INTRODUCTION
Nematode infection is a major disease in ruminants (Perry and Randolph, Reference Perry and Randolph1999), and Teladorsagia circumcincta is the dominant species affecting sheep in the UK. Not only does nematode infection in farmed animals adversely affect welfare, it also imposes constraints on the efficiency of livestock production and is the most costly endemic disease affecting domestic sheep populations (Nieuwhof and Bishop, Reference Nieuwhof and Bishop2005). Control measures currently depend on regular anthelmintic treatments, which are simple and cost-effective. However, the appearance of anthelmintic resistance in parasite populations (Yue et al. Reference Yue, Coles and Blake2003; Bartley et al. Reference Bartley, Jackson, Jackson and Sargison2004; Sargison et al. Reference Sargison, Jackson, Bartley, Wilson, Stenhouse and Penny2007) means that alternative methods of nematode control are needed before sheep farming in its present form becomes unsustainable (Sayers and Sweeney, Reference Sayers and Sweeney2005; Stear et al. Reference Stear, Doligalska and Donskow-Schmelter2007).
Control methods include grazing management (Githigia et al. Reference Githigia, Thamsborg and Larsen2001; Niven et al. Reference Niven, Anderson and Vizard2002), protein supplementation (Coop and Kyriazakis, Reference Coop and Kyriazakis2001; Houdijk et al. Reference Houdijk, Kyriazakis, Jackson, Huntley and Coop2005) and selective breeding for host resistance (Bishop and Stear, Reference Bishop and Stear1997; Stear et al. Reference Stear, Bishop, Mallard and Raadsma2001). While there are no commercially available vaccines, in the future, vaccination could help to control nematodes (Knox et al. Reference Knox, Redmond, Newlands, Skuce, Pettit and Smith2003; Hein and Harrison, Reference Hein and Harrison2005).
No single control measure is likely to be totally satisfactory, and a combination of methods may be needed for efficient control. However, exploring all the potential combinations through experimentation would be a complex and costly procedure. Assessment of combinations of control methods requires models that integrate knowledge of the immune response to infection with knowledge of the parasite life cycle on pasture. The development of such a model provides a step towards the assessment of 2 groups of control strategies: those that act to reduce host exposure to the parasite (e.g. grazing management), and those that aim to improve the host response to infection (e.g. supplementary feeding, selective breeding).
The model presented here incorporates our understanding of nematode fecundity and establishment, and also considers density-dependent constraints on worm growth and egg production. Model predictions are tested against data collected in the field (Stear et al. Reference Stear, Abuagob, Benothman, Bishop, Innocent, Kerr and Mitchell2006). This simple model of immune effector mechanisms incorporated with a model of the parasite life cycle on pasture reproduces observed patterns of faecal egg counts across a season.
MATERIALS AND METHODS
Model overview
The model used data obtained over 4 years (1992–1995) from populations of straightbred Scottish Blackface sheep from a commercial upland farm in Scotland. The major effector mechanisms in nematode infections of sheep are IgA-mediated suppression of fecundity and mast cell-mediated control of parasite numbers. The data were used to estimate the relationships between IgA activity and worm length (Henderson and Stear, Reference Henderson and Stear2006), and between worm length and fecundity (Stear and Bishop, Reference Stear and Bishop1999). Although mast cell degranulation has been associated with a reduction in the number of T. circumcincta (Stear et al. Reference Stear, Bishop, Doligalska, Duncan, Holmes, Irvine, McCririe, McKellar, Sinski and Murray1995c), the development of type I hypersensitivity reactions in sheep is not very well understood, and we have used the results of a recent meta-analysis of parasite establishment (Gaba et al. Reference Gaba, Gruner and Cabaret2006).
The model by Bishop and Stear (Reference Bishop and Stear1997) was adapted to incorporate empirical data to model the interaction between developing immunity and parasite fecundity and establishment. The model can be summarized by 2 equations, one describing worm numbers in the host, and the other the number of infective larvae contaminating the pasture. The model updates daily, and lambs are assumed to be born on day t=0.
