The population dynamics of infection

In principle, the sorts of conclusions that were drawn in Chapter 10 regarding the population dynamics of predator-prey and herbivore-plant interactions can be extended to parasites and hosts. Parasites harm individual hosts, which they use as a resource. The way in which this affects their populations varies with the densities of both parasites and hosts and with the details of the interaction. In particular, infected and uninfected hosts can exhibit compensatory reactions that may greatly reduce the effects on the host population as a whole. Theoretically, a range of outcomes can be predicted: varying degrees of reduction in host-population density, varying levels of parasite prevalence and various fluctuations in abundance.

With parasites, however, there are particular problems. One difficulty is that parasites often cause a reduction in the 'health' or 'morbidity' of their host rather than its immediate death, and it is therefore usually difficult to disentangle the effects of the parasites from those of other factors with which they interact (see Section 12.5). Another problem is that even when parasites cause a death, this may not be obvious without a detailed postmortem examination (especially in the case of microparasites). Also, the biologists who describe themselves as parasitologists have in the past tended to study the biology of their chosen parasite without much consideration of the effects on whole host populations; while ecologists have tended to ignore parasites. Plant pathologists and medical and veterinary parasitologists, for their part, generally study parasites with known severe effects that live typically in dense and aggregated populations of hosts, paying little attention to the more typical effects of parasites in populations of 'wildlife' hosts. Elucidation of the role of parasites in host-population dynamics is one of the major challenges facing ecology.

Here, we begin by looking at the dynamics of infection within host populations without considering any possible effects on the total abundance of hosts. This 'epidemiological' approach (Anderson, 1991) has especially dominated the study of human disease, where total abundance is usually considered to be determined by a whole spectrum of factors and is thus effectively independent of the prevalence of any one infection. Infection only affects the partitioning of this population into susceptible (uninfected), infected and other classes. We then take a more 'ecological' approach by considering the effects of parasites on host abundance in a manner much more akin to conventional predator-prey dynamics.

12.6.1 The basic reproductive rate and the transmission threshold

In all studies of the dynamics of parasite populations or the spread of infection, there are a number of particularly key concepts. The first is the basic reproductive rate, R0. For microparasites, because infected hosts are the unit of study, this is defined as the average number of new infections that would arise from a single infectious host introduced into a population of susceptible hosts. For macroparasites, it is the average number of established, reproductively mature offspring produced by a mature parasite throughout its life in a population of uninfected hosts.

The transmission threshold, which must be crossed if an infection is to spread, is then given by the condition R0 = 1. An infection will eventually die out for R0 < 1 (each present infection or parasite leads to less than one infection or parasite in the future), but an infection will spread for R0 > 1.

Insights into the dynamics of infection can be gained by considering the various determinants of the basic reproductive rate. We do this in some detail for directly transmitted microparasites, and then deal more briefly with related issues for indirectly transmitted microparasites, and directly and indirectly transmitted macroparasites.

12.6.2 Directly transmitted microparasites: R0 and the critical population size

For microparasites with direct, density-dependent transmission (see Section 12.4.3), R0 can be said to increase with: (i) the average period of time over which an infected host remains infectious, L; (ii) the number of susceptible individuals in the host population, S, because greater numbers offer more opportunities for transmission of the parasite; and (iii) the transmission coefficient, P (see Section 12.4.3). Thus, overall:

Note immediately that by this definition, the greater the number of susceptible hosts, the higher the basic reproductive rate of the infection (Anderson, 1982).

The transmission threshold can now be expressed in terms of a critical population size, ST, where, because R0 = 1 at that threshold:

effects on health or morbidity

R0, the basic reproductive rate the transmission threshold the critical population size ...

In populations with numbers of susceptibles less than this, the infection will die out (R0 < 1). With numbers greater than this the infection will spread (R0 > 1). (ST is often referred to as the critical community size because it has mostly been applied to human 'communities', but this is potentially confusing in a wider ecological context.) These simple considerations allow us to make sense of some very basic patterns in the dynamics of infection (Anderson, 1982; Anderson & May, 1991).

