The dynamics of nematode infections of farmed ruminants

Abstract

SUMMARYIn 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 Ro 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 Q, is less than one the parasite population cannot maintain itself in the host population, and in the long term becomes extinct ; but if Qo is greater than one the parasite can invade the host population. An alternative quantity, ME), 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.

A simple parasite model with complicated dynamics

“…Hence our results should also hold for these models, although the complicated behaviour could occur at biologically unrealistic parameter values. The approximation P0 may be justified when the rate at which acquired immunity is lost is small in comparison to typical rates of parasite dynamics (see Roberts and Grenfell, 1991;Roberts and Heesterbeek, 1995, where "0.01 was assumed). Typically the immunity of animals tends to increase with their age, and animals that are constantly exposed to parasites will retain their immunity.…”

“…The between-year dynamics are determined by the discrete map between the levels of pasture parasite contamination in consecutive years. This model was extended by Roberts and Heesterbeek (1995) to include heterogeneities in host and parasite, and the development of parasite strains resistant to chemotherapy.…”

“…The equations used by Roberts and Grenfell (1991) and Roberts and Heesterbeek (1995) to model the dynamics of nematode populations in farmed ruminants were dḑ t "! ( # H)¸#q (r)HA…”

“…However, we are interested in the situation where periodic perturbations occur; that is whenever t is an integer we reset y and z to zero, but x is continuous. This procedure is used to represent the introduction of new immunologically naive animals onto contaminated pasture at the beginning of each grazing season, replacing the existing animals that had acquired immunity (see Heesterbeek and Roberts, 1995;Roberts, 1995;Roberts and Heesterbeek, 1995). Instead of determining the existence and stability of steady states we are now interested in the existence and stability of dynamic equilibria.…”

Threshold quantities for helminth infections

“…In [18] and [20], P was taken as the threshold property for the nonautonomous periodic version of the ruminant/nematode system described in Example 2 of Section 2. These authors considered a model for nematode infections of ruminants with seasonal transmission, and analysed the system described as Example 2 above, but with p, fl and q periodic functions of time.…”