The calculation of the worm burden on any particular day, M t, (Equation (1)) can be separated into 2 terms. The first term, M t−1(1−m 3), describes the number of worms surviving at day t given the worm burden at t-1 and an adult mortality rate of m 3. The second term, I t−jE t, describes the new infection, and is the product of the number of infective larvae ingested, I, and the nematode establishment, E. Here, j is the pre-patent period, that is the period before infection in the host becomes evident, and captures the time interval from larval ingestion to maturity of the adult parasite. The number of infective larvae ingested on any day, I t, is directly related to the larval availability on pasture on that day, L t in Equation (2).
Larval availability on pasture, L, is also the sum of 2 terms; one describing the daily mortality of infective larvae, which occurs with rate m 2, and the other describing the appearance of new infective larvae on pasture. The product M t−un t−u is the number of eggs produced per lamb, where n is the fecundity of the adult worms. Here, fecundity is specified per adult worm; the actual number of eggs produced per female worm would be double this value, assuming a 1:1 sex ratio. We also take into account the ewes’ deposition of eggs on pasture, denoted by S. The life stages from the egg up to, but not including, the infective stage die at a daily rate m 1. The number of days spent in these stages is denoted by u. The units of L are larvae per lamb.
Host immunity
We assume that the immune response is driven by the ingestion of infective larvae, with a delay between first exposure and the initiation of the response of z days. The increase in total response on day t is assumed to be directly proportional to the number of larvae ingested on day t-z, with constant of proportionality ρ. Pre-existing activity stimulated by previously ingested larvae is assumed to decay exponentially with half-life τ. Thus, IgA activity, which mediates the immunological control of parasite fecundity, is given by Equation (3), where τ 1 is the half-life of IgA activity and ρ 1 is the immune response related to IgA production.
Establishment
The available data do not allow us to relate the development of mast cell responses to establishment (which we have defined as the percentage of ingested larvae that develop into adult worms). We have therefore modelled an establishment control factor (ECF) whose activity has the half-life τ 2 and a response factor ρ 2 as shown in Equation (4).
We assume that the establishment decreases over time as the lambs’ immune systems develop. To simulate this decrease in establishment, we model an ECF that increases as host immunity develops. By using the meta-analysis published by Gaba et al. (Reference Gaba, Gruner and Cabaret2006), we can estimate the range over which establishment can be expected to vary in pasture-fed lambs as immunity develops. From this we have produced a function for establishment dependent on the acquisition of immunity in the host as shown in Equation (5), where E early is the parasite establishment for lambs newly introduced to infected pasture, and E late is the establishment at the end of the grazing season.
Nematode fecundity
The results from Stear et al. (Reference Stear, Bishop, Doligalska, Duncan, Holmes, Irvine, McCririe, McKellar, Sinski and Murray1995c) were used to fit a function for worm length, l, in cm based on IgA activity and adult worm burden (M), as shown in Equation (6).
Similarly, results from Stear and Bishop (Reference Stear and Bishop1999) were used to produce a relationship for fecundity, n (eggs worm−1 day−1), based on worm length, shown in Equation (7). As an example, for a typical range of worm lengths from 0·7 cm to 1·2 cm the number of eggs produced daily ranges from approximately 17 to 228.
Combining Equations (6) and (7), allows us to express the fecundity of adult nematodes in terms of worm burdens and IgA activity in the host. Coefficients are listed in Table 2.
Nematode mortality
The model includes 3 mortality rates to cover 3 periods of the nematode life cycle. These 3 periods are defined by the changes in environment as nematodes transition from faeces to herbage to host. Mortality in the life stages from egg to L2 is captured by m 1, mortality in the L3 stage by m 2, and mortality in the L4 and adult stages by m 3.
Lamb live weight
SAS statistical software was used to fit the mean live weight of lambs weighed at 28-day intervals throughout a season (Bishop et al. Reference Bishop, Bairden, McKellar, Park and Stear1996) to the Gompertz equation (8) following Macciotta et al. (Reference Macciotta, Cappio-Borlino, Pulina and Saxton2004).
