New Operational Matrices for Solving Fractional Differential Equations on the Half-Line

In this paper, the fractional-order generalized Laguerre operational matrices (FGLOM) of fractional derivatives and fractional integration are derived. These operational matrices are used together with spectral tau method for solving linear fractional differential equations (FDEs) of order ν (0 < ν < 1) on the half line. An upper bound of the absolute errors is obtained for the approximate and exact solutions. Fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on fractional-order generalized Laguerre functions and compare them with other methods. Several numerical examples are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.     

On modeling two immune effectors two strain antigen interaction

In this paper we consider the fractional order model with two immune effectors interacting with two strain antigen. The systems may explain the recurrence of some diseases e.g. tuberculosis (TB). The stability of equilibrium points are studied. Numerical solutions of this model are given. Using integer order system the system oscillates. Using fractional order system the system converges to a stable internal equilibrium. Ulam-Hyers stability of the system has been studied.     

Numerical simulation of time fractional dual-phase-lag model of heat transfer within skin tissue during thermal therapy

In this paper, we investigated the thermal behavior in living biological tissues using time fractional dual-phase-lag bioheat transfer (DPLBHT) model subjected to Dirichelt boundary condition in presence of metabolic and electromagnetic heat sources during thermal therapy. We solved this bioheat transfer model using finite element Legendre wavelet Galerkin method (FELWGM) with help of block pulse function in sense of Caputo fractional order derivative. We compared the obtained results from FELWGM and exact method in a specific case, and found a high accuracy. Results are interpreted in the form of standard and anomalous cases for taking different order of time fractional DPLBHT model. The time to achieve hyperthermia position is discussed in both cases as standard and time fractional order derivative. The success of thermal therapy in the treatment of metastatic cancerous cell depends on time fractional order derivative to precise prediction and control of temperature. The effect of variability of parameters such as time fractional derivative, lagging times, blood perfusion coefficient, metabolic heat source and transmitted power on dimensionless temperature distribution in skin tissue is discussed in detail. The physiological parameters has been estimated, corresponding to the value of fractional order derivative for hyperthermia treatment therapy.     

Fractional modeling dynamics of HIV and CD4<Superscript>+</Superscript> T-cells during primary infection

In this paper, we introduce fractional-order into a model of HIV-1 infection of CD4⁺ T cells. We study the effect of the changing the average number of viral particles N with different sets of initial conditions on the dynamics of the presented model. Generalized Euler method (GEM) will be used to find a numerical solution of the HIV-1 infection fractional order model.     

Fractional Poisson–Nernst–Planck Model for Ion Channels I: Basic Formulations and Algorithms

In this work, we propose a fractional Poisson–Nernst–Planck model to describe ion permeation in gated ion channels. Due to the intrinsic conformational changes, crowdedness in narrow channel pores, binding and trapping introduced by functioning units of channel proteins, ionic transport in the channel exhibits a power-law-like anomalous diffusion dynamics. We start from continuous-time random walk model for a single ion and use a long-tailed density distribution function for the particle jump waiting time, to derive the fractional Fokker–Planck equation. Then, it is generalized to the macroscopic fractional Poisson–Nernst–Planck model for ionic concentrations. Necessary computational algorithms are designed to implement numerical simulations for the proposed model, and the dynamics of gating current is investigated. Numerical simulations show that the fractional PNP model provides a more qualitatively reasonable match to the profile of gating currents from experimental observations. Meanwhile, the proposed model motivates new challenges in terms of mathematical modeling and computations.     

On the Order of the Fractional Laplacian in Determining the Spatio-Temporal Evolution of a Space-Fractional Model of Cardiac Electrophysiology

Space-fractional operators have been used with success in a variety of practical applications to describe transport processes in media characterised by spatial connectivity properties and high structural heterogeneity altering the classical laws of diffusion. This study provides a systematic investigation of the spatio-temporal effects of a space-fractional model in cardiac electrophysiology. We consider a simplified model of electrical pulse propagation through cardiac tissue, namely the monodomain formulation of the Beeler-Reuter cell model on insulated tissue fibres, and obtain a space-fractional modification of the model by using the spectral definition of the one-dimensional continuous fractional Laplacian. The spectral decomposition of the fractional operator allows us to develop an efficient numerical method for the space-fractional problem. Particular attention is paid to the role played by the fractional operator in determining the solution behaviour and to the identification of crucial differences between the non-fractional and the fractional cases. We find a positive linear dependence of the depolarization peak height and a power law decay of notch and dome peak amplitudes for decreasing orders of the fractional operator. Furthermore, we establish a quadratic relationship in conduction velocity, and quantify the increasingly wider action potential foot and more pronounced dispersion of action potential duration, as the fractional order is decreased. A discussion of the physiological interpretation of the presented findings is made.     

