Passage time,resilience, and structure of compartmental models

The relation between the graphical structure and two measures of stability of a compartmental model is studied. The two are resilience as measured by deviation from equilibrium throughout the history of the system and mean first passage time from a central compartment back to itself. A number of examples are studied and a theory developed for models whose graphical structure is in the form of an advanced rosette. The resilience, known to increase with complexity, is shown to increase as well with uneveness in length of food chains. The mean first passage time, on the other hand, depends only on the number of food chains and their average length.     

Complexity of compartmental models

A family of complexity indices for compartmental models is introduced. These are analogous to the diversity indices used to study ecological communities. Each index is shown to be related to the eigenvalues of the model, and in one case (the Shannon index) is equal to the negative of the sum of all eigenvalues. Relations between these indices and equivalent models as well as intrinsic complexity are investigated. In each case, large values of the complexity index lead to greater stability, but not necessarily conversely.     

Some equivalent compartmental models

Compartmental models with strongly connected digraphs always have a stable equilibrium solution. Each such model may be reduced by a sequence of matrix operations to a mammilary system with the same equilibrium solution. It may also be reduced by a sequence of operations on the digraph. It may then be further reduced to a model whose digraph is a 2-cycle with the same mean first passage time m. If this m is taken as an indicator of the relative stability, then the latter is independent of the complexity as measured by the number of arcs per vertex. In a number of special cases the index m is related to the eigenvalues of the matrix. It is shown to have a simple relation to other time parameters as well.     

On stochastic compartmental models

The statistical averaging of compartmental models with parameters being random processes is derived for the case of vanishing input and uniformly bounded input. The difference of resulting models is discussed.     

Structural properties of compartmental models

This article treats structural properties of the inverse of a compartmental matrix and how they relate to properties of coefficients of the transfer function of the compartmental system. Newly formulated conditions are presented for certain of these parameters to be zero or positive. Also results are given on the interdependence of transfer function coefficients and how this relates to the identifiability problem. Answers to some questions raised in the recent literature about coefficient dependence are discussed.     

Size identifiability of compartmental models

The problem of determining the size of a compartmental model from the impulse response is studied. A number of examples are presented to show that the size cannot always be so determined. Sufficient conditions for this determination are given and then shown to hold almost surely for irreducible matrices.     

On complex eigenvalues of compartmental models

Conditions under which the eigenvalues of the matrix of a compartmental model have a nonzero imaginary part are studied. Inequalities for the total imaginary part are obtained. The effect of excretions and combinations of cycles on this imaginary part are studied.     

Analysis of compartmental models of ligand-induced endocytosis

Kinetic models have played a pivotal role in the study of ligand-induced endocytosis. However, an analysis that suggests a systematic way to validate such models is lacking. The current work analyses the base model of ligand-induced endocytosis for three widely used experimental protocols. In protocol I cells initially devoid of ligand are incubated in ligand solution, whereas protocols II and III are desorption experiments in which an initial pool of surface or internalized ligand-receptor complexes, respectively, are released into an elution medium that is initially devoid of ligand. A short-time analysis of protocol I using successive substitutions yielded a corrected pre-factor for the In/Sur plot introduced by Wiley and Cunningham (Cell 25 (1981) 433). In contrast, neglecting the variation in receptor numbers yielded an approximation of protocol I that is valid for long times (e.g. tens of minutes). Similarly, the low cell-concentration limits of protocols II and III are derived by neglecting the concentration of free ligand. The simplicity of these approximations provides a simple and reliable method for estimating the parameters governing ligand kinetics, while their definitive nature implies that they can be used to verify the validity of the base model. This analysis also provides insight on the fast endocytosis and recycling limit of protocol III.     

Stochastic compartmental models with banching and immigrant particles

A multicompartmental model in which particles enter the system from the environment and reproduce according to a Markov branching process has been considered. Explicit expressions have been obtained for the mean vector and the correlation structure for the numbers of particles in different compartments in different time points of the system. Growth rates of the mean vector and some special cases of the system are also discussed.     