The weight, W, increases in accordance with a growth rate, μ, which is itself modified by a decay parameter, κ. The Gompertz equation is preferred over the generalized logistic equation for fitting growth rates, as it overcomes the possibly inappropriate symmetry of the latter. To fit the data, an intercept, ϕ, was subtracted from all the weights to ensure the curve went through the origin. The coefficients are listed in Table 2, and at the end of the season a lamb weighs approximately 27 kg. We do not include production penalty related to worm burden in our calculation of weight as it is beyond the scope of this paper, and the relationship between production costs of worm burden and increased immunity is unclear. Additionally, the purpose of the growth curve is simply to act as a proxy for intake and faecal output.
Maternal deposition of eggs on pasture and initial larval availability
Infection in the lambs is initiated by larvae that have survived on the pasture from the previous season, L 0, and by the deposition of nematode eggs onto pasture by ewes, S. Though adult sheep with a fully developed immune system usually have very low faecal egg counts (Bishop and Stear, Reference Bishop and Stear2001), periparturient ewes exhibit greatly increased egg counts (Jackson et al. Reference Jackson, Jackson and Williams1988). To model this effect, we assume an initial rate of egg deposition by the ewes at t=0 that decreases linearly to zero after three months.
Ingestion of larvae and faecal egg counts
The larval availability on pasture is only one factor affecting the number of larvae ingested by a lamb, the others being herbage consumption and the amount of herbage per sheep on the pasture. The number of larvae ingested, I, is given in Equation (9).
The stocking density of lambs on pasture, D, is measured in lambs ha−1. The herbage density, H, is measured in kg dry matter (DM) ha−1.
We assume that the amount of herbage consumed, Q, is a function of live weight, and therefore the mass consumed in kilograms is given by Equation (10), with the coefficients listed in Table 2.
The growth curve was also used to predict the mass of faeces produced daily by the developing lambs, on the assumption that the amount of faeces produced, f, is 20 g for every kg of lamb body weight. Therefore, the faecal egg count, c, of a lamb on any particular day is given by Equation (11).
Anthelmintic treatment
Sheep are usually treated with anthelmintics to control nematode infections. Here, we assume that treatment begins at t=29 and continues at 28-day intervals, and is 100% effective. We also assume that treatment is only given to lambs, and ewes remain untreated.
Comparison with observed faecal egg counts
Data from Stear et al. (Reference Stear, Abuagob, Benothman, Bishop, Innocent, Kerr and Mitchell2006) together with further unpublished data collected at the same time were used as a basis for testing the model. These data were collected from Scottish Blackface sheep over a number of years (1992–1995). Faecal samples were taken at 28-day intervals, and all lambs were treated with a broad-spectrum anthelmintic immediately after sampling.
Parameters
Table 1 summarizes the parameter values. The mortality rate of 0·23 for the pre-infective nematode life stages, m 1, was obtained from Learmount et al. (Reference Learmount, Taylor, Smith and Morgan2006), and derived from data published by Niezen et al. (Reference Niezen, Charleston, Hodgson, Miller, Waghorn and Robertson1998). These data were collected from various plant species in New Zealand, in an environment experiencing similar temperature ranges and levels of moisture to the regions of Scotland.
Summary of parameters and variables, and their values used to produce the output shown in Fig. 1a
Summary of coefficients, their values and related parameters
For the mortality rate of the infective stage, m 2, the value of 0·008 was obtained from Kao et al. (Reference Kao, Leathwick, Roberts and Sutherland2000) with reference to data from Gibson and Everett (Reference Gibson and Everett1972). It is interesting to note the marked difference between this mortality rate and that of the earlier life stages. The relatively high value of 0·23 compared to 0·008 may be due to susceptibility to predation by fungi and habitat disturbance in the earlier life stages. When the nematode leaves the faeces, it also undergoes physiological changes that make it more resilient.
For parasites that have been ingested and become established in the host, we use the value of 0·0307 for the daily mortality rate (m 3), obtained from Kao et al. (Reference Kao, Leathwick, Roberts and Sutherland2000) with data from Hong et al. (Reference Hong, Michel and Lancaster1986).
The maternal contribution to larval availability is set to an initial value of 250 000 eggs deposited per lamb per day. In other words, assuming each ewe gives birth to twins, 500 000 eggs per ewe per day. If we assume an average Blackface ewe weighs 50 kg and the production of 20 g of faeces per kg as with lambs, then this equates to a faecal egg count of 500 eggs per gram (epg).
