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1989 | Buch

Computer Models of Individual Living Cells in Cell Populations Kapitel lesen Erstes Kapitel lesen

Mathematical Approaches to Problems in Resource Management and Epidemiology

Proceedings of a Conference held at Ithaca, NY, Oct. 28–30, 1987

herausgegeben von: Carlos Castillo-Chavez, Simon A. Levin, Christine A. Shoemaker

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Biomathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

Increasingly, mathematical methods are being used to advantage in addressing the problems facing humanity in managing its environment. Problems in resource management and epidemiology especially have demonstrated the utility of quantitative modeling. To explore these approaches, the Center of Applied Mathematics at Cornell University organized a conference in Fall, 1987, with the objective of surveying and assessing the state of the art. This volume records the proceedings of that conference. Underlying virtually all of these studies are models of population growth, from individual cells to large vertebrates. Cell population growth presents the simplest of systems for study, and is of fundamental importance in its own right for a variety of medical and environmental applications. In Part I of this volume, Michael Shuler describes computer models of individual cells and cell populations, and Frank Hoppensteadt discusses the synchronization of bacterial culture growth. Together, these provide a valuable introduction to mathematical cell biology.

Inhaltsverzeichnis

Frontmatter

Cell Population Dynamics

Frontmatter
Computer Models of Individual Living Cells in Cell Populations
Abstract
The biosynthetic capacity of an individual cell is dependent on its structure. The response of large population of cells reflects the aggregated response of individual cells. Individual cell’s differ from one and another. The use of population balance equations to describe the dynamic response of populations to perturbations in their environment is computationally difficult when both the structure of individual cells and their distribution within the population are important. We circumvent these computational problems by building highly structured models of individual cells and then using a finite-representation technique to model the whole population. Application of this technique to predicting protein production from recombinant DNA is described.
Michael L. Shuler
Synchronization of Bacterial Culture Growth
Abstract
Synchronization of cell doubling times due to alternating starvation-nutrition cycles is studied here using a method based on nonlinear Lexis diagrams and the assumption that the cell cycle has three phases, pre-replication, replication and post-replication, the middle of which is always of fixed length once started.
F. C. Hoppensteadt

