Showing papers in "Bellman Prize in Mathematical Biosciences in 1999"
TL;DR: A SEIR model for the transmission of an infectious disease that spreads in a population through direct contact of the hosts is studied and a unique endemic equilibrium state is shown to be globally asymptotically stable in the interior of the feasible region.
Abstract: A SEIR model for the transmission of an infectious disease that spreads in a population through direct contact of the hosts is studied The force of infection is of proportionate mixing type A threshold sigma is identified which determines the outcome of the disease; if sigma 1, the infected fraction persists and a unique endemic equilibrium state is shown, under a mild restriction on the parameters, to be globally asymptotically stable in the interior of the feasible region Two other threshold parameters sigma' and sigma are also identified; they determine the dynamics of the population sizes in the cases when the disease dies out and when it is endemic, respectively
580 citations
TL;DR: Two simple models are used to examine the relative importance of different stages of infection and different chronic levels of virus to the spreading of the disease and suggest that a small subset of infected people may be responsible for a disproportionate number of infections.
Abstract: Recent studies of HIV RNA in infected individuals show that viral levels vary widely between individuals and within the same individual over time. Individuals with higher viral loads during the chronic phase tend to develop AIDS more rapidly. If RNA levels are correlated with infectiousness, these variations explain puzzling results from HIV transmission studies and suggest that a small subset of infected people may be responsible for a disproportionate number of infections. We use two simple models to study the impact of variations in infectiousness. In the first model, we account for different levels of virus between individuals during the chronic phase of infection, and the increase in the average time from infection to AIDS that goes along with a decreased viral load. The second model follows the more standard hypothesis that infected individuals progress through a series of infection stages, with the infectiousness of a person depending upon his current disease stage. We derive and compare threshold conditions for the two models and find explicit formulas of their endemic equilibria. We show that formulas for both models can be put into a standard form, which allows for a clear interpretation. We define the relative impact of each group as the fraction of infections being caused by that group. We use these formulas and numerical simulations to examine the relative importance of different stages of infection and different chronic levels of virus to the spreading of the disease. The acute stage and the most infectious group both appear to have a disproportionate effect, especially on the early epidemic. Contact tracing to identify superspreaders and alertness to the symptoms of acute HIV infection may both be needed to contain this epidemic.
256 citations
TL;DR: This review article examines four most important formulations, focusing on important practical issues closely linked with the distribution of the number of mutants, including the probability generating functions, moments, computational methods and asymptotics.
Abstract: The Luria-Delbruck mutation model has been mathematically formulated in a number of ways. This review article examines four most important formulations, focusing on important practical issues closely linked with the distribution of the number of mutants. These issues include the probability generating functions, moments (cumulants), computational methods and asymptotics. This review emphasizes basic principles which not only help to unify existing results but also allow for a few useful extensions. In addition, the review offers a historical perspective and some new explanations of divergent moments.
155 citations
TL;DR: Numerical simulations of a one-dimensional lamella reveal that even this extremely simplified model is capable of producing several typical features of cell motility, including periodic 'ruffle' formation, protrusion-retraction cycles, centripetal flow and cell-substratum traction forces.
Abstract: The motion of amoeboid cells is characterized by cytoplasmic streaming and by membrane protrusions and retractions which occur even in the absence of interactions with a substratum. Cell translocation requires, in addition, a transmission mechanism wherein the power produced by the cytoplasmic engine is applied to the substratum in a highly controlled fashion through specific adhesion proteins. Here we present a simple mechano-chemical model that tries to capture the physical essence of these complex biomolecular processes. Our model is based on the continuum equations for a viscous and reactive two-phase fluid model with moving boundaries, and on force balance equations that average the stochastic interactions between actin polymers and membrane proteins. In this paper we present a new derivation and analysis of these equations based on minimization of a power functional. This derivation also leads to a clear formulation and classification of the kinds of boundary conditions that should be specified at free surfaces and at the sites of interaction of the cell and the substratum. Numerical simulations of a one-dimensional lamella reveal that even this extremely simplified model is capable of producing several typical features of cell motility. These include periodic 'ruffle' formation, protrusion-retraction cycles, centripetal flow and cell-substratum traction forces.
