Showing papers in "Bellman Prize in Mathematical Biosciences in 1989"

Similarity transformation approach to identifiability analysis of nonlinear compartmental models.

Abstract: Through use of the local state isomorphism theorem instead of the algebraic equivalence theorem of linear systems theory, the similarity transformation approach is extended to nonlinear models, resulting in finitely verifiable sufficient and necessary conditions for global and local identifiability. The approach requires testing of certain controllability and observability conditions, but in many practical examples these conditions prove very easy to verify. In principle the method also involves nonlinear state variable transformations, but in all of the examples presented in the paper the transformations turn out to be linear. The method is applied to an unidentifiable nonlinear model and a locally identifiable nonlinear model, and these are the first nonlinear models other than bilinear models where the reason for lack of global identifiability is nontrivial. The method is also applied to two models with Michaelis-Menten elimination kinetics, both of considerable importance in pharmacokinetics, and for both of which the complicated nature of the algebraic equations arising from the Taylor series approach has hitherto defeated attempts to establish identifiability results for specific input functions.

Biological growth and spread modeled by systems of recursions. I. Mathematical theory.

Abstract: We study the asymptotic behavior of solutions to a system of recursions u in+1 = Qi[mu n], i = 1, ..., k. The vector operator Q has the origin theta and a positive vector beta as fixed points and is defined for vector-valued functions bounded between theta and gamma where gamma greater than or equal to beta. In addition, Q is order-preserving, commutes with translation, and is continuous in the topology of uniform convergence on compact subsets. Let theta less than or equal to pi much less than beta, and suppose that for all pi much less than alpha much less than beta, Q(n) alpha]----beta as n----infinity. If u0 much greater than pi on a sufficiently large ball and has bounded support, then un propagates with a speed c*(xi) in the direction of the unit vector xi as n----infinity. In certain cases, c*(xi) can be calculated explicitly. The results generalize those of a scalar equation studied by Weinberger.

Biological growth and spread modeled by systems of recursions. II. biological theory

Abstract: The mathematical theory developed in Part I is applied to a selection-migration model in population genetics with sex-linked locus and to the host-vector or venereal disease epidemic model. In both models, a constant c*(xi) is found for each unit vector xi. The mathematical results imply that under certain initial conditions, the frequency of the advantageous gene in the male and female gametic outputs or the epidemic will spread at a speed c*(xi) in the direction xi as time goes to infinity. Time is measured in discrete nonoverlapping generations. In most cases, we can find a formula for c*(xi).

Like-with-like preference and sexual mixing models.

Abstract: Two new general methods for incorporating like-with-like preference into one-sex mixing models in epidemiology are presented. The first is a generalization of the preferred mixing equation, while the second comprises a transformation of a general preference function for partners of similar sexual activity levels. Both methods satisfy the constraints implicit in a mixing model. The behavior of the transformation preference method is illustrated, and it is compared with the standard proportionate mixing model.

A Green's function method for analysis of oxygen delivery to tissue by microvascular networks.

Abstract: A theoretical model is formulated for analyzing oxygen delivery from an arbitrary network configuration of cylindrical microvessels to a finite region of tissue. In contrast to models based on the classical Krogh cylinder approach, this model requires no a priori assumptions concerning the extent of the tissue region supplied with oxygen by each vessel segment. Steady-state conditions are assumed, and oxygen consumption in the tissue is assumed to be uniform. The nonlinear dissociation characteristics of oxyhemoglobin are taken into account. A computationally efficient Green's function approach is used, in which the tissue oxygen field is expressed in terms of the distribution of source strengths along each segment. The utility of the model is illustrated by analyses of oxygen delivery to a cuboidal tissue region by a single segment and by a six-segment network. It is found that the fractional contribution of the proximal segments to total oxygen delivery increases with decreasing flow rate and metabolic rate.

Mathematical models of tumor growth. IV. Effects of a necrotic core.

Abstract: Two mathematical models for the control of the growth of a tumor by diffusion of mitotic inhibitor are presented. The inhibitor production rate is taken to be uniform in a necrotic core for the first model and in the nonnecrotic region for the second model. Regions of stable and unstable growth are determined, and conclusions are drawn about the limiting peripheral widths of stable tissue growth for both models. Comparisons of the results from the two models indicate that the models are sensitive to the source distributions of inhibitor production.

