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Modelling of parasite populations: gastrointestinal nematode models
veterinary parasitology ELSEVIER
VeterinaryParasitology54 (1994) 127-143
Modelling of parasite populations: gastrointestinal nematode models G. Smith a'*, B.T.
Grenfell b
aSchool of Veterinary Medicine, University of Pennsylvania, New Bolton Center, 382 W. Street Road, Kennett Square, PA 19348, USA bDepartment of Zoology, University of Cambridge, Cambridge CB2 3EJ, UK
Abstract
This paper surveys models of nematode parasites of veterinary importance. A distinction is drawn between generic models which are usually simple formulations applicable to whole classes of parasite and specific models which are often more complex and designed to address questions concerning a particular species. Most of the models considered employ a deterministic framework. Four main groups are considered: generic models of trichostrongylid infection of domestic ruminants, specific models of trichostrongylid infection of domestic ruminants, specific models of experimental laboratory infections of rodents, and a specific model of nematode infections in wildlife. Keywords: Modelling
1. Introduction
Roberts and Heesterbeck (1994) cite a number of reasons why we should want to model the dynamics of infectious diseases. Among the most important are that models provide insight into the mechanisms underlying observed patterns and enable the modeller to conduct thought experiments concerning the efficacy of plausible disease control strategies. Roberts and Heesterbeck also grudgingly mentioned that models may sometimes be useful for predicting future trends, but only under very constrained circumstances. Each of these various modelling objectives will be illustrated in the context of a historical review of nematode models of veterinary importance. We shall consider two kinds of model: generic and spe* Correspondingauthor. 0304-4017/94/$07.00 © 1994 ElsevierScienceB.V. All rights reserved SSD10304-4017 (94)03081-7
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cific. Generic models are used to tease out generalisations about the dynamics of the parasite-host interaction. Generic models refer not to any particular parasite but rather to an assemblage of similar species. The structure of generic models is quite deliberately kept as simple as possible. This facilitates the analysis of their behaviour and obviates the possibility that extraneous biological details may obscure the more important processes. As we shall see, the usual technique for analysing the behaviour of generic models is to increase the complexity of the model in small increments and to examine the significance of that change in stepwise fashion. Specific models, however, frequently contain a great deal of biological detail. They are designed to address particular questions about the dynamics or control of a particular infection and are imbued with a spurious credibility by reason of their "biological realism". Their complex structure also hampers the analysis of model behaviour. Often the best we can do is to generalise from repeated numerical simulations. Nevertheless, specific models have a long and honourable history in the veterinary literature and, prudently applied, they can be very useful indeed. We shall consider the generic models first, tracing their development over the last 30 years. Next, we shall describe some specific models. These fall into three main categories: models for the trichostrongylid nematode parasites of cattle and sheep, models of experimental nematode infections in laboratory animals, and finally models for nematode infections of wildlife species.
2. Generic models 2.1. Stochastic models
TaUis and Leyton (1966) described a stochastic model for nematode infections in sheep. Although the primary objective of the paper was to show how generic models of infections could be formulated, the authors did explore the relationship between their assumptions regarding the uptake and maturation of female worms and the distribution of eggs in the faeces. They were concerned with describing the probabilities that a parasite was ingested during the small interval, x, and entered the mature class during the small interval, y. To keep things simple, they assumed that neither probability was affected by the number of parasites already in the host (i.e. no host immune response) and were able to show that if both infection and maturation were random, Poisson processes, the distribution of parasites in the hosts would have a Poisson distribution but the eggs in the faeces would have a Neyman Type A (aggregated, clumped) distribution. This was important because it demonstrated that an aggregated distribution of eggs in the faeces does not necessarily imply an aggregated distribution of parasites in the hosts. Nevertheless, TaUis and Leyton were perfectly well aware that the infective stages on pasture were themselves distributed in an aggregated fashion (Tallis and Donald, 1970) and that, in consequence, the distribution of mature parasites was unlikely to be random. In a sequel to their original paper, Tallis and
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Leyton ( 1969 ) compared the consequences of placing hosts on pastures in which the distribution of infective stages was random or aggregated. Not surprisingly, when the distribution of infective stages was assumed to be aggregated, the equilibrium frequency distribution of mature parasites turned out to be a negative binomial. Subsequently, the negative binomial distribution has proved to be a very useful, phenomenological mimic of the aggregated distributions that characterise the distribution of nematodes in the parasitic phase of the life cycle (Barger, 1985 ). We note in passing, of course, that there are other factors which give rise to aggregated parasite distributions in addition to heterogeneities in the availability of infective stages. Not least are differences between hosts that may affect infection rate or resistance to parasites (Anderson and May, 1991 ). 2.2. Deterministic models
Stochastic models are notoriously difficult to handle and this pioneering effort was followed by a more tractable deterministic model developed by Gordon et al. (1970). This basic architecture of the model was mapped on to the life cycle of a trichostrongylid nematode parasite of sheep. An abbreviated version will serve to illustrate the principle points of the paper. Infective larvae (L) are ingested by the host population at a rate 23. Ingested larvae (I) die at a rate/zl and mature at a rate 2 ~. The mature parasites (A) produce infective stages at a rate 22 and die at a rate/z2. The infective larvae on the pasture die at a rate/t3. Thus, the rate of change with respect to time in the abundance of each stage is given by tmA3 L -
(ILl "~-~l )I
ttm~.lI--fl2 A
dL
( 1)
= ~ 2 A - - (/.t3 +,~,3)L
In the initial formulation all the development and mortality rates were assumed to be constant. Of particular interest were the conditions that must pertain if the parasite is to persist in the host population. Gordon et al. approached this question by means of a sophisticated stability analysis but the essentials can be captured much more simply. If the parasite is to persist, there must always be viable infective stages (L). The best way of ensuring this is to maintain the rate of change in the number of infective stages ( d L / d t ) at zero or above (a negative rate of change implies a decrease in the value of L). For this to be true the values of A and L at equilibrium can be found by setting the equations for I and M respectively to zero. The condition then becomes ,~,2A (/z3 +Y3)L >~1
(2)
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212223
,/-/2(#3+23) (#1 +21)
>~1
(3)
Gordon et al. noted that if the ratio on the left hand side of Eq. (3) was less than 1 then parasite numbers would eventually decline to zero. If the ratio was exactly equal to 1 then a precarious balance was maintained in which parasites would persist at some fixed level depending upon the initial conditions, and if the ratio were greater than 1, parasite numbers would increase indefinitely. The ratio on the left hand side of Eq. (3) is now known as the basic reproduction ratio (R0). It has a straightforward biological interpretation. The number of infective stages produced per day per parasite is 22. The average lifespan of a mature parasite is given by the reciprocal of the parasite mortality ( 1/#2). Thus the total number of infective stages produced per parasite over its average lifespan is ;t 2/#2. Of these only a fraction, 2 3/(#3 + '~3 ), survive to be ingested and a fraction, ;t 1/(#1 +21 ) survive to become mature parasites. Thus, the number of infective larvae produced during the lifetime of a single parasite that survive to become mature parasites is given by the product of 22/#2, 23/(#3 + '~3 ) and 21/(#1 + 21 ), which is the basic reproduction ratio. Substituting realistic starting values for the parameters of the model described in Eq. ( 1 ) leads quickly to the conclusion that the basic reproduction ratio for trichostrongylids is greater than 1. Nevertheless, parasite numbers do not continue to increase indefinitely. Gordon et al. argued that the host immune system reduced the values of 21, 22 and 23 and increased the values off1, #2 and #3. This would eventually reduce the value of the basic reproduction ratio to values to below 1 and parasite numbers would then decline. Gordon et al. supposed that the hosts possessed an "information dam" into which antigenic information flowed at a net rate proportional to the intensity of infection and leaked away at a constant instantaneous rate. The parameters of the model were deemed to be functions of the accumulated amount of antigenic information and thus modulated by the hosts' experience of infection. Gordon et al. did not specify the nature of the functions linking the amount of information in the "information dam" other than to note in general terms that 2i would be expected to decrease with increasing antigenic information and/~ would increase. It was left to Roberts and Grenfell ( 1991, 1992 ) to supply plausible functions. Their generic model for trichostrongylid infections in ruminants utilised a parameter called the "level of acquired immunity" (r) (Anderson and May, 1991 ) which was essentially the same as the "antigenic information" defined by Gordon et al. (1970) except that it depended on the cumulative number of infective larvae ingested by the host rather than on the current worm burden
dr
d t _ f l L - ar
(4)
Here, fl is the transmission parameter (equivalent to 23 in Eq. ( 1 ) above) and represents the rate of fade of immunological memory. The pasture larval contamination (L) and mature parasites (A) were modelled as follows:
G. Smith, B.T. Grenfell / Veterinary Parasitology 54 (1994) 127-143
~tt= P ( r ) p L - # ( r ) A
131
(5)
~t =q2 ( r )A - p L - flL The model includes no immature parasitic phase, p being the probability that an ingested larva survives to become mature. This probability, the death rate of mature worms (/t) and the rate of egg production of mature worms (2) were each functions of the level of acquired immunity (r). The eggs were assumed to have a probability, q, of developing to infective larvae and these larvae were assumed to die at a rate, r. In the initial analysis, q and r were assumed to be constant (i.e. climatic effects were ignored). The basic reproduction ratio for Eqs. (4) and ( 5 ) was given by
R
o
fl2(O)p(O)q =
~
(6)
When Ro < 1 the infection cannot persist. Roberts and Grenfell were interested in accounting for the midsummer rise in pasture larval contamination that is so characteristic of the intensive calf management systems of northern temperate Europe. They found that Eqs. (4) and (5) generated a midsummer rise even in the absence of climatic effects provided that the mean number of parasites (A) and level of acquired immunity (r) were reduced to zero on an annual basis. This device simulated the annual replacement of stock with naive animals. When climatic sensitivity was subsequently incorporated into the model Roberts and Grenfell noted that this annual resetting of the system was still the predominant cause of the midsummer rise. They concluded that discrete management perturbations control the pattern of infection whereas continuous (climatic) perturbations modify the magnitude of the absolute worm burden. This is a particularly clear example of the way in which generic models can clarify which factors are responsible for particular aspects of observed epidemiological patterns. It should be noted that such models can be extended to examine various kinds of problems. For example, Grenfell ( 1988 ) used a model very similar to that represented by Eq. (5) to investigate the impact of ruminant parasitism on the entire grazing system. The required extension was an equation representing the dynamics of the available herbage (V) dV d-~ = G ( V ) - c ( V ) .H
(7)
The function, G(V), is the net growth rate of plant biomass. G(V) increases with plant biomass (V) until shading effects cause it to decline to very low levels (Noy-Meir, 1975 ). The function, c(V), is the per capita rate of herbage ingestion by animals in the gazing herd. The value of c(V) increases sigmoidally with plant biomass and eventually saturates at some fixed level (Cmax) (Fig. 1 ). The net rate
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(..)