149 citations
TL;DR: An approximation is derived for the quasi-stationary distribution of the stochastic logistic epidemic in the intricate case where the transmission factor R0 lies in the transition region near the deterministic threshold value 1.
Abstract: An approximation is derived for the quasi-stationary distribution of the stochastic logistic epidemic in the intricate case where the transmission factor R0 lies in the transition region near the deterministic threshold value 1. An approximation for the expected time to extinction from quasi-stationarity in the same parameter region is also given. Mathematics subject classification: 60J27; 92D30
126 citations
TL;DR: Models for the spread of an SIS epidemic among a population consisting of m households, each containing n individuals, are considered and their behaviour is analysed under the practically relevant situation when m is large and n small.
Abstract: Models for the spread of an SIS epidemic among a population consisting of m households, each containing n individuals, are considered and their behaviour is analysed under the practically relevant situation when m is large and n small. A threshold parameter R ∗ is determined. For the stochastic model it is shown that the epidemic has a non-zero probability of taking off if and only if R ∗ >1 , and the extension to unequal household sizes is also considered. For the deterministic model, with households of size 2, it is shown that if R ∗ ⩽1 then the epidemic dies out, whilst if R ∗ >1 the epidemic settles down to an endemic equilibrium. The usual basic reproductive ratio R0 does not provide a good indicator for the behaviour of these household epidemic models unless the household size n is large.
120 citations
TL;DR: Cannibalism can be a destabilizing force in a predator-prey system if the mortality rate of juveniles is high and/or the recruitment rate to the mature population is low, but a loss of stability by a Hopf bifurcation will take place as the level of cannibalism increases.
Abstract: The dynamics of a predator-prey system, where the predator has two stages, a juvenile stage and a mature stage, are modelled by a system of three ordinary differential equations. The mature predators prey on the juvenile predators in addition to the prey. If the mortality rate of juveniles is low and/or the recruitment rate to the mature population is high, then there is a stable equilibrium with all three population sizes positive. On the other hand, if the mortality rate of juveniles is high and/or the recruitment rate to the mature population is low, then the equilibrium will be stable for low levels of cannibalism, but a loss of stability by a Hopf bifurcation will take place as the level of cannibalism increases. Numerical studies indicate that a stable limit cycle appears. Cannibalism can therefore be a destabilizing force in a predator-prey system.
110 citations
TL;DR: An epidemiological model consisting of a linear chain of three cocirculating influenza A strains that provide hosts exposed to a given strain with partial immune cross-protection against other strains is reduced to a six-dimensional kernel capable of showing self-sustaining oscillations at relatively high levels of cross- protection.
Abstract: We analyze an epidemiological model consisting of a linear chain of three cocirculating influenza A strains that provide hosts exposed to a given strain with partial immune cross-protection against other strains. In the extreme case where infection with the middle strain prevents further infections from the other two strains, we reduce the model to a six-dimensional kernel capable of showing self-sustaining oscillations at relatively high levels of cross-protection. Dimensional reduction has been accomplished by a transformation of variables that preserves the eigenvalue responsible for the transition from damped oscillations to limit cycle solutions.
103 citations
TL;DR: Using the method of coincidence degree and Lyapunov functional, a set of easily verifiable sufficient conditions are derived for the existence of at least one strictly positive periodic solution of periodic n-species Lotka-Volterra competition systems with several deviating arguments.
Abstract: In this paper, we study the existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems. By using the method of coincidence degree and Lyapunov functional, a set of easily verifiable sufficient conditions are derived for the existence of at least one strictly positive (componentwise) periodic solution of periodic n-species Lotka-Volterra competition systems with several deviating arguments and the existence of a unique globally asymptotically stable periodic solution with strictly positive components of periodic n-species Lotka-Volterra competition system with several delays. Some new results are obtained. As an application, we also examine some special cases of the system we considered, which have been studied extensively in the literature. Some known results are improved and generalized.
