A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.     

A stochastic model of a cell population with quiescence

A cell population in which cells are allowed to enter a quiescent (nonproliferating) phase is analyzed using a stochastic approach. A general branching process is used to model the population which, under very mild conditions, exhibits balanced exponential growth. A formula is given for the asymptotic fraction of quiescent cells, and a numerical example illustrates how convergence toward the asymptotic fraction exhibits a typical oscillatory pattern. The model is compared with deterministic models based on semigroup analysis of systems of differential equations.     

A stochastic computer model for simulating population growth

A model is described for investigating the interactions of age-specific birth and death rates, age distribution and density-governing factors determining the growth form of single-species populations. It employs Monte Carlo techniques to simulate the births and deaths of individuals while density-governing factors are represented by simple algebraic equations relating survival and fecundity to population density. In all respects the model's behavior agrees with the results of more conventional mathematical approaches, including the logistic model andLotka's Law, which predicts a relationship betwen age-specific rates, rate of increase and age distribution. Situations involving exponential growth, three different age-independent density functions affecting survival, three affecting fecundity and their nine combinations were tested. The one function meeting the assumptions of the logistic model produced a logistic growth curve embodying the correct values or r_m and K. The others generated sigmoid curves to which arbitrary logistic curves could be fitted with varying success. Because of populational time lags, two of the functions affecting fecundity produced overshoots and damped oscillations during the initial approach to the steady state. The general behavior of age-dependent density functions is briefly explored and a complex example is described that produces population fluctuations by an egg cannibalism mechanism similar to that found in the flour beetle Tribolium. The model is free of inherent time lags found in other discrete time models yet these may be easily introduced. Because it manipulates separate individuals, the model may be combined readily with the Monte Carlo simulation models of population genetics to study eco-genetic phenomena.     

A study of the electromyogram using a population stochastic model of skeletal muscle

The features of the electromyogram (EMG) are studied using a population model of skeletal muscle based on the differing properties and the independent activation of motor units (MUs). It is shown, both analytically and by computer simulation, that: (a) The power spectrum of the EMG is determined by the distribution of filtering and firing properties of the active MUs. (b) A tendency towards a rhythmical grouping of action potentials is to be expected from a set of asynchronous MUs firing semiregularly at similar rates; the grouped electrical activity has a phase-lead over the force output of the set of about 180°. A unified explanation of the properties of the muscle force waveform and the electromyogram, in terms of asynchronous activity of MUs, is proposed. The explanation covers the relationship and the differences between the two signals.     

An age-dependent stochastic model of population growth

A stochastic model of population growth is treated using the Bellman-Harris theory of agedependent stochastic branching processes. The probability distribution for the population size at any time and the expectation are obtained when it is assumed that there is probability (1−σ), 0≤σ<1, of the organism dividing into two at the end of its lifetime, and probability σ that division will not take place.     

A study of the muscle force waveform using a population stochastic model of skeletal muscle

A population stochastic model based on the differing properties and the independent activation of motor units is used to describe the production of force in the contracting skeletal muscle. Detailed force predictions of the model concerning a hand muscle are obtained by computer simulation. General features of the force signal are established analyticaly on the basis of the general properties of the neuromuscular system which the population model takes into account. The results show that the asynchronous activity of motor units and the distribution of their filtering and firing properties at various levels of muscle contraction are esponsible, at least partially, for the main features of the muscle force waveform, including tremor.     

A stochastic model for the development of an AIDS epidemic in a heterosexual population.

A non-age-dependent model, describing the evolution of a bisexual population, is developed in this paper and applied to projecting an AIDS epidemic in a heterosexual population. Included in the formulation are frequency- and non-frequency-dependent rules of partnership formation as well as five states of HIV disease, affecting the probability of infection per sexual contact. Results from computer experiments, designed to study the development of an AIDS epidemic in a heterosexual population fed by single males with a 50% prevalence of HIV infection prior to becoming active in heterosexual partnerships, are reported. In these experiments, the only source of HIV infection for females was sexual contacts with infected males within partnerships. Data on the probability of infection per sexual contact with an infected partner and the number of sexual contacts per month were incorporated into the model. However, the numbers used for the initial population of singles, couples, and those becoming sexually active per month were hypothetical. Even though the prevalence of HIV infection among males entering heterosexual partnerships was high, after 30 years the projected prevalence of HIV infection among females ranged from about 10 to 15% depending in part on the expected duration of partnerships and on whether the frequency- or non-frequency-dependent model was used. In these experiments, solutions of the embedded, nonlinear, deterministic equations for the incidence of HIV infection and the cumulative number of deaths due to AIDS proved to be good measures of central tendency for the sample functions of the stochastic population process.     