Resource Management

Frontmatter
Biological Resource Modeling—A Brief Survey
Abstract
The use of mathematical models in natural resource management has increased greatly over the past two or three decades, both in response to the perceived need for a more careful shepherding of the renewable resource base upon which life itself depends, and also as a result of improvements in analytical and computational techniques. In this review I first describe two classical resource models, one designed for fisheries and the other for forestry. In subsequent sections I describe some of the major directions in which these models have been extended to become more realistic and more useful. I conclude with a call to resource modelers to extend their outlook to encompass global resource issues.
Colin W. Clark
Mathematical Modeling in Plant Biology: Implications of Physiological Approaches for Resource Management
Abstract
I provide a brief overview of mathematical models that have been developed for particular plant physiological processes, with emphasis on the difficulties involved in taking these upscale to deal with whole plant, crop and forest growth analyses. As the issues addressed by physiologists are often highly reductionist in nature, I point out the gap which has developed between our detailed knowledge of certain physiological processes and our general ignorance of appropriate ways to integrate these processes over whole plant or canopy scales. The importance of accurate integration for crop and forest management is discussed. Finally, I review models for the spread of plant pathogens, and indicate how these may be modified to take account of the spatial nature of plant infection and the continuum of resistance types within a natural population.
Louis J. Gross
Economics, Mathematical Models and Environmental Policy
Abstract
This paper briefly reviews several models of externality which provide the theoretical basis of environmental economics. An externality may be defined as a situation where the output or action of a firm or individual affects the production possibilities or welfare of another firm or individual who has no direct control over the initial level of the output or activity. Pollution, resulting from the disposal of residual wastes, is a classic example of externality.
Three static models examine the optimality conditions for (1) a two-person externality, (2) a many-person externality (where the externality takes the form of a “pure public bad”), and (3) a two-plant polluter. In the case of a two-person externality negotiation between the affected parties may lead to the optimal level for the externality regardless of the initial assignment of property rights. In the many- person case, environmental policies, such as direct controls or economic incentives, may be required to achieve an optimal allocation of resources. Economic incentives may take the form of per unit taxes on emissions or transferable discharge rights. In the third model it is shown how a tax can induce optimal (least cost) treatment from a two-plant polluter.
Two dynamic models examine the cases where (1) a pollution stock may accumulate or degrade according to rates of discharge and biodegradation and (2) a toxic residual must be transported from sites where it is generated to sites where it may be safely stored. The latter problem poses environmental risks from spills in transit or leakage at storage sites.
While radioactive and toxic wastes are likely to continue to be regulated by direct controls some of the more “benign” residuals are suitable for regulation by economics incentives. Effluent taxes in France and the Netherlands, transferable discharge permits on the Fox River in Wisconsin, transferable stove permits in Telluride, Colorado and the EPA’s emission-offset policy are indications that economic incentives will play a greater role in the future management of environmental quality.
Jon M. Conrad
Stochastic Nonlinear Optimal Control of Populations: Computational Difficulties and Possible Solutions
Abstract
Computing optimal control policies for the stochastic nonlinear systems associated with population dynamics is computationally and mathematically difficult. We discuss the characteristics of population management problems arising in forestry, fisheries and pest management and how these characteristics affect the selection of optimal control algorithms and the associated computations. We compare the advantages and disadvantages of differential dynamic programming and stochastic dynamic programming The computational issues associated with linear or nonlinear interpolation for stochastic dynamic programming are discussed. We conclude that for very large-scale nonlinear problems like those arising in forestry, differential dynamic programming would seem to be the most effective method. Most pest control problems require stochastic dynamic programming and nonlinear interpolation. Because of the high degree of parameter uncertainty for fisheries models, these problems require a stochastic procedure like dynamic programming, but in many cases linear interpolation will be adequate.
Christine A. Shoemaker, Sharon A. Johnson
Optimal Evolution of Tree-Age Distribution for a Tree Farm
Abstract
We characterize here the evolution of tree age distribution under optimal forest management. The object is to maximize the discounted sum of the yield in each period, this being an increasing, concave function of the timber harvested. For a family of simple models, it is found that tree age distributions always evolve into either a ‘Faustmann uniform distribution’ or ‘Faustmann cycles’ where all trees are cut at the Faustmann age.
Contrary to the conclusions from the optimal growth literature, cyclical orbits can be optimal under any small but positive discount rate. Furthermore, such cyclicality is structurally stable, due to the reacheability constraints as implied by the underlying vintage asset structure.
Henry Wan Jr.

