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Parasite—host coevolution R. M. May and R. M. Anderson Parasitology / Volume 100 / Supplement S1 / June 1990, pp S89 ­ S101 DOI: 10.1017/S0031182000073042, Published online: 06 April 2009

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Parasite-host coevolution R. M. MAY* and R M. ANDERSON Department of Pure and Applied Biology, Imperial College, London SW7 2BB Key words: parasite—host coevolution, virulence, population dynamics, population genetics. INTRODUCTION

In this paper we wish to develop three themes, each having to do with evolutionary aspects of associations between hosts and parasites (with parasite defined broadly, to include viruses, bacteria and protozoans, along with the more conventionally defined helminth and arthropod parasites). The three themes are: the evolution of virulence; the population dynamics and population genetics of host—parasite associations; and invasions by, or 'emergence' of, new parasites. Much has been written on the first two of these three themes over the past decade and more, and we have little that is new to say here. But older misconceptions — that 'successful' parasites evolve to be harmless to the host, or the implicit assumption that genetic polymorphisms maintained by interactions between host and parasite populations will always be stable - persist. We therefore thought it appropriate to give a brief account of the first two themes in the context of the present volume; we present a bald summary of what we believe to be the central facts and issues, with a signposted guide to the underlying literature. The advent of human immunodeficiency virus, HIV, the causative agent of AIDS, along with other apparently new, or newly significant, infections of human and other animals has focused attention on the ecological and evolutionary circumstances surrounding the emergence of a new parasite. On an evolutionary time-scale such questions are, of course, not new. Essentially all directly-transmitted viral and bacterial infections of childhood have emerged and established themselves in human populations only over the past 10000 years or so (such infections typically require populations in excess of a few hundred thousand for endemic maintenance, and such population aggregations did not exist before the agricultural revolution). T h e questions, however, can seem different when the invasion or emergence of the new parasite is in our own time. Our third section therefore reviews some recent work on the emergence, establishment and spread of new parasites. In particular, we summarize the fragmentary * Reprint requests to Dr R. M. May, Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS. Parasitology (1990), 100, S89-S101

Printed in Great Britain

evidence that is currently available about the emergence of H I V / A I D S . Molecular taxonomy suggests that HIV first appeared in human populations (probably in Africa) over 100 years ago. HIV has been tentatively identified in stored samples of blood serum dating back about 30 years, and it is probably unreasonable to expect earlier evidence even if HIV was then present at relatively low levels in rural Africa. A puzzle remains. Given anything like current doubling rates of HIV seroprevalence in Africa, it is very hard to see how HIV could have arisen in rural regions over a century ago, without the epidemiological processes of exponential growth having long ago brought the incidence of A I D S to conspicuous levels. Vague ideas abut H I V 'smouldering in rural areas' cannot easily be reconciled with traditional epidemiological models. By focusing on how heterogeneities of various kinds can affect the invasion and early spread of an infectious agent, we show that simple estimates of doubling times and rates of spread (based on homogeneous models) can in some circumstances be misleading. Specifically, complexities associated with transmission of infection within and between many weakly communicating villages or other relatively small groups (and, to a lesser extent, complexities associated with infectiousness over the long and variable incubation interval from HIV infection to the onset of AIDS disease) can confound simple estimates. The work is new, and we consequently present it in more detail than the rest of the paper (with the mathematical details banished to Appendices). Although it may seem a bit technical, we believe this work is interesting, partly because it suggests a possible reconciliation of epidemiological data with a century-old origin of HIV, but more importantly because it highlights the complications attendant upon spatial and temporal heterogeneity. Such heterogeneities are often not reckoned with in simple studies of the evolution and ecology of host-parasite associations - it does, after all, make sense to crawl before you walk - but we think they will increasingly need to be dealt with.

EVOLUTION OF VIRULENCE

The conventional wisdom about parasite harmlessness continues to be set out in many text books in biology, medicine and veterinary science. T h e belief rests on both theoretical and empirical grounds. T h e

R. M. May and R. M. Anderson theoretical arguments are usually flawed, being openly based on group-selectionist arguments about what is good for the parasite species (some representative quotations are collected in May & Anderson, 1983). The empirical arguments are more equivocal. Allison (1982), for example, gave a collection of case studies in support of the view that parasitic infections (in the broad sense defined in the Introduction section) are relatively harmless to those animal hosts with which they have had a longestablished relationship; the same parasites have a serious impact on newly introduced host species, or when the parasites themselves are introduced into new regions. Against this are the facts that for many parasites (especially those of invertebrate hosts) death of the host is an integral part of the transmission cycle, or that transmission efficiency may be enhanced by modifying the host's behaviour in such a way as typically to shorten its life. These contending verbal arguments can be brought into sharper focus by explicit definition of the 'basic reproductive rate', Ro, of the parasite (Anderson & May, 1979, 1981, 1982). In essentials, Ro measures the average number of offspring surviving to the reproductive stage that each reproducing parasite is capable of producing, under natural circumstances but not limited by effects due to parasite density. Thus defined, Ro is the quantity that natural selection will tend to maximize for individual parasites. If the parasite is established within a host and producing transmission stages at some fixed rate (per day, say), then - all other things being equal — Rg will clearly be maximized by the host surviving as long as possible. But rarely are all other things equal, and more generally the harm done to a host, by damage to internal organs or adverse modification of behaviour or in other ways, will be directly related to the production of transmission stages of the parasite. That is, maximizing Ro involves trade-offs between production of transmission stages (which ideally should be high) and damage to the host (which ideally should be small). If the life-history of a particular parasite is such that high fecundity can be realised in a host-harmless way, then the conventional wisdom will pertain. But, in general, things will be more complex, and the evolutionary trajectory may be toward some intermediate grade of virulence, or even (as in the case of rabies and many infections of invertebrates) toward high virulence. For a more analytic discussion, it is useful to make a distinction between microparasites and macroparasites (Anderson & May, 1979). Microparasites, which are typified by most viral, bacterial and many protozoan parasites, are essentially defined as those for which it makes sense to partition the host population into a relatively small number of categories (for instance, susceptible, infected-andinfectious, recovered-and-immune). The basic re-