Consider first the kinds of population in which we might expect to find different sorts of infection. If microparasites are highly infectious (large Ps), or give rise to long periods of infectiousness (large Ls), then they will have relatively high R0 values even in small populations and will therefore be able to persist there (ST is small). Conversely, if parasites are of low infectivity or have short periods of infectious-ness, they will have relatively small R0 values and will only be able to persist in large populations. Many protozoan infections of vertebrates, and also some viruses such as herpes, are persistent within individual hosts (large L), often because the immune response to them is either ineffective or short lived. A number of plant diseases, too, like club-root, have very long periods of infectiousness. In each case, the critical population size is therefore small, explaining why they can and do survive endemically even in small host populations.

On the other hand, the immune responses to many other human viral and bacterial infections are powerful enough to ensure that they are only very transient in individual hosts (small

L), and they often induce lasting immunity. Thus, for example, a disease like measles has a critical population size of around 300,000 individuals, and is unlikely to have been of great importance until quite recently in human biology. However, it generated major epidemics in the growing cities of the industrialized world in the 18th and 19th centuries, and in the growing concentrations of population in the developing world in the 20th century. Around 900,000 deaths occur each year from measles infection in the developing world (Walsh, 1983).

12.6.3 Directly transmitted microparasites: the epidemic curve

The value of R0 itself is also related to the nature of the epidemic curve of an infection. This is the time series of new cases following the introduction of the parasite into a population of hosts. Assuming there are sufficient susceptible hosts present for the parasite to invade (i.e. the critical population size, ST, is exceeded), the initial growth of the epidemic will be rapid as the parasite sweeps through the population of susceptibles. But as these susceptibles either die or recover to immunity, their number, S, will decline, and so too therefore will R0 (Equation 12.5). Hence, the rate of appearance of new cases will slow down and then decline. And if S falls below ST and stays there, the infection will disappear - the epidemic will have ended. Two examples of epidemic curves, for Legionnaires' disease in Spain and for foot-and-mouth disease in the UK, are shown in Figure 12.16.

Not surprisingly, the higher the initial value of R0, the more rapid will be the rise in the epidemic curve. But this will also lead

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Figure 12.16 (a) An epidemic curve for an outbreak of Legionnaires' disease in Murcia, a municipality in southeastern Spain, in 2001. (After Garcia-Fulgueiras et al., 2003.) (b) An epidemic curve for an outbreak of foot-and-mouth disease (mostly affecting cattle and sheep) in the United Kingdom in 2001. Infected premises (farms) are shown, since infection was transmitted from farm to farm, and once infected, all the stock on that farm were destroyed. (After Gibbens & Wilesmith, 2002.)

Figure 12.17 (a) Reported cases of measles in England and Wales from 1948 to 1968, prior to the introduction of mass vaccination. (b) Reported cases of pertussis (whooping cough) in England and Wales from 1948 to 1982. Mass vaccination was introduced in 1956. (After Anderson & May, 1991.)

Figure 12.17 (a) Reported cases of measles in England and Wales from 1948 to 1968, prior to the introduction of mass vaccination. (b) Reported cases of pertussis (whooping cough) in England and Wales from 1948 to 1982. Mass vaccination was introduced in 1956. (After Anderson & May, 1991.)

to the more rapid removal of susceptibles from the population and hence to an earlier end to the epidemic: higher values of R0 tend to give rise to shorter, sharper epidemic curves. Also, whether the infection disappears altogether (i.e. the epidemic simply ends) depends very largely on the rate at which new susceptibles either move into or are born into the population, since this determines how long the population remains below ST. If this rate is too low, then the epidemic will indeed simply end. But a sufficiently rapid input of new susceptibles should prolong the epidemic, or even allow the infection to establish endemically in the population after the initial epidemic has passed.