A comparison of variant theories of intact biochemical systems. I. enzyme-enzyme interactions and biochemical systems theory

Abstract: The need for a well-structured theory of intact biochemical systems becomes increasingly evident as one attempts to integrate the vast knowledge of individual molecular constituents, which has been expanding for several decades. In recent years, several apparently different approaches to the development of such a theory have been proposed. Unfortunately, the resulting theories have not been distinguished from each other, and this has led to considerable confusion with numerous duplications and rediscoveries. Detailed comparisons and critical tests of alternative theories are badly needed to reverse these unfortunate developments. In this paper we (1) characterize a specific system involving enzyme-enzyme interactions for reference in comparing alternative theories, and (2) analyze the reference system by applying the explicit S-system variant within biochemical systems theory (BST), which represents a fundamental framework based upon the power-law formalism and includes several variants. The results provide the first complete and rigorous numerical analysis within the power-law formalism of a specific biochemical system and further evidence for the accuracy of the explicit S-system variant within BST. This theory is shown to represent enzyme-enzyme interactions in a systematically structured fashion that facilitates analysis of complex biochemical systems in which these interactions play a prominent role. This representation also captures the essential character of the underlying nonlinear processes over a wide range of variation (on average 20-fold) in the independent variables of the system. In the companion paper in this issue the same reference system is analyzed by other variants within BST as well as by two additional theories within the same power-law formalism—flux-oriented and metabolic control theories. The results show how all these theories are related to one another.

Strategies for representing metabolic pathways within biochemical systems theory: reversible pathways.

Abstract: The search for systematic methods to deal with the integrated behavior of complex biochemical systems has over the past two decades led to the proposal of several theories of biochemical systems. Among the most promising is biochemical systems theory (BST). Recent comparisons of this theory with several others that have recently been proposed have demonstrated that all are variants of BST and share a common underlying formalism. Hence, the different variants can be precisely related and ranked according to their completeness and operational utility. The original and most fruitful variant within BST is based on a particular representation, called an S-system (for synergistic and saturable systems), that exhibits many advantages not found among alternative representations. Even within the preferred S-system representation there are options, depending on the method of aggregating fluxes, that become especially apparent when one considers reversible pathways. In this paper we focus on the paradigm situation and clearly distinguish the two most common strategies for generating an S-system representation. The first is called the “reversible” strategy because it involves aggregating incoming fluxes separately from outgoing fluxes for each metabolite to define a net flux that can be positive, negative, or zero. The second is the “irreversible” strategy, which involves aggregating forward and reverse fluxes through each reaction to define a net flux that is always positive. This second strategy has been used almost exclusively in all variants of BST. The principal results of detailed analyses are the following: (1) All S-system representations predict the same changes in dependent concentrations for a given change in an independent concentration. (2) The reversible strategy is superior to the irreversible on the basis of several criteria, including accuracy in predicting steady-state flux, accuracy in predicting transient responses, and robustness of representation. (3) Only the reversible strategy yields a representation that is able to capture the characteristic feature of amphibolic pathways, namely, the

Use of implicit methods from general sensitivity theory to develop a systematic approach to metabolic control. II. Complex systems.

Abstract: In the accompanying paper (Cascante et al., this issue) we have used general sensitivity theory to develop a matrix algebra that, in the case of sequential reactions, directly relates global and local properties of a given system. In complex biochemical systems this direct relationship is not possible due to the existence of linear dependencies among fluxes and among metabolite concentrations (conserved aggregate concentrations in BST or moiety-conserved concentrations in MCT). In this paper our matrix algebra is applied to conserved cycles and branched pathways, and it is shown that with minor modifications it again relates global properties to the local properties of the enzymes in the system. In the case of conserved cycles, elasticities become modified due to the existence of linear dependencies among the concentration variables in the cycle. In branched pathways, new matrix elements involving ratios of fluxes appear. With these modifications, one can show that the so-called theorems of metabolic control theory specific to these types of pathways are special cases of more general relationships. Rules for the construction of matrices relating global and local properties are given that apply to an arbitrary system of cycles and branches. The implicit approach developed in these papers, which is a generalization of that used in MCT, allows one to make more direct comparisons with the general explicit approach originally developed in BST.

Use of implicit methods from general sensitivity theory to develop a systematic approach to metabolic control. I. Unbranched pathways.

Abstract: It is shown that metabolic control theory (MCT), is its present form, is a particular case of general sensitivity theory, which studies the effects of parameter variations on the behavior of dynamic systems. It has been shown that metabolic control theory is obtained from this more general theory for the particular case of steady-state and linear relationships between velocities and enzyme concentrations. In such conditions the relationships between elasticities and flux control coefficients are easily obtained. These relationships are in the form of a matrix product constructed in a priori form. Relationships between combined response coefficients and concentration control coefficients are presented. The use of implicit methodology from general sensitivity theory provides a generalization of MCT, which is applied to unbranched pathways. For this particular case, provided the matrices have been properly constructed, the matrix of global properties (flux and concentration control coefficients) can be obtained by inversion of the matrix of local properties (elasticities). The theorems of MCT (concentration summation, flux summation, flux connectivity, and concentration connectivity) applicable for unbranched pathways are directly obtained by inspection of the matrix product. With these results, the present theoretical basis of MCT is extended with a more structured framework that allows a wider range of application. The results make clearer the relatedness of MCT to the more general approach provided by biochemical systems theory (BST).

