A note on quasi-stationary distributions of birth–death processes and the SIS logistic epidemic

For Markov processes on the positive integers with the origin as an absorbing state, Ferrari, Kesten, Martínez and Picco studied the existence of quasi-stationary and limiting conditional distributions by characterizing quasi-stationary distributions as fixed points of a transformation Φ on the space of probability distributions on {1, 2, …}. In the case of a birth–death process, the components of Φ(ν) can be written down explicitly for any given distribution ν. Using this explicit representation, we will show that Φ preserves likelihood ratio ordering between distributions. A conjecture of Kryscio and Lefèvre concerning the quasi-stationary distribution of the SIS logistic epidemic follows as a corollary.

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Quasi-stationary distributions for birth-death processes with killing

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/jamsa/2006/84640 ◽

2006 ◽

Vol 2006 ◽

pp. 1-15 ◽

Cited By ~ 6

Author(s):

Pauline Coolen-Schrijner ◽

Erik A. van Doorn

Keyword(s):

Orthogonal Polynomials ◽

Stationary Distribution ◽

Transition Probabilities ◽

The State ◽

Death Process ◽

Stationary Distributions ◽

Absorbing State ◽

Quasi Stationary Distribution ◽

Birth Death

The Karlin-McGregor representation for the transition probabilities of a birth-death process with an absorbing bottom state involves a sequence of orthogonal polynomials and the corresponding measure. This representation can be generalized to a setting in which a transition to the absorbing state (killing) is possible from any state rather than just one state. The purpose of this paper is to investigate to what extent properties of birth-death processes, in particular with regard to the existence of quasi-stationary distributions, remain valid in the generalized setting. It turns out that the elegant structure of the theory of quasi-stationarity for birth-death processes remains largely intact as long as killing is possible from only finitely many states. In particular, the existence of a quasi-stationary distribution is ensured in this case if absorption is certain and the state probabilities tend to zero exponentially fast.

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Domain of Attraction of the Quasistationary Distribution for Birth-and-Death Processes

Journal of Applied Probability ◽

10.1017/s0021900200013152 ◽

2013 ◽

Vol 50 (01) ◽

pp. 114-126 ◽

Cited By ~ 1

Author(s):

Hanjun Zhang ◽

Yixia Zhu

Keyword(s):

Probability Measure ◽

Domain Of Attraction ◽

Death Process ◽

Extinction Time ◽

Decay Parameter ◽

Birth And Death Processes ◽

Absorbing State ◽

Positive Integers ◽

Initial Distributions ◽

Birth Death

We consider a birth–death process {X(t),t≥0} on the positive integers for which the origin is an absorbing state with birth coefficients λ n ,n≥0, and death coefficients μ n ,n≥0. If we define A=∑ n=1 ∞ 1/λ n π n and S=∑ n=1 ∞ (1/λ n π n )∑ i=n+1 ∞ π i , where {π n ,n≥1} are the potential coefficients, it is a well-known fact (see van Doorn (1991)) that if A=∞ and S<∞, then λ C >0 and there is precisely one quasistationary distribution, namely, {a j (λ C )}, where λ C is the decay parameter of {X(t),t≥0} in C={1,2,...} and a j (x)≡μ1 -1π j xQ j (x), j=1,2,.... In this paper we prove that there is a unique quasistationary distribution that attracts all initial distributions supported in C, if and only if the birth–death process {X(t),t≥0} satisfies bothA=∞ and S<∞. That is, for any probability measure M={m i , i=1,2,...}, we have lim t→∞ℙ M (X(t)=j∣ T>t)= a j (λ C ), j=1,2,..., where T=inf{t≥0 : X(t)=0} is the extinction time of {X(t),t≥0} if and only if the birth–death process {X(t),t≥0} satisfies both A=∞ and S<∞.

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Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes

Advances in Applied Probability ◽

10.2307/1427670 ◽

1991 ◽

Vol 23 (4) ◽

pp. 683-700 ◽

Cited By ~ 91

Author(s):

Erik A. Van Doorn

Keyword(s):

State Space ◽

Spectral Representation ◽

Transition Probabilities ◽

Initial Distribution ◽

Death Process ◽

Finite Support ◽

Stationary Distributions ◽

Absorbing State ◽

Initial Distributions ◽

Birth Death

For a birth–death process (X(t), ) on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1, ·· ·} is an irreducible class, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t→∞ of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth–death process and a duality concept for birth–death processes.

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Limiting Conditional Distributions for Birthdeath Processes

Advances in Applied Probability ◽

10.2307/1427866 ◽

1997 ◽

Vol 29 (1) ◽

pp. 185-204 ◽

Cited By ~ 22

Author(s):

M. Kijima ◽

M. G. Nair ◽

P. K. Pollett ◽

E. A. Van Doorn

Keyword(s):

Differential Equations ◽

Transition Probabilities ◽

Death Process ◽

Transition Rates ◽

Conditional Distributions ◽

Stationary Distributions ◽

Birth Death

In a recent paper [16], one of us identified all of the quasi-stationary distributions for a non-explosive, evanescent birth-death process for which absorption is certain, and established conditions for the existence of the corresponding limiting conditional distributions. Our purpose is to extend these results in a number of directions. We shall consider separately two cases depending on whether or not the process is evanescent. In the former case we shall relax the condition that absorption is certain. Furthermore, we shall allow for the possibility that the minimal process might be explosive, so that the transition rates alone will not necessarily determine the birth-death process uniquely. Although we shall be concerned mainly with the minimal process, our most general results hold for any birth-death process whose transition probabilities satisfy both the backward and the forward Kolmogorov differential equations.

