A NOTE ON QUASI-STATIONARY DISTRIBUTIONS OF BIRTH-DEATH PROCESSES AND THE SIS LOGISTIC EPIDEMIC

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1 (January 8, 2003) A NOTE ON QUASI-STATIONARY DISTRIBUTIONS OF BIRTH-DEATH PROCESSES AND THE SIS LOGISTIC EPIDEMIC DAMIAN CLANCY, University of Liverpoo PHILIP K. POLLETT, University of Queensand Abstract For Marov processes on the positive integers with the origin as an absorbing state, Ferrari, Kesten, Martinez and Picco [4] studied the existence of quasi-stationary and imiting conditiona distributions by characterizing quasi-stationary distributions as fixed points of a transformation Φ on the space of probabiity distributions on {, 2,...}. In the case of a birth-death process, one can write down the components of Φ(ν) expicity for any given distribution ν. Using this expicit representation, we wi show that Φ preserves ieihood ratio ordering between distributions. A conjecture of Kryscio and Lefèvre [7] concerning the quasi-stationary distribution of the SIS ogistic epidemic foows as a coroary. KEYWORDS: LIKELIHOOD RATIO ORDERING; STOCHASTIC ORDERING; LIMITING CONDITIONAL DISTRIBUTION AMS 2000 SUBJECT CLASSIFICATION: PRIMARY 60J27 SECONDARY 60J80. Finite state-space birth-death processes Consider a birth-death process {X(t) : 0 t < } on the finite state space {0,,..., N}. Denote by λ i, µ i the birth rate and death rate, respectivey, from state i. Assume that λ i > 0 for i =, 2,..., N, µ i > 0 for i =, 2,..., N, λ N = λ 0 = 0 and µ 0 = 0. Thus the state space consists of a communicating cass {, 2,..., N} and a singe absorbing state 0, which wi amost surey be reached within finite time. This process has a unique imiting conditiona distribution q = (q, q 2,..., q N ), that is q i = im t Pr (X(t) = i X(t) > 0), () whatever the distribution of the initia state X(0) (Darroch and Seneta [3]). This distribution is aso quasi-stationary in that if Pr(X(0) = i) = q i, then Pr(X(t) = i X(t) > 0) = q i for a t > 0, and, it is the unique soution of the equations λ i q i (λ i + µ i ) q i + µ i+ q i+ = µ q q i (i =, 2,..., N), (2) Posta address: Department of Mathematica Sciences, University of Liverpoo, Liverpoo, L69 7ZL, Engand. E-mai address: d.cancy@iv.ac.u Posta address: Department of Mathematics, The University of Queensand, Queensand 4072, Austraia. E-mai address: pp@maths.uq.edu.au

2 2 Damian Cancy and Phiip K. Poett where we adopt the convention that q 0 = q N+ = 0. In order to study quasi-stationary distributions of this and more genera processes, Ferrari et a. [4] defined a map Φ as foows. For a given distribution ν = (ν, ν 2,..., ν N ), suppose that whenever the process hits state 0, it immediatey restarts in state j {, 2,..., N} with probabiity ν j. The process restarted in this way has a finite, irreducibe state space, and hence a unique stationary distribution ρ = (ρ, ρ 2,..., ρ N ). The map is then defined by Φ(ν) = ρ. Since q is a quasi-stationary distribution, we have Φ(q) = q. Indeed it is cear that q is the unique fixed point of the map Φ. In the present case of a birth-death process on a finite state space, we can specify the Φ expicity. If we are given ν, then ρ = Φ(ν) satisfies λ i ρ i (λ i + µ i ) ρ i + µ i+ ρ i+ = µ ρ ν i (i =, 2,..., N), (3) with ρ 0 = ρ N+ = 0. Foowing Nåse [8] (see aso Nåse [0]) we set f i = µ i ρ i λ i ρ i for i = 2, 3,..., N. Equations (3) then become Thus we have f 2 = µ ρ ( ν ), f i+ = f i µ ρ ν i (i = 2, 3,..., N ). f i = µ ρ a i (i = 2, 3,..., N), (4) where a i = N j=i ν j. On substituting (4) into the definition of f i and soving for ρ i, we find that {( i ) } a j ρ i = ρ µ µ i j= =j λ µ (i =, 2,..., N) (5) where ρ is chosen so that ρ is a proper distribution. Thus, the components of ρ = Φ(ν) are given expicity by (5). Now, given any two distributions ν () and ν (2) on {, 2,..., N}, the ieihood ratio ordering LR is defined by ν () LR ν (2) if, for i j N, ν () i ν (2) j ν () j ν (2) i (6) (see, for exampe, Kijima and Seneta [6]). Using formua (5), we can show that Φ preserves ieihood ratio ordering between distributions. Theorem. For any two distributions ν () and ν (2) on {, 2,..., N}, ν () LR ν (2) Φ ( ν ()) LR Φ ( ν (2)). Proof. Write ρ () = Φ ( ν ()), ρ (2) = Φ ( ν (2)), and for i N set b i = i = (λ /µ ). From (5) we have ρ () i = ρ () µ µ i = b i a () and ρ (2) j = ρ (2) µ µ j = b j a (2),