0.30 0.24
-
-
..
t..-
(.9 09
or"
0.18 0.12 0.06 0.00
,/i//
/
,,,
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--
~••
///
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Plant a b u n d a n c e (V) Fig. 1. Diagram to show the impact of parasitism on the stability of a grazing season. The plant dynamics are represented by G (solid line). Uninfected herbivore consumption rates (C) at moderate stocking densities (dashed line) intersect the plant dynamics curve in three places. Each intersection represents a potential equilibrium. The infected herbivore consumption curve (dotted/dashed line) intersects the plant dynamics curve only once. The arrow indicates the effect of parasitism on the herbivore consumption curve (Grenfell, 1988 ).
of herbage consumption varies with the stocking rate, H. The dynamics of this grazing system depend upon the relative shape of the curves for G(V) and c(V). The points where these two curves intersect represent all the possible end points (equilibria) towards which the model forever tends. As might be expected, at low stocking rates there is a single stable equilibrium with high plant biomass; at very high stocking rates the system collapses to a very low equilibrium plant biomass. Paradoxically, at intermediate stocking rates the G (V) and c (V) curves intersect in three places, indicating three possible equilibria. Only two of these equilibria are stable: the lowest and highest, respectively. In this case, the eventual fate of the system depends critically on the initial conditions. The introduction of parasites into the grazing herd has two effects: it reduces per capita food intake (thus affecting c(V) ) and by reducing the proportion of the pasture sampled each day reduces the transmission parameter, fl (thus affecting the infection rate). The overall consequence of this density dependent constraint on consumption is to linearise (flatten) the sigmoidally shaped function representing c(V). As a result, the G(V) and c(V) curves may intersect in only one place, an equilibrium with high plant biomass. The grazing system is effectively stabilised by parasitism. It follows that parasite control may act as a destabilising influence with unpredictable results. Though this particular model assumed that the host population was a domestic population maintained at a constant stocking rate, Grenfell (1992a) showed that the same result also pertained in the case of naturally fluctuating herbivore populations. 3. Specific models as forecasting tools
Among the very first specific nematode models of veterinary interest were models intended to help forecast the risk of infection in the coming year. The
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models described by Gettinby and his co workers (Gettinby et al., 1979; Gettinby and Paton, 1981; Paton et al., 1984; Paton, 1987) were detailed mimics of the life cycle of trichostrongylid nematode parasites of cattle and sheep and have been reviewed and evaluated elsewhere (Smith and Galligan, 1988; Smith, 1989). A somewhat earlier model for the same system adopted a very different approach. Ollerenshaw et al. (1978) were interested in helping farmers control parasitic gastroenteritis in lambs and ewes. To this end, they devised a system of forecasting the intensity of disease (measured by combining a variety of indices). The probable intensity of disease was indicated by a "disease rating", D, where
D=O.3A+O.12M+O.25J+O.5Jy-6.16
(8)
This equation was derived for northern temperate Europe and incorporated several climatic parameters: parameter A was the return to field capacity date in the previous autumn, and M, J, and Jy were the number of wet days in May, June, and July, respectively, in the current year. The correlation between the disease rating and a subjective classification of actual disease occurrence over the 20 years considered was 0.97. The post hoc rationale for this system was that a dry summer in the previous grazing season would delay the peak pasture larval count which would, in turn, cause a high proportion of these larvae to overwinter either on pasture or as arrested forms in ewes retained for breeding. This would facilitate a surge of infection at the beginning of the next grazing season especially if the spring and summer were wet. Thomas and Starr ( 1978 ) described a less convincing system (based on only 9 years of data) for forecasting the timing of the peak pasture larval count. Both methods involved a simple empirical model linking climate with some index of parasite abundance. Neither group of workers was interested in explaining why the midsummer rise in pasture contamination occurred, instead they sought to identify factors which altered its magnitude and timing. Though most workers in the field would accept that weather patterns markedly alter pasture larval contamination, the causal inference remains unproven. Nevertheless, Ollerenshaw et al. (1978 ) found an excellent correlation between climate and the risk of disease, and, while correlation does not imply a causation, the investigation of mechanism implicit in the generic modelling work of Roberts and Grenfell (1992) and our experience with a dozen or so specific models for trichostrongylid infections (see below) suggests that the idea is at least plausible.
4. Specific models for the control of trichostrongylid infections of ruminants
4.1. Basic model structure Smith (1989, 1994) has reviewed the biological ideas that underpin the specific models for trichostrongylid nematode infections of ruminants. This is a well trodden field of research, Smith (1994) lists 13 distinct models. All of them were designed to help evaluate and devise disease control strategies and all of them are
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crammed full of biological detail. Part of their interest lies in the fact that each model is a concise historical record of the information available at the time of model building and a mirror to the various schools of thought about which were the important processes with regard to the natural regulation and control of parasite abundance. An illuminating example is the model for Trichostrongylus colubriformis infections in sheep described by Dobson et al. (1990a). An elegant series of trickle infection experiments revealed that the establishment of adult worms declined with the host's experience of infection (Dobson et al., 1990b). The pattern of decline was sigmoidal, the exact shape of the curve depending on the infection rate and age of the host. Smith (1994) has since suggested that a declining sigmoidal pattern of establishment is a characteristic of all the common trichostrongylid infections of cattle and sheep. However, the T. colubriformis example described by Dobson et al. (1990b) is the only one in which there is any obvious relationship between the rate of infection and the rate of decline in establishment. In the absence of any precise information about the immunological processes leading to this decrease in establishment in other trichostrongylids, Smith (1988 ) and Smith and Galligan (1988 ) settled for a simple empirical logistic model, i.e. e x
P = ( l + e x)
(9)
where p is proportional establishment and x is a linear function of the duration of infection (t).
x=a+bt
(10)
The constant a determined the initial value ofp and the constant b determined its rate of decline. Dobson et al. (1990a) also used a logistic function but chose to interpret their results in terms of a threshold level of adult worm load before a substantial host immune response is generated. They noted that the logistic model was only an approximate mimic of the threshold argument developed by Dineen ( 1963 ) and Dineen et al. ( 1965 ) but accepted all the same the cumbersome formulation that this interpretation obliged them to adopt. In their model, the parameter x becomes
x=b. T+b.ET-b.t
( 11 )
where T is the time taken for the threshold burden to be exceeded, E T is a parameter which determines the rate of decline in establishment once Thas elapsed and b is a constant. T and E T are themselves functions (of the infection rate and host age, respectively). The important point is that, although the establishment data could just as easily have been modelled using a straightforward empirical extension of Eq. (10), Dobson et al. (1990a) used an existing precept to guide their choice of function. There was no discussion of alternative models that fitted less well. This was quite proper and is characteristic of almost all the models described in this paper but does little to enhance our understanding of what is going
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on. This is important because models are not necessarily unique explanations of particular data sets 4.2. S o m e problems with trichostrongylid models
The principal difficulty with models of this sort is the problem of model validation. Since they are intended to be used to compare existing or proposed disease control strategies they should at least generate patterns which experienced field workers would accept as being within normal limits (Smith and Guerrero, 1993 ) and place tested control strategies in the correct rank order with respect to some index of efficacy (Smith et al., 1987a). Unfortunately, it is very difficult to compare model output with field data. Models are sensitive to the initial conditions of the simulation (e.g. stocking density, initial levels of pasture contamination, immune status of the hosts) and these are frequently only imperfectly known or recorded. It is unreasonable to expect precise one-to-one correspondence between a single study and its corresponding model output. One can sometimes combine similar studies in the same bioagricultural area and so get an impression of the reasonable range of epidemiological patterns, but such data are hard to come by. A partial solution is to test elements of the model against the results of experimental studies (see, for example, Dobson et al., 1990a) but this still does not guarantee that all the model components will interact in an appropriate way nor that the model embodies all the important processes. Another difficulty common to all the specific models oftrichostrongylid infections is that as more biological detail is included it become progressively more difficult to model several crucial elements. One of these is the anamnestic response. The host immune response to infection with trichostrongylid nematodes is rarely complete and wanes during periods when the infection rate is low (during periods when animals are housed or maintained on dry lots, for example). This waning response is an explicit component of the generic models outlined above, but only rarely does it feature in the otherwise detailed specific models for trichostrongylid infections, most of which assume that the host's immunological memory is infinitely long. Exceptions include Coyne's model for Haemonchus contortus (Coyne, 1991 ). A further problem with the host's response to infection arises because not every host harbours the same number of parasites. When parasite demography in a single host is determined in part by the intensity of infection in that host, the dynamics of the infection over an entire population of hosts depend upon the parasite frequency distribution. Based upon work by May (1977) and Anderson and May ( 1978 ), Smith et al. (1987b) suggested a method of dealing with the problem when parasite fecundity was a function of the host's experience of infection and Smith and Guerrero (1993) extended the same method to deal with those instances where parasite abundance was regulated by host-mediated parasite mortality. Unfortunately, all of these formulations assumed that the constraint on parasite population increase depended upon the current parasite frequency distribution and ignored the probability that increased parasite mortality or decreased parasite fecundity was a function of the whole history of
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infection, including all past parasite frequency distributions. In the event, Smith and Guerrero (1993) suggested that parasite frequency distributions could be legitimately ignored provided the parasite frequency distribution was not very aggregated and sensitivity analyses of the behaviour of a model for Ostertagia ostertagi infections in cattle would seem to support that conclusion (Grenfell et al., 1987).