Pasture intake increases in line with lamb growth. Here we assume that consumption increases from zero at birth to a maximum of about 1·5 kg DM of pasture when growth begins to level off at about 24 weeks of age. We modelled pasture intake increasing from birth.
A value of 35 lambs ha−1 was used for the stocking density, D. Herbage density, H, was given the value 1200 kg ha−1 (Waller et al. Reference Waller, Dobson, Donald and Thomas1981). For the parameters relating directly to the parasite, we used a value of 21 days for the development time from egg to infective stage, u (Urquhart et al. Reference Urquhart, Armour, Duncan, Dunn and Jennings1996), and a value of 14 days for the pre-patent period, j (Stear et al. Reference Stear, Bairden, Bishop, Duncan, Karimi, McKellar and Murray1995a). Stocking density is one parameter in the model that we consider free, in that it is not dictated by biological constraints but rather by farming practice. Therefore, this is one parameter that is likely to vary from farm to farm.
For parameterization of the immune response, data from Henderson (Reference Henderson2002) were used to obtain an estimate of 8·1 days for the half-life of plasma IgA (τ 1). This same value was used for the response controlling establishment in the model (τ 2). To fit the model output to the empirical data, a best-fit analysis in which a range of values for the IgA- and establishment-related response parameters (ρ 1 and ρ 2, respectively) were trialled to find those that provided the minimum value of the sum of the squares between simulated and measured faecal egg counts at the 4 measurement points between June and September. It was not our intention to fit the majority of parameters to the empirical data, but rather to match the qualitative increase and decline of faecal egg counts through a season. Although the observed data showed a rise in faecal egg counts in October, this is believed to be due to eggs produced by parasite species other than T. circumcincta, therefore this point was excluded from the analysis. Data points for measured IgA values and estimated establishment were also included in the analysis. Three data points (August, September and October) were taken from Strain et al. (Reference Strain, Bishop, Henderson, Kerr, McKellar, Mitchell and Stear2002), and were used to obtain a least squares best-fit output in combination with faecal egg counts and establishment as described above. Based on data from Gaba et al. (Reference Gaba, Gruner and Cabaret2006), we set the establishment in naive lambs to be 0·4 (E early) and used a single data-point estimate of 0·1 in September with which to obtain the best-fit output by varying E late.
RESULTS
The model reproduced the observed magnitude and time-course of faecal egg counts (Fig. 1a), excluding the October measurement when other parasite species are assumed to be contributing to this number. In a simulation where anthelmintics were not administered, the peak geometric mean egg count was increased by approximately two thirds, and egg count declined more slowly in the second half of the season (Fig. 1b).
(a) A comparison of output from the model with geometric means of empirical faecal egg count data (X): May 3, Jun 142, Jul 255, Aug 73, Sep 79, Oct 181. Parameters used are given in Table 1. Periods of zero egg counts follow treatments with anthelmintic. End-of-season establishment is 0·070. (b) The progression of faecal egg counts through the season when no anthelmintic treatment is given. Parameters used are given in Table 1. End-of season establishment is 0·015.
The breakdown of the contributions made to total larvae on pasture over time shows that the contribution from the ewes dominates the larval availability until late in the season (Fig. 2a). Although the ewes were removed at 12 weeks (indicated by the vertical line), their contribution (dotted line) to the total larval availability (solid line), exceeds that of the lambs’ (dashed line) throughout the season as a consequence of the large numbers of eggs deposited by the ewes and the length of time eggs, pre-infective and then infective larvae are able to survive on pasture. As the ewes were not subject to anthelmintic treatments (treatment times indicated by triangles), there is relatively little impact of the treatment on the level of infective larvae on pasture.
(a) Breakdown of the contributions made to total larvae (solid line) on pasture from ewes (dotted line), lambs (dashed line) and over-wintered larvae (dash-dotted line). Triangles mark points of anthelmintic treatment, and the vertical dashed line marks the point at which ewes no longer deposit any new eggs on pasture. (b) Faecal egg counts obtained when using the parameters in Table 1, but assuming no deposition of eggs by ewes. End-of-season establishment is 0·273.
