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Compartmental models with multiple sources of stochastic variability: The one-compartment models with clustering

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Abstract

Previous compartmental models have introduced variability either at the particle or at the replicate level. This paper integrates both types of variability through the concept of clustering. The paper develops two different, general clustered models, each with time-dependent hazard rates for the clusters and for the particles within the clusters, and each with random initial number and sizes of clusters. The coefficient of variation of the total number of particles,CV[X(t)], for either model is shown to be bounded below, under very broad conditions, by the coefficient of variation of the initial number of clusters,CV[c(0)]. This high relative variability of the clustered models makes them potentially very useful in kinetic modeling. In many applications, binding and clustering are common phenomena, and two applications of the models to such phenomena are breifly outlined.

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Literature

  • Bernard, S. R. 1977. “An Urn Model Study of Variability within a Compartment.”Bull. Math. Biol. 39, 463–470.

    Article  MATH  MathSciNet  Google Scholar 

  • Ellis, W. C., J. H. Matis and C. Lascano. 1979. “Quantitating Ruminal Turnover.”Fedn Proc. Fedn Am. Socs exp. Biol. 38, 2702–2706.

    Google Scholar 

  • Evans, E. W., G. R. Pearce, J. Burnett and S. L. Pillinger. 1973. “Changes in some Physical Characteristics of the Digesta in the Reticulo-Rumen of Cows Fed Once Daily.”Br. J. Nutr. 29, 357–376.

    Article  Google Scholar 

  • Gross, A. J. and V. A. Clark. 1975.Survival Distributions: Reliability Applications in the Biomedical Sciences. New York: Wiley.

    MATH  Google Scholar 

  • Johnson, N. L. and S. Kotz. 1969.Discrete Distributions. Boston: Houghton Mifflin.

    MATH  Google Scholar 

  • Matis, J. H. 1972. “Gamma Time-Dependency in Blaxter's Compartmental Model.”Biometrics 28, 597–602.

    Article  Google Scholar 

  • Matis, J. H. and B. C. Patten. 1980. “Environ Analysis of Linear Compartmental Systems: the Static, Time Invariant Case.”Int. Statist. Inst. Bull., (to appear).

  • — and B. C. Tolley. 1979. “Compartmental Models with Multiple Sources of Variability: the One-Compartment, Time-Invariant Hazard Rate Case.”Bull. Math. Biol. 41, 491–515.

    Article  MATH  MathSciNet  Google Scholar 

  • — and —. 1980. “On the Stochastic Modeling of Tracer Kinetics.”Fedn Proc. Fedn Am. Socs exp. Biol. 39, 104–109.

    Google Scholar 

  • — and T. E. Wehrly. 1979a “Stochastic Models of Compartmental Systems.”Biometrics 35, 199–220.

    Article  MATH  Google Scholar 

  • — and —. 1979b. “An Approach to a Compartmental Model with Multiple Sources of Stochasticity for Modeling Ecological Systems.” InCompartmental Analysis of Ecosystem Models. Eds. J. H. Matis, B. C. Patten and G. C. White, pp. 195–222. Burtonsville, MD: International Co-operative Publishing House.

    Google Scholar 

  • McInnis, B. C., S. A. El-Asfouri and S. A. Kapadia. 1979. “On Stochastic Compartmental Modeling.”Bull. Math. Biol. 41, 611–613.

    Article  MathSciNet  Google Scholar 

  • Patten, B. C., R. W. Bosserman, J. T. Finn and W. G. Cale. 1976. “Propagation of Cause in Ecosystems.” InSystems Analysis and Simulation in Ecology, Ed. B. C. Patten, Vol. 4, pp. 457–579. New York: Academic Press.

    Google Scholar 

  • Purdue, P. 1979. “Stochastic Compartmental Models: a Review of the Mathematical Theory with Ecological Applications.” InCompartmental Analysis of Ecosystem Models, Ed. J. H. Matis, B. C. Patten and G. C. White, pp. 223–260. Burtonsville, MD: International Co-operative Publishing House.

    Google Scholar 

  • — 1981. “Variability in a Single Compartment System: A Note on S. R. Bernard's Model.”Bull. Math. Biol. 43, 111–116.

    Article  MATH  MathSciNet  Google Scholar 

  • Rescigno, A. and J. H. Matis. 1981. “On the Relevance of Stochastic Compartmental Models to Pharmacokinetic Systems.”Bull. Math. Biol. 43, 245–247.

    Article  MathSciNet  Google Scholar 

  • Troelsen, J. E. and J. B. Campbell. 1968. “Voluntary Consumption of Forage by Sheep and Its Relation to the Size and Shape of Particles in the Digestive Tract.”Anim. Prod. 10, 289–296.

    Article  Google Scholar 

  • Wise, M. E. 1979. “The Need for Rethinking on Both Compartments and Modeling.” InCompartmental Analysis of Ecosystem Models, Ed. J. H. Matis, B. C. Patten and G. C. White, pp. 279–293. Burtonsville, MD: International Co-operative Publishing House.

    Google Scholar 

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Authors and Affiliations

  1. Institute of Statistics, Texas A&M University, 77843, College Station, TX, USA

    James H. Matis & Thomas E. Wehrly

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  1. James H. Matis
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  2. Thomas E. Wehrly
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Matis, J.H., Wehrly, T.E. Compartmental models with multiple sources of stochastic variability: The one-compartment models with clustering. Bltn Mathcal Biology 43, 651–664 (1981). https://doi.org/10.1007/BF02458415

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  • DOI: https://doi.org/10.1007/BF02458415

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