100 citations
TL;DR: This paper model the effect of RTI drugs and PI drugs as inhibition rates of cell infection and infectious virus production, respectively, based on the biological mechanisms of these two different types of drugs.
Abstract: Recently, potent combination antiviral therapies consisting of reverse transcriptase inhibitor (RTI) drugs and protease inhibitor (PI) drugs, have been developed which can rapidly suppress HIV below the limit of detection. Two phases of plasma viral decay after initiation of treatment have been observed in clinical studies. Some researchers have suggested that the viral decay rates may reflect the potency (efficacy) of antiviral therapies. In this paper we model the effect of RTI drugs and PI drugs as inhibition rates of cell infection and infectious virus production, respectively, based on the biological mechanisms of these two different types of drugs. Through rigorous mathematical derivation, we show that the two viral decay rates are monotone functions of the treatment effects of these antiviral therapies. We derive approximation formulas for the relationships between viral decay rates and treatment effects. Computer simulations show that the approximation formulas approximate the true values very well. These formulas may be used to study what factors really affect the viral decay rates. The results in this paper provide a theoretical justification for using both viral decay rates for evaluation of the treatment efficacy of antiviral therapies.
90 citations
TL;DR: A time-continuous version of the first-order difference equation model of cocaine use introduced by Everingham and Rydell is set up and it is suggested that drug prevention can temper drug prevalence and consumption, but that drug treatment's effectiveness depends critically on the stage in the epidemic in which it is employed.
Abstract: We set up a time-continuous version of the first-order difference equation model of cocaine use introduced by Everingham and Rydell [S.S. Everingham, C.P. Rydell, Modeling the Demand for Cocaine, MR-332-ONDCP/A/DPRC, RAND, Santa Monica, CA, 1994] and extend it by making initiation an endogenous function of prevalence. This function reflects both the epidemic spread of drug use as users 'infect' non-users and Musto's [D.F. Musto, The American Disease: Origins of Narcotic Control, Oxford University, New York, 1987] hypothesis that drug epidemics die out when a new generation is deterred from initiating drug use by observing the ill effects manifest among heavy users. Analyzing the model's dynamics suggests that drug prevention can temper drug prevalence and consumption, but that drug treatment's effectiveness depends critically on the stage in the epidemic in which it is employed. Reducing the number of heavy users in the early stages of an epidemic can be counter-productive if it masks the risks of drug use and, thereby, removes a disincentive to initiation. This strong dependence of an intervention's effectiveness on the state of the dynamic system illustrates the pitfalls of applying a static control policy in a dynamic context.
TL;DR: Knowing the induced velocity field makes it possible to consider the energetic needs of fish swimming in that school and to describe the break up of schools due to oxygen depletion, which allows us to estimate maximum school sizes.
Abstract: Schooling behavior is a challenging topic in the context of animal aggregation. It is also of economic importance for the estimation and conservation of stock sizes. An individual based movement model will be developed, taking into account energetic advantages of schooling. This model is a cellular automaton with a hexagonal grid. The latter considers the geometry of a school, where fish swim in a diamond-shape configuration in order to take advantage of the velocity, induced by the tail strokes of preceding fish. Furthermore, knowing the induced velocity field makes it possible to consider the energetic needs of fish swimming in that school and to describe the break up of schools due to oxygen depletion. This allows us to estimate maximum school sizes.
TL;DR: In the simulations the addition of adult pertussis booster vaccinations every 10 yr is beneficial in reducing adult incidence, but causes only modest reductions in the incidence in infants and young children, suggesting that a careful cost effectiveness analysis is needed before implementation of an adult pertussedis vaccination program.
Abstract: An expanded pertussis (whooping cough) vaccination program which includes adult boosters every 10 yr is studied using computer simulations of two models. These age-structured pertussis transmission models include waning of both infection-acquired and vaccine-induced immunity, and vaccination of children corresponding to the vaccination coverage since 1940. Adult vaccinations cause a larger boost in the immunity level in the second model than in the first model. In the simulations the addition of adult pertussis booster vaccinations every 10 yr is beneficial in reducing adult incidence, but causes only modest reductions in the incidence in infants and young children. These simulations suggest that a careful cost effectiveness analysis is needed before implementation of an adult pertussis vaccination program.
