Showing papers in "Bellman Prize in Mathematical Biosciences in 2002"
TL;DR: A precise definition of the basic reproduction number, R0, is presented for a general compartmental disease transmission model based on a system of ordinary differential equations and it is shown that, if R0<1, then the disease free equilibrium is locally asymptotically stable; whereas if R 0>1,Then it is unstable.
Abstract: A precise definition of the basic reproduction number, Ro, is presented for a general compartmental disease transmission model based on a system of ordinary dierential equations. It is shown that, if Ro 1, then it is unstable. Thus,Ro is a threshold parameter for the model. An analysis of the local centre manifold yields a simple criterion for the existence and stability of super- and sub-threshold endemic equilibria for Ro near one. This criterion, together with the definition of Ro, is illustrated by treatment, multigroup, staged progression, multistrain and vectorhost models and can be applied to more complex models. The results are significant for disease control.
7,106 citations
TL;DR: A generalized form of the logistic growth curve is introduced which incorporates additional growth models which are markedly different from the Logistic growth and its variants, at least in their mathematical representation.
Abstract: A variety of growth curves have been developed to model both unpredated, intraspecific population dynamics and more general biological growth. Most successful predictive models are shown to be based on extended forms of the classical Verhulst logistic growth equation. We further review and compare several such models and calculate and investigate properties of interest for these. We also identify and detail several previously unreported associated limitations and restrictions. A generalized form of the logistic growth curve is introduced which is shown incorporate these models as special cases. The reported limitations of the generic growth model are shown to be addressed by this new model and similarities between this and the extended growth curves are identified. Several of its properties are also presented. We furthermore show that additional growth characteristics are accommodated by this new model, enabling previously unsupported, untypical population dynamics to be modelled by judicious choice of model parameter values alone.
871 citations
TL;DR: It is shown that when the drug efficacy is less than perfect the estimated value of the loss rate of productively infected T cells, delta, is increased when data is fit with delay models compared to the values estimated with a non-delay model.
Abstract: Models of HIV-1 infection that include intracellular delays are more accurate representations of the biology and change the estimated values of kinetic parameters when compared to models without delays. We develop and analyze a set of models that include intracellular delays, combination antiretroviral therapy, and the dynamics of both infected and uninfected T cells. We show that when the drug efficacy is less than perfect the estimated value of the loss rate of productively infected T cells, δ, is increased when data is fit with delay models compared to the values estimated with a non-delay model. We provide a mathematical justification for this increased value of δ. We also provide some general results on the stability of non-linear delay differential equation infection models.
445 citations
TL;DR: Thresholds, equilibria, and their stability are found for SIQS and SIQR epidemiology models with three forms of the incidence and the endemic equilibrium is an unstable spiral for some parameter values and periodic solutions arise by Hopf bifurcation.
Abstract: Thresholds, equilibria, and their stability are found for SIQS and SIQR epidemiology models with three forms of the incidence. For most of these models, the endemic equilibrium is asymptotically stable, but for the SIQR model with the quarantine-adjusted incidence, the endemic equilibrium is an unstable spiral for some parameter values and periodic solutions arise by Hopf bifurcation. The Hopf bifurcation surface and stable periodic solutions are found numerically.
320 citations
TL;DR: The seminal paper by Daniel Bernoulli published in 1766 is put into a new perspective and a new formula for the basic reproduction number is derived which involves the average force of infection, the average case fatality and the life expectancy at the time of infection.
Abstract: The seminal paper by Daniel Bernoulli published in 1766 is put into a new perspective. After a short account of smallpox inoculation and of Bernoulli's life, the motivation for that paper and its impact are described. It determines the age-specific equilibrium prevalence of immune individuals in an endemic potentially lethal infectious disease. The gain in life expectancy after elimination of this cause of death can be explicitly expressed in terms of the case fatality and the endemic prevalence of susceptibles. D'Alembert developed in 1761 an alternative method for dealing with competing risks of death, which is also applicable to non-infectious diseases. Bernoulli's formula for the endemic prevalence of susceptibles has so far escaped attention. It involves the lifetime risk of the infection, the force of infection and the life expectancy at birth. A new formula for the basic reproduction number is derived which involves the average force of infection, the average case fatality and the life expectancy at the time of infection. One can use this estimate to assess the gain in life expectancy if only a fraction of the population is immunized.
