Kinetic theory of age-structured stochastic birth-death processes.

Reply to evolutionary flux of canonical microRNAs and mirtrons in Drosophila.

To analyze the change of droplets number caused by collision and coalescence in gas/droplet two-phase flows, a stochastic model was established, in which the collision probability of two drops with different sizes was derived and the stochastic process theory was introduced to describe the change of drops number when the phenomenon of collision and coalescence takes place. By considering the immigration and emigration and collision of the droplets located at a fixed region, it was found that the number of drops is a birth death process, and the relevant model equation was derived. The stationary distribution and its mean value were obtained by analyzing the number characteristic of the birth-death process. These results are helpful to explain the mass exchange between drop classes in gas/drop two phase flows. Keywordsgas/drop two-phase flows; droplet; collision and coalescence; birth-death process

This paper considers the cycle maximum in birth–death processes as a stepping stone to characterisation of the asymptotic behaviour of the maximum number of customers in single queues and open Kelly–Whittle networks of queues. For positive recurrent birth–death processes we show that the sequence of sample maxima is stochastically compact. For transient birth–death processes we show that the sequence of sample maxima conditioned on the maximum being finite is stochastically compact.We show that the Markov chain recording the total number of customers in a Kelly–Whittle network is a birth–death process with birth and death rates determined by the normalising constants in a suitably defined sequence of closed networks. Explicit or asymptotic expressions for these normalising constants allow asymptotic evaluation of the birth and death rates, which, in turn, allows characterisation of the cycle maximum in a single busy cycle, and convergence of the sequence of sample maxima for Kelly–Whittle networks of queues.

1. A graphical model is described with which one can calculate for any reproductive schedule values for population growth rate, birth rate, death rate, egg ratio, and age structure. Variations are discussed to describe different kinds of life history. 2. Examples are described to show how various population parameters are interrelated and how they vary with reproductive schedule. Discrepancies that arise in deriving instantaneous coefficients from counts of a population growing by discrete steps are noted.

This thesis is centered on the theory of stochastic processes and their applications in biological systems characterized by a complex environment. Three case studies have been modeled by the use of the three fundamental tools of stochastic processes: the master equation (ME), the stochastic differential equation (SDE) and the partial differential equation (PDE). The principal approach here applied to deal with complexity is the characterization of the system by means of probability distributions describing each a parameter of the model or the introduction of fractional order derivatives to include non-local and memory effects maintaining the linearity in the equations. In Chapter 1 we briefly review the theory of stochastic processes. In Chapter 2 we derive a birth-death process master equation to test if Long Interspersed Elements (LINEs) can be modeled according to the neutral theory of biodiversity. In Chapter 3 we derive a model of anomalous diffusion based on a Langevin approach in which anomalous behavior arise in the asymptotic intermediate state as a consequence of the heterogeneity of the system, from the superposition of Ornstein-Uhlenback processes. In Chapter 4 we propose an extension of the cable equation, used to describe anomalous diffusion phenomena as the signal conduction in spiny dendrites, by introducing a Caputo time fractional derivative.

A standard reaction–diffusion equation consists of two additive terms, a diffusion term and a reaction rate term. The latter term is obtained directly from a reaction rate equation which is itself derived from known reaction kinetics, together with modelling assumptions such as the law of mass action for well-mixed systems. In formulating a reaction–subdiffusion equation, it is not sufficient to know the reaction rate equation. It is also necessary to know details of the reaction kinetics, even in well-mixed systems where reactions are not diffusion limited. This is because, at a fundamental level, birth and death processes need to be dealt with differently in subdiffusive environments. While there has been some discussion of this in the published literature, few examples have been provided, and there are still very many papers being published with Caputo fractional time derivatives simply replacing first order time derivatives in reaction–diffusion equations. In this paper, we formulate clear examples of reaction–subdiffusion systems, based on; equal birth and death rate dynamics, Fisher–Kolmogorov, Petrovsky and Piskunov (Fisher–KPP) equation dynamics, and Fitzhugh–Nagumo equation dynamics. These examples illustrate how to incorporate considerations of reaction kinetics into fractional reaction–diffusion equations. We also show how the dynamics of a system with birth rates and death rates cancelling, in an otherwise subdiffusive environment, are governed by a mass-conserving tempered time fractional diffusion equation that is subdiffusive for short times but standard diffusion for long times.

