SOME STATE SPACE MODELS OF AIDS EPIDEMIOLOGY IN HOMOSEXUAL POPULATIONS - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 46

S people are healthy people but can contract HIV to become I people
through sexual contact and/or IV drug contact with I people or A people
(AIDS patients) or through contact with HIV-contaminated blood. I people
are people who have contracted HIV and can pass the HIV to S people
through sexual contact or IV drug contact with S people. According to
the 1993 AIDS case denition 4 by the Center of Disease Control (CDC)
at Atlanta, GA, an I person will be classied as a clinical AIDS patient
when this person develops AIDS symptoms and/or when his/her CD4 +
T-cell counts fall below 200=mm 3 . In this section we will illustrate how to
develop a discrete time stochastic model for the HIV epidemic with variable
infection duration in these populations. (With no loss of generality we will
let month be the time unit unless otherwise stated.)
To start the AIDS epidemic, we assume that at time t 0 = 0, a few HIV
were introduced into the population to start the HIV epidemic so that with
probability one, I(0; 0) > 0 and I(u; 0) = 0 if u > 0.
Let S(t) denote the number of S people at time t, A(t) the number of
new AIDS cases during the month [t; t + 1) and I(u; t) the number of I
people who have contracted HIV at time tu (tu). (We refer u as the
infection duration of I people and denote by I(u) infective people with infec-
tion duration in [u; u + 1).) When time is discrete, we are then entertaining
a multi-dimensional stochastic processfS(t); I(u; t); u = 0; 1; : : : ; t; A(t)g
with discrete time and discrete state space. This is basically a Markov pro-
cess with discrete state space and with discrete time; however, the number
of state variables increases as time increases. For this stochastic process,
the traditional approaches from most texts are too complicated and can
hardly lead to useful results. For deriving useful results, we will thus use
an alternative approach through stochastic dierence equations.
3.1. The Stochastic Dierence Equations for the State
Variables
To develop a stochastic model for the above stochastic process, let p S (t)
be the probability that a S person will contract HIV to become an I(0)
person during [t; t + 1) and (u; t) the probability that an I(u) person will
develop AIDS symptoms to become a clinical AIDS patient during [t; t+1).
Let d S (t) be the probability that a S person will die during [t; t + 1) and
d I (u; t) the probability that an I(u) person will die during [t; t+1). Further,
we make the following assumptions:
(1) We may assume that p S (t) and (u; t) are deterministic functions 12 .