4.3. Anthelmintic resistance Although we have been careful to point out what seem to us to be the major problems with the specific models of trichostrongylid infections, the reader should not conclude that we believe such models are without value. The more recent models are demonstrably good mimics of the patterns observed in the field (Grenfell et al., 1987; Smith and Guerrero, 1993 ). They can and have been used to great effect to demonstrate to farmers and ranchers why one disease control strategy works better than another and there are instances in which they have suggested innovative methods of parasite control that could not have arisen from the study of the behaviour of generic models. A single example will suffice. There are three models that have dealt with the problem of anthelmintic resistance amongst trichostrongylids. Two of them are specific models (Gettinby et al., 1989; Barnes and Dobson, 1990), the third is a generic model (Smith, 1990). All three models used the same simple genetic framework to simulate the spread of resistant phenotypes but only the specific models incorporated realistic details with respect to grazing management practices. A study of the behaviour of the uncluttered generic model quickly led to generalisations about the consequences of particular anthelmintic regimens. The specific models were less helpful in this regard. For example, Gettinby et al. (1989) concluded on the basis of their simulations that selection for resistance can be expected to be slower with less efficacious drugs. In fact, this is only true when the drug is incapable of killing the heterozygotes. This would be the case when the allele for resistance is completely dominant over the allele for susceptibility. As the degree of dominance lessens, and drugs are able to kill at least some of the heterozygotes, the time to significant resistance increases. Over this range, selection for resistance is slowest for those dosages which kill the largest fraction ofheterozygotes (Fig. 2 ) (Smith, 1990). However, the shortcomings of the generic model were revealed in a comparative study by I.A. Barger (personal communication, 1992 ). He compared its behaviour with that of the specific model described by Barnes and Dobson (1990). As might be expected, both models gave qualitatively identical results when parasite control was confined to the administration of anthelmintics. In particular both models demonstrated the superiority of a strategy in which different drugs were administered simultaneously rather than in various kinds of rotation. Selection for resistance was slowest when mixtures of drugs were used. However, Barger was able to extend the analysis using the more detailed specific model and show that strategies which combined judicious movement from one pasture to
G. Smith, B.T. Grenfell / Veterinary Parasitology 54 (1994) 127-143
Decreasingdrug efficacyimpedes the spreadof resistance
o 1,00 0.80
c
•-
~
0.60
"~ v
0.40
~
137
Increasingdrug ,~ efficacyimpedes / { the spreadof t / resistance
0.20
0.00 u_
0.00
0.20
0.40
0.60
0.80
1.00
Fractionkilledby drug Fig. 2. To show the effect of increasing anthelmintic efficacy on the frequency of the allele for resistance (measured over an arbitrary time period) (Smith, 1990).
another with the administration of a single drug were also more effective than drug rotation.
5. Specific models for experimental infections in the laboratory
Heligmosomoidespolygyrus is a well studied nematode parasite of rodents. In particular, it has been the subject of a number of modelling studies which provide an excellent illustration of the way in which models can provide insight into the mechanisms that generate the observed patterns of parasite and host abundance. For example, Keymer and Hiorns ( 1986 ) experimentally infected outbred laboratory mice with a known number of Heligmosomoidespolygyrus. There was no evidence that worm survival or fecundity was regulated in those mice that were infected on just a single occasion. The demographic parameters estimated from this primary infection experiment allowed Keymer and Hiorns to generate a simulation of the expected result when the mice were repeatedly infected with Heligmosomoides polygyrus. The expected and observed results were very different, thus providing unequivocal evidence that some regulatory process was operating in repeated infections (Fig. 3 (a)). Further laboratory work showed that this regulatory response was modulated by protein intake (Slater and Keymer, 1986), the prevalence of infection in a population of mice on a low plane of nutrition continuing to increase long after the prevalence of infection in a better fed population of hosts had begun to diminish (Fig. 3 (b)). The entire system was modelled in detail by Berding et al. (1987). The resulting model is interesting because of the method used to represent acquired immunity. Berding et al. ( 1987 ) made the following assumptions: the antigenic information (I) which triggers the immune response (/~z) depended on the number (L) of tissue dwelling larvae summed over the previous T days t
I= [ L(i)d[ t--T
(12)
G. Smith, B.T. Grenfell / Veterinary Parasitology 54 (I994) 127-143
138
b.
///~~
100 300240180
/ / / / f
/ ~
-
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I
40 12o
/
o JLo.
oo
t
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.
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~
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Fig. 3. Experimental infections with Heligmosomoidespolygyrus. (a) Comparison of observed and predicted mean worm burden in mice infected with 50 larvae per week for 22 weeks: e, observed results; dashed line, predicted burdens assuming no host response; solid line, predicted burdens based on Eqs. ( 12)-(14) (Keymer and Hiorns, 1986; Berding et al., 1987). (b) Prevalence of infection in naturally infected mice on high (dashed line) and low (solid line ) planes of protein nutrition. (Berding et al., 1987 ). (c) Predicted frequencies of mouse genotypes in the presence of Heligmosomoides polygyrus infection. Instantaneous rate of parasite-induced host mortality (per host per parasite per week) was 0.0006 (genotype AA), 0.00094 (genotype aa), and 0.00044 (genotype Aa) (mortality rates estimated from data in Si-Kwang, 1966 ). Model was a simple extension of the Anderson and May (1978 ) generic model for parasite induced host mortality given an aggregated macroparasitic infection. In cases of heterozygote advantage a stable polymorphism eventually results (Smith et al., 1984).
They further assumed that the strength of the immune response depended on the nutritional status of the host, that the immune system exhibits threshold behaviour and that its activity saturates at some maximum level. This led to the expression aI2 l/2 - ( f l + 1 2 )
( 13 )
in which a was the maximum value of/z2 and fl was an index of the sensitivity of the system. The expression for I should be compared with the expressions used by Gordon et al. (1970) and Roberts and Grenfell ( 1991 ) (Eq. ( 4 ) ) . One of the principal differences is that Berding et al. (1987) assumed that the immunological memory lasted for a fixed period, T, whereas Gordon et al. (1970) and Roberts and Grenfell ( 1991 ) imagined a more plastic response with a mean duration of 1/a.
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139
The entire model as set out by Berding et al. (1987) was
d•t•
= 2 - ~D . L
(14)
~dt = ~ D ' L - (It1 + It2)M where L is the mean number of tissue dwelling larvae, M is the mean number of mature worms, 2 is the infection rate, D is the proportion of larvae which become mature worms after a developmental time delay of 1/~,/zl is the intrinsic death rate of mature worms and/t2 (defined above) is the additional death rate due to the host's immune response. This very simple model proved to be an excellent mimic of the repeated experimental infection shown in Fig. 3 (a). The model represented by Eq. (14) included no reference to the host population because it was designed to model repeated moderate infections of a group of hosts maintained on a high plane of nutrition. Under these conditions the infection is well tolerated and there is no parasite-induced host mortality. However, Slater and Keymer (1986) showed that natural infections in mice maintained without protein supplement caused substantial host mortality. Classic generic model studies by Anderson and May ( 1978 ) showed that density dependent parasite induced host mortality could potentially regulate host population abundance and it is interesting to note that the introduction of Heligmosomoides polygyrus into free-running laboratory mouse colonies reduced mouse population abundance to an equilibrium level nearly one-twentieth the size of the original uninfected equilibrium level (Scott, 1987). The relationship between rodents and Heligmosomoides polygyrus is presumed to be long standing in evolutionary terms and so the question arises as to why is there still substantial host mortality in populations that are otherwise adequately fed? In fact, the question itself is flawed because it assumes that the evolutionary path with be toward a less pathogenic relationship. This is not necessarily the case. Smith et al. (1984) noted that homozygous host genotypes for which Heligmosomoides polygyrus was highly pathogenic would coexist in the long term with better adapted genotypes provided the crosses between the two genotypes were better able to survive the infection than either of the homozygotes (Fig. 3 (c)). Anderson and May (1982) argued more generally that the interplay between pathogenicity and transmissibility left room for "many coevolutionary paths to be followed with many end points".
6. Specific models of nematode infections in wildlife populations 6.1. Trichostrongylus tenuis in the Red Grouse Trichostrongylus tenuis is found in the caecum of the Red Grouse, a territorial game bird that feeds almost exclusively on ling heather. Long term data sets for
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grouse populations in the north of England and Scotland indicate that grouse population abundance cycles with a period of between 4 and 5 years. There is considerable dispute about what causes these cycles (Cherfas, 1990). Some workers suggest that differential aggression between kin and non-kin coupled with inversely density-dependent breeding success may be the explanation (Moss and Watson, 1991 ). Others suggest that T. tenuis is implicated (Dobson and Hudson, 1992, 1994). The argument that the parasite might cause the observed cycles is based on an extension of the generic models set out by Anderson and May ( 1978 ). The principal additions are an equation for the free living stages (W) of the nematode and an equation for those ingested larvae (A) that undergo a phase of temporarily arrested development. The remaining equations model host (H) and developing parasite (P) abundance, respectively
dH/dt= ( a - b ) H - (a+f)P dW/dt=2P-TW-flwg
( 15 )
dA/ dt=eflWH- (PA + b+ O) w - a P ~
( 17)
(16)
The grouse are assumed to have an intrinsic rate of increase ( a - b) which is modified by infections as a result of parasite-induced host deaths ( a ) and a reduction in fecundity (6). The free living stages die at a rate, 7, and are ingested at a rate fill. A proportion (g) of these enters the arrested state for a constant period (1/0). All parasites are subject to intrinsic mortality (fli) as well as mortality due to natural host mortality (b) and parasite-induced host mortality ( a ) . The parasites that survive to maturity give rise to free-living larvae at a rate 2 (Dobson and Hudson, 1992 ). A formal stability analysis of the model indicates that it will settle to a stable equilibrium (i.e. host abundance constant) provided a/6> k, where k is an inverse measure of the degree of parasite aggregation (Dobson and Hudson, 1992 ). In fact, this condition is rarely met in the field and the substitution of realistic parameters for a, 6 and k lead to oscillations that may either escalate or gradually disappear depending on the values of the other parameters in the model. However, the oscillations observed in the field are sustained (at least over the last 50 years) and so the model represented by Eqs. (15 ) - ( 18 ) is still imperfect. One modification which leads to sustained (but complicated) cycles is the inclusion of seasonal grouse breeding cycles (Dobson and Hudson, 1994). This device more than doubled the number of equations in the model but revealed that the seasonal birth of uninfected, susceptible chicks was sufficient to convert the expanding or damped cycles of the original model into stable limit cycles. This is an interesting example with which to finish. It shows how models can provide tentative explanations for observed phenomena (cycling grouse populations), and how model development characteristically
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proceeds in a stepwise manner, from simple to more complex, in order to investigate the impact of each new feature on the property under investigation. It also shows how generic models with established behaviours can form the basis for more complicated specific models whose behaviour has yet to be analysed.
