ON THE EXTINCTION OF POPULATION-SIZE-DEPENDENT BISEXUAL GALTON-WATSON PROCESSES

In this article, the population-size-dependent bisexual Galton-Watson processes are considered. Under some suitable conditions on the mating functions and the offspring distribution, existence of the limit of mean growth rate per mating unit is proved. And based on the limit, a criterion to identify whether the process admits ultimate extinct with probability one is obtained.

EXTINCTION OF POPULATION-SIZE-DEPENDENT BRANCHING CHAINS IN RANDOM ENVIRONMENTS

We consider a population-size-dependent branching chain in a general random environment.We give suffcident conditions for certain extinction and for non-certain extinction.The chain exhibits different asymptotic according to supk,θmk,θ1, mk,θn→1 as k →∞, n→∞, infk,θmk,θ1.

ON THE BASIC REPRODUCTION NUMBER OF GENERAL BRANCHING PROCESSES

Under a very general condition （TNC condition） we show that the spectral radius of the kernel of a general branching process is a threshold parameter and hence plays a role as the basic reproduction number in usual CMJ processes. We discuss also some properties of the extinction probability and the generating operator of general branching processes. As an application in epidemics, in the final section we suggest a generalization of SIR model which can describe infectious diseases transmission in an inhomogeneous population.

Bisexual Galton-Watson Branching Processes in Random Environments

In this paper, we consider a bisexual Galton-Watson branching process whose offspring probability distribution is controlled by a random environment proccss. Some results for the probability generating functions associated with the process are obtained and sufficient conditions for certain extinction and for non-certain extinction are established.

Coexistence of nearly neutral species

Aims The neutral theory of biodiversity provides a powerful framework for modeling macroecological patterns and interpreting species assemblages.However,there remain several unsolved problems,including the effect of relaxing the assumption of strict neutrality to allow for empirically observed variation in vital rates and the‘problem of time’—empirically measured coexistence times are much shorter than the prediction of the strictly neutral drift model.Here,we develop a nearly neutral model that allows for differential birth and death rates of species.This model provides an approach to study species coexistence away from strict neutrality.Methods Based on Moran’s neutral model,which assumes all species in a community have the same competitive ability and have identical birth and death rates,we developed a model that includes birth–death trade-off but excludes speciation.This model describes a wide range of asymmetry from strictly neutral to nearly neutral to far from neutral and is useful for analyzing the effect of drift on species coexistence.Specifically,we analyzed the effects of the birth–death trade-off on the time and probability of species coexistence and quantified the loss of biodiversity(as measured by Simpson’s diversity)due to drift by varying species birth and death rates.Important Findings We found(i)a birth–death trade-off operating as an equalizing force driven by demographic stochasticity promotes the coexistence of nearly neutral species.Species near demographic trade-offs(i.e.fitness equivalence)can coexist even longer than that predicted by the strictly neutral model;(ii)the effect of birth rates on species coexistence is very similar to that of death rates,but their compensatory effects are not completely symmetric;(iii)ecological drift over time produces a march to fixation.Trade-off-based neutral communities lose diversity more slowly than the strictly neutral community,while non-neutral communities lose diversity much more rapidly;and(iv)nearly neutral systems have substantially shorter time of coexistence than that of neutral systems.This reduced time provides a promising solution to the problem of time.

Stochastic and spatio-temporal analysis of the Middle East Respiratory Syndrome outbreak in South Korea, 2015

South Korea was free of the Middle East Respiratory Syndrome(MERS)until 2015.The MERS outbreak in South Korea during 2015 was the largest outbreak of the Coronavirus outside the Middle East.The major characteristic of this outbreak is inter-or intra-hospital transmission.This recent MERS outbreak in South Korea is examined and assessed in this paper.The main objectives of the study is to characterize the pattern of the MERS outbreak in South Korea based on a basic reproductive ratio,the probability of ultimate extinction of the disease,and the spatio-temporal proximity of occurrence between patients.The survival function method and stochastic branching process model are adapted to calculate the basic reproductive ratio and the probability of ultimate extinction of the disease.We further investigate the occurrence pattern of the outbreak using a spatio-temporal autocorrelation function.