## Mathematical models of hostparasite population dynamics

These examples of parasites and herbivorous insects in quasi-natural ecosystems have led to useful questions and suggestions, but they have also indicated how difficult it is to understand fully what is going on. One way forward is to apply mathematical modelling. I start by outlining briefly a model which has substantially advanced our understanding of the relationship between animals and their parasites (Box 8.3). This can

Box 8.3. Simple models of the population biology of parasites and diseases and their hosts, and of herbivorous insects.

Based on Anderson & May (1979,1986); Crawley (1983); Swinton & Anderson (1995). The letters are defined below.

(1) Diseases ofaaimals) causal parasite transmitted from animal to animal by contact.

The rate of increase in the number of infected individual animals is dJ/dt - pSl-cI - /(pS-c) (8.3) If animals can become immune to the disease c-b + 0 + v (8.4) otherwise c-b + B (8.5)

The disease will maintain itself in the population when d//dtiO

i.e. whenfJSac, i.e. whenSac/p

Therefore, the critical number of susceptible host individuals above which disease will establish, is

(2) Parasite or insect can survive outside the host, and can in this way infect new hosts.

Meaning of letters:

b—death rate of host individuals from non-disease causes c—rate at which infected individuals cease to be infected, by recovery or death

/—death rate of propagules 1—number of infected host individuals

5—number of uninfected, susceptible host individuals

ST— critical number of susceptible individuals, below which the disease fails to persist t—time p—transmission coefficient of disease in the population

6—death rate of infected individuals caused by disease ( - virulence) v—rate at which infected individuals recover and become immune X—rate at which propagules produced be applied, after some modifications, to diseases of plants and to insects eating plants.

In the model, at any time the total population of host individuals consists of S uninfected susceptibles, I infected, and, in some species (e.g. mammals), also some immune. The rate of transmission of the infection through the population depends on how many infected individuals there are, to provide a source of infection, and also on how many susceptibles there are ready to become infected. This rate of transmission thus equals PSJ. The number of infected individuals increases by spread of the parasite to new individuals, but it decreases by the death of infected individuals and perhaps by the recovery of others; it is the balance between these increases and decreases that determines whether the parasite survives in the population. Equations 8.3-8.6 in Box 8.3 set this out, showing, by simple algebra, that there is a critical population size of uninfected susceptible individuals, ST, below which the parasite will not survive and above which it will survive,- this is defined by Equation 8.6. ST can alternatively be expressed as a critical population density, if I and S in the preceding equations are densities, i.e. numbers of individuals per unit area.

Up to now we have considered diseases of animals which require contact between individuals for their transmission. Rabies is an example, considered later. Many diseases, however, have a stage that survives free of the host and is involved in transfer. For example, many fungal diseases of plants disperse by spores; viruses that attack insects may survive on plant surfaces until another host insect comes along; insects that attack other insects or plants may have a motile adult stage. Equation 8.7 modifies Equation 8.6 to take account of the formation and death of the propagule stage. This would be different, however, if the propagule is carried by a vector or alternate host. Can the disease The equations in Box 8.3 are very simple, but they can give us some persist in the important messages. The idea of a critical host population density, below populationI which the disease will die out, is a very important one. Equations 8.6 and

8.7 show that a parasite can maintain itself in a sparser population if it is more effective at spreading (larger p), which is perhaps not surprising. Less obvious is that a disease that kills its host more quickly (higher virulence, larger 0) needs a denser population to survive. The quicker the animals die, the fewer infectives there are to spread the disease; or the more rapidly a herbivorous insect kills its host plants, the less chance that there will be other, live plants waiting for it to move on to. Earlier I described a nematode worm that can infect moose, and which usually kills them. Yet the worm is absent from most moose populations: only when white-tailed deer carry it into moose areas are the moose infected and killed. The explanation is probably that the worm kills moose too rapidly, so that in the absence of deer as a carrier there are soon few susceptibles left. In terms of Equation 8.6 0 is high so ST is high, i.e. the worm can only maintain itself in a moose population if the density of animals is high. In deer, because 0 = 0, ST is probably much lower.