7. Discussion
The models discussed here were selected to show the kind and variety of problems that can be usefully addressed by modelling. There are many styles of modelling, but the cynic might aver that those modellers most removed from field work tend to write the simplest models. The proponents of simple generic modelling approaches argue, correctly, that there is nothing intrinsically better about writing a model rich in biological detail. It is a matter of experience that much of what one sees in the field can be explained in terms of the operation of just a few significant mechanisms, the rest is often unimportant clutter. Nevertheless, there is also nothing wrong with complexity if that is what is necessary to address the problem at hand. The grouse model and the models dealing with anthelmintic resistance are good examples of that. The rational strategy is start simply, evaluate model behaviour at each stage, and add complications in a stepwise fashion until the model is adequate to the task. In terms of future work, we believe there is considerable scope for refining generic models of the epidemiology of nematode infections of wildlife host populations. There are a number of pertinent problems. For example, does seasonality in host and parasite demography have the same profound effects on macroparasite dynamics as it does in some microparasitic systems. In human measles, for instance, the seasonal aggregation of children in schools forced dramatic (possibly chaotic) fluctuations in incidence in developed countries in the prevaccination era (Grenfell, 1992b; Bolker and Grenfell, 1993 ). The same effect can be generated in macroparasitic models with seasonal variations in host or parasite population parameters as long as the natural time scale of the parasite dynamics (roughly the period in which a population perturbed from equilibrium cycles back to it in the absence of forcing) is near the forcing frequency (Roberts and Grenfell, 1992; Grenfell et al., unpublished data, 1993). In real systems this damping rate appears to be many years (Roberts and Grenfell, 1992), so that host-nematode interactions may be too "sluggish" to exhibit complex dynamics via this simple form of resonance. This and other areas (most notably host immunity and host-parasite coevolution ) are likely to be growth areas in modelling. Though we stress that such models should never stray far from their empirical base.
References Anderson, R.M. and May, R.M., 1978. Regulation and stability of host parasite interactions. J. Anim. Ecol., 85: 373-398.
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Anderson, R.M. and May, R.M., 1982. Coevolution of hosts and parasites. Parasitology, 85:411-426. Anderson, R.M. and May, R.M., 1991. Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford. Barger, I.A., 1985. The statistical distribution of trichostrongylid nematodes in grazing lambs. Int. J. Parasitol., 15: 645-649. Barnes, E.H. and Dobson, R.J., 1990. Population dynamics of Trichostrongylus colubriformis in sheep: a computer model to simulate grazing systems and the evolution of anthelmintic resistance. Int. J. Parasitol., 20:823-831. Berding, C., Keymer, A.E., Murray, J.D. and Slater, A.F.G., 1987. The population dynamics of acquired immunity to helminth infection: experimental and natural transmission. J. Theor. Biol., 126: 167-182. Bolker, B.M. and Grenfell, B.T., 1993. Chaos and biological complexity in measles dynamics. Proc. R. Soc. London, Ser. B, 251: 75-81. Cherfas, J., 1990. Feathers fly in grouse population dispute. Science, 247: 32. Coyne, M.J., 1991. Field and experimental studies of Haemonchus contortus in lambs. Ph.D. Thesis, University of Pennsylvania. Dineen, J.K., 1963. Immunological aspects of parasitism. Nature, 197: 268-269. Dineen, J.K., Donald, A.D., Wagland, B.M. and Turner, J.H., 1965. The dynamics of the host parasite relationship. II. The response of sheep to primary and secondary infection with Nematodirus spathiger. Parasitology, 55: 163-171. Dobson, A.P. and Hudson, P.J., 1992. Regulation and stability of a free-living host-parasite system: Trichostrongylus tenuis in red grouse. II. Population models. J. Anim. Ecol., 61: 487-498. Dobson, A.P. and Hudson, P.J., 1994. The population biology of Trichostrongylus tenuis in red grouse, Lagopus lagopus scoticus. In: M.E. Scott and G. Smith (Editors), Ecological Epidemiology of Infectious and Parasitic Disease. Academic Press, New York, pp. 309-317. Dobson, R.J., Donald, A.D., Barnes, E.H. and Waller, P.J., 1990a. Population dynamics of Trichostrongylus colubriformis in sheep: model to predict the worm population over time as a function of infection rate and host age. Int. J. Parasitol., 20: 365-373. Dobson, R.J., Waller, P.J. and Donald, A.D., 1990b. Population dynamics of Trichostrongylus colubriformis in sheep: the effect of infection rate on the establishment of infective larvae and parasite fecundity. Int. J. Parasitol., 20: 347-352. Gettinby, G. and Paton, G., 1981. The role of temperature in predicting the pattern of bovine Ostertagia infection on pasture. J. Theor. Biol., 6: 395-402. Gettinby, G., Bairden, K., Armour, J. and Benitez-Usher, C., 1979. A prediction model for bovine ostertagiasis. Vet. Rec., 105: 57-59. Gettinby, G., Soutar, A., Armour, J. and Evans, P., 1989. Anthelmintic resistance and the control of ovine ostertagiasis: a drug action model for genetic selection. Int. J. Parasitol., 19: 369-376. Gordon, G., O'Callaghan, M. and Tallis, G.M., 1970. A deterministic model for the life cycle of a class of internal parasites of sheep. Math. Biosci., 8: 209-226. Grenfell, B.T., 1988. Gastrointestinal nematode parasites and the stability and productivity of intensive ruminant grazing systems. Phil. Trans. R. Soc., London, Ser. B, 321:541-563. Grenfell, B.T., 1992a. Parasitism and the dynamics of ungulate grazing systems. Am. Nat., 139: 907929. Grenfell, B.T., 1992b. Chance and chaos in measles dynamics. J. R. Stat. Soc., B, 54: 383-398. Grenfell, B.T., Smith, G. and Anderson, R.M., 1987. A mathematical model of the population of Ostertagia ostertagi in calves and yearlings. Parasitology, 95:389-406. Keymer, A.E. and Hiorns, R.W., 1986. Heligmosomoides polygyrus (Nematoda): the dynamics of primary and repeated infection in outbred mice. Proc. R. Soc. London, B, 229: 47-67. May, R.M., 1977. Dynamical aspects of host-parasite associations: Croften's model revisited. Parasitology, 75: 259-275. Moss, R. and Watson, A., 1991. Population cycles and kin selection in red grouse, Lagopus lagopus scoticus. Ibis, 133 (Suppl.): 113-120. Noy-Meir, I., 1975. The stability of grazing systems: an application of predator prey graphs. J. Ecol., 63: 417-427.
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Ollerenshaw, C.B., Graham, E.G. and Smith, L.P., 1978. Forecasting the incidence of parasitic gastroenteritis in lambs in England and Wales. Vet. Rec., 103:461-465. Paton, G., 1987. A model for predicting parasitic gastroenteritis in lambs in mixed nematode infections. Res. Vet. Sci., 43: 67-71. Paton, G., Thomas, R.J. and Waller, P.J., 1984. A prediction model for parasitic gastroenteritis in lambs. Int. J. Parasitol., 14: 439-445. Roberts, M.G. and Grenfell, B.T., 1991. The population dynamics of nematode infections of ruminants: periodic perturbations as a model for management. IMA J. Math. Appl. Med. Biol., 8: 8393. Roberts, M.G. and Grenfell, B.T., 1992. The population dynamics of nematode infections in ruminants: the effect of seasonality in the free-living stages. IMA J. Math. Appl. Med. Biol., 9:29-41. Roberts, M.G. and Heesterbeck, J., 1994. Bluffyour way into epidemiological models. Trends Microbiol., in press. Scott, M., 1987. Regulation of mouse colony abundance by Heligmosomoides polygyrus. Parasitology, 95: 111-124. Si-Kwang, L., 1966. Genetic influence on resistance of mice to Nematospiroides dubius. Exp. Parasitol., 18:311-319. Slater, A.F.G. and Keymer, A.E., 1986. Epidemiology of Heligmosomoides polygyrus in mice: experiments on natural transmission. Parasitology, 93: 177-187. Smith, G., 1988. The population biology of the parasitic phase of Haemonchus contortus. Parasitology, 96: 105-195. Smith, G., 1989. The population biology of the parasitic phase of Ostertagia circumcincta. Int. J. Parasitol., 19: 385-393. Smith, G., 1990. A mathematical model for the evolution of anthelmintic resistance in a direct lifecycle nematode parasite. Int. J. Parasitol., 7:913-921. Smith, G., 1994. Population biology of the parasitic phase of trichostrongylid nematode parasites of cattle and sheep. Int. J. Parasitol., 24: 167-178. Smith, G. and Galligan, D.T., 1988. Mathematical models of the population biology of Ostertagia ostertagi and Teladorsagia circumcincta, and the economic evaluation of disease control strategies. Vet. Parasitol., 27: 73-83. Smith, G. and Guerrero, J., 1993. Mathematical models for the population biology of Ostertagia ostertagi and the significance of aggregated parasite distributions. Vet. Parasitol., 46: 243-257. Smith, G., Grenfell, B.T. and Kerboeuf, D., 1984. The population dynamics of Heligmosomoides polygyrus (Nematospiroides dubius) in mice: the consequences of parasite induced host mortality and the effect of the host response on parasite survival and fecundity. In Proc. of the British Society for Parasitology, Bristol, 4-6 April 1984, Parasitology. (Abstr.) Smith, G., Grenfell, B.T., Anderson, R.M. and Beddington, J., 1987a. Population biology of Ostertagia ostertagi and anthelmintic strategies against ostertagiasis in calves. Parasitology, 95: 407-420. Smith, G., Grenfell, B.T. and Anderson, R.M., 1987b. Regulation of Ostertagia ostertagi populations in calves: density dependent control of fecundity. Parasitology, 95: 373-388. Tallis, G.M. andDonald, A.D., 1970. Further models for the distribution on pasture of infective larvae of the strongyloid nematode parasites of sheep. Math. Biosci., 7:179-190. Tallis, G.M. and Leyton, M.K., 1966. A stochastic approach to the study of parasite populations. J. Theor. Biol., 13: 251-260. Tallis, G.M. and Leyton, M.K., 1969. Stochastic models of helminthic parasites in the definitive host. 1. Math. Biosci., 4: 39-48. Thomas, R.J. and Start, J.R., 1978. Forecasting the peak of gastrointestinal nematode infection in lambs. Vet. Rec., 103: 465-468.
