Mathematics in Biology and Medicine

R.A. Fisher [F] and the famous troika of Kolmogorov, Petrovskii, and Piscunov [KPP] studied diffusion models in population dynamics as early as 1936. However, the first really systematic attempt at a critical examination of the role of diffusion in population biology was begun by J.G. Skellam in his 1951 paper [S1] and essentially completed in his 1973 paper [S2]. Skellam’s work fits my definition of the truly classic, that is, work which is often cited but seldom read. It is my object in this lecture to pay homage to J.G. Skellam and to expound some of his ideas. In particular, I want to discuss how the form of the diffusion operator in a diffusion model is determined by the behaviour patterns of the dispersing animals. My hope is that some members of the “we want the beasties to move so lets slap on a Laplacian” — school of diffusion modeling will be inspired to read Skellam and mend their ways. An elegant exposition of Skellam’s ideas (and a great deal more) can be found in Okubo’s monograph [O]. My presentation will differ somewhat from both [S2] and [O] because I want to emphasize the underlying discrete process and delay as long as possible the passage to the diffusion approximation.

In realistic models in population ecology individuals are distinguished from one another according to relevant quantities such as age, weight, amount of toxic substances accumulated in the body etc. (Streifer, 1974). The state of the individual (i-state) is given by the values of these quantities, whereas the state of the population (p-state) is given by the distribution (or density) function describing the number of individuals within each i-state.

Mathematical models describing the evolution of age-structured populations have received increasing attention in recent years, both for the biological interest and for the mathematical one. In fact age dependent fertility and mortality rates are among the most basic parameters in the theory of population dynamics and demography; and the mathematical problems arising, are interesting and challenging in their own rights.

Let u = u(t,x) be the density distribution of individuals over x ∈ ℝ and w = w(t,x) their mean flux. Then without birth and death the simple conservation law holds (1)$${\partial _t}u + {\partial _X}w = 0$$. Modelling dispersion by Ficks law would result in (2)$$w = - {\mu _0}(u) \cdot {\partial _x}u,\;{\mu _0}(u) \geqslant 0$$ and (1) would be the usual diffusion equation. In contrast, modelling aggregation would require the opposite sign of μ₀ leading to an ill-posed problem for (1) in general.

In the Fisher — Wright — Haidane model from population genetics (see Crow and Kimura [6], Edwards [7], Hadeler [9], [10]) the mean fitness of a population of diploid individuals increases as time evolves and remains constant only when the population is in equilibrium. In mathematical terms this means that the mean fitness is a global Ljapunov function, which ensures that any solution of the underlying differential equation or difference equation, respectively, approaches the set Eof stationary solutions. Both in the continuous time and the discrete time version of the Fisher-Wright-Haldane model the set Egenerally is the union of linear submanifolds of the state space (see Hughes and Seneta [15] and Aulbach and Hadeler [5]) and so the following problem arises. IF A SOLUTION APPROACHES A CONTINUUM C OF STATIONARY SOLUTIONS, DOES IT THEN CONVERGE TO A POINT ON C OR MAY IT IGNORE THE STATIONARY FLOW ON C BY CREEPING ALONG C FOR ALL FUTURE TIME?As far as the concrete equations of the Fisher-Wright-Haldane model are concerned thi s question has found partial answers by Feller [8], an der Heiden [12], and Aulbach and Hadeler [5]. Meanwhile the general answer has been given by Losert and Akin [17].

We present some results concerning an initial and boundary value problem for a system of first order linear partial differential equations. This problem comes from modelling the evolution of a homogeneous cell population, where the cells may replicate and are. submitted to the action of some mutagenic agent.

The secretion of a slimy mucus from the foot, besides being mechanically essential for the locomotion of many molluscs, seems to constitute a system of external storage of information used for orientation towards biologically significant goals such as hiding places, sexual partners or prey. Field observations and laboratory experiments have shown that many gastropod species are able to interpret chemical information contained in their mucous trails. In some cases the trail-following mechanism can involve the recognition of “personal” trails and the detection of trail polarization (Wells & Buckley, 1972; Cook & Cook, 1975).

Let X(t) denote the size of a certain population at time t **2267 0. It is assumed that at times when no catastrophes occur, the population grows according to the differential equation (1)$$ X'(t)=\alpha(X(t))$$. The hazard function for the occurrence of a catastrophe is β(X(t)), i.e. (2)$$\Pr\{no catastrophe occurs in the interval({T_{1}},{T_{2}})\}=\exp(-\int\limits_{{{T_{2}}}}^{{{T_{1}}}}{\beta(X(s))ds)}$$. If a catastrophe or downward jump takes place at time T, then it is assumed that (3)$$\Pr\{X(T)\leqq y|X(T-)=x\}=h(x,y)$$, where h is a given function. The problem will be to study the distribution of X(t), especially as t → ∞.

