Ann. Zool. Fennici 40: 115–130 ISSN 0003-455X Helsinki 30 April 2003 © Finnish Zoological and Botanical Publishing Board 2003

Extinction thresholds in –parasite dynamics

Anne Deredec & Franck Courchamp*

Ecologie, Systématique & Evolution, UMR CNRS 8079, Université Paris-Sud XI, Bâtiment 362, F-91405 Orsay Cedex, France (*e-mail: [email protected])

Received 13 Jan. 2003, revised version received 5 Feb. 2003, accepted 22 Feb. 2003

Deredec, A. & Courchamp, F. 2003: Extinction thresholds in host–parasite dynamics. — Ann. Zool. Fennici 40: 115–130.

In this paper, we review the main thresholds that can infl uence the population dynamics of host–parasite relationships. We start by considering the thresholds that have infl u- enced the conceptualisation of theoretical . The most common threshold involving parasites is the host population invasion threshold, but persistence and infec- tion thresholds are also important. We recap how the existence of the invasion thresh- old is linked to the nature of the term in theoretical studies. We examine some of the main thresholds that can affect host population dynamics including the Allee effect and then relate these to parasite thresholds, as a way to assess the dynamic consequences of the interplay between host and parasite thresholds on the fi nal out- come of the system. We propose that overlooking the existence of parasite and host thresholds can have important detrimental consequences in major domains of applied ecology, including in epidemiology, conservation biology and biological control.

Introduction the would not spread and would dis- appear, and the few infected hosts would die, The interplay between parasite species and their leaving a remaining healthy and uninfected host hosts constitutes a biological process that is both population. The existence of such a threshold has ubiquitous and of major ecological importance since been both widely used for theoretical stud- (Anderson 1982b, Shigesada 1997). On the level ies and confi rmed by empirical data and is now of population dynamics, these interactions are almost considered an intrinsic property of host– of great signifi cance, as they infl uence the num- microparasite systems. In addition, this threshold bers of both species involved. Pioneering work has since been shown to be of major importance in epidemiological modelling emphasized the in the ecology and in the epidemiology of animal consequence of the size of host populations as and human health. a main factor in disease propagation (Kermack Concurrently, the host population can be & McKendrick 1927, Bailey 1975, Anderson & affected by its own size. In particular, small May 1978, 1979, 1981, May & Anderson 1978, sized host populations may be prone to extinc- 1979). One particularly important feature is a tion due to intrinsic dynamic properties that are critical size or density of the host population, independent of parasitism. Indeed, several types below which the disease cannot persist, a concept of dynamic processes, including the Allee effect, introduced by the theoretical works of Kermack increase the extinction probability of small popu- and McKendrick (1927). Below this threshold, lations. In many cases, these processes engender 116 Deredec & Courchamp • ANN. ZOOL. FENNICI Vol. 40 a critical size or density threshold below which Thresholds in parasite dynamics populations are almost invariably doomed. Criti- cal thresholds for the persistence of populations The thresholds we discuss in this paper can be have been studied mostly through mathematical defi ned as a point of a dynamical system where models, given the extreme diffi culty in accu- any quantitative change leads to a qualitative rately studying extinction processes through alteration of the system behaviour. Classically, empirical approaches. Thresholds for both para- thresholds are encountered in parasite dynam- sites and hosts have been considered extensively, ics and can play a role at various stages. For but almost exclusively as separate subjects. instance, studies have suggested and shown Herein, we discuss the interplay between these thresholds for the invasion (i.e. spread) and the thresholds, and investigate the dynamic proc- persistence of parasites in a population as well as esses that drive host and parasite populations. for the infection of individual organisms. For the Much energy has been expended in debates sake of clarity, the thresholds discussed in this concerning standard defi nition of terms (see for review are briefl y defi ned in Table 1. example, McCallum et al. 2001). Therefore, after briefl y reviewing the current state of knowledge on parasite thresholds for invasion and persist- Invasion thresholds ence, we present the two classical transmission terms that have been the cause of much confu- The fi rst type of threshold central to epidemiol- sion and the basis of debate (see for example, ogy concerns invasion, which is the introduction McCallum et al. 2001, De Jong et al. 2002, and the subsequent increase of a parasite in the McCallum et al. 2002). This is important as one host population. The concept of threshold as per- of these transmission terms yields a threshold taining to the spread of infection was introduced below which the parasite cannot invade the by Kermack and McKendrick (1927), and since population, while the other does not. Hence has been at the core of many studies concerning the need to clarify the appropriate conditions the regulation of either hosts or their diseases. for the use of different transmission terms in According to this ground-breaking theory, the models of host–parasite dynamics. We also con- introduction of a few infectious individuals into sider the extinction thresholds that concern the a community of susceptible hosts will result host population, and the interplay of these two in the spread of disease only if the susceptible related types of thresholds. Finally, we discuss individuals occur at or greater than a certain the possible applications of the intermingling of critical density or number. Simple epidemiologi- parasitism and host population size thresholds in cal models including removal (by immunisation ecology. or mortality) and transmission terms generate a

Table 1. The main types of threshold employed in studies of the dynamics of host–parasite relationships, and the unit and species they refer to.

