Assessing the heterogeneity of a disease spread through a community

A multivariate non-parametric test and a semi-parametric regression model via counting process are proposed for detecting the heterogeneity of a disease spread through a community. The infection rates are allowed to depend on time in an arbitrary manner. Infectious data usually are not completely observed, nevertheless only partial information of the epidemic is needed for the suggested methods. The testing procedures and the associated methods of analysis are illustrated with reference to epidemics of respiratory disease on the Island of Tristan da Cunha in the South Atlantic     

Nonlinear models of diffusion on a finite space

Summary The basic convergence theorems for finite state Markov chains are extended to the nonlinear case. An operatorT inl ₁ of a finite space with counting measure is called nonexpansive if Tf-Tg₁f-g₁ holds for allf, g. It is shown that, for anyf, there exists an integer p>=1 such thatT ^pnf converges. Sufficient conditions forp=1 are given. In the case of continuous parameter nonexpansive semigrous {T _t, t>=0},T _tf converges fort.The main tool is a geometric theorem on isometriesS of compact subsets of the abovel ₁: It is shown that any orbit underS is finite.The exponential speed of convergence does not extend from the Markov chain case to nonlinearT.This research has been done during a visit of M.A.A. to the University of Göttingen. The principal results were announced in C.R. Math. Rep. Acad. Sci. Canada Vol. VIII, 1. Feb. 1986The research of this author is supported in part by an N.S.E.R.C.-Grant     

Nonlinear diffusion with a bounded stationary level surface

We consider nonlinear diffusion of some substance in a container (not necessarily bounded) with bounded boundary of class C²$C^2$. Suppose that, initially, the container is empty and, at all times, the substance at its boundary is kept at density 1. We show that, if the container contains a proper C²$C^2$-subdomain on whose boundary the substance has constant density at each given time, then the boundary of the container must be a sphere. We also consider nonlinear diffusion in the whole RN${\mathbb{R}}^N$ of some substance whose density is initially a characteristic function of the complement of a domain with bounded C²$C^2$ boundary, and obtain similar results. These results are also extended to the heat flow in the sphere SN${\mathbb{S}}^N$ and the hyperbolic space HN${\mathbb{H}}^N$.     

Nonlinear diffusion equation and Liesegang rings

Doklady Mathematics -     

Nonlinear diffusion of impurities in semiconductors

We study the concentration-dependent diffusion of dopant impurities into semiconductors. In particular, we examine the two-dimensional diffusion in the vicinity of a mask. Numerical solutions are obtained for dopant diffusion with fixed-total-concentration and with constant-surface-concentration. For the fixed-total-concentration case, we also obtain approximate power series solutions. Our numerical and approximate results are compared with analytical and numerical results obtained by other investigators.     

Nonlinear wave propagation through a stratified atmosphere

We examine the propagation of sound waves through a stratified atmosphere. The method of multiple scales is employed to obtain an asymptotic equation which describes the evolution of sound waves in an atmosphere with spatially dependant density and entropy fields. The evolution equation is an inviscid Burger-like equation which contains quadratic and cubic nonlinearities, and a curvature term all of which are functions of the space variables. A model equation is derived when the modulations of the signal in a direction transverse to the direction of propagation become significant.     

Flow through a porous medium with mass removal and diffusion

A one-dimensional model of filtration through a saturated porous medium with a mutual action between the solid matrix and the flow is investigated. The substances removed from the porous soil are both particles transported by the liquid flow and substances which diffuse in the fluid. Received June 30, 1997     

Global dynamics of a spatial heterogeneous viral infection model with intracellular delay and nonlocal diffusion

In this paper, we propose a spatial heterogeneous viral infection model, where heterogeneous parameters, the intracellular delay and nonlocal diffusion of free virions are considered. The global well-posedness, compactness and asymptotic smoothness of the semiflow generated by the system are established. It is shown that the principal eigenvalue problem of a perturbation of the nonlocal diffusion operator has a principal eigenvalue associated with a positive eigenfunction. The principal eigenvalue plays the same role as the basic reproduction number being defined as the spectral radius of the next generation operator. The existence of the unique chronic-infection steady state is established by the super-sub solution method. Furthermore, the uniform persistence of the model is investigated by using the persistence theory of infinite dimensional dynamical systems. By setting the eigenfunction as the integral kernel of Lyapunov functionals, the global threshold dynamics of the system is established. More precisely, the infection-free steady state is globally asymptotically stable if the basic reproduction number is less than one; while the chronic-infection steady state is globally asymptotically stable if the basic reproduction number is larger than one. Numerical simulations are carried out to illustrate the effects of intracellular delay and diffusion rate on the final concentrations of infected cells and free virions, respectively.     

Nonlinear reflecting diffusion process,and the propagation of chaos and fluctuations associated

A nonlinear diffusion satisfying a normal reflecting boundary condition is constructed and a result of propagation of chaos for a system of interacting diffusing particles with normal reflecting boundary conditions is proven. Then a gaussian limit for the fluctuation field which is defined in L₀²(B) of a Wiener type space B is obtained. The covariance of the gaussian limit is computed in terms of a Hilbert-Schmidt operator on L₀²(B).     

