On the Fokker-Planck equation in the stochastic theory of mortality: II

This is the continuation of part I, which was published in the September, 1963, issue ofThe Bulletin. Section 5 treats the special case in which the left absorbing barrier recedes to −∞, leaving essentially only one barrier at a finite distance Λ (>0) from the origin. The eigenfunctions are now parabolic cylinder functions. The limiting cases Λ→+∞ and Λ→0 are also considered. Though meaningless for practical applications to our problem, they are of interest, mathematically, because the Green’s function for the solution of the Fokker-Planck equation assumes a particularly simple form. In section 6 we study, by means of an example, how the “force of mortality” may vary with time before attaining its final asymptotic value. Section7, still dealing with only one absorbing barrier, shows that our results for “strong homeostasis” are identical with those derived by Chandrasekhar for the escape of particles through a potential barrier in the limiting case of quasi-static flow. Precise conditions are given for the validity of both the quasi-static and the Smoluchowski approximations to the Fokker-Planck equation. Finally, in section 8, a brief mention is made of Gevrey’s method for the solution of parabolic partial differential equations.     

On the stochastic theory of compartments: I. A single-compartment system

A stochastic model is developed for a compartment with a single time-dependent input, and generalized to include inputs from several sources. With the number of particles of a given molecular species in the compartment as the random variable, the mean, variance and third central moment of this variable are calculated from its generating function, and compared with previous results. The behavior of the calculated moments is discussed, and the possibility of applying the model to chemical and biological systems is considered.     

On the stochastic theory of compartments: II. Multi-compartment systems

A stochastic model is developed for a system of interconnected compartments. The generating function of the random variable of any compartment can be constructed from a flow graph involving the expectations of the random variables of all compartments of the system.     

On the stochastic theory of compartments: III. General time-dependent reversible systems

General formulation of stochastic single- and multi-compartment reversible systems with time-dependent transitions is made. The correspondence between the stochastic mean and the deterministic value is established in case of time-dependence and it is shown how the consequence of this can be utilized to compute the distribution and the moments of each individual compartment of the system. A two-compartment reversible system previously proposed by Cardenas and Matis (1975a) is analyzed on the basis of the theory.     

On the stochastic theory of compartments: The leaving process of the two-compartment systems

In this paper three stochastic models are developed for a class of two-compartment systems to analyse the randomness of the leaving process of the particles in the system. Results in closed form for the distribution of the leaving process of the particles in the system are given both for general and exponential sojourn time distributions and also in association with forward recurrence time distributions with and without Poisson input.     

Studies in stochastic learning theory: I path-dependent linear models for experimenter-subject controlled experiments

The derivation of learning models relative to choice behavior in experimenter-subject controlled experiments with two outcomes (right or wrong) is considered from the point of view that any such model must satisfy a criterion of optimality. The criterion adopted for investigation, termed optimal asymptotic behavior, is that of the subject asymptotically learning which of the alternatives has the greater probability of being correct. A class of path-dependent linear models is posed as possible candidates. It is shown that no members of this class satisfy the criterion although two of them approach it by making a learning parameter small enough. The possible implications of this are discussed.     

A theory of stochastic harvesting in stochastic environments

We investigate how model populations respond to stochastic harvesting in a stochastic environment. In particular, we show that the effects of variable harvesting on the variance in population density and yield depend critically on the autocorrelation of environmental noise and on whether the endogenous dynamics of the population display over- or undercompensation to density. These factors interact in complicated ways; harvesting shifts the slope of the renewal function, and the net effect of this shift will depend on the sign and magnitude of the other influences. For example, when environmental noise exhibits a positive autocorrelation, the relative importance of a variable harvest to the variance in density increases with overcompensation but decreases with undercompensation. For a fixed harvesting level, an increasing level of autocorrelation in environmental noise will decrease the relative variation in population density when overcompensation would otherwise occur. These and other intricate interactions have important ramifications for the interpretation of time series data when no prior knowledge of demographic or environmental details exists. These effects are important whenever the harvesting rate is sufficiently high or variable, conditions likely to occur in many systems, whether the harvesting is caused by commercial exploitation or by any other strong agent of density-independent mortality.     

