Population dynamics in echinococcosis and cysticercosis: mathematical model of the life-cycle of Echinococcus granulosus

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Abstract
SUMMARY: A mathematical model of the life-cycle of Echinococcus granulosus in dogs and sheep in New Zealand is constructed and used to discuss previously published experimental and survey data. The model is then used to describe the dynamics of transmission of the parasite, and the means by which it may be destabilized. It is found that under the conditions that prevailed in New Zealand during the late 1950s, at the time of surveys of this parasite, the dog–sheep life-cycle was not regulated by any effective density-dependent constraint. In contrast there was evidence for an effective acquisition of immunity to reinfection by cattle. The long time to maturity of the cyst in sheep, together with the practice of feeding aged sheep to dogs, provides a time delay in the intermediate host. By comparison, the time to maturity of the adult stage in dogs is short, but it is of sufficient magnitude to be a key factor in the destabilization of the system by a regular dog-dosing programme. The model used to describe the life-cycle is a linear integrodifferential equation of the Volterra type. Such equations are intrinsically unstable in that a small perturbation in parameters can drive a previous equilibrium solution to zero. At the time of the surveys, the value of the basic reproductive rate, R0 was close to 1, and it has since been reduced below 1 by control measures.