Biomedical Engineering Reference
In-Depth Information
The state space model was originally proposed by Kalman and his
associates in the early 1960's for engineering control and communication
7
.
Since then it has been successfully used as a powerful tool in aero-space
research, satellite research and military missile research. It has also been
used by economists in econometrics research and time series research
5
for
solving many dicult problems which appear to be extremely dicult from
other approaches. It was rst proposed by Tan and his associates for AIDS
research and for cancer research (see [9, 10, 15, 13, 14, 16, 17, 18]). Appar-
ently state space models can be extended to other diseases as well, including
heart disease and tuberculosis
11
.
2.2. A General Bayesian Procedure for Estimating
Unknown Parameters and State Variables via State
Space Models
Applying the state space models, Tan and his associates
9;10;11;15;16;17
have
developed a general Bayesian procedure to estimate simultaneously the un-
known parameters and the state variables. These procedures would combine
information from three sources: (1) previous information and experiences
about the parameters in terms of the prior distribution of the parameters,
(2) biological information via the stochastic system equations of the stochas-
tic system, and (3) information from observed data via the statistical model
from the system.
The general Bayesian procedure is given and illustrated in detail in
Tan
10
, Chapter 9 and will be used to derive estimates of the unknown
parameters and the state variables in state space models, see Section 4.3.
3. Stochastic Models of AIDS Epidemic in Homosexual and
Bisexual Populations
In US and Western countries such as Europe and Australia, the CDC (Cen-
ter of Disease Control and Prevention, Atlanta, Georgia, USA) had reported
that most of the AIDS cases were observed in homosexual and bisexual men
(about 60%) and IV drug users (about 30%); other avenues such as hetero-
sexual transmission were not common in these countries.
To illustrate how to develop state space models for the AIDS epidemic,
we will thus consider a large population of homosexual and bisexual men
who are at risk for AIDS. In this population, under risk for AIDS, then one
can identify three types of people in the population: S people (susceptible
people), I people (infective people) and AIDS patients (A people).
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