A general mathematical model of a vector-borne disease involving two vertebrate host species and one insect vector species is described. The model is easily extended to other situations involving more than two hosts and one vector species. The model, which was developed from the single-host model for malaria described by Aron & May (1982), is applied to the African trypanosomiases and allows for incubation and immune periods in the two host species and for variable efficiency of transmission of different trypanosome species from the vertebrates to the vectors and vice versa. Equations are derived for equilibrium disease prevalence in each of the species involved. Model predictions are examined by 3-dimensional phase-plane analysis, which is presented as a simple extension of the 2-dimensional phase-plane analysis of the malaria model. Parameter values appropriate for the African trypanosomiases are derived from the literature, and a typical West African village situation is considered, with 300 humans, 50 domestic animals and an average population of 5000 tsetse flies. The model predicts equilibrium prevalences of Trypanosoma vivax, T. congolense and T. brucei of 47·0, 45·8 and 28·7% respectively in the animal hosts, 24·2, 3·4 and 0·15% in the tsetse vectors, and a 7·0% infection of humans with human-infective T. brucei. The contribution to the basic rate of reproduction of the human-infective T. brucei is only 0·11 from the human hosts and 2·54 from the animal hosts, indicating that in the situation modelled human sleeping sickness cannot be maintained in the human hosts alone. The animal reservoir is therefore crucial in determining not only the continued occurrence of the disease in humans, but its prevalence in these hosts as well. The effect of changing average fly density on equilibrium disease prevalences is examined, together with the effect of seasonal changes in fly numbers on disease incidence. In a seasonal situation changes in fly mortality rates affect both future population size and infection rate. Peak disease incidence lags behind peak fly numbers, and that in the less favoured host lags behind that in the more favoured host. Near the threshold fly density for disease transmission disease incidence is more changeable than at higher fly densities and may even exceed equilibrium prevalence at the same average fly density (because most hosts are susceptible at the time that fly numbers begin their annual increase). The implications of the model for disease control are discussed. Identifying the precise role of the animal reservoir may suggest that treating such animals will achieve a greater reduction of human sleeping sickness than direct treatment of the humans alone. Statistically significant results of control campaigns may also be more easily shown by monitoring the non-human reservoirs. The model provides a means by which a correct perspective view can be obtained of the complex epidemiology and epizootiology of the African trypanosomiases.
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