On the theory of blood-tissue exchanges: I. Fundamental equations

A mathematical analysis of the absorption of an inert gas by a heterogeneous system ofn phases, e.g., a limb consisting ofn tissues, is presented. The total uptake of gas, ϕ(t), up to timet is given in terms of arterial concentration, cardiac delivery, blood volume, and the volume, permeability, and partition coefficient of each tissue. The theory predicts how the uptake curve should change in shape under a variety of physiological conditions, and how from the numerical values of the constants the values of certain tissue constants, e.g. permeabilities, may be obtained. The material in this article should be construed only as the personal opinion of the writers and not as representing the opinion of the Navy Department officially.     

On the stochastic theory of compartments: A semi-Markov approach for the analysis of theK-compartmental systems

In this paper a semi-Markov process approach is developed to analyse stochastic compartmental systems using straightforward probabilistic arguments. Explicit expressions for several characteristics of thek-compartmental systems with a Poisson process input are derived and various models found in the literature arising from biological applications are generalised here using the semi-Markov process technique.     

On the stochastic theory of compartments: Solution forn-compartment systems with irreversible,time-dependent transition probabilities

The bivariate distribution of a two-compartment stochastic system with irreversible, time-dependent transition probabilities is obtained for any point in time. The mean and variance of the number of particles in any compartment and the covariance between the number of particles in each of the two compartments are exhibited and compared to existing results. The two-compartment system is then generalized to ann-compartment catenary and to ann-compartment mammillary system. The multivariate distributions of these two systems are obtained under two sets of initial conditions: (1) the initial distribution is known; and (2) the number of particles in each compartment of the system at timet=0 is determined. The moments of these distributions are also produced and compared with existing results.     

On the stochastic interpretation of a pathological cell renewal system,I: Theoretical basis

The population constituted by parenchymal liver cells is normally a static or slowly expanding one. This is in contrast with “renewal systems,” or cell populations which turn over more or less rapidly. In experimental situations tending to produce cirrhosis of the liver the rate of cell renewal is appreciable. The situation then corresponds to the definition of a “renewal system.” A repeating unit of the liver lattice structure is proposed in which liver cells are assumed to be sequestrated by anatomical constraints on cell migration, and the order of magnitude of its population determined. The characteristics of the simple birth-and-death process are reviewed to show how it leads to greater fluctuation than does a pure birth process with equivalent expected net growth. A hypothetical birth-and-death process is proposed for the renewal of the cell population of the liver unit. The rate of cell proliferation is placed at a level comparable to that found in reports of experimental cirrhosis taken from the literature. This example of a birth-and-death process leads to the prediction of appreciable fluctuation in the population of the liver unit. It is suggested that fluctuations of this kind may account for some of the morphological features of cirrhosis and lead to a new definition of “nodular regeneration.”     

Analysis of stochastic strategies in bacterial competence: a master equation approach

Competence is a transiently differentiated state that certain bacterial cells reach when faced with a stressful environment. Entrance into competence can be attributed to the excitability of the dynamics governing the genetic circuit that regulates this cellular behavior. Like many biological behaviors, entrance into competence is a stochastic event. In this case cellular noise is responsible for driving the cell from a vegetative state into competence and back. In this work we present a novel numerical method for the analysis of stochastic biochemical events and use it to study the excitable dynamics responsible for competence in Bacillus subtilis. Starting with a Finite State Projection (FSP) solution of the chemical master equation (CME), we develop efficient numerical tools for accurately computing competence probability. Additionally, we propose a new approach for the sensitivity analysis of stochastic events and utilize it to elucidate the robustness properties of the competence regulatory genetic circuit. We also propose and implement a numerical method to calculate the expected time it takes a cell to return from competence. Although this study is focused on an example of cell-differentiation in Bacillus subtilis, our approach can be applied to a wide range of stochastic phenomena in biological systems.     