Nonlinear sloshing and passage through resonance in a shallow water tank

This paper is concerned with the effect of slowly changing the length of a tank on the nonlinear standing waves (free vibrations) and resonant forced oscillations of shallow water in the tank. The analysis begins with the Boussinesq equations. These are reduced to a nonlinear differential-difference equation for the slow variation of a Riemann invariant on one end. Then a multiple scale expansion yields a KdV equation with slowly changing coefficients for the standing wave problem, which is reduced to a KdV equation with a variable dispersion coefficient. The effect of changing the tank length on the number of solitons in the tank is investigated through numerical solutions of the variable coefficient KdV equation. A KdV equation which is “periodically” forced and slowly detuned results for the passage through resonance problem. Then the amplitude-frequency curves for the fundamental resonance and the first overtone are given numerically, as well as solutions corresponding to multiple equilibria. The evolution between multiple equilibria is also examined.     

基于元胞自动机的植物病虫害传播的计算机仿真

利用元胞自动机方法建立植物病虫害传播的数学模型。在此基础上,分别对两种不同病虫害来源的情况进行仿真。仿真结果表明,在参数给定的情况下,无论病虫来源于自身还是外界,植物病虫害的传播均在一定时间后达到稳定状态,不同状态元胞占有率相近;相同参数下,同病虫来源于自身相比,植物病虫从外界入侵时,植物被感染的变化率较低,病虫害传播路径较有规律,有利于病虫害源的确定和病虫害的治理。     

Nonlinear sloshing and passage through resonance in a shallow water tank

This paper is concerned with the effect of slowly changing the length of a tank on the nonlinear standing waves (free vibrations) and resonant forced oscillations of shallow water in the tank. The analysis begins with the Boussinesq equations. These are reduced to a nonlinear differential-difference equation for the slow variation of a Riemann invariant on one end. Then a multiple scale expansion yields a KdV equation with slowly changing coefficients for the standing wave problem, which is reduced to a KdV equation with a variable dispersion coefficient. The effect of changing the tank length on the number of solitons in the tank is investigated through numerical solutions of the variable coefficient KdV equation. A KdV equation which is “periodically” forced and slowly detuned results for the passage through resonance problem. Then the amplitude-frequency curves for the fundamental resonance and the first overtone are given numerically, as well as solutions corresponding to multiple equilibria. The evolution between multiple equilibria is also examined.Received: December 16, 2003     

Group theoretic approach for solving the problem of diffusion of a drug through a thin membrane

The transformation group theoretic approach is applied to study the diffusion process of a drug through a skin-like membrane which tends to partially absorb the drug. Two cases are considered for the diffusion coefficient. The application of one parameter group reduces the number of independent variables by one, and consequently the partial differential equation governing the diffusion process with the boundary and initial conditions is transformed into an ordinary differential equation with the corresponding conditions. The obtained differential equation is solved numerically using the shooting method, and the results are illustrated graphically and in tables.     

Nonlinear stochastic time-fractional slow and fast diffusion equations on Rd

This paper studies the nonlinear stochastic partial differential equation of fractional orders both in space and time variables: $\left({?}^{\beta }+\frac{\nu }{2}{(?\Delta )}^{\alpha ∕2}\right)u(t,x)=I_t^{\gamma }\left[\rho \left(u(t,x)\right)\stackrel{?}{W}(t,x)\right],t>0,x\in {\mathbb{R}}^d,$where $\stackrel{?}{W}$ is the space–time white noise, $\alpha \in (0,2]$, $\beta \in (0,2)$, $\gamma \ge 0$ and $\nu >0$. Fundamental solutions and their properties, in particular the nonnegativity, are derived. The existence and uniqueness of solution together with the moment bounds of the solution are obtained under Dalang’s condition: $d<2\alpha +\frac{\alpha }{\beta }min(2\gamma ?1,0)$. In some cases, the initial data can be measures. When $\beta \in (0,1]$, we prove the sample path regularity of the solution.     

Nonlinear singular sturm-liouville problems and an application to transonic flow through a nozzle

We consider a class of singular Sturm-Liouville problems with a nonlinear convection and a strongly coupling source. Our investigation is motivated by, and then applied to, the study of transonic gas flow through a nozzle. We are interested in such solution properties as the exact number of solutions, the location and shape of boundary and interior layers, and nonlinear stability and instability of solutions when regarded as stationary solutions of the corresponding convective reaction-diffusion equations. Novel elements in our theory include a priori estimate for qualitative behavior of general solutions, a new class of boundary layers for expansion waves, and a local uniqueness analysis for transonic solutions with interior and boundary layers.     

Unsteady convective diffusion of a solute in a Hagen-Poiseuille flow through a tube with permeable wall

Influence of Interphase Mass Transfer (IMT) on the unsteady convective diffusion in a fluid flow through a tube surrounded by a porous medium is examined against the background of no IMT. The three coefficients namely exchange coefficient, convection coefficient, and dispersion coefficient are evaluated asymptotically at large-time. The exchange coefficient exists due to IMT. All-time analysis is made analytically when there is no IMT. The mean concentration distribution is measured at a point inside and outside the slug. The peak of mean concentration is higher than that of pure convection and it is further enhanced with increase of porous parameter. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)