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productive rate for microparasites Ro, may accordingly be defined as the average number of secondary infections produced by an infected individual, in a population where virtually all are susceptible. Conversely macroparasites, which are typified by most helminth and arthropod parasites, are essentially those for which the parasite's vital rates and the harm done to hosts depend on the number of parasites within an individual host, so that mathematical models must deal not with a few categories of host, but rather with the full distribution of parasites among hosts. In this case, Ro is essentially the average number of sexually mature parasites produced by each adult parasite, when the parasite population is at sufficiently low density for densitydependent effects to be negligible. Both biological and mathematical aspects of these definitions are discussed more fully elsewhere (Anderson & May, 1979). For a microparasite, Ro may in the simplest case be expressed as Ro = A(N)/(a + b + v).

(1)

Here A(N) represents the rate of production of successfully established transmission stages (which in general will be a function of host population density, N, because this will affect the probability of establishment in a new host), a is the diseaseinduced death rate of hosts (henceforth called the 'virulence'), b is the host death rate from all other causes, and V is the recovery rate. From the simple equation (1), it is obvious that if transmission, A, and recovery, v, are independent of parasite virulence, a, then Ro is maximized by having a -> 0; this represents a harmless, avirulent parasite. The problem, as stated more generally above, is that A and v are typically dependent on (that is, functionally related to) a. For example, if A were to increase linearly with increasing a, then Rg is maximized by having ct^- oo, corresponding to extreme virulence. In this event, natural selection would proceed down this perverse road, possibly even to the point of extinguishing the host population and thence the parasite; natural selection maximizes the reproductive success of individuals, not the 'good of the species'. What are needed are factual data about the functional relationships among A, v and a, derived from the life-history details of specific host-parasite associations. In no single instance do we have enough information to determine the likely evolutionary trajectory dictated by maximizing Ro in equation (1). The closest we can come is for the story of the Australian rabbit and the myxoma virus. This example has been analysed in detail by May & Anderson (1983). The simple model of equation (1), when combined with such data as are available about the relevant functional relationships, predicts evolution of the virus to a grade of virulence intermediate between the highly virulent strains originally intro-

Parasite-host coevolution

duced and the low virulence strains that have appeared in the field (but which have remained at low frequency). This prediction seems to accord with events both in Australia and in the UK; see May & Anderson (1983). Things get more complicated if a single host is likely to harbour two or more strains of the parasite. In this situation, a virulent strain can come to predominate, even though its i?0-value is lower than a less virulent strain (which clearly would prevail when ' superinfection' was irrelevant). T o comprehend this, contrast a strain that produces, say, 10 transmission stages/week and (by so doing) kills the host after an average of 2 weeks with a more virulent strain producing 15 transmission stages/week but killing the host after 1 week. The first strain has Ro = 20, and the second Ro = 15. If the infection was at low density, or if superinfection by a second strain was not possible, the first strain would be selected for. But doubly-infected hosts will die after 1 week, producing 15 transmission stages of strain 2 and only 10 of strain 1 (which no longer realises its full Rovalue of 20). Thus, if superinfection is common, the virulent strain 2 will come to predominate, despite its lower i?0-value. This situation has been analysed in game-theoretic detail by Bremmerman & Pickering (1983). The above discussion has focused on the parasite. More generally, consideration of the coevolutionary trajectories of host-parasite associations involves trade-offs not only between transmission and virulence, but also among the costs and benefits of host resistance. These and other issues are dealt with in the growing literature on this subject: see the survey by Levin et al. (1982), and also May & Anderson (1983), Levin & Pimentel (1981), Bremmerman (1980), Seger (1988) and Seger & Hamilton (1988). The fundamental point to emerge from all this work is simply that theory does not dictate that parasites should evolve toward a commensal association with their hosts, but rather that many coevolutionary trajectories are possible, depending on the details of the parasite's life-history. In view of the great range of situations found in nature, this theoretical pluralism is just as well. CHAOS AND THE DYNAMICS AND GENETICS OF HOST-PARASITE ASSOCIATIONS

The interactions between hosts and parasites - be they microparasites or macroparasites — are essentially one among many particular kinds of preypredator associations. As such, there is increasing recognition that the dynamics of the association may be stable levels of host and parasite density, or it may be stable limit cycles (in which both amplitude and period of the cycle are set by the biological parameters characterizing the association), or even deterministic chaos (in which simple, deterministic

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equations produce apparently random fluctuations; for general accounts of chaos, see May (1976), Gleick (1987) and Stewart (1989)). In the parasitological literature, however, it is still often assumed that host—parasite associations will result in steady average levels of infection within a population, or at most in regularly cyclic oscillations. But Crofton's (1971a, b) basic model for the dynamics of a host—macroparasite association yields exceedingly complex dynamics, complete with regimes of chaotic behaviour (May, 1977, 1979). These properties of Crofton's pioneering model do not seem to be widely appreciated. By the same token, the correspondingly simple model for a host—microparasite association in discrete time (that is, where the host population has discrete, nonoverlapping generations, as in Crofton's model) has only chaotic solutions, although for small growth rates of the host population the fluctuations are approximately periodic (May, 1985). These dynamical complexities carry through to models of host-microparasite or host-macroparasite associations in continuous time (where hosts are born at any time of the year, as in human populations). Thus it seems increasingly likely that the irregular 2-year cycles in the incidence of measles in the USA and the UK before the advent of immunization is not just a roughly 2-year prey—predator ('Lotka-Volterra') cycle with superimposed random noise, but rather that the irregular, quasiperiodic time-series of weekly or monthly case notifications is generated by a low-dimensional chaotic attractor (Schaffer & Kot, 1986; Schaffer, 1987; Sugihara & May, 1990). Once such host-parasite interactions are embedded in more complex multi-species systems, the potential for dynamical complexity is even greater. Thus, simple 3-species interactions among a host, an insect parasitoid, and a pathogen can result in two alternative states (with the system converging on one or the other, depending on the starting values), either or both of which may be chaotic (May & Hassell, 1988). A simple model for the interaction between HIV and the immune system (essentially a predator-prey interaction) can lead to chaotic fluctuations when an opportunistic infection subsequently arises (a second predator is introduced); this simple model mimics many observed features of infection with HIV in humans (Anderson, 1990). The essential point here is that the interactions between hosts and infectious agents tend to give rise to cyclic patterns of overshoot and overcompensation, just as in other prey-predator systems. Whether these tendencies are damped out to produce stable levels of association, or whether they manifest themselves as stable limit cycles or even as the apparently random fluctuations of deterministic chaos, depends on the biological details of particular associations. This spectrum of dynamical possi-