Numerical integration of the master equation in some models of stochastic epidemiology

The processes by which disease spreads in a population of individuals are inherently stochastic. The master equation has proven to be a useful tool for modeling such processes. Unfortunately, solving the master equation analytically is possible only in limited cases (e.g., when the model is linear), and thus numerical procedures or approximation methods must be employed. Available approximation methods, such as the system size expansion method of van Kampen, may fail to provide reliable solutions, whereas current numerical approaches can induce appreciable computational cost. In this paper, we propose a new numerical technique for solving the master equation. Our method is based on a more informative stochastic process than the population process commonly used in the literature. By exploiting the structure of the master equation governing this process, we develop a novel technique for calculating the exact solution of the master equation--up to a desired precision--in certain models of stochastic epidemiology. We demonstrate the potential of our method by solving the master equation associated with the stochastic SIR epidemic model. MATLAB software that implements the methods discussed in this paper is freely available as Supporting Information S1.     

On Darwin's 'metaphysical notebooks'. I: teleology and the project of a theory.

Huxley's essay On the Reception of the 'Origin of Species' brings us close to the issue of cause and of why- and how-questions in the understanding of the living world. The present contribution, which is divided into two parts, reviews the problem of Teleology as conceived by Huxley and re-examines Darwin as the author who revealed the existence of a 'foundations problem' in the explanation of an entire realm of nature, i.e., the problem of explaining such realm in terms of its own, specific legality, or iuxta sua propria principia. In the first part the enquiry is mainly focused on the secularization of natural history after Paley; in the second part it is mainly focused on the desubjectivization of the inquiry into natural history after Erasmus Darwin and Lamarck. The second part will be published in the next issue of Rivista di Biologia/Biology Forum. In the first part below an analysis is made of Notebooks M and N. The author disputes the correctness of conceiving them only as the works where Darwin envisages the 'metaphysical' themes later to become the subject of The Expression of the Emotions. He suggests to conceive of them also as the works where Darwin defines the terms of the general project of his own, peculiar evolutionary theory. The author then outlines the intellectual progress of Darwin from the inosculation to the transmutation hypotheses. Darwin's reading of Malthus appears to be analytically decisive, because it offers him the vintage point to attack the metaphysical and theological citadels on the morphological side. Darwin is thus able to re-consider Erasmus' comprehensive zoonomic project, by displacing it, however, from the old idea of the scala naturae to the new one of the "coral of life", and by emphasising the distinction between "the fittest" and "the best" vs. the tradition of Natural Theology.     

Common source epidemics I: A stochastic model

A discrete time stochastic model is formulated for the spread of a disease which is transmitted to an uninfected but susceptible individual through an environmental source and not through contact (either direct or indirect) with infected individuals. The model incorporates both exposure and infection components. The exposure component includes consideration of the introduction of an infectious agent into the environment and the subsequent diffusion of the agent. It also includes time and location patterns for visits by individuals in the target population to the affected environment. The infection component incorporates physiological responses of exposed individuals to the infectious agent. The goal of the model is to provide a method for developing a predicted epidemic curve. Comments are given on an application of the model to the study of an outbreak of toxoplasmosis in Atlanta, Georgia, in 1977. This work was partially supported by BRSG Grant S07 RR0731 awarded by the Biomedical Research Support Grant Program, Division of Research Resources, National Institutes of Health.     

Comments on the approximate solution of the diffusion equation: I

The approximation method of N. Rashevsky is discussed and reviewed. It is shown that in addition to theexplicit assumptions and approximations there is involved the assumption that the rate of metabolism is the same at every point in the cell and that theaverage rate of metabolism is different from zero. An expression is given for the error in the approximate method when the rate of metabolism is any function of the concentration. It is also shown that a solution in theform of that obtained by the approximate method is not possible if the generalized laws of diffusion are assumed to apply. A portion of this work was performed while the author was a research participant, Oak Ridge Institute of Nuclear Studies, assigned to the Mathematics Panel, Oak Ridge National Laboratory.