The extended individual host model

The flock model above has output variables of mean parasite burden, mean immunity level and pasture larval density. We now consider a group of n individual hosts, each of which might possess different pasture consumption rates, larval establishment probabilities, and parasite mortality and fecundity rates. The basis for this difference between individuals is probably genetic and we would expect some correlation between the latter three traits. For example, a host with a lower probability of larval establishment than another with the same level of immunity would also be expected to have higher parasite mortality and lower parasite fecundity rates. It has been conjectured that much of the observed variation in parasite burden between hosts is attributable to host heterogeneity (Grenfell et al. 1995; Cornell, Isham & Grenfell, 2004). The following model will enable us to explore the consequences of allowing some of the most important host characteristics to vary individually. The corresponding DEs to (1) become:

where the subscript i runs from 1 to n for the n hosts. We use l to indicate number of infective larvae and

to stand for the larval intake rate by the ith host at time t. This is a slightly different interpretation for β(t) than used before so we have indicated this by inserting a tilde, (β(t) has units of m² week⁻¹ whereas

will have units of week⁻¹). Each host carries its own adult parasite burden A_i and immune status r_i.

The establishment rate probability p_i(r_i), adult parasite mortality μ_i(r_i) and fecundity λ_i(r_i) depend on the individual host's immune status and may also vary for each host, even when those hosts have the same immune status. One simplifying assumption would be to amend this so that individual hosts are distinguished only by differences in their immune status. In that case we are dealing with a group of genetically identical hosts (clones). However, differences in immune status could still arise through varying larval ingestion rates or immunity fade rates. In dealing with clones we can drop the first subscript i in the adult parasite parameters p_i(r_i), μ_i(r_i) and λ_i(r_i) so that each individual will have the same adult parasite establishment probability, mortality and fecundity as other hosts with the same immunity status.

Parameter values

Here we present the functional forms used for the parameters appearing in Table 2.

The probability that an ingested larva develops into an adult parasite decreases as host immunity increases, and for the ith host (i=1 to n) we assume this is given by

A graph of this function for various values of k_i is shown in Fig. 1A. The value of p₀ (probability of an ingested larva developing into an adult parasite in an immunologically naïve host with r=0) can vary from 0·35 to 0·77 depending on parasite species (Kao et al. 2000).

The parameter k_i controls how rapidly the probability of an ingested larvae developing successfully to an adult parasite in host i declines with that host's increasing immune status.

Fig. 1. The variation amongst individuals of the 3 host–parasite traits considered in the model. The most resistant host is represented by the solid curves and the least resistant host by the dashed-dotted curves. Two other hosts with intermediate resistance are also shown (dotted and dashed curves).

The mortality rate of adult parasites increases as host immunity increases, and for the ith host (i=1 to n) we assume this is given by

A graph of this function for various values of m_i is shown in Fig. 1B. The value of μ₀ is the common mortality rate when all hosts have reached full immunity. The parameter m_i controls the point of inflection occurring in the function defined by equation (4). Kao et al. (2000) use constant values for the mortality rate ranging from 0·01 d⁻¹ to 0·11 d⁻¹ depending on parasite species.

The fecundity rate of adult parasites decreases as host immunity increases, and for the ith host (i=1 to n) we assume this is given by

A graph of this function for various values of f_i is shown in Fig. 1C. The value of λ₀ is the common fecundity rate for immunological naïve hosts with r=0. The parameter f_i controls the point of inflection occurring in the function defined by equation (5). Kao et al. (2000) used constant values for the fecundity rate ranging from 150 eggs d⁻¹ to 6300 eggs d⁻¹ (per adult female) depending on parasite species.

For the purposes of the present model the pasture consumption rate is assumed not to vary between hosts but will increase with age. Consumption of pasture is zero for host age less than 2 weeks and thereafter is given by the relationship of Leathwick et al. (1992)

where t is the age of the host in weeks and the units of β are m² week⁻¹. To convert this parameter for use in the individual model we need to divide β by the grazing area a as noted previously. We have assumed the pasture cover is uniform at 0·2 kg DM m⁻² throughout the year so consumption of 1·45 kg DM d⁻¹ is equivalent to grazing 50·75 m² week⁻¹.

Because the host immunity level r is being modelled by cumulative larval intake, its value ranges from zero (immunologically naïve) to millions (full immunity). Thus the values for parameters k_i, m_i and f_i appearing in (3)–(5) above will be of the order 10⁻⁶, 10⁶ and 10⁶ respectively to ensure a reasonable range of response as the immunity level increases from zero to its final value. For the purpose of demonstrating the effect of individual host variability we choose (i) k_i to vary uniformly from 23×10⁻⁶ (most resistant) to 5×10⁻⁶ (least resistant), (ii) m_i to vary uniformly from 0·1×10⁶ (most resistant) to 4·6×10⁶ (least resistant) and (iii) f_i to vary uniformly from 0·1×10⁶ (most resistant) to 1·9×10⁶ (least resistant).

By most resistant we mean that at a given level of immunity this host has the lowest probability of ingested larvae developing to adult parasite, and the highest mortality and lowest fecundity rates for its adult parasites. For large groups of hosts the variation in these parameters is more likely to be normally distributed (Niezen, Waghorn & Charleston, 1998; Harrison et al. 1999; Leathwick et al. 1999; Hong, Michel & Lancaster, 1986; Sutherland et al. 1997). We chose a uniform distribution with small host numbers for illustrative purposes. The other parameter values used are given in Table 2 and are constant in producing the results in Figs 2 and 3 and during the sensitivity analysis of the next section.

Sensitivity analysis

Because there is considerable variation in the values for k_i, m_i and f_i and uncertainty in their field measurement, a sensitivity analysis was undertaken. Sensitivity analyses assist in identifying those parameters responsible for greatest perturbations in the model output and whose values must be known most precisely. To do this in a systematic fashion, one parameter set was varied at a time using the base values prescribed for k_i, m_i and f_i in the preceding section. Ten hosts were modelled varying each set of k_i, m_i and f_i by ±50%. For example, to test the sensitivity of the model to variation in establishment of ingested larvae, we allowed k_i to vary uniformly from 34·5×10⁻⁶ to 7·5×10⁻⁶ (+50%) and then from 11·5×10⁻⁶ to 2·5×10⁻⁶ (−50%) with m_i and f_i as prescribed before. The results are shown in Figs 2A through 2C. In these figures and the remainder of this section, the 10 hosts start with zero worm burdens and graze a 1 ha field which initially has 10⁶ infective larvae on it. For clarity of presentation we have shown worm burdens for only the most and least resistant hosts.

Fig. 2. Results from the parameter sensitivity analysis for host–parasite traits in a group of 10 hosts. Graphs A to C (column 1), D to F (column 2) and G to I (column 3) are from varying the parameters controlling establishment probability, mortality rate and fecundity rate respectively. The solid curves show results when default parameter values for the 10 hosts were used. The dotted and dashed curves resulted from varying the default parameter set by minus and plus 50% respectively.

The effect of varying k_i on pasture larval density, worm burdens for the most and least resistant host, and the host immunity are shown in Fig. 2A and C respectively. There are considerable differences in the peak values of larval density and worm burdens. Also note that the worm burden in the least resistant host is sustained for much longer; the adult parasites in this host being largely responsible for the secondary peak in larval density seen after day 200, when climatic conditions are again conducive to larval establishment from their deposited eggs. The primary peak in larval density lags the peak in worm burden for the most resistant host, but of course every host is contributing to the larvae on pasture, so the timing of these peaks is not sequential as in the identical host case. Because we are assuming the larval intake and immunity fade rates for each host are the same, in Fig. 2C the immunity status for each host will remain identical if they start with the same value.

We now reset k_i to the default values and vary the mortality rate parameter m_i. The results are shown in Fig. 2D–F. The most striking difference here is that the peak values of pasture larval density and worm burdens show almost no change as m_i varies (Fig. 2D and E). However, both the pasture larval densities and worm burdens for the least resistant host remain at different levels through to 365 days, whereas for varying k_i the values came together by that time. The host immunities in Fig. 2F also remain very similar for considerably longer here (about 150 days). This would go some way to explaining why the peak worm burdens remain at approximately the same level in Fig. 2E.

Finally we examine the sensitivity of the results to varying the fecundity parameter f_i. These results are displayed in Fig. 2G–I. There are larger dif-ferences in the value of the second peak of pasture larval density (Fig. 2G), which is not entirely unexpected as this would be the output most directly affected by changes in fecundity. There is no difference in the value or timing for the peak of the worm burdens (Fig. 2H) and the difference in worm burden, though again sustained through to day 365, is not as great as when varying m_i. The changes in host immunity are intermediate compared to the two previous cases (Fig. 2I): a divergence is apparent about day 120 which then grows as before through to day 365.

In terms of worm burden, clearly the most sensitive parameter is k_i associated with establishment of ingested larvae as adult parasites. To some extent this reflects the functional form we have chosen to represent this process. However, when the establishment probability is plotted as a function of time and compared with the corresponding graph from running the simulation model of Leathwick et al. (1992), there is very little difference in the result. This indicates that the functional form we have chosen can be justified on the basis of field data.

The three parameters considered in this section are unlikely to vary independently within a group of hosts. For example, we would expect individuals with a high value of k_i to have both lower values for m_i and f_i. Thus, when comparing two groups, increasing k_i for each individual within that group should also be accompanied by decreasing values for both m_i and f_i for every individual. This would tend to reinforce the trends already seen in Fig. 2.

Comparison with flock model

To demonstrate clearly the connection between the individual and flock models, we consider first the simplest case of n indistinguishable hosts grazing on an area a. That is, we have n genetically identical individuals which are also grazing in an identical manner. The DE in equation (2c) shows that the immune status of every host will remain identical if they started with the same immune status. Thus we can drop the subscript i from the variables A and r in equations (2b) and (2c), and equation (2a) becomes

On dividing this DE through by the grazing area a we get

where H is the host density n/a and L is the pasture larval density l/a. Thus the DEs are identical with the flock model in equations (1a–c) as long as we take

(which also makes their units agree).

We next examine the case where individuals are identical except for one varying host parasite trait and compare with the flock model. This differs from the sensitivity analysis of the preceding section in that only one host parasite trait is allowed to vary whilst the other two traits are held constant (at the flock value) between hosts. For the sensitivity analysis each host had three different parameter values describing the three parasite traits.

There are two obvious ways in which we can define an average value for the varying host parasite trait so that comparison can be made with the flock model. The first is to use the median value of the parasite parameter. With μ(r) and λ(r) held fixed at the flock values, consider a group of ten individual hosts with k_i varying uniformly from 23×10⁻⁶ (most resistant) to 5×10⁻⁶ (least resistant). The median value of k_i for this group of ten hosts is

, and we define the median establishment probability as

Alternatively, we could use the mean establishment probability, defined as

This differs considerably from the median probability, particularly at higher levels of immunity. Note that these definitions do not actually depend on the parameter k_i being uniformly distributed. The critical assumption is that the form of p(r) is identical for each individual and that host heterogeneity can be described by variation in k_i.

It is straightforward to show from the governing DEs of the individual model that if we use this mean establishment probability as the group average, and define the mean worm burden as

, then the outcomes from the individual and flock models will be identical (see Appendix). This result is not true if we use the median establishment probability

as the group average.

Now hold p(r) and μ(r) fixed at the flock values and vary the fecundity parameter f_i only. Equation (2b) shows that in this case every host has the same worm burden over time if all hosts started with the same burden. Using this result in equation (2a) renders it identical to equation (1a) as long as we define the group average fecundity as

. Again, this result is not true were we to use the median fecundity as the group average.

Lastly, hold p(r) and λ(r) fixed at the flock values and vary the mortality parameter m_i only. Equation (2b) shows that worm burden will now be different for each host. Furthermore, there is now no definition of a group average mortality for adult worms that will make this DE agree with its flock counterpart in (1b).

As already noted when we first introduced the individual model, the parameters describing the host parasite traits are unlikely to vary independently, and are more likely to be related with one another. Thus, when comparing hosts, one that has a lower probability for establishment of ingested larvae is also likely to have a higher adult parasite mortality and lower adult parasite fecundity. When the parameters are varied at the same time in this manner, (i.e. high k_i implies both low m_i and low f_i for the ith individual) there is a large degree of over-dispersion in the distribution of the adult worm burden. In other words, a plot of flock fraction against worm burden would not be a Poisson distribution but would be significantly skewed, with most of the flock having low or moderate burdens whilst a small fraction of the flock possesses very high burden. This is because the above host–parasite traits interact strongly to produce far greater burdens than if the traits were not related. To demonstrate this, the contribution to the total worm burden in a flock of 10 individuals by the 2 least resistant hosts is shown in Fig. 3. For comparative purposes, also plotted in the figure is the same contribution by the two least resistant hosts when the host–parasite traits are varied singly (i.e. are assumed to be not related). As before, we assume the flock is grazing on a 1 ha field which initially contains 10⁶ larvae and use the default host parasite parameters from the sensitivity analysis section. When each of the parasite parameters is varied, the other two parameters are set at the mean flock value as defined above. From Fig. 3 we see also that host variation in mortality rate of adult parasites has the greatest effect on the proportion of the parasite population present in susceptible individuals. However, it is only when the parasite parameters are all allowed to vary that the fraction of total worm burden carried by the two least resistant hosts of the flock reaches a value nearing 80%. This provides further confidence in the parameter values used as these figures are supported by field measurements where parasite populations are generally found to be over-dispersed with the majority of hosts generally carrying few worms and a few are heavily infected and harbour the majority of the total parasite population (Barger, 1985; Sréter et al. 1994).

Fig. 3. Demonstration of the over-dispersed worm burden distribution that occurs when the host–parasite traits vary simultaneously for 10 hosts (solid curve). Also shown for comparative purposes are the distributions when these traits vary singly. The dotted, dashed and dashed-dotted curves show the results from varying only the mortality rate, establishment probability and fecundity rate respectively.