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PubMed1994

On some formulas in a partnership model from the perspective of a semi-Markov process.

Many deterministic models of sexually transmitted diseases, as well as population models in general, contain elements of stochastic or statistical reasoning. An example of such a model is that of Dietz and Hadeler (1988) concerning sexually transmitted diseases in which there is partnership formation and dissolution. Among the interesting formulas in this paper, which enter into the analysis of the model, are those for the expected number of partners a male or female has during a lifetime. To a probabilist such formulas suggest the possibility that some stochastic process may be constructed so as to yield these formulas as well as others that may be of interest. The principal purpose of this paper is to demonstrate that such a stochastic process does indeed exist in the form of a three state semi-Markov process in continuous time with stationary laws of evolution and with a one-step density matrix determined by four parameters which were interpreted as constant latent risk functions in the classical theory of competing risks. This construction of a semi-Markov process not only provides a framework for the systematic derivation of the formulas of Dietz and Hadeler but also suggests pathways for extensions to the age-dependent case.

Female↗

PubMed1989

Pharmacokinetics from a dynamical systems point of view.

The pharmacological action of many drugs depends on several variables at the same time and therefore will be dominated by an attractor of a dimension greater than zero. The pharmacokinetic behavior is likely to be dominated by a zero dimensional point attractor so that it is highly predictable. Pharmacokinetics is discussed from a dynamical systems point of view, whereby the transport of drugs in the tissues and organs is considered a stochastic process characterized by density functions of transit times and blood flows. In the body, the tissues and organs are arranged in parallel, in series, and in a feedback-loop fashion. Consequently, the single-pass transport of drugs through the body is again a stochastic process characterized by the density function of total body transit times, the cardiac output, and the total body extraction. The drug molecules, however, may pass through the body several times before ultimately leaving the system by metabolism or excretion. As a result, the body may be regarded as a positive feedback system with the pulmonary circulation (and its tissues) as the forward transfer function and the systemic circulation (with all its tissues) as the feedback transfer function. Consequently, the total body transport function (closed loop) is again a stochastic process characterized by a density function of total body residence times. The relationship between the body transit time distribution and the body residence distribution is determined by the feedback-loop arrangement, the cardiac output, and the extraction ratio which can easily be written in the Laplace domain. The pharmacokinetic parameters logically follow from the systems approach. They are the cardiac output, the mean transit time, the extraction ratio, the clearance, the volume of distribution in steady state, the mean residence time, and the average number of recirculations. The dynamic systems approach in pharmacokinetics has been illustrated with some examples notably with caffeine.

Absorption↗

PubMed1992

On inherited fertility in biological systems: a model of correlated fluctuations in the stochastic branching process.

A new evolutionary model with hereditary modes considered as correlated fluctuations of fertility has been proposed. It has been demonstrated that the model allows the global statistical properties of the system to be evaluated, e.g. the ensemble average and the probability of extinction. The results obtained show the increase of instability of a population with the enhancement of inheritance efficiency. The existence of at least an exponential stratification in the population has also been shown. Possible applications of the present model are discussed.

Animals↗

PubMed1982

A hybrid computer model of stochastic activity of the neurone.

The authors describe a hybrid computer neurone model intended for the investigation of stochastic transformations effected by neurones in correlation to some of their physiological parameters. The model is designed so as to give the best possible characterization of the internal dynamics of a neurone with minimum limiting conditions. It allows the generation of suitable input stochastic processes or operates with input processes obtained experimentally in the living neurone and it carries out basic statistical tests of the output stochastic process.

Computers, Hybrid↗

PubMed1989

Ordered appearance of antigenic variants of African trypanosomes explained in a mathematical model based on a stochastic switch process and immune-selection against putative switch intermediates.

Antigenic variation of African trypanosomes results from the periodic activation of a single new variant cell surface glycoprotein (VSG) gene out of a repertoire of about a 1000 VSG genes. In spite of the apparently random genetic basis of the process of antigenic variation, the relapsing parasitemias are characterized by an as yet unexplained order of appearance of major VSG variants. Here we mathematically test hypotheses concerning the blood-based parasitemia. In our model the antigenic switches occur at random at the DNA level. A variable proportion of the switches has a short intermediate phase in which two different VSGs simultaneously occur on the cell surface. We show that, in a theoretical population of 230 single expressor variants in an immunocompetent or in an immunodeficient host, it is not possible to explain the ordered appearance of variants by affecting the growth coefficients of single expressors or double expressors or by affecting the antigen switch probabilities. Rather, a realistic parasitemia can be obtained if the majority of switches has a double expressor switch-intermediate phase and if the double expressors have a differential susceptibility to the immune control. This study is significant in providing a theoretical basis for the ordered appearance of variants and in explaining previously unresolved discrepancies between the rate of appearance of new variants in culture and in vivo. In addition, testable predictions as to the development of the infections, switch rate of variants, fraction of double expressors, and parasite mortality coefficients are generated.