281 citations
TL;DR: The numerical simulations seem to suggest that the PVS is slightly more efficient than the continuous vaccination strategy, and it is demonstrated that the above criteria for the LAS guarantee also the GAS.
Abstract: The problem of the applicability of the pulse vaccination strategy (PVS) for the stable eradication of some relevant general class of infectious diseases is analyzed in terms of study of local asymptotic stability (LAS) and global asymptotic stability (GAS) of the periodic eradication solution for the SEIR epidemic model in which is included the PVS. Demographic variations due or not to diseased-related fatalities are also considered. Due to the non-triviality of the Floquet's matrix associate to the studied model, the LAS is studied numerically and in this way it is found a simple approximate (but analytical) sufficient criterion which is an extension of the LAS constraint for the stability of the trivial equilibrium in SEIR model without vaccination. The numerical simulations also seem to suggest that the PVS is slightly more efficient than the continuous vaccination strategy. Analytically, the GAS of the eradication solutions is studied and it is demonstrated that the above criteria for the LAS guarantee also the GAS.
279 citations
TL;DR: It is suggested that heterogeneity in the epidemic will affect the phylogenetic distance distribution of the disease-causing organisms and the small world lattices are investigated, and the effects are even stronger.
Abstract: We consider a spatial model related to bond percolation for the spread of a disease that includes variation in the susceptibility to infection. We work on a lattice with random bond strengths and show that with strong heterogeneity, i.e. a wide range of variation of susceptibility, patchiness in the spread of the epidemic is very likely, and the criterion for epidemic outbreak depends strongly on the heterogeneity. These results are qualitatively different from those of standard models in epidemiology, but correspond to real effects. We suggest that heterogeneity in the epidemic will affect the phylogenetic distance distribution of the disease-causing organisms. We also investigate small world lattices, and show that the effects mentioned above are even stronger.
214 citations
TL;DR: The approach employs classical linear mixed models and operates on a gene-by-gene basis and simultaneously considers the data across all chips in an experiment, which can accommodate complex experiments involving many kinds of treatments and test for their effects at the probe level.
Abstract: We outline and describe steps for a statistically rigorous approach to analyzing probe-level Affymetrix GeneChip data. The approach employs classical linear mixed models and operates on a gene-by-gene basis. Forgoing any attempts at gene presence or absence calls, the method simultaneously considers the data across all chips in an experiment. Primary output includes precise estimates of fold change (some as low as 1.1), their statistical significance, and measures of array and probe variability. The method can accommodate complex experiments involving many kinds of treatments and can test for their effects at the probe level. Furthermore, mismatch probe data can be incorporated in different ways or ignored altogether. Data from an ionizing radiation experiment on human cell lines illustrate the key concepts.
208 citations
TL;DR: A threshold parameter R(*) governing whether or not global epidemics can occur, the probability that a global epidemic occurs and the mean proportion of initial susceptibles ultimately infected by aglobal epidemic are all determined.
