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Why apply ecological laws to epidemiology?
Opinion
Why apply ecological laws to epidemiology? Serge Morand1,2 and Boris Krasnov3 1 Institut des Sciences de l’Evolution, CNRS, De´partement Ge´ne´tique Environnement, CC065, Universite´ Montpellier 2, 34095, Montpellier cedex 05, France 2 UR AGIRs CIRAD, Campus International de Baillarguet, 34398 Montpellier cedex 05, France 3 Mitrani Department of Desert Ecology, Jacob Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede-Boqer Campus, 84990 Midreshet Ben-Gurion and Ramon Science Center, P.O. Box 194, 80600 Mizpe Ramon, Israel
Using ecological laws, or rules, is a useful strategy for epidemiological observations. The application of Taylor’s power law to epidemiology and evolutionary ecology of parasites is exemplified here. Taylor’s power law takes the form of s2 = amb, where s2 is the variance in population abundance, m is the mean abundance of the population, a represents a constant parameter and b represents an index of spatial heterogeneity. Although Taylor’s power law reflects the aggregation of parasite (or pathogen) individuals among host population, the values of b could reflect regulation processes in host– parasite systems. Illustrations are given showing how b value is linked to various epidemiological situations: pathogen emergence, the impact of vaccination or the level of host immune defence. The search for rules and laws The search for patterns, rules and laws has been central to ecology, from its foundation to the recent development of macroecology [1–3]. Kleiber’s law, Bergman’s rule and Rapoport’s rule are, among others, famous examples [4,5]. In epidemiology, a science that focuses primarily on risk analysis, the search for patterns does not have such weighty importance. However, the renewal of interest in studies on scaling in ecology has the correlative effect of reviving interest in epidemiology. We aim to show the application of Taylor’s power law to epidemiology and the evolutionary ecology of parasites. Taylor’s power law corresponds to a consistent and universal ecological observation that variance in abundance of a species increases with its mean abundance in accordance with a simple power law [6]. This relationship takes the form of s2 = amb, where s2 is the temporal or spatial variance in population abundance at a given geographical location, m is the temporal or spatial mean abundance of the population, a represents a constant parameter and b represents an index of spatial heterogeneity. Abundance and variance values are fitted to the power function, which gives estimates of a and b by using log transformation. The parameters a and b are not assumed to be the properties of individual local populations, but, rather, they describe patterns over a much larger spatial scale [7], which suggests the existence of constraints on Corresponding author: Morand, S. ([email protected]).
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the temporal variation in abundance of local populations [8]. Taylor’s power law has been observed in many free-living organisms [9,10] and in parasites [11,12]. The application of Taylor’s power relationship to parasitology and epidemiology has gained growing interest [11–13] (Figure 1a); one way that this can be seen is the use of comparative data in the predictions of epidemiological models [14–18] (Box 1). Taylor’s power law is a measure of aggregation of parasites within their hosts Aggregation commonly is observed in parasite individuals among host populations [19]. The aggregated distribution of parasites is caused by a variety of factors such as heterogeneities in host populations and/or infection pressure; the factors can act independently or together and might increase or decrease the observed level of parasite aggregation [20]. Although the mechanisms that cause aggregation still are debated [21], the consequences in terms of host–parasite regulation are better explored thanks to the epidemiological theory [22–24]. Low-level aggregation theoretically is associated with the destabilization of host–parasite population dynamics, whereas high-level aggregation is associated with stability [25]. However, although all parasites appear to be aggregated, significant differences are observed in the degree of aggregation between parasite species or populations; the causes of these differences are not well understood. There are several ways to measure parasite aggregation, the most familiar being the variance of abundance to mean abundance ratio, or the k value of the negative binomial distribution [26]. Aggregation also can be estimated by exponent b of Taylor’s power relationship, linking the variance of parasite abundance to the mean abundance [14,16,27]. The value of b is easy to estimate through regression analysis by using log transformation. The biological meaning of Taylor’s power law, where b < 2, is that more abundant populations of a species are less variable than would be expected from simple statistical properties (see, for example, Ref. [9]). Although early studies suggested that values of b could reflect regulation processes in host–parasite systems [28], it is only recently that more thorough investigations have been done [13].
1471-4922/$ – see front matter ß 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pt.2008.04.003
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Figure 1. Relationships between the logarithms of variance of abundance, V(m), and abundance, m, are fitted to a power function (a) in nematodes parasitizing mammals (reproduced, with permission, from Ref. [41]) and (b) case reports of measles in 366 communities in England and Wales. The data are subdivided between the prevaccination era (1944–1966, green dots), the 80% vaccination era (1980–1990, red dots) and the 90% vaccination era (1990–1997, blue dots). The solid black line shows the expected results for Poisson distributions, where V = m. The inset graph gives the total number of reported cases in England and Wales, and the bars indicate the three vaccination eras. Reproduced, with permission, from Ref. [11]. (c) The relationship between log(variance) and log(mean) for outbreak sizes, where the slope b = 2.25. Data are published reports of numbers of clinical cases during outbreaks of several pathogens: Clostridium botulinum, Staphylococcus aureus, Bacillus cereus, Campylobacter spp., human echovirus, Salmonella spp. (3 data points), Clostridium perfringens, Escherichia coli O157 (2 data points), Shigella spp., small round structured virus, Listeria spp., Ebola virus, Cryptosporidium spp., Trichinella spiralis, measles virus, human polio virus and Vibrio cholerae. Reproduced, with permission, from Ref. [31].