Threshold Unit Species of relevance

Invasion Number/density of hosts necessary for Parasites parasite spread in host population Persistence Number/density of hosts necessary for Parasites parasite maintenance in host population Infection Parasite infectious dose necessary for Parasites individual host infection of and/or reproduction within an individual host Extinction Number/density of hosts necessary for Hosts; for parasites, invasion or host population continued existence persistence are often used instead Eradication Same as extinction threshold Same as extinction threshold. More often used when species elimination is the objective ANN. ZOOL. FENNICI Vol. 40 • Thresholds in host–parasite dynamics 117 threshold density of susceptibility to infection. will be able to persist in lower density popula- For densities below this critical value, any initial tions. However, diseases characterised by a low trace of infection will be removed at a faster rate transmission effi ciency may be able to persist than it can build up (Bailey 1964). This threshold even in relatively low density host populations, can be defi ned on the basis of another important provided that the expected lifespan of infected concept in epidemiology: the basic reproductive hosts is long (e.g., natural and parasite-induced , noted R0. This concept is central to mortalities are both low and the average duration the analysis of the population dynamics of host– of the infection is long, as is the case for immu- parasite interactions and is defi ned as the number nodefi ciency ; Anderson & May 1981). of individuals contracting infection from a single Similarly to the R0 of microparasitic models, infectious individual, introduced in a completely a threshold quantity has been introduced for susceptible host population (Knell et al. 1998a). models of macroparasitic (helminths),

A recent review on this epidemiological param- and by analogy has been named Q0 or basic eter is given by Heesterbeek (2002). For classi- reproduction quotient (Roberts 1995). This cally transmitted microparasites, this parameter value can be defi ned as the expected number of can be calculated by multiplying the rate at which reproductively mature parasites produced by one new infections are produced by a single infec- adult parasite during its lifetime, in the absence tious individual within the time period during of density dependent constraints (Heesterbeek & which that individual remains infectious (Knell et Roberts 1995). A parasite population can invade al. 1998a). Infection can spread in the population a host population only if Q0 is higher than one. if one contagious individual, during its infectious This threshold quantity is basic to the under- period, infects more than one susceptible indi- standing of parasite population dynamics, includ- vidual, i.e. when R0 > 1. As the infection rate may ing the development of methods of control in depend on the density of susceptible individuals order to protect host populations (Roberts 1995). in the population, the condition in which R0 > 1 can be interpreted as a case when the host popu- lation exceeds a threshold density (Anderson & Persistence thresholds May 1979, May & Anderson 1979, May et al. 1981). For this reason, invasion may not be the Invasion addresses the events that occur in the best term for this threshold, nor is the sometimes short term immediately after the introduction of used “invasibility threshold”: it connotes a notion an infection. However, infection does not neces- of initiation, while here invasion simply means sarily imply that a parasite will be able to persist the potential to spread further (and is still relevant over the long term (Gubbins et al. 2000). The even if most of the individuals are infected). persistence threshold is a host density thresh- Obviously, the threshold density also old that conditions infection dynamics. This depends on the characteristics of the parasite. threshold is greater than the invasion threshold For example, a laboratory experiment comparing and hence it is more diffi cult to reach (Bolker the spread of two has shown that the & Grenfell 1996). But, whereas the invasion threshold density was considerably less for one threshold concerns the spread of infection, the species, indicating that it would be able to persist persistence threshold concerns the durability of in populations with lower densities (Knell et al. the infection. The persistence threshold has been 1998a). In fact, the precise value of this thresh- given a number of equivalent defi nitions, includ- old depends simultaneously on life history traits ing the population size below which infection of both the host and parasite. Microparasites with tends to die out in the troughs between epidem- low transmission effi ciency, such as or ics and above which infection will persist (Bar- smallpox, or with relatively high pathogenicity tlett 1957, Bolker & Grenfell 1996, Grenfell & (i.e. inducing high mortality rates) in general Harwood 1997, Keeling & Grenfell 1997). will persist only in high-density populations The persistence threshold has been central of hosts (Anderson & May 1981). Conversely, to the study of certain categories of parasites, in microparasites with high transmission effi ciency particular morbilliviruses, such as measles and 118 Deredec & Courchamp • ANN. ZOOL. FENNICI Vol. 40