We consider the nonlinear diffusion equation (1.1)$${p_{t}}={p_{{xx}}}-\omega {p_{x}}-\lambda p+f\left({I\left(p\right)} \right)p in\left({0,\ell}\right)\times\left({0,\infty}\right),$$ where $$I\left( p\right)\equiv I\left(p\right)\left({x,t}\right)=\int\limits_{0}^{x}{\left({k+p\left({\xi ,t}\right)}\right)}d\xi.$$ Here κ ≥ 0, λ > 0, ω ≥ 0, and 0 < ℓ ≤ ∞ are given constants, f: [0, ∞) → (0, ∞) is a decreasing Lipschitz function such that lim_r→∞ f(r) = 0, and p: [0, ℓ] × [0, ∞) → [0, ∞) is the unknown. Imposed are the boundary condition (1.2)$${p_{x}}\left({0,t}\right)=\omega p(0,t)$$, (1.3)$$\left\{{_{{\mathop{{\lim }}\limits_{{x \to\infty}}p\left({x,t}\right)= 0if\ell=\infty }}^{{{p_{x}}\left({\ell,t}\right)=\omega p\left({\ell,t}\right)ifell< \infty,}}}\right.$$ for all t > 0 and the initial condition (1.4)$$p=\left({x,0}\right)={p_{0}}\left(x\right)forx \in\left({0,\ell}\right).$$

Aphids are among the most conspicuous and important pests in the greenhouses and in the fields. As such, they have been the subject of intensive studies and various attempts have been made to develop models of their and their predators’ population dynamics (see, e.g., Barlow, 1982; Barlow and Dixon, 1980 and references therein).

We consider two differential systems, suggested by two different ecological models, describing two mutually interacting populations subject to depletion effects (due to harvesting or to predation), cf. [l,2], which share certain “switching” features.

Reaction-diffusion models in population dynamics have been widely investigated in recent years [6]. In particular, quasilinear models have been recently suggested [11] to deal with situations where diffusivity depends on crowding (see also [7]). In the case of two interacting species such models have the general form (1)$$\begin{array}{*{20}{c}} {{u_t} = {\Delta _y}\left( {u,v} \right) + u\;f\left( {x,u,v} \right)} \\ {{v_t} = {\Delta _j}\left( {u,v} \right) + v\;g\left( {x,u,v} \right)} \\ \end{array}$$ the solution satisfying suitable initial and boundary conditions in a bounded domain (here t is the time variable, x the space variable, Δ the diffusion operator, u=u(t,x), v=v(t,x) are population densities, see [6,7,11]).

In order to understand the problem of “chromosomal speciation” (White, 1978) the clarification and quantification of the processes and mechanisms involved as well as theoretical models are required in explaining the onset, fixation, spreading and accumulation of chromosome mutations and their role in reproductive isolation.

We examine the possibility of predator-mediated coexistence of all species with model ecosystems of Volterra type. We are concerned with dynamics of a biological community of three competing species to which one predator is added. Stability at equilibria increases when a predator is included. Oscillatory coexistence in limit cycles or in chaotic motions is also possible with a predator. Stable coexistence is enhanced if a predator prefers a dominant competitor. Even when coexistence of species in the subcommunities is impossible, four species can coexist.

We examine a mathematical model arising from the study of the interaction of algae with light. In fact, let us consider a population of algae suspended in water in sea or in artificial tanks. If n=n(z;t) measures the suspected concentration of the algae that, at time t are suspended in water at depth z (-∞<-b≤z≤0, b is the maximum depth of water), we conjecture the following evolution for n (1a)$$\begin{array}{*{20}{c}} {{u_t} = {\Delta _y}\left( {u,v} \right) + u\;f\left( {x,u,v} \right)} \\ {{v_t} = {\Delta _j}\left( {u,v} \right) + v\;g\left( {x,u,v} \right)} \\ \end{array}$$ In the first term of right hand side of (1a), $$\frac{1}{\tau }$$ represents the death rate of algae, whereas the second term of (1a) represents the growth of the algae concentration due to the absorption of light. This growth is thought to be proportional to the product of algae-concentration with a known function of total radiant energy flux (see equation (1b)). The third term in (1a) is a removal term. The radiant energy satisfies the Boltzmann equation [1].

An important consequence of mass immunization programmes is their tendency to increase the average age at which an individual typically acquires an infection over that which pertained before vaccination. As emphasized in a number of recent papers, such an increase in the average age at infection may, under certain circumstances, result in an increased incidence of infection among older age classes of the community than was the case before the implementation of immunization (Knox, 1980; Dietz, 1981; Cvjetanovic, Grab and Dixon, 1982; Hethcote, 1983; Anderson and May, 1983a). This observation is a cause for concern if the effects of infection are typically more severe among adults than among children.