Transition to quorum sensing in an Agrobacterium population: A stochastic model

Understanding of the intracellular molecular machinery that is responsible for the complex collective behavior of multicellular populations is an exigent problem of modern biology. Quorum sensing, which allows bacteria to activate genetic programs cooperatively, provides an instructive and tractable example illuminating the causal relationships between the molecular organization of gene networks and the complex phenotypes they control. In this work we—to our knowledge for the first time—present a detailed model of the population-wide transition to quorum sensing using the example of Agrobacterium tumefaciens. We construct a model describing the Ti plasmid quorum-sensing gene network and demonstrate that it behaves as an “on–off” gene expression switch that is robust to molecular noise and that activates the plasmid conjugation program in response to the increase in autoinducer concentration. This intracellular model is then incorporated into an agent-based stochastic population model that also describes bacterial motion, cell division, and chemical communication. Simulating the transition to quorum sensing in a liquid medium and biofilm, we explain the experimentally observed gradual manifestation of the quorum-sensing phenotype by showing that the transition of individual model cells into the “on” state is spread stochastically over a broad range of autoinducer concentrations. At the same time, the population-averaged values of critical autoinducer concentration and the threshold population density are shown to be robust to variability between individual cells, predictable and specific to particular growth conditions. Our modeling approach connects intracellular and population scales of the quorum-sensing phenomenon and provides plausible answers to the long-standing questions regarding the ecological and evolutionary significance of the phenomenon. Thus, we demonstrate that the transition to quorum sensing requires a much higher threshold cell density in liquid medium than in biofilm, and on this basis we hypothesize that in Agrobacterium quorum sensing serves as the detector of biofilm formation.     

A stochastic model for PSA levels: behavior of solutions and population statistics

This paper investigates the partial differential equation for the evolving distribution of prostate-specific antigen (PSA) levels following radiotherapy. We also present results on the behavior of moments for the evolving distribution of PSA levels and estimate the probability of long-term treatment success and failure related to values of treatment and disease parameters. Results apply to a much wider range of parameter values than was considered in earlier studies, including parameter combinations that are patient specific.     

A plea for stochastic population dynamics

It is argued that biological populations are finite and consisting of individuals with varying life span and reproduction, and that they should be thus modelled. Modern probability theory provides tools for this.     

A stochastic SIR model with contact-tracing: large population limits and statistical inference

This paper is devoted to the presentation and study of a specific stochastic epidemic model accounting for the effect of contact-tracing on the spread of an infectious disease. Precisely, one considers here the situation in which individuals identified as infected by the public health detection system may contribute to detecting other infectious individuals by providing information related to persons with whom they have had possibly infectious contacts. The control strategy, which consists of examining each individual who has been able to be identified on the basis of the information collected within a certain time period, is expected to efficiently reinforce the standard random-screening-based detection and considerably ease the epidemic. In the novel modelling of the spread of a communicable infectious disease considered here, the population of interest evolves through demographic, infection and detection processes, in a way that its temporal evolution is described by a stochastic Markov process, of which the component accounting for the contact-tracing feature is assumed to be valued in a space of point measures. For adequate scalings of the demographic, infection and detection rates, it is shown to converge to the weak deterministic solution of a PDE system, as a parameter n, interpreted as the population size, roughly speaking, becomes larger. From the perspective of the analysis of infectious disease data, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as maximum likelihood estimation. We state preliminary statistical results in this context. Eventually, relations of the model with the available data of the HIV epidemic in Cuba, in which country a contact-tracing detection system has been set up since 1986, is investigated and numerical applications are carried out.     

A stochastic SIR model with contact-tracing: large population limits and statistical inference

This paper is devoted to the presentation and study of a specific stochastic epidemic model accounting for the effect of contact-tracing on the spread of an infectious disease. Precisely, one considers here the situation in which individuals identified as infected by the public health detection system may contribute to detecting other infectious individuals by providing information related to persons with whom they have had possibly infectious contacts. The control strategy, which consists of examining each individual who has been able to be identified on the basis of the information collected within a certain time period, is expected to efficiently reinforce the standard random-screening-based detection and considerably ease the epidemic. In the novel modelling of the spread of a communicable infectious disease considered here, the population of interest evolves through demographic, infection and detection processes, in a way that its temporal evolution is described by a stochastic Markov process, of which the component accounting for the contact-tracing feature is assumed to be valued in a space of point measures. For adequate scalings of the demographic, infection and detection rates, it is shown to converge to the weak deterministic solution of a PDE system, as a parameter n, interpreted as the population size, roughly speaking, becomes larger. From the perspective of the analysis of infectious disease data, this approximation result may serve as a key tool for exploring the asymptotic properties of standard inference methods such as maximum likelihood estimation. We state preliminary statistical results in this context. Eventually, relations of the model with the available data of the HIV epidemic in Cuba, in which country a contact-tracing detection system has been set up since 1986, is investigated and numerical applications are carried out.     

A stochastic model for a progressive chronic disease

For many progressive chronic diseases, there exist useful prognostic indicators for the course of the disease and the survival of the patient. The evolution of such an indicator is modelled as a monotone transformation of a pure birth process with killing. Explicit formulas are derived for the probability distribution of this process at an arbitrary time, the distribution of the first-passage times, the joint distribution of the survival time and the maximum of the process, and the marginals of this joint distribution. In two examples, the general formulas are evaluated in closed form.     

