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     

Analytical solution of the cable equation with synaptic reversal potentials for passive neurons with tip-to-tip dendrodendritic coupling

A passive cable model is presented for a pair of electrotonically coupled neurons in order to investigate the effects of tip-to-tip dendrodendritic gap junctions on the interaction between excitation and either pre or postsynaptic inhibition. The model represents each dendritic tree by a tapered equivalent cylinder attached to an isopotential soma. Analytical solution of the cable equation with synaptic reversal potentials is considered for each neuron to yield a system of Volterra integral equations for the voltage. The solution to the system of linear integral equations (expressed as a Neumann series) is used to determine the current spread within the two coupled neurons, and to re-examine the sensitivity of the soma potentials (in particular) to the coupling resistance for various loci of synaptic inputs. The model is actually posed generally, so that active as well as passive properties could be considered. In the active case, a system of non-linear integral equations is derived for the voltage.     

A nonlinear integral equation for visual impedance

The Hartline-Ratliff equation is a linear integral equation of the second kind and is employed in modeling inhibitory networks. Saturation of the inhibiting elements is commonly modeled as a function whose form is sigmoid; however, the resulting integral equation is nonlinear. Whenever the unknown function within the integral is hypothesized to be a nondecreasing nonlinear function, the Hartline-Ratliff equation becomes a nonlinear integral equation of the Hammerstein type. We present existence and uniqueness theorems for a Hammerstein equation which represents a further generalization of the Hartline-Ratliff equation.     

Some types of relaxation oscillations as models of all-or-none phenomena

After some general remarks the results of the approximation method of Kryloff and Bogoliuboff are applied to show the existence of a threshold for a simple non-linear differential equation of a special form.     

On oscillatory responses of the Limulus retina

Published observations of the dynamic properties of lateral and self-inhibition in the Limulus retina lead to a non-linear integral equation for the response of ommatidia located near the center of a uniformly illuminated region. Coleman and Renninger (1976, 1978) showed that when the excitation is constant in time and the sum of the inhibitory coefficients for the illuminated region exceeds a critical value, the integral equation has a stable periodic solution describing a sustained, spatially synchronized, oscillatory response in which bursts of activity alternate with silent periods. Such spatially synchronized bursting has been observed in the Limulus retina in situ by Barlow and Fraioli (1978), using the preparation of Barlow and Kaplan (1971). Employing experimental data on the temporal dependence of lateral and self-inhibition, which were then available only for the excised eye, Coleman and Renninger calculated a value of 0.34 s for the period p of the bursting response, which is significantly above the range, 0.11–0.20 s, of values of p observed for the Limulus eye in situ. Brodie et al. (1978) have recently published measurements of the temporal dependence of lateral and self-inhibition for the in situ preparation. Here we show that when the kernel functions in Coleman and Renninger's integral equation are chosen in accord with these new data, the periodic solutions of the equation have a period of approximately 0.13s, which is in the range (0.11–0.20 s) required for agreement with experiment. Other properties of the periodic solutions, i.e., their general form and the threshold levels of inhibition required for their existence, are also in accord with published observations of the behavior of the retina in situ.     

Transient behavior of the single loop solute cycling countercurrent multiplier

Transient behavior of a single loop solute cycling countercurrent multiplier is described by a Volterra type integral equation similar to that describing circulation of an indicator in the systemic circulation. Solution of the equation is given for pumping from ascending to descending flow proportional to concentration in ascending flow and no back leak. This exact solution is compared with an approximate solution of the integral equation and with a solution in which the flow system is represented by a finite Markov chain. Agreement between the Markov approximation and the exact solution is excellent.     

Random Search Algorithm for Solving the Nonlinear Fredholm Integral Equations of the Second Kind

In this paper, a randomized numerical approach is used to obtain approximate solutions for a class of nonlinear Fredholm integral equations of the second kind. The proposed approach contains two steps: at first, we define a discretized form of the integral equation by quadrature formula methods and solution of this discretized form converges to the exact solution of the integral equation by considering some conditions on the kernel of the integral equation. And then we convert the problem to an optimal control problem by introducing an artificial control function. Following that, in the next step, solution of the discretized form is approximated by a kind of Monte Carlo (MC) random search algorithm. Finally, some examples are given to show the efficiency of the proposed approach.     

Bionomics dynamic model of a class of competition systems ☆

Differential equation problem is an important research topic in the international academia. In accordance with certain ecological phenomena, previous research was conducted based on simple observational and statistical data. But this approach does not effectively study the essence of the ecological phenomena. Recently, one dynamic approach has been proposed for the study of ecology in the international academia. According to this approach, first of all, the ecology is reduced to the differential equation model which represents the essential phenomenon, and then the dynamic law and rules of mathematics and biology will be studied. Currently, an extensive research is conducted on the differential equation problem. This paper primarily explores a type of competitive ecological model, which is a system of differential equation with infinite integral. we first study the existence of positive periodic solution to this model, and then present sufficient conditions for the global attractivity of positive periodic solutions.     

A statistical approach to the theory of the central nervous system

A “probabilistic” rather than a “deterministic” approach to the theory of neural nets is developed. Neural nets are characterized by certain parameters which give the probability distributions of different kinds of synaptic connections throughout the net. Given a “state” of the net (i.e., the distribution of firing neurons) at a given moment, an equation for the state at the next moment of quantized time is deduced. Certain very special cases involving constant distributions are solved. A necessary condition for a steady state is deduced in terms of an integral equation, in general non-linear.     