If an individual is successfully vaccinated he or she is protected against the corresponding infection during the duration of the induced immunity. The vaccination of a certain proportion in a population is of obvious benefit to those covered. But there is also some effect to the rest of the community: the risk of acquiring an infection is reduced. For certain kinds of infections this also results in a reduction of the incidence of disease. Some infections can only lead to disease in higher age groups, like rubella which may cause congenital malformations if the mother gets infected during early pregnancy. In this case a reduction in infection risk results in an increase of age at infection. This may lead to an increase of the total incidence of disease in the population as has been pointed out by Knox (1980), Dietz (1981) and Anderson & May (1983). This example shows that an evaluation of a vaccination strategy has to take into account the consequences on the disease incidence in the total population.

The classical epidemic models derived from the Kermack-McKendrick model consider the prevalence of the disease only. The host population is partitioned into classes of susceptibles S, infectious I, and recovered R. The transition between these classes is described by ordinary differential equations. The following equations take into account the recruitment of new individuals into the class of susceptibles by birth and a death rate depending on the class. The equations read (1)$$\begin{gathered} \dot s = - \beta SI + \gamma R - {\delta _1}S + \left( {{\delta _1}S + {\delta _2}I + {\delta _3}R} \right) \hfill \\ \dot I = \beta SI - \alpha I - {\delta _2}I \hfill \\ \dot R = \alpha I - \gamma R - {\delta _3}R \hfill \\ \end{gathered}$$ Of course these equations are independent of δ₁, and P = S + I + R is an invariant. Thus the equations simplify to (2)$$\begin{gathered} <Emphasis Type="Bold">\dot s = - \beta SI + \left( {\gamma + {\delta _3}} \right)\;\left( {P - S} \right) - \left( {\gamma - {\delta _2} + {\delta _3}} \right)I \hfill \\ \dot I = \beta SI - \left( {\alpha + {\delta _2}} \right)I \hfill \\</Emphasis> <Emphasis Type="Bold">\end{gathered}</Emphasis>$$ The “natural” parameters of the system are β, γ + δ₃, α + δ₂, α + γ + δ₃. In the following we consider only the endemic case γ + δ₃ > 0. The state space of the system (2) is the triangle T = {S ≧ 0, I ≧ 0, S + I ≦ P} which is positively invariant with respect to the flow.

This article deals with modeling an epidemic by utilizing the information of the times of occurrences and removals of the infectives, through a bivariate counting process. This approach incorporates the epidemic features into the model more flexibly than the Markov chain approach. It also provides a convenient framework for statistical analysis. After a brief discussion of the general epidemic model (a Markov Chain) we will present a point process model for epidemics and discuss some statistical inference problems.

The computer simulation of multistate epidemiological models is presented and the conditions for performing simulation experiments are discussed. A general purpose interactive software package for continuous simulation developed on this basis is implemented. Successful tests were made with several models of infectious diseases. The need for the introduction of sensitivity analysis into simulation experiments is emphasized.

A multistate epidemiological model is constructed on the ground of natural history of the disease. The transfers from one class to another are governed by the mathematical relationship between classesof population of the dynamics of hepatitis B is expressed in the system of equations. All rates of transfers are calculated on a daily basis. The model is used for simulating the natural course of the infection as well as the effects of various public health interventions, such as passive and active immunization and/or application of appropriate sanitary and hygienic measures. The model is evaluated through above simulations of actual and hypothetical situations.

The proposition that infectious diseases act as important selective forces for maintaining polymorphisms in host populations is an intriguing part of theories of evolution. Moreover, a proposition of considerable epidemiological importance is that host genetic variability is partially responsible for the maintenance of infectious disease in populations.

This paper is concerned with the analysis of the discrete time version of the general epidemic model. Both deterministic and stochastic models are discussed. Moreover, using an heuristic argument an approximating system is proposed for the case where the population contains initially many susceptibles and little infectives.

The purpose of this note is to demonstrate the need of using models with explicit age-structure when dealing with questions of control of virus transmission.

In an earlier report [2] a general model for common source epidemics was developed and subsequently [3] specialized to an outbreak of toxoplasmosis. The model for infection developed in [4] can be used to refine the results in [3] and provide a stronger physiological basis for the study. In this paper we use epidemiological data collected in the toxoplasmosis outbreak, in particular the daily onset data, to estimate the experimental parameters of the model. We then validate the model with respect to its predictive ability.