Abstract: This paper is concerned with a general stochastic model for susceptible→infective→removed epidemics, among a closed finite population, in which during its infectious period a typical infective makes both local and global contacts. Each local contact of a given infective is with an individual chosen independently according to a contact distribution ‘centred’ on that infective, and each global contact is with an individual chosen independently and uniformly from the whole population. The asymptotic situation in which the local contact distribution remains fixed as the population becomes large is considered. The concepts of local infectious clump and local susceptibility set are used to develop a unified approach to the threshold behaviour of this class of epidemic models. In particular, a threshold parameter R* governing whether or not global epidemics can occur, the probability that a global epidemic occurs and the mean proportion of initial susceptibles ultimately infected by a global epidemic are all determined. The theory is specialised to (i) the households model, in which the population is partitioned into households and local contacts are chosen uniformly within an infective’s household; (ii) the overlapping groups model, in which the population is partitioned in several ways, with local uniform mixing within the elements of the partitions; and (iii) the great circle model, in which individuals are equally spaced on a circle and local contacts are nearest-neighbour.
208 citations
TL;DR: Approximations of quasi-stationary distributions and of times to extinction are derived for stochastic versions of SI, SIS, SIR, and SIRS models, and conditions for validity of the approximations are given.
Abstract: Stochastic models are established and studied for several endemic infections with demography. Approximations of quasi-stationary distributions and of times to extinction are derived for stochastic versions of SI, SIS, SIR, and SIRS models. The approximations are valid for sufficiently large population sizes. Conditions for validity of the approximations are given for each of the models. These are also conditions for validity of the corresponding deterministic model. It is noted that some deterministic models are unacceptable approximations of the stochastic models for a large range of realistic parameter values.
180 citations
TL;DR: Recent Bayesian methods for the analysis of infectious disease outbreak data using stochastic epidemic models are reviewed and rely on Markov chain Monte Carlo methods.
Abstract: Recent Bayesian methods for the analysis of infectious disease outbreak data using stochastic epidemic models are reviewed. These methods rely on Markov chain Monte Carlo methods. Both temporal and non-temporal data are considered. The methods are illustrated with a number of examples featuring different models and datasets.
TL;DR: The concept and impact of optimal average cluster or generalized household size on TB dynamics is discussed and the potential impact of the results on the spread of TB is discussed.
Abstract: Models that incorporate local and individual interactions are introduced in the context of the transmission dynamics of tuberculosis (TB). The multi-level contact structure implicitly assumes that individuals are at risk of infection from close contacts in generalized household (clusters) as well as from casual (random) contacts in the general population. Epidemiological time scales are used to reduce the dimensionality of the model and singular perturbation methods are used to corroborate the results of time-scale approximations. The concept and impact of optimal average cluster or generalized household size on TB dynamics is discussed. We also discuss the potential impact of our results on the spread of TB.
TL;DR: The stochastic stability properties of the deterministic model for the epidemics induced by virulent phages on bacteria in marine environment are investigated both analytically and numerically suggesting that the Deterministic model is robust with respect to Stochastic perturbations.
Abstract: In this paper we extend the deterministic model for the epidemics induced by virulent phages on bacteria in marine environment introduced by Beretta and Kuang [Math. Biosci. 149 (1998) 57], allowing random fluctuations around the positive equilibrium. The stochastic stability properties of the model are investigated both analytically and numerically suggesting that the deterministic model is robust with respect to stochastic perturbations.
TL;DR: This paper considers stochastic epidemics among a population partitioned into households, with mixing locally within households and globally throughout the population, and considers optimality in terms of the cost of the vaccination program.
Abstract: This paper considers stochastic epidemics among a population partitioned into households, with mixing locally within households and globally throughout the population. The two levels of mixing have important implications for the threshold behaviour of the epidemic and consequently for the form and construction of optimal vaccination policies. Optimality is considered in terms of the cost of the vaccination program, the form of which is general enough to include costs of the vaccine itself, its administration, travel to and/or contact with the households. New explicit results are obtained by a constructive method which explain the form of optimal vaccination policies. Numerical studies are presented which exemplify the results discussed.
TL;DR: This paper considers data sets in which a categorical or continuous response is recorded, along with gene expression, on a given number of experimental samples, and presents a dimension reduction strategy that allows it to overcome under-resolution.
