Parametric dependence in model epidemics. II: Non-contact rate-related parameters

In a previous paper, we discussed the bifurcation structure of SEIR equations subject to seasonality. There, the focus was on parameters that affect transmission: the mean contact rate, β₀, and the magnitude of seasonality, ? _B . Using numerical continuation and brute force simulation, we characterized a global pattern of parametric dependence in terms of subharmonic resonances and period-doublings of the annual cycle. In the present paper, we extend this analysis and consider the effects of varying non-contact-related parameters: periods of latency, infection and immunity, and rates of mortality and reproduction, which, following the usual practice, are assumed to be equal. The emergence of several new forms of dynamical complexity notwithstanding, the pattern previously reported is preserved. More precisely, the principal effect of varying non-contact related parameters is to displace bifurcation curves in the β₀?? _B parameter plane and to expand or contract the regions of resonance and period-doubling they delimit. Implications of this observation with respect to modeling real-world epidemics are considered.     

Parametric dependence in model epidemics. I: contact-related parameters

One of the interesting properties of nonlinear dynamical systems is that arbitrarily small changes in parameter values can induce qualitative changes in behavior. The changes are called bifurcations, and they are typically visualized by plotting asymptotic dynamics against a parameter. In some cases, the resulting bifurcation diagram is unique: irrespective of initial conditions, the same dynamical sequence obtains. In other cases, initial conditions do matter, and there are coexisting sequences. Here we study an epidemiological model in which multiple bifurcation sequences yield to a single sequence in response to varying a second parameter. We call this simplification the emergence of unique parametric dependence (UPD) and discuss how it relates to the model's overall response to parameters. In so doing, we tie together a number of threads that have been developing since the mid-1980s. These include period-doubling; subharmonic resonance, attractor merging and subduction and the evolution of strange invariant sets. The present paper focuses on contact related parameters. A follow-up paper, to be published in this journal, will consider the effects of non-contact related parameters.     

Parametric dependence in model epidemics. II: Non-contact rate-related parameters

In a previous paper, we discussed the bifurcation structure of SEIR equations subject to seasonality. There, the focus was on parameters that affect transmission: the mean contact rate, β(0), and the magnitude of seasonality, ε(B). Using numerical continuation and brute force simulation, we characterized a global pattern of parametric dependence in terms of subharmonic resonances and period-doublings of the annual cycle. In the present paper, we extend this analysis and consider the effects of varying non-contact-related parameters: periods of latency, infection and immunity, and rates of mortality and reproduction, which, following the usual practice, are assumed to be equal. The emergence of several new forms of dynamical complexity notwithstanding, the pattern previously reported is preserved. More precisely, the principal effect of varying non-contact related parameters is to displace bifurcation curves in the β(0)-ε(B) parameter plane and to expand or contract the regions of resonance and period-doubling they delimit. Implications of this observation with respect to modeling real-world epidemics are considered.     

A generalized compartmental model to estimate the fibre mass in the ruminoreticulum: 1. Estimating parameters of digestion

Parameters related to the microbial digestion of nutrients in the ruminoreticulum have been estimated by fitting mathematical models to degradation profiles generated from kinetic studies. In the present paper, we propose a generalized compartmental model of digestion (GCMD) based on implicit theoretical concepts and the gamma probability density function to estimate fibre digestion parameters. The proposed model is consistent to a broader compartmental model presented in a companion paper that integrates aspects of fibre digestion and passage. Different versions of the GCMD were generated by increasing the integer order of time dependency of the gamma function. These versions were fitted to 192 published fibre degradation profiles that were obtained using an in vitro fermentation technique. The quality of fit was evaluated based on the frequency of minimum sum of squares of errors (SSE), the number of runs of signs of residuals, and its likelihood probability calculated according to the Akaike's Information Criterion. The likelihood of the proposed model was also compared to a discrete lag time model (DLT), which is commonly used to interpret fibre degradation profiles. The GCMD had superior quality of fit compared to the DLT and was considered more likely in describing 68.75% of the profiles evaluated. Only 9.38% of the degradation profiles that were fitted to the DLT model had a lower SSE. Even though the degradation profiles studied were generated by incubating feed samples up to 96 h, the true asymptotic limit of fibre degradation can only be achieved by long-term fermentations. This fact leads to questioning the uniformity of the potentially digestible fibre fraction and a further approach based on GCMD-type model was used to account for its heterogeneous nature.     