12.6.4 Directly transmitted microparasites: cycles of infection dynamic patterns of different types of parasite

This leads us naturally to consider the longer term patterns in the dynamics of different types of endemic infection. As described above, the immunity induced by many bacterial and viral infections reduces S, which reduces R0, which therefore tends to lead to a decline in the incidence of the infection itself. However, in due course, and before the infection disappears altogether from the population, there is likely to be an influx of new susceptibles into the population, a subsequent increase in S and R0, and so on. There is thus a marked tendency with such infections to generate a sequence from 'many susceptibles (R0 high)', to 'high incidence', to 'few susceptibles (R0 low)', to 'low incidence', to 'many susceptibles', etc. - just like any other predator-prey cycle. This undoubtedly underlies the observed cyclic incidence of many human diseases, with the differing lengths of cycle reflecting the differing characteristics of the diseases: measles with peaks every 1 or 2 years (Figure 12.17a), pertussis (whooping cough) every 3-4 years (Figure 12.17b), diphtheria every 4-6 years, and so on (Anderson & May, 1991).

By contrast, infections that do not induce an effective immune response tend to be longer lasting within individual hosts, but also tend not to give rise to the same sort of fluctuations in S and R0. Thus, for example, protozoan infections tend to be much less variable (less cyclic) in their prevalence.

12.6.5 Directly transmitted microparasites: immunization programs

Recognizing the importance of critical population sizes also throws light on immunization programs, in which susceptible hosts are rendered nonsusceptible without ever becoming diseased (showing clinical symptoms), usually through exposure to a killed or attenuated pathogen. The direct effects here are obvious: the immunized individual is protected. But, by reducing the number of susceptibles, such programs also have the indirect effect of reducing R0. Indeed, seen in these terms, the fundamental aim of an immunization program is clear - to hold the number of susceptibles below ST so that R0 remains less than 1. To do so is said to provide 'herd immunity'.

In fact, a simple manipulation of Equation 12.5 gives rise to a formula for the critical proportion of the population, pc, that needs to be immunized in order to provide herd immunity (reducing R0 to a maximum of 1, at most). If we define S0 as the typical number of susceptibles prior to any immunization and note that ST is the number still susceptible (not immunized) once the program to achieve R0 = 1 has become fully established, then the proportion immunized is:

The formula for ST is given in Equation 12.6, whilst that for S0, from Equation 12.5, is simply R0/(3L, where R0 is the basic reproductive rate of the infection prior to immunization. Hence:

Eradication

Eradication

Rubella Smallpox

Measles

Rubella Smallpox

Persistence

Figure 12.18 The dependence of the critical level of vaccination coverage required to halt transmission, pc, on the basic reproductive rate, R0, with values for some common human diseases indicated. (After Anderson & May, 1991.)

This reiterates the point that in order to eradicate a disease, it is not necessary to immunize the whole population - just a proportion sufficient to bring R0 below 1. It also shows that this proportion will be higher the greater the 'natural' basic reproductive rate of the disease (without immunization). This general dependence of pc on R0 is illustrated in Figure 12.18, with the estimated values for a number of human diseases indicated on it. Note that smallpox, the only disease where in practice immunization seems to have led to eradication, has unusually low values of R0 and pc.

12.6.6 Directly transmitted microparasites: frequency-dependent transmission

Suppose, however, that transmission is frequency dependent (see Section 12.4.3), as it is likely to be, for example, with sexually transmitted diseases, where transmission occurs after an infected individual 'seeks out' (or is sought out by) a susceptible individual. Then there is no longer the same dependence on the number of susceptibles, and the basic reproductive rate is simply given by:

12.6.7 Crop pathogens: macroparasites viewed as microparasites

Most of plant pathology has been concerned with the dynamics of diseases within crops, and hence with the spread of a disease within a generation. Moreover, although most commonly studied plant pathogens are macroparasites in the sense we have defined them, they are typically treated like microparasites in that disease is monitored on the basis of some measure of disease severity -often, the proportion of the population infected (i.e. prevalence). We refer to yc as the proportion affected by lesions at time t, and hence (1 — yt) is the proportion of the population without lesions and thus susceptible to infection. It is also usually necessary with plant pathogens to take explicit account of the latent period, length p, between the time when a lesion is initiated and the time when it becomes infectious (spore-forming) itself, in which state it remains for a further period l. Hence, the proportion of the population affected by infectious lesions at time t is (yt—p — yt—p—l). The rate of increase in the proportion of a plant population affected by lesions (Vanderplank, 1963; Zadoks & Schein, 1979; Gilligan, 1990) may thus be given by:

which is essentially a PSJ formulation, with D the plant patho-logists' version of a transmission coefficient. This gives rise to S-shaped curves for the progress of a disease within a crop that broadly match the data derived from many crop-pathogen systems (Figure 12.19).