Animals↗

PubMed1985

Statistics and quantum bumps in arthropod photoreceptors.

Discrete waves of depolarizing membrane potential in arthropod photoreceptors, called quantum bumps, appear to result from single-photon absorptions of the visual pigment. Statistical analysis of bump records suggest a model for bump occurrence in dark-adapted receptors at low levels of illumination. This model assumes that a photon that isomerizes a visual pigment molecule can trigger a stochastic process that can produce no more than one bump under normal conditions, and that the stochastic processes triggered by different isomerized visual pigment molecules are independent of each other.

Animals↗

PubMed1998

Stochastic effects in a model of nematode infection in ruminants.

We illustrate the importance of stochastic effects in population models of biological systems and demonstrate a number of analytic and simulation-based approaches that can usefully be applied to such models. In so doing, we compare the stochastic approach to the more usual deterministic one. The model studied represents the gastrointestinal infection of ruminants by nematodes when the hosts maintain a fixed density. The incorporation of a feedback mechanism, which accounts for the immune response of the infected animals, results in a highly nonlinear model; similar forms of nonlinearity are a feature of many plausible models in population biology. In the absence of an analytic solution to the full stochastic model we explore a number of approximations and compare them to simulations of the full stochastic process. We explore three modes of behaviour of the system. In the endemic regime the stochastic system fluctuates widely around the non-zero fixed points of the deterministic model. In the managed regime, where the system is subject to external periodic perturbation, stochastic effects are negligible. Finally, we find that in a regime in which the deterministic model predicts the long-term persistence of oscillations the stochastic model shows that extinction can occur. Of the approximation procedures we consider, the Normal approximation to the full stochastic process is the most generally applicable, and it is also the most accurate in the light of simulation results. Local linearization provides reasonably accurate prediction of the variance-covariance structure, and a transfer function approach allows calculation of the time-lagged auto- and cross-correlations in the endemic regime. Linearization of the stochastic updates themselves results in poor prediction of the population variances.

Animals↗

PubMed1986

Evidence that the process of murine melanoma metastasis is sequential and selective and contains stochastic elements.

Malignant neoplasms are heterogeneous for many different biological characteristics, including invasion and metastasis. The pathogenesis of metastasis involves a series of sequential steps which must be completed by metastatic cells. In the present study we examined the metastatic behavior of three highly metastatic and three nonmetastatic subpopulations isolated from the K-1735 melanoma syngeneic to the C3H/HeN mouse. Cells were labeled with [125I]iodo-2'-deoxyuridine, and their initial organ distribution, fate, and production of experimental metastases were determined. Highly metastatic cells survived in lung parenchyma to produce metastases, whereas nonmetastatic cells did not. However, even with the highly metastatic cells only 2% of the original inoculum was responsible for the final production of metastases. The results support the concept that the fate of tumor cells released into the bloodstream is determined by sequential and selective events and introduces a third regulatory factor. Cells endowed with metastatic properties have a higher probability of forming metastases than cells not so endowed, but this probability is not 100%. Hence, metastasis should be considered as a sequential, selective, and stochastic process.

Animals↗

PubMed1996

Stochastic comparison of processes generated by random interruptions of monotone functions and related results.

Many stochastic processes considered in applied probability models, and, in particular, in reliability theory, are processes of the following form: Shocks occur according to some point process, and each shock causes the process to have a random jump. Between shocks the process increases or decreases in some deterministic fashion. In this paper we study processes for which the rate of increase or decrease between shocks depends only on the height of the process. For such processes we find conditions under which the processes can be stochastically compared. We also study 'hybrid' processes in which periods of increase and periods of decrease alternate. A further result yields a stochastic comparison of processes that start with a random jump, rather than processes in which there is at the beginning some random delay time before the first jump.