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Box 1. Taylor’s power law, epidemiological models and the prevalence–abundance relationship Epidemiological models [29] predict that the probability distribution of parasite numbers per host individual, being overdispersed, determines the relationship between the prevalence of infection P(t) and the mean abundance of parasites m(t) at any time, t, as: k mðtÞ PðtÞ ¼ 1 1 þ ; (1) k where k is the parameter of the negative binomial distribution inversely indicating the degree of aggregation. The parameter k can be estimated with parameters a and b of Taylor’s power law [9]. This ecological law states that mean abundance (m) and variance of abundance [V(m)] of an organism are related as V(m) = amb. Values of k can be estimated [42] as: 1=k ¼ mðb2Þ ð1=mÞ or, using the moment estimate of Elliot (1977): V ðmÞ =½V ðmÞ m; k ¼ m2 n
(2)
(3)
where n is host sample size. Comparisons of observed and predicted values of prevalence were done for several host parasite systems: intestinal nematodes of mammals (Figure I), ectoparasitic monogeneans of fish and ticks of rodents (Figure II).
Figure I. The graph illustrates the comparison of observed (white dots) and simulated (black dots in line) values of prevalence in relation to parasite abundance, using Equation 1, in the case of gut nematodes of mammals. Reproduced, with permission, from Ref. [14].
Figure II. The graph illustrates the relationship between observed and expected prevalence, using Equation 1, and with different k estimation, using Equation 2 or Equation 3, in the case of the larval Ixodes ricinus parasitizing the rodent Apodemus agrarius. Reproduced, with permission, from Ref. [17].
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Taylor’s law and epidemiology Taylor’s power law has been applied to validate the prediction of the relationship between prevalence and abundance given by the epidemiological models developed for macroparasites [29] (Box 1). The prevalence–abundance relationship is equivalent to the occupancy–abundance relationship of the ecologists. The basic epidemiological models explicitly take into account the aggregative nature of macroparasites, and the distribution of macroparasites among their hosts is generally well described by negative binomial distribution. Simulation studies and comparative analyses have shown that, knowing the power relationship between variance of abundance and abundance of parasites, the observed prevalence of infected hosts is well predicted by the epidemiological models (Box 1) [14–16]. Therefore, in epidemiology, as in ecology, variance–meanabundance and occupancy–abundance patterns that characterize the spatial distributions of species (in patches for free-living organisms or in hosts for parasites) are connected [30]. In addition to being used to test the prediction of epidemiological modelling in a comparative way, Taylor’s power law was used by Keeling and Grenfell [11] to explore the basic relationship between abundance and variability in a specific case of infectious disease. They showed that power-law scaling occurs in measles case reports in England and Wales, and vaccination affects the slope b of the power law (Figure 1b). A decrease of the b coefficient has been found to be associated with an increasing level of vaccination, and the value of b changed from around 1.7 when there was little or no vaccination to 1.2 when 90% of the human population was vaccinated. When the level of vaccination approached the measles eradication threshold, the number of human cases was expected to be random, with b equalling 1 (i.e. following a Poisson distribution) (Figure 1b). Moreover, Keeling and Grenfell [11] developed simple demographic models showing that the power law is an increasing function of the amount of stochasticity in the epidemiological system. More recently, Woolhouse et al. [31] have shown that epidemic outbreak sizes tend to be overdispersed and more than 50% of the number of outbreaks involves less than ten cases, whereas less than 10% of outbreaks involves 100 cases or more. This pattern was supported by data for a variety of infectious diseases, such as E. coli in Scotland, Ebola virus, and Staphylococcus aureus [31]. The mean and variance of outbreak sizes obey Taylor’s power law with exponent b > 2 (Figure 1c), meaning that outbreaks concern infectious diseases with a great amount of stochasticity with low regulation processes. The use of Taylor’s law in evolutionary ecology of parasitism A high level of aggregation leads to high mortality in heavily infected hosts, although the pathogenecity is highly variable between and among parasite species [28], and random distribution of parasites reduces their mating opportunities [32]. Thus, parasites face a trade-off between being too aggregated and being too randomly distributed [19,27], suggesting that the pattern of distribution evolves to an optimum, together with the evolution
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Figure 2. Relationships between parameter b of Taylor’s power law and (a) host specificity of fleas, estimated by the number of rodent species exploited and (b) species richness of flea assemblages. Reproduced, with permission, from Ref [13], with independent contrasts.