R. M. May and R. M. Anderson bilities should be kept in mind whenever we are studying a host-parasite system, be it in a purely descriptive way, or analytically with the help of mathematical models or computer simulations. Turning from the dynamics to the genetics of host—parasite associations, we recall Haldane's (1949) observation that diseases have been the main selective force acting on humans at least since the agricultural revolution, and probably much longer. In his paper, Haldane also observed that infections tend to spread more easily in high-density host populations. Thus, in a situation where there are several host genotypes, each particularly susceptible to its own genotypes of parasites, there will ensue frequency-dependent selection; at any one time, those host genotypes that are common will be at a selective disadvantage (because the parasites afflicting them spread relatively easily), while rarer host genotypes enjoy a selective advantage (being relatively free of parasites by virtue of host rarity). Haldane suggested such frequency-dependent selection could be a mechanism helping to maintain genetic polymorphism in both host and parasite populations, and he even gave a simple model to illustrate this (unfortunately, the typographic errors in Haldane's paper are so numerous that the mathematical model is incomprehensible to us). T h e idea that frequency-dependent selection can maintain genetic polymorphisms was subsequently developed (for instance, by Haldane & Jayakar (1963) in general terms, and by Gillespie (1975) in a specific host—microparasite context). Essentially all this work, however, deals only with the static properties of the population genetics of host-parasite associations, implicitly assuming that if a polymorphism can exist then it will persist stably at a constant level. But, as emphasized by Hamilton (1980) and May & Anderson (1983), the dynamics of gene frequencies in host—parasite associations are such that cyclic or chaotic fluctuations arise naturally and easily. That is, the non-linearities inherent in the interactions between different genotypes of hosts and different genotypes of parasites indeed seem likely to maintain genetic polymorphisms in many situations, but the polymorphisms so maintained may be stable, or they may vary cyclicly or in a chaotic, apparently random, manner from generation to generation. These ideas have been developed in detail elsewhere (May & Anderson, 1983; Seger 1988, Seger & Hamilton, 1988). Fig. 1 gives a representative illustration, showing chaotic fluctuations over time in the frequency of the allele A, in a 2-allele host polymorphism; this illustration is based on a very simple, and completely deterministic, model for the epidemiology and genetics of a host—pathogen association. Hamilton (1980, 1982) has gone further to suggest that the existence of such chaotic fluctuations in gene frequency (produced by parasites, pathogens

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CD O" CD

CD

c O

10

Time, t

20

30

Fig. 1. Illustrating the chaotic dynamical behaviour of the gene frequency, pt, of the allele A (in a simple 2allele polymorphism), as a function of time (or generation), t, that can arise once the selective forces exerted by pathogens depend both on gene frequency and on host density. The figure is for a haploid host, and shows successive iterates of pt, as generated by the appropriate first-order difference equation, with epidemiological considerations determining the fitness functions. (Data from May & Anderson (1983), where the parameter values and other details are given.) or other selective agents) may play an important part in the evolution and maintenance of sexual reproduction, giving rise to effects that offset the factor-of-two cost often associated with having two sexes. T h e essential message here is, first, that interactions between hosts and parasites tends to promote genetic polymorphism in both populations and, second, that such polymorphisms may be steady, or cyclic, or chaotic. T h e first of these facts is fairly widely recognized, but the second is not. Yet the potential existence of chaotic polymorphisms cuts across the currently pervasive assumption that polymorphisms - if they exist - will be roughly steady over time and space or, at worst, will be regularly cyclic. This assumption is often made implicitly, without even recognizing that it is an assumption and that a wider range of dynamical possibilities can easily arise. Viewed in this light, the findings by Burdon (1990) and Forsythe et al. (1988) of variable patterns of polymorphism in neighbouring populations (of plants afflicted by a pathogen, and humans afflicted by malaria, respectively) are not surprising. We need more of these well-designed kinds of studies, looking systematically at dynamic and genetic aspects of populations that are structured in time and space. INVASION OR EMERGENCE OF NEW VIRAL AND OTHER PARASITES

T h e emergence of HIV as a new and lethal parasite of humans has quickened interest in the ecological