Computational optimal control for the time fractional convection-diffusion-reaction system

This paper proposes a numerical approximation method for computational optimal control of a time fractional convection-diffusion-reaction system. The proposed method involves discretizing the spatial domain by finite element method, approximating the admissible controls by control parameterization, and then obtaining an optimal parameter selection problem which can be solved by numerical optimization algorithms such as sequential quadratic programming. Specifically, an implicit finite difference method is employed to solve the time fractional system, and the sensitivity method for gradient computation in integer order optimal control problems is adjusted to the fractional order case. Simulation results demonstrate the validity and accuracy of the proposed numerical approximation method.     

Optimal control of hybrid variable-order fractional coronavirus (2019-nCov) mathematical model; numerical treatments

A novel coronavirus is a serious global issue and has a negative impact on the economy of Egypt. According to the publicly reported data, the first case of the novel corona virus in Egypt was reported on 14 February 2020. Total of 96753 cases were recorded in Egypt from the beginning of the pandemic until the eighteenth of August, where 96, 581 individuals were Egyptians and 172 were foreigners. Recently, many mathematical models have been considered to better understand coronavirus infection. Most of these models are based on classical integer-order derivatives which can not capture the fading memory and crossover behavior found in many biological phenomena. Therefore, we study the coronavirus disease in this paper by exploring the dynamics of COVID-19 infection using new variable-order fractional derivatives. This paper presents an optimal control problem of the hybrid variable-order fractional model of Coronavirus. The variable-order fractional operator is modified by an auxiliary parameter in order to satisfy the dimensional matching between the both sides of the resultant variable-order fractional equations. Existence, uniqueness, boundedness, positivity, local and global stability of the solutions are proved. Two control variables are considered to reduce the transmission of infection into healthy people. To approximate the new hybrid variable-order operator, Grünwald-Letnikov approximation is used. Finite difference method with a hybrid variable-order operator and generalized fourth order Runge-Kutta method are used to solve the optimality system. Numerical examples and comparative studies for testing the applicability of the utilized methods and to show the simplicity of these approximation approaches are presented. Moreover, by using the proposed methods we can concluded that, the model given in this paper describes well the confirmed real data given by WHO about Egypt.     

The characteristics of epidemics and invasions with thresholds

In this paper we report the development of a highly efficient numerical method for determining the principal characteristics (velocity, leading edge width, and peak height) of spatial invasions or epidemics described by deterministic one-dimensiohal reaction-diffusion models whose dynamics include a threshold or Allee effect. We prove that this methodology produces the correct results for single-component models which are generalizations of the Fisher model, and then demonstrate by numerical experimentation that analogous methods work for a wide class of epidemic and invasion models including the S-I and S-E-I epidemic models and the Rosenzweig-McArthur predator-prey model. As examplary application of this approach we consider the atto-fox effect in the classic reaction-diffusion model of rabies in the European fox population and show that the appropriate threshold for this model is within an order of magnitude of the peak disease incidence and thus has potentially significant effects on epidemic properties. We then make a careful re-parameterisation of the model and show that the velocities calculated with realistic thresholds differ surprisingly little from those calculated from threshold-free models. We conclude that an appropriately thresholded reaction-diffusion model provides a robust representation of the initial epidemic wave and thus provides a sound basis on which to begin a properly mechanistic modelling enterprise aimed at understanding the long-term persistence of the disease.     

Algebraic determination of the evolutionary stable germination fraction

In this article, we consider a bet-hedging model with density dependence for germination behavior of dormant seeds. We show how to compute accurate values of the evolutionary stable germination fraction using a method based on expansion around the average seed bank size. The expansion is made up to the fourth order and is expressed in terms of the first four moments of the distribution of the year-to-year variation in the average yield per adult plant. This method leads to a simple algebraic equation, which can be solved without recurring to simulations of the seed bank dynamics. We first show the analytical derivation under general assumptions about the effect of density dependence. We then test our results with a numerical analysis where an explicit form of the density dependence has been taken.     

On fractional order models for Hepatitis C

In this paper we present a fractional order generalization of Perelson et al. basic hepatitis C virus (HCV) model including an immune response term. We argue that fractional order equations are more suitable than integer order ones in modeling complex systems which include biological systems. The model is presented and discussed. Also we argue that the added immune response term represents some basic properties of the immune system and that it should be included to study longer term behavior of the disease.     

A general class of pattern mixture models for nonignorable dropout with many possible dropout times

Summary .   In this article we consider the problem of fitting pattern mixture models to longitudinal data when there are many unique dropout times. We propose a marginally specified latent class pattern mixture model. The marginal mean is assumed to follow a generalized linear model, whereas the mean conditional on the latent class and random effects is specified separately. Because the dimension of the parameter vector of interest (the marginal regression coefficients) does not depend on the assumed number of latent classes, we propose to treat the number of latent classes as a random variable. We specify a prior distribution for the number of classes, and calculate (approximate) posterior model probabilities. In order to avoid the complications with implementing a fully Bayesian model, we propose a simple approximation to these posterior probabilities. The ideas are illustrated using data from a longitudinal study of depression in HIV-infected women.     

