基于Riccati-Bernoulli辅助常微分方程的Davey-Stewartson方程的行波解

Riccati-Bernoulli辅助常微分方程方法可以用来构造非线性偏微分方程的行波解.利用行波变换,将非线性偏微分方程化为非线性常微分方程, 再利用Riccati-Bernoulli方程将非线性常微分方程化为非线性代数方程组, 求解非线性代数方程组就能直接得到非线性偏微分方程的行波解.对Davey-Stewartson方程应用这种方法, 得到了该方程的精确行波解.同时也得到了该方程的一个Backlund变换.所得结果与首次积分法的结果作了比较.Riccati-Bernoulli辅助常微分方程方法是一种简单、有效地求解非线性偏微分方程精确解的方法.     

Burgers方程的一类自相似解

利用李群理论中的伸缩变换群,将二阶非线性偏微分方程-Burgers方程化为一类Riccati方程和三类二阶非线性常微分方程,从而Riccati方程和这三类二阶非线性常微分方程给出了Burgers方程的自相似解的表现形式.     

深水表面波可积发展方程的行波解与分支

本文研究了small-aspect-ratio波方程和深水表面波可积发展方程的行波解问题.利用微分方程定性理论的方法,分析了行波系统的相图分支,获得了孤立波解的精确表达式.     

深水表面波可积发展方程的行波解与分支（英文）

本文研究了small-aspect-ratio波方程和深水表面波可积发展方程的行波解问题.利用微分方程定性理论的方法,分析了行波系统的相图分支,获得了孤立波解的精确表达式.     

MKdV和FPU方程的李点对称

本文研究了离散微分方程的李对称问题.利用差分方程的延拓方法和交换流方法,我们求得了若干重要的差分方程、微分差分方程的李对称,推广了对称性分析法在连续微分方程讨论时的结果.     

Volterra 积分微分大系统的稳定性

近20年来大系统理论已得到了较大的发展,对其稳定性的研究已引起了人们的足够重视,出版的大量论文与专著中已涉及到微分方程、积分方程、泛函微分方程、随机与抽象微分方程、偏微与复微分方程以及差分方程所定义的大系统.虽然近年来对积分微分方程的研究也引起了不少学者的注意,但很少涉及到此类方程所定义的大系统.本文     

变质量非Четаев型非完整系统在相空间的Lie对称性与守恒量

在相空间引入无限小群变换,研究变质量非Четаев型非完整系统的Lie对称和守恒量.利用系统运动微分方程在无限小群变换下的不变性建立Lie对称的确定方程和限制方程,得到Lie对称的结构方程和守恒量,并举例说明结果的应用.     

Degasperis-Procesi方程的类孤子新解

利用辅助方程与函数变换相结合的方法,构造了Degasperis-Procesi(D-P)方程的无穷序列类孤子新解.首先,通过两种函数变换,把D-P方程化为常微分方程组.然后,利用常微分方程组的首次积分,把D-P方程的求解问题化为几种常微分方程的求解问题.最后,利用几种常微分方程的Bcklund变换等相关结论,构造了D-P方程的无穷序列类孤子新解.这里包括由Riemannθ函数、Jacobi椭圆函数、双曲函数、三角函数和有理函数组成的无穷序列光滑孤立子解、尖峰孤立子解和紧孤立子解.     

时间测度上具有时滞基于半比率的捕食者-食饵扩散系统的周期解

在时间测度上研究一类具有时滞和基于半比率且有功能性反应的两种群捕食者-食饵扩散系统,利用Mawhin重合度理论建立了这类系统的周期解存在的一个充分性判据.从而使这一类系统的连续与离散情形即相应的微分方程和差分方程的周期解存在性问题得到了统一研究.     

随机Volterra积分方程的广义样本解

本文研究了随机积分方程的广义样本解.利用随机微分方程转换为带参数常微分方程的方法,给出了一类随机Volterra积分方程的广义样本解,这类方程在许多应用领域是常见的.     

Modelling circumcision and condom use as HIV/AIDS preventive control strategies

We present a sex-structured model for heterosexual transmission of HIV/AIDS with explicit incubation period for modelling the effect of male circumcision as a preventive strategy for HIV/AIDS. The model is formulated using integro-differential equations, which are shown to be equivalent to delay differential equations with delay due to incubation period. The threshold and equilibria for the model are determined and stabilities are examined. We extend the model to incorporate the effects of condom use as another preventive strategy for controlling HIV/AIDS. Basic reproductive numbers for these models are computed and compared to assess the effectiveness of male circumcision and condom use in a community. The models are numerically analysed to assess the effects of the two preventive strategies on the transmission dynamics of HIV/AIDS. We conclude from the study that in the continuing absence of a preventive vaccine or cure for HIV/AIDS, male circumcision is a potential effective preventive strategy of HIV/AIDS to help communities slow the development of the HIV/AIDS epidemic and that it is even more effective if implemented jointly with condom use. The study provides insights into the possible community benefits that male circumcision and condom use as preventive strategies provide in slowing or curtailing the HIV/AIDS epidemic.     