Parasite-host convolution

and evolutionary criteria for success of such novelties. HIV is, moreover, only the most conspicuous of a range of viral and other parasites of humans and other animals that have recently emerged, or invaded new regions. Other examples surveyed at a recent meeting on ' Emerging Viruses' include several kinds of haemorrhagic fever (Marburg virus, Ebola virus, Lassa fever, hantaan virus, Seoul virus, and others), Kyasanur Forest disease in India, O'Nyong-nyong in Uganda, and Rocio encephalitis in Brazil (Miller, 1989). At the same time, dengue fever has dramatically expanded its range and produced a more serious set of symptoms (dengue haemorrhagic fever), and new viral diseases have also been observed in wild and domestic animals. Much of the discussion about the emergence of new parasites (broadly defined, as ever) focuses narrowly on the molecular mechanisms underlying the pathogenicity of the organism. For instance, Rosqvist, Skurnik & Wolf-Watz (1988) have shown that a single point-mutation may be important in determining the virulence of Yersinia pestis, the bacterium causing bubonic plague. Rosqvist et al. (1988) based their work on elucidation of the genetic determination of virulence in Y. pseudotuberculosis, which is a close relative of Y. pestis. Building on earlier work which implicated two outer-membrane proteins (invasin, which is encoded chromosomally, and Yopl, which is encoded by a plasmid) in mediating the invasion of mammalian cell cultures by Y. pseudotuberculosis, these authors showed that mutations in one or other of the genes encoding for these proteins have little effect on the virulence of Y. pseudotuberculosis in mice, but that mutations in both genes result in substantially heightened virulence. Y. pestis apparently expresses neither of these two proteins, consistent with its much greater virulence relative to Y. pseudotuberculosis and with its inability to grow in mammalian cell cultures. Rosqvist et al. (1988) sequenced the yopA gene from Y. pseudotuberculosis and Y. pestis, and found only 15 nucleotide differences among some 1200 base pairs. One of these differences, however, was a one-base deletion that throws off the reading-frame in Y. pestis. In confirmation of their ideas, the authors introduced the functional gene for Yopl from Y. pseudotuberculosis into Y. pestis, and observed a corresponding reduction in the virulence of Y. pestis. They concluded that a single point-mutation could have played an important part in affecting the virulence of the plague organism Y. pestis and, consequently, in triggering plague epidemics. Crucial though such molecular insights are to our understanding of the emergence of new parasites, they are only part of the story. As Lenski (1988) has emphasized, 'mutations alone cannot drive epidemics.... There remain equally perplexing questions concerning the selective pressures that [are] responsible for the increase in the frequency of

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hypervirulent strains, once they appear by mutation'. It is one thing - albeit an important t h i n g to understand the molecular basis for pathogenicity, but it is something quite different to understand why the transformed organism does or does not have a basic reproductive rate, Ro, in excess of unity in a particular place at a particular time. This criterion, Ro > 1, must be fulfilled if a parasite is to invade and establish itself. If Ro > 1, each case produces, on average, more than one subsequent infection, and a 'seed' of infection will spark an exponentiating chain of epidemic spread. As time goes by, the epidemic will slow down as the pool of susceptibles is consumed, so that the effective reproductive rate of the infection must be discounted in proportion t o the fraction of the host population remaining susceptible; the epidemic may eventually burn itself out, leaving the host population to await the slow replenishment of susceptibility by births and a new epidemic triggered by future invasion or mutation, or it may exhibit slowly damped oscillations to an eventual state of endemic infection. On the other hand, if Ro < 1 each infection produces fewer than one progeny, and the parasite simply cannot invade o r establish itself. Since the parasite's transmission efficiency, A(N), will tend to be greater for higher host density, the criterion Ro > 1 usually translates into an equivalent threshold criterion, N > NT. Here NT is the threshold host density, above which parasite invasion and establishment is possible, and below which it is not. Thus any discussion of the ecology and evolution of an emerging or invading parasite must reckon with population-level considerations of Ro and NT, as well as with molecular-level considerations of the mechanisms whereby the parasite affects the host. By definition, Ro (and NT) generally involve both biological and social factors. It follows that emergence of a new parasite may be triggered by mutations that result, for example, in enhanced transmission (and thus increased Ro), but equally emergence may be triggered by changes in the ecological setting (independent of any changes in the interaction between individual hosts and individual parasites) that, for example, cause greater crowding (thus increasing Ro, or bringing N above NT). All this has been discussed much more fully, in the context of invading pathogens in general, and H I V / A I D S in particular, by Anderson & May (1986). The remainder of this section is devoted to aspects of the emergence of HIV, as a special case of the above ideas. This discussion of HIV motivates the following two sections, dealing with how estimates of the rate of spread of infection may be complicated by spatial heterogeneity, or by a long and variable period of infectiousness; the new material in the two sections may, however, have wider applicability. PAR s

R. M. May and R. M. Anderson The molecular history of HIV A rough picture of the evolutionary history of the various forms of the HIV virus can be obtained from appropriate studies of molecular sequences. Nucleotide substitutions in retroviral (RNA) genomes occur about a million times faster than in the nuclear DNA of higher genomes, essentially because there seem to be no 'proof-reading' mechanisms. To reconstruct the phylogeny of HIV-1 and HIV-2 in relation to other groups of primate lentiviruses (SIVMAC in macaques, SIVAGM in African green monkeys), Sharp & Li (1988) (see also Li, Tanimura & Sharp, 1988), Yokoyama, Chung & Gojobori (1988) and others have examined the more conserved parts of the genome, such as the pol gene. They conclude that HIV-1, HIV-2, and SIVAGM diverged from each other somewhere around 140-160 years ago, and that the divergence between HIV-2 and SIVMAC is more recent, around 30 years ago. The phylogenies similarly assigned to the various strains of HIV-1 and HIV-2 (where divergences are around 0 to 20 years old) accord with phylogenies inferred from purely epidemiological arguments, which engenders some confidence in the estimates about divergences at higher taxonomic levels. In short, and although all such estimates must be hedged with many caveats, the ticking of molecular clocks currently suggests that HIV first appeared in humans well over a century ago. Doubling times for HIV seroprevalence If HIV emerged and established itself in human populations over a century ago, then it must have had Ro > 1 over that span. It could perhaps be that the HIV virus, or something like it, has appeared in human populations many times in the past, as a result of mutations, but with Ro < 1 ; in all such events, the virus will have failed to invade and maintain itself. We can speculate that increasing population density in Central Africa and other countries, coupled perhaps with increasing rates of acquiring new sexual partners in relatively recent times, may only in the past century or so have brought about a situation where human ecology and human behaviour result in Ro > 1 for a virus with the life-history characteristics of HIV. As discussed more fully elsewhere (May & Anderson, 1979; Anderson & May, 1982), the general questions here are not new. Most directly transmitted viral and bacterial infections of humans - measles, smallpox, chickenpox, and so on - require population aggregates of a few hundred thousand for the infection to remain endemic (Ro > 1). Such infections thus must have emerged only as human numbers climbed above threshold values in the wake of the agricultural revolution. Once a new infectious agent has appeared, by

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mutation or otherwise, in an ecological and evolutionary setting where it has Ro > 1, the prevalence of infection will at first tend to increase exponentially: y(t)/A

^exp(At).