Life Tables↗

PubMed1997

A stochastic model for the analysis of bivariate longitudinal AIDS data.

We present a model for multivariate repeated measures that incorporates random effects, correlated stochastic processes, and measurement errors. The model is a multivariate generalization of the model for univariate longitudinal data given by Taylor, Cumberland, and Sy (1994, Journal of the American Statistical Association 89, 727-736). The stochastic process used in this paper is the multivariate integrated Ornstein-Uhlenbeck (OU) process, which includes Brownian motion and a random effects model as special limiting cases. This process is an underlying continuous-time autoregressive order [AR(1)] process for the derivatives of the multivariate observations. The model allows unequally spaced observations and missing values for some of the variables. We analyze CD4 T-cell and beta-2-microglobulin measurements of the seroconverters at multiple time points from the Los Angeles section of the Multicenter AIDS Cohort Study. The model allows us to investigate the relationship between CD4 and beta-2-microglobulin through the correlations between their random effects and their serial correlation. The data suggest that CD4 and beta-2-microglobulin follow a bivariate Brownian motion process. The fit of the model implies that an increase in beta-2-microglobulin is associated with a decrease in future CD4 but not vice versa, agreeing with immunologic postulates about the relationship between these two variables.

Acquired Immunodeficiency Syndrome↗

PubMed1992

Graphical representation of survival curves associated with a binary non-reversible time dependent covariate.

The use of time dependent covariates has allowed for incorporation into analysis of survival data intervening events that are binary and non-reversible (for example, heart transplant, initial response to chemotherapy). We can represent this type of intervening event as a three-state stochastic process with a starting state (S), an intervening state (I), and an absorbing state (D), which usually represents death. In this paper we present three procedures for calculating survivorship functions which attempt to display the prognostic significance of the time dependent covariate. The first method compares survival from baseline for the two possible paths through the stochastic process; the second method compares overall survival to survival with state I removed from the process; and, the third method compares survival for those already in state I at a landmark time x to those in state S at time x who will never enter state I. We develop discrete hazard estimates for the survival curves associated with the three methods. Two examples illustrate how these methods can yield different results and in which situations one might employ each of the three methods. Extensions to applications with reversible binary time dependent covariates and models with both baseline and time dependent covariates are suggested.

Data Interpretation, Statistical↗

PubMed1998

Does the covariance structure matter in longitudinal modelling for the prediction of future CD4 counts?

We investigate the importance of the assumed covariance structure for longitudinal modelling of CD4 counts. We examine how individual predictions of future CD4 counts are affected by the covariance structure. We consider four covariance structures: one based on an integrated Ornstein-Uhlenbeck stochastic process; one based on Brownian motion, and two derived from standard linear and quadratic random-effects models. Using data from the Multicenter AIDS Cohort Study and from a simulation study, we show that there is a noticeable deterioration in the coverage rate of confidence intervals if we assume the wrong covariance. There is also a loss in efficiency. The quadratic random-effects model is found to be the best in terms of correctly calibrated prediction intervals, but is substantially less efficient than the others. Incorrectly specifying the covariance structure as linear random effects gives too narrow prediction intervals with poor coverage rates. Fitting using the model based on the integrated Ornstein-Uhlenbeck stochastic process is the preferred one of the four considered because of its efficiency and robustness properties. We also use the difference between the future predicted and observed CD4 counts to assess an appropriate transformation of CD4 counts; a fourth root, cube root and square root all appear reasonable choices.

Acquired Immunodeficiency Syndrome↗

PubMed1993

Optimal radiation beam profiles considering uncertainties in beam patient alignment.

The often large uncertainties that exist in beam patient alignment during radiation therapy may require modification of the incident beams to ensure an optimal delivered dose distribution to the target volume. This problem becomes increasingly severe when the required dose distribution of the incident beams becomes more heterogeneous. A simple analytical formula is derived for the case when the fraction number is high, and the desired relative dose variations are small. This formula adjusts the fluence distribution of the incident beam so that the resultant dose distribution will be as close as possible to the desired one considering the uncertainties in beam patient alignment. When sharp dose gradients are important, for instance at the border of the target volume, the problem is much more difficult. It is shown here that, if the tumor is surrounded by organs at risk, it is generally best to open up the field by about one standard deviation of the positional uncertainty--that is sigma/2 on each side of the target volume. In principle it is simultaneously desirable to increase the prescribed dose by a few per cent compared to the case where the positional uncertainty is negligible, in order to compensate for the rounded shoulders of the delivered dose distribution. When the tissues surrounding the tumor no longer are dose limiting even larger increases in field size may be advantageous. For more critical clinical situations the positional uncertainty may even limit the success of radiotherapy. In such cases one generally wants to create a steeper dose distribution than the underlying random Gaussian displacement process allows. The problem is then best handled by quantifying the treatment outcome under the influence of the stochastic process of patient misalignment. Either the coincidence with the desired dose distribution, or the expectation value of the probability of achieving complication-free tumor control is maximized under the influence of this stochastic process. It is shown that the most advantageous treatment is to apply beams that are either considerably widened or slightly widened and over flattened near the field edges for small and large fraction numbers respectively.

