Accounting for the freeliving stage

We will now develop, but not analyze, a model that accounts for the free-living stage of the parasite in more detail. It is based on, but not identical to, the work of Hudson et a/. (1992a, b) and Hudson and Dobson (1997) which has all of the details, and additional models.

Until now, we have assumed that the transmission of adult parasites from hosts to hosts occurred with a very short intermediate stage, but now allow for a longer intermediate stage, understood to be the free-living eggs or larvae. The dynamical system that we consider is thus expanded to include hosts, H(t), adult parasites, P(t), and the free-living stage, which we denote by W(t) to follow the notation in Dobson and Hudson (1992).

The dynamics of the hosts do not change, but the dynamics of the parasites do. We assume that the free-living stage, denoted by W(t), is produced by adult parasites at constant per capita rate 1, have intrinsic natural mortality rate 7 and are captured by hosts according to a type II functional response. In such a case, the dynamics of the free-living stage are dW H

The dynamics of the adult parasites still involve production and mortality. Mortality is unchanged from the previous case, but production is now determined by the uptake of the free-living stage and so the adult dynamics become dP cWH _ a(k + 1)P2

Dobson and Hudson (1992) develop models that include larvae with arrested development (hypobiosis), show how to find the basic reproductive rate of the disease and analyze the dynamic properties of the model; I refer you to them for the details.

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