The relation between Krogh and compartmental transport models

The meaning of compartmental concentration is intuitively simple in well-stirred compartment models with uniform concentration but not so in the general case. This meaning may be better understood by relating the general compartmental model to a spatially explicit one such as the Krogh cylinder model. We show that compartmental equations result directly by spatial averaging of the Krogh partial differential equations. Intercompartmental transport is usually stated as flux is proportional to the capillary-tissue concentration difference or $\text{J = ? (}\text{C}{}_{\mathrm{c}}\text{-}\text{C}{}_{\mathrm{T}}\text{)}$. We verify this simple equation by showing that flux in the Krogh model is also approximately first order and then derive the transport coefficient, ?, in terms of the geometry and diffusivity in the Krogh model. A relation between capillary and venous concentration or, for a gas, partial pressure is needed. For the highly diffusable, metabolic gas C0₂ this relation is $\text{P}{}_{\mathrm{c}}\text{= P}{}_{\mathrm{v}}\text{?}\text{M}\text{20K}{}_{\mathrm{c}}$ that is, capillary and venous partial pressures differ by one half the metabolic generation.     

Testing,tracing and isolation in compartmental models

Existing compartmental mathematical modelling methods for epidemics, such as SEIR models, cannot accurately represent effects of contact tracing. This makes them inappropriate for evaluating testing and contact tracing strategies to contain an outbreak. An alternative used in practice is the application of agent- or individual-based models (ABM). However ABMs are complex, less well-understood and much more computationally expensive. This paper presents a new method for accurately including the effects of Testing, contact-Tracing and Isolation (TTI) strategies in standard compartmental models. We derive our method using a careful probabilistic argument to show how contact tracing at the individual level is reflected in aggregate on the population level. We show that the resultant SEIR-TTI model accurately approximates the behaviour of a mechanistic agent-based model at far less computational cost. The computational efficiency is such that it can be easily and cheaply used for exploratory modelling to quantify the required levels of testing and tracing, alone and with other interventions, to assist adaptive planning for managing disease outbreaks.     

Parameter space boundaries for unidentifiable compartmental models

Methods for dealing with unidentifiable compartmental models are first reviewed, emphasizing the parameter interval analysis and exhaustive modeling approaches. More general methods are presented for generating the set of all nonnegative parameter solutions that localize the parameters within bounded regions of parameter space and extend previously published parameter bounding strategies. Each point of these regions is an equivalent solution of the parameter identification problem. If a point on the boundary is selected, at least one of the parameters vanishes and an equivalent submodel is obtained. This property shows the close relationship between the exhaustive modeling and parameter interval analysis approaches.     

Indistinguishability and identifiability analysis of linear compartmental models.

Two compartmental model structures are said to be indistinguishable if they have the same input-output properties. In cases in which available a priori information is not sufficient to specify a unique compartmental model structure, indistinguishable model structures may have to be generated and their attributes examined for relevance. An algorithm is developed that, for a given compartmental model, investigates the complete set of models with the same number of compartments and the same input-output structure as the original model, applies geometrical rules necessary for indistinguishable models, and test models meeting the geometrical criteria for equality of transfer functions. Identifiability is also checked in the algorithm. The software consists of three programs. Program 1 determines the number of locally identifiable parameters. Program 2 applies several geometrical rules that eliminate many (generally most) of the candidate models. Program 3 checks the equality between system transfer functions of the original model and models being tested. Ranks of Jacobian matrices and submatrices and other criteria are used to check patterns of moment invariants and local identifiability. Structural controllability and structural observability are checked throughout the programs. The approach was successfully used to corroborate results from examples investigated by others.     

Equilibrium points for nonlinear compartmental models.

Equilibrium points for nonlinear autonomous compartmental models with constant input are discussed. Upper and lower bounds for the steady states are derived. Theorems guaranteeing existence and uniqueness of equilibrium points for a large collection of system are proved. New information relating to mean residence times is developed. Asymptotic results and a section on stability are included. A recursive process is discussed that generates iterates that converge to steady states for certain types of models. An interesting range of models are included as examples. An attempt is made to provide general qualitative theory for such nonlinear compartmental systems.     