Phocine Distemper (Grenfell et al. 1992, unable to persist within the population (Ander- Grenfell & Harwood 1997, Keeling & Grenfell son 1982b). Using the example of helminth 1997, Swinton et al. 1998, Keeling & Grenfell infections, Heesterbeek and Roberts (1995) sug- 1999). In this context, this threshold has also gested that there were threshold quantities that been called the Critical Community Size (CCS), determine whether or not an introduced infection as used by Bartlett (1957) in a measles epidem- will persist in a population of susceptible hosts. ics study. This threshold is related to the obser- However, the distinction between invasion and vation that the chance of infection “fade-out” persistence thresholds appears to be clearer in (where fadeout is defi ned as a given period of microparasitic than in macroparasitic models. time without new infections), or random extinc- The thresholds for the survival of the parasite tion between , decreases as population populations are separated into invasion and per- size increases (Bartlett 1957). The recurrence of sistence thresholds, but both can also be called epidemics is facilitated by the continual infl ux parasite extinction or eradication thresholds (Nee of susceptible individuals, but is prevented by 1994, Bascompte & Rodriguez-Trelles 1998). the random extinction of the infection. These Another type of threshold that concerns parasites extinctions are more likely in smaller communi- which is linked less to their population dynamics ties (Bartlett 1957, Black 1966). For example, it is called the infection threshold. has been estimated that a human community of around 250 000 individuals or more is necessary to maintain measles (Bartlett 1957). Infection thresholds The criteria for persistence thresholds are said to be generally more diffi cult to analyse The last type of parasite threshold that can be than those of invasion thresholds (Gubbins found in epidemiological studies concerns the et al. 2000). The CCS was fi rst determined infection of individual hosts. For many micro- theoretically and then confi rmed by analyses parasites, there is a tolerance level below which of epidemiological data. The value of the CCS the hostʼs immune system can resist succumbing may be governed by deterministic factors, or by to the infection, and above which an infection is one or another kind of stochastic considerations, contracted. In such models, the infection depends including so-called fade-out and epi- on the total exposure of individuals to the dis- demic fade-out (Anderson & May 1991). Several ease, a term including the amount of infectivity. epidemiological parameters may be important In various experiments the probability of infec- for the determination of the CCS, including the tion was observed to increase sigmoidally with transmission term, the host life expectancy and increasing parasite dose. In a simulation model, the parasite latent period (Dietz 1982). Which- Regoes (2002) showed that the parasite popula- ever factor has the greatest infl uence will depend tion can establish an infection only if its founder on the details specifi c to the host–microparasite population size (the infection dose) exceeds an relationship. For example, microparasitic infec- invasion threshold, which the author called an tions with very short duration and low transmis- Allee effect. In this model, the point of infl ec- sion effi ciency will require a large population of tion of a sigmoidal pattern describes well the hosts in which the rate of birth of new suscepti- infection threshold below which any parasite ble individuals can replace those individuals lost concentrations challenging the host are too low to infection (i.e., the threshold density for the to avoid extinction. The threshold concentration host population will be large). that is needed by a parasite to overcome the host The CCS concept has been used principally immunity and successfully invade the population in the case of measles and Phocine Distemper is determined by the balance between infectious- Virus, but it has applications in other systems. ness and virulence of the parasite. This has been Although it is said to be of less epidemiological described in various theoretical and empirical importance relative to microparasitic infections, models (see references in Regoes et al. 2002). macroparasite models also exhibit a critical Dose dependency in the rates of infection host density below which the parasite will be has also been addressed in the context ANN. ZOOL. FENNICI Vol. 40 • Thresholds in host–parasite dynamics 119 of macroparasitic infections (e.g., May 1977, We have focused here on animal species as Dushoff 1996). For sexually reproducing para- hosts, but the criteria for parasite invasion and sites, epidemiology is characterised fi rst by the persistence thresholds have also been studied infection, then by sexual reproduction within the in the context of plant-parasite interactions host. Because many of these parasites must fi nd (Gubbins et al. 2000). These thresholds are of mates within the host, there can be an equivalent considerable practical importance in botanical Allee effect for the parasites in this system. epidemiology, in particular for the prediction of There is a persistence threshold below which the yield loss, the deployment of chemical control macroparasite density is too low for mates to fi nd and the implementation of biological control each other within the infected host (Anderson by introduced microorganisms (Gubbins et al. & May 1991). The resulting mating failure can 2000). hinder macroparasite spread in the population. Nevertheless, despite the widespread occur- In addition, the parasitic load of an individual rence of thresholds in epidemiological models, is signifi cantly related to its infectiousness, lead- there is an important class of exceptions: infec- ing to a second type of Allee effect. If parasites tions transmitted by intimate contact within a are too scarce or too few, the immune defences defi ned group of individuals, as is the case with of the host have a greater chance to combat and sexually transmitted diseases. Under these cir- overcome the parasites. Both processes create cumstances, the threshold for infection persist- an unstable breakpoint phenomenon that is pro- ence depends on the average number of sexual portional to the degree of parasite aggregation partners per individual, rather than the host den- within the host population (Anderson & May sity of classical models described above (Ander- 1991), with resultant repercussions at the level of son & May 1991). both host and parasite populations.

Modelling parasite transmission Other thresholds Transmission is a key process in host–parasite The theory of thresholds has been much devel- interactions and thus occupies a core position in oped since the pioneering work of Kermack and host–parasites models. Whereas for macropara- McKendrick (1927), and it would be diffi cult sites the parasite dynamics are explicitly taken to list all existing models. However, there are into account, the modelling of microparasitic several interesting extensions of this threshold infection dynamics usually considers only the theorem that deserve greater attention. For host population and the different stages accord- example, models involving an intermediate host ing to its infection status. In the simplest classi- or a , as in the case of malaria (although cal microparasitic studies, hosts are divided into this was initiated in their fi rst paper, Kermack two compartments: susceptible and infected, & McKendrick 1927) and models taking into with transmission being the main link between account stochasticity or the spatial dimension them. Mathematically, the transmission term is (Bailey 1964, 1975) are of special interest. the number of susceptible individuals becoming Another interesting extension is the considera- infected per time unit. tion of hosts as “habitat patches” of a The transmission process may be described metapopulation (Swinton et al. 1998), which as follows. The number of susceptible indi- leads to the generalization that the CCS is a viduals contracting the infection depends on (i) critical metapopulation distribution of popula- the number of contagious individuals, (ii) the tion numbers across a particular metapopulation number of conspecifi cs encountered by a conta- structure (Swinton et al. 1998). This was elu- gious individual, (iii) the fraction of susceptible cidated in a study on Phocine Distemper Virus individuals among all encountered individuals persistence, which is affected by a mixing effect and (iv) the proportion of these encounters that that arises from a patchiness in some harbour result in successful transmission; (see Getz & seal Phoca vitulina populations. Pickering 1983, De Jong et al. 1995, McCallum 120 Deredec & Courchamp • ANN. ZOOL. FENNICI Vol. 40