Abstract: The analysis of global gene expression data from microarrays is breaking new ground in genetics research, while confronting modelers and statisticians with many critical issues. In this paper, we consider data sets in which a categorical or continuous response is recorded, along with gene expression, on a given number of experimental samples. Data of this type are usually employed to create a prediction mechanism for the response based on gene expression, and to identify a subset of relevant genes. This defines a regression setting characterized by a dramatic under-resolution with respect to the predictors (genes), whose number exceeds by orders of magnitude the number of available observations (samples). We present a dimension reduction strategy that, under appropriate assumptions, allows us to restrict attention to a few linear combinations of the original expression profiles, and thus to overcome under-resolution. These linear combinations can then be used to build and validate a regression model with standard techniques. Moreover, they can be used to rank original predictors, and ultimately to select a subset of them through comparison with a background 'chance scenario' based on a number of independent randomizations. We apply this strategy to publicly available data on leukemia classification.
TL;DR: The extent to which the forms of the infectivity and recovery functions affect the possibility of backward bifurcations is shown, and the SIR and SI models examined do not exhibit this behavior.
Abstract: We consider models for a disease with acute and chronic infective stages, and variable infectivity and recovery rates, within the context of a vaccination campaign. Models for SIRS and SIS disease cycles exhibit backward bifurcations under certain conditions, which complicate the criteria for success of the vaccination campaign by making it possible to have stable endemic states when R 0 . We also show the extent to which the forms of the infectivity and recovery functions affect the possibility of backward bifurcations. SIR and SI models examined do not exhibit this behavior.
TL;DR: It is shown that reactivity is necessary for Turing instability in multispecies systems of reaction-diffusion equations, integrodifference equations, coupled map lattices, and systems of ordinary differential equations.
Abstract: The theory of spatial pattern formation via Turing bifurcations - wherein an equilibrium of a nonlinear system is asymptotically stable in the absence of dispersal but unstable in the presence of dispersal - plays an important role in biology, chemistry and physics. It is an asymptotic theory, concerned with the long-term behavior of perturbations. In contrast, the concept of reactivity describes the short-term transient behavior of perturbations to an asymptotically stable equilibrium. In this article we show that there is a connection between these two seemingly disparate concepts. In particular, we show that reactivity is necessary for Turing instability in multispecies systems of reaction-diffusion equations, integrodifference equations, coupled map lattices, and systems of ordinary differential equations.
TL;DR: An epidemiological model of Mycobacterium tuberculosis infection is presented to investigate the effects of host genetics and demographic factors on epidemic TB and results show that changes in transmission parameters, the fraction of the population genetically susceptible to infection, and demographics strongly affect TB prevalence and incidence rates.
Abstract: There is wide variation in endemic tuberculosis (TB) levels between countries and we seek to identify possible causes of these differences. In this study we present an epidemiological model of Mycobacterium tuberculosis infection to investigate the effects of host genetics and demographic factors on epidemic TB. We discuss the general framework for this approach and present analytical results to identify important parameters affecting steady-state prevalence and incidence rates of TB disease. We then use numerical simulations of our model to observe the effects of a genetically susceptible subpopulation on TB disease dynamics at the population level. Finally, we simulate infection within a genetically heterogeneous population in two demographic settings: India (a typical population with high TB prevalence) and the USA (a typical population with low TB prevalence). Results show that changes in transmission parameters, the fraction of the population genetically susceptible to infection, and demographic factors strongly affect TB prevalence and incidence rates.
TL;DR: Some modern trends in statistical analysis of microarray data with a special focus on statistical classification (pattern recognition) and variable selection are discussed, and the utility of some distances between random vectors and their nonparametric estimates obtained from gene expression data are considered.