TL;DR: A mathematical model of microbial growth for limiting nutrient in a plug flow reactor which accounts for the colonization of the reactor wall surface by the microbes is formulated and studied analytically and numerically and can be viewed as a model of the large intestine or of the fouling of a commercial bio-reactor or pipe flow.
Abstract: A mathematical model of microbial growth for limiting nutrient in a plug flow reactor which accounts for the colonization of the reactor wall surface by the microbes is formulated and studied analytically and numerically. It can be viewed as a model of the large intestine or of the fouling of a commercial bio-reactor or pipe flow. Two steady state regimes are identified, namely, the complete washout of the microbes from the reactor and the successful colonization of both the wall and bulk fluid by the microbes. Only one steady state is stable for any particular set of parameter values. Sharp and explicit conditions are given for the stability of each, and for the long term persistence of the bacteria in the reactor.
TL;DR: An analysis is presented that converts in vitro gas production models to an effective first-order rate constant that can be used directly in rumen systems models to predict the extent of ruminal carbohydrate digestion.
Abstract: Physiological systems models for ruminant animals are used to predict the extent of ruminal carbohydrate digestion, based on rates of intake, digestion, and passage to the lower tract. Digestion of feed carbohydrates is described in these models by a first-order rate constant. Recently, an in vitro gas production technique has been developed to determine the digestion kinetics in batch fermentation, and nonlinear mathematical models have been fitted to the cumulative gas production data from these experiments. In this paper, we present an analysis that converts these gas production models to an effective first-order rate constant that can be used directly in rumen systems models. The analysis considers the digestion of an incremental mass of substrate entering the rumen. The occurrence of passage is represented probabilistically, and integration through time gives the total mass of substrate and total rate of digestion in the rumen. To demonstrate the analysis, several gas production models are fitted to a sample data set for corn silage, and the effective first-order rate constants are calculated. The rate constants for digestion depend on ruminal passage rate, an interaction that arises from the nonlinearity of the gas production models.
TL;DR: Both the theoretical basis and numerical accuracy of these and related models are investigated, consisting of an ordinary differential equation with 'effective rate coefficients' incorporating reaction and transport parameters.
Abstract: Optical biosensors, including the BIACORE, provide an increasingly popular method for determining reaction rates of biomolecules. In a flow chamber, with one reactant immobilized on a chip on the sensor surface, a solution containing the other reactant (the analyte) flows through the chamber. The time course of binding of the reactants is monitored. Scientists using the BIACORE to understand biomolecular reactions need to be able to separate intrinsic reaction rates from the effects of transport in the biosensor. For a model to provide a useful basis for such an analysis, it must reflect transport accurately, while remaining simple enough to couple with a routine for estimating reaction rates from BIACORE data. Models have been proposed previously for this purpose, consisting of an ordinary differential equation with 'effective rate coefficients' incorporating reaction and transport parameters. In this paper we investigate both the theoretical basis and numerical accuracy of these and related models.
TL;DR: Two modelling frameworks for studying dynamic anistropy in connective tissue are presented, motivated by the problem of fibre alignment in wound healing and it is shown that the first model predicts patterns of alignment on macroscopic length scales that are lost in a continuum model of the cell population.
Abstract: We present two modelling frameworks for studying dynamic anistropy in connective tissue, motivated by the problem of fibre alignment in wound healing. The first model is a system of partial differential equations operating on a macroscopic scale. We show that a model consisting of a single extracellular matrix material aligned by fibroblasts via flux and stress exhibits behaviour that is incompatible with experimental observations. We extend the model to two matrix types and show that the results of this extended model are robust and consistent with experiment. The second model represents cells as discrete objects in a continuum of ECM. We show that this model predicts patterns of alignment on macroscopic length scales that are lost in a continuum model of the cell population.