Frequency-dependent Density-dependent transmission term f(N) = 1 transmission term f(N) = N often assumed in chemical kinetics studies (e.g., Kermack & McKendrick 1927). According to the Mass Action principle, individuals, like molecules, would be mixed homogeneously in space and every encounter would have the same This part of the curve mimics occurrence probability. Thus, the number of con- the frequency-dependent transmission term specifi cs encountered by an individual depends on the population density: f(N) is proportional to N. This assumption leads to a transmission term called Mass Action transmission: bSI and has been mostly used to describe air-borne diseases. The Mass Action term is also called den- sity-dependent (Getz & Pickering 1983, Thrall Parasite transmission rate et al. 1993, Antonovics et al. 1995) since the This part of the curve mimics the density-dependent probability of an individual becoming infected transmission term is a function of the density of infectious indi- Host population density viduals. This name is distinguished from the Fig. 1. Representation of the density-dependent and frequency-dependent transmission term: bSI/N, frequency-dependent terms (f(N), see text, thin lines) where f(N) = 1 (see Fig. 1). Up to now, the according to population density, and of a non-linear frequency-dependent term, also called the Pro- transmission term (thick-line curve) that varies from portionate Mixing term (Dietz & Schenzle 1985, N to 1 with population density and mimics density- dependent (Mass Action) transmission at low density Hethcote & Vanark 1987), has been mostly used and frequency-dependent (Proportionate Mixing) trans- in cases of sexually transmitted and vector-borne mission at high density. diseases, where an individual is supposed to have a constant number of potentially infectious contacts (either direct: sexual, or indirect: via a et al. 2001, for quite similar partitions of the vector). Indeed, in sexually transmitted diseases transmission term). For the sake of simplicity, the number of infectious encounters depends on infected and contagious individuals are often the number of sexual partners, which in turn assumed to be the same, in both their life his- is related more to the species mating system tory traits and in their behaviour, and are noted than to its population density (McCallum et al. I. In addition, the disease is assumed not to alter 2001). Similarly, for vector-borne diseases, the the behaviour of infected individuals. In this transmission rate of the pathogen depends on the scenario, an infected individual encounters as probability that the vector has previously been in many other individuals as a susceptible indi- contact with infected hosts, hence it depends on vidual does. The relative number of susceptible the proportion of infected hosts in the population individuals to all individuals encountered by an (McCallum et al. 2001). infected host equals the proportion of suscepti- However, De Jong et al. (1995) pointed ble individuals in the population: S/N. The rate out that “density-dependence” was not the best of transmission success following an encounter description of the Mass Action assumption since with a is often assumed to S and I sometimes designate numbers instead of be a constant: b. Thus transmission is described densities. They demonstrated that in such a case, by (bSI/N)f(N), where f(N) designates the if density is kept constant, Mass Action should number of contacts with other individuals per be characterised by bSI/N. Consequently they unit of time per individual. The key issue in the called the previously named Mass Action term transmission term is how to best model point (ii) “pseudo mass action”, while they considered the above, that is f(N) (De Jong et al. 2002). frequency-dependent as the “true mass action” Historically the transmission was assumed term. Considering that this new terminology to follow the Mass Action principle. This prin- had brought confusion, McCallum et al. (2001) ciple is a model that describes an ideal situation recommended building models based on density ANN. ZOOL. FENNICI Vol. 40 • Thresholds in host–parasite dynamics 121 rather than on numbers, thereby restricting the transmission term. In 1979, Anderson and May density-dependent term for the Mass Action (1979) found a good fi t between empirical data assumption. Although we concur with the call and the Mass Action transmission term of their for harmonisation, we remain unconvinced that simple model. But several more recent studies population density is always more biologically have highlighted a lack of concordance between relevant (or easier to measure) than population this modelled transmission term and real data, size. This concept is fundamental, as the relation- including tests by Dwyer (1991, 1993), De Jong ship between density and population size may et al. (1995), Begon et al. (1999), Knell et al. vary in several different ways, each of which can (1996, 1998b) and DʼAmico et al. (1996); but differentially affect how Mass Action is trans- note that the frequency-dependent term also lated. For instance, density and population size often did not provide an adequate fi t of the data may vary simultaneously, or one may remain (see discussions in McCallum et al. 2001, Fenton constant while the other varies with the occupied et al. 2002). Perhaps it is best to consider a more area. We suggest that the fi rst requirement in any fl exible non-linear transmission term, where, modelling exercise should be to defi ne precisely depending on the host-population size, trans- whether population size or density is being con- mission changes its behaviour, from density- to sidered (McCallum et al. 2001). frequency-dependent. Such a transmission term It should be noted, however, that the terms could be more appropriate for populations with frequency- and density-dependent transmission changing sizes, either decreasing (as studied in simply represent two ends of a continuum of the context of biological control or conservation transmission types. The pattern of transmission biology) or increasing (as studied in the context is likely to shift along this continuum accord- of invading populations). Among the different ing to the combination of densities studied and transmission terms that have been previously the scale of observation (Fenton et al. 2002). proposed, several exhibited this pattern (see In host–parasite systems, population size is a references in McCallum et al. 2001: table 1), dynamic variable, because pathogens often cause and a test of non-linear transmission term fi t- signifi cant host mortality. In this case, systems ness with experimental data seems convincing that can be well described by a density-depend- in this regard (see Fenton et al. 2002). However, ent term at low host population densities may be despite the advantage of being relevant for both more correctly described by a frequency-depend- low and high densities, such varying terms have ent term when the host population is larger. This two drawbacks. First, they introduce non-lineari- is due to the fact that, for very large population ties, making analytical studies markedly more sizes or densities, the number of new encounters diffi cult or even impossible. Second, they imply made by a single individual cannot increase lin- the choice of a function of f(N) and therefore early with population size, either because over- the models become more complex, while one crowding reduces the movements of individuals, may prefer to keep them simple (McCallum et or because a related increase of occupied habitat al. 2002). prevents its total occupation by individuals. The choice of which transmission term is In fact, when density is suffi ciently large, the used to describe the dynamics of host–parasite encounter function saturates: f(N) becomes a relationships appears to be crucial, as the exist- constant independent of population density (i.e. ence of thresholds depends on this term. Ker- frequency-dependent). In these cases, a non- mack and McKendrickʼs demonstration (1927) linear function for f(N) that equals from 1 to of a threshold was based on the study of models N according to density would provide a suitable using a density-dependent transmission term. In transmission term for all densities exhibited by contrast, Getz and Pickering (1983) showed that, a population (Hochberg 1991, Barlow 2000, depending on what exact transmission term is Fenton et al. 2002; see also Fig. 1). employed, a threshold might or might not exist. Only a few studies have assessed the Particularly, if the transmission is expressed as a applicability of simple models with frequency-dependent term, mathematical models available data, in particular with regards to the yield no threshold (but see below). Indeed, there 122 Deredec & Courchamp • ANN. ZOOL. FENNICI Vol. 40