The role of CD4 T cells in immune system activation and viral reproduction in a simple model for HIV infection

CD4 T cells play a fundamental role in the adaptive immune response including the stimulation of cytotoxic lymphocytes (CTLs). Human immunodeficiency virus (HIV) which infects and kills CD4 T cells causes progressive failure of the immune system. However, HIV particles are also reproduced by the infected CD4 T cells. Therefore, during HIV infection, infected and healthy CD4 T cells act in opposition to each other, reproducing virus particles and activating and stimulating cellular immune responses, respectively. In this investigation, we develop and analyze a simple system of four ordinary differential equations that accounts for these two opposing roles of CD4 T cells. The model illustrates the importance of the CTL immune response during the asymptomatic stage of HIV infection. In addition, the solution behavior exhibits the two stages of infection, asymptomatic and final AIDS stages. In the model, a weak immune response results in a short asymptomatic stage and faster development of AIDS, whereas a strong immune response illustrates the long asymptomatic stage. A model with a latent stage for infected CD4 T cells is also investigated and compared numerically with the original model. The model shows that strong stimulation of CTLs by CD4 T cells is necessary to prevent progression to the AIDS stage.     

Asymptotic properties of an HIV/AIDS model with a time delay

A mathematical model for HIV/AIDS with explicit incubation period is presented as a system of discrete time delay differential equations and its important mathematical features are analysed. The disease-free and endemic equilibria are found and their local stability investigated. We use the Lyapunov functional approach to show the global stability of the endemic equilibrium. Qualitative analysis of the model including positivity and boundedness of solutions, and persistence are also presented. The HIV/AIDS model is numerically analysed to asses the effects of incubation period on the dynamics of HIV/AIDS and the demographic impact of the epidemic using the demographic and epidemiological parameters for Zimbabwe.     

A nonlinear HIV/AIDS model with contact tracing

A nonlinear mathematical model is proposed and analyzed to study the effect of contact tracing on reducing the spread of HIV/AIDS in a homogeneous population with constant immigration of susceptibles. In modeling the dynamics, the population is divided into four subclasses of HIV negatives but susceptibles, HIV positives or infectives that do not know they are infected, HIV positives that know they are infected and that of AIDS patients. Susceptibles are assumed to become infected via sexual contacts with (both types of) infectives and all infectives move with constant rates to develop AIDS. The model is analyzed using the stability theory of differential equations and numerical simulation. The model analysis shows that contact tracing may be of immense help in reducing the spread of AIDS epidemic in a population. It is also found that the endemicity of infection is reduced when infectives after becoming aware of their infection do not take part in sexual interaction.     

SWEIA艾滋病毒传染模型及应用

人类免疫缺陷病毒(HIV)是一种严重威胁生命的病毒,感染艾滋病毒患者一般经历四个阶段:i)艾滋病毒阴性的窗口期(W);ii)阳性的无症状潜伏期(E);iii)有症状期(Ⅰ);以及iv)移除阶段(A).为深入研究艾滋病传播过程,建立SWEIA艾滋病毒传染模型,定义基本再生数,分析无病与地方病平衡点的存在性和局部稳定性,根据2004至2015年中国艾滋病患者数据,采用遗传算法对SWEIA模型中参数进行估计.通过对基本再生数敏感性分析以及模型数值随参数不同而产生的变化,揭示艾滋病窗口期的接触率是影响艾滋病流行的主要原因之一.     

A nonlinear AIDS epidemic model with screening and time delay

A nonlinear mathematical model to study the effect of time delay in the recruitment of infected persons on the transmission dynamics of HIV/AIDS is proposed and analyzed. In modeling the dynamics, the population is divided into four subclasses: the susceptibles, the HIV positives or infectives that do not know they are infected, the HIV positives that know they are infected and the AIDS patients. Susceptibles are assumed to become infected via sexual contacts with (both types of) infectives. The model is analyzed using stability theory of delay differential equations. Both the disease-free and the endemic equilibria are found and their stability is investigated. It is shown that the introduction of time delay in the model has a destabilizing effect on the system and periodic solutions can arise by Hopf bifurcation. Numerical simulations are also carried out to investigate the influence of key parameters on the spread of the disease, to support the analytical conclusion and to illustrate possible behavioral scenario of the model.     

A mathematical approach to HIV infection dynamics

In order to obtain a comprehensive form of mathematical models describing nonlinear phenomena such as HIV infection process and AIDS disease progression, it is efficient to introduce a general class of time-dependent evolution equations in such a way that the associated nonlinear operator is decomposed into the sum of a differential operator and a perturbation which is nonlinear in general and also satisfies no global continuity condition. An attempt is then made to combine the implicit approach (usually adapted for convective diffusion operators) and explicit approach (more suited to treat continuous-type operators representing various physiological interactions), resulting in a semi-implicit product formula. Decomposing the operators in this way and considering their individual properties, it is seen that approximation–solvability of the original model is verified under suitable conditions. Once appropriate terms are formulated to describe treatment by antiretroviral therapy, the time-dependence of the reaction terms appears, and such product formula is useful for generating approximate numerical solutions to the governing equations. With this knowledge, a continuous model for HIV disease progression is formulated and physiological interpretations are provided. The abstract theory is then applied to show existence of unique solutions to the continuous model describing the behavior of the HIV virus in the human body and its reaction to treatment by antiretroviral therapy. