A stochastic approach to the study of parasite populations

Abstract

The aim of this paper is to develop a general framework for building stochastic models to describe some features of parasite populations. Starting from the few basic biological assumptions outlined below, it is shown that input mechanisms, whereby the parasite gains entrance to the host, can be defined in probabilistic terms. Once in the host, the female parasite is allowed to “mature” and produce offspring according to given probability laws. Moreover, the host is supposed to react to control the rate of maturation and/or the rate of production of offspring. The overall accumulation of parasites, male and female, in the host is also considered. In the sequel, a host is defined to be any organism which is subjected to a burden or infection of “lesser” organisms which we will call parasites. The biological assumptions concerning the relationships between host and parasite are outlined below. 1. (i) The parasite gains entrance to the host, either orally, intradermally or otherwise. Entry may be as a continuous stream or in the form of administered doses. 2. (ii) Once in the host the female enters a period of maturation at the completion of which she is capable of producing offspring. 3. (iii) Each parasite in the host has an “antigenic information trajectory”. This term is used to describe the phenomenon that at any fixed time, the parasite is releasing information to the host to the effect that he, the parasite, is there. It is further assumed that antigenic information is additive, in the sense that the information emitted by a number of parasites is the sum of the individual antigenic informations. The offspring are assumed to produce no relevant antigenic information. 4. (iv) Each host responds in his own way to the build up of antigenic information. He responds by, in some way, controlling or otherwise affecting the rate of maturation of the parasites and/or the rate of reproduction of the females. A deliberate attempt has been made to keep the treatment as general as possible. Unfortunately, this may lead to some obscurities and, in order to demonstrate the ideas, examples will be discussed in conjunction with the general development. In one case we consider sheep as the host, worms as the parasite and eggs of the female worm as the offspring. The sheep are given a massive dose of larvae (immature worms) at zero time and the task is to describe the total egg output of the worm population in the sheep as a function of time. As further examples we examine some highly simplified continuous models. The purpose of these exercises is purely illustrative and the components of the models have been chosen for mathematical convenience. Nevertheless, it is hoped that the resulting models are not too unrealistic. Details leading to the biological assumptions used in this paper may be found in Dineen (1963 a, b), Donald, Dineen, Turner & Wagland (1964) and Dineen, Donald, Wagland & Offner (1965), Dineen, Donald, Wagland & Turner (1965).