Frequency-dependent Density-dependent On the contrary, increased density exacerbates transmission term transmission term Host mortality density related mortality, and infected individu- term als die too rapidly to transmit infection. To our knowledge the upper threshold of frequency- dependent transmission models has not been pre- viously discussed in the literature. This is partly because some models used a constant mortality rate, resulting in the absence of this upper thresh- old, or because this threshold could not exist in natural conditions. Note that for any form of

Parasite transmission rate density-dependent mortality (i.e. any increasing curve — not necessary represented by a straight line, as shown in Fig. 2), the results are qualita- Host population density tively similar, as mortality still intersects once

Threshold below which Threshold above which for each transmission term, although usually at infection cannot spread infection cannot spread different values.

Fig. 2. Interactions between the mortality (thick line) and the two classical transmission (thin lines) terms of mathematical models. If the transmission term is higher Thresholds in host dynamics than the term for mortality, then the infection spreads in a healthy population. The intersection between the two Although more neglected than parasite thresh- functions, when it exists, represents the threshold for olds over the past half-century, thresholds that infection spread. If transmission is density-dependent (respectively, frequency-dependent), values of popu- directly concern the host population, often called lation density above (resp., below) this threshold are extinction or eradication thresholds, are also of those facilitating the spread of the disease. tremendous importance. Indeed, much recent sci- entifi c energy has been expended on the study of the specifi c or exacerbated dynamical processes is some interplay between transmission and of small populations. For example, environ- mortality terms (Fig. 2). Infection spreads in a mental variability, whether due to catastrophic healthy population if the transmission term is events or more minor, random changes in the greater than the term for mortality (combining local environment, may hasten the extinction of natural and disease induced mortality), otherwise a population that is not composed of enough indi- the infected hosts will die too rapidly to further viduals. Demographic stochasticity also weakens transmit the disease. Thus infection spreads only small populations. The small yearly variations in when the transmission curve is higher than the the numbers of deaths and births may be incon- mortality curve, and the intersection between sequential when populations are large, but they them constitutes a threshold. In case of a den- may have a considerable impact on small popu- sity-dependent transmission, a threshold exists lations. Similarly, ordinary variations in the pri- above which infection spreads and under which mary sex-ratio may have dramatic consequences it disappears (Fig. 2). In the case of a frequency- if the population is small (e.g., Clout & Merton dependent transmission term, there may be a dif- 1998). Small populations are also sensitive to ferent threshold below which infection spreads effects that are not directly related to dynamics, and above which it cannot spread. A non-linear including genetic drift and inbreeding depression transmission term would yield both thresholds, (Lande 1995). The most important consequence in between which infection may spread, and out- is that all these processes may contribute towards side which the host population is protected from a reduction in population size, thus leading to infection. In the case of the frequency-dependent extinction (Caughley 1994). Even though these model, the number of contacts between individu- phenomena are exacerbated in cases of small als is assumed to be constant: infection does not population size, they do not always translate into require a minimum population size to spread. size or density thresholds under which popula- ANN. ZOOL. FENNICI Vol. 40 • Thresholds in host–parasite dynamics 123 tions are doomed. Nevertheless, the interplay of are characterized by reduced growth rates that these processes, amongst others, has prompted nonetheless remain positive as density decreases. population biologists to utilize the interesting In these cases, there is no deterministic threshold. and useful concept of minimum viable popula- In contrast, a strong Allee effect occurs when the tion (MVP). By considering extinction as an reduction of the growth rate drops below zero inherently probabilistic phenomenon, conserva- and becomes negative under a given density tion biologists have attempted to estimate the threshold. When a population density falls under minimal size necessary for a given population this threshold, that population will continue to to have a particular likelihood of persisting for a decrease and enter an extinction vortex. certain length of time (Soulé 1996). The concept There are three main biological processes of MPV links many interacting factors, including that provoke an Allee effect: the reduced repro- population dynamics, environmental variations ductive effi ciency at low density (low probabil- (randomness, stochasticity, catastrophes), genet- ity that mates, gametes, or pollinators/fl owers ics, habitat quality, metapopulation structure and will encounter each other), the failure of an fragmentation (Soulé 1996). MVP does not pro- adequate modifi cation of the habitat (e.g., of the duce an extinction threshold and it is important soil, in the case of plants), and the decrease of not to mistake components of viability analysis, benefi cial social interactions (as in the case of such as genetic and demographic criteria, that can cooperative species). The Allee effect seems to yield extinction thresholds, for the MPV itself. be widely spread. It occurs in numerous species, ranging from insects (Fauvergue et al. 1995) to large mammals (Larkin et al. 2002). Although Allee effect empirical demonstration of this effect remains uncommon, many theoretical studies have dem- An important phenomenon exhibiting a signifi - onstrated its heavy impact on population dynam- cant extinction threshold is the Allee effect (for ics (Dennis 1989, Lewis & Kareiva 1993, Amar- recent reviews, see for example Courchamp et al. asekare 1998, Courchamp et al. 1999b, Gyllen- 1999a, Stephens & Sutherland 1999, Stephens et berg et al. 1999, Wang et al. 1999, Courchamp et al. 1999; A. Deredec & F. Courchamp unpubl.). al. 2000a, Courchamp et al. 2000b, Brassil 2001, While small populations usually benefi t from a Keitt et al. 2001, South & Kenward 2001, Wang high growth rate due to the lack of intraspecifi c & Kot 2001, Dennis 2002, Etienne et al. 2002, competition, certain small populations are subject Ferdy 2002, Wang et al. 2002). to Allee effects. In these cases, the populations are inhibited by low survival and/or reproduc- tion, and thus are characterized by low to nega- Spatial thresholds tive growth rates. On a formal level, there is a distinction between component Allee effects (that While population biologists have focused on the is, those positive relationships between any trait minimum population size or density, community of fi tness of the individuals and population den- ecologists have mainly focused on the minimum sity or size) and the resulting demographic Allee size of areas necessary for system viability (Soulé effect (the ensuing decrease of population growth 1996). The concept of an extinction threshold is rate at low size or density, Stephens et al. 1999). also used in spatial ecology, where it refers to the The demographic Allee effect may be of vary- minimum amount of habitat required for a popu- ing strength, and may or may not be characterized lation of a particular species to persist in a land- by an extinction threshold. Although sometimes scape (Lande 1987, Hanski et al. 1996, Hanski & mistaken as a necessary outcome of the Allee Ovaskainen 2000, Fahrig 2002). Metapopulation effect, the existence of an extinction threshold theory has been used to estimate the minimum is only one possible (albeit frequently encoun- amount of suitable habitat (MASH) necessary tered) consequence of the Allee effect. Wang for a population to persist (Hanski & Gilpin (2001, 2002) and Brassil (2001) judiciously 1991). The related Minimum Area Requirement suggested that weak Allee effects are those that (MAR) is a spatial equivalent of MPV (Soulé 124 Deredec & Courchamp • ANN. ZOOL. FENNICI Vol. 40

