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Multitype branching processes conditioned on non-extinction
We mentioned in the introduction how much the study of the extinction of populations is of a great interest in biology. Conditioning on non-extinction can notably lead to a stationary behavior of the process. We point out that branching processes can also result in populations with stable sizes if one allows an immigration component or assume an increasing number of ancestors. But conditioning on extinction or non-extinction provides in addition a lot of information about the evolution of the population before extinction, right before extinction, or in case of a very late extinction. We review in Section 2.1 some of the historical results related to this topic, and from Section 2.3 we focus on two kinds of conditioning for multitype BGWc and Feller diffusion processes, for which we provide a systematic study (which is new as far as multitype BGWc processes are concerned). The first conditioning consists in studying the asymptotic behavior of the process Xt under the condition that it is not extinct at time t + θ, for some θ > 0, but does eventually die out. This conditional limit distribution is a generalization of the Yaglom limit, obtained for θ = 0, and is the subject of Section 2.3. The second object of interest, studied in Section 2.4, is the so-called Q-process associated with the multitype branching processes (BGWc or Feller diffusion), i.e. the process “conditioned on not being extinct in the distant future and on being extinct in the even more distant future”, as described in [AthNey72]. We thus always consider population which are doomed to become extinct. We know from Subsection 1.1.3 and Subsection 1.2.3 that this is the case for subcritical and critical processes, but for the sake of completeness we chose to extend the study to supercritical processes with a positive risk of extinction, which is done thanks to Section 2.2.
Historical introduction: various ways of conditioning
The pioneer historical results on survival chances and conditional population size given non-extinction are due to Kolmogorov and Yaglom. In 1938 Kolmogorov provided the asymptotical survival probability of a single-type BGW process. Denoting by m and σ2 the mean value and variance of the offspring distribution f , and assuming the existence of a third-order moment (later proved to be unnecessary), he showed that in the subcritical case m < 1, for all x ∈ N ([Kolm38]), Px (Xn > 0) n→∞ Cxmn, (2.1.1) where C is some positive constant, while in the critical case m = 1, for 0 < σ2 < ∞, Px (Xn > 0) 2x n→∞ . (2.1.2) σ2n In both cases the probability of survival thus decreases to 0 as n tends to infinity, but much slower in the critical case that in the subcritical case.
Yaglom limit and quasi-stationary distributions
About ten years later, Yaglom proved under moment restriction the fundamental result that in the subcritical case, the distribution of (Xn|Xn > 0) converges to a proper distribution ([Yag47]). His proof was later simplified and the moment restriction removed in [SenVer66] and [Jof67]. The current usual formulation is the following: if m < 1, then for each x ∈ N , nlim Px (Xn = y | Xn > 0) = ν(y), (2.1.3) →∞ where ν is a probability measure on N which is independent of x, now referred to as the Yaglom distribution associated with the process (Xn)n>0. The generating function H(r) = ∞ ν(x)rx, x=1 r ∈ [0, 1], of this distribution satisfies the non-linear implicit equation H f = 1 m + mH. (2.1.4) In the critical case, Yaglom proved that (again under a third moment assumption, removed later) for every x ∈ N , assuming that 0 < σ2 < ∞, Xn > y | Xn > 0 = e 2y nlim Px . (2.1.5) σ2 n →∞ Hence the rescaled process Xnn converges in distribution conditionally on non-extinction to an exponential law with parameter σ22 . Corresponding results of (2.1.3) and (2.1.5) for single-type BGWc processes then followed, first in [Sew51] and later in [Con67], the latter using embedding arguments. It then extended to the field of multitype branching processes, first for BGW processes ([JofSpit67]) and later on for BGWc processes ([Sew75], see Proposition 2.3.3 in this work).
It can be proved that the Yaglom distribution as defined by (2.1.3) is a quasi-stationary distri-bution ([SenVer66]), in the sense that it is a stationary distribution for the dynamics conditioned on non-extinction, i.e.