of life-history traits. Moreover, because the hosts might invest in several defence mechanisms to control or reduce the level of infection, the optimal distribution of parasites should reveal the coevolutionary adaptive processes that link parasite virulence and host defence. These hypotheses are far from being explored. One of the most important properties of any parasite is its host specificity (i.e. the number of different host species a parasite can exploit). Krasnov et al. [13] hypothesized that host specificity should affect the variability of parasite abundance through the differential exploitation of different host species. The slope of Taylor’s power relationship, repeatable within a parasite species (fleas in this example), was shown to decrease with a decrease of flea host specificity (Figure 2a). The repeatability within species and also among years suggests that aggregating datasets or any
kind of data manipulation does not seem to affect the estimation of the b coefficient of Taylor’s power law [13]. For a given parasite abundance, high parasite population variability is associated with a small host range. This is a pattern similar to what is known in ecology: high spatial variability is associated with a small range size. In ecology it also is known that greater temporal variation of a population is likely to lead to extinction [33]. The results of Krasnov et al. [13] suggest that specific parasites potentially are more at risk of extinction because of the high variability of their population size, which might represent a cost of specialization. The value of parameter b of Taylor’s power law has been assumed to be dependent on some population-regulation processes. Interspecific competition is one of them. Kilpatrick and Ives’ simulation study [8] concerning free-living 307
Opinion organisms showed that the slope of Taylor’s power relationship is effectively dependent on the strength of the interspecific competition. Low values of b are associated with increased strength of competition. The study by Krasnov et al. [13] demonstrated that the value of b in a flea species decreases with flea community size, reflecting the intensity and/or efficiency of population regulation processes (Figure 2b). This is either because of interspecific competition, as experimentally demonstrated, or because it is indirectly mediated by the host (i.e. apparent competition) through immune defence [16]. Finally, Krasnov et al. [13] found that intraspecific aggregation is positively correlated with several host factors representing variation in body mass, burrow complexity and mass-independent BMR (basal metabolic rate). Heavily infested individuals are characteristically larger host species that possess more complicated burrows and/or host species with higher mass-independent BMRs. The relationship between high BMR and parasite aggregation might suggest an increased exposure to parasites because a high BMR is associated with increased activity [34] and high parasite diversity [35]. It also could suggest that hosts with higher BMR invest more in immune response, which is supposed to be costly in terms of energetic demands [35,36]. Investment in immunity affects aggregation of parasites [21], as does vaccination [11]. However, there is no study that has specifically tested the relationship between slope b and the investment in immune defence or behavioural defence, such as grooming in mammals or preening in birds. Future perspectives More theoretical and empirical studies are needed for a general comprehension of the mechanisms and processes that lead us to observe Taylor’s power law in epidemiological data (Box 2). First, mathematical models can help because they can connect the microscopic description of the processes (i.e. individual interaction) to the macroscopic pattern (i.e. the emergence of an empirical law). Important contributions of the network theory by integrating the consequences of heterogeneity for population and community dynamics are in development in ecology [37] and, more specifically, epidemiology [38–40]. Second, and as mentioned above, a link between host defence and the value of b of Taylor’s power law is expected [21]. Few empirical studies have questioned this. Boag et al. [12] showed that myxomatosis has a consistent impact by lowering the degree of aggregation for helminth parasites in all age groups of rabbit. Myxomatosis has been associated with the suppression of immune responses in the infected rabbits, and the breakdown in immunity of rabbits because of myxomatosis could explain the decreases in aggregation observed (low b). Third, Taylor [6] emphasized that b value is a species-specific attribute, and this was confirmed by Krasnov et al. [16] for rodent fleas. Boag et al. [12], by contrast, emphasized that values of b are not stable, but aggregation is a dynamic phenomenon. Experimental manipulations and measurements of heritability of the b value of Taylor’s power law in various host–parasite systems have to be conducted. 308
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Box 2. Values of b coefficient of the power law and the dynamics and ecology of infection Demographic models showed that Taylor’s power law is an increasing function of the amount of stochasticity in the epidemiological system [11]. The available literature is scarce, but some patterns of b variation seem to emerge in relation to the dynamics and ecology of infection and transmission: Values of coefficient b of Taylor’s power law change in relation to the epidemiological dynamics of infection, with higher b values in epidemic infections (i.e. outbreaks) and lower b values in endemic ones [31]. Vaccination, by reducing the epidemiological stochasticity and the number of outbreaks, decreases b values [11]. Type of parasites (micro- versus macroparasites), mode of transmission (direct or indirect using vectors) and lifestyle (ectoversus endoparasites) do not affect the b value [14–17], although a comparative analysis should be performed to confirm these patterns. Increasing host specificity (i.e. the number of different host species a parasite can infect) is linked with a decrease in b values [13]. This pattern illustrates that an increase in the number of exploited host species decreases the stochasticity in the transmission dynamics. The structure of both parasite communities affects the coefficient b; b values decrease as the parasite community size increases [13]. This supports the hypothesis that large parasite communities are less stochastic than poor-species communities. The intensity and/or efficiency of parasite-population-regulation processes leads to lowering b values [13,28], although the mechanisms could be varied. Finally, any kind of heterogeneity – spatial or temporal, extrinsic or intrinsic – affects the values of b, which increased with the increase of heterogeneity.
Finally, we hope that this article will convince readers that epidemiology has much to gain by integrating theoretical ecology. Acknowledgements This is publication no. 614 of the Mitrani Department of Desert Ecology and no. 249 of the Ramon Science Center.