TL;DR: A cellular automaton model, including lateral inhibition of an autocatalytic morphogen, as well as a genetic switch that differentiates tissue into substrate-depleting vessels, is presented, yielding isotropic morphogenesis.
Abstract: We present a cellular automaton model, including lateral inhibition of an autocatalytic morphogen, as well as a genetic switch that differentiates tissue into substrate-depleting vessels. This model yields isotropic morphogenesis, including: dichotomous and lateral branching, blind vessel ends, and closed loops due to anastosmosis. The algorithm consists of a list of simple rules describing the essential biophysical features, permitting comfortable programming and fast computations. Depending on the choice of the substrate s, the model is applicable to leaf veins (s is auxin), insect trachea (s is CO2) or neovascularization (s is an angiogenesis factor). Sequential addition of rules can be correlated to evolutionary steps in leaf morphogenesis.
TL;DR: An application of several approaches aimed at determining of the non-stationarities in the signals and testing whether non-linear dynamics exists is reported, suggesting that no one approach taken alone is the best for the aims.
Abstract: Most of the physiological signals (EEG, ECG, blood flow, human gait, etc.) characterize by complex dynamics including both non-stationarities and non-linearities. These time series resemble red noise with long-range correlation and 1/(f beta) power spectrum. A question arises as to how to distinguish the characteristics of the process underlying the signal dynamics from the properties of the observed time series. The classical methods to determine possible non-linear (chaotic) dynamics (e.g. correlation dimension) often fail in such signals because of relatively short data records containing stochastic components and non-stationarities. We report an application of several approaches, aimed at (1) determining of the non-stationarities in the signals and (2) testing whether non-linear dynamics exists. Assessment of the intrinsic correlation properties of the dynamic process and distinguishing the same from external trends was performed using singular spectra and detrended fluctuation analysis. The existence of non-linear dynamics was tested by correlation dimension (modified algorithm of re-embedding) and by correlation integrals of real and surrogate data. The correlation integrals of real signal and surrogate data sets were statistically compared using Kolmogorov-Smirnov (K-S) test. The procedures were tested on EEG and laser-Doppler (LD) blood flow. Our suggestion is that no one approach taken alone is the best for our aims. Instead, a battery of methods should be used.
TL;DR: The mathematical results of Arino et al. (1995) are extended to the case in which cell death is present, in cells with telomeres above and below the critical threshold of length, generally with differing probabilities.
Abstract: Shortening of chromosome ends, known as telomeres, is one of the supposed mechanisms of cellular aging and death. We provide a probabilistic analysis of the process of loss of telomere ends. The first work concerned with that issue is the paper by Levy et al. [J. Molec. Biol. 225 (1992) 951–960]. Their deterministic model reproduced the observed frequencies of viable cells in the in vitro experiments. Arino et al. [J. Theor. Biol. 177 (1995) 45–57] reformulated the model of Levy et al. (1992) in the terms of branching processes with denumerable type space. In the present paper, the mathematical results of Arino et al. (1995) are extended to the case in which cell death is present, in cells with telomeres above and below the critical threshold of length, generally with differing probabilities. Both exact and asymptotic results are provided, as well as a discussion of biological relevance of the results.
TL;DR: The schemes are formulated for the numerical solution of size-dependent population models that discretize size by means of a natural grid, which introduces a discrete dynamics.
Abstract: We formulate schemes for the numerical solution of size-dependent population models. Such schemes discretize size by means of a natural grid, which introduces a discrete dynamics. The schemes are analysed and optimal rates of convergence are derived. Some numerical experiments are also reported to demonstrate the predicted accuracy of the schemes.
TL;DR: A qualitative method to validate and monitor the structure of a non-linear model with respect to experimental data, under some hypotheses, and it is shown that the usual moving average of the outputs follows this transition graph.
Abstract: We present in this paper a qualitative method to validate and monitor the structure of a non-linear model with respect to experimental data, under some hypotheses. This method is broadly independent of the analytical formulation of the model and depends only on the qualitative structure (the signs of the Jacobian matrix). The temporal sequences of the extrema of a filtered experimental signal are compared with the transitions allowed by a graph. In particular, Re show that the usual moving average of the outputs follows this transition graph. We apply this method to compare models of algal growth in a bioreactor with experimental data, (C) 1999 Elsevier Science Inc. All rights reserved.