Modeling malaria vaccines. II: Population effects of stage-specific malaria vaccines dependent on natural boosting.

Abstract: Population effects of malaria vaccination programs will depend on the stage specificity of the vaccine, its duration of effectiveness, whether it is responsive to natural boosting, the proportion vaccinated, and the preexisting endemic conditions. This paper develops models of infection-blocking (sporozoite), disease-modifying (merozoite), and transmission-blocking (gametic) vaccines. It explores numerically their different effects on prevalence of infection and disease when utilized in different types of immunization programs at various levels of coverage. Simulations show that possible qualitative consequences of malaria vaccination programs include decreased prevalence of infection and disease and decreased prevalence of infection without a corresponding decrease in prevalence of disease. Epidemics, either one-time or cyclical, could occur. These effects could be accompanied by changes in the age distribution of disease. Finally, vaccination could contribute to elimination of transmission. The duration of effectiveness of the malaria vaccine relative to the duration of natural immunity could have important consequences for the unvaccinated. The problem of predicting a threshold for elimination of transmission is discussed.

A comparison of variant theories of intact biochemical systems. II. Flux-oriented and metabolic control theories.

Abstract: In the past two decades, several theories, all ultimately based upon the same power-law formalism, have been proposed to relate the behavior of intact biochemical systems to the properties of their underlying determinants. Confusion concerning the relatedness of these alternatives has become acute because the implications of these theories have never been compared. In the preceding paper we characterized a specific system involving enzyme-enzyme interactions for reference in comparing alternative theories. We also analyzed the reference system by using an explicit variant that involves the S-system representation within biochemical systems theory (BST). We now analyze the same reference system according to two other variants within BST. First, we carry out the analysis by using an explicit variant that involves the generalized mass action representation, which includes the flux-oriented theory of Crabtree and Newsholme as a special case. Second, we carry out the analysis by using an implicit variant that involves the generalized mass action representation, which includes the metabolic control theory of Kacser and his colleagues as a special case. The explicit variants are found to provide a more complete characterization of the reference system than the implicit variants. Within each of these variant classes, the S-system representation is shown to be more mathematically tractable and accurate than the generalized mass action representation. The results allow one to make clear distinctions among the variant theories.

Diffusion-mediated persistence in two-species competition Lotka-Volterra model.

Abstract: We consider a system composed of two Lotka-Volterra patches connected by diffusion. Each patch has two competitors. Conditions for persistence of the system are given. It is proved that the system can be made persistent under appropriate diffusion coefficients ensuring the instability of boundary equilibria, even if each species is not persistent within each patch. The choice of the coefficients depends closely on the patch dynamics without diffusion.

The effect of a random initial value in neural first-passage-time models.

Abstract: The effect of a random initial value is examined in several stochastic integrate-and-fire neural models with a constant threshold and a constant input. The three models considered are approximations of Stein's model, namely: (1) a leaky integrator with deterministic trajectories, (2) a Wiener process with drift, and (3) an Ornstein-Uhlenbeck process. For model 1, different distributions for the initial value lead to commonly observed interspike interval distributions. For model 2, a discrete and a uniform distribution for the initial value are examined along with some parameter estimation procedures. For model 3, with a truncated normal distribution for the initial value, the coefficient of variation is shown to be greater than 1, and as the threshold becomes large the first-passage-time distribution approaches an exponential distribution. The relationships among the models and between them and previous models are also discussed, along with the robustness of the model assumptions and methods of their verification. The effects of a random initial value are found to be most pronounced at high firing rates.

A mathematical model of facultative mutualism with populations interacting in a food chain

Abstract: Facultative mutualism with populations interacting in a food chain is modeled by a system of four autonomous ordinary differential equations. Two cases are considered: mutualism with the prey and mutualism with the first predator. In both cases persistence and extinction criteria are developed in terms of the invariant flows on the boundaries.

Modeling malaria vaccines. I: New uses for old ideas.