3 Quasi-stationary distributions of birth-death processes 3 where a (r) j = N =j ν(r) ρ () i ρ (2) j ρ () j ρ (2) i ρ () ρ (2). Therefore, for i j N, = µ2 µ j µ i = µ2 µ j µ i ( = b i a () = =i+ = b i b j ( b j a (2) a () = b j a () a(2) a () a (2) the ast equaity hoding because b i b j = b j b i. Now, for <, ), = b i a (2) ) a () a(2) a () a (2) = = r= r= ν () r s= s= ν (2) s r= ν () r s= ν (2) s ( ) ν () r ν s (2) ν s () ν r (2) 0, and so ρ () i ρ (2) j ρ () j ρ (2) i, as required. Denoting by e i the distribution assigning unit mass to state i, we have the foowing. Coroary. The quasi-stationary distribution q satisfies Φ (e ) LR q LR Φ (e N ). Proof. Since e LR q LR e N, we have Φ (e ) LR Φ (q) LR Φ (e N ), but Φ (q) = q and so the resut foows. Since ieihood ratio ordering impies the usua stochastic ordering (see the discussion foowing Definition 2 of Kijima and Seneta [6]), the above coroary provides a more immediate proof of Theorem of Keison and Ramaswamy [5], that Φ (e ) ST q ST Φ (e N ). Notice that the distribution Φ (e ) has a particuary simpe form, since in this case the ony non-zero term in the summation on the right-hand side of equation (5) is the j = term. 2. The SIS ogistic epidemic The SIS ogistic epidemic mode is a birth-death process on {0,,..., N} with λ i = λi (N i) /N; µ i = µi (i = 0,,..., N) where λ, µ > 0. For this mode, equation (5) becomes ρ i = i(n i)! j= { ( ) } i j T (N j)! a j ρ, (7) N

4 4 Damian Cancy and Phiip K. Poett where T = λ/µ. Kryscio and Lefèvre [7], and ater Nåse [8] and Nåse [9], used the stationary distributions of two approximating processes to study the quasi-stationary distribution of the SIS ogistic epidemic mode. Firsty, they considered the process with refection at state 0, which is equivaent to having the process restart at state whenever absorption at 0 occurs. Thus the stationary distribution of this approximating process is our Φ (e ). It provides a ower bound for the quasi-stationary distribution q in the sense of ieihood ratio ordering. As an upper bound for q, one coud use the distribution Φ (e N ). However, this does not have the particuary simpe form of Φ (e ), since now a terms in the summation on the right-hand side of equation (7) are non-zero. Instead, Kryscio and Lefèvre [7] considered a process with the same birth rates λ i = λi(n i)/n, but with death rates modified to µ i = µ(i ). The stationary distribution of this process, which we denote here by m = (m, m 2,..., m N ), was found to be m i = (N )! (N i)! ( T N ) i m (i =, 2,..., N). (8) They conjectured, but did not prove, that q ST m. Using Theorem, we are now abe to prove this conjecture as foows. The process giving rise to the stationary distribution m is not an SIS epidemic restarted whenever it hits state 0. However, the map Φ given by (5) can readiy be inverted; indeed, given any distribution ρ the vaues ν, ν 2,..., ν N such that Φ (ν) = ρ are given immediatey by equation (3). Hence, by soving Φ(ν) = m, it may be possibe to determine an initia distribution ν for which the restarted SIS epidemic has stationary distribution m. Putting ρ i = m i in (3), and using the origina rates λ i = λi(n i)/n and µ i = µi, we find that ν i = (N )! (N i)! ( T N ) i ( T ( i N )) (i =, 2,..., N). (9) In order that ν i 0 for a i, we require T + /(N ) (note that N i= ν i = is assured by virtue of (3)). Thus, certainy in the sub-critica (or non-endemic) case T <, the distribution m can be obtained as the stationary distribution of a restarted SIS epidemic. This ends some theoretica support to Kryscio and Lefèvre s empirica observation that m provides a good approximation to q in the sub-critica case. Now, for i j N, with ν given by (9) and m given by (8), m i ν j m j ν i = ((N )!)2 (T/N) i+j m (j i) 0, (N i)!(n j)! so that m LR ν. That is, Φ (ν) LR ν. Appying Theorem repeatedy gives Φ n (ν) LR Φ n (ν) LR LR m LR ν. On etting n we obtain Φ n (ν) q; this is true for any finite-state absorbing chain (see Section 5 of Ferrari et a. [4]). Thus q LR m, which impies that q ST m, as conjectured by Kryscio and Lefèvre [7].