1996). Spatially explicit models, or those using rates (and possible parasite elimination) but also migration rates, usually predict a higher extinc- to high risks of extinction. In fact, the fate of tion threshold, suggesting that when dispersal infected populations that are subject to an Allee is spatially constrained, more habitat is required effect depends upon the respective position of the for population persistence (Bascompte & Sole Allee limit and the infection threshold (Fig. 3). 1996, Pagel & Payne 1996, Hill & Caswell Stochastic considerations aside, if the Allee limit 1999). Depending on whether their approach is is below the infection threshold, a decrease in essentially deterministic or stochastic, model- density could help eliminate the parasite, without ling studies depict the extinction threshold as the endangering population survival. If the Allee limit minimum amount of habitat below which either is above the infection threshold, however, the the equilibrium population size is null, or the host population is unlikely to get rid of the dis- probability of long-term population survival is ease by decreasing in size (e.g. due to the action lower than one (Fahrig 2002). Empirical studies of the parasite), because the fi rst encountered use a somewhat different defi nition of threshold: threshold on the declining curve is fatal to the an amount of habitat below which probability of host but not to the parasite. Obviously, the closer occupancy declines precipitously (Fahrig 2002). the two thresholds are to each other, the less per- The main factors thought to determine the spatial tinent is the question of their relative position, extinction threshold of a given organism are its especially in presence of stochastic variations. reproductive rate, its rate of emigration from Environmental stochasticity will increase not the habitat, habitat fragmentation and survival only the risk of passing under a threshold but also rate of the organism in non-habitat area (Fahrig the chances of recovering after passing below it. 2001). The value of this threshold is increased by Two points concerning these thresholds merit Allee effects caused by the diffi culty of fi nding a further discussion. First, parasites may debilitate mate, edge effects resulting from the fi nite extent hosts in such a way that a signifi cant part of of regions containing suitable habitat, and the the population is effectively removed from the infl uence of stochastic fl uctuations in life history viable population (i.e. for reproduction or social parameters, usually caused by regional environ- interactions), even if those debilitated individu- mental variations (Lande 1987). als remain alive. In this case, a simple census of There are of course interactions among these total individuals present in the population may dynamic processes; for example, environmental falsely give the appearance that the population stochasticity may push some populations below is further away from the Allee limit than it actu- the Allee limit, or perhaps even rescue other ally is. Second, the dynamics of both small-sized populations that have fallen below it. Similarly, populations and infected populations may act on density and migration play different roles for two different time scales, thus reducing the value the Allee limit and for spatial thresholds, often of making direct comparisons. Environmental or revealing complex relationships. But interactions demographic stochasticity, in addition to some that hold the most interest are certainly those Allee effects that impact survival, generally between host and parasite thresholds, since these have immediate effects. In contrast, dynamic two types of threshold have opposite infl uences processes related to the genetic consequences on the host population: one is detrimental (e.g., of small population size and Allee effects that Allee limit) and the other is benefi cial (e.g. para- affect reproduction will have effects on a greater site invasion or persistence thresholds), because time scale (at least one generation). Meanwhile, it contributes to its resistance to disease. parasites will often have immediate effects. The differences on the temporal scale between these processes will be reduced in species with short Interplay between host and generation times (such as insects). parasite thresholds Thresholds play an important role in several domains of applied ecology related to popula- In case of populations subject to an Allee effect, tion dynamics, where recognizing the interplay low density is synonymous to low infection between thresholds of parasites and of hosts ANN. ZOOL. FENNICI Vol. 40 • Thresholds in host–parasite dynamics 125

A Infection

Disease eradication Host population size P Parasite extinction threshold H is higher than host extinction threshold

Time Fig. 3. Importance of the B relative position of the host Infection and the parasite extinction thresholds for the persist- ence of one or the other species. — A: the parasite threshold is higher than Population that of the host, and thus extinction a decline will eradicate the parasite. — B: the host extinction threshold is the highest, and a decline will Host population size H doom the host population. Host extinction threshold is higher than Situation A is sought in epi- P parasite extinction threshold demiology and conserva- tion biology, while situation B is sought in biological control. Time may turn out to be crucial. In particular, we now gies against epidemics in all three categories. briefl y highlight three major fi elds in which Apart from the use of quarantine, which is rarely parasite thresholds, host thresholds, or both are possible in natural populations, the only way particularly signifi cant: epidemiology, conserva- to eradicate epidemics consists in driving the tion biology and biological control. infection below its invasion or its persistence thresholds. This is accomplished by decreasing the density of susceptible individuals below the Animal and human health parasite invasion threshold, such that the aver- age number of effective transmissions produced The focus of studies on the dynamics of para- by an infected individual is below one, which sites may be divided into three main categories: prevents infection from