Branching processes with immigration in atypical random environment

Motivated by a seminal paper of Kesten et al. (Ann. Probab., 3(1), 1–31, 1975) we consider a branching process with a conditional geometric offspring distribution with i.i.d. random environmental parameters An, n ≥ 1 and with one immigrant in each generation. In contrast to above mentioned paper we assume that the environment is long-tailed, that is that the distribution F of $\xi _{n}:=\log ((1-A_{n})/A_{n})$ ξ n : = log ( ( 1 − A n ) / A n ) is long-tailed. We prove that although the offspring distribution is light-tailed, the environment itself can produce extremely heavy tails of the distribution of the population size in the n th generation which becomes even heavier with increase of n. More precisely, we prove that, for all n, the distribution tail $\mathbb {P}(Z_{n} \ge m)$ ℙ ( Z n ≥ m ) of the n th population size Zn is asymptotically equivalent to $n\overline F(\log m)$ n F ¯ ( log m ) as m grows. In this way we generalise Bhattacharya and Palmowski (Stat. Probab. Lett., 154, 108550, 2019) who proved this result in the case n = 1 for regularly varying environment F with parameter α > 1. Further, for a subcritical branching process with subexponentially distributed ξn, we provide the asymptotics for the distribution tail $\mathbb {P}(Z_{n}>m)$ ℙ ( Z n > m ) which are valid uniformly for all n, and also for the stationary tail distribution. Then we establish the “principle of a single atypical environment” which says that the main cause for the number of particles to be large is the presence of a single very small environmental parameter Ak.

1035COVID-19: Boom or bust?

Background In the final weeks of 2019, a SARS-CoV-2 virus slipped furtively from animal to human in China. As of March 13, there have been 1,34,918 confirmed cases, out of which 4,990 is the death count. We are predicting extinction or explosion of the virus from the current realization of a Galton Watson process. Methods Based on the region wise reported number of cases, total was calculated. The observed offspring distribution was found by calculating the difference between the total number of cases in consecutive days. Hence the distribution modelled using Sequential Probability Ratio Tests (SPRT) to predict whether extinction or explosion will occur for the current realization of the process. Kolmogorov-Smirnov test was performed on the data to check the distribution of fit. Results We assume conservative approach of SPRT. The geometric distribution fits to the data taken from January 2020 to March 12, 2020. The SPRT on the offspring distribution predicts extinction of the disease if the number of cases reported on a new day are less than 58 then the disease will extinct, and will explode if more than 9,990 cases. Conclusions Our results show that if COVID-19 transmission is established, understanding the effectiveness of control measures in different settings will be crucial for understanding the likelihood that transmission can eventually be effectively mitigated. Key messages Our analysis highlights the value of recording individual cases and analyzing geographically heterogeneous data of COVID-19. Our results also have implications for estimation of transmission dynamics using the number of exported cases from a specific area.

The $k$-Cut Model in Deterministic and Random Trees

The \(k\)-cut number of rooted graphs was introduced by Cai et al. as a generalization of the classical cutting model by Meir and Moon. In this paper, we show that all moments of the \(k\)-cut number of conditioned Galton-Watson trees converge after proper rescaling, which implies convergence in distribution to the same limit law regardless of the offspring distribution of the trees. This extends the result of Janson. Using the same method, we also show that the \(k\)-cut number of various random or deterministic trees of logarithmic height converges in probability to a constant after rescaling, such as random split-trees, uniform random recursive trees, and scale-free random trees.

On the tail of the branching random walk local time

Consider a critical branching random walk on $$\mathbb Z^d$$ Z d , $$d\ge 1$$ d ≥ 1 , started with a single particle at the origin, and let L(x) be the total number of particles that ever visit a vertex x. We study the tail of L(x) under suitable conditions on the offspring distribution. In particular, our results hold if the offspring distribution has an exponential moment.

Inference of person-to-person transmission of COVID-19 reveals hidden super-spreading events during the early outbreak phase

Coronavirus disease 2019 (COVID-19) was first identified in late 2019 in Wuhan, Hubei Province, China and spread globally in months, sparking worldwide concern. However, it is unclear whether super-spreading events occurred during the early outbreak phase, as has been observed for other emerging viruses. Here, we analyse 208 publicly available SARS-CoV-2 genome sequences collected during the early outbreak phase. We combine phylogenetic analysis with Bayesian inference under an epidemiological model to trace person-to-person transmission. The dispersion parameter of the offspring distribution in the inferred transmission chain was estimated to be 0.23 (95% CI: 0.13–0.38), indicating there are individuals who directly infected a disproportionately large number of people. Our results showed that super-spreading events played an important role in the early stage of the COVID-19 outbreak.

A unifying approach to branching processes in a varying environment

Branching processes $(Z_n)_{n \ge 0}$ in a varying environment generalize the Galton–Watson process, in that they allow time dependence of the offspring distribution. Our main results concern general criteria for almost sure extinction, square integrability of the martingale $(Z_n/\mathrm E[Z_n])_{n \ge 0}$, properties of the martingale limit W and a Yaglom-type result stating convergence to an exponential limit distribution of the suitably normalized population size $Z_n$, conditioned on the event $Z_n \gt 0$. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.

Non-intersection of transient branching random walks

Let G be a Cayley graph of a nonamenable group with spectral radius $$\rho < 1$$ ρ < 1 . It is known that branching random walk on G with offspring distribution $$\mu $$ μ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring $${\overline{\mu }}$$ μ ¯ satisfies $$\overline{\mu }\le \rho ^{-1}$$ μ ¯ ≤ ρ - 1 . Benjamini and Müller (Groups Geom Dyn, 6:231–247, 2012) conjectured that throughout the transient supercritical phase $$1<\overline{\mu } \le \rho ^{-1}$$ 1 < μ ¯ ≤ ρ - 1 , and in particular at the recurrence threshold $${\overline{\mu }} = \rho ^{-1}$$ μ ¯ = ρ - 1 , the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost surely on the event that the walk survives forever. This is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely. We prove this conjecture, along with several other related conjectures made by the same authors. A central contribution of this work is the introduction of the notion of local unimodularity, which we expect to have several further applications in the future.

Very fat geometric Galton-Watson trees

Let τn be a random tree distributed as a Galton-Watson tree with geometric offspring distribution conditioned on {Zn = an} where Zn is the size of the nth generation and (an, n ∈ ℕ*) is a deterministic positive sequence. We study the local limit of these trees τn as n →∞ and observe three distinct regimes: if (an, n ∈ ℕ*) grows slowly, the limit consists in an infinite spine decorated with finite trees (which corresponds to the size-biased tree for critical or subcritical offspring distributions), in an intermediate regime, the limiting tree is composed of an infinite skeleton (that does not satisfy the branching property) still decorated with finite trees and, if the sequence (an, n ∈ ℕ*) increases rapidly, a condensation phenomenon appears and the root of the limiting tree has an infinite number of offspring.