Infectious Diseases

Frontmatter
Mathematical Models of Infectious Diseases in Multiple Populations
Abstract
Some infectious diseases can exist in several populations or species and transmit from one to another. After a short review of several mathematical models for this phenomenon, it is pointed out that in the model of Holt and Pickering (1985), where the disease-caused mortality is taken into account, all equilibria can be unstable simultaneously and a periodic solution can appear through Hopf bifurcation.
Some infectious diseases can or have to transmit from one species to another (Dobson and May, 1986). For example, malaria infection is caused by single-celled blood parasites, protozoa, in the genus Plasmodia. The disease also appears in birds, reptiles and monkeys. The life cycle of the human malaria parasite involves a phase of asexual growth and development in man and a phase of sexual proliferation in the mosquito. Plague, schistosomiasis, and typhus are some other examples which involve more than one host species. Section 1 is a brief review of the malaria model of Ross (1911), which is perhaps the first mathematical model for an infectious disease in more than one population. In this model and some other host-vector relationship models, only the interpopulation transmission is taken into account. Section 2 points out that a heterogeneous population model of Hethcote (1978) can be considered as a model of infectious disease in n populations, whose population sizes are assumed to be constants. Section 3 introduces a model of Holt and Pickering (1985), where the disease-induced mortality is included and the population sizes are no longer assumed to be constants. This section also shows that all equilibria in this model can be unstable simultaneously and a stable periodic solution can occur through Hopf bifurcation.
Wei-min Liu
Epidemic Models in Populations of Varying Size
Abstract
We consider simple compartmental models for infectious diseases with exposed and infective periods of fixed length. In particular, we examine the effect of incorporating nonlinear population dynamics into such models on threshold phenomena and the stability of endemic equilibria.
Fred Brauer
Stability and Thresholds in Some Age-Structured Epidemics
Abstract
Age-structured models for diseases that can be transmitted both horizontally and vertically are derived and analyzed. The relation between these models and the catalytic curve models of epidemics is explicitly given. The possibility of using the catalytic curve to deduce information about the age-dependent contact rate and to identify the presence of vertical transmission is demonstrated. For certain age-dependent forms of the force of infection terms, explicit endemic threshold criteria are derived, and the stability of the steady states is determined.
Stavros Busenberg, Kenneth Cooke, Mimmo Iannelli
Multiple Time Scales in the Dynamics of Infectious Diseases
Abstract
For an age-structured SIR-model with constant host life span, the dominant eigenvalues of the endemic equilibrium are computed using asymptotic expansions of the stability equation. To the first order the eigenvalues are purely imaginary and the real part of the second order term is negative showing that the endemic equilibrium is always stable. In an age-structured epidemic model with two interacting viral strains that confer partial cross-immunity, purely imaginary eigenvalues exists for some parameter values indicating the possibility of a limit cycle appearing through a Hopf bifurcation.
Viggo Andreasen
A Distributed-Delay Model for the Local Population Dynamics of a Parasitoid-Host System
Abstract
A data-based model for the population dynamics of the parasitoid-host system involving Urolepis rufipes Ashmead and Musca domestica L. is introduced and analyzed. The model uses distributed delays to account for variable development times of parasitoid larvae, and assumes that recruitment of susceptible hosts is unaffected by local dynamics. Using an arbitrary form for the functional response by parasitoids to the abundance of parasitized and unparasitized pupae, we compute the range of fly and parasitoid life history parameters for which the parasitoids persist. The model converges to a stable equilibrium whether or not the parasitoids persist, and the equilibrium level of hosts can be forced to arbitrarily low values by extremely efficient parasitoids. The model is intended to describe the dynamics of a single patch in a system where host dispersal greatly exceeds parasitoid dispersal, and may have implications for biological control under these circumstances.
Fred Adler, Lincoln Smith, Carlos Castillo-Chavez

Acquired Immunodefiency Syndrome (AIDS)

Frontmatter
A Model for HIV Transmission and AIDS
Abstract
The model formulated for transmission of HIV (the AIDS virus) and the subsequent progression to AIDS is a system of nonlinear differential equations. They describe the infection process by interactions within and between risk groups such as homosexual men, bisexual men, female prostitutes, intravenous drug abusers and heterosexually active men and women. The progression to AIDS after infection is modeled by a sequence of stages. A modification of this model is being used to study the transmission of HIV infection and the incidence of AIDS in risk groups in the United States.
Herbert W. Hethcote
The Role of Long Periods of Infectiousness in the Dynamics of Acquired Immunodeficiency Syndrome (AIDS)
Abstract
Single and multiple group models for the spread of HIV (human immunodeficiency virus) are introduced. Partial analytical results for these models are presented for two specific cases. First for a model for which the duration of infectiousness has a negative exponential distribution and second for a model for which all individuals remain infectious for a fixed length of time.
Carlos Castillo-Chavez, Kenneth Cooke, Wenzhang Huang, Simon A. Levin
The Effect of Social Mixing Patterns on the Spread of AIDS
Abstract
Mathematical models of the transmission of the AIDS virus can help us better understand the spread of the AIDS epidemic and prepare for the future. Model explorations can indicate which factors the epidemic is most sensitive to and provide guidance in designing interventions, educational programs and social behavior studies. We explore the sensitivity of a transmission model to different social mixing patterns. This model continuously distributes a homosexual community according to sexual partner change rates and can account for infectivity and conversion times that vary with time since infection. An acceptance function determines which partners are acceptable to an individual and defines the mixing between groups with different partner change rates. We find that if people only select partners with very similar behavior the epidemic grows much slower than if they are not as discriminating. Therefore, understanding social mixing patterns may be one of the most urgent tasks if we are to anticipate the future. We also find that the epidemic is sensitive to variable infectivity and conversion times.
James M. Hyman, E. Ann Stanley
Possible Demographic Consequences of HIV/AIDS Epidemics: II, Assuming HIV Infection does not Necessarily Lead to AIDS
Abstract
It seems likely that mortality associated with HIV/AIDS infections, transmitted horizontally by heterosexual contacts among adults and vertically to the offspring of infected mothers, will have significant demographic effects in Africa, and possibly in other developing countries. Such demographic effects may, indeed, also be significant in the long run among subgroups (such IV-drug abusers), and possibly more generally, in developed countries.
Robert M. May, Roy M. Anderson, Angela R. McLean