VeterinaryParasitology54 (1994) 127-143
Modelling of parasite populations: gastrointestinal nematode models G. Smith a'*, B.T.
Grenfell b
aSchool of Veterinary Medicine, University of Pennsylvania, New Bolton Center, 382 W. Street Road, Kennett Square, PA 19348, USA bDepartment of Zoology, University of Cambridge, Cambridge CB2 3EJ, UK
Abstract
This paper surveys models of nematode parasites of veterinary importance. A distinction is drawn between generic models which are usually simple formulations applicable to whole classes of parasite and specific models which are often more complex and designed to address questions concerning a particular species. Most of the models considered employ a deterministic framework. Four main groups are considered: generic models of trichostrongylid infection of domestic ruminants, specific models of trichostrongylid infection of domestic ruminants, specific models of experimental laboratory infections of rodents, and a specific model of nematode infections in wildlife. Keywords: Modelling
1. Introduction
Roberts and Heesterbeck (1994) cite a number of reasons why we should want to model the dynamics of infectious diseases. Among the most important are that models provide insight into the mechanisms underlying observed patterns and enable the modeller to conduct thought experiments concerning the efficacy of plausible disease control strategies. Roberts and Heesterbeck also grudgingly mentioned that models may sometimes be useful for predicting future trends, but only under very constrained circumstances. Each of these various modelling objectives will be illustrated in the context of a historical review of nematode models of veterinary importance. We shall consider two kinds of model: generic and spe* Correspondingauthor. 0304-4017/94/$07.00 © 1994 ElsevierScienceB.V. All rights reserved SSD10304-4017 (94)03081-7
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cific. Generic models are used to tease out generalisations about the dynamics of the parasite-host interaction. Generic models refer not to any particular parasite but rather to an assemblage of similar species. The structure of generic models is quite deliberately kept as simple as possible. This facilitates the analysis of their behaviour and obviates the possibility that extraneous biological details may obscure the more important processes. As we shall see, the usual technique for analysing the behaviour of generic models is to increase the complexity of the model in small increments and to examine the significance of that change in stepwise fashion. Specific models, however, frequently contain a great deal of biological detail. They are designed to address particular questions about the dynamics or control of a particular infection and are imbued with a spurious credibility by reason of their "biological realism". Their complex structure also hampers the analysis of model behaviour. Often the best we can do is to generalise from repeated numerical simulations. Nevertheless, specific models have a long and honourable history in the veterinary literature and, prudently applied, they can be very useful indeed. We shall consider the generic models first, tracing their development over the last 30 years. Next, we shall describe some specific models. These fall into three main categories: models for the trichostrongylid nematode parasites of cattle and sheep, models of experimental nematode infections in laboratory animals, and finally models for nematode infections of wildlife species.
2. Generic models 2.1. Stochastic models
TaUis and Leyton (1966) described a stochastic model for nematode infections in sheep. Although the primary objective of the paper was to show how generic models of infections could be formulated, the authors did explore the relationship between their assumptions regarding the uptake and maturation of female worms and the distribution of eggs in the faeces. They were concerned with describing the probabilities that a parasite was ingested during the small interval, x, and entered the mature class during the small interval, y. To keep things simple, they assumed that neither probability was affected by the number of parasites already in the host (i.e. no host immune response) and were able to show that if both infection and maturation were random, Poisson processes, the distribution of parasites in the hosts would have a Poisson distribution but the eggs in the faeces would have a Neyman Type A (aggregated, clumped) distribution. This was important because it demonstrated that an aggregated distribution of eggs in the faeces does not necessarily imply an aggregated distribution of parasites in the hosts. Nevertheless, TaUis and Leyton were perfectly well aware that the infective stages on pasture were themselves distributed in an aggregated fashion (Tallis and Donald, 1970) and that, in consequence, the distribution of mature parasites was unlikely to be random. In a sequel to their original paper, Tallis and
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Leyton ( 1969 ) compared the consequences of placing hosts on pastures in which the distribution of infective stages was random or aggregated. Not surprisingly, when the distribution of infective stages was assumed to be aggregated, the equilibrium frequency distribution of mature parasites turned out to be a negative binomial. Subsequently, the negative binomial distribution has proved to be a very useful, phenomenological mimic of the aggregated distributions that characterise the distribution of nematodes in the parasitic phase of the life cycle (Barger, 1985 ). We note in passing, of course, that there are other factors which give rise to aggregated parasite distributions in addition to heterogeneities in the availability of infective stages. Not least are differences between hosts that may affect infection rate or resistance to parasites (Anderson and May, 1991 ). 2.2. Deterministic models
Stochastic models are notoriously difficult to handle and this pioneering effort was followed by a more tractable deterministic model developed by Gordon et al. (1970). This basic architecture of the model was mapped on to the life cycle of a trichostrongylid nematode parasite of sheep. An abbreviated version will serve to illustrate the principle points of the paper. Infective larvae (L) are ingested by the host population at a rate 23. Ingested larvae (I) die at a rate/zl and mature at a rate 2 ~. The mature parasites (A) produce infective stages at a rate 22 and die at a rate/z2. The infective larvae on the pasture die at a rate/t3. Thus, the rate of change with respect to time in the abundance of each stage is given by tmA3 L -
(ILl "~-~l )I
ttm~.lI--fl2 A
dL
( 1)
= ~ 2 A - - (/.t3 +,~,3)L
In the initial formulation all the development and mortality rates were assumed to be constant. Of particular interest were the conditions that must pertain if the parasite is to persist in the host population. Gordon et al. approached this question by means of a sophisticated stability analysis but the essentials can be captured much more simply. If the parasite is to persist, there must always be viable infective stages (L). The best way of ensuring this is to maintain the rate of change in the number of infective stages ( d L / d t ) at zero or above (a negative rate of change implies a decrease in the value of L). For this to be true the values of A and L at equilibrium can be found by setting the equations for I and M respectively to zero. The condition then becomes ,~,2A (/z3 +Y3)L >~1
(2)
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212223
,/-/2(#3+23) (#1 +21)
>~1
(3)
Gordon et al. noted that if the ratio on the left hand side of Eq. (3) was less than 1 then parasite numbers would eventually decline to zero. If the ratio was exactly equal to 1 then a precarious balance was maintained in which parasites would persist at some fixed level depending upon the initial conditions, and if the ratio were greater than 1, parasite numbers would increase indefinitely. The ratio on the left hand side of Eq. (3) is now known as the basic reproduction ratio (R0). It has a straightforward biological interpretation. The number of infective stages produced per day per parasite is 22. The average lifespan of a mature parasite is given by the reciprocal of the parasite mortality ( 1/#2). Thus the total number of infective stages produced per parasite over its average lifespan is ;t 2/#2. Of these only a fraction, 2 3/(#3 + '~3 ), survive to be ingested and a fraction, ;t 1/(#1 +21 ) survive to become mature parasites. Thus, the number of infective larvae produced during the lifetime of a single parasite that survive to become mature parasites is given by the product of 22/#2, 23/(#3 + '~3 ) and 21/(#1 + 21 ), which is the basic reproduction ratio. Substituting realistic starting values for the parameters of the model described in Eq. ( 1 ) leads quickly to the conclusion that the basic reproduction ratio for trichostrongylids is greater than 1. Nevertheless, parasite numbers do not continue to increase indefinitely. Gordon et al. argued that the host immune system reduced the values of 21, 22 and 23 and increased the values off1, #2 and #3. This would eventually reduce the value of the basic reproduction ratio to values to below 1 and parasite numbers would then decline. Gordon et al. supposed that the hosts possessed an "information dam" into which antigenic information flowed at a net rate proportional to the intensity of infection and leaked away at a constant instantaneous rate. The parameters of the model were deemed to be functions of the accumulated amount of antigenic information and thus modulated by the hosts' experience of infection. Gordon et al. did not specify the nature of the functions linking the amount of information in the "information dam" other than to note in general terms that 2i would be expected to decrease with increasing antigenic information and/~ would increase. It was left to Roberts and Grenfell ( 1991, 1992 ) to supply plausible functions. Their generic model for trichostrongylid infections in ruminants utilised a parameter called the "level of acquired immunity" (r) (Anderson and May, 1991 ) which was essentially the same as the "antigenic information" defined by Gordon et al. (1970) except that it depended on the cumulative number of infective larvae ingested by the host rather than on the current worm burden
dr
d t _ f l L - ar
(4)
Here, fl is the transmission parameter (equivalent to 23 in Eq. ( 1 ) above) and represents the rate of fade of immunological memory. The pasture larval contamination (L) and mature parasites (A) were modelled as follows:
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~tt= P ( r ) p L - # ( r ) A
131
(5)
~t =q2 ( r )A - p L - flL The model includes no immature parasitic phase, p being the probability that an ingested larva survives to become mature. This probability, the death rate of mature worms (/t) and the rate of egg production of mature worms (2) were each functions of the level of acquired immunity (r). The eggs were assumed to have a probability, q, of developing to infective larvae and these larvae were assumed to die at a rate, r. In the initial analysis, q and r were assumed to be constant (i.e. climatic effects were ignored). The basic reproduction ratio for Eqs. (4) and ( 5 ) was given by
R
o
fl2(O)p(O)q =
~
(6)
When Ro < 1 the infection cannot persist. Roberts and Grenfell were interested in accounting for the midsummer rise in pasture larval contamination that is so characteristic of the intensive calf management systems of northern temperate Europe. They found that Eqs. (4) and (5) generated a midsummer rise even in the absence of climatic effects provided that the mean number of parasites (A) and level of acquired immunity (r) were reduced to zero on an annual basis. This device simulated the annual replacement of stock with naive animals. When climatic sensitivity was subsequently incorporated into the model Roberts and Grenfell noted that this annual resetting of the system was still the predominant cause of the midsummer rise. They concluded that discrete management perturbations control the pattern of infection whereas continuous (climatic) perturbations modify the magnitude of the absolute worm burden. This is a particularly clear example of the way in which generic models can clarify which factors are responsible for particular aspects of observed epidemiological patterns. It should be noted that such models can be extended to examine various kinds of problems. For example, Grenfell ( 1988 ) used a model very similar to that represented by Eq. (5) to investigate the impact of ruminant parasitism on the entire grazing system. The required extension was an equation representing the dynamics of the available herbage (V) dV d-~ = G ( V ) - c ( V ) .H
(7)
The function, G(V), is the net growth rate of plant biomass. G(V) increases with plant biomass (V) until shading effects cause it to decline to very low levels (Noy-Meir, 1975 ). The function, c(V), is the per capita rate of herbage ingestion by animals in the gazing herd. The value of c(V) increases sigmoidally with plant biomass and eventually saturates at some fixed level (Cmax) (Fig. 1 ). The net rate
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(..)
