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Extinction Thresholds in Host–Parasite Dynamics

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Extinction Thresholds in Host–Parasite Dynamics Ann. Zool. Fennici 40: 115–130 ISSN 0003-455X Helsinki 30 April 2003 © Finnish Zoological and Botanical Publishing Board 2003 Extinction thresholds in host–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 epidemiology. 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 transmission 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 infection 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 infection rate, 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 viruses; 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 infections (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-
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