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