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Domain of Attraction of the Quasistationary Distribution for Birth-and-Death Processes

Journal of Applied Probability ◽

10.1239/jap/1363784428 ◽

2013 ◽

Vol 50 (1) ◽

pp. 114-126 ◽

Cited By ~ 1

Author(s):

Hanjun Zhang ◽

Yixia Zhu

Keyword(s):

Probability Measure ◽

Domain Of Attraction ◽

Death Process ◽

Extinction Time ◽

Decay Parameter ◽

Birth And Death Processes ◽

Absorbing State ◽

Positive Integers ◽

Initial Distributions ◽

Birth Death

We consider a birth–death process {X(t),t≥0} on the positive integers for which the origin is an absorbing state with birth coefficients λn,n≥0, and death coefficients μn,n≥0. If we define A=∑n=1∞ 1/λnπn and S=∑n=1∞ (1/λnπn)∑i=n+1∞ πi, where {πn,n≥1} are the potential coefficients, it is a well-known fact (see van Doorn (1991)) that if A=∞ and S<∞, then λC>0 and there is precisely one quasistationary distribution, namely, {aj(λC)}, where λC is the decay parameter of {X(t),t≥0} in C={1,2,...} and aj(x)≡μ1-1πjxQj(x), j=1,2,.... In this paper we prove that there is a unique quasistationary distribution that attracts all initial distributions supported in C, if and only if the birth–death process {X(t),t≥0} satisfies bothA=∞ and S<∞. That is, for any probability measure M={mi, i=1,2,...}, we have limt→∞ℙM(X(t)=j∣ T>t)= aj(λC), j=1,2,..., where T=inf{t≥0 : X(t)=0} is the extinction time of {X(t),t≥0} if and only if the birth–death process {X(t),t≥0} satisfies both A=∞ and S<∞.

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Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes

Advances in Applied Probability ◽

10.1017/s0001867800023880 ◽

1991 ◽

Vol 23 (04) ◽

pp. 683-700 ◽

Cited By ~ 26

Author(s):

Erik A. Van Doorn

Keyword(s):

State Space ◽

Spectral Representation ◽

Transition Probabilities ◽

Initial Distribution ◽

Death Process ◽

Finite Support ◽

Stationary Distributions ◽

Absorbing State ◽

Initial Distributions ◽

Birth Death

For a birth–death process (X(t), ) on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1, ·· ·} is an irreducible class, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t→∞ of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth–death process and a duality concept for birth–death processes.

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Limiting Conditional Distributions for Birthdeath Processes

Advances in Applied Probability ◽

10.1017/s0001867800027841 ◽

1997 ◽

Vol 29 (01) ◽

pp. 185-204

Author(s):

M. Kijima ◽

M. G. Nair ◽

P. K. Pollett ◽

E. A. Van Doorn

Keyword(s):

Differential Equations ◽

Transition Probabilities ◽

Death Process ◽

Transition Rates ◽

Conditional Distributions ◽

Stationary Distributions ◽

Birth Death

In a recent paper [16], one of us identified all of the quasi-stationary distributions for a non-explosive, evanescent birth-death process for which absorption is certain, and established conditions for the existence of the corresponding limiting conditional distributions. Our purpose is to extend these results in a number of directions. We shall consider separately two cases depending on whether or not the process is evanescent. In the former case we shall relax the condition that absorption is certain. Furthermore, we shall allow for the possibility that the minimal process might be explosive, so that the transition rates alone will not necessarily determine the birth-death process uniquely. Although we shall be concerned mainly with the minimal process, our most general results hold for any birth-death process whose transition probabilities satisfy both the backward and the forward Kolmogorov differential equations.

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Extinction Probability in A Birth-Death Process with Killing

Journal of Applied Probability ◽

10.1017/s0021900200000152 ◽

2005 ◽

Vol 42 (01) ◽

pp. 185-198 ◽

Cited By ~ 4

Author(s):

Erik A. Van Doorn ◽

Alexander I. Zeifman

Keyword(s):

Rate Of Convergence ◽

Lower Bounds ◽

Transition Probabilities ◽

Upper And Lower Bounds ◽

Death Process ◽

Logarithmic Norm ◽

Absorbing State ◽

The Common ◽

Absorption Probabilities ◽

Birth Death

We study birth-death processes on the nonnegative integers, where {1, 2,…} is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator.

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Quasi-stationary distributions and one-dimensional circuit-switched networks

Journal of Applied Probability ◽

10.1017/s0021900200116821 ◽

1987 ◽

Vol 24 (04) ◽

pp. 965-977 ◽

Cited By ~ 4

Author(s):

Ilze Ziedins

Keyword(s):

Communication Networks ◽

Stationary Distribution ◽

Death Process ◽

Birth And Death Process ◽

Stationary Distributions ◽

Switched Networks ◽

One Dimensional ◽

Special Cases ◽

Quasi Stationary Distribution

We discuss the quasi-stationary distribution obtained when a simple birth and death process is conditioned on never exceeding K. An application of this model to one-dimensional circuit-switched communication networks is described, and some special cases examined.

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