THE MCKENDRICK-PARTIAL-DIFFERENTIAL-EQUATION AND ITS USES IN EPIDEMIOLOGY AND POPULATION STUDY

One way of modeling the evolution in time of an age-structured populat ion is to set up the evolution process as a partial differential equat ion in which time and age are the independent variables. The resulting equation, known as the McKendrick equation, has received attention re cently from mathematicians. Some advantages of the PDE are that it can easily be adapted to include more detail in the model, including expl icit time-dependence in the coefficients and even some nonlinear effec ts. The initial-boundary conditions for the McKendrick equation, impos ed by the population model, are not the standard side conditions one s ees in PDE theory for an evolution equation. In the simplest case, the problem reduces to a well-known model in demography, the Lotka integr al equation. In this paper, we explain the solution of the McKendrick model and compare the McKendrick equation with other common models for age-structured populations (the Leslie matrix and the difference equa tion, as well as the integral equation) in several ways. The approache s differ in their suitability for computation, their ease of generaliz ation, and their adaptability to different demographic objectives and other biological applications. With small intervals of age and time al l forms are identical, but if the intervals are finite, differences wi ll appear in the numerical results. The structure of solutions of the partial differential equation contributes to better understanding and computation of population models.