Abstract: Lack of adequate statistical methods for the analysis of microarray data remains the most critical deterrent to uncovering the true potential of these promising techniques in basic and translational biological studies. The popular practice of drawing important biological conclusions from just one replicate (slide) should be discouraged. In this paper, we discuss some modern trends in statistical analysis of microarray data with a special focus on statistical classification (pattern recognition) and variable selection. In addressing these issues we consider the utility of some distances between random vectors and their nonparametric estimates obtained from gene expression data. Performance of the proposed distances is tested by computer simulations and analysis of gene expression data on two different types of human leukemia. In experimental settings, the error rate is estimated by cross-validation, while a control sample is generated in computer simulation experiments aimed at testing the proposed gene selection procedures and associated classification rules.
TL;DR: It is predicted that, although an active response can successfully control tumor growth via the deposition of large amounts of collagen, this alone is insufficient for capsule formation, and it is shown that transcapsular spread or invasion of the tumor may be due to the production by the tumor cells of proteases and their subsequent action.
Abstract: In this paper, a mathematical modeling framework is presented which describes the growth, encapsulation, and transcapsular spread of solid tumors. The model is based on the physical forces and cellular interactions involved in tumorigenesis and is used to test and compare the active (foreign body hypothesis) and passive (expansive growth hypothesis) hypotheses of capsule formation, such investigations being ideally suited to our mechanical model. The model simulations lead us to predict that, although an active response can successfully control tumor growth via the deposition of large amounts of collagen, this alone is insufficient for capsule formation. In contrast, a solely passive responsive is capable of producing an encapsulated tumor with minimal accumulation of connective tissue within the tumor. When both responses are active, a denser capsule forms and there is a significant increase in connective tissue within the tumor. Using a modified version of the model, in which tumor cells are assumed to produce degradative proteases at a rate which depends on the pressure they experience, it is also possible to show that transcapsular spread or invasion of the tumor may be due to the production by the tumor cells of proteases and their subsequent action.
TL;DR: This paper generalizes the previous results to density-dependent positive noise intensities of very general form so that they also become independent from the way environmental fluctuations affect population growth rates.
Abstract: In a previous paper [Math. Biosci. 156 (1999) 1], we have studied quite general stochastic differential equation models for the growth of populations subjected to harvesting in a random environment. We have obtained conditions for non-extinction and for the existence of stationary distributions (as well as expressions for such distributions) similar to conditions for non-extinction and for the existence of a stable equilibrium in the corresponding deterministic model. The models were quite general, considering density-dependent natural growth functions and harvesting policies of very general form, so that our results would be model independent and provide minimal requirements for the choice of a wise density-dependent harvesting policy. Those models, however, although quite general on all other respects, have a serious limitation. In fact, the ways environmental fluctuations affect the population per capita growth rate are poorly known and those models only considered two possible ways, namely the noise intensity could be constant or proportional to that rate. To overcome this limitation, in this paper we generalize the previous results to density-dependent positive noise intensities of very general form so that they also become independent from the way environmental fluctuations affect population growth rates.
TL;DR: The asynchronous exponential growth is proved by demonstrating that the strongly continuous linear semi-group associated with the partial differential equations of the model is positive, irreducible, and eventually compact.
Abstract: A model of a proliferating cell population is analyzed. The model distinguishes individual cells by cell age, which corresponds to phase of the cell cycle. The model also distinguishes individual cells by proliferating or quiescent status. The model allows cells to transit between these two states at any age, that is, any phase of the cell cycle. The model also allows newly divided cells to enter quiescence at cell birth, that is, cell age 0. Sufficient conditions are established to assure that the cell population has asynchronous exponential growth. As a consequence of this asynchronous exponential growth the population stabilizes in the sense that the proportion of the population in any age range, or the fraction in proliferating or quiescent state, converges to a limiting value as time evolves, independently of the age distribution and proliferating or quiescent fractions of the initial cell population. The asynchronous exponential growth is proved by demonstrating that the strongly continuous linear semigroup associated with the partial differential equations of the model is positive, irreducible, and eventually compact.