This paper is concerned with two problems in connection with exponential ergodicity for birth-death processes on a semi-infinite lattice of integers. The first is to determine from the birth and death rates whether exponential ergodicity prevails. We give some necessary and some sufficient conditions which suffice to settle the question for most processes encountered in practice. In particular, a complete solution is obtained for processes where, from some finite statenonwards, the birth and death rates are rational functions ofn. The second, more difficult, problem is to evaluate the decay parameter of an exponentially ergodic birth-death process. Our contribution to the solution of this problem consists of a number of upper and lower bounds.

When cancers or bacterial infections establish, small populations of cells have to free themselves from homoeostatic regulations that prevent their expansion. Trait evolution allows these populations to evade this regulation, escape stochastic extinction and climb up the fitness landscape. In this study, we analyse this complex process and investigate the fate of a cell population that underlies the basic processes of birth, death and mutation. We find that the shape of the fitness landscape dictates a circular adaptation trajectory in the trait space spanned by birth and death rates. We show that successful adaptation is less likely for parental populations with higher turnover (higher birth and death rates). Including density- or trait-affecting treatment we find that these treatment types change the adaptation dynamics in agreement with a geometrical analysis of fitness gradients. Treatment strategies that simultaneously target birth and death rates are most effective, but also increase evolvability. By mapping physiological adaptation pathways and molecular drug mechanisms to traits and treatments with clear eco-evolutionary consequences, we can achieve a much better understanding of the adaptation dynamics and the eco-evolutionary mechanisms at play in the dynamics of cancer and bacterial infections.

We discuss connections between birth and death processes and orthogonal polynomials. We survey certain spectral properties of orthogonal polynomials and spectra of birth and death processes. Birth and death processes with linear birth and death rates are studied in some detail. We also mention some results concerning birth and death processes whose transition rates are rational functions.

An increase in the average age of dengue hemorrhagic fever (DHF) cases has been reported in Thailand. The cause of this increase is not known. Possible explanations include a reduction in transmission due to declining mosquito populations, declining contact between human and mosquito, and changes in reporting. We propose that a demographic shift toward lower birth and death rates has reduced dengue transmission and lengthened the interval between large epidemics. Using data from each of the 72 provinces of Thailand, we looked for associations between force of infection (a measure of hazard, defined as the rate per capita at which susceptible individuals become infected) and demographic and climactic variables. We estimated the force of infection from the age distribution of cases from 1985 to 2005. We find that the force of infection has declined by 2% each year since a peak in the late 1970s and early 1980s. Contrary to recent findings suggesting that the incidence of DHF has increased in Thailand, we find a small but statistically significant decline in DHF incidence since 1985 in a majority of provinces. The strongest predictor of the change in force of infection and the mean force of infection is the median age of the population. Using mathematical simulations of dengue transmission we show that a reduced birth rate and a shift in the population's age structure can explain the shift in the age distribution of cases, reduction of the force of infection, and increase in the periodicity of multiannual oscillations of DHF incidence in the absence of other changes. Lower birth and death rates decrease the flow of susceptible individuals into the population and increase the longevity of immune individuals. The increase in the proportion of the population that is immune increases the likelihood that an infectious mosquito will feed on an immune individual, reducing the force of infection. Though the force of infection has decreased by half, we find that the critical vaccination fraction has not changed significantly, declining from an average of 85% to 80%. Clinical guidelines should consider the impact of continued increases in the age of dengue cases in Thailand. Countries in the region lagging behind Thailand in the demographic transition may experience the same increase as their population ages. The impact of demographic changes on the force of infection has been hypothesized for other diseases, but, to our knowledge, this is the first observation of this phenomenon. Please see later in the article for the Editors' Summary.