Abstract: Starting from a modification of the model of malaria transmission developed for the Garki project, this paper develops a model containing variables relevant to the simulation of malaria vaccination programs. Modifications include (1) integration of maintenance of immunity dependent on boosting and the possibility of loss immunity; (2) introduction of a boosting factor distinct from susceptibility to infection; (3) reinterpretation of the epidemiological compartments of positive immunes and nonimmunes in terms of severity of disease rather than just infection; (4) interpretation of the different stage-specific levels of immunity; (5) discrimination between different susceptibilities for the immune and nonimmune classes; (6) reformulation of the expression for acquisition of immunity to be biologically more acceptable. Simulations using the Garki model, Nedelman's modification of it, and our Basic model compare the similarities and differences in the predictive behavior of the models. Simulations using the Basic model reproduce observed periodic fluctuations of malaria attributed to the interplay of transmission-blocking immunity and loss of immunity in the absence of boosting in areas of unstable malaria transmission.

An epidemiological model for direct and indirect transmission of typhoid fever

Abstract: The modified SIS epidemiological model considers the usual direct transmission (short cycle) and indirect transmission (long cycle) of typhoid fever. Thresholds are determined, and the equilibrium points are shown to be globally stable. Local stability of the equilibrium points is shown in the corresponding model with vaccines. After estimating parameters using current statistical data for typhoid fever in Chile, computer simulations are used to obtain the numerical behavior of this disease and to estimate the effect of several control policies.

Conditions for uniqueness of limit cycles in general predator-prey systems

Abstract: Using a transformation to a generalized Lienard system, theorems are presented that give conditions under which unique limit cycles for generalized ecological systems, including those of predator-prey form, exist The generalized systems contain those studied by Rosenzweig and MacArthur (1963); Hsu, Hubbell, and Waltman (1978); Kazarinnoff and van den Driessche (1978); Cheng (1981); Liou and Cheng (1987); and Kuang and Freedman (1988) Although very similar in approach to the result presented by Kuang and Freedman, the conditions presented here are of simpler form and in terms of the original (untransformed) functions The results also apply to more general growth terms for the prey as shown in the examples provided In particular, an immigration term is allowable

Analysis of fractal ion channel gating kinetics: kinetic rates, energy levels, and activation energies.

Abstract: Ion channels in the cell membranes of the corneal endothelium, hippocampal neurons, and fibroblasts, and gramicidin channels in lipid bilayers have open and closed times that can be fit, in whole or part, by power law distributions. That is, the gating is self-similar when viewed at different time scales. Hence, kinetic processes at slow and fast time scales are not independent but rather are interrelated. To study how such a relationship can arise we analyze a closed ⇌ open channel with the fractal dimension for leaving the closed state D CO ≈2 and the fractal dimension for leaving the open state D OC ≈1. This special case can be analyzed because it can be represented by equivalent Markov processes. We show that it is equivalent to Markov chains with forward and backward kinetic rate constants approximately equal at each stage, and forming an approximate geometric progression along the different stages. These kinetic rates determine the energy levels and activation energy barriers separating those levels. We find that there are many conformational states (substates) separated by high activation energy barriers. This is similar to the energy structure found for globular proteins such as myoglobin. However, the novel feature reported here is that the activation energy barriers are not independent but are interrelated and form an arithmetic progression. Because of this relationship the fast processes across the low activation energy barriers are linked to slow processes across the high activation energy barriers.

Persistence of an infectious disease in a subdivided population.

Abstract: The transmission dynamics of a communicable disease in a subdivided population where the spread among groups follows the proportionate mixing model while the within-group transmission can correspond to prefered mixing, proportionate mixing among subgroups, or mixing between social and nonsocial subgroups, is analyzed. It is shown that the threshold condition for the disease to persist is that either (i) the disease can persist within at least one group through intragroup contacts, or—if (i) does not hold—(ii) the intergroup transmission is sufficiently high. The among-group transmission is computed as an average where each subgroup's reproductive number is weighted according to its intragroup activity level squared and the total number of cases that one infectious individual will cause through intragroup contacts. The model thus allows for a study of the relative importance of communitywide disease transmission and of disease transmission within geographically or socially separate groups.

Successful invasion of a food web in a chemostat.

Abstract: A food web in a chemostat is considered in which an arbitrary number of competitor populations compete for a single, essential, nonreproducing, growth-limiting substrate, and an arbitrary number of predator populations prey on some or all of the competitor populations. Although any number of predator populations may prey on the same competitor population, each predator population preys on only one competitor population. The dynamics of substrate uptake is modeled by Lotka-Volterra or Michaelis-Menten (Holling type I or II), but the dynamics of competitor uptake is restricted to Lotka-Volterra. Based on certain parameters, the model predicts the asymptotic survival or extinction of each of the different populations and suggests how competitor and/or predator populations could successfully invade the chemostat with or without causing a diverse ecosystem to crash. Similarly, it suggests how the elimination of certain populations could result in a more diverse or less diverse system.