(2)

Here y(t) is the fraction infected at time t, A represents the initial ' seed' of infection (z) = y(fi)), and A is the initial growth rate in the prevalence of infection. In this early growth phase, levels of seroprevalence or other measures of the prevalence of infection will increase in compound-interest fashion, with a doubling time, td, that is trivially related to A: td 2^ (In T)lA.

(3)

Simple models, in which the host population is treated as homogeneously mixed, give a further relation between A and Ro which enables us to write

Here D is the characteristic duration of infectiousness. Specifically, for simple models of the transmission dynamics of HIV, Ro = ficD and A = flc—\/D (with /? the probability that an infected individual will transmit infection to a sexual- or needle-sharing partner over the duration of the relationship, and c the appropriately averaged rate of acquiring new partners). More details of this have been given, for example, by May & Anderson (1988). In recent years, the doubling times for HIV seroprevalence among the general population in large cities in Central Africa (as estimated from serological studies among pregnant women) is 2-3 years or less. It is inconceivable that HIV could have been doubling on that time scale since its estimated time of emergence over a century ago. This problem is usually dealt with by observing that doubling times may have been much longer in the past than over the last few decades (essentially because c, the rate of acquiring new partners, plausibly had lower average values in the past, resulting in smaller A and longer td), coupled with the observation that significant mortality from AIDS could possibly have been lost amidst the noise in rural health statistics. This explanation does not, however, stand up to analytic examination. Unless Ro is close to unity, equation (4) shows that td will be less than, or close to, the characteristic duration of infectiousness, D. But current estimates suggest that at most D ~ 6-7 years, and that a more appropriate value in estimates of td may be the duration of the early phase of peak infectiousness, resulting in D — 1 year or less (May & Anderson, 1988; Anderson, 1989). Taking the largest value of D to estimate td ~ 7 years, we would still have 20 doublings over the 140 or more years that Sharp & Li (1988) suggest HIV has been present in the human population; this represents an amplification by a factor 106 of the initial seed.

Parasite-host coevolution The only way to resolve this dilemma, within a homogeneous epidemiological framework, is to assume Ro happens to have remained extremely close to unity for most of the past century or more. If, for example, Ro = 107, then equation (4) shows that ta will be 10 times longer than D. Such an improbable numerical coincidence in the value of Ro could conceivably have arisen and been maintained, but it is an unsatisfactory explanation. Even with this additional factor of 10 or so, we still need D close to the upper limit of the range of estimates to get doubling times sufficiently long that the prevalence of HIV/AIDS would not long ago have risen to noticeable levels. We thus have a puzzle. History, as written in the molecular sequences, suggests that HIV first appeared in human populations more than 100 years ago. But conventional epidemiological models make it very hard to explain why HIV/AIDS, once it had emerged, could have taken so long to rise to levels that compelled attention, unless by some miracle Ro happened to be poised on the razor's edge fractionally above unity for a century or so. The next two sections suggest some possible answers.

SPATIAL HETEROGENEITY AND THE SPREAD OF INFECTION

In analysing the emergence and spread of a new parasite in rural regions of Central Africa or elsewhere, we must recognize that populations are not distributed homogeneously in space, but rather are clustered in many villages or other local aggregations. The probability for transmission of infection between individuals in the same village will typically be much greater than that between individuals in different villages. Hoppensteadt (1975) pioneered the formal analysis of models incorporating such spatial heterogeneity, and May & Anderson (1984) initiated exploration of the implications for the design of optimal programmes of immunization and for criteria of overall eradication of infection in systems of 'cities and villages' (for later elaboration of this work, see Hethcote & Van Ark, (1987), Andreasen & Christiansen (1990) and Diekmann et al. (1990)). All this work, however, deals with endemic infections; that is, with equilibrium states. We now show how spatial heterogeneity can affect the early spread of a newly emerged parasite. The presentation in the text is confined to an outline of the assumptions underlying a simple model and the conclusions that follow. Mathematical details are sketched in Appendix A. Consider a rural population made up of a large number, n (n P 1), of roughly equally sized villages. In this population, the average per capita birth and death rates are a and b, respectively; the overall population is thus growing exponentially at the rate r = a — b. On average, members of this population

S95 acquire new sexual partners at a rate c. The probability that an individual in village i will choose any one partner from village j, pti, is assumed to be ptj = K(Stj + e), where Stt is the Kronecker delta function (Stj = 1 if i = j, and St] = 0 otherwise) and K is the normalization constant K = 1/(1+we), ensuring that £)p(j = 1- This form for the 'who mixes with whom' matrix, p(f, corresponds to assuming the probability that an individual will choose a partner from any given village other than his or her own, relative to the probability of choosing the 'girl or boy next door', is e/(l +e); we assume e t-ept).

(6)

Here the rate coefficients p and p' are defined to be p = Pc-(a + v), p' = /lc/(\+ne)-(a

(7) + v).

(8)

These expressions can be understood intuitively, in biological terms. For the overall system defined above, the basic reproductive rate of the HIV virus is Ro = fic/(v + a).

