SOME STATE SPACE MODELS OF AIDS EPIDEMIOLOGY IN HOMOSEXUAL POPULATIONS - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 49

variables, 1 the collection of all incidence of HIV infection and HIV in-
cubation, 2 the collection of all the death probabilities and 3 the pa-
rameter for immigration and recruitment. Let f S (j; t) be the probabil-
ity of (R S (t) = j) and f I (j; u; t) the probability of (R I (u; t) = j). Let
P rfX(t + 1)jX(t); gdenote the conditional probability density function
of X(t + 1) given X(t). Using results in Section (2.1), the conditional prob-
ability distribution P rfXjX(0)gof X given X(0) is

t M 1

Y

P rfXjX(0)g= P rfX(j + 1)jX(j); g (5)

j=0

and

S(t)
X

S(t)
i

S(t)i
I(0; t + 1)

d S i (1d S ) S(t)i P rfX(t + 1)jX(t); g=f

i=0

p S (t)
1d S

] I(0;t+1) [1 p S (t)
1d S

] S(t)iI(0;t+1)

[

f S [a i (t); t]H(i; I(u; t); u = 0; : : : ; t)g; (6)

where

I(u;t) I(u;t)j 1

Y

t

X X

I(u; t)
j 1 ; j 2

[(u)] j 1 H(i : I(u; t); u = 0; : : : ; t) = f

u=0 j 1 =0 j 2 =0

[d I (u)] j 2 [1(u)d I (u)] I(u;t)j 1 j 2
f I (b j (u; t); u; t)g;

(7)

where a i (t) = Max(0; S(t+1)S(t)+I(0; t+1)+i) , b j (u; t) = Max(0; I(u+
1; t + 1)I(u; t) + j 1 + j 2 ).

4. A State Space Model of AIDS Epidemic in Homosexual
Populations

Given AIDS incidence data of homosexual and bisexual men, in this section
we develop a state space model for AIDS epidemic in this population.
In the state space model, the state variables are X(t) =fS(t); I(u; t); u =
0; 1; : : : ; tgand the stochastic system model is given by the stochastic dif-
ference equations (1)-(4) and the probability distribution of these state
variables in (3.2). The observation model is a statistical model based on
AIDS incidence data which relate the observed AIDS incidence to A(t).