Abstract Four measures of the intensity of disease in a community are defined and discussed, viz. the incidence rate, the point, cumulative and period prevalence rates. In general, prevalence rates are more readily available for chronic diseases, but their use for making inferences about incidence of disease can lead to difficulties. A deterministic mathematical approach to the problem of relating incidence rates to the three types of prevalence rates is described. Using the integral calculus, integral equations are developed which may be simplified under certain strong assumptions. It is proved that the period prevalence rate is the product of average incidence and average duration in the sick population. A set of differential equations, obtained by simply differentiating the integral equations, was developed and solved under certain simplifying assumptions. In particular, a simple result emerged linking the cumulative (and point) prevalence rate to the annual incidence rates. The equations derived above were applied to results of various cerebrovascular disease morbidity studies. In addition, simple Gompertz and Makeham type curves were fitted to the incidence rates of these studies. The computed prevalence rates were compared with the actual, published rates, and the accuracy of the fit was discussed. The Poisson Process is introduced and its suitability for representing the flow of incident cases for a disease is discussed. It is found that this assumption implies that prevalence also forms a Poisson Process, but with modified parameters. Formulae for relating the three expected prevalence rates to the force of incidence are established, and even noted to be closely related to the equations proven in the deterministic approach. It is suggested that simple models, such as those described herein, may be used to test the consistency of incidence and prevalence data, to provide insight into the flow of new cases of a disease into a community, and to provide information about subsidiary variables like the average duration in the sick population.
Read full abstract