一个离散单种群动力系统的超临界Flip分支和Hoph分支(英文)

文中考虑一个非线性的具有年龄阶段结构的单种群离散模型,致力于揭示该系统的动力学行为,说明了系统在分支阈值附近会出现超临界Flip分支和Hoph分支.与一些文献不同的是文中用数学工具给出证明过程而不是用数值模拟的结果说明.     

The analysis of group truncated binary data with random effects: injury severity in motor vehicle accidents

The analysis of group truncated binary data has been previously considered by O'Neill and Barry (1995b, Biometrics 51, 533-541), where the analysis assumed that responses within each group were independent. In this paper, we consider the analysis of such data when there is group-level heterogeneity. A generalized linear mixed model is hypothesized to model the response and maximum likelihood estimates are derived for the truncated case. A score test is derived to test for heterogeneity. Finally, the method is applied to a set of traffic accident data.     

A fractional derivative model to describe arterial viscoelasticity

Arterial viscoelasticity can be described with a complex modulus (E*) in the frequency domain. In arteries, E* presents a power-law response with a plateau for higher frequencies. Constitutive models based on a combination of purely elastic and viscous elements can be represented with integer order differential equations but show several limitations. Recently, fractional derivative models with fewer parameters have proven to be efficient in describing rheological tissues. A new element, called "spring-pot", that interpolates between springs and dashpots is incorporated. Starting with a Voigt model, we proposed two fractional alternative models with one and two spring-pots. The three models were tested in an anesthetized sheep in a control state and during smooth muscle activation. A least squares method was used to fit E*. Local activation induced a vascular constriction with no pressure changes. The E* results confirmed the steep increase from static to dynamic values and a plateau in the range 2-30 Hz, coherent with fractional model predictions. Activation increased E*, affecting its real and imaginary parts separately. Only the model with two spring-pots correctly followed this behavior with the best performance in terms of least squares errors. In a context where activation separately modifies E*, this alternative model should be considered in describing arterial viscoelasticity in vivo.     

Quantile Function Models for Quantal Response Analysis

The usual analysis of quantal response data occurring in diverse fields such as economics, medicine, psychology and toxicology use probit and logit models or their extensions with generalized least squares or the principle of likelihood as the method of statistical inference. The symmetric alternative models lead to practically comparable results and the choice of model or method is determined by considerations of familiarity and computational convenience. Recent attempts at improvement involve larger parametric families of tolerance distributions and employ the method of maximum likelihood in analysis. In this paper we consider models with the tolerance distributions based upon the Tukey-lambda distributions which are described in terms of their quantile functions. The likelihood methods for fitting the models and testing their adequacies are developed and illustrated using classical data due to BLISS (1935) and ASHFORD and SMITH (1964).     

The identification of the periodic behaviour in an epidemic model

In this paper we study a method for the identification of the unknown parameter of the periodic function and also the first component of the state vector, in a mathematical model which describes the evolution of some diseases with an oro-fecal transmission.To solve the identification problem we use a numerical method to integrate the differential equations system, which reproduces the stability properties of the above mentioned continuous system.The numerical methods which we propose can be applied also to a spatial semi discretization of the reaction-diffusion model which is a diffusive generalization of the system that we consider in this paper.Finally, through an analysis on both the continuous and the discrete system we also obtain a necessary condition on the experimental data in order that a periodic trajectory of the system exists.Work supported by: Progetto Finalizzato Controllo Malattie da Infezione-CNR and by Ministero Pubblica Istruzione     

Stability analysis and optimal vaccination of an SIR epidemic model

Almost all mathematical models of diseases start from the same basic premise: the population can be subdivided into a set of distinct classes dependent upon experience with respect to the relevant disease. Most of these models classify individuals as either a susceptible individual S, infected individual I or recovered individual R. This is called the susceptible-infected-recovered (SIR) model. In this paper, we describe an SIR epidemic model with three components; S, I and R. We describe our study of stability analysis theory to find the equilibria for the model. Next in order to achieve control of the disease, we consider a control problem relative to the SIR model. A percentage of the susceptible populations is vaccinated in this model. We show that an optimal control exists for the control problem and describe numerical simulations using the Runge-Kutta fourth order procedure. Finally, we describe a real example showing the efficiency of this optimal control.     

Respiratory sinus arrhythmia from two coupled pacemakers.

We reproduce global features of respiratory sinus arrhythmia (RSA), a prominent source of heart rate variability, from two signals coupled in alternate fashion so dominance periodically switches back and forth between them. We consider two different possibilities for this coupling and illustrate our method with numerical simulations that we contrast with the corresponding results from real data. We interpret our findings within the context of the two-pacemaker model of the heartbeat, an alternative to the single-pacemaker mechanism of pulse generation in the orthodox conduction model.     

A diffusive-convective model for the dynamics of population-toxicant interactions: some analytical and numerical results

In this paper we consider a diffusive-convective model for the dynamics of a population living in a polluted environment. Threshold results are given concerning the effect of the toxicant on the living population. Some analytic results are proved and numerical experiments give suggestions in more general cases.