References 1 Brown, J.H. (1995) Macroecology. University of Chicago Press 2 Gaston, K.J. and Blackburn, T. (2000) Pattern and process in macroecology. Blackwell 3 Lange, M. (2005) Ecological laws: what would they be and why would they matter? Oikos 110, 394–403 4 Meiri, S. and Dayan, T. (2003) On the validity of Bergmann’s rule. J. Biogeogr. 30, 331–351 5 Rapoport, E.H. (1982) Areography. Geographical Strategies of Species. Pergamon Press 6 Taylor, L.R. (1961) Aggregation, variance, and the mean. Nature 189, 732–735 7 Maurer, B.A. and Taper, M.L. (2002) Connecting geographical distributions with population processes. Ecol. Lett. 5, 223–231 8 Kilpatrick, A.M. and Ives, A.R. (2003) Species interactions can explain Taylor’s power law for ecological time series. Nature 422, 65–68 9 Taylor, L.R. et al. (1980) Variance and the large-scale spatial stability of aphids, moths and birds. J. Anim. Ecol. 49, 831–854 10 Anderson, R.M. et al. (1982) Variability in the abundance of animal and plant species. Nature 296, 245–248 11 Keeling, M. and Grenfell, B. (1999) Stochastic dynamics and a power law for measles variability. Philos. Trans. R. Soc. Lond. B Biol. Sci. 354, 769–776 12 Boag, B. et al. (2001) Patterns of parasite aggregation in the wild European rabbit (Oryctolagus cuniculus). Int. J. Parasitol. 31, 1421– 1428 13 Krasnov, B.R. et al. (2006) Host specificity, parasite community size and the relation between abundance and its variance. Evol. Ecol. 25, 75–91
Opinion 14 Morand, S. and Gue´gan, J-F. (2000) Abundance and distribution of parasitic nematodes: ecological specialisation, phylogenetic constraints or simply epidemiology? Oikos 55, 563–573 15 Sˇimkova´, A. et al. (2002) Abundance–prevalence relationship of gill congeneric ectoparasites: testing for core-satellite hypothesis and ecological specialisation. Parasitol. Res. 88, 682–686 16 Krasnov, B.R. et al. (2005) Distribution of fleas (Siphonaptera) among small mammals: mean abundance predicts prevalence via simple epidemiological model. Int. J. Parasitol. 35, 1097–1101 17 Stanko, M. et al. (2006) Relationship between host abundance and parasite distribution: inferring regulating mechanisms from census data. J. Anim. Ecol. 75, 575–583 18 Stanko, M. et al. (2007) Simple epidemiological model predicts the relationships between prevalence and abundance in ixodid ticks. Parasitology 134, 59–68 19 Anderson, R.M. and Gordon, D.M. (1982) Processes influencing the distribution of parasite numbers within host populations with special emphasis on parasite-induced host mortalities. Parasitology 85, 373–398 20 Wilson, K. et al. (2001) Heterogeneities in macroparasite infections – patterns and processes. In The Ecology of Wildlife Diseases (Hudson, A. et al., eds), pp. 6–44, Oxford University Press 21 Galvani, A.P. (2003) Immunity, antigenic heterogeneity, and aggregation of helminth parasites. J. Parasitol. 89, 232–241 22 Anderson, R.M. and May, R.M. (1978) Regulation and stability of hostparasite population interactions I. Regulatory processes. J. Anim. Ecol. 47, 219–247 23 May, R.M. and Anderson, R.M. (1978) Regulation and stability of hostparasite population interactions II. Destabilizing processes. J. Anim. Ecol. 47, 249–267 24 Rosa`, R. and Pugliese, A. (2002) Aggregation, stability, and oscillations in different models for host–macroparasite interactions. Theor. Popul. Biol. 61, 319–334 25 Dobson, A.P. and Hudson, P.J. (1992) Regulation and stability of a freeliving host–parasite system: Trichostrongylus tenuis in red grouse. II. Population models. J. Anim. Ecol. 61, 487–498 26 Scott, M.E. (1987) Temporal changes in aggregation: a laboratory study. Parasitology 94, 583–595
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27 Shaw, D.J. and Dobson, A.P. (1995) Patterns of macroparasite abundance and aggregation in wildlife populations: a quantitative review. Parasitology 111, S111–S127 28 Madhavi, R. and Anderson, R.M. (1985) Variability in the susceptibility of the fish host, Poecilia reticulata, to infection with Gyrodactylus bullatarudis (Monogenea). Parasitology 91, 531–544 29 Anderson, R.M. and May, R.M. (1985) Helminth infections of humans: mathematical models, population dynamics, and control. Adv. Parasitol. 24, 1–101 30 He, F. and Gaston, K.J. (2003) Occupancy, spatial variance, and the abundance of species. Am. Nat. 162, 366–375 31 Woolhouse, M.E.J. et al. (2001) Population biology of multihost pathogens. Science 292, 1109–1112 32 May, R.M. and Woolhouse, M.E.J. (1993) Biased sex ratios and parasite mating probabilities. Parasitology 107, 287–295 33 Goodman, D. (1987) The demography of chance extinction. In Viable Populations for Conservation (Soule´, M.E., ed.), pp. 11–34, Cambridge University Press 34 Gregory, R.D. et al. (1996) Helminth parasite richness among vertebrates. Biodivers. Conserv. 5, 985–997 35 Morand, S. and Harvey, P.H. (2000) Mammalian metabolism, longevity and parasite species richness. Proc. Biol. Sci. 267, 1999–2003 36 Lochmiller, R.I. and Deerenberg, C.B. (2000) Trade-offs in evolutionary ecology: just what is the cost of immunity? Oikos 88, 87–98 37 Bascompte, J. (2007) Networks in ecology. Basic Appl. Ecol. 8, 485–490 38 Barthe´lemy, M. et al. (2005) Dynamical patterns of epidemic outbreaks in complex heterogeneous networks. J. Theor. Biol. 235, 275–288 39 Eubank, S. et al. (2004) Modelling disease outbreaks in realistic urban social networks. Nature 429, 180–184 40 Roy, M. and Pascual, M. (2006) On representing network heterogeneities in the incidence rate of simple epidemic models. Ecol. Complex 3, 80–90 41 Poulin, R. and Morand, S. (2000) Parasite body size and interspecific variation in levels of aggregation among nematodes. J. Parasitol. 86, 642–647 42 Perry, J.N. and Taylor, L.R. (1986) Stability of real interacting populations in space and time: implications, alternatives and negative binomial kc. J. Anim. Ecol. 55, 1053–1068
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Why apply ecological laws to epidemiology? Serge Morand1,2 and Boris Krasnov3 1 Institut des Sciences de l’Evolution, CNRS, De´partement Ge´ne´tique Environnement, CC065, Universite´ Montpellier 2, 34095, Montpellier cedex 05, France 2 UR AGIRs CIRAD, Campus International de Baillarguet, 34398 Montpellier cedex 05, France 3 Mitrani Department of Desert Ecology, Jacob Blaustein Institutes for Desert Research, Ben-Gurion University of the Negev, Sede-Boqer Campus, 84990 Midreshet Ben-Gurion and Ramon Science Center, P.O. Box 194, 80600 Mizpe Ramon, Israel