Population diffusion in a two-patch environment.

Abstract: A model of a single-species population diffusing in a two-patch environment is proposed. It is shown that there exists a positive, monotonic, continuous steady-state solution with continuous flux, in the cases of both reservoir and no-flux boundary conditions, that is asymptotically stable. In the case of patches with equal carrying capacities, it is shown that the uniform steady state is globally asymptotically stable.

Sampling properties of the asymptotic behavior of age- or stage-grouped population models.

Abstract: The discrete-time linear recurrent models proposed by Leslie in 1945 and Usher in 1966 for age-or stage-grouped populations are discussed with emphasis on the random nature, due to sampling variations, of their well-known asymptotic behavior. The statistical properties of the estimated asymptotic multiplication rate, stage, or age stable structures and mean generation time are inspected by both a theoretical approach and a simulation procedure. Illustrative case studies provide some order of magnitudes of the sampling bias and variance of these statistics.

A density-dependent Leslie matrix model

Abstract: A density-dependent Leslie matrix model introduced in 1948 by Leslie is mathematically analyzed. It is shown that the behavior is similar to that of the constant Leslie matrix. In the primitive case, the density-dependent Leslie matrix model has an asymptotic distribution corresponding to the logistic equation. However, in the imprimitive case, the asymptotic distribution is periodic, with period depending on the imprimitivity index.

Two predators feeding on two prey species: a result on permanence.

Abstract: For biological populations the precise asymptotic behavior of the corresponding dynamic system is probably less important than the question of extinction and survival of species. An ecological differential equation is called permanent if there exists some level k >0 such that if the number x i (0) of species i at time 0 is positive for i =1,2, … , n then x i ( t )> k for all sufficiently large times t Characterizations for permanence in a four-species prey-predator system modeled by the Lotka-Volterra equation are presented. The method used is based on a combination of two well-known approaches to dealing with permanence. An interesting feature is the occurrence of heteroclinic cycles.

On the formulation of discrete-time epidemic models.

Abstract: Jacquez constructed a properly posed, more general model for the Reed-Frost epidemic process by assuming independent behaviors for the susceptibles and introducing the generating function for the number of contacts per person. An alternative approach is proposed here that relies on similar hypotheses for the infectives and allows the usual chain-binomial structure of the infection process to be extended. For this new model, the derivation of the final size and the threshold phenomenon becomes much simpler. A detailed analysis and its generalization to heterogeneous populations and continuous-time models will be the subject of a forthcoming paper.

Weak ergodicity of population evolution processes

Abstract: The weak ergodic theorems of mathematical demography state that the age distribution of a closed population is asymptotically independent of the initial distribution. In this paper, we provide a new proof of the weak ergodic theorem of the multistate population model with continuous time. The main tool to attain this purpose is a theory of multiplicative processes, which was mainly developed by Garrett Birkhoff, who showed that ergodic properties generally hold for an appropriate class of multiplicative processes. First, we construct a general theory of multiplicative processes on a Banach lattice. Next, we formulate a dynamical model of a multistate population and show that its evolution operator forms a multiplicative process on the state space of the population. Subsequently, we investigate a sufficient condition that guarantees the weak ergodicity of the multiplicative process. Finally, we prove the weak and strong ergodic theorems for the multistate population and resolve the consistency problem.

Hopf bifurcation and stable limit cycle behavior in the spread of infectious disease, with special application to fox rabies.

Abstract: The problem of disease spreading in any population is considered, and necessary and sufficient conditions are obtained for the occurrence of stable limit cycles

A survival model and estimation of time to tumor.

Abstract: In this paper we introduce a stochastic model of survival distribution, where the mortality intensity is a function of the accumulated effect of an individual's continuous exposure to toxic material in the environment (absorbing coefficient) and his biological reaction to the toxin absorbed (discharging coefficient). Formulas for the density function, the distribution function, and the expectation of lifetime are presented. The paper also includes special cases where there is a change in exposure level or exposure is discontinued or exposure is discrete in time. The model is then applied to the NCTR's serial sacrifice experimental study on mice fed 2-AAF, including some mice whose feeding was discontinued. The random variable here is the time to tumor. The chi-square test shows a good fit of the model to the data (P = 0.365). In addition to the parameters and their standard errors, estimates are computed for the expectation, variance, and percentiles of time to tumor, and for the age-specific cancer incidence rates. Confidence intervals for the parameters are also given.