TL;DR: A mathematical model of a simple food web consisting of one predator and two prey populations in a chemostat is analyzed and multistability phenomena of coexistence of steady and periodic states at the same operating conditions are found.
Abstract: We analyze a mathematical model of a simple food web consisting of one predator and two prey populations in a chemostat. Monod's model is employed for the dependence of the specific growth rates of the two prey populations on the concentration of the rate-limiting substrate and a generalization of Monod's model for the dependence of the specific growth rate of the predator on the concentrations of the prey populations. We use numerical bifurcation techniques to determine the effect of the operating conditions of the chemostat on the dynamics of the system and construct its operating diagram. Chaotic behavior resulting from successive period doublings is observed. Multistability phenomena of coexistence of steady and periodic states at the same operating conditions are also found.
TL;DR: The RLC formalism generalizes previous models, fits current data adequately and facilitates mechanistically based extrapolations from high-dose experiments to the much lower doses of interest for most applications.
Abstract: Ionizing radiation produces DNA double strand breaks (DSBs) in chromosomes. For densely ionizing radiation, the DSBs are not spaced randomly along a chromosome: recent data for size distributions of DNA fragments indicate break clustering on kbp–Mbp scales. Different DSB clusters on a chromosome are typically made by different, statistically independent, stochastically structured radiation tracks, and the average number of tracks involved can be small. We therefore model DSB positions along a chromosome as a stationary Poisson cluster process, i.e. a stochastic process consisting of secondary point processes whose locations are determined by a primary point process that is Poisson. Each secondary process represents a break cluster, typically consisting of 1–10 DSBs in a comparatively localized stochastic pattern determined by chromatin geometry and radiation track structure. Using this Poisson cluster process model, which we call the randomly located clusters (RLC) formalism, theorems are derived for how the DNA fragment-size distribution depends on radiation dose. The RLC dose-response relations become non-linear when the dose becomes so high that DSB clusters from different tracks overlap or adjoin closely. The RLC formalism generalizes previous models, fits current data adequately and facilitates mechanistically based extrapolations from high-dose experiments to the much lower doses of interest for most applications.
TL;DR: A diffusive-convective model for the dynamics of a population living in a polluted environment and threshold results are given concerning the effect of the toxicant on the living population are considered.
Abstract: In this paper we consider a diffusive-convective model for the dynamics of a population living in a polluted environment Threshold results are given concerning the effect of the toxicant on the living population Some analytic results are proved and numerical experiments give suggestions in more general cases
TL;DR: An SIS model for a heterosexually transmitted disease with core and non-core compartments and a generalized recovery function P(t) is analyzed and exhibits R0 threshold behavior and leads to discussions of stability with respect to choice of P( t), and of the effects of allowing recruitment between core andnon-core groups.
Abstract: An SIS model for a heterosexually transmitted disease with core and non-core compartments and a generalized recovery function P(t) is analyzed. It exhibits R0 threshold behavior and leads to discussions of stability with respect to choice of P(t), and of the effects of allowing recruitment between core and non-core groups.
TL;DR: This work presents a time discrete model for a structured population in which it can distinguish two processes of a general nature and whose corresponding time scales are very different from each other.
Abstract: In this work we extend approximate aggregation methods to deal with a very general linear time discrete model. Approximate aggregation consists in describing some features of the dynamics of a general system in terms of the dynamics of a reduced system governed by a few global variables. We present a time discrete model for a structured population (i.e., the population is subdivided in subpopulations) in which we can distinguish two processes of a general nature and whose corresponding time scales are very different from each other. We transform the general system to make the global variables appear and obtain the reduced system. These global variables are, for each subpopulation, a certain linear combination of the corresponding state variables. We show that, under quite general conditions, the asymptotic behavior of the reduced system can be known in terms of the corresponding behavior for the reduced system. The general method is applied to aggregate a multiregional Leslie model in which the demographic process is supposed to be fast with respect to migration.