(9)

This equation (9) differs from the approximate expression Ro = jicD = fic/v given earlier, because the complications associated with births and other deaths have been included. It might seem that the disease-associated death rate, v, in the approximate expression should be replaced by the augmented death rate v + b, but closer consideration shows that the excess of births over deaths in a growing population must also be reckoned with, resulting in the factor v + a in equation (9) (for a detailed discussion of the dynamics of infection in a nonstationary population, see May & Anderson (1985, 1988) and Anderson, May & McLean, 1988). The overall value of Ro, equation (9), may be partitioned 7-2

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R. M. May and R. M. Anderson into a component arising from contacts within the same village, Rol = Ro/(\+ne), and a component from contacts between villages, R02 = neR0/(\ +ne). If the basic reproductive rate for HIV exceeds unity from intra-village contacts alone, Rol > 1, things are relatively straightforward. In this case, we see from equations (8) and (7) that both p > 0 and p > 0, so that both yt(t) and yf(t) (t' 4= 1) grow exponentially once the infection is seeded in the village labelled 1. Unless Rol is only fractionally above unity, the seroprevalence doubling times will not be very different from those estimated by treating this concatenation of loosely-coupled subpopulations as if it were a homogeneous population. But if R01 < 1 (with Ro = R01+R02 > 1), a more complicated situation arises. Now p' < 0, although p > 0. It follows from equation (5) that y^t) will at first decrease below the initial value A. Only after some time (roughly T~ (\nn)/p) will the second term in equation (5) assert itself, and lead to exponential growth of y^t)/A above unity. While this is happening, seroprevalance levels in other villages are increasing, on average, from values that are very small early on. Fig. 2 illustrates this situation, for a set of parameter values that are a bit extreme — but not wholly unrealistic — as a metaphor for HIV in Africa. What is happening here is that levels of infection are initially falling in the focal village 1, because the basic reproductive rate is below unity for solely intra-village contacts (Rol < 1). At the same time, infection is trickling out to other villages, such that overall Ro > I. In each newly infected village, infection will at first tend to die out (/?01 < 1), but each local decrease must be seen against a rising tide of diffusion of infection to more and more villages. Eventually this diffuse background rises to levels where the combination of infections originating within a village plus infections imported from other villages makes the process selfsustaining in each individual village, and the seroprevalence levels really take off-as illustrated in Fig. 2. The situation illustrated in Fig. 2 does require that the epidemiological and demographic parameters be such that Rol < 1 while R0l+R02> 1 (which can fairly easily happen if ne exceeds unity). Although restrictive, this constraint is nothing like the exquisitely delicate requirement that Ro lies marginally above unity, which is essentially the only way to produce sufficiently long doubling times in a homogeneous system. Although indicative, the above model is ultimately a nonsense. It treats infection levels as continuous variables, even though yt{t) is initially small and at first gets smaller, while all other yt(t) are initially zero and become finite when the first infected individual arrives. Any accurate study must necessarily be based on demographically stochastic models. The rough analysis above suggests, however, that such

25

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75

100

125

Time, f (years)

Fig. 2. The solid curve shows the fraction infected in the village where HIV first appeared (as a ratio to the initial fraction infected, A), yx/A, as a function of the time, t, since the infection first appeared. The dashed curve shows the overall average value of the corresponding fraction infected, y,/A {i' #= 1), in any other village. This illustration is for the model defined in the text and Appendix A, with the parameters having the values /? = 0-1, c = 2/year, v = 0-1/year, and a = 005/year (which are not unreasonable estimates of these parameters in the early stages of the epidemic). For this parameter combination, p = 005/year and p' = — 005/year, corresponding to the infection's basic reproductive rate being below unity for within-village transmission alone (Rol < 1), although above unity overall (Ro > 1). The consequent initial decline in the levels of infection within any one 'seeded' village, against a rising tide of diffusion of infection among villages, is discussed in the text.

stochastic studies will tend to see initial chains of infection stuttering to extinction in any one village, while at the same time throwing off sparks to ignite new chains in other villages. We conjecture that, although each newly-sparked chain will tend to extinguish itself (Rol < 1), the overall number of ignition points will steadily increase over time (Ro > 1), until eventually a conflagration is lit. That is, we recognize the ideas encapsulated in Fig. 2 are based on a shaky (continuous) approximation to what ultimately must be analysed as discrete events, but we nevertheless believe these ideas give a qualitatively reliable account of the epidemiological complexities inherent in the kinds of spatial heterogeneities that arise in rural regions of developing countries. LONG AND VARIABLE INFECTIOUS PERIOD AND THE SPREAD OF INFECTION

A'lost of the above discussion of the emergence of HIV is based on simple models in which infected individuals remain infectious throughout an incubation period of average duration D — \/v (before they develop terminal AIDS). Current evidence suggests, however, that there may be two relatively

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brief episodes of infectiousness, the first immediately following initial infection with HIV and the second attendant upon the onset of AIDS Related Complex, ARC, and AIDS itself, separated by a relatively long 'silent period' during which the infected individual is essentially uninfectious (for reviews see May & Anderson (1988) and Anderson (1988)). Specifically, we may write the average durations of the initial and final phase of infection as Z), = 1 /vy and D3 = \/v3, respectively, with corresponding transmission probabilities during these episodes being fix and /?3; the intervening uninfectious phase has an average duration D2 = \/v2 (and fl2 = 0). Characteristic values for HIV may be D1 ~ 0-5-1 year, D2 a 6-7 years, and D3 — 1—2 years. There is also some suggestion that individuals might intrinsically be more infectious, per unit time, in the final phase than in the initial one, /?3 > fil. Such a pattern of varying infectiousness, over the long and variable incubation interval from infection with HIV to death from AIDS, can introduce significant complications into the epidemiological analyses. In particular, the basic reproductive rate of HIV, Ro, can be partitioned into two components, R = R