Using ecological laws, or rules, is a useful strategy for epidemiological observations. The application of Taylor’s power law to epidemiology and evolutionary ecology of parasites is exemplified here. Taylor’s power law takes the form of s2 = amb, where s2 is the variance in population abundance, m is the mean abundance of the population, a represents a constant parameter and b represents an index of spatial heterogeneity. Although Taylor’s power law reflects the aggregation of parasite (or pathogen) individuals among host population, the values of b could reflect regulation processes in host– parasite systems. Illustrations are given showing how b value is linked to various epidemiological situations: pathogen emergence, the impact of vaccination or the level of host immune defence. The search for rules and laws The search for patterns, rules and laws has been central to ecology, from its foundation to the recent development of macroecology [1–3]. Kleiber’s law, Bergman’s rule and Rapoport’s rule are, among others, famous examples [4,5]. In epidemiology, a science that focuses primarily on risk analysis, the search for patterns does not have such weighty importance. However, the renewal of interest in studies on scaling in ecology has the correlative effect of reviving interest in epidemiology. We aim to show the application of Taylor’s power law to epidemiology and the evolutionary ecology of parasites. Taylor’s power law corresponds to a consistent and universal ecological observation that variance in abundance of a species increases with its mean abundance in accordance with a simple power law [6]. This relationship takes the form of s2 = amb, where s2 is the temporal or spatial variance in population abundance at a given geographical location, m is the temporal or spatial mean abundance of the population, a represents a constant parameter and b represents an index of spatial heterogeneity. Abundance and variance values are fitted to the power function, which gives estimates of a and b by using log transformation. The parameters a and b are not assumed to be the properties of individual local populations, but, rather, they describe patterns over a much larger spatial scale [7], which suggests the existence of constraints on Corresponding author: Morand, S. ([email protected]).
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the temporal variation in abundance of local populations [8]. Taylor’s power law has been observed in many free-living organisms [9,10] and in parasites [11,12]. The application of Taylor’s power relationship to parasitology and epidemiology has gained growing interest [11–13] (Figure 1a); one way that this can be seen is the use of comparative data in the predictions of epidemiological models [14–18] (Box 1). Taylor’s power law is a measure of aggregation of parasites within their hosts Aggregation commonly is observed in parasite individuals among host populations [19]. The aggregated distribution of parasites is caused by a variety of factors such as heterogeneities in host populations and/or infection pressure; the factors can act independently or together and might increase or decrease the observed level of parasite aggregation [20]. Although the mechanisms that cause aggregation still are debated [21], the consequences in terms of host–parasite regulation are better explored thanks to the epidemiological theory [22–24]. Low-level aggregation theoretically is associated with the destabilization of host–parasite population dynamics, whereas high-level aggregation is associated with stability [25]. However, although all parasites appear to be aggregated, significant differences are observed in the degree of aggregation between parasite species or populations; the causes of these differences are not well understood. There are several ways to measure parasite aggregation, the most familiar being the variance of abundance to mean abundance ratio, or the k value of the negative binomial distribution [26]. Aggregation also can be estimated by exponent b of Taylor’s power relationship, linking the variance of parasite abundance to the mean abundance [14,16,27]. The value of b is easy to estimate through regression analysis by using log transformation. The biological meaning of Taylor’s power law, where b < 2, is that more abundant populations of a species are less variable than would be expected from simple statistical properties (see, for example, Ref. [9]). Although early studies suggested that values of b could reflect regulation processes in host–parasite systems [28], it is only recently that more thorough investigations have been done [13].
1471-4922/$ – see front matter ß 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.pt.2008.04.003
Opinion
Trends in Parasitology
Vol.24 No.7
Figure 1. Relationships between the logarithms of variance of abundance, V(m), and abundance, m, are fitted to a power function (a) in nematodes parasitizing mammals (reproduced, with permission, from Ref. [41]) and (b) case reports of measles in 366 communities in England and Wales. The data are subdivided between the prevaccination era (1944–1966, green dots), the 80% vaccination era (1980–1990, red dots) and the 90% vaccination era (1990–1997, blue dots). The solid black line shows the expected results for Poisson distributions, where V = m. The inset graph gives the total number of reported cases in England and Wales, and the bars indicate the three vaccination eras. Reproduced, with permission, from Ref. [11]. (c) The relationship between log(variance) and log(mean) for outbreak sizes, where the slope b = 2.25. Data are published reports of numbers of clinical cases during outbreaks of several pathogens: Clostridium botulinum, Staphylococcus aureus, Bacillus cereus, Campylobacter spp., human echovirus, Salmonella spp. (3 data points), Clostridium perfringens, Escherichia coli O157 (2 data points), Shigella spp., small round structured virus, Listeria spp., Ebola virus, Cryptosporidium spp., Trichinella spiralis, measles virus, human polio virus and Vibrio cholerae. Reproduced, with permission, from Ref. [31].