TL;DR: It is shown that the equilibrium level of ATL-cell proliferation is higher when the HTLV-I infection of T cells is chronic than when it is acute, and a unique endemic equilibrium is globally stable in the interior of the feasible region.
Abstract: Mathematical analysis is carried out that completely determines the global dynamics of a mathematical model for the transmission of human T-cell lymphotropic virus I (HTLV-I) infection and the development of adult T-cell leukemia (ATL). HTLV-I infection of healthy CD4(+) T cells takes place through cell-to-cell contact with infected T cells. The infected T cells can remain latent and harbor virus for several years before virus production occurs. Actively infected T cells can infect other T cells and can convert to ATL cells, whose growth is assumed to follow a classical logistic growth function. Our analysis establishes that the global dynamics of T cells are completely determined by a basic reproduction number R(0). If R(0) 1, HTLV-I infection becomes chronic, and a unique endemic equilibrium is globally stable in the interior of the feasible region. We also show that the equilibrium level of ATL-cell proliferation is higher when the HTLV-I infection of T cells is chronic than when it is acute.
TL;DR: The paper presents a method of constructing confidence intervals for mutation rates using data from fluctuation experiments, inspired by a rediscovery of a little-known, not fully developed method of Lea and Coulson.
Abstract: This paper aims at removing certain long-standing impediments to more effective and widespread use of fluctuation analysis. The paper presents a method of constructing confidence intervals for mutation rates using data from fluctuation experiments. The method was inspired by a rediscovery of a little-known, not fully developed method of Lea and Coulson; substantial modifications have been made both to enhance computational efficiency and to widen the scope of the original method's applicability. A computer package named SALVADOR is presented that can be used for Monte Carlo simulation, for point and interval estimation of mutation rates, and for exploration of various hypotheses spawned by the directed mutation controversy. In addition to the maximum likelihood method, methods of considerable historical interest are also examined and included in SALVADOR to help the reader compare and assess some of the most popular methods for estimating mutation rates.
TL;DR: It is found, under reasonably general assumptions, that a unique evolutionarily stable state for virulence, alpha(*), exist for both models, however, the pattern of the invasibility plots depends on the shape of the trade-off (between virulence and transmissibility, or superinfection rates) functions, and on the host demography.
Abstract: Classical models of parasite competition show that coexistence is impossible if different strains give complete cross-immunity. However, parasite coexistence is possible if some of the model assumptions are changed. For instance, coexistence is impossible if density-dependence operates only in hosts' fertility, but surprisingly becomes possible if hosts' mortality is density-dependent. Parasite strains can also coexist if a host already infected with one strain may become infected by another strain (superinfection). I examine here if these reasons for coexistence carry over to evolutionary timescales: in other words, suppose that potentially a continuum of parasite strains may arise by mutations; will evolution arrive at a halt? in that case, will only one or several strains persist? The paradigm and methods of adaptive dynamics are used in this study. It is found, under reasonably general assumptions, that a unique evolutionarily stable state for virulence, alpha(*), exist for both models. However, the pattern of the invasibility plots depends on the shape of the trade-off (between virulence and transmissibility, or superinfection rates) functions, and on the host demography. In many cases, the state alpha(*) is evolutionarily stable only with respect to small mutations, not to larger ones; hence, evolutionary dynamics will bring virulence to alpha(*) only if mutations are sufficiently small; for larger mutations, evolutionary dynamics are more complex and still mainly unresolved.
TL;DR: It is shown that the new model may have a bifurcation at which the unique endemic equilibrium changes the stability and stable periodic solutions exist, quite different from the simpler models.
Abstract: New models for schistosomiasis are developed. These models incorporate several realistic features including drug treatment for human hosts, an infection age in snail hosts, density-dependent birth rate of snails, distribution of schistosomes within human hosts, and disease-induced mortality in both human and snail hosts. The qualitative and quantitative mathematical properties of the models are studied, their biological consequences and some control strategies are discussed, and the results of the new models are compared with those of simpler models. It is shown that the new model may have a bifurcation at which the unique endemic equilibrium changes the stability and stable periodic solutions exist. This is quite different from the simpler models. Explicit thresholds of treatment rate are established above which the infection will be controlled under certain levels. Evaluations of cost-effectiveness are also discussed by analyzing the sensitivity of the mean number of parasites per person to changes of other parameters.