0.30 0.24
-
-
..
t..-
(.9 09
or"
0.18 0.12 0.06 0.00
,/i//
/
,,,
4
--
~••
///
"~
0.00
0.25
0.50
0.75
1.00
Plant a b u n d a n c e (V) Fig. 1. Diagram to show the impact of parasitism on the stability of a grazing season. The plant dynamics are represented by G (solid line). Uninfected herbivore consumption rates (C) at moderate stocking densities (dashed line) intersect the plant dynamics curve in three places. Each intersection represents a potential equilibrium. The infected herbivore consumption curve (dotted/dashed line) intersects the plant dynamics curve only once. The arrow indicates the effect of parasitism on the herbivore consumption curve (Grenfell, 1988 ).
of herbage consumption varies with the stocking rate, H. The dynamics of this grazing system depend upon the relative shape of the curves for G(V) and c(V). The points where these two curves intersect represent all the possible end points (equilibria) towards which the model forever tends. As might be expected, at low stocking rates there is a single stable equilibrium with high plant biomass; at very high stocking rates the system collapses to a very low equilibrium plant biomass. Paradoxically, at intermediate stocking rates the G (V) and c (V) curves intersect in three places, indicating three possible equilibria. Only two of these equilibria are stable: the lowest and highest, respectively. In this case, the eventual fate of the system depends critically on the initial conditions. The introduction of parasites into the grazing herd has two effects: it reduces per capita food intake (thus affecting c(V) ) and by reducing the proportion of the pasture sampled each day reduces the transmission parameter, fl (thus affecting the infection rate). The overall consequence of this density dependent constraint on consumption is to linearise (flatten) the sigmoidally shaped function representing c(V). As a result, the G(V) and c(V) curves may intersect in only one place, an equilibrium with high plant biomass. The grazing system is effectively stabilised by parasitism. It follows that parasite control may act as a destabilising influence with unpredictable results. Though this particular model assumed that the host population was a domestic population maintained at a constant stocking rate, Grenfell (1992a) showed that the same result also pertained in the case of naturally fluctuating herbivore populations. 3. Specific models as forecasting tools
Among the very first specific nematode models of veterinary interest were models intended to help forecast the risk of infection in the coming year. The
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models described by Gettinby and his co workers (Gettinby et al., 1979; Gettinby and Paton, 1981; Paton et al., 1984; Paton, 1987) were detailed mimics of the life cycle of trichostrongylid nematode parasites of cattle and sheep and have been reviewed and evaluated elsewhere (Smith and Galligan, 1988; Smith, 1989). A somewhat earlier model for the same system adopted a very different approach. Ollerenshaw et al. (1978) were interested in helping farmers control parasitic gastroenteritis in lambs and ewes. To this end, they devised a system of forecasting the intensity of disease (measured by combining a variety of indices). The probable intensity of disease was indicated by a "disease rating", D, where
D=O.3A+O.12M+O.25J+O.5Jy-6.16
(8)
This equation was derived for northern temperate Europe and incorporated several climatic parameters: parameter A was the return to field capacity date in the previous autumn, and M, J, and Jy were the number of wet days in May, June, and July, respectively, in the current year. The correlation between the disease rating and a subjective classification of actual disease occurrence over the 20 years considered was 0.97. The post hoc rationale for this system was that a dry summer in the previous grazing season would delay the peak pasture larval count which would, in turn, cause a high proportion of these larvae to overwinter either on pasture or as arrested forms in ewes retained for breeding. This would facilitate a surge of infection at the beginning of the next grazing season especially if the spring and summer were wet. Thomas and Starr ( 1978 ) described a less convincing system (based on only 9 years of data) for forecasting the timing of the peak pasture larval count. Both methods involved a simple empirical model linking climate with some index of parasite abundance. Neither group of workers was interested in explaining why the midsummer rise in pasture contamination occurred, instead they sought to identify factors which altered its magnitude and timing. Though most workers in the field would accept that weather patterns markedly alter pasture larval contamination, the causal inference remains unproven. Nevertheless, Ollerenshaw et al. (1978 ) found an excellent correlation between climate and the risk of disease, and, while correlation does not imply a causation, the investigation of mechanism implicit in the generic modelling work of Roberts and Grenfell (1992) and our experience with a dozen or so specific models for trichostrongylid infections (see below) suggests that the idea is at least plausible.
4. Specific models for the control of trichostrongylid infections of ruminants
4.1. Basic model structure Smith (1989, 1994) has reviewed the biological ideas that underpin the specific models for trichostrongylid nematode infections of ruminants. This is a well trodden field of research, Smith (1994) lists 13 distinct models. All of them were designed to help evaluate and devise disease control strategies and all of them are
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crammed full of biological detail. Part of their interest lies in the fact that each model is a concise historical record of the information available at the time of model building and a mirror to the various schools of thought about which were the important processes with regard to the natural regulation and control of parasite abundance. An illuminating example is the model for Trichostrongylus colubriformis infections in sheep described by Dobson et al. (1990a). An elegant series of trickle infection experiments revealed that the establishment of adult worms declined with the host's experience of infection (Dobson et al., 1990b). The pattern of decline was sigmoidal, the exact shape of the curve depending on the infection rate and age of the host. Smith (1994) has since suggested that a declining sigmoidal pattern of establishment is a characteristic of all the common trichostrongylid infections of cattle and sheep. However, the T. colubriformis example described by Dobson et al. (1990b) is the only one in which there is any obvious relationship between the rate of infection and the rate of decline in establishment. In the absence of any precise information about the immunological processes leading to this decrease in establishment in other trichostrongylids, Smith (1988 ) and Smith and Galligan (1988 ) settled for a simple empirical logistic model, i.e. e x
P = ( l + e x)
(9)
where p is proportional establishment and x is a linear function of the duration of infection (t).
x=a+bt
(10)
The constant a determined the initial value ofp and the constant b determined its rate of decline. Dobson et al. (1990a) also used a logistic function but chose to interpret their results in terms of a threshold level of adult worm load before a substantial host immune response is generated. They noted that the logistic model was only an approximate mimic of the threshold argument developed by Dineen ( 1963 ) and Dineen et al. ( 1965 ) but accepted all the same the cumbersome formulation that this interpretation obliged them to adopt. In their model, the parameter x becomes
x=b. T+b.ET-b.t
( 11 )
where T is the time taken for the threshold burden to be exceeded, E T is a parameter which determines the rate of decline in establishment once Thas elapsed and b is a constant. T and E T are themselves functions (of the infection rate and host age, respectively). The important point is that, although the establishment data could just as easily have been modelled using a straightforward empirical extension of Eq. (10), Dobson et al. (1990a) used an existing precept to guide their choice of function. There was no discussion of alternative models that fitted less well. This was quite proper and is characteristic of almost all the models described in this paper but does little to enhance our understanding of what is going
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on. This is important because models are not necessarily unique explanations of particular data sets 4.2. S o m e problems with trichostrongylid models
The principal difficulty with models of this sort is the problem of model validation. Since they are intended to be used to compare existing or proposed disease control strategies they should at least generate patterns which experienced field workers would accept as being within normal limits (Smith and Guerrero, 1993 ) and place tested control strategies in the correct rank order with respect to some index of efficacy (Smith et al., 1987a). Unfortunately, it is very difficult to compare model output with field data. Models are sensitive to the initial conditions of the simulation (e.g. stocking density, initial levels of pasture contamination, immune status of the hosts) and these are frequently only imperfectly known or recorded. It is unreasonable to expect precise one-to-one correspondence between a single study and its corresponding model output. One can sometimes combine similar studies in the same bioagricultural area and so get an impression of the reasonable range of epidemiological patterns, but such data are hard to come by. A partial solution is to test elements of the model against the results of experimental studies (see, for example, Dobson et al., 1990a) but this still does not guarantee that all the model components will interact in an appropriate way nor that the model embodies all the important processes. Another difficulty common to all the specific models oftrichostrongylid infections is that as more biological detail is included it become progressively more difficult to model several crucial elements. One of these is the anamnestic response. The host immune response to infection with trichostrongylid nematodes is rarely complete and wanes during periods when the infection rate is low (during periods when animals are housed or maintained on dry lots, for example). This waning response is an explicit component of the generic models outlined above, but only rarely does it feature in the otherwise detailed specific models for trichostrongylid infections, most of which assume that the host's immunological memory is infinitely long. Exceptions include Coyne's model for Haemonchus contortus (Coyne, 1991 ). A further problem with the host's response to infection arises because not every host harbours the same number of parasites. When parasite demography in a single host is determined in part by the intensity of infection in that host, the dynamics of the infection over an entire population of hosts depend upon the parasite frequency distribution. Based upon work by May (1977) and Anderson and May ( 1978 ), Smith et al. (1987b) suggested a method of dealing with the problem when parasite fecundity was a function of the host's experience of infection and Smith and Guerrero (1993) extended the same method to deal with those instances where parasite abundance was regulated by host-mediated parasite mortality. Unfortunately, all of these formulations assumed that the constraint on parasite population increase depended upon the current parasite frequency distribution and ignored the probability that increased parasite mortality or decreased parasite fecundity was a function of the whole history of
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infection, including all past parasite frequency distributions. In the event, Smith and Guerrero (1993) suggested that parasite frequency distributions could be legitimately ignored provided the parasite frequency distribution was not very aggregated and sensitivity analyses of the behaviour of a model for Ostertagia ostertagi infections in cattle would seem to support that conclusion (Grenfell et al., 1987).