Fitting Models to Data

Frontmatter
Fitting Mathematical Models to Biological Data: A Review of Recent Developments
Abstract
A common problem is to fit a theoretical model, y = f(x,β) + error, relating a measured response, y to a measured vector of predictors, x, and an unknown parameter vector, β. Ordinary least—squares is an appropriate fitting method when the variation of y about the model, i.e., the “error”, is normally distributed with a conditional variance, given x, that is constant. In many biological problems, y is nonnegative, right—skewed, and has a conditional variance that is an increasing function of f(x,β). In this paper, two methods for fitting models to such data are discussed. The methods are (1) transformation of the response and the model and (2) weighted least—squares. The rationale behind these methods, estimation techniques, and statistical inference (testing and confidence intervals) are discussed.
David Ruppert
Inverse Problems for Distributed Systems: Statistical Tests and ANOVA
Abstract
In this note we outline some recent results on the development of a statistical testing methodology for inverse problems involving partial differential equation models. Applications to problems from biology and mechanics are presented. The statistical tests are based on asymptotic distributional results for estimators and residuals in a least squares approach.
H. T. Banks, B. G. Fitzpatrick
Small Models are Beautiful: Efficient Estimators are Even More Beautiful
Abstract
There is always a conflict between fidelity to nature and the simplicity of statistical models. This conflict becomes especially acute when data are noisy or uninformative, as is often the case for problems in resource management. In such situations, the most effective model must sacrifice fidelity in order to obtain efficient estimators. This point is illustrated for a simple linear model, which is solved completely. An application from fisheries management also illustrates the virtues of simple models. It also shows how equally simple models can differ sub stantially in their estimation efficiency.
D. Ludwig

Dynamic Properties of Population Models

Frontmatter
Inferring the Causes of Population Fluctuations
Or, chaos meets data in population ecology
Abstract
Many natural populations fluctuate in abundance, in ways that appear random to the eye and to some standard time-series analyses. This paper reviews the methods, and results, of recent attempts to determine if the fluctuations are chaotic rather than random. Early “indirect” methods (fit a model to the data, and see if the model is chaotic) found no evidence of chaos, but the results are sensitive to the choice of model and parameter-estimation methods. “Direct” methods (based on reconstructing trajectories in time-delay coordinates) appear to reveal “strange” (i.e., fractal) attractors that are a hallmark of chaotic dynamics. Reconstruction is model-free, but offers no way of objectively evaluating one’s visual impression of the attractor. Fractal dimension calculations, using a new maximum-likelihood method for short time-series (Ellner 1988), can be used to test for fractal structure (non-integer dimension) if there’s enough data. For some measles incidence data, dimension calculations suggest an attractor with dimension between 2 and 3, supporting the visual impression of a strange attractor underlying the fluctuations.
Stephen Ellner
Stochastic Growth Models: Recent Results and Open Problems
Abstract
In this paper we will describe some recent work in interacting particle systems and state some open problems. We will discuss three models: Richardson’s model, an epidemic model and the contact process.
Richard Durrett
Use Differential Geometry with the Secret Ingredient: Gradients!
Abstract
Amongst the programs for various applications, I would like to insert a sales pitch. The mathematical device I will describe has already served me well and I hope to convince you of its general usefulness.
Ethan Akin
Obstacles to Modelling Large Dynamical Systems
Abstract
This paper describes some of the difficulties in bringing modern dynamical systems theory to bear upon the simulation of dynamical phenomena in complex models of natural processes.
John Guckenheimer

Erratum

Erratum to: Mathematical Modeling in Plant Biology: Implications of Physiological Approaches for Resource Management
Louis J. Gross
Erratum to: Economics, Mathematical Models and Environmental Policy
Jon M. Conrad
Backmatter
Metadaten
Titel
Mathematical Approaches to Problems in Resource Management and Epidemiology
herausgegeben von
Carlos Castillo-Chavez
Simon A. Levin
Christine A. Shoemaker
Copyright-Jahr
1989
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-642-46693-9
Print ISBN
978-3-540-51820-4
DOI
https://doi.org/10.1007/978-3-642-46693-9