In the progress of such infections, plant pathologists recognize three phases.

1 The 'exponential' phase, when, although the disease is rarely detectable, rapid acceleration of parasite prevalence occurs. This is therefore the phase in which chemical control would be most effective, but in practice it is usually applied in phase 2. The exponential phase is usually considered arbitrarily to end at y = 0.05; about the level of infection at which a nonspecialist might detect that an epidemic was developing (the perception threshold).

2 The second phase, which extends to y = 0.5. (This is sometimes confusingly called the 'logistic' phase, although the whole curve is logistic.)

3 The terminal phase, which continues until y approaches 1.0. In this phase chemical treatment is virtually useless - yet it is at this stage that the greatest damage is done to the yield of a crop.

Here, there is apparently no threshold population size and such infections can therefore persist even in extremely small populations (where, to a first approximation, the chances of sexual contact for an infected host are the same as in large populations).

On the other hand, some crop diseases are not simply transmitted by the passive spread of infective particles from one host to another. For example, the anther smut fungus, Ustilago violacea, is spread between host plants of white campion, Silene alba, by

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Figure 12.19 'S-shaped' curves of the progress of diseases through crops from an initial inoculum to an asymptotic proportion of the total population infected. (a) Puccinia recondita attacking wheat (cultivar Morocco) and triticale (a crop derived from the hybridization of wheat and rye) in 1983 and 1984. (b) Fusarium oxysporum attacking tomatoes in experiments comparing untreated and sterilized soil and untreated and artificially heated soil. (After Gilligan, 1990, in which the original data sources and methods of curve-fitting may be found.)

pollinating insects that adjust their flight distances to compensate for changes in plant density, such that the rate of transmission is effectively independent of host density (Figure 12.20a). However, this rate decreases significantly with the proportion of the population that is susceptible: transmission is frequency dependent (Figure 12.20b), favoring, as we have seen, persistence of the disease even in low-density populations. Of course, this is really just another case of frequency-dependent transmission in a sexually transmitted disease - except that sexual contact here is indirect rather than intimate.

12.6.8 Other classes of parasite

For microparasites that are spread from one host to another by a vector more generally (where the vector does not compensate for changes in host density as in the above example), the life cycle characteristics of both the host and vector enter into the calculation of R0. In particular, the transmission threshold

(R0 = 1) is dependent on a ratio of vector : host numbers. For a disease to establish itself and spread, that ratio must exceed a critical level - hence, disease control measures are usually aimed directly at reducing the numbers of vectors, and are aimed only indirectly at the parasite. Many virus diseases of crops, and vector-transmitted diseases of humans and their livestock (malaria, onchocerciasis, etc.), are controlled by insecticides rather than chemicals directed at the parasite; and the control of all such diseases is of course crucially dependent on a thorough understanding of the vector's ecology.

The effective reproductive rate of a directly transmitted macroparasite (no intermediate host) is directly related to the length of its reproductive period within the host (i.e. again, to L) and to its rate of reproduction (rate of production of infective stages). Both of these are subject to density-dependent constraints that can arise either because of competition between the parasites, or commonly because of the host's immune response (see Section 12.3.8). Their intensity varies with the distribution of the parasite population between its hosts and, as we have seen, vector-borne infections directly transmitted macroparasites

Figure 12.20 Frequency-dependent transmission of a sexually transmitted disease. The number of spores of Ustilago violacea deposited per flower of Silene alba (log10(x + 1) transformed) where spores are transferred by pollinating insects. (a) The number is independent of the density of susceptible (healthy) flowers in experimental plots (P > 0.05) (and shows signs of decreasing rather than increasing with density, perhaps as the number of pollinators becomes limiting). (b) However, the number decreases with the frequency of susceptibles (P = 0.015). (After Antonovics & Alexander, 1992.)