Emergent heterogeneity in declining tuberculosis epidemics

Tuberculosis is a disease of global importance: over 2 million deaths are attributed to this infectious disease each year. Even in areas where tuberculosis is in decline, there are sporadic outbreaks which are often attributed either to increased host susceptibility or increased strain transmissibility and virulence. Using two mathematical models, we explore the role of the contact structure of the population, and find that in declining epidemics, localized outbreaks may occur as a result of contact heterogeneity even in the absence of host or strain variability. We discuss the implications of this finding for tuberculosis control in low incidence settings.     

Public avoidance and epidemics: insights from an economic model

In this paper, we present a mathematical model of infectious disease transmission in which people can engage in public avoidance behavior to minimize the likelihood of acquiring an infection. The framework employs the economist's theory of utility maximization to model people's decision regarding their level of public avoidance. We derive the reproductive number of a disease which determines whether an endemic equilibrium exists or not. We show that when the contact function exhibits saturation, an endemic equilibrium must be unique. Otherwise, multiple endemic equilibria that differ in disease prevalence can coexist, and which one the population gets to depends on initial conditions. Even when a unique endemic equilibrium exists, people's preferences and the initial conditions may determine whether the disease will eventually die out or become endemic. Public health policies that increase the recovery rate or encourage self-quarantine by infected people can be beneficial to the community by lowering disease prevalence. However, it is also possible for these policies to worsen the situation and cause prevalence to rise since these measures give people less incentive to engage in public avoidance behavior. We also show that implementing policies that result in a higher level of public avoidance behavior in equilibrium does not necessarily lower prevalence and can result in more infections.     

Estimating the kernel parameters of premises-based stochastic models of farmed animal infectious disease epidemics using limited, incomplete, or ongoing data

Three different estimators are presented for the types of parameters present in mathematical models of animal epidemics. The estimators make use of the data collected during an epidemic, which may be limited, incomplete, or under collection on an ongoing basis. When data are being collected on an ongoing basis, the estimated parameters can be used to evaluate putative control strategies. These estimators were tested using simulated epidemics based on a spatial, discrete-time, gravity-type, stochastic mathematical model containing two parameters. Target epidemics were simulated with the model and the three estimators were implemented using various combinations of collected data to independently determine the two parameters.     

The effect of integral conditions in certain equations modelling epidemics and population growth

Summary Models of epidemics that lead to delay differential equations often have subsidiary integral conditions that are imposed by the interpretation of these models. The neglect of these conditions may lead to solutions that behave in a radically different manner from solutions restricted to obey them. Examples are given of such behavior, including cases where periodic solutions may occur off the natural set defined by these conditions but not on it. A complete stability analysis is also given of a new model of a disease propagated by a vector where these integral conditions play an important role.This work was partially supported by N.S.F. Grant MCS 7903497     

A generalized compartmental model to estimate the fibre mass in the ruminoreticulum: 2. Integrating digestion and passage