4.3. Anthelmintic resistance Although we have been careful to point out what seem to us to be the major problems with the specific models of trichostrongylid infections, the reader should not conclude that we believe such models are without value. The more recent models are demonstrably good mimics of the patterns observed in the field (Grenfell et al., 1987; Smith and Guerrero, 1993 ). They can and have been used to great effect to demonstrate to farmers and ranchers why one disease control strategy works better than another and there are instances in which they have suggested innovative methods of parasite control that could not have arisen from the study of the behaviour of generic models. A single example will suffice. There are three models that have dealt with the problem of anthelmintic resistance amongst trichostrongylids. Two of them are specific models (Gettinby et al., 1989; Barnes and Dobson, 1990), the third is a generic model (Smith, 1990). All three models used the same simple genetic framework to simulate the spread of resistant phenotypes but only the specific models incorporated realistic details with respect to grazing management practices. A study of the behaviour of the uncluttered generic model quickly led to generalisations about the consequences of particular anthelmintic regimens. The specific models were less helpful in this regard. For example, Gettinby et al. (1989) concluded on the basis of their simulations that selection for resistance can be expected to be slower with less efficacious drugs. In fact, this is only true when the drug is incapable of killing the heterozygotes. This would be the case when the allele for resistance is completely dominant over the allele for susceptibility. As the degree of dominance lessens, and drugs are able to kill at least some of the heterozygotes, the time to significant resistance increases. Over this range, selection for resistance is slowest for those dosages which kill the largest fraction ofheterozygotes (Fig. 2 ) (Smith, 1990). However, the shortcomings of the generic model were revealed in a comparative study by I.A. Barger (personal communication, 1992 ). He compared its behaviour with that of the specific model described by Barnes and Dobson (1990). As might be expected, both models gave qualitatively identical results when parasite control was confined to the administration of anthelmintics. In particular both models demonstrated the superiority of a strategy in which different drugs were administered simultaneously rather than in various kinds of rotation. Selection for resistance was slowest when mixtures of drugs were used. However, Barger was able to extend the analysis using the more detailed specific model and show that strategies which combined judicious movement from one pasture to
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Decreasingdrug efficacyimpedes the spreadof resistance
o 1,00 0.80
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~
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"~ v
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Increasingdrug ,~ efficacyimpedes / { the spreadof t / resistance
0.20
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Fractionkilledby drug Fig. 2. To show the effect of increasing anthelmintic efficacy on the frequency of the allele for resistance (measured over an arbitrary time period) (Smith, 1990).
another with the administration of a single drug were also more effective than drug rotation.
5. Specific models for experimental infections in the laboratory
Heligmosomoidespolygyrus is a well studied nematode parasite of rodents. In particular, it has been the subject of a number of modelling studies which provide an excellent illustration of the way in which models can provide insight into the mechanisms that generate the observed patterns of parasite and host abundance. For example, Keymer and Hiorns ( 1986 ) experimentally infected outbred laboratory mice with a known number of Heligmosomoidespolygyrus. There was no evidence that worm survival or fecundity was regulated in those mice that were infected on just a single occasion. The demographic parameters estimated from this primary infection experiment allowed Keymer and Hiorns to generate a simulation of the expected result when the mice were repeatedly infected with Heligmosomoides polygyrus. The expected and observed results were very different, thus providing unequivocal evidence that some regulatory process was operating in repeated infections (Fig. 3 (a)). Further laboratory work showed that this regulatory response was modulated by protein intake (Slater and Keymer, 1986), the prevalence of infection in a population of mice on a low plane of nutrition continuing to increase long after the prevalence of infection in a better fed population of hosts had begun to diminish (Fig. 3 (b)). The entire system was modelled in detail by Berding et al. (1987). The resulting model is interesting because of the method used to represent acquired immunity. Berding et al. ( 1987 ) made the following assumptions: the antigenic information (I) which triggers the immune response (/~z) depended on the number (L) of tissue dwelling larvae summed over the previous T days t
I= [ L(i)d[ t--T
(12)
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138
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Fig. 3. Experimental infections with Heligmosomoidespolygyrus. (a) Comparison of observed and predicted mean worm burden in mice infected with 50 larvae per week for 22 weeks: e, observed results; dashed line, predicted burdens assuming no host response; solid line, predicted burdens based on Eqs. ( 12)-(14) (Keymer and Hiorns, 1986; Berding et al., 1987). (b) Prevalence of infection in naturally infected mice on high (dashed line) and low (solid line ) planes of protein nutrition. (Berding et al., 1987 ). (c) Predicted frequencies of mouse genotypes in the presence of Heligmosomoides polygyrus infection. Instantaneous rate of parasite-induced host mortality (per host per parasite per week) was 0.0006 (genotype AA), 0.00094 (genotype aa), and 0.00044 (genotype Aa) (mortality rates estimated from data in Si-Kwang, 1966 ). Model was a simple extension of the Anderson and May (1978 ) generic model for parasite induced host mortality given an aggregated macroparasitic infection. In cases of heterozygote advantage a stable polymorphism eventually results (Smith et al., 1984).
They further assumed that the strength of the immune response depended on the nutritional status of the host, that the immune system exhibits threshold behaviour and that its activity saturates at some maximum level. This led to the expression aI2 l/2 - ( f l + 1 2 )
( 13 )
in which a was the maximum value of/z2 and fl was an index of the sensitivity of the system. The expression for I should be compared with the expressions used by Gordon et al. (1970) and Roberts and Grenfell ( 1991 ) (Eq. ( 4 ) ) . One of the principal differences is that Berding et al. (1987) assumed that the immunological memory lasted for a fixed period, T, whereas Gordon et al. (1970) and Roberts and Grenfell ( 1991 ) imagined a more plastic response with a mean duration of 1/a.
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The entire model as set out by Berding et al. (1987) was
d•t•
= 2 - ~D . L
(14)
~dt = ~ D ' L - (It1 + It2)M where L is the mean number of tissue dwelling larvae, M is the mean number of mature worms, 2 is the infection rate, D is the proportion of larvae which become mature worms after a developmental time delay of 1/~,/zl is the intrinsic death rate of mature worms and/t2 (defined above) is the additional death rate due to the host's immune response. This very simple model proved to be an excellent mimic of the repeated experimental infection shown in Fig. 3 (a). The model represented by Eq. (14) included no reference to the host population because it was designed to model repeated moderate infections of a group of hosts maintained on a high plane of nutrition. Under these conditions the infection is well tolerated and there is no parasite-induced host mortality. However, Slater and Keymer (1986) showed that natural infections in mice maintained without protein supplement caused substantial host mortality. Classic generic model studies by Anderson and May ( 1978 ) showed that density dependent parasite induced host mortality could potentially regulate host population abundance and it is interesting to note that the introduction of Heligmosomoides polygyrus into free-running laboratory mouse colonies reduced mouse population abundance to an equilibrium level nearly one-twentieth the size of the original uninfected equilibrium level (Scott, 1987). The relationship between rodents and Heligmosomoides polygyrus is presumed to be long standing in evolutionary terms and so the question arises as to why is there still substantial host mortality in populations that are otherwise adequately fed? In fact, the question itself is flawed because it assumes that the evolutionary path with be toward a less pathogenic relationship. This is not necessarily the case. Smith et al. (1984) noted that homozygous host genotypes for which Heligmosomoides polygyrus was highly pathogenic would coexist in the long term with better adapted genotypes provided the crosses between the two genotypes were better able to survive the infection than either of the homozygotes (Fig. 3 (c)). Anderson and May (1982) argued more generally that the interplay between pathogenicity and transmissibility left room for "many coevolutionary paths to be followed with many end points".
6. Specific models of nematode infections in wildlife populations 6.1. Trichostrongylus tenuis in the Red Grouse Trichostrongylus tenuis is found in the caecum of the Red Grouse, a territorial game bird that feeds almost exclusively on ling heather. Long term data sets for
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grouse populations in the north of England and Scotland indicate that grouse population abundance cycles with a period of between 4 and 5 years. There is considerable dispute about what causes these cycles (Cherfas, 1990). Some workers suggest that differential aggression between kin and non-kin coupled with inversely density-dependent breeding success may be the explanation (Moss and Watson, 1991 ). Others suggest that T. tenuis is implicated (Dobson and Hudson, 1992, 1994). The argument that the parasite might cause the observed cycles is based on an extension of the generic models set out by Anderson and May ( 1978 ). The principal additions are an equation for the free living stages (W) of the nematode and an equation for those ingested larvae (A) that undergo a phase of temporarily arrested development. The remaining equations model host (H) and developing parasite (P) abundance, respectively
dH/dt= ( a - b ) H - (a+f)P dW/dt=2P-TW-flwg
( 15 )
dA/ dt=eflWH- (PA + b+ O) w - a P ~
( 17)
(16)
The grouse are assumed to have an intrinsic rate of increase ( a - b) which is modified by infections as a result of parasite-induced host deaths ( a ) and a reduction in fecundity (6). The free living stages die at a rate, 7, and are ingested at a rate fill. A proportion (g) of these enters the arrested state for a constant period (1/0). All parasites are subject to intrinsic mortality (fli) as well as mortality due to natural host mortality (b) and parasite-induced host mortality ( a ) . The parasites that survive to maturity give rise to free-living larvae at a rate 2 (Dobson and Hudson, 1992 ). A formal stability analysis of the model indicates that it will settle to a stable equilibrium (i.e. host abundance constant) provided a/6> k, where k is an inverse measure of the degree of parasite aggregation (Dobson and Hudson, 1992 ). In fact, this condition is rarely met in the field and the substitution of realistic parameters for a, 6 and k lead to oscillations that may either escalate or gradually disappear depending on the values of the other parameters in the model. However, the oscillations observed in the field are sustained (at least over the last 50 years) and so the model represented by Eqs. (15 ) - ( 18 ) is still imperfect. One modification which leads to sustained (but complicated) cycles is the inclusion of seasonal grouse breeding cycles (Dobson and Hudson, 1994). This device more than doubled the number of equations in the model but revealed that the seasonal birth of uninfected, susceptible chicks was sufficient to convert the expanding or damped cycles of the original model into stable limit cycles. This is an interesting example with which to finish. It shows how models can provide tentative explanations for observed phenomena (cycling grouse populations), and how model development characteristically
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proceeds in a stepwise manner, from simple to more complex, in order to investigate the impact of each new feature on the property under investigation. It also shows how generic models with established behaviours can form the basis for more complicated specific models whose behaviour has yet to be analysed.