+R

(101

Here Rol = /?, cDl is the average number of infections produced by an infected individual during the initial phase of infection, and R03 = fi3cD3 is the corresponding reproductive rate of HIV for the final phase. Estimation of the initial doubling rate for HIV seroprevalence is now messier: by extending the basic model for the transmission dynamics we can show that the growth rate, A, no longer obeys the simple equation (4), but instead is given by the cubic equation 1 =

(11)

| 6

! 4 en c S 2 3 o Q

Fig. 3. Illustrating how the doubling time for HIV seropositivity, td, depends on the portion of the basic reproductive rate of HIV that derives from 'first phase' infections, Rol, when infectiousness varies over the long and variable interval between initial infection with HIV and death from AIDS. Specifically, the figure is based on the model defined in Appendix B, in which infected individuals initially produce new infections at a rate fit c for a period of average duration Z),, and later, after the onset of ARC and AIDS, produce infections at a rate /?3c over an average interval D3; the two infectious phases are separated by a silent interval of average duration D2. The doubling time then follows from equation (3) with A given by the cubic equation (11). Here the parameters have the plausible values Dy = 05, D2 = 6, D3 = 2 years. The overall reproductive rate of HIV is set at Ro (= filcDl+/33cD3) = 2. As discussed more fully in the text, when R0l = /?, cDl is significantly below unity, then td is essentially determined by D2, whereas when Rol is above unity td is determined by D1. Thus relatively modest changes in Rol (derived from changes in c) can result in td changing by an order-of-magnitude or more.

then for v2 -4 vlt v3 the growth rate A is given approximately by /j ~ v2(R0 —1)/(1 —Rol).

The derivation of this result is sketched in Appendix B. In a manner reminiscent of the preceding section, two situations can now be distinguished if- as seems to be the case for HIV - the average duration of the intervening silent phase is significantly longer than either of the two infectious phases; that is, if D2 is significantly greater than Dx and D3 (v2 significantly less than v1 and v3). If Rol > 1, so that HIV can maintain itself and spread on the basis of 'first-phase' infections alone, equation (11) has the approximate solution (for v2 < vlt v3) A ~ v^R^ — 1).

(12)

That is, the characteristic scale of A is set roughly by vx (and the corresponding scale of td is set roughly by

DJ. Alternatively, if Rol < 1 (but Ro = R01 + R03 > 1),

(13)

That is, the characteristic scale of ta is now set by D2, which for HIV is almost an order-of-magnitude larger than Dv The approximate results, equations (12) and (13), are derived, and their limitations discussed, in Appendix B. Fig. 3 shows how the early doubling time for HIV seroprevalence, td, can change markedly as Rol increases from 0 to 2, assuming the overall value of the basic reproductive rate is held constant at Ro = 2, and choosing representative values for Du D2 and D3 (0-5, 6 and 2 years). Again, these analytical and numerical results have a simple biological explanation. If Rol > 1, then the HIV epidemic can spread by first-phase infections alone, and such infections will dominate the early dynamics; finalphase infections, produced much later in the history of any infectious individual, are severely discounted in this fast-growing epidemic. But if RQ < 1, the infection cannot maintain itself or spread without the

R. M. May and R. M. Anderson contributions from final-phase infections, which results in the doubling time being keyed to the average time taken to reach this later phase (roughly D2). Rates of acquiring new sexual partners, c, in large cities in Central Africa may well be significantly higher over the past few decades than such rates have typically been over the past century in rural regions of Africa. Thus it is reasonable to assume that overall values of Ro for HIV may have been significantly higher in recent times. Other things being equal, high values of Ro imply shorter doubling times. But if this were the only thing happening, we believe it would be hard to explain average doubling rates for HIV in Africa over the past century being slower than current ones by anything much more than a factor of 2 to 4 or so; in models with constant infectiousness, linear changes in Ro imply essentially linear changes in td. But the situation sketched in this section, and in particular the contrast between equations (12) and (13), suggest that with such variable infectiousness a relatively small change in behaviour, and thence in c, that carries Rol from below unity to above unity (always with an overall Ro > 1) can be non-linearly amplified to produce more than a factor of 10 reduction in the doubling time (essentially, a change of order D1/D2). The suggestion therefore is that African regions may have had Rn < 1 but Ro > 1 for a century and more, but that modest levels of behavioural change may relatively recently have pushed R01 > 1, with the result that the doubling time for HIV may have dropped sharply, from a decade or so to a year or so. As in the previous section, this does require the relevant parameters to lie within particular ranges. These requirements are admittedly a bit special, but - as mentioned before - not as wildly restrictive as the homogeneous models' requirement that Ro lie infinitesimally above unity for 100 years. By combining the considerations of this section with those of spatial heterogeneity in the previous section, we can thus construct a scenario in which HIV could take a century between its first appearance in human populations and its rise to noticeable levels. This scenario does require that some parameter combinations lie in, or change around, restricted regions, but it requires nothing like the delicate numerical coincidence demanded by simple, homogeneous models. CONCLUSIONS

The first two parts of this paper review earlier work. One theme was that parasite associations do not necessarily evolve toward avirulence, but rather may evolve along many possible trajectories, constrained by the trade-offs among production of transmission stages by the parasite, parasite virulence, and the cost of host resistance; such trade-offs will take