spreading. Reducing parasites affecting human populations, those the amount of susceptible individuals can be affecting wild or feral animals and those affect- achieved by their removal, or culling (Lafferty ing farmed animals. The dynamics of the host & Gerber 2002). For human populations or population (i.e. constant, density dependent, etc.) threatened species, where culling is not possible, is quite different for each of these three cases. the strategy consists in reducing the fraction of Yet, the existence of infection thresholds has the susceptible population by immunizing them resulted in the development of common strate- against the disease (e.g., Anderson 1994, Gren- 126 Deredec & Courchamp • ANN. ZOOL. FENNICI Vol. 40 fell & Harwood 1997). Much theoretical work possibly close to extinction thresholds. An objec- has been conducted to compare the effectiveness tive of most conservation programs is to increase of these two different options, for example in population size in order to keep it as far as pos- rabies (Anderson 1982a, Barlow 1996) or bovine sible above their critical size. However, as we tuberculosis (White & Harris 1995, Smith & have seen, an increasing population can push Cheeseman 2002). Theory predicts that the the population above the extinction threshold of Critical Community Size will increase as a vac- one or several parasite species, and thus generate cination program is undertaken in the population new and potentially disastrous infections. (Dietz 1982, Anderson & May 1991, Grenfell & Parasitism has a large impact in biological Harwood 1997) and this has been confi rmed by conservation (Minchella & Scott 1991, Viggers the success of some programs. The et al. 1993, Gulland 1995, McCallum & Dobson history of public health initiatives has indeed 1995, Saether et al. 1996, Woodroffe 1999, Sasal shown that vaccination has a tremendous poten- et al. 2000, Lafferty & Gerber 2002). Diseases tial, although some diseases, such as measles, may be a signifi cant cause of population collapse continue to resist eradication efforts (e.g., Bolker (Crawley 1992, Lafferty & Gerber 2002) and & Grenfell 1996, Keeling & Grenfell 1999). population extinction, in particular of small ones However, these strategies of disease control (Woodroffe 1999). For example, parasites may may present some risks, particularly in cases keep populations at a size where they are vulner- where possible host thresholds have not been able to the host population thresholds mentioned taken into account. For example, as quarantine above. Disease may also compromise conserva- forbids interactions between certain individuals tion projects such as reintroductions, transloca- and the rest of the population, it can reduce the tions or restocking programmes (Viggers et al. effective size of threatened populations, thereby 1993). These activities illustrate particularly well pushing it closer or even under its extinction the contrasting effects of population increase on threshold. Culling must also be employed with parasite and host extinction thresholds, and the caution, for if it is employed carelessly, it could resultant potential dangers to the population. push the host population under both the parasite The enlargement of populations may favour the and the host extinction thresholds. Vaccination spread of infection, either of already existing can be an option in farmed animals but there parasites that were kept below invasion thresh- is no evidence that it is effective in protecting olds, or of new parasites that were introduced threatened mammal populations, partly because with the reintroduction activities (Lafferty & an understandable unwillingness to leave some Gerber 2002). The debilitation and stress related unvaccinated animals as control prevents effec- to captivity and subsequent release into a new tiveness monitoring (Woodroffe 1999, Lafferty & environment also increase the risk of epidemics Gerber 2002). Moreover, vaccination has some- among the reintroduced or reinforced popula- times been rejected by conservationists, because tions (Scott 1988, Lafferty & Gerber 2002). it was too expensive, logistically diffi cult (espe- The old controversy between creating a cially when multiple doses are necessary) or Single Large reserve Or Several Small ones considered unsafe for the animals (Woodroffe (SLOSS) is pertinent in the case of parasites: 1999). The latter was the case, for example, with a single large population would be more easily Canine Distemper Virus vaccines in programmes threatened by epidemics, while several small to protect Ethiopian wolves following the debate populations would be susceptible to the threat over the problems encountered by African wild of extinction, because they will be closer to their dogs and black-footed ferrets (Woodroffe 1999). extinction thresholds. The same type of predica- ment concerns the establishment of connections of habitats (and thus of populations) by corri- Conservation biology dors: while they can protect small populations from extinction by acting as artifi cial enhancers Conservation programs involve endangered of population sizes, at the same time the cor- populations that are often declining and thus ridors render them more vulnerable to existing ANN. ZOOL. FENNICI Vol. 40 • Thresholds in host–parasite dynamics 127 parasites, or to parasites initially present in only tion below this threshold. Again, this is possible, one population (Hess 1994, Hess 1996, McCal- in theory, only if the host extinction threshold is lum & Dobson 2002). The quarantine strategy above the parasite extinction threshold. Criteria mentioned above is one possible solution for the choice of a predator or parasitoid spe- advocated in this context (Hess 1996). Because cies as a pest control agent should thus include increasing population size can save populations consideration of the population size of optimal from extinction thresholds while also making effi ciency and possibly extinction thresholds, as them more susceptible to parasites that have well as possible parasite thresholds that could been concurrently (and unwittingly) unleashed, hinder their effi cacy as control agents (Hopper conservation managers have much to gain in & Roush 1993). paying greater attention to extinction thresholds.

