SOME STATE SPACE MODELS OF AIDS EPIDEMIOLOGY IN HOMOSEXUAL POPULATIONS - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 48

variables:

S(t + 1) = S(t) + R S (t)F S (t)D S (t);
= S (t) + S(t)f1[p S (t) + d S (t)]g+ " S (t + 1);

(1)

I(0; t + 1) = F S (t) = S(t)p S (t) + " I (0; t + 1); (2)

I(u + 1; t + 1) = I(u; t) + R I (u; t)F I (u; t)D I (u; t)
= I (u; t) + I(u; t)f1[(u) + d I (u; t)]g
+ " I (u + 1; t + 1); u = 0; : : : ; t; (3)

X

t

X

t

A(t + 1) = F I (u; t) = I(u; t)(u) + " A (t + 1); (4)

u=0 u=0

where the random noises "(t + 1) = [" S (t + 1); " I (u; t + 1); u = 0; 1; : : : ; t +
1; " A (t + 1)] T are derived by subtracting the conditional means from the
respective random variables in the above equations and are given by:

" S (t) = [R S (t) S (t)][F S (t)S(t)p S (t)]
[D S (t)S(t)d S ];
" I (0; t + 1) = [F S (t)S(t)p S (t)];
" I (u + 1; t + 1) = [R I (u; t) I (u; t)][F I (u; t)I(u; t)(u)]
[D I (u; t)I(u; t)d I (u)]; u = 0; 1; : : : ; t

X

t

" A (t + 1) = [F I (u; t)I(u; t)(u)]:

u=1

In equations (1)-(4), given X(t) the random noises "(t) have expectation
zero. It follows that the expected value of these random noises is 0. Using the
basic formulae Cov(X; Y ) = EfCov[(X; Y )jZ]g+ Cov[E(XjZ); E(YjZ)], it
is also obvious that elements of "(t) are uncorrelated with elements of X(t)
as well as with elements of "() for all t 6= . Further, these random noises
are linear combinations of negative binomial, binomial and multinomial
random variables. Hence one may readily derive variances, covariances and
higher moments and cumulants of these random noises.

3.2. The Probability Distributions of
X(t) = fS(t); I(r; t); r = 0; 1; : : : ; tg

Let X =fX(1); : : : ; X(t M )g, where t M is the last time point and =
f 1 ; 2 ; 3 g, where 1 =fp S (t); (t); t = 0; 1; : : : ; t M g, 2 =fd S ; d I (u),
u = 0; : : : ; t M gand 3 = !. Then X is the collection of all the state