In the absence of deposition of eggs by ewes, faecal egg counts were substantially lower and rose gradually during the full course of the grazing season (Fig. 2b).
Immune response
IgA activity (Fig. 3a) peaks later in the season followed by a slight decline. As we assume that IgA production is a function of the quantity of larvae ingested, there is no sharp fall in IgA following anthelmintic treatment. The decline in establishment over the season is shown in Fig. 3b.
The progress of IgA activity through the season as driven by exposure to infective larvae through pasture grazing (solid line), and the decline in parasite establishment (dashed line). X indicates empirical IgA data for comparison. These plots were obtained using the parameters in Table 1.
Sensitivity analyses
A series of sensitivity analyses were performed for several of the model parameters; one parameter was varied across a range of values while all other parameter values were kept as in Table 1.
Mortality rate of pre-infective larvae, m 1
For m 1, at the first peak in June we observe a large effect on faecal egg counts from a relatively small change in mortality rate (Fig. 4a). We can see that lower mortality rates lead to higher faecal egg counts. Interestingly, however, at the end of the season it was actually the higher mortality rates that produced the higher egg counts. This difference is due to the rate of acquisition of host immunity; when lambs are exposed to large numbers of larvae early on, the rate of establishment and parasite growth will decline more rapidly. Consequently, faecal egg count will reduce more rapidly. With a mortality rate of 0·20, the faecal egg counts do not exhibit the typical mid-season peak, with the peak being a month earlier.
Sensitivity analyses of various model parameters: (a) mortality rate of pre-infective stages, m1; (b) mortality rate of infective stage, m2; (c) adult mortality rate, m3; (d) maternal deposition of eggs on pasture, S; (e) pasture stocking density, D; (f) pasture herbage density, H.
Mortality rate of infective larvae, m 2
In contrast to mortality of pre-infective larvae, the mortality rate of the infective larval stage, m 2, can be varied over a much wider range without a major impact on faecal egg counts (Fig. 4b). Similarly to m 1, at the end of the season the lowest faecal egg count is derived from the lowest mortality rate.
Mortality rate of established adults, m 3
The mortality of established adult worms, m 3, also has little influence on faecal egg counts. For example, in July, when we observe the season peak in faecal egg counts, there is less than a doubling in egg count when m 3 is decreased 100-fold (Fig. 4c). In our model, the development of immunity is related to the number of larvae ingested. Therefore, the reduced number of adult nematodes has no effect on the development of immunity.
Maternal deposition of eggs on pasture, S
The initial number of eggs deposited by ewes was allowed to vary, but declined to zero over the same period. Early in the season, greater deposition of eggs by ewes lead to higher faecal egg counts in the lambs (Fig. 4d). However, higher early infection lowers faecal egg counts in mid-season, and by October the peaks are the reverse of those seen in June. Although this model only considers a single season, this pattern could generate robust year-to-year dynamics.
Stocking density, D
High stocking densities lead to initially high faecal egg counts, but we observe ultimately lower faecal egg counts at the end of the season and higher egg counts when stocking density is lowered (Fig. 4e).
Herbage density, H
Increasing herbage density decreases infection potential, as there will be fewer infective larvae per kg DM of herbage for any given number of larvae on pasture (this is assuming that larvae are evenly distributed along the herbage length, and do not migrate towards the tip). This is reflected in the faecal egg counts generated by the model (Fig. 4f). With the lower value of 500, the faecal egg counts do not show the typical mid-season peak, but instead peak a month earlier.
Establishment and faecal egg counts
To examine the relationship between establishment and faecal egg counts, the end-of-season establishment was recorded for all of the simulations shown in Fig. 4. The plot of end-of-season establishment against end-of-season faecal egg count (Fig. 5) shows how increasing egg counts are related to increased establishment up to a plateau level of around 0·15 establishment and 145 epg. After this point, egg counts appear to fall with increasing establishment because of reduced intake leading to both lower egg counts and reduced stimulation of the immune response. The corresponding establishment rates for m 3 in Fig. 4 have not been included in Fig. 5, as this parameter is related to the already established adult worms and so has no direct effect on establishment.
