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T, A(t)}'. Then under some general conditions, {X(t),t G T = ( 0 , 1 , . . ,oo)} is a multi-dimensional Markov chain with discrete time. This type of Markov chains in HIV epidemic has been developed by Tan and his associates in terms of chain binomial and chain multinomial distributions; see [7-12]. 2. }. Then the random behavior and properties of this Markov chain is characterized by the transition probabilities which is defined by pij(s,t) = P{X(t) = j\X(s) = i} for t > s. Prom these transition The Transition Probabilities and Computation 39 probabilities, obviously, we have: pij(s,s) = 5^, where 5^ is the Kronecker's 6 defined by da = 1 and S^ — 0 if i ^ j .

Let {S(t),I(u,t)} denote the numbers of S people and of I(u) people at the ith month respectively and A(t) the total number of AIDS cases including those died from AIDS by the ith month. Put X(t) = {S(t),I(u,t),u = 0 , 1 , . . ,t, A(t)}'. Then under some general conditions, {X(t),t G T = ( 0 , 1 , . . ,oo)} is a multi-dimensional Markov chain with discrete time. This type of Markov chains in HIV epidemic has been developed by Tan and his associates in terms of chain binomial and chain multinomial distributions; see [7-12].

The Transition Probabilities and Computation 43 The characteristic function (x) of Q is 4>{x) = \Q- xl4\ = (-x) ^ [ - - x \~X 1 I 8 V2 \ \ \ - X = -G-^)fG^)(i-)-fnG^)} =iG_:c)G":c)(4x2~2x~i)=oHence the eigenvalues of Q are {Ai = ^,A2 = \,\3 = \(1 + V§) = ^ei,A 4 = ^(1 — y/5) = 162}- Since all eigenvalues are real and distinct, Q can be expressed in terms of spectral expansion. The spectrum matrices of Q are: Ei = 1 (Q - X2I4)(Q - X3I4)(Q - A4/4) (Ax - A2)(A! - A3)(A! ) 4 V 4 2 / X 1 i / 4 £1 4 £1 4 1 J and £4 = -(Q-A1/4)(Q-A274)(Q-A3/4) (A 4 -A 1 )(A 4 -A 2 )(A 4 -A 3 )' 1 l(' + !

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