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Box 1. Taylor’s power law, epidemiological models and the prevalence–abundance relationship Epidemiological models [29] predict that the probability distribution of parasite numbers per host individual, being overdispersed, determines the relationship between the prevalence of infection P(t) and the mean abundance of parasites m(t) at any time, t, as: k mðtÞ PðtÞ ¼ 1 1 þ ; (1) k where k is the parameter of the negative binomial distribution inversely indicating the degree of aggregation. The parameter k can be estimated with parameters a and b of Taylor’s power law [9]. This ecological law states that mean abundance (m) and variance of abundance [V(m)] of an organism are related as V(m) = amb. Values of k can be estimated [42] as: 1=k ¼ mðb2Þ ð1=mÞ or, using the moment estimate of Elliot (1977): V ðmÞ =½V ðmÞ m; k ¼ m2 n
(2)
(3)
where n is host sample size. Comparisons of observed and predicted values of prevalence were done for several host parasite systems: intestinal nematodes of mammals (Figure I), ectoparasitic monogeneans of fish and ticks of rodents (Figure II).
Figure I. The graph illustrates the comparison of observed (white dots) and simulated (black dots in line) values of prevalence in relation to parasite abundance, using Equation 1, in the case of gut nematodes of mammals. Reproduced, with permission, from Ref. [14].
Figure II. The graph illustrates the relationship between observed and expected prevalence, using Equation 1, and with different k estimation, using Equation 2 or Equation 3, in the case of the larval Ixodes ricinus parasitizing the rodent Apodemus agrarius. Reproduced, with permission, from Ref. [17].
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Taylor’s law and epidemiology Taylor’s power law has been applied to validate the prediction of the relationship between prevalence and abundance given by the epidemiological models developed for macroparasites [29] (Box 1). The prevalence–abundance relationship is equivalent to the occupancy–abundance relationship of the ecologists. The basic epidemiological models explicitly take into account the aggregative nature of macroparasites, and the distribution of macroparasites among their hosts is generally well described by negative binomial distribution. Simulation studies and comparative analyses have shown that, knowing the power relationship between variance of abundance and abundance of parasites, the observed prevalence of infected hosts is well predicted by the epidemiological models (Box 1) [14–16]. Therefore, in epidemiology, as in ecology, variance–meanabundance and occupancy–abundance patterns that characterize the spatial distributions of species (in patches for free-living organisms or in hosts for parasites) are connected [30]. In addition to being used to test the prediction of epidemiological modelling in a comparative way, Taylor’s power law was used by Keeling and Grenfell [11] to explore the basic relationship between abundance and variability in a specific case of infectious disease. They showed that power-law scaling occurs in measles case reports in England and Wales, and vaccination affects the slope b of the power law (Figure 1b). A decrease of the b coefficient has been found to be associated with an increasing level of vaccination, and the value of b changed from around 1.7 when there was little or no vaccination to 1.2 when 90% of the human population was vaccinated. When the level of vaccination approached the measles eradication threshold, the number of human cases was expected to be random, with b equalling 1 (i.e. following a Poisson distribution) (Figure 1b). Moreover, Keeling and Grenfell [11] developed simple demographic models showing that the power law is an increasing function of the amount of stochasticity in the epidemiological system. More recently, Woolhouse et al. [31] have shown that epidemic outbreak sizes tend to be overdispersed and more than 50% of the number of outbreaks involves less than ten cases, whereas less than 10% of outbreaks involves 100 cases or more. This pattern was supported by data for a variety of infectious diseases, such as E. coli in Scotland, Ebola virus, and Staphylococcus aureus [31]. The mean and variance of outbreak sizes obey Taylor’s power law with exponent b > 2 (Figure 1c), meaning that outbreaks concern infectious diseases with a great amount of stochasticity with low regulation processes. The use of Taylor’s law in evolutionary ecology of parasitism A high level of aggregation leads to high mortality in heavily infected hosts, although the pathogenecity is highly variable between and among parasite species [28], and random distribution of parasites reduces their mating opportunities [32]. Thus, parasites face a trade-off between being too aggregated and being too randomly distributed [19,27], suggesting that the pattern of distribution evolves to an optimum, together with the evolution
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Figure 2. Relationships between parameter b of Taylor’s power law and (a) host specificity of fleas, estimated by the number of rodent species exploited and (b) species richness of flea assemblages. Reproduced, with permission, from Ref [13], with independent contrasts.