End-of-season faecal egg counts from Fig. 4 plotted against their corresponding end-of-season establishment. (Values for m3 have been omitted, as this parameter has no direct effect on establishment).
Response to potential control strategies
By varying certain parameters of the model we were able to explore the potential effects of applying particular strategies to control the infection of lambs by nematodes. We chose to look at 4 different control strategies. Reducing the stocking density is one means by which lambs will be exposed to reduced numbers of infective larvae, and this was simulated simply by lowering the stocking density parameter, D. Reducing the number of eggs deposited by ewes, either by treating them with anthelmintics or removing them from the pasture earlier, will also expose the lambs fewer larvae, and this method of control is simulated by varying parameter S. Another way to reduce the exposure of lambs is to remove infective larvae from the pasture sooner by increasing their mortality rate (m 3). An alternative to reducing exposure to infection is to increase the immune response, e.g. by nutritional supplementation or selective breeding, and this is related to the parameters ρ 1 and ρ 2.
From these simulations, control strategies aimed at reducing exposure (related to parameters S, D and m 2) could lead to increased end-of-season faecal egg counts, except in the case of extreme reduction in stocking density (Fig. 6). However, strategies to improve the immune response could lead to reduced faecal egg counts in all cases.
The results of varying the values of 5 parameters from those given in Table 1 to simulate the effects of applying particular control strategies. The percentage change is a decrease for parameters S and D, and an increase for m 1, ρ 1 and ρ 2. End-of-season faecal egg counts are shown in (a) and establishment in (b).
DISCUSSION
We have modelled the course of nematode infection in newborn lambs over the grazing season. In particular, we have explicitly modelled the role of IgA and density dependence in regulating parasite fecundity and used a recent meta-analysis to parameterize the rate of parasite establishment. We chose to restrict the model to lambs because T. circumcincta infection is most prominently a problem in developing lambs and not in adults, except for ewes during the periparturient period; additionally, we were interested in modelling the developing immune system rather than immunologically mature animals.
Although we have set stocking density at a realistic value for the modelled flocks, this could be regarded as a free parameter of the model that can be adjusted to capture much of the variation between flocks. This model produces a good qualitative fit to empirical faecal egg count data, with a relatively sharp rise in egg counts seen early in the season, peaking mid-season, and then declining steadily throughout the remainder of the season.
In an earlier study, Anderson and May (Reference Anderson and May1978) discussed several approaches to modelling the dynamics of host-parasite interactions. Many more models have investigated the interactions between nematode parasites and their ruminant hosts since the review by Smith and Grenfell (Reference Smith and Grenfell1985) of the population biology of Ostertagia ostertagi, and these have been thoroughly reviewed (Roberts, Reference Roberts1995; Cornell, Reference Cornell2005). More recently, Learmount et al. (Reference Learmount, Taylor, Smith and Morgan2006) have described a model that includes flock dynamics and environmental parameters, and Louie et al. (Reference Louie, Vlassoff and Mackay2007) and Vagenas et al. (Reference Vagenas, Bishop and Kyriazakis2007a,Reference Vagenas, Bishop and Kyriazakisb) present models that consider the interaction between parasitism and sheep growth. Acquired immunity in the host has also been modelled, where it is assumed to increase over time as a consequence of exposure to infective larvae, and manifests itself as a combination of constraints on establishment, fecundity and the survival of adult parasites (Anderson and May, Reference Anderson and May1985; Roberts and Grenfell, Reference Roberts and Grenfell1991). Although the importance of host-acquired immunity has long been recognized, there are few models addressing in detail the immuno-epidemiology of farmed ruminants. However, we now have a clearer understanding of the development of acquired immunity and the mechanisms involved in within-host regulation of parasite burden, length and fecundity (Stear et al. Reference Stear, Bishop, Doligalska, Duncan, Holmes, Irvine, McCririe, McKellar, Sinski and Murray1995c, Reference Stear, Park and Bishop1996). Detailed analyses show that there are 3 key features of the host response in sheep. First, immunity is acquired in response to exposure and develops in 2 stages, with lambs initially regulating worm growth and fecundity, and then worm number (Seaton et al. Reference Seaton, Jackson, Smith and Angus1989). Second, variation among hosts in IgA production and local eosinophilia accounts, in a statistical sense, for most of the variation in worm growth and fecundity (Strain et al. Reference Strain, Bishop, Henderson, Kerr, McKellar, Mitchell and Stear2002; Stear et al. Reference Stear, Bairden, Innocent, Mitchell, Strain and Bishop2004; Henderson and Stear, Reference Henderson and Stear2006). Third, much of the variation in worm number in deliberate infections is associated with variation in IgE activity and the number of degranulated mucosal mast cells (globule leucocytes, GL) which affect establishment and mortality (Stear et al. Reference Stear, Bishop, Doligalska, Duncan, Holmes, Irvine, McCririe, McKellar, Sinski and Murray1995c).