The dynamics of many infectious diseases cannot be properly described by a mathematical model, if it does not take the structure of the host population into account. Typical examples are venereal diseases, where the population has to be divided into several subgroups according to sex, symptomatic and asymptomatic infection etc., and diseases also involving intermediate hosts.

The behavior of predators, both natural and human, must be influenced to some extent by the need to obtain information pertaining to the location and abundance of their prey. If prey items are uniformly distributed at random throughout the entire environment, then no particular search strategy is of any avail. (If predation depletes the prey in localized areas, then the uniform distribution is upset, and strategy may become important.)

Logistic and Gompertz models of population growth, with, an extra terato allow for fishing under constant quotas, constant effort, and mixed policies, are considered Stochastic fluctuations of environmental and fishing conditions are added as noise terms. Ultimate extinction occurs with probability one for constant quotas or mixed policies, but not for constant moderate effort policies. For constant effort policies, we show that the fishing efforts that maximize the expected yield are close to the ones obtained in the deterministic models if the noise fluctuations are small. Conditions on the effort in order to control the probability of the population size dropping below some critical threshold are studied. Maximum likelihood techniques of parameter estimation based on yield data are developed for constant effort models.

Mathematical models in conjunction with dynamic input-output data are increasingly used in quantitative studies of endocrine-metabolic systems both in physiology and clinical medicine /1/.The purpose of modeling includes understanding, estimation of internal (non-measurable) parameters, diagnosis and control. Depending on the purpose of the study, models of different complexity are developed, and an essential role is played by identification and validation, respectively the determination of structure and parameter values and the assessment of whether or not the postulated model is adequate for its intended purpose.

The electrocardiographic forward problem consists in the prediction of the potential distribution on the body surface given the geometry and conductivity of the cardiac tissue and the “thorax and the knowledge of the cardiac electric events, i.e., electric changes occurring in the cardiac cells during the depolarization and repolarization processes and the activation sequence of the cardiac tissue. We shall investigate a model concerning the potential generated only in the depolarization phase and, as in the majority of attempts made to solve the forward problem, we confine ourselves to the ventricular depolarization which is the origin of the QRS complex in the electrocardiogram. We denote by H the ventricular heart muscle and in the fol lowing we shall refer only to this part of the heart. During this phase of the heart beat a “thin” layer of cardiac cells undergoes the depolarization process, i.e., a change of the intracellular electric potential w of biochemical origin occurs in a very “short” time interval from a resting value u_r to a plateau value u_a with u_r <u_a; moreover since during a heart beat any cell depolarizes only once and in the QRS phase the repolarization effects are negligible, the following assumptions are usually made: (1.1)The depolarization of a cell is an instantaneous process and it is the same for all ventricular cells (i.e. u_a and u_r are constant in H).(1.2)At any time instant t of the QRS heart beat complex, setting Ha = {x H: w(x)= u_a}, the activated heart tissue, Hr = {x H: w(x) = u_r}, the resting heart tissue, we have $$\bar H = {{\bar H}^a}$$$${{\bar H}^r}$$ and the so called excitation wavefront surface S = ∂Ha ∂Hr is a “regular” surface.

Simple enzyme systems, with interacting diffusion and reaction, are modeled by O.D.E.s involving several parameters and showing, according to the parameter values, multiple steady statesand/or periodic solutions. The 2-parameter continuation of singular points gives some insight into the behaviour of these systems. In particular it yields isola’s.

Cell cycle events (initiation of DNA replication, length of the G₂ phase, occurrence of mitosis and of cell division) follow temporal patterns which are characteristic of each cell type and growth condition, thus suggesting the existence of complex control mechanisms /1,2/. While considerable progress as been made in the description of macromolecular syntheses which characterize different growth conditions, our knowledge on the regulatory mechanisms that coordinate growth with nuclear and cell division is still poor. Clearly the reductionistic approach, which has been so useful for the description of the molecular components of a cell and of the more simple regulatory systems (regulation of enzymatic pathways, control of gene expression in bacteria), is not adequate for the study of highly integrated systems, such those controlling growth and cell division, for which it seems necessary to develop an integrated approach. Mathematical models may well be useful for this undertaking. In fact, they allow quantitative description of both the interrelations among the relevant variables of the phenomenon and the dynamics of the events under consideration. Many mathematical models have been developed in order to describe the dynamics of the cycle events, and their analytical solutions and the simulations, which are often required for a more accurate analysis of their predictions, may help to better understand the regulatory mechanisms of cell proliferation. Comparison of the predictions of the model with experimental results allows verification of the validity of the assumed functional or causal links.

The first transient and steady-state phase transition curves of a model of a molluscan neurone are obtained numerically, and discussed in relation to experimental results.