Models used to predict digestibility and fill of the dietary insoluble fibre (NDF) treat the ruminoreticular particulate mass as a single pool. The underlying assumption is that escape of particles follows first-order kinetics. In this paper, we proposed and evaluated a model of two ruminoreticular sequential NDF pools. The first pool is formed by buoyant particles (raft pool) and the second one by fluid dispersed particles (escapable pool) ventrally to the raft. The transference of particles between these two pools results from several processes that reduce particles buoyancy, assuming the gamma distribution. The exit of escapable pool particles from the ruminoreticulum is exponentially distributed. These concepts were evaluated by comparing ruminoreticular NDF masses as 43 and 27 means from cattle and sheep, respectively, to the same predicted variable using single- and two-pools models. Predictions of the single-pool model were based on lignin turnover and the turnover associated to the descending phase of the elimination of Yb-labelled forage particles in the faeces of sheep. Predictions of the two-pool model were obtained by estimating fractional passage rates associated to the ascending and descending phases of the same Yb excretion profiles in sheep faeces. All turnovers were scaled to the power 0.25 of body mass for interspecies comparisons. Predictions based on lignin turnover (single pool) and the two-pool model presented similar trends, accuracies and precisions. The single-pool approach based solely on the descending phase of the marker yielded biased estimates of the ruminoreticular NDF mass.     

Stability in a two host epidemic model

An epidemic model is derived for a two host infectious disease. It is shown that if a non-trivial equilibrium solution exists, it is globally stable. This result is also proved for a similar one host model.     

A theory of drug tolerance and dependence I: a conceptual analysis

A mathematical model of drug tolerance and its underlying theory is presented. The model extends a first approach, published previously. The model is essentially more complex than the generally used model of homeostasis, which is demonstrated to fail in describing tolerance development to repeated drug administrations. The model assumes the development of tolerance to a repeatedly administered drug to be the result of a regulated adaptive process. The oral detection and analysis of exogenous substances is proposed to be the primary stimulus for the mechanism of drug tolerance. Anticipation and environmental cues are in the model considered secondary stimuli, becoming primary only in dependence and addiction or when the drug administration bypasses the natural-oral-route, as is the case when drugs are administered intravenously. The model considers adaptation to the effect of a drug and adaptation to the interval between drug taking autonomous tolerance processes. Simulations with the mathematical model demonstrate the model's behavior to be consistent with important characteristics of the development of tolerance to repeatedly administered drugs: the gradual decrease in drug effect when tolerance develops, the high sensitivity to small changes in drug dose, the rebound phenomenon and the large reactions following withdrawal in dependence. The mathematical model verifies the proposed theory and provides a basis for the implementation of mathematical models of specific physiological processes. In addition, it establishes a relation between the drug dose at any moment, and the resulting drug effect and relates the magnitude of the reactions following withdrawal to the rate of tolerance and other parameters involved in the tolerance process. The present paper analyses the concept behind the model. The next paper discusses the mathematical model.     

On the uniqueness of the positive solution of an integral equation which appears in epidemiological models

 In this paper, we show that the positive solution of a non-linear integral equation which appears in classical SIR epidemiological models is unique. The demonstration of this fact is necessary to justify the correctness of any approximate or numerical solution. The SIR epidemiological model is used only for simplicity. In fact, the methods used can be easily extended to prove the existence and uniqueness of the more involved integral equations that appear when more biological realities are considered. Thus the inclusion of a latent class (SLIR models) and models incorporating variability in the infectiousness with duration of the infection and spatial distribution lead to integral equations to which the results derived in this paper apply immediately. Received: 7 May 1999     

A theory of drug tolerance and dependence II: the mathematical model

The preceding paper presented a model of drug tolerance and dependence. The model assumes the development of tolerance to a repeatedly administered drug to be the result of a regulated adaptive process. The oral detection and analysis of exogenous substances is proposed to be the primary stimulus for the mechanism of drug tolerance. Anticipation and environmental cues are in the model considered secondary stimuli, becoming primary in dependence and addiction or when the drug administration bypasses the natural-oral-route, as is the case when drugs are administered intravenously. The model considers adaptation to the effect of a drug and adaptation to the interval between drug taking autonomous tolerance processes. Simulations with the mathematical model demonstrate the model's behaviour to be consistent with important characteristics of the development of tolerance to repeatedly administered drugs: the gradual decrease in drug effect when tolerance develops, the high sensitivity to small changes in drug dose, the rebound phenomenon and the large reactions following withdrawal in dependence. The present paper discusses the mathematical model in terms of its design. The model is a nonlinear, learning feedback system, fully satisfying control theoretical principles. It accepts any form of the stimulus-the drug intake-and describes how the physiological processes involved affect the distribution of the drug through the body and the stability of the regulation loop. The mathematical model verifies the proposed theory and provides a basis for the implementation of mathematical models of specific physiological processes.     