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aggregation of the parasites is the most common condition. This means that a very large proportion of the parasites exist at high densities where the constraints are most intense, and this tightly controlled density dependence undoubtedly goes a long way towards explaining the observed stability in prevalence of many helminth infections (such as hookworms and roundworms) even in the face of perturbations induced by climatic change or human intervention (Anderson, 1982).

Most directly transmitted helminths have an enormous reproductive capability. For instance, the female of the human hookworm Necator produces roughly 15,000 eggs per worm per day, whilst the roundworm Ascaris can produce in excess of 200,000 eggs per worm per day. The critical threshold densities for these parasites are therefore very low, and they occur and persist endemically in low-density human populations, such as hunter-gatherer communities.

Density dependence within hosts also plays a crucial role in the epidemiology of indirectly transmitted macroparasites, such as schistosomes. In this case, however, the regulatory constraints can occur in either or both of the hosts: adult worm survival and egg production are influenced in a density-dependent manner in the human host; but also, production of infective stages by the snail (intermediate host) is virtually independent of the number of (different) infective stages that penetrate the snail. Thus, levels of schistosome prevalence tend to be stable and resistant to perturbations from outside influences.

The threshold for the spread of infection depends directly on the abundance of both humans and snails (i.e. a product as opposed to the ratio that was appropriate for vector-transmitted microparasites). This is because transmission in both directions is by means of free-living infective stages. Thus, since it is inappropriate to reduce human abundance, schistosomiasis is often controlled by reducing snail numbers with molluscicides in an attempt to depress R0 below unity (the transmission threshold). The difficulty with this approach, however, is that the snails have an enormous reproductive capacity, and they rapidly recolonize aquatic habitats once molluscicide treatment ceases. The limitations imposed by low snail numbers, moreover, are offset to an important extent by the long lifespan of the parasite in humans (L is large): the disease can remain endemic despite wide fluctuations in snail abundance.

12.6.9 Parasites in metapopulations: measles

With host-parasite dynamics, as with other areas of ecology, there is increasing recognition that populations cannot be seen as either homogeneous or isolated. Rather, hosts are usually distributed amongst a series of subpopulations, linked by dispersal between them, and which together comprise a 'metapopulation' (see Section 6.9). Thus, since the argument has already been made (see Section 12.4.1) that each host supports a subpopulation and a host population supports a metapopulation of parasites, host-parasite systems are typically metapopulations of metapopulations.

Such a perspective immediately changes our view of what is required of a host population if it is to support a persistent population of parasites. This is apparent from an analysis of the dynamics of measles in 60 towns and cities in England and Wales from 1944 to 1994: 60 subpopulations comprising an overall metapopulation (Figure 12.21) (Grenfell & Harwood, 1997). Taken as a whole, the metapopulation displayed regular cycles in the number of measles cases and measles was ever-present (Figure 12.21a), at least before widespread vaccination (c. 1968). But amongst the individual subpopulations, only the very largest were not liable to frequent 'stochastic fade-out' (disappearance of the disease when a few remaining infectious individuals fail to pass indirectly transmitted macroparasites

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Figure 12.21 (a) The weekly measles notifications for 60 towns and cities in England and Wales, combined, are shown below for the period 1944-94. The vertical line indicates the start of mass vaccination around 1968. The data for the individual towns (town size on the vertical axis) are displayed above as a dot for each week without a measles notification. (b) Persistence of measles in these towns and cities in the prevaccination era (1944-67) as a function of population size. Persistence is measured inversely as the number of 'fade-outs' per year, where a fade-out here is defined as a period of three or more weeks without notification, to allow for the underreporting of cases. (After Grenfell & Harwood, 1997.)

it on), especially during the cycle troughs: the idea of a critical population size of around 300,000-500,000 is therefore well supported (Figure 12.21b). Thus, patterns of dynamics may be apparent, and persistence may be predictable, in a metapopulation taken as a whole. But in the individual subpopulations, especially if they are small, the patterns of dynamics and persistence are likely to be far less clear. The measles data set is unusual in that we have information both for the metapopulation and individual subpopulations. In many other cases, it is almost certain that the principle is similar but we have data only for the metapopulation (and do not appreciate the number of fade-outs in smaller parts of it), or we have data only for a subpopulation (and do not appreciate its links to other subpopulations within the larger metapopulation).

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