Dose-Response Relationship, Radiation↗

PubMed1987

Weak convergence of a sequence of stochastic difference equations to a stochastic ordinary differential equation.

We consider a sequence of discrete parameter stochastic processes defined by solutions to stochastic difference equations. A condition is given that this sequence converges weakly to a continuous parameter process defined by solutions to a stochastic ordinary differential equation. Applying this result, two limit theorems related to population biology are proved. Random parameters in stochastic difference equations are autocorrelated stationary Gaussian processes in the first case. They are jump-type Markov processes in the second case. We discuss a problem of continuous time approximations for discrete time models in random environments.

Genetics, Population↗

PubMed1967

Neuronal spike trains and stochastic point processes. II. Simultaneous spike trains.

The statistical analysis of two simultaneously observed trains of neuronal spikes is described, using as a conceptual framework the theory of stochastic point processes.The first statistical question that arises is whether the observed trains are independent; statistical techniques for testing independence are developed around the notion that, under the null hypothesis, the times of spike occurrence in one train represent random instants in time with respect to the other. If the null hypothesis is rejected-if dependence is attributed to the trains-the problem then becomes that of characterizing the nature and source of the observed dependencies. Statistical signs of various classes of dependencies, including direct interaction and shared input, are discussed and illustrated through computer simulations of interacting neurons. The effects of nonstationarities on the statistical measures for simultaneous spike trains are also discussed. For two-train comparisons of irregularly discharging nerve cells, moderate nonstationarities are shown to have little effect on the detection of interactions.Combining repetitive stimulation and simultaneous recording of spike trains from two (or more) neurons yields additional clues as to possible modes of interaction among the monitored neurons; the theory presented is illustrated by an application to experimentally obtained data from auditory neurons.A companion paper covers the analysis of single spike trains.

Action Potentials↗

PubMed1998

Two mechanisms underlie processing of stochastic motion stimuli.

We have constructed "limited lifetime" stochastic motion stimuli using Gabor functions instead of dots, thereby controlling the local attributes of spatial frequency and orientation. Human psychophysical data for direction discrimination using these stimuli reveal two qualitatively distinct kinds of processing. For small displacements, direction discrimination performance as a function of displacement is scaled with spatial frequency in a manner consistent with a linear filtering motion mechanism. Motion perception for relatively large displacements is not directly related to the spatial frequency, and is consistent with a nonlinear process which signals motion of contrast envelopes.

Discrimination, Psychological↗

PubMed1996

Two-state stochastic models for memory in ion channels.

AIM: To study quantitatively the memory existing in ion channels. METHODS: Stochastic processes were used to model 2 categories of memory (short-term and long-term) by persisting in the standpoint of two-state, instead of multiple states, but with different transition mechanism. RESULTS: A two-state Markov process with constant transition intensities well fitted the short-term memory and a two-state Markov process within a kind of random environment well fitted the long-term memory. Statistical procedures for parameter estimation were proposed and demonstrated with 2 real examples on the channels of PC12 cells. CONCLUSION: The memory in ion channels can be quantitatively modelled as stochastic process with 2 states.

Adrenal Gland Neoplasms↗

PubMed1991

Stochastic differential equations, their interpretation and application.

The work is recommended to readers with some, maybe heuristic, imagine about stochastic processes that want to meet the notion stochastic differential equation and its interpretation. The notions like Brownian motion and stochastic integral with interpretations in concrete situations in areas of biology and medicine are discussed. The questions are related to mathematical modelling and they may be interpreted in connection with stochastic signal filtering and optimal queuing theory.

Stochastic Processes↗