Stochastic Model of Dengue: Analysing the Probability of Extinction and LLN

In this article, we develop and analyze a continuous-time Markov chain (CTMC) model to study the resurgence of dengue. We also explore the large population asymptotic behavior of probabilistic model of dengue using the law of large numbers (LLN). Initially, we calculate and estimate the probabilities of dengue extinction and major outbreak occurrence using multi-type Galton-Watson branching processes. Subsequently, we apply the LLN to examine the convergence of the stochastic model towards the deterministic model. Finally, theoretical numerical simulations are conducted exploration to validate our findings. Under identical conditions, our numerical results demonstrate that dengue could vanish in the stochastic model while persisting in the deterministic model. The highlighting of the law of large numbers through numerical simulations indicates from what population size a deterministic model should be considered preferable.

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.

A note on asymptotic behavior of Galton-Watson branching processes in random environments

In this paper, we investigate Galton-Watson branching processes in random environments. In the case where the environmental process is a Markov chain which is positive recurrent or has a transition matrix Q (θ,α) such that sup_θ Q (θ,α)> 0 for some α, we prove that the model has the asymptotic behavior being similar to that of Galton-Watson branching processes. In other case where the environments are non-stationary independent, the sufficient conditions are obtained for certain extinction and uncertain extinction for the model.

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.

On the extinction route of a stochastic population model under heteroclinic bifurcation

The noise-induced transition of the augmented Lotka-Volterra system is investigated under vanishingly small noise.Populations will ultimately go extinct because of intrinsic noise,and different extinction routes may occur due to the Freidlin-Wentzell large deviation theory.The relation between the most probable extinction route(MPER)and heteroclinic bifurcation is studied in this paper.The MPERs and the quasi-potentials in different regimes of parameters are analyzed in detail.Before the bifurcation,the predator goes extinct,and the prey will survive for a long time.Then,the heteroclinic bifurcation changes the MPER wherein both species go extinct.The heteroclinic cycle plays a role in transferring the most probable extinction state.Moreover,the analyses of the weak noise limit can contribute to predicting the stochastic behavior under finite small noise.Both the heteroclinic bifurcation and the rotational deterministic vector field can reduce the action necessary for the MPER.

A new approach in analyzing extinction probability of Markov branching process with immigration and migration

We use a new approach to consider the extinction properties of the Markov branching process with immigration and migration recently discussed by Li and Liu [Sci. China Math., 2011, 54： 1043-1062]. Some much better explicit expressions are obtained for the extinction probabilities of the subtle super-interacting case.

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.

General collision branching processes with two parameters

A new class of branching models,the general collision branching processes with two parameters,is considered in this paper.For such models,it is necessary to evaluate the absorbing probabilities and mean extinction times for both absorbing states.Regularity and uniqueness criteria are firstly established.Explicit expressions are then obtained for the extinction probability vector,the mean extinction times and the conditional mean extinction times.The explosion behavior of these models is investigated and an explicit expression for mean explosion time is established.The mean global holding time is also obtained.It is revealed that these properties are substantially different between the super-explosive and sub-explosive cases.

Age-dependent branching processes in random environments

We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ 0, ξ 1,…) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξ n ) on ?+, and reproduce independently new particles according to a probability law p(ξ n ) on ?. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean E ξ Z(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.

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.

Markov branching processes with immigration-migration and resurrection

We consider a modified Markov branching process incorporating with both state-independent immigration-migration and resurrection. The effect of state-independent immigration-migration is firstly in- vestigated in detail. The explicit expressions for the extinction probabilities and mean extinction times are presented. The ergodicity and stability properties of the process incorporating with resurrection structure are then investigated. The conditions for recurrence, ergodicity and exponential ergodicity are obtained. An explicit expression for the equilibrium distribution is also presented. As a preparation, the criteria for regularity and uniqueness for such structure are firstly established.

Generalized Markov interacting branching processes

We consider a very general interacting branching process which includes most of the important interacting branching models considered so far. After obtaining some key preliminary results, we first obtain some elegant conditions regarding regularity and uniqueness, Then the extinction vector is obtained which is very easy to be calculated. The mean extinction time and the conditional mean extinction time are revealed.The mean explosion time and the total mean life time of th, processes are also investigated and resolved.

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.