5 Quasi-stationary distributions of birth-death processes 5 If T > + /(N ), then ν given by (9) has one or more stricty negative eements. However, it remains true that m i ν j m j ν i 0 for j i, which is sufficient for the proof of Theorem to remain vaid, and we can sti concude that q LR m. It foows that the components q i of the quasi-stationary distribution are non-increasing in i for i Nx, where x is the stabe equiibrium point of the deterministic SIS ogistic epidemic mode, given by x = max {0, (/T )}. To see this, notice that m i+ m i = ( ) i ( ( (N )! T N ) ) i m, (N i)! N T so that for i Nx, m i+ m i. Since q LR m, we have q i+ /q i m i+ /m i, for i Nx. In particuar, if T + /(N ), then q q 2 q 3 q N. 3. Infinite state-space birth-death processes The resuts of Section carry over to the infinite-state case with some minor adjustments. Consider the birth-death process on {0,,...} with λ 0 = µ 0 = 0 and λ i, µ i > 0 for i. Hence there is a singe absorbing state 0, which is accessibe from the irreducibe cass C = {, 2,... }. For a given distribution ν = (ν, ν 2,... ), suppose that whenever the process hits state 0, it immediatey restarts in state j C with probabiity ν j. The new process has an irreducibe state space, but it might not have a stationary distribution. However, if a stationary distribution ρ = (ρ, ρ 2,... ) exists, it uniquey satisfies λ i ρ i (λ i + µ i )ρ i + µ i+ ρ i+ = µ ρ ν i (i ). (0) If, as we sha assume, the origina process is absorbed with probabiity in finite mean time, the existence of a stationary distribution is guaranteed (see for exampe Theorem of Asmussen []), and we may then write ρ = Φ(ν). As before q = (q, q 2,... ) is a quasistationary distribution if and ony if Φ(q) = q (Theorem 3. of van Doorn []), but now this fixed point need not be unique. Indeed there in a one-parameter famiy of quasi-stationary distribution indexed by ρ in a finite interva [0, α], α 0 (Theorem 2 of Cavender [2]). However, (5) hods good for a i with a i = j=i ν j, and so too does Theorem. Coroary hods for any quasi-stationary distribution, in particuar the imiting conditiona distribution (), which exists for a initia states X(0) if and ony if α > 0 (Theorem 4. of van Doorn []). Reationship (9) of Cavender [2] says precisey that Φ (e ) LR q, and this is used to prove his Proposition 5, that Φ (e ) ST q. Our coroary provides a consideraby simper proof of these resuts. Simiary, Theorem 4 of Kijima and Seneta [6] states that Φ (e ) s q, where s denotes strong ieihood ratio ordering, which differs from the usua ieihood ratio ordering in that the inequaity in (6) is strict. Again our coroary aows these resuts to be proved more directy.

6 6 Damian Cancy and Phiip K. Poett Acnowedgements The support of the Austraian Research Counci (Grant No. A ) is gratefuy acnowedged. References [] Asmussen, S. (987). Appied Probabiity and Queues. Wiey, New Yor. [2] Cavender, J. (978). Quasistationary distributions of birth-death processes. Adv. App. Probab. 0, [3] Darroch, J. and Seneta, E. (967). On quasi-stationary distributions in absorbing continuous-time finite Marov chains. J. App. Probab. 4, [4] Ferrari, P., Kesten, H., Martínez, S. and Picco, P. (995). Existence of quasi-stationary distributions. A renewa dynamic approach. Ann. Probab. 23, [5] Keison, J. and Ramaswamy, R. (984). Convergence of quasistationary distributions in birth-death processes. Stochastic Process. App. 8, [6] Kijima, M. and Seneta, E. (99). Some resuts for quasistationary distributions of birth-death processes. J. App. Probab. 28, [7] Kryscio, R. and Lefèvre, C. (989). On the extinction of the S-I-S stochastic ogistic epidemic. J. App. Probab. 27, [8] Nåse, I. (996). The quasi-stationary distribution of the cosed endemic SIS mode. Adv. App. Probab. 28, [9] Nåse, I. (999). On the quasi-stationary distribution of the stochastic ogistic epidemic. Math. Biosci. 56, [0] Nåse, I. (200). Extinction and quasi-stationarity in the Verhust ogistic mode. J. Theor. Bio. 2, 27. [] van Doorn, E. (99). Quasi-stationary distributions and convergence to quasistationarity of birth-death processes. Adv. App. Probab. 23,