It thus appears that ν is a left eigenvector of the transition matrix for the eingenvalue m. Al-though the Yaglom limit is uniquely defined by property (2.1.3), it is not the only quasi-stationary distribution, and consequently not the only conditional limit distribution. It was proved indeed in [SenVer66] that there exists a continuous range of quasi-stationary distributions {να, 0 < α 6 1} such that for each 0 < α 6 1, να is a left eigenvector for the eigenvalue mα (the superscript α denotes here the usual exponentiation):
This paper of Seneta and Vere-Jones actually dealt with the problem of quasi-stationarity for more general absorbing Markov chains, and initiated many other works (see e.g. [Fer95], and [Gos01] for an application on state-dependent branching processes). The equivalent theory for diffusion processes started with [Man61] and was then developed by many authors. In our direction of research, we shall quote the results of Lambert who dedicated his paper [Lamb07] on CB processes, and proved that in the subcritical case ρ := ψ′(0+) < 0 (see (1.2.7) for a definition of the branching mechanism ψ), there exists a family of quasi-stationary distributions {νγ , 0 < γ 6 ρ}, i.e. satisfying Pνγ (Xt ∈ A | Xt > 0) = νγ (A). (2.1.10)
The Yaglom distribution then corresponds to ν ρ. For a recent work on this topic, we refer e.g. to [Cat09], dealing with the existence, uniqueness and domain of attraction of quasi-stationary distributions for a large class of diffusion models arising from population dynamics.
Quasiextinction probabilities
There are many possible ways to generalize the study of the Yaglom limit. A first idea is to condition the population on not being too small, instead of conditioning on non-extinction. Seneta and Vere-Jones proved for example in [SenVer68] that for a subcritical Jiˇrina process (i.e. a discrete-time continuous-state branching process), for each ε > 0, exists and is a nondegenerate law on [ε, +∞[. Similar inquiries were done for CB processes, with a first ecological application in [Gin82], where this distributions are called quasiextinction probabilities.
Extinction in a close future
Alternatively, instead of conditioning the process on non-extinction at the present time as in (2.1.3), it might be interesting to condition in addition this process on extinction in a close future, and look at the asymptotic distribution. It was proved in [Sen67] that for single-type BGW processes one obtains in the critical case a nondegenerate result: if p1 > 0, for every fixed integer k > 0.
Table of contents :
1 Multitype branching processes
1.1 Continuous-time multitype Bienaym´e-Galton-Watson process
1.1.1 Preliminaries: generating functions and infinitesimal generator
1.1.2 Moments and the Perron-Frobenius Theorem
1.1.3 Extinction probability
1.1.4 Some basic examples
1.2 Multitype Feller diffusion process
1.2.1 Definitions and preliminaries
1.2.2 Associated martingale problem and SDE
1.2.3 Extinction of the process
2 Branching processes conditioned on non-extinction
2.1 Historical introduction: various ways of conditioning
2.2 Multitype branching processes forced to extinction
2.2.1 BGWc process forced to extinction
2.2.2 Feller diffusion process forced to extinction
2.3 Yaglom-type limits
2.3.1 Yaglom-type limits for the multitype BGWc process
2.3.2 Yaglom-type limits for the multitype Feller diffusion process
2.4 Q-process
2.4.1 Q-process associated with the multitype BGWc process
2.4.2 Q-process associated with the multitype Feller diffusion process
3 Commutativity results
3.1 Commutativity of the long-time limits
3.1.1 Long-time limits of the conditioned BGWc process
3.1.2 Long-time limits of the Feller diffusion process
3.2 Commutativity between rescaling and conditioning
3.2.1 Rescaled BGWc process
3.2.2 Scaling limit of the BGWc process conditioned on extinction
3.2.3 Scaling limit of the Yaglom-type distributions
3.2.4 Scaling limit of the Q-process
3.2.5 Scaling limit of the time asymptotic of the Q-process
3.3 Overview of the commutativity results
4 Risk analysis for vanishing branching populations
4.1 Stochastic branching model
4.2 Estimation of the unknown parameter
4.2.1 A CLSE with asymptotic properties, as |X0| → ∞
4.2.2 A CLSE with asymptotic properties, as n → ∞
4.2.3 An explicit estimator with asymptotic properties, as n → ∞
4.2.4 Comparison of the estimators and illustration of the asymptotic
4.3 Study of the very late extinction case
4.3.1 Q-process associated with the model
4.3.2 CLSE for the Q-process
5 BSE epidemic in Great-Britain
5.1 The epidemic model
5.1.1 Description
5.1.2 Theoretical results
5.1.3 The geometrical case
5.1.4 Origin of the model
5.2 Prediction of the disease spread
5.2.1 Estimation of the infection parameter
5.2.2 Prediction of the incidences of cases and infected cattle
5.2.3 Prediction of the extinction time
5.2.4 Prediction of the total size of the epidemic
5.3 Prediction of the disease spread in case of a very late extinction
5.3.1 Estimation of the infection parameter
5.3.2 Prediction of the disease spread