Properties of the transfer functions of compartmental models

This paper is concerned with the nonlinear system of algebraic equations relating the positive parameters of a linear time-invariant compartmental model to its transfer function coefficients. The general form that these equations must take is shown, and simple necessary conditions for the existence of positive solutions are given. An immediate use of these conditions is the development of necessary conditions for a polynomial with positive coefficients to have negative roots. A method is then outlined which triangularizes the system and reduces the complete solution problem to one of finding and counting roots of a polynomial. Sufficient conditions for the existence of real and positive solutions are demonstrated.     

Mechanics of compartmental models of the chest wall

Standard methods for describing the mechanical properties of a linear elastic system are applied to the two- and three-compartment models of the chest wall. The compliance matrix and the experiments required to determine the entries in this matrix and thereby to describe the mechanical properties of the relaxed chest wall are described. The effective forces exerted by external loads and muscle tension are defined. The formal theory is used to identify relations among variables. From the definition of effective force, it follows that the ratio of the forces exerted by the diaphragm on the rib cage and abdomen is the same as the ratio of the dependence of diaphragm length on rib cage and abdominal volumes. As an example of relations among variables that follow from the symmetry of the compliance matrix, it is shown that the change of gastric pressure caused by raising pleural pressure is related to the change in lung volume caused by changing stomach volume.     

Metabolism of human apolipoproteins A-I and A-II: compartmental models

The metabolism of radioiodinated apolipoproteins (apo) A-I and A-II have been examined using the techniques of compartmental modeling. The model for apoA-I contains two plasma compartments decaying at different rates. One component of apoA-I has a residence time of 3.8 days and the second has a residence time of 6.1 days. In contrast, the apoA-II model has only one plasma component, with a residence time of 5.5 days, which decays through two distinct pathways. Twenty-seven percent of apoA-II decays through a pathway that takes 1.1 days longer to reach the urine than the remaining 73% which decays through the more direct path. These differences in the metabolism exist in both male and female populations. Comparison of fasting and nonfasting concentrations of apoA-I revealed that apoA-I concentration was elevated 0.5 standard deviations in the nonfasting samples while there was no significant difference in the apoA-II concentrations. The fasting apoA-I concentrations were found to be less stable over the study period when compared to fasting apoA-II concentrations. These findings are interpreted as indicating that apoA-I and apoA-II each have a separate metabolism which overlaps when they are present on the same lipoprotein particle. Furthermore, these findings are consistent with the concept that apoA-I metabolism is influenced more by perturbations such as dietary modulation.     

Hidden oscillations in generalized linear mammaillary compartmental models

Oscillations due to complex eigenvalues, known to exist but difficult to detect, are sometimes totally hidden in the output of compartmental models, i.e., none of their modes appear in the output. An example is constructed of a class of linear compartmental models with complex eigenvalues, which have oscillating modes appearing in the output for some single-pool-input/single-pool-output (SpISpO) configurations, while for other such configurations all oscillations are totally hidden in the output. To generate the example, generalized mammillary compartmental models are defined in which a central pool exchanges with peripheral submodels called clusters, through individual connector pools, and their transfer functions are calculated corresponding to all SpISpO configurations. When such a model is repetitive, i.e., when it has identical peripheral clusters, and the input or the output is in the central pool, then it is zero-state equivalent, up to a multiplicative constant, with a reduced model having one peripheral cluster only. We analyze the visibility of an eigenvalue, i.e., whether or not the modes associated with it appear in the output, for repetitive generalized mammillary models. Sufficient conditions are given for such models to have oscillating modes appearing in the impulse response for some input/output configurations, while for other such configurations all oscillations are totally hidden, i.e., none appear in the output. A particularly interesting example is presented of a class of linear models with complex eigenvalues satisfying these conditions. This class has the structure of nonlinear models used to describe the process of protein synthesis and turnover.     

On linear stochastic compartmental models in discrete time

This note presents a general time-dependent study of linear stochastic compartmental models in discrete time. The transient distribution of the state of the system is obtained by adapting methods used in the continuous time analysis. Covariance functions with and without a time lag are then deduced by a simple probabilistic argument. Results are derived in the Markov case and are partly extended to the semi-Markov case.