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A Statistical Model and Estimation of Disease Rates as Functions of Age and Time

Abstract
Estimating incidence, remission, and mortality rates as functions of age and time is an important public health problem. We model disease rates as smooth with respect to both age and time. This is combined with an ordinary differential equation for the healthy and infected population, where date of birth (cohort) is fixed and age is the independent variable. The result is a multivariate Gauss-Markov random field model for the healthy population, infected population, and the disease rates. The corresponding inverse covariance matrix (precision matrix) is sparse. We model some standard public health measurements in terms of this random field. An offset parameter controls the level at which the residual transitions between being lognormal and normal. This enables the combination of measurement values that are zero with measurements that are lognormal for large values. The freely available nonlinear programming solver Ipopt, which takes advantage of the sparsity, is used to obtain the maximum a posteriori estimate for the healthy population, infected population, and the disease rates as a function of age and cohort. A simulation example is used to demonstrate that a small amount of prevalence data can recover the shape of the incidence rate as a function of age and time. A diabetes example is used to demonstrate how one applies this model to real world public health data.

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Input File: dismod_pde.omh