A stochastic population model for Lepidium papilliferum (Brassicaceae), a rare desert ephemeral with a persistent seed bank

Population viability analysis (PVA) is a valuable tool for rare plant conservation, but PVA for plants with persistent seed banks is difficult without reliable information on seed bank processes. We modeled the population dynamics of the Snake River Plains ephemeral Lepidium papilliferum using data from an 11-yr artificial seed bank experiment to estimate age-specific vital rates for viability loss and germination. We related variation in postgermination demographic parameters to annual variation in precipitation patterns and used these relationships to construct a stochastic population model using precipitation driver variables. This enabled us to incorporate realistic levels of environmental variability into the model. A model incorporating best estimates for parameter values resulted in a mean trajectory for seed bank size that remained essentially stable through time, although there was a measurable risk of extinction over a 100-yr period for the study population under this scenario. Doubling the annual seed viability loss rate resulted in near-certain extinction, as did increasing first-year germination to 100%, showing the importance of the persistent seed bank. Interestingly, increasing environmental variance substantially decreased the risk of extinction, presumably because this plant relies on extremely good years to restock the persistent seed bank, while extremely bad years have little impact. If every year were average in this desert environment, the species could not persist. Simulated effects of livestock trampling resulted in greatly increased extinction risk, even over time frames as short as 15 years.     

A stochastic model for the mutation-selection balance in an infinite asexual population with a genome of fixed size

A stochastic model is presented which describes the evolution of a genome of a haploid species in an infinite population. The genome is a finite set of elements. The elements are divided into different classes according to their effect on the fitness of the organism. Repeated mutations of the genome elements are permitted, in particular positive mutations are introduced. The distribution of the deleterious elements in the genome with respect to the impact on the fitness is given after the replication step. The steady state is fully described including the distribution and the fitness.     

Dynamics of a stochastic heroin epidemic model with bilinear incidence and varying population size

We investigate a stochastic heroin epidemic model with bilinear incidence and varying population size.Sufficient criteria for the extinction of the drug abusers and the existence of ergodic stationary distribution for the model are established by constructing suitable stochastic Lyapunov functions.By analyzing the sensitivity of the threshold of spread,we obtain that prevention is better than cure.Numerical simulations are carried out to confirm the analytical results.     

A stochastic SIR model on a graph with epidemiological and population dynamics occurring over the same time scale

Journal of Mathematical Biology - We define and study an open stochastic SIR (Susceptible–Infected–Removed) model on a graph in order to describe the spread of an epidemic on a cattle...     

A simple model of a Gyrodactylus population

Experimental populations of 20 Gyrodactylus alexanderi Mizelle &; Kritsky, 1967, on 19 isolated Gasterosteus aculeatus at 15°C increased for 2 weeks to a mean of 61, then decreased in 2 further weeks to a mean of 9. Fish that lost their fluke infestations were refractory to further infestation for about 3 weeks.The chief factors affecting fluke abundance were measured, including reproduction and mortality rates of flukes on fish, rate of shedding by the fish, mortality rates of flukes while off fish, and the rate of reattachment of the flukes. Data on these individual factors were combined to form a simple deterministic model which simulated the population changes on isolated fish. This was later made more realistic by the introduction of a random variable. When the model was tested in a multiple-host situation it predicted results close to those observed experimentally.     

A stochastic two-phase growth model

This article proposes a stochastic growth model that starts as a Yule process and is subsequently joined with a Prendiville process when the population attains certain prescribed critical size. In other words, the model assumes exponential growth in an early stage and logistic growth later on to reflect growth retardation caused by overcrowding. In the case that the population starts with a single unit, closed form expressions are given for the distribution of the population size and for the mean and variance functions of the process. Numerical solutions are briefly discussed for the process that starts with more than one unit.     

A stochastic model of animal growth

The plasticity of growth of animals in time, due to the resilience of their response to the ways they can be fed, suggests the difficulty of describing growth by a stochastic model in the time domain. A model is presented which avoids this difficulty by describing growth as a Markov process in the food-consumed domain, assuming that, at conception, (1) the maximum mature weight as a number α of biomass units of mass μ, and (2) the probability B of production of a biomass unit per unit of food consumed, are specified. Constancy of α, μ and B, as the animal feeds, is the basis of the proposed Markov process. The mean growth from infancy to maturity in the food-consumed domain is then the old law of diminishing returns empirically formulated first by Spillman (1924) for cattle and swine, and confirmed by Titus, Jull &; Hendricks (1934) for fowl, and by Parks (1972) across species from mice to steers. The solution also leads to the possibility that the distribution of weights in a population of growing animals of the same species, is related to the distribution of mature weights among the individuals. An experiment by Lister &; McCance (1967) with well-fed and severely undernourished pigs, shows the stability of growth in the foodconsumed domain compared to the plasticity in the time domain. Other implications of the model are discussed.