A reduced mathematical model of the acute inflammatory response: I. Derivation of model and analysis of anti-inflammation

The acute inflammatory response, triggered by a variety of biological or physical stresses on an organism, is a delicate system of checks and balances that, although aimed at promoting healing and restoring homeostasis, can result in undesired and occasionally lethal physiological responses. In this work, we derive a reduced conceptual model for the acute inflammatory response to infection, built up from consideration of direct interactions of fundamental effectors. We harness this model to explore the importance of dynamic anti-inflammation in promoting resolution of infection and homeostasis. Further, we offer a clinical correlation between model predictions and potential therapeutic interventions based on modulation of immunity by anti-inflammatory agents.     

Temperature dependence of two parameters in a photosynthesis model

The temperature dependence of the photosynthetic parameters V_cmax, the maximum catalytic rate of the enzyme Rubisco, and J_max, the maximum electron transport rate, were examined using published datasets. An Arrehenius equation, modified to account for decreases in each parameter at high temperatures, satisfactorily described the temperature response for both parameters. There was remarkable conformity in V_cmax and J_max between all plants at T_leaf < 25 °C, when each parameter was normalized by their respective values at 25 °C (V_cmax0 and J_max0), but showed a high degree of variability between and within species at T_leaf > 30 °C. For both normalized V_cmax and J_max, the maximum fractional error introduced by assuming a common temperature response function is < ± 0·1 for most plants and < ± 0·22 for all plants when T_leaf < 25 °C. Fractional errors are typically < ± 0·45 in the temperature range 25–30 °C, but very large errors occur when a common function is used to estimate the photosynthetic parameters at temperatures > 30 °C. The ratio J_max/V_cmax varies with temperature, but analysis of the ratio at T_leaf = 25 °C using the fitted mean temperature response functions results in J_max0/V_cmax0 = 2·00 ± 0·60 (SD, n = 43).     

Fish kairomones, its benefits and detriments: A model based study both from releaser and acceptor perspective

Kairomones have been documented as an infochemicals to convey information between individuals in aquatic system. However, whether the effect of fish kairomones on acceptor (zooplankton) is beneficial or detrimental is a debatable and unanswered issue. This may be due to lack of feasibility of experimentation. In this study, we theoretically explore how fish kairomones affect the aquatic food chain and provide possible explanation of such different behaviors of kairomones. To do this, we propose two hypotheses and formulate two simple mathematical models which resemble more realistic scenario synergetic with natural complex system. Our study suggests that vertical migration helps zooplankton species for proper conservation of its abundance by avoiding unnecessary over predation.     

The species-area relation I. A model for discrete sampling

A simple mathematical model describing the species-area relation was developed. This paper dealt with the case that discrete random samples are combined. Modelling was made on the assumption that the occurrence probability of a species in a quadrat has a continuous density distribution. The model, given by the equation (6), holds only for a particular size of quadrat (i.e. the characteristic area). More general form applicable to the quadrats the size of which is near to the characteristic area was represented by the equation (9). Validity of the model was examined for the data of plant and insect communities, and it was concluded that the observation can be predicted by the model unless the size of sampling unit considerably differs from the characteristic area. The uniformity of specific density (i. e. the number of species per quadrat) and the size of characteristic area were discussed as being important in an understanding of community structure.     