A model for the spatial spread of an epidemic

Summary We set up a deterministic model for the spatial spread of an epidemic. Essentially, the model consists of a nonlinear integral equation which has an unique solution. We show that this solution has a temporally asymptotic limit which describes the final state of the epidemic and is the minimal solution of another nonlinear integral equation. We outline the asymptotic behaviour of this minimal solution at a great distance from the epidemic's origin and generalize D. G. Kendall's pandemic threshold theorem (1957).     

Solitary waves in prestressed elastic tubes

In this work, we studied the propagation of non-linear waves in a pre-stressed thin elastic tube filled with an inviscid fluid. In the analysis, analogous to the physiological conditions of the arteries, the tube is assumed to be subject to a uniform pressureP ₀ and a constant axial stretch ratio λ_z. In the course of blood flow it is assumed that a large dynamic displacement is superimposed on this static field. Furthermore, assuming that the displacement gradient in the axial direction is small, the non-linear equation of motion of the tube is obtained. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the long-wave approximation is investigated. It is shown that the governing equations reduce to the Korteweg-deVries equation which admits a solitary wave solution. The result is discussed for some elastic materials existing in the literature.     

The mathematical theory of lateral impedance of visual activity and an application of Markov processes

The mathematical relationship describing recurrent lateral inhibition is expressed as a linear operator equation. Under quite general conditions, the operator equation is shown to have a unique nonnegative solution. It is also shown that the linear operator for recurrent impedance is representable as an integral operator and that, when in application to physiological models it is interpreted as recurrent inhibition, the corresponding linear equation assumes a form more general than the well known Hartline-Ratliff equation. Finally, we introduce a class of impedance operators based on the probabilistic theory of Markov processes, solve the corresponding linear integral equation, and apply the theoretical properties of the solution to the analysis of physiological and psychophysical phenomena.     

Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis

The existence of spatially localized solutions in neural networks is an important topic in neuroscience as these solutions are considered to characterize working (short-term) memory. We work with an unbounded neural network represented by the neural field equation with smooth firing rate function and a wizard hat spatial connectivity. Noting that stationary solutions of our neural field equation are equivalent to homoclinic orbits in a related fourth order ordinary differential equation, we apply normal form theory for a reversible Hopf bifurcation to prove the existence of localized solutions; further, we present results concerning their stability. Numerical continuation is used to compute branches of localized solution that exhibit snaking-type behaviour. We describe in terms of three parameters the exact regions for which localized solutions persist.     

Integral Equation Solution for Biopotentials of Single Cells

A Fredholm integral equation of the second type is developed for the biopotentials of single cells. Two singularities arise in the numerical solution of this integral equation and methods for handling them are presented. The problem of a spherical cell in an applied uniform field is used to illustrate the technique.     

Two-dimensional Fourier representation used in the bioelectric forward problem.

The objective of this paper is the application of two-dimensional discrete Fourier transformation for solving the integral equation of the bioelectric forward problem. Therefore, the potential, the source term, and the integral equation kernel are assumed to be sampled at evenly spaced intervals. Thus the continuous functions of the problem domain can be expressed by their two-dimensional discrete Fourier transform in the spatial frequency domain. The method is applied to compute the surface potential generated by an eccentric dipole in a homogeneous spherical conducting medium. The integral equation for the potential is solved in the spatial frequency domain and the value of the potential at the sampling points is obtained from inverse Fourier transformation. The solution of the presented method is compared to both, an analytic solution and a solution gained from applying the boundary element method. Isoparametric quadrilateral boundary elements are used for modeling the spherical volume conductor in the boundary element solution, while in the two-dimensional Fourier transformation method the volume conductor is represented by a parametric boundary surface approximation.     

EXISTENCE OF MULTIPLE PERIODIC SOLUTIONS TO A NONLINEAR INTEGRAL EQUATION FROM THE THEORY OF EPIDEMICS

Thenonlinearintegralequation(1.1)isageneralizedmodelforthespreadofdiseaseswithseasonaldependence.Inthispaper,wehaveprovedtheexistenceofatleastthreenontrivialnonnegativeperiodicsolutionstothisequation.     

Analysis of a nonlinear diffusion problem With Michaelis-Menten kinetics by an integral equation method

A mathematical model for oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics is analyzed by means of an intergral equation method. It is shown that an integral equation formulation can be used to obtain a numerical solution associated with this boundary and initial value problem. Through an illustrative numerical calculation we are able to obtain an accurate solution for both the steady and transient problems. Finally, a comparison is made with the numerical solution of McElwain and the variational solution of Anderson and Arthurs for the steady state and Lin's result concerning the unsteady state.     

林分随机生长模型与Richard模型

本文从一般的假设出发,很自然导出连续状态空间的树木直径分布生长方程,证明此方程解的存在性,并给出一个例子说明Richards方程是直径分布生长方程的一个特解的二阶矩．     

Analytical solutions for global geodesic acoustic modes in tokamak plasmas

Analytical solutions for global geodesic acoustic modes in the plasma of a tokamak with circular concentric magnetic surfaces are obtained. In the framework of ideal magnetohydrodynamics, an integral equation for eigenvalues (dispersion relation) taking into account toroidal coupling between electrostatic perturbations and electromagnetic perturbations with the poloidal mode number |m| = 2 is derived. In the absence of such coupling, the dispersion relation yields only the standard continuous spectrum. The existence of a global geodesic acoustic mode is analyzed for equilibria with both on-axis and off-axis maxima of the local geodesic acoustic frequency. The analytical results are compared with results of numerical calculations.     

分数阶微分方程边值问题正解的唯一性

分数阶微分方程在生物科学和其他领域发挥着重要作用.考虑来类分数阶微分方程边值问题正解的这来性,得到的结证不仅可以保证正解的存在这来,而且可以用于构造迭代序列来逼近这个解.