structure (i.e. set of age classes or plant organs) that depend on seasonal change are called here the host's phenological cycle. Phenological cycle and GT may differ in such species as perennial plants that require several years to complete a generation but have seasonal changes in plant organs that repeat every year. It is assumed that host population structure and community dynamics are unaffected by environments that do not change over a very long period (>ten host generations) or that change quickly over short time intervals, such as diurnally. The identification of temporal scale depends on the influence of the environment on the organisms and on the relative durations of the life cycles of the pathogen and host (Table 1). Either hosts have unstable population structures because seasonal or other environmental changes and disturbances significantly affect the host, or they have stable population structures that are unaffected by their environments and pathogens. Humans are the best example of the latter group. Most hosts have unstable population structures; plants and poikilothermic animals are included in this group. These hosts can be further subdivided into three categories (column 1, Table 1): those with phenological cycles that occur more than once per generation, those with cycles that match the GT, and those host populations that are unstable but are not affected phenologically by climate. Many birds and mammals fit this third category (see fox example).
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Temporal scalesfor nine ecological scenarios consisting of a variety of host phenological cycles (pc) and generation times of the host (hgt) and pathogen (pgt) Host population structure and phenologyt
Minimum temporal unit based on relative duration of generation times:l:
A. Unstable population structure (1) pc