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different forms for different parasite life-histories. The second theme was that interactions between hosts and parasites may result in steady populations and stable genetic polymorphisms, but they can just as naturally produce regular oscillations or apparently random, 'chaotic' fluctuations in population sizes and/or gene frequencies; this has implications that are only just beginning to be explored. The third part of our paper deals with the emergence of, or invasions by, new parasites. We emphasize that it is not enough for mutation or other processes to produce a new pathogen, but that the ecological setting must be such that the pathogen can maintain itself and spread (Ro > 1). By way of specific illustration of general ideas, we took up the apparent paradox that molecular evidence suggests HIV has been present in human populations for over a century, yet it appears to have reached noticeable levels only recently. We show how the effects of spatial heterogeneity, and of temporal variability in infectiousness, can suggest possible answers to the paradox. Whether or not these ideas are relevant to the emergence of HIV, they serve to highlight the way heterogeneities of various kinds can confound ecological and evolutionary conclusions based on inappropriately homogeneous models. We have benefitted from helpful conversations with M. P. Hassell, P. Harvey, A. E. Keymer, A. P. Dobson, T. R. E. Southwood and C. H. Watts (who also produced the figures for us). The work was supported in part by The Royal Society (R.M.M.), by the Medical Research Council and the Overseas Development Agency (R. M. A), and by the National Environmental Research Council through its Interdisciplinary Research Centre at Silwood Park. APPENDIX A

Suppose an overall population is divided among n villages, with N((t) individuals in village i at time t. Extending the epidemiological and demographic model derived for the transmission dynamics of HIV within an initially growing population by May & Anderson (1988) (see also May, Anderson & McLean, 1988, 1989), we may write (A.I) Yv

(A.2)

Here Xt and Yi are the number of susceptible and infected-and-infectious individuals, respectively, in the j'th village (Xt + Yt = A^), a and b are the average per capita birth and disease-free death rates, and v is the rate at which infected individuals move on to develop AIDS and die (so that the incubation time is of average duration D = \/v). In this basic model, the 'force of infection' in the ith village, A,., is given simply by A, =

ij

Y,/N,.

(A.3)

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Here, as defined in the text, /? is the transmission probability, c is the average rate of acquiring new partners, and pi} is the probability that an individual in village i will choose any one partner from village j (and Y}/Nf represents the probability that any one such partner will be infectious). This system of equations can be simplified somewhat by writing Xt(t) = N,(t) xt(t) and Yt(i) = N((t)yt(i) (so that x(+y( = 1 for all /). Noting from equations (A.I) and (A. 2) that dNJdt = Nt[a — b — vyt], we can reduce equations (A.2) and (A.3) to a set of n non-linear equations for the n quantities y,(t):

This expression leads immediately from equations (A.9) and (A.10) to equation (6) in the text. APPENDIX B

Suppose we have the kind of variable infectiousness described in the main text, within the framework of an otherwise homogeneous, simple model (Anderson et al. 1986). Neglecting demographic effects (that is, treating the epidemic as spreading in a closed population), we then have: dX/dt = = AX-v1 Yu

dyjdt =

-yt)]yt.

(A.4)

The initial condition is y^O) = A, yt(0) = 0 for all t*l. In the early phase of the epidemic, the fraction infected in any village is significantly less than unity, and equation (A.4) may be approximated by the linear set of first-order differential equations dyjdt =

(A.5)

Here S(i is the Kronecker delta function defined in the text. This set of n linear equations can be solved formally, but to get an explicit expression some simplifying assumptions must be made. We put ptj = K(Stj + e).

(A. 6)

Here K = 1/(1 + ne) is a normalization constant. The biological significance of the assumption is discussed in the text. Equation (A.5) now reads dyjdt =

(A.7)

Here y(t) is the average value of y^t), defined as y = (.Zyd/nA couple of tricks now produce a quick answer (which can alternatively be obtained by turning the handle on the Laplace transform machinery). First, sum equation (A.7) over all i to get an equation for y: dy/dt = [cpK{\ + we) - a - v] y.

(A.8)

The initial condition is y(0) = A/n, and so we have simply y(t) = (A/n)ept.

(A.9)

Here p = cfi—a — v is as defined in equation (7). Substituting equation (A.9) into equation (A.7) with i = l, we get a straightforward differential equation for yt{t). This integrates to give

with p — cfiK—a — v as defined in equation (8). On average, all yt{t) (i #= 1) are the same: y,=y-{yx/n).

(A.11)

dYJdt = v1 Yl-viY2,

(B.I) (B.2) (B.3) (B.4)

Here X{t) is the number of susceptibles at time t, Yx is the number of infected individuals in the first phase of infectiousness (having transmission probability /?,), Y2 the number in the uninfectious ' silent phase' (/?2 = 0), and Y3 the number in the second (ARC/AIDS) phase of infectiousness (with transmission probability /?3); the total population is N = X+ Yj-t- Y2+ Y3. The rate constants vx, v2, v3 are as discussed in the text, and the force of infection, A, is given by A = c[/31Y1+/33Y3]/N.

(B.5)

In the initial stages of the epidemic, we may regard Y( (i = l, 2, 3) as small and linearize equations (B.1)-(B.4) in the usual way. In the resulting linear, first-order differential equations, the time dependence may as usual be factored out as exp {At), to get AY1 = (fi1c-vl)Y1+03cY3,

(B.6)

AY2 = vlYl-v2Y2,

(B.7)

= v2Y2-v3Y3.

(B.8)

This set of homogeneous equations has a non-trivial solution if, and only if, the appropriate determinant vanishes, which leads to a cubic equation for A: (B.9) With the routine definitions Rol = /?x c/v1 and R02 = fl3c/v3, equation (B.9) is equation (10) of the main text. In practice, for HIV we have v2 roughly an orderof-magnitude smaller than vt and v3 (that is, D2 is around 10 years, while D, and D3 are of order 1 year). This provides the basis for the approximate results of equations (12) and (13) for Rol significantly above and below unity, respectively; these limiting results are illustrated in Fig. 3, and their biological significance is elaborated in the main text. More specifically, if R01 > 1 we have A ~ vt, and the second term inside the square brackets in

R. M. May and R. M. Anderson

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equation (B.9) will typically be smaller than the first by a factor of order v2/vl. Expanding equation (B.9) on that basis, for Rol > 1 we get

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( B l 0 )

The first term in this approximation scheme gives equation (12). Conversely, if Rol < 1, we have A ~ v2. Systematic expansion of equation (B.9) on this basis gives

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Parasite-host coevolution.

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