一类含CF分数阶导数传染病模型的求解

 2022-05-25 09:05

论文总字数:37257字

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\begin{center}{\kaishu \zihao{2}{一类含}\LARGE{CF}\zihao{2}{分数阶导数传染病模型的求解}}\end{center}

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\begin{center}{\kaishu\zihao{4} 摘\ \ \ \ 要}

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\addcontentsline{toc}{chapter}{摘\ \ \ \ 要} {\kaishu \ \

本文研究了Caputo-Fabrizio 分数阶导数下满足Beddington-Deangelis 功能反应式的HIV病毒感染CD$_{4}$ T细胞的数学模型。CF分数阶导数是2015 年提出的带有非奇异核的分数阶导数。数学模型的解析解通过同伦分析变换法得到。本文求解方程使用的是同伦分析变换法(HATM),是一种基于同伦分析法的求解强非线性微分方程的方法,此方法的核心步骤依次是:拉普拉斯变换方程,构造非线性算子,求解关于同伦系数$p$的级数展开式,得到解析解。本文将首先介绍同伦分析法的基础知识,随后给出用CF 分数阶导数描述的HIV 病毒感染CD$_{4}$ T 细胞的微分方程,最后利用同伦分析变换法得出的低阶近似解并与常规ODE45 数值解进行比较。在最后,本文给出了一些任意阶导数的数值解与表格。}

\vskip 1cm \noindent{\kaishu 关键词: \ CF分数阶导数,\ 同伦分析法, \

Beddington-Deangelis功能反应, \ 拉普拉斯变换, \ HIV模型 }

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\begin{center}{\rm Solution of a Class of Infectious Disease Model with CF Fractional Derivative }\end{center}

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\begin{center}{\rm\zihao{4} Abstract}

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In this work, mathematical model of CD$_{4}$ T-cells during HIV infection with Beddington-Deangelis functional response is studied under Caputo-Fabrizio fractional derivative. Caputo-Fabrizio fractional derivative is a fractional derivative without singular kernel that has been proposed in 2015. The analytical solution of the mathematical model is derived by applying the homotopy analysis transform method{HATM}.The homotopy analysis transformation method is used to solve the equation in this paper. It is a method based on homotopy analysis to solve strongly nonlinear differential equations. The core steps of this method are: Laplace transformation equation, construction of non-linear operators, solving homotopy series of homotopy coefficient p, and obtaining analytical solutions. First , the essay introduces the elements of the homotopy analysis method, then the differential equations of CD$_{4}$ T-cells during HIV infection described by CF fractional derivative are given. Finally, the low-order approximate solutions obtained by homotopy analysis transform method are used to compare with the conventional ODE45 numerical solution. Ultimately some numerical solutions and tables of derivatives of arbitrary order are presented.

\vskip 0.8cm \noindent{\rm Key Words:\ CF fractional derivative, \ Homotopy analysis method, \

Beddington-Deangelis functional response, \ Laplace transform, \ HIV model}

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%东~南~大~学~毕~业~设~计~报~告}\hspace{0.5cm}{\bf 第一章 \qquad 引言}

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%微型计算机中压缩数据的存储、空间探索中的微型机械、显微外科工具以及现代通讯都是植根于微系统技术的广大应用中的一部分.

长期以来,数学模型被用于研究各类传染病的动态变化并帮助医疗工作者更好的拟定疾病的预防与控制。很多论文作者在无延迟到有延迟的情况下,使用微分方程研究了HIV 病毒感染CD$_{4}$ T 细胞模型的动态特性,并且都获得了关于HIV-1 感染模型的不少研究成果$^{\cite{CG1}\cite{ES1}\cite{LG1}}$。 本文研究的HIV病毒攻击CD$_{4}$ T 细胞的数学模型,是一种具有Beddington-Deangelis 功能反应的HIV-1型感染数学模型,此类带有Beddington-Deangelis 功能反应的大规模传染病模型在2010 年后陆续得到了众多国内科研者的关注$^{\cite{FH1}}$。

本文研究模型的创新点主要在于将普通的带有Beddington-Deangelis 功能反应的微分方程与CF 分数阶导数(Caputo-Fabrizio fractional derivative)结合,并将其用于求解HIV模型。由于CF 分数阶导数的核是光滑的,用它处理时间变量时适合进行Laplace变换,并且在处理一类非局部的系统时,具有能够描述研究对象异质性和波动性的能力。CF分数阶导数在定义时对时间与空间使用了两种表示方式。在针对时间变量时,通常分数阶导数解中出现的实数幂将转化为整数幂,并在公式和计算中进行了一些简化,而最传统的分数阶导数,如黎曼- 刘维尔分数阶导数,Caputo 分数阶导数均不具备这种优良的性质。

由于上述CF分数阶导数的优点,在常规同伦分析方法中的线性算子选取为拉普拉斯算子,能够在非局部系统中拥有优良的性质,将实数幂转化为整数幂,同时也让计算和公式得到简化。这也是我们选取CF分数阶导数的核心原因,正是HIV 病毒感染CD$_{4}$ T 细胞模型的特性,使得我们选择使用CF 分数阶导数进行研究,从而在HAM 法中选择HATM法计算该模型的近似解。

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本论文的结构如下:

第二章将对同伦分析法、CF分数阶导数、同伦导数等相关知识进行简要介绍。

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