of life-history traits. Moreover, because the hosts might invest in several defence mechanisms to control or reduce the level of infection, the optimal distribution of parasites should reveal the coevolutionary adaptive processes that link parasite virulence and host defence. These hypotheses are far from being explored. One of the most important properties of any parasite is its host specificity (i.e. the number of different host species a parasite can exploit). Krasnov et al. [13] hypothesized that host specificity should affect the variability of parasite abundance through the differential exploitation of different host species. The slope of Taylor’s power relationship, repeatable within a parasite species (fleas in this example), was shown to decrease with a decrease of flea host specificity (Figure 2a). The repeatability within species and also among years suggests that aggregating datasets or any
kind of data manipulation does not seem to affect the estimation of the b coefficient of Taylor’s power law [13]. For a given parasite abundance, high parasite population variability is associated with a small host range. This is a pattern similar to what is known in ecology: high spatial variability is associated with a small range size. In ecology it also is known that greater temporal variation of a population is likely to lead to extinction [33]. The results of Krasnov et al. [13] suggest that specific parasites potentially are more at risk of extinction because of the high variability of their population size, which might represent a cost of specialization. The value of parameter b of Taylor’s power law has been assumed to be dependent on some population-regulation processes. Interspecific competition is one of them. Kilpatrick and Ives’ simulation study [8] concerning free-living 307
Opinion organisms showed that the slope of Taylor’s power relationship is effectively dependent on the strength of the interspecific competition. Low values of b are associated with increased strength of competition. The study by Krasnov et al. [13] demonstrated that the value of b in a flea species decreases with flea community size, reflecting the intensity and/or efficiency of population regulation processes (Figure 2b). This is either because of interspecific competition, as experimentally demonstrated, or because it is indirectly mediated by the host (i.e. apparent competition) through immune defence [16]. Finally, Krasnov et al. [13] found that intraspecific aggregation is positively correlated with several host factors representing variation in body mass, burrow complexity and mass-independent BMR (basal metabolic rate). Heavily infested individuals are characteristically larger host species that possess more complicated burrows and/or host species with higher mass-independent BMRs. The relationship between high BMR and parasite aggregation might suggest an increased exposure to parasites because a high BMR is associated with increased activity [34] and high parasite diversity [35]. It also could suggest that hosts with higher BMR invest more in immune response, which is supposed to be costly in terms of energetic demands [35,36]. Investment in immunity affects aggregation of parasites [21], as does vaccination [11]. However, there is no study that has specifically tested the relationship between slope b and the investment in immune defence or behavioural defence, such as grooming in mammals or preening in birds. Future perspectives More theoretical and empirical studies are needed for a general comprehension of the mechanisms and processes that lead us to observe Taylor’s power law in epidemiological data (Box 2). First, mathematical models can help because they can connect the microscopic description of the processes (i.e. individual interaction) to the macroscopic pattern (i.e. the emergence of an empirical law). Important contributions of the network theory by integrating the consequences of heterogeneity for population and community dynamics are in development in ecology [37] and, more specifically, epidemiology [38–40]. Second, and as mentioned above, a link between host defence and the value of b of Taylor’s power law is expected [21]. Few empirical studies have questioned this. Boag et al. [12] showed that myxomatosis has a consistent impact by lowering the degree of aggregation for helminth parasites in all age groups of rabbit. Myxomatosis has been associated with the suppression of immune responses in the infected rabbits, and the breakdown in immunity of rabbits because of myxomatosis could explain the decreases in aggregation observed (low b). Third, Taylor [6] emphasized that b value is a species-specific attribute, and this was confirmed by Krasnov et al. [16] for rodent fleas. Boag et al. [12], by contrast, emphasized that values of b are not stable, but aggregation is a dynamic phenomenon. Experimental manipulations and measurements of heritability of the b value of Taylor’s power law in various host–parasite systems have to be conducted. 308
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Box 2. Values of b coefficient of the power law and the dynamics and ecology of infection Demographic models showed that Taylor’s power law is an increasing function of the amount of stochasticity in the epidemiological system [11]. The available literature is scarce, but some patterns of b variation seem to emerge in relation to the dynamics and ecology of infection and transmission: Values of coefficient b of Taylor’s power law change in relation to the epidemiological dynamics of infection, with higher b values in epidemic infections (i.e. outbreaks) and lower b values in endemic ones [31]. Vaccination, by reducing the epidemiological stochasticity and the number of outbreaks, decreases b values [11]. Type of parasites (micro- versus macroparasites), mode of transmission (direct or indirect using vectors) and lifestyle (ectoversus endoparasites) do not affect the b value [14–17], although a comparative analysis should be performed to confirm these patterns. Increasing host specificity (i.e. the number of different host species a parasite can infect) is linked with a decrease in b values [13]. This pattern illustrates that an increase in the number of exploited host species decreases the stochasticity in the transmission dynamics. The structure of both parasite communities affects the coefficient b; b values decrease as the parasite community size increases [13]. This supports the hypothesis that large parasite communities are less stochastic than poor-species communities. The intensity and/or efficiency of parasite-population-regulation processes leads to lowering b values [13,28], although the mechanisms could be varied. Finally, any kind of heterogeneity – spatial or temporal, extrinsic or intrinsic – affects the values of b, which increased with the increase of heterogeneity.
Finally, we hope that this article will convince readers that epidemiology has much to gain by integrating theoretical ecology. Acknowledgements This is publication no. 614 of the Mitrani Department of Desert Ecology and no. 249 of the Ramon Science Center.