When a duplex DNA molecule is covalently closed into a circle, conservation of the antiparallel chemical (5′–3′) orientations of the two strands dictates that the resulting ring molecule must have the topological structure of two closed, interlinked circles. This fixes the value of the molecular linking number Lk. This Lk is an integer topological invariant which measures the number of times either strand would have to be passed through the other before they could be physically disentangled (provided the molecule is not knotted). Its value remains fixed, independent of geometric deformation of the molecule, as long as both strands are covalently closed. Lk can only be varied by the introduction of transient strand breaks. A class of enzymes called DNA topoisomerases has been found whose function is to regulate Lk in this manner, suggesting a physiological function for this activity.1,2

Flow microfluorometry is a valuable tool for cell kinetics investi gations [1]. Several mathematical methods are available to evaluate the fraction of cells in cycle phases from a flow-cytometrie (FCM) histogram [2]. A few methods allow to reconstruct the DNA synthesis rate from a histogram measured in balanced exponential growth [3, 4] or from histograms measured during desynchronization [5]. In a recent paper [6], an expression was given to determine the rate of DNA synthesis from a single histogram in balanced growth or from a couple of histograms when the S-phase influx has locally an exponential behaviour. With the aim of evaluating the reliability of the method, in the present paper the rate of DNA synthesis is reconstructed from couples of histograms in saturated stages of growth, obtained by computer simulation.

In this paper we study an assemblage of N cells in which a chemical reaction takes place whose generation term is described by a function g(λ,c_j) where λ stands for a control parameter and c_j denotes the concentration of a substrate in the j-th cell. The cells communicate via membranes M (compare Fig.1) admitting diffusive transport with diffusion coefficients D_j. The j-th cell may be connected to an outside reservoir which feeds constant concentration α into the system through a membrane with the diffusion constant E_j. If a cell is not connected to the reservoir we put E_j=0. Next we introduce the new variable x_j=α-c_j (j = 1, ... N) and define the function f(λ, x)=g(λ, α-x). With this notation the steady states of our assemblage of cells are the solutions of the system (1a)$$({D_2} + {E_1}){x_1} - {D_2}{x_2} = f(\lambda ,{x_1})$$(1b)$$- {D_j}{x_{j - 1}} + ({D_j} + {D_{j + 1}} + {E_j}){x_j} - {D_{j + 1}}{x_{j + 1}} = f(\lambda ,{x_j})\;(j = 2, \ldots ,N - 1)$$(1c)$$- {D_N}{x_{N - 1}} + ({D_N} + {E_N}){x_N} = f(\gamma ,{x_N})$$.

As well known, most types of cells undergo a growth and proliferation cycle, along which, the following three main phases can be identified: the G_0/1 phase, during which the cell either rests or synthesizes RNA and proteinsthe S phase, in which DNA content increases from the resting value up to twice thatthe G₂+M phase, in which DNA content is twice the resting value, cell completes its growth and finally divides in two daughter cells (mythosis).

Body surface mapping is a new electrocardiographic method which consists in displaying the distribution of equipotential lines on the entire surface of the torso at a number of time instants during the cardiac cycle. It is widely recognised that cardiac electromaps provide more information on normal and abnormal intracardiac electrical events, than can be obtained from traditional 12-leads electrocardiograms.

A method that allows a statistical evaluation of the protein coding capacity of any sequenced DNA fragment is presented.The method, applied to a limited number of protein coding gene sequences, allows the identification of some hypothetical protein genes which are coded by the the DNA strand complementary to the coding DNA strand. The finding that some proteins are indeed encoded on opposite DNA strands and that the two genes have complementary codon sequences, led us to investigate a much larger sample of DNA sequences, included in our DNA data bank.The results of this exhaustive investigation demonstrate the existence of a new class of protein coding gene sequences that are coded by the complementary DNA strand.

The non-linear behaviour of a series array of two membranes has been first investigated by Kedem and Katchalsky (1) and by Patlak, Goldstein and Hoffman (2). This study has been subsequently resumed by several different Authors (3–11) and an extension to a series array of n membranes has been attempted by Ludwikòw (12), strictly following the approach by Kedem and Katchalsky.

Metabolic systems are composed of many biochemical reactions and transport processes with very different rates. Their dynamic interaction produces a temporal organization whose salient feature is the existence of a distinct time hierarchy. Only a subset of the dynamic variables of the system moves with a velocity comparable to the characteristic time scale of the whole metabolic system, while others move very quickly and produce a dynamic “rapid substructure”. The mathematical description of this situation leads to differential equations with small parameters as multipliers of the derivatives of fast components (cf. Reich and Sel’kov 1981).