Fig. 8.4 Predictions of abundance of plants (V) and herbivorous insects (P|. In graph (b) insect multiplication ability g 10 times as high as in (a). From Crawley (1983).

The models can be extended to predict how much the parasite or herbivore reduces its host's abundance, and also whether this abundance is stable or subject to cycles; but I do not present the equations here. Crawley (1983) presented predictions, from a whole family of models of this type, of the abundance of insect herbivores and their host plants. Figure 8.4 shows two examples. In the absence of the insects plant abundance would stabilize at 1000. The graphs show the predicted time-course of abundance of the insect and the plant, for two different values of g, which controls the rate of insect multiplication:

where P is the number of insects, V is the abundance of plants (measured as numbers or biomass), and /is death rate of the insects.

In Fig. 8.4(a), with the lower value of g, plant abundance is reduced to a stable level, but only slightly below the level of 1000 attained in the absence of insects. If the insects have higher multiplication ability, g (part (b)), not surprisingly they increase in abundance compared with (a), and the plants are reduced more. However, the prediction is that both cycle in abundance. The insects increase so fast that they kill almost all the plants; many insects cannot then find a live plant to feed on, and so the insect population crashes; then the remaining few plants increase; and so on. This insect would probably not be considered an effective biological control agent. This has interesting similarities to real insect outbreaks, e.g. the spruce budworm (Fig. 8.2). However, in real cycles of

Fox density in absence of rabies (number km-2)

Fig. 8.5 Predictions by model of Anderson et al. (1981) of population dynamics of foxes and rabies, (a) Abundance of foxes when rabies present; (b) percentage reduction in fox abundance caused by rabies; (c) percentage of foxes infected at any time. ST is the critical fox density below which rabies does not persist. Reprinted with permission; copyright Macmillan Magazines Limited.

Fox density in absence of rabies (number km-2)

Fig. 8.5 Predictions by model of Anderson et al. (1981) of population dynamics of foxes and rabies, (a) Abundance of foxes when rabies present; (b) percentage reduction in fox abundance caused by rabies; (c) percentage of foxes infected at any time. ST is the critical fox density below which rabies does not persist. Reprinted with permission; copyright Macmillan Magazines Limited.

herbivorous insects that have been investigated, the causes always seem to be more complex.

Rabies in foxes A similar model has been applied to rabies in foxes in Europe (Ander son et al. 1981; Anderson 1982). The disease is caused by a virus, which spreads between animals by direct contact. It occurs in fox populations in much of central Europe and the USA and can be spread to humans, usually via domestic dogs and cats. In humans the disease has very unpleasant symptoms and is usually fatal. From research on foxes and rabies it is possible to derive values for the key variables in the model. Figure 8.5 shows predictions by the model of the percentage of foxes infected and the effect of the disease on fox numbers, for a range of initial (pre-rabies) fox densities. The model predicts that for any initial (disease-free) fox density within the range 1-9 per km2, rabies will maintain the population near the critical density (ST), which is predicted to be about 1 per km2. Above a disease-free density of about 9 per km2 the model predicts cycles, rather like those in Fig. 8.4(b). Outbreaks of rabies and consequent cycles of fox abundance, with peaks every 3-5 years, have been recorded in various parts of Europe and North America. Within the range where rabies and fox densities are stable, the percentage of foxes infected at any one time is predicted to be usually only 2-4% (Fig. 8.5(c)), which is in reasonable agreement with field observations that have shown the figure to be usually 3-7% where the disease is stable and endemic. In spite of this, the reduction in the fox population can be large (Fig. 8.5(b)). Rabies shows that a disease can be the most important single factor controlling the population density of its host, even if only a few per cent of the population are infected at any time.

Many more complex variants of these models have been presented and their predictions explored (Crawley 1983; Briggs et al. 1995). A principal aim has been to predict situations in which the host's abundance will be greatly reduced but in a stable manner. Although more complex models aim for greater realism, it is often difficult to know whether the predictions are in fact realistic.

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