References 1 Brown, J.H. (1995) Macroecology. University of Chicago Press 2 Gaston, K.J. and Blackburn, T. (2000) Pattern and process in macroecology. Blackwell 3 Lange, M. (2005) Ecological laws: what would they be and why would they matter? Oikos 110, 394–403 4 Meiri, S. and Dayan, T. (2003) On the validity of Bergmann’s rule. J. Biogeogr. 30, 331–351 5 Rapoport, E.H. (1982) Areography. Geographical Strategies of Species. Pergamon Press 6 Taylor, L.R. (1961) Aggregation, variance, and the mean. Nature 189, 732–735 7 Maurer, B.A. and Taper, M.L. (2002) Connecting geographical distributions with population processes. Ecol. Lett. 5, 223–231 8 Kilpatrick, A.M. and Ives, A.R. (2003) Species interactions can explain Taylor’s power law for ecological time series. Nature 422, 65–68 9 Taylor, L.R. et al. (1980) Variance and the large-scale spatial stability of aphids, moths and birds. J. Anim. Ecol. 49, 831–854 10 Anderson, R.M. et al. (1982) Variability in the abundance of animal and plant species. Nature 296, 245–248 11 Keeling, M. and Grenfell, B. (1999) Stochastic dynamics and a power law for measles variability. Philos. Trans. R. Soc. Lond. B Biol. Sci. 354, 769–776 12 Boag, B. et al. (2001) Patterns of parasite aggregation in the wild European rabbit (Oryctolagus cuniculus). Int. J. Parasitol. 31, 1421– 1428 13 Krasnov, B.R. et al. (2006) Host specificity, parasite community size and the relation between abundance and its variance. Evol. Ecol. 25, 75–91
Opinion 14 Morand, S. and Gue´gan, J-F. (2000) Abundance and distribution of parasitic nematodes: ecological specialisation, phylogenetic constraints or simply epidemiology? Oikos 55, 563–573 15 Sˇimkova´, A. et al. (2002) Abundance–prevalence relationship of gill congeneric ectoparasites: testing for core-satellite hypothesis and ecological specialisation. Parasitol. Res. 88, 682–686 16 Krasnov, B.R. et al. (2005) Distribution of fleas (Siphonaptera) among small mammals: mean abundance predicts prevalence via simple epidemiological model. Int. J. Parasitol. 35, 1097–1101 17 Stanko, M. et al. (2006) Relationship between host abundance and parasite distribution: inferring regulating mechanisms from census data. J. Anim. Ecol. 75, 575–583 18 Stanko, M. et al. (2007) Simple epidemiological model predicts the relationships between prevalence and abundance in ixodid ticks. Parasitology 134, 59–68 19 Anderson, R.M. and Gordon, D.M. (1982) Processes influencing the distribution of parasite numbers within host populations with special emphasis on parasite-induced host mortalities. Parasitology 85, 373–398 20 Wilson, K. et al. (2001) Heterogeneities in macroparasite infections – patterns and processes. In The Ecology of Wildlife Diseases (Hudson, A. et al., eds), pp. 6–44, Oxford University Press 21 Galvani, A.P. (2003) Immunity, antigenic heterogeneity, and aggregation of helminth parasites. J. Parasitol. 89, 232–241 22 Anderson, R.M. and May, R.M. (1978) Regulation and stability of hostparasite population interactions I. Regulatory processes. J. Anim. Ecol. 47, 219–247 23 May, R.M. and Anderson, R.M. (1978) Regulation and stability of hostparasite population interactions II. Destabilizing processes. J. Anim. Ecol. 47, 249–267 24 Rosa`, R. and Pugliese, A. (2002) Aggregation, stability, and oscillations in different models for host–macroparasite interactions. Theor. Popul. Biol. 61, 319–334 25 Dobson, A.P. and Hudson, P.J. (1992) Regulation and stability of a freeliving host–parasite system: Trichostrongylus tenuis in red grouse. II. Population models. J. Anim. Ecol. 61, 487–498 26 Scott, M.E. (1987) Temporal changes in aggregation: a laboratory study. Parasitology 94, 583–595
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27 Shaw, D.J. and Dobson, A.P. (1995) Patterns of macroparasite abundance and aggregation in wildlife populations: a quantitative review. Parasitology 111, S111–S127 28 Madhavi, R. and Anderson, R.M. (1985) Variability in the susceptibility of the fish host, Poecilia reticulata, to infection with Gyrodactylus bullatarudis (Monogenea). Parasitology 91, 531–544 29 Anderson, R.M. and May, R.M. (1985) Helminth infections of humans: mathematical models, population dynamics, and control. Adv. Parasitol. 24, 1–101 30 He, F. and Gaston, K.J. (2003) Occupancy, spatial variance, and the abundance of species. Am. Nat. 162, 366–375 31 Woolhouse, M.E.J. et al. (2001) Population biology of multihost pathogens. Science 292, 1109–1112 32 May, R.M. and Woolhouse, M.E.J. (1993) Biased sex ratios and parasite mating probabilities. Parasitology 107, 287–295 33 Goodman, D. (1987) The demography of chance extinction. In Viable Populations for Conservation (Soule´, M.E., ed.), pp. 11–34, Cambridge University Press 34 Gregory, R.D. et al. (1996) Helminth parasite richness among vertebrates. Biodivers. Conserv. 5, 985–997 35 Morand, S. and Harvey, P.H. (2000) Mammalian metabolism, longevity and parasite species richness. Proc. Biol. Sci. 267, 1999–2003 36 Lochmiller, R.I. and Deerenberg, C.B. (2000) Trade-offs in evolutionary ecology: just what is the cost of immunity? Oikos 88, 87–98 37 Bascompte, J. (2007) Networks in ecology. Basic Appl. Ecol. 8, 485–490 38 Barthe´lemy, M. et al. (2005) Dynamical patterns of epidemic outbreaks in complex heterogeneous networks. J. Theor. Biol. 235, 275–288 39 Eubank, S. et al. (2004) Modelling disease outbreaks in realistic urban social networks. Nature 429, 180–184 40 Roy, M. and Pascual, M. (2006) On representing network heterogeneities in the incidence rate of simple epidemic models. Ecol. Complex 3, 80–90 41 Poulin, R. and Morand, S. (2000) Parasite body size and interspecific variation in levels of aggregation among nematodes. J. Parasitol. 86, 642–647 42 Perry, J.N. and Taylor, L.R. (1986) Stability of real interacting populations in space and time: implications, alternatives and negative binomial kc. J. Anim. Ecol. 55, 1053–1068
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