SOME STATE SPACE MODELS OF AIDS EPIDEMIOLOGY IN HOMOSEXUAL POPULATIONS - Quantitative Medical Data Analysis Using Mathematical Tools and Statistical Techniques - page 47

As in the literature [2, 9], we further assume thatfd s (t) = d s ; (u; t) =
(u); d I (u; t) = d I (u)g.
(2) As in the literature 2, 9], we assume that there are no reverse tran-
sition from I to S and from AIDS cases to I.
(3) Because of the awareness of AIDS, we assume that there are no
immigration and recruitment of A people and that there are no sexual and
IV contacts between S people and AIDS patients.
Let R S (t) denote the number of immigrants and recruitment of S peo-
ple during [t; t + 1) and R I (u; t) the number of immigrants and recruitment
of I(u) people during [t; t + 1). For dealing with immigration and recruit-
ment, in what follows we will assume that the R S (t) given S(t) and the
R I (u; t) given I(u; t) are negative binomial random variables with param-
etersfS(t); !g(0 < ! < 1) andfI(u; t); !g, respectively, unless other-
wise stated. ( We note that dierent distributions with the same mean
numbers give similar estimates of the state variables and the HIV infec-
tion, the HIV incubation distributions and the death rates.) Then the
conditional means and the conditional variances of these variables are
given by E[R S (t)jS(t)] = S (t) = S(t)!=(1!), E[R I (u; t)jI(u; t)] =
I (u; t) = I(u; t)!=(1!), Var[R S (t)jS(t)] = 2 (t) = S(t)!=(1!) 2
and VarR I (u; t) = I (u; t) = I(u; t)!=(1!) 2 .
To derive stochastic equations for the state variables, denote by:
I(0; t + 1) = F S (t) = Number of S!I(0) during [t; t + 1),
F I (u; t) = Number of I(u)!A during [t; t + 1),
D S (t) = Number of death of S people during [t; t + 1),
D I (u; t) = Number of death of I(u) people during [t; t + 1).
Assume that R S (t) and R I (u; t) are independently distributed of each
other and of the other random variables. Then, the conditional distribution
of [F S (t); D S (t)] given S(t) is multinomial with parametersfS(t); p S (t); d S g
(i.e. F S (t)jS(t)MLfS(t); p S (t); d S g) independently of the immi-
gration and recruitment process. Similarly, [F I (u; t); D I (u; t)]jI(u; t)
MLfI(u; t); (u); d I (u)gindependently of the other state variables and the
immigration and recruitment processes. Then, under assumptions (1)-(3)
given above, we have the following stochastic equations for the state