The maternal contribution to the total larval availability on pasture dominates throughout the season. Indeed, the lambs do not begin to make any contribution at all until the end of June, a delay that is due to a combination of the nematode development time, u, and the pre-patent period, j, coupled with the small initial ingestion rates at the start of the season. The maternal deposition of eggs on the pasture is an important means by which the concentration of larvae on pasture is maintained before the lambs themselves begin to make a significant contribution, which supports previous observations (Urquhart et al. Reference Urquhart, Armour, Duncan, Dunn and Jennings1996).
Our model is robust to perturbations in the majority of the parameters used in that it displays a characteristic increase and decline in faecal egg counts through a season, possibly because of early priming of the immune system; a particular characteristic of this model is the importance of early infection rate in determining the ultimate faecal egg count. As the immune response is stimulated by the ingestion of infective larvae, then the sooner a lamb is exposed to infection, and the greater the number of larvae, then the more rapidly parasite establishment and growth will be suppressed. This mechanism has a regulatory effect on host worm burdens, and acts to constrain faecal egg counts later in the season even if the initial infection rate is relatively high. The tendency for initially high faecal egg counts to produce lower egg counts later in the year was observed by Stear et al. (Reference Stear, Bairden, Duncan, Gettinby, McKellar, Murray and Wallace1995b), and suggests that the increased deposition of eggs by ewes during the periparturient period may be useful in stimulating the development of immune responses.
The robustness of this host-parasite system may be a consequence of the long coevolution between host and parasites. Nematodes of the subfamily Ostertaginiae are widely distributed among the bovidae and cervidae, which suggests that these nematodes and their ancestors have been infecting sheep and their ancestors for millions of years. Indeed, it may be that such an evolved system may act to reduce selective pressures on both host and parasite. These characteristics of the system also suggest that the control of nematode infection will be challenging.
We have considered the effects of applying particular control strategies to this model system by varying parameters that may respond to such strategies. These have shown that methods such as anthelmintic treatment of ewes or those that would increase the mortality of infective larvae could actually lead to higher faecal egg counts at the end of the season. This is because these control measures reduce exposure levels and ultimately delay the development of the immune system. Moderate reduction in lamb stocking density also produces higher end-of-season egg counts, but there is a significant decline in egg counts at very low stocking density. By varying the parameters directly related to the immune response, we see a consistent reduction in faecal egg counts with improved response. We also see that the response related to the suppression of establishment has a much larger effect in reducing egg counts than does the response controlling nematode fecundity. These results suggest that the most effective method of controlling nematode infection in lambs would be to improve the immune response, either with programmes of selective breeding or by nutritional supplementation.
The simple model presented here does not incorporate observed heterogeneity in individual response to infection, in worm burdens and in faecal egg counts. However, to fully assess the relative merits of combinations of control measures, such as selective breeding or vaccination, will require an individual-based model. We have also made the simplifying assumption that control of establishment and fecundity develop at the same rate; a more sophisticated model that allows for differential development rates may shift the dominant source of robustness from immune priming to density dependence.
In summary, we have constructed a realistic but relatively simple model of nematode infection in sheep based on the immune mechanisms associated with protection. This model emulates the pattern of faecal egg counts seen in the field, and is remarkably robust to perturbations in most of the observed parameters.
ACKNOWLEDGEMENTS
We are grateful to R. R. Kao for his comments on an earlier draft.
FINANCIAL SUPPORT
We thank the BBSRC for funding (grant number BB/F015313/1).
