James H. Matis

On some stochastic formulations and related statistical moments of pharmacokinetic models

This paper presents the deterministic and stochastic model for a linear compartment system with constant coefficients, and it develops expressions for the mean residence times (MRT) and the variances of the residence times (VRT) for the stochastic model. The expressions are relatively simple computationally, involving primarily matrix inversion, and they are elegant mathematically, in avoiding eigenvalue analysis and the complex domain. The MRT and VRT provide a set of new meaningful response measures for pharmacokinetic analysis and they give added insight into the system kinetics. The new analysis is illustrated with an example involving the cholesterol turnover in rats.

ON APPROXIMATING THE MOMENTS OF THE EQUILIBRIUM DISTRIBUTION OF A STOCHASTIC LOGISTIC MODEL

The classical Verhulst-Pearl logistic equation is a widely used deterministic population growth model. Bartlett, Gower, and Leslie (1960, Biometrika 47, 1-11) develop an analogous stochastic logistic model. They obtain the equilibrium distribution of the population size numerically, and derive elegant approximations for the mean, variance, and skewness of this equilibrium distribution for the model. This paper derives new approximations for these moments. The approximations are simple analytical expressions and in general are more accurate than the Bartlett et al. approximations. The approximations are illustrated with a growth model for Africanized honey bee populations and also with well-known examples in the literature.

The water environs of Okefenokee swamp: An application of static linear environ analysis

Abstract A static water budget model is constructed for the Okefenokee Swamp watershed (swamp plus upland portions) and decomposed into partition units called environs. Flow, storage, intercompartmental transfer and residence time analyses of these units are performed by environ analysis to determine qualitative and quantitative characteristics of Okefenokee ecosystem hydrology. The analysis indicates two relatively discrete hydrologic subsystems within the watershed. One consists of swamp surface water, driven by precipitation over the swamp. The other consists of upland sirface water plus subsurface water in both swamp and uplands, whose source is precipitation on the uplands. Little exchange occurs between these two subsystems.

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.

Generalized stochastic compartmental models with Erlang transit times.

This paper considers the use of semi-Markov process models with Erlang transit times for the description of compartmental systems. The semi-Markov models seem particularly useful for systems with nonhomogeneous “poorly-stirred” compartments. The paper reviews the Markov process models with exponential transit times, and illustrates the application of such models, describing the clearance of calcium in man. The semi-Markov model with Erlang transit times is then developed, and the solutions for its concentration-time curves and residence time moments are given. The use of semi-Markov models is illustrated with the same calcium data, and the results from the two models are compared. The example demonstrates that these semi-Markov models are physiologically more realistic than standard models and may befitted to pharmacokinetic data using readily available software.

Marker concentration patterns of labelled leaf and stem particles in the rumen of cattle grazing bermuda grass ( Cynodon dactylon ) analysed by reference to a raft model

Large (>1600 microm), ingestively masticated particles of bermuda grass (Cynodon dactylon L. Pers.) leaf and stem labelled with 169Yb and 144Ce respectively were inserted into the rumen digesta raft of heifers grazing bermuda grass. The concentration of markers in digesta sampled from the raft and ventral rumen were monitored at regular intervals over approximately 144 h. The data from the two sampling sites were simultaneously fitted to two pool (raft and ventral rumen-reticulum) models with either reversible or sequential flow between the two pools. The sequential flow model fitted the data equally as well as the reversible flow model but the reversible flow model was used because of its greater application. The reversible flow model, hereafter called the raft model, had the following features: a relatively slow age-dependent transfer rate from the raft (means for a gamma 2 distributed rate parameter for leaf 0.0740 v. stem 0.0478 h(-1)), a very slow first order reversible flow from the ventral rumen to the raft (mean for leaf and stem 0.010 h(-1)) and a very rapid first order exit from the ventral rumen (mean of leaf and stem 0.44 h(-1)). The raft was calculated to occupy approximately 0.82 total rumen DM of the raft and ventral rumen pools. Fitting a sequential two pool model or a single exponential model individually to values from each of the two sampling sites yielded similar parameter values for both sites and faster rate parameters for leaf as compared with stem, in agreement with the raft model. These results were interpreted as indicating that the raft forms a large relatively inert pool within the rumen. Particles generated within the raft have difficulty escaping but once into the ventral rumen pool they escape quickly with a low probability of return to the raft. It was concluded that the raft model gave a good interpretation of the data and emphasized escape from and movement within the raft as important components of the residence time of leaf and stem particles within the rumen digesta of cattle.

On the time-dependent reversible stochastic compartmental model—II. A class of n-compartment systems

This paper discusses the solution of a general n-compartment system with time dependent transition probabilities utilizing the technique described by Cardenas and Matis (1975) (hereafter abbreviated (CM)). In addition, the cumulant generating function is derived for a special class of reversible n-compartment systems where the time-dependent intensity coefficients corresponding to the migration and death rates are some multiple of each other. The immigration rates can be any integrable function of time. The moments are also obtained and the solution to the two-compartment system is presented explicitly. The solution is illustrated with a linear and a periodic function which forms have been widely reported in the literature.

On the time-dependent reversible stochastic compartmental model—I. The general two-compartment system

The distribution of the two-compartment, reversible system with time-dependent transitions is proposed and verified. Inasmuch as the required probabilities cannot, in general, be expressed in closed form, a method of approximating these probabilities is described. An example with specific inverse functions of time is presented.

A simple saddlepoint approximation for the equilibrium distribution of the stochastic logistic model of population growth

Abstract The deterministic logistic model of population growth and its notion of an equilibrium ‘carrying capacity’ are widely used in the ecological sciences. Leading texts also present a stochastic formulation of the model and discuss the concept and calculation of an equilibrium population size distribution. This paper describes a new method of finding accurate approximating distributions. Recently, cumulant approximations for the equilibrium distribution of this model were derived [Biometrics 52 (1996) 980], and separately a simple saddlepoint (SP) method of approximating distributions using exact cumulants was presented [J. Math. Appl. Med. Biol. 15 (1998) 41]. This paper proposed using the SP method with the new approximate cumulants, which are readily obtained from the assumed birth and death rates. The method is shown to be quite accurate with three test cases, namely on a classic model proposed by Pielou [Mathematical Ecology, New York, Wiley, p. 304] and on two African bee models proposed previously by the authors [Biometrics 52 (1996) 980; Theor. Popul. Biol. 53 (1998) 16]. Because the new method is also relatively simple to apply, it is expected that its use will lead to a more widespread utilization of the stochastic model in ecological modeling.

Describing the spread of biological populations using stochastic compartmental models with births

This paper derives new models for describing the spread of biological populations in space and time from classical birth-death-migration processes. The spatial aspect is incorporated using compartmental analysis and is developed for two spatial areas (or compartments). The exact bivariate distributions for such processes are intractable; hence approximating distributions are constructed by matching cumulants. A basic Markovian model with exponential waiting times between births is investigated first. The individual effects of swarming, multiple births, and Erlang distributed waiting times, all of which enhance the biological realism, are investigated. A full model which includes all of these effects is then studied. The models are illustrated with observed data on the spread of the Africanized honey bee in French Guiana. A full model with swarming, with an average of 2.64 colonies per swarming episode, and with waiting times following an Erlang distribution with shape parameter 5 is found to provide the best description of the observed data. The methodology is very general and should have broad application for other biological population models involving dispersal and growth.