TL;DR: It appears that the region in the control parameter space where a predator can invade increases with its growth rate, which implies that short food chains with moderate growth rate are liable to be invaded by fast growing invaders which consume the top predator.
Abstract: We study the invasion of a top predator into a food chain in a chemostat. For each trophic level, a bioenergetic model is used in which maintenance and energy reserves are taken into account. Bifurcation analysis is performed on the set of nonlinear ordinary differential equations which describe the dynamic behaviour of the food chain. In this paper, we analyse how the ability of a top predator to invade the food chain depends on the values of two control parameters: the dilution rate and the concentration of the substrate in the input. We investigate invasion by studying the long-term behaviour after introduction of a small amount of top predator. To that end we look at the stability of the boundary attractors; equilibria, limit cycles as well as chaotic attractors using bifurcation analysis. It will be shown that the invasibility criterion is the positiveness of the Lyapunov exponent associated with the change of the biomass of the top predator. It appears that the region in the control parameter space where a predator can invade increases with its growth rate. The resulting system becomes more resistant to further invasion when the top predator grows faster. This implies that short food chains with moderate growth rate of the top predator are liable to be invaded by fast growing invaders which consume the top predator. There may be, however, biological constraints on the top predator's growth rate. Predators are generally larger than prey while larger organisms commonly grow slower. As a result, the growth rate generally decreases with the trophic level. This may enable short food chains to be resistant to invaders. We will relate these results to ecological community assembly and the debate on the length of food chains in nature.
TL;DR: A state space model for the HIV pathogenesis under treatment by anti-viral drugs and a much stronger effect of the treatment within the first 10 to 20 h than that predicted by the deterministic model is developed.
Abstract: In this paper we have extended the model of HIV pathogenesis under treatment by anti-viral drugs given by Perelson et al. [A.S. Perelson et al., Science 271 (1999) 1582] to a stochastic model. By using this stochastic model as the stochastic system model, we have developed a state space model for the HIV pathogenesis under treatment by anti-viral drugs. In this state space model, the observation model is a statistical model based on the observed numbers of RNA virus copies over different times. For this model we have developed procedures for estimating and predicting the numbers of infectious free HIV and non-infectious free HIV as well as the numbers of different types of T cells through extended Kalman filter method. As an illustration, we have applied the method of this paper to the data of patient Nos. 104, 105 and 107 given by Perelson et al. [A.S. Perelson et al., Science 271 (1999) 1582] under treatment by Ritonavir. For these individuals, it is shown that within two weeks since treatment, most of the free HIV are non-infectious, indicating the usefulness of the treatment. Furthermore, the Kalman filter method revealed a much stronger effect of the treatment within the first 10 to 20 h than that predicted by the deterministic model.
TL;DR: This work transforms the system to make the global variables explicit, and justifies the quick derivation of the aggregated system, applied to aggregate a multiregional Leslie model with density dependent migration rates.
Abstract: The aim of this work is to develop an approximate aggregation method for certain non-linear discrete models Approximate aggregation consists in describing the dynamics of a general system involving many coupled variables by means of the dynamics of a reduced system with a few global variables We present discrete models with two different time scales, the slow one considered to be linear and the fast one non-linear because of its transition matrix depends on the global variables In our discrete model the time unit is chosen to be the one associated to the slow dynamics, and then we approximate the effect of fast dynamics by using a sufficiently large power of its corresponding transition matrix In a previous work the same system is treated in the case of fast dynamics considered to be linear, conservative in the global variables and inducing a stable frequency distribution of the state variables A similar non-linear model has also been studied which uses as time unit the one associated to the fast dynamics and has the non-linearity in the slow part of the system In the present work we transform the system to make the global variables explicit, and we justify the quick derivation of the aggregated system The local asymptotic behaviour of the aggregated system entails that of the general system under certain conditions, for instance, if the aggregated system has a stable hyperbolic fixed point then the general system has one too The method is applied to aggregate a multiregional Leslie model with density dependent migration rates