Several ligands exist that strongly modify complex cellular functions by means of reversible interactions with cell membrane receptors, which undergo a reorganization of their surface distribution. As typical “in vitro” examples, we recall the action of the nerve growth factor on PC12 cells (1), the action of dimeric IgE on basophils (2), the action of lectins on lymphocytes (3, 4) and the action of parathyroid hormone (PTH) on osteoblast-like cells (5). In the last three cases (as well as in many others), the external ligands have no effect if Ca++ is not present in the medium. Moreover, it has recently been shown that a lectin (phytohaemagglutinin (PHA)) induces a rise in free calcium inside lymphocytes a few seconds after it has been added to the culture (6). In accordance with these results, it has been suggested (4, 5) that, as a consequence of the ligand-receptor interactions, changes both in the Ca++ binding at the cell surface and in the Ca++ fluxes through the cell membrane can occur, that are mainly due to an increase in the clustering of receptors. In recent years, it has been shown that similar phenomena can also be caused by weak exogenous electric fields (7, 8) which can interfere with the action of some ligands too (5, 9).

One purpose of mathematical models in cell biology is to illustrate and support theoretical arguments, e.g. evaluating alternative modes of regulation in complex systems. Since it is not aimed to mimic the natural phenomena in great detail, a highly simplified mathematical description (e.g. by cutting down the system’s dimensionality) is quite excusable. At the end of the spectrum of approximations, a model provides just an abstract metaphor of reality.

The purpose of this paper is to present a new model for the growth and age structure of a population of a species which reproduces by asymmetric division, i.e. cell reproduction where one clearly can distinguish between the mother and the daughter cell after division as opposed to division patterns where each fission produces two equal daughter cells. The cells of such populations are thus naturally divided into different classes according to the number of divisions they have undergone. A typical example of an organism which divides asymmetrically is Sacehavomyees oevevisiae, which reproduces by budding. After each cell separation a bud scar is left on the wall of the mother cell. These scars can be counted using electron microscopy. We say that a cell with i offsprings and hence i bud scars belongs to the ith scar class. It is supposed that cells of different scar classes behave in a different way, for instance with respect to cell growth and metabolism activity.

The contraction of the heart is triggered by an electric signal going throughout the myocardium, the action potential. At the level of one cell, this phenomenon is due to membrane currents carried by various ions whose conductance is time and voltage dependent. For the nerve axon, these currents have been analyzed and modelled by Hodgkin and Huxley. Since, precise data have become available for the cardiac cells. In this paper we present simulations of the propagation of an action potential. We are interested in the conducting tissue between atria and ventricular myocardium; the Hiss bundle. We use the model of Purkinje cell action potential published by McAllister-Noble-Tsien [4] (MNT).

Periodic solutions of the Hodgkin-Huxley excitation equations bifurcate from equilibria at super- or sub-critical Hopf bifurcations as the applied current density I, maximal specific K+-conductance ḡ_K, the extracellular Ca2+ activity [Ca2+]_o and the extracellular K+ activity [K+]_o or Nernst potential for K+, V_K, are varied as bifurcation parameters. Sections through the bifurcation surface produced when (V_K, ḡ_K, I) is a bifurcation parameter are presented.

The purpose of this paper is to present a family of differential equations yielding a unified scheme which describes in a concise way various phenomena of ontogenetic organization in the central nervous system (C.N.S.).

Flux rates of metabolic and transport processes in whole organs may be determined by conventional compartmental system analysis (1,2). However, this approach is not consistent with the events in real organs characterized by continuous variation of tracer concentrations with space. When the events under study are comparable in rate with transit times or recirculation, their adequate description is only possible by using distributed model systems.

This paper uses branching processes to model the cross-linking of identical particles by multiple ligand-types. We derive gel points and mole- and weight-average cluster sizes for Binomial and Poisson bonding.In Immunology, this model might apply to cross-linking by antibodies specific to different antigenic sites. It represents a refinement of the Goldberg-Watson theory of immune complex formation and makes predictions readily tested by experiment.The model makes the undesirable assumption that no intramolecular bonding occurs. Relaxation of this assumption is mathematically challenging and is of interest to polymer chemists.

Many investigations in studying primary and secondary structures in Biology require theoretical statistical (that is enumerative) work. We solve one of these problems: enumerate secondary structures of single-stranded nucleic acids (RNA, tRNA, etc…) having a given complexity. This parameter has been introduced for energy computation purpose in order to predict the most stable secondary structure. The method relies on the (non-classical) use of non-commutative variables. Some orthogonal polynomials appear. The final solution shows a relationship between the parameter complexity and another parameter appearing in Hydrography and Botanic.