7. Discussion
The models discussed here were selected to show the kind and variety of problems that can be usefully addressed by modelling. There are many styles of modelling, but the cynic might aver that those modellers most removed from field work tend to write the simplest models. The proponents of simple generic modelling approaches argue, correctly, that there is nothing intrinsically better about writing a model rich in biological detail. It is a matter of experience that much of what one sees in the field can be explained in terms of the operation of just a few significant mechanisms, the rest is often unimportant clutter. Nevertheless, there is also nothing wrong with complexity if that is what is necessary to address the problem at hand. The grouse model and the models dealing with anthelmintic resistance are good examples of that. The rational strategy is start simply, evaluate model behaviour at each stage, and add complications in a stepwise fashion until the model is adequate to the task. In terms of future work, we believe there is considerable scope for refining generic models of the epidemiology of nematode infections of wildlife host populations. There are a number of pertinent problems. For example, does seasonality in host and parasite demography have the same profound effects on macroparasite dynamics as it does in some microparasitic systems. In human measles, for instance, the seasonal aggregation of children in schools forced dramatic (possibly chaotic) fluctuations in incidence in developed countries in the prevaccination era (Grenfell, 1992b; Bolker and Grenfell, 1993 ). The same effect can be generated in macroparasitic models with seasonal variations in host or parasite population parameters as long as the natural time scale of the parasite dynamics (roughly the period in which a population perturbed from equilibrium cycles back to it in the absence of forcing) is near the forcing frequency (Roberts and Grenfell, 1992; Grenfell et al., unpublished data, 1993). In real systems this damping rate appears to be many years (Roberts and Grenfell, 1992), so that host-nematode interactions may be too "sluggish" to exhibit complex dynamics via this simple form of resonance. This and other areas (most notably host immunity and host-parasite coevolution ) are likely to be growth areas in modelling. Though we stress that such models should never stray far from their empirical base.
References Anderson, R.M. and May, R.M., 1978. Regulation and stability of host parasite interactions. J. Anim. Ecol., 85: 373-398.
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Anderson, R.M. and May, R.M., 1982. Coevolution of hosts and parasites. Parasitology, 85:411-426. Anderson, R.M. and May, R.M., 1991. Infectious Diseases of Humans: Dynamics and Control. Oxford University Press, Oxford. Barger, I.A., 1985. The statistical distribution of trichostrongylid nematodes in grazing lambs. Int. J. Parasitol., 15: 645-649. Barnes, E.H. and Dobson, R.J., 1990. Population dynamics of Trichostrongylus colubriformis in sheep: a computer model to simulate grazing systems and the evolution of anthelmintic resistance. Int. J. Parasitol., 20:823-831. Berding, C., Keymer, A.E., Murray, J.D. and Slater, A.F.G., 1987. The population dynamics of acquired immunity to helminth infection: experimental and natural transmission. J. Theor. Biol., 126: 167-182. Bolker, B.M. and Grenfell, B.T., 1993. Chaos and biological complexity in measles dynamics. Proc. R. Soc. London, Ser. B, 251: 75-81. Cherfas, J., 1990. Feathers fly in grouse population dispute. Science, 247: 32. Coyne, M.J., 1991. Field and experimental studies of Haemonchus contortus in lambs. Ph.D. Thesis, University of Pennsylvania. Dineen, J.K., 1963. Immunological aspects of parasitism. Nature, 197: 268-269. Dineen, J.K., Donald, A.D., Wagland, B.M. and Turner, J.H., 1965. The dynamics of the host parasite relationship. II. The response of sheep to primary and secondary infection with Nematodirus spathiger. Parasitology, 55: 163-171. Dobson, A.P. and Hudson, P.J., 1992. Regulation and stability of a free-living host-parasite system: Trichostrongylus tenuis in red grouse. II. Population models. J. Anim. Ecol., 61: 487-498. Dobson, A.P. and Hudson, P.J., 1994. The population biology of Trichostrongylus tenuis in red grouse, Lagopus lagopus scoticus. In: M.E. Scott and G. Smith (Editors), Ecological Epidemiology of Infectious and Parasitic Disease. Academic Press, New York, pp. 309-317. Dobson, R.J., Donald, A.D., Barnes, E.H. and Waller, P.J., 1990a. Population dynamics of Trichostrongylus colubriformis in sheep: model to predict the worm population over time as a function of infection rate and host age. Int. J. Parasitol., 20: 365-373. Dobson, R.J., Waller, P.J. and Donald, A.D., 1990b. Population dynamics of Trichostrongylus colubriformis in sheep: the effect of infection rate on the establishment of infective larvae and parasite fecundity. Int. J. Parasitol., 20: 347-352. Gettinby, G. and Paton, G., 1981. The role of temperature in predicting the pattern of bovine Ostertagia infection on pasture. J. Theor. Biol., 6: 395-402. Gettinby, G., Bairden, K., Armour, J. and Benitez-Usher, C., 1979. A prediction model for bovine ostertagiasis. Vet. Rec., 105: 57-59. Gettinby, G., Soutar, A., Armour, J. and Evans, P., 1989. Anthelmintic resistance and the control of ovine ostertagiasis: a drug action model for genetic selection. Int. J. Parasitol., 19: 369-376. Gordon, G., O'Callaghan, M. and Tallis, G.M., 1970. A deterministic model for the life cycle of a class of internal parasites of sheep. Math. Biosci., 8: 209-226. Grenfell, B.T., 1988. Gastrointestinal nematode parasites and the stability and productivity of intensive ruminant grazing systems. Phil. Trans. R. Soc., London, Ser. B, 321:541-563. Grenfell, B.T., 1992a. Parasitism and the dynamics of ungulate grazing systems. Am. Nat., 139: 907929. Grenfell, B.T., 1992b. Chance and chaos in measles dynamics. J. R. Stat. Soc., B, 54: 383-398. Grenfell, B.T., Smith, G. and Anderson, R.M., 1987. A mathematical model of the population of Ostertagia ostertagi in calves and yearlings. Parasitology, 95:389-406. Keymer, A.E. and Hiorns, R.W., 1986. Heligmosomoides polygyrus (Nematoda): the dynamics of primary and repeated infection in outbred mice. Proc. R. Soc. London, B, 229: 47-67. May, R.M., 1977. Dynamical aspects of host-parasite associations: Croften's model revisited. Parasitology, 75: 259-275. Moss, R. and Watson, A., 1991. Population cycles and kin selection in red grouse, Lagopus lagopus scoticus. Ibis, 133 (Suppl.): 113-120. Noy-Meir, I., 1975. The stability of grazing systems: an application of predator prey graphs. J. Ecol., 63: 417-427.
G. Smith, B.T. Grenfell / Veterinary Parasitology 54 (1994) 127-143
143
Ollerenshaw, C.B., Graham, E.G. and Smith, L.P., 1978. Forecasting the incidence of parasitic gastroenteritis in lambs in England and Wales. Vet. Rec., 103:461-465. Paton, G., 1987. A model for predicting parasitic gastroenteritis in lambs in mixed nematode infections. Res. Vet. Sci., 43: 67-71. Paton, G., Thomas, R.J. and Waller, P.J., 1984. A prediction model for parasitic gastroenteritis in lambs. Int. J. Parasitol., 14: 439-445. Roberts, M.G. and Grenfell, B.T., 1991. The population dynamics of nematode infections of ruminants: periodic perturbations as a model for management. IMA J. Math. Appl. Med. Biol., 8: 8393. Roberts, M.G. and Grenfell, B.T., 1992. The population dynamics of nematode infections in ruminants: the effect of seasonality in the free-living stages. IMA J. Math. Appl. Med. Biol., 9:29-41. Roberts, M.G. and Heesterbeck, J., 1994. Bluffyour way into epidemiological models. Trends Microbiol., in press. Scott, M., 1987. Regulation of mouse colony abundance by Heligmosomoides polygyrus. Parasitology, 95: 111-124. Si-Kwang, L., 1966. Genetic influence on resistance of mice to Nematospiroides dubius. Exp. Parasitol., 18:311-319. Slater, A.F.G. and Keymer, A.E., 1986. Epidemiology of Heligmosomoides polygyrus in mice: experiments on natural transmission. Parasitology, 93: 177-187. Smith, G., 1988. The population biology of the parasitic phase of Haemonchus contortus. Parasitology, 96: 105-195. Smith, G., 1989. The population biology of the parasitic phase of Ostertagia circumcincta. Int. J. Parasitol., 19: 385-393. Smith, G., 1990. A mathematical model for the evolution of anthelmintic resistance in a direct lifecycle nematode parasite. Int. J. Parasitol., 7:913-921. Smith, G., 1994. Population biology of the parasitic phase of trichostrongylid nematode parasites of cattle and sheep. Int. J. Parasitol., 24: 167-178. Smith, G. and Galligan, D.T., 1988. Mathematical models of the population biology of Ostertagia ostertagi and Teladorsagia circumcincta, and the economic evaluation of disease control strategies. Vet. Parasitol., 27: 73-83. Smith, G. and Guerrero, J., 1993. Mathematical models for the population biology of Ostertagia ostertagi and the significance of aggregated parasite distributions. Vet. Parasitol., 46: 243-257. Smith, G., Grenfell, B.T. and Kerboeuf, D., 1984. The population dynamics of Heligmosomoides polygyrus (Nematospiroides dubius) in mice: the consequences of parasite induced host mortality and the effect of the host response on parasite survival and fecundity. In Proc. of the British Society for Parasitology, Bristol, 4-6 April 1984, Parasitology. (Abstr.) Smith, G., Grenfell, B.T., Anderson, R.M. and Beddington, J., 1987a. Population biology of Ostertagia ostertagi and anthelmintic strategies against ostertagiasis in calves. Parasitology, 95: 407-420. Smith, G., Grenfell, B.T. and Anderson, R.M., 1987b. Regulation of Ostertagia ostertagi populations in calves: density dependent control of fecundity. Parasitology, 95: 373-388. Tallis, G.M. andDonald, A.D., 1970. Further models for the distribution on pasture of infective larvae of the strongyloid nematode parasites of sheep. Math. Biosci., 7:179-190. Tallis, G.M. and Leyton, M.K., 1966. A stochastic approach to the study of parasite populations. J. Theor. Biol., 13: 251-260. Tallis, G.M. and Leyton, M.K., 1969. Stochastic models of helminthic parasites in the definitive host. 1. Math. Biosci., 4: 39-48. Thomas, R.J. and Start, J.R., 1978. Forecasting the peak of gastrointestinal nematode infection in lambs. Vet. Rec., 103: 465-468.