TL;DR: This work analyzes two different second order models for anaerobic waste water treatment processes using two data sets obtained from different experimental setups, and shows that proving structural identifiability of the mathematical models with currently used methods fails.
Abstract: Biochemical reactions can often be formulated mathematically as ordinary differential equations. In the process of modeling, the main questions that arise are concerned with structural identifiability, parameter estimation and practical identifiability. To clarify these questions and the methods how to solve them, we analyze two different second order models for anaerobic waste water treatment processes using two data sets obtained from different experimental setups. In both experiments only biogas production rate was measured which complicates the analysis considerably. We show that proving structural identifiability of the mathematical models with currently used methods fails. Therefore, we introduce a new, general method based on the asymptotic behavior of the maximum likelihood estimator to show local structural identifiability. For parameter estimation we use the multiple shooting approach which is described. Additionally we show that the Hessian matrix approach to compute confidence intervals fails in our examples while a method based on Monte Carlo Simulation works well.
TL;DR: The pure oncogenetic tree model is extended by introducing false positive and false negative observations and it is shown that addition of the error model significantly improves the ability of the model to describe the data.
Abstract: Human solid tumors are believed to be caused by a sequence of genetic abnormalities arising in the tumor cells. The understanding of these sequences is extremely important for improving cancer treatment. Models for the occurrence of the abnormalities include linear structure and a recently proposed tree-based structure. In this paper we extend the pure oncogenetic tree model by introducing false positive and false negative observations. We state conditions sufficient for the reconstruction of the generating tree. As an example we analyze a comparative genomic hybridization data set and show that addition of the error model significantly improves the ability of the model to describe the data.
TL;DR: The authors extend their previous work on compartmental systems without lags to show that, for discrete lags and for a very large class of pdfs of continuous lags, compartmental system with lags are equivalent to larger compartmental Systems without lag.
Abstract: Dynamic models of many processes in the biological and physical sciences give systems of ordinary differential equations called compartmental systems. Often, these systems include time lags; in this context, continuous probability density functions (pdfs) of lags are far more important than discrete lags. There is a relatively complete theory of compartmental systems without lags, both linear and non-linear [SIAM Rev. 35 (1993) 43]. The authors extend their previous work on compartmental systems without lags to show that, for discrete lags and for a very large class of pdfs of continuous lags, compartmental systems with lags are equivalent to larger compartmental systems without lags. Consequently, the properties of compartmental systems with lags are the same as those of compartmental systems without lags. For a very large class of compartmental systems with time lags, one can show that the time lags themselves can be generated by compartmental systems without lags. Thus, such systems can be partitioned into a main system, which is the original system without the lags, plus compartmental subsystems without lags that generate the lags. The latter may be linear or non-linear and may be inserted into main systems that are linear or non-linear. The state variables of the compartmental lag subsystems are hidden variables in the formulation with explicit lags.
TL;DR: The article indicates with three vignettes that non-linear compartment models in the formulation of biochemical systems theory are viable candidates for post-genomic models-of-processes.
Abstract: As we are entering the post-genomic era, models-of-data, such as mining and filtering methods for gene sequences and microarrays and the clustering of co-expressed genes, must be complemented with models-of-processes that explain relationships between genomic information and phenomena at biochemical and physiological levels. Many of these models will have the structure of compartment models, whose conceptualization, identification and analysis will fundamentally benefit from the seminal work of John Jacquez. The article indicates with three vignettes that non-linear compartment models in the formulation of biochemical systems theory are viable candidates for post-genomic models-of-processes.