Growth is one of the most evident features of life. We know that it has been studied mathematically for at least several thousand years (in Savageau 1979). Nevertheless, there are still more open questions than answered ones. The growth of cell populations can be examined under many different aspects (reviewed in Voit and Dick 1983 a). The study of growth in one or more dimensions brings up questions about allometric growth and thus the formation of patterns in a very general sense. Both are phenomena that are tightly connected to differentiation and all kinds of control mechanisms. A cell population can be considered as an entity — for instance a tissue or an organism — that increases in volume or length, or as a set of proliferating individuals such as protozoa. Cell population growth obviously depends on the cell cycle whose many details are subject to intensive research. The cell cycle in turn — and therefore cell population growth — is based on genetical, biochemical, and physiological mechanisms. Particular motivation in researching cell population growth stars from the hope to understand cancer and to provide a more effective cancer therapy.

The staging process is not a new concept. The process existed even before the creation of man. Formation of the heavenly body, evolution of living things, advancement of civilization and of sciences, as well as the development of social structure, religious faith, political systems, etc. are all by stages. At the micro level, metamorphosis in biology and development of a fetus are good examples, each of which follows a definite staging process. This concept became eminent in recent years in survival analysis when it was recognized that development of many chronic conditions is by stages and. patients in different stages are subject to different chances of dying. It was in the study of survival analysis of such patients that a stochastic model of the staging process was formulated. We shall briefly review the staging process in terms of survival and death of chronic patients, and present its applications in carcinogenesis, fertility and epidemics.

Most of the present compartmental modeling and analysis found in the applied literature is based on a classical deterministic formulation (1–4). An alternative stochastic formulation has developed rapidly in the mathematical modeling literature (5–7), but the practical application of this formulation to experimental data analysis has developed at a much slower pace. Two recent reviews have now shown that the stochastic model may be useful in practical applications for several reasons related to the statistical analysis of data (8,9).

Dynamic studies with radioactive tracers require measurements of instantaneous tracer activity X(t) as a function of time; this is usually done either with a counter set for a sequence of short intervals of time, or with a ratemeter.

This paper is concerned with the problem of a priori identifiability of linear time-invariant compartmental systems for experimental configurations with two separate input and two output. We will check for unique identifiability for a particular class of models which often occur in both biomedical and ecological studies /1/.

Compartmental and noncompartmental linear models are widely used in physiological and clinical studies (e.g. endocrinology, metabolism, pharmacokinetics) for the interpretation of kinetic data obtained from linearising input-output experiments /1/. Recently some bases for choice between these two approaches have been discussed for the case where only one compartment is accessible for test input and output /2/. Both modeling strategies have been applied for studying the individual kinetics of ketone bodies (KB), i.e. of acetoacetate (AcAc) and 3hydroxybutyrate (βOHB), in man /3,4/. This is a situation where two pools are accessible and a two input-four output tracer experiment has been proved to provide an adequate data base. In this paper we discuss the relative merits of compartmental vs noncompartmental modeling by using the ketone body system as a prototype. The example allows us to evidence the structural (i.e. in terms of system connectivity) conditions for which noncompartmental modeling is inappropriate and to discuss some (superimposed) ambiguities which arise from computational problems when a noncompartmental model is numerically quantified from the data.

Bellman-Äström (1970) [2] have stated the question of identifiability for time-invariant linear compartmental systems in general input-output configurations. Di Stefano (1977) [8], Eisenfeld (1979) [9], Cobelli and all (1979) [3] have shown that the controllability, the observability and the minimality are not in general necessary and/or sufficient conditions for the identifiability of such systems.

We present convergence arguments for algorithms developed to estimate spatially and/or time dependent coefficients and boundary parameters in general transport (diffusion, advection, sink/source) models in a bounded domain Ω ⊂ R2. A brief summary of numerical results obtained using the algorithms is given.

Delay differential or differential difference or functional differential equations arise in models of biological phenomena when the time delays occurring in these phenomena are taken into account. This will be illustrated below with some predator-prey models. If there are equilibrium states in these equations, it is important to determine sets of parameter values for which these states are stable (locally or globally), and critical parameter values at which stability may be lost. Often the loss of stability may be associated with Hopf bifurcation and the onset of oscillatory behavior. The local stability problem is usually analyzed by linearization around the equilibrium point. For an equation with a single discrete time lag T this could lead to a delay differential equation of retarded type of the form [1]$$ \sum\limits_{k = 0}^n {{a_k}\frac{{{d^k}u(t)}}{{d{t^k}}} + \sum\limits_{k = 0}^m {{b_k}} } \frac{{{d^k}u(t - T)}}{{d{t^k}}} = 0 $$ Associated with this equation is the characteristic equation [2]$$ P(z) + Q(z){e^{{T_z}}} = 0 $$ where $$ P(z) = \sum\limits_{k = 0}^n {{a_{{k^{{z^k}}}}}} ,\quad Q(z) = \sum\limits_{k = 0}^m {{b_{{k^{{z^k}}}}}} $$ It is well-known that the necessary and sufficient condition for uniform asymptotic stability is that all roots of the characteristic equation lie in the left half-plane, Re z < 0. However, it is a difficult mathematical problem to determine when this condition will be satisfied.