A mathematical model of hematopoiesis: II. Cyclical neutropenia

Cyclical neutropenia is a dynamical disease of the hematopoietic system marked by an oscillation in circulating leukocyte (e.g. neutrophil) numbers to near zero levels and then back to normal. This oscillation is also mirrored in the platelets and reticulocytes which oscillate with the same period. Cyclical neutropenia has an animal counterpart in the grey collie. Using the mathematical model of the hematopoietic system of Colijn and Mackey [A mathematical model of hematopoiesis: I. Periodic chronic myelogenous leukemia. Companion paper to the present paper.] we have determined what parameters are necessary to mimic laboratory and clinical data on untreated grey collies and humans, and also what changes in these parameters are necessary to fit data during treatment with granulocyte colony stimulating factor (G-CSF). Compared to the normal steady-state values, we found that the major parameter changes that mimic untreated cyclical neutropenia correspond to a decreased amplification (increased apoptosis) within the proliferating neutrophil precursor compartment, and a decrease in the maximal rate of re-entry into the proliferative phase of the stem cell compartment. For the data obtained during G-CSF treatment, good fits were obtained only when parameters were altered that would imply that G-CSF led to higher amplification (lower rate of apoptosis) in the proliferating neutrophil precursors, and a elevated rate of differentiation into the neutrophil line.     

A computational model of hemodynamic parameters in cortical capillary networks

The analysis of hemodynamic parameters and functional reactivity of cerebral capillaries is still controversial. To assess the hemodynamic parameters in the cortical capillary network, a generic model was created using 2D voronoi tessellation in which each edge represents a capillary segment. This method is capable of creating an appropriate generic model of cerebral capillary network relating to each part of the brain cortex because the geometric model is able to vary the capillary density. The modeling presented here is based on morphometric parameters extracted from physiological data of the human cortex. The pertinent hemodynamic parameters were obtained by numerical simulation based on effective blood viscosity as a function of hematocrit and microvessel diameter, phase separation and plasma skimming effects. The hemodynamic parameters of capillary networks with two different densities (consistent with the variation of the morphometric data in the human cortical capillary network) were analyzed. The results show pertinent hemodynamic parameters for each model. The heterogeneity (coefficient variation) and the mean value of hematocrits, flow rates and velocities of the both network models were specified. The distributions of blood flow throughout the both models seem to confirm the hypothesis in which all capillaries in a cortical network are recruited at rest (normal condition). The results also demonstrate a discrepancy of the network resistance between two models, which are derived from the difference in the number density of capillary segments between the models.     

Lipid peroxidation in mitochondrial inner membranes. I. An integrative kinetic model

An integrative mathematical model was developed to obtain an overall picture of lipid hydroperoxide metabolism in the mitochondrial inner membrane and surrounding matrix environment. The model explicitly considers an aqueous and a membrane phase, integrates a wide set of experimental data, and unsupported assumptions were minimized. The following biochemical processes were considered: the classic reactional scheme of lipid peroxidation; antioxidant and pro-oxidant effects of vitamin E; pro-oxidant effects of iron; action of phospholipase A₂, glutathione-dependent peroxidases, glutathione reductase and superoxide dismutase; production of superoxide radicals by the mitochondrial respiratory chain; oxidative damage to proteins and DNA. Steady-state fluxes and concentrations as well as half-lives and mean displacements for the main metabolites were calculated. A picture of lipid hydroperoxide physiological metabolism in mitochondria in vivo showing the main pathways is presented. The main results are:(a) perhydroxyl radical is the main initiation agent of lipid peroxidation (with a flux of 10⁻⁷Ms⁻¹); (b) vitamin E efficiently inhibits lipid peroxidation keeping the amplification (kinetic chain length) of lipid peroxidation at low values (10); (c) only a very minor fraction of lipid hydroperoxides escapes reduction via glutathione-dependent peroxidases; (d) oxidized glutathione is produced mainly from the reduction of hydrogen peroxide and not from the reduction of lipid hydroperoxides.