Data relating to birth and death were collected from throughout the state of Kerala, India, in a health survey conducted by the Kerala Sastra Sahitya Parishad, a voluntary organization. In this study, the authors analyze birth and death rates as calculated from the sample of 9,940 households (57,665 persons), with respect to other variables such as region, religion, and socioeconomic status. In order to study the effect of socioeconomic factors on birth and death rates, a socioeconomic status rating (SES rating) was developed, taking into account such factors as income, education, housing conditions, and land ownership. Socioeconomic status was found to have a definite influence on birth and death rates, with higher socioeconomic status resulting in lower birth and death rates. This effect was independent of such confounding variables as age structure of the population, religion, and region. The higher risk of mortality among the poorer households can partly be explained by the material deprivation: the higher birth rates could be the result of poorer educational attainments.

Circumstances in connection with the recent census have again directed my attention to the laws which govern human life. I have long been of the opinion that the old ideas that birth rates and death rates had no biological relationship beyond the obvious ones, that many infants mean more deaths, etc., usually found stated in public health text books, were based on a very imperfect induction. On one aspect of this I published a paper a number of years ago in the Transactions of the Royal Philosophical Society of Glasgow (1), and last summer Sir Shirley Murphy(2) read a paper on another aspect of the same subject before the Sanitary Congress held at York. But I have hitherto refrained from publishing theories because I believe that until quantitative measures are applied no scientific results worthy of discussion can be obtained. Now that such seem to be possible, I propose to discuss in a series of papers the different relationships which I have investigated. No mathematics will be introduced in the earlier papers, but the results of all will be summarised and dealt with in their mathematical and physico-chemical relationships in a concluding communication. The first paper relates to the connection between the “corrected” death rates and the “true” death rates as found by constructing a life table.

Cancer evolution and progression are shaped by cellular interactions and Darwinian selection. Evolutionary game theory incorporates both of these principles, and has been proposed as a framework to understand tumour cell population dynamics. A cornerstone of evolutionary dynamics is the replicator equation, which describes changes in the relative abundance of different cell types, and is able to predict evolutionary equilibria. Typically, the replicator equation focuses on differences in relative fitness. We here show that this framework might not be sufficient under all circumstances, as it neglects important aspects of population growth. Standard replicator dynamics might miss critical differences in the time it takes to reach an equilibrium, as this time also depends on cellular turnover in growing but bounded populations. As the system reaches a stable manifold, the time to reach equilibrium depends on cellular death and birth rates. These rates shape the time scales, in particular, in coevolutionary dynamics of growth factor producers and free-riders. Replicator dynamics might be an appropriate framework only when birth and death rates are of similar magnitude. Otherwise, population growth effects cannot be neglected when predicting the time to reach an equilibrium, and cell-type-specific rates have to be accounted for explicitly.

Persson’s theory of purely normal elastic rough surface contact: A tutorial based on stochastic process theory

In recent years, the proportion of new energy power generation in the power supply has increased yearly. However, the random volatility generated by new energy generation poses a new challenge to the stability of the power system. In terms of stability analysis, the traditional methods based on the deterministic system can not fully describe the influence of random fluctuation. In this case, this paper introduces the power grid stability analysis method considering random influence. Based on the original model, this paper introduces the random fluctuation simulated by the Gaussian process, establishes the power grid model described by nonlinear stochastic differential equations, and gives the numerical solution format of the model with the help of the Euler method. Secondly, compared with the traditional method of quantifying the size of the escaping state of the system, this method considers the influence of the random variable on the stability of the system at the same time. Therefore, this paper establishes the stability analysis system of the power grid and further considers the influence of power grid topology and system parameters on system stability. By analyzing the Brass paradox, it is found that it is more reasonable to quantify the stability of the system with exit time. Finally, this paper uses the stability analysis system to study the stability of the high voltage power grid in Shandong Province and gives suggestions to improve the stability.