In all biochemical processes of practical interest such as fermentation of industries, pharmaceuticals and waste treatment, the reaction rates are influenced by diffusion and the equations are nonlinear [11,12], Several interacting populations related to ecology [10] as well as problems of chemical engineering [14] also yield nonlinear elliptic equations as a by-product. Since constructive proofs of existence, which can also provide numerical procedures for the computation of solutions, are of greater value than nonconstructive proofs, the method of upper and lower solutions coupled with the monotone iterative method has been, in recent years, employed to prove existence of multiple solutions of a variety of nonlinear problems [8].

Everyone would certainly agree that mathematics is becoming an indispensable tool of investigation in the biological sciences. What is probably less evident is that in turn mathematicians have been often guided in their investigations by needs and indications deeply rooted in the realm of biology, to the point that a number of by now well developed fields of mathematics would have probably remained unexplored without the pressing demands posed by their colleagues engaged in the biological discovery. The theory of reaction-diffusion equations is an example of a branch of mathematics whose origin relies on questions and problems of biological nature. A second example is provided by the theory of (stochastic) singular diffusion equations developed by W. Feller in order to account for some new and somewhat “strange” features exhibited by simple equations derived to describe the growth of a population subject to random effects. A third example is offered by the revamped interest of mathematicians towards the so-called first passage time problems for random processes with a view to working out algorithms and effective numerical procedures for the evaluation of neuronal firing distributions and population extinction probabilities.

In the context of various models appearing in biology, genetics and medicine, the concepts of exchangeability and its extension partial exchangeability are discussed. It is pointed out that these concepts appear either as a basic assumption in some models or are inherent in the composition of a population (or subpopulations). That is, when the population is homogeneous as a whole or can be partitioned to several subgroups which are homogeneous within themselves. Also extensions of some of the existing results in the literature are carried out.

In previous papers (Beretta et al, 1979; Beretta and Lazzari, 1983) we obtained asymptotic stability criteria concerning positive equilibria of chemical networks with mass action kinetics.

The purpose of the present note is twofold, namely to present some results (obtained by the authors in [3]) on existence, uniqueness and attractivity of stationary solutions of certain quasilinear systems, and to show how such results apply to some models of mathematical biology.

In the present paper some results recently obtained by the authors on the asymptotic behaviour of the principal eigenvalue of certain operators are summarized and discussed.

The connection between positive feedback and instability has been widely observed in network theory. In this paper several points are made regarding this connection from the point of view of developing a criteria for determining which models from a class of possible models are unstable or admit an exchange of stabilities. The criteria may be used as a preliminary test of model design.

In this paper we show how Lyapunov functions may be used to investigate qualitative properties (recurrence, existence of invariant sets, etc.) for a class of stochastic diffusion models which describe interaction between two populations.

In this note we collect some results (see [3], [4]) concerning a Cauchy problem for a non linear and non local partial differential equation arising in a model of the contracting mechanism of the cardiac muscle. We refer to a model that has been introduced by physiologists and mathematicians of the University of Pavia, starting from the sliding filament theory δf Huxley (see [1], [2], [5], [6]).

In view of the long and fruitful partnership between mathematics and physics, it was only natural that the first applications of mathematics in biology should take theoretical physics as a model. Indeed, Lotka (1924) entitled his pioneering work Elements of Physical (not, as in the 1956 reprint, Mathematical) Biology, claiming as his intention the ‘application of physical principles and methods in the contemplation of biological systems’.

For a long time, the major goal of biological and medical research was a more and more detailed analysis of the elementary parts of an organism. Now, in some of the simpler organisms like Escherichia coli, a very large number of proteins (Bloch et al., 1980) and many other compounds are known and the question arises how these elements work and interact in an integrated system. On a theoretical level, the first approach in answering this question has been a linear systems analysis that is based on the multitude of simple but powerful methods of linear mathematics. However, biological and medical phenomena often are quite nonlinear, and in most cases their underlying processes cannot simply be studied separately and then superimposed again, because these processes may influence each other. That is, a biological system as a whole frequently reacts differently than the sun of all its components. Such a system is called “cooperative” (if you like Latin) or “synergistic” (if you prefer Greek).