Conclusion Biological control Nee (1994) states that the extinction threshold Biological control is the use of living organisms for a population can be understood simply as the as pest control agents. One common action of unused amount of its limiting resource. This rep- biological control is to introduce a natural enemy resents susceptible individuals in epidemiology, into the population targeted for reduction or prey in the case of biological control by preda- elimination (Waage & Greathead 1988). Here, tors or parasitoids, and habitat or the number parasites play a double role: they are important of available mates in conservation biology. The control agents, but they can also be hindrances focus of attention of these major domains of to the control program if they affect the control applied ecology is driven by the same concern: agent, either a predator or a parasitoid. It has the eradication of species (Nee 1994, Bascompte been argued that ecological theory has not been & Rodriguez-Trelles 1998). It has long been rec- useful in guiding the choice of natural enemies ognised that extinction or eradication thresholds or in developing release plans (Murdoch & are of tremendous importance whether one is Briggs 1996). In particular, until very recently interested in the control of parasites (epidemiol- there were no theoretical rules for decisions ogy, conservation biology) or in the control by concerning the number and size of released bio- parasites (biological control). What has perhaps logical control agents, in the context of avoiding been overlooked in the past is that the opposite effects of interacting host and parasite thresholds extinction thresholds (Grevstad 1999). may have important and fascinating implications As control agents, parasites are exploited for the study of host parasite dynamics. for their potential to spread in the pest popula- tion. This propagation is possible only if the host population exceeds the infection threshold. Acknowledgments Whereas this could be easily achieved during the fi rst stages of the epidemics, it may become less The authors wish to thank Marc Girondot and Matthew God- propitious as the host population is reduced and frey and three anonymous referees for helpful comments. the parasite extinction threshold is approached. As feral cat density on Marion Island decreased in response to the introduction of Feline Pan- References leucopenia Virus, the effi ciency of this control agent decreased and stabilised at a level of about Amarasekare, P. 1998: Allee effects in metapopulation 20% of the original feral cat population (van dynamics. — Am. Nat. 152: 298–302. Rensburg et al. 1987, Bester et al. 2000). If the Anderson, R. M. 1982a: Fox rabies. — In: Anderson, R. M. host population is sensitive to its own extinction (ed.), Population dynamics of infectious diseases. Theory and applications: 242–261. Chapman & Hall, London. threshold, due to an Allee effect, then the objec- Anderson, R. M. 1982b: Population dynamics of infectious tive of the introduced parasite is more likely to diseases. Theory and applications. — Chapman & Hall, succeed, as it must only reduce the host popula- London. 128 Deredec & Courchamp • ANN. ZOOL. FENNICI Vol. 40

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