This paper generalizes the two-dimensional approximation of models of macroparasites on homogeneous populations developed by Anderson & May (1978), focusing on how the dispersion (the variance to mean ratio) of the equilibrium distribution of parasites on hosts is related to the stability of the equilibrium. We show in the approximate system that the equilibrium is stabilized not by aggregation, but by dispersion which increases as a function of the mean. Computer simulations indicate, however, that this analysis fails to capture properly the dynamics of the full system, raising the question of whether any two-dimensional system could produce an adequate approximation. We discuss the relevance of our results to several empirical studies which have examined the relation of dispersion to the mean.

In this paper the dynamics and control of nematode parasites of farmed ruminants are discussed via a qualitative analysis of a differential equation model. To achieve this a quantity, ‘the basic reproduction quotient’ (Q0), whose definition coincides with previous definitions of R0 for macroparasites, but extends to models with periodic time-varying transition rates between parasite stages or management interventions, is introduced. This quantity has the usual threshold property: if Q0 is less than one the parasite population cannot maintain itself in the host population, and in the long term becomes extinct; but if Q0 is greater than one the parasite can invade the host population. An alternative quantity, R(E), that is often easier to calculate is also introduced, and shown to have the same threshold property. The use of these two quantities in analysing models for the dynamics of nematodes in complex situations is then demonstrated, with reference to the dynamics of mixed parasite species in one host; the effects of breeding host animals for resistance to parasitism; and the development of parasite strains that are resistant to chemotherapy. Five examples are discussed using parameters for the dynamics of nematode infections in sheep, and some statements on control policies are derived.

We use results from a simulation-based model of nematode infection of sheep to refine the parameters in a simpler generic model of host–parasite population dynamics. These parameters describe the following host–parasite traits: probability of establishment of ingested larvae, mortality rate of adult parasites, and fecundity of adult female parasites. This simple model is then extended by allowing those parameters to vary amongst individual hosts. A sensitivity analysis is performed to determine which parameters have most influence on host parasite burden. The establishment parameter has the greatest effect on the peak value of parasite burden whilst the other two parameters have more effect on the duration of the burden. A comparison is made with results from the flock model after discussion of the definition of an average host. By allowing these parameters to vary simultaneously within the individual hosts we are able to reproduce the over-dispersed distribution of adult parasites frequently seen in nematode infections of sheep flocks.

We review the literature on parameter values relevant to the epidemiology of strongyle nematode infections of domestic sheep. Information is subdivided by parasite genus, country of origin and climate type. While field observations have been made in a large number of countries, the bulk of studies under controlled conditions have been conducted in Australia, New Zealand and the UK. For these countries, experiments and parameters are interpreted in terms of a previously published model of nematode dynamics, and are used to calculate the basic reproduction number. Average values range from less than 6 for Haemonchus contortus in New Zealand and a winter rainfall region of Australia, to more than 16 for Ostertagia circumcincta in New Zealand and the UK. Additional considerations of the effects of climate and the annual replacement of host stock show that for conditions favourable for parasite transmission this is a robust indicator of parasite epidemiology. When climate variation and annual replacement are added to the model, it is shown to reasonably describe the qualitative behaviour of an experimental data set, indicating it to be a useful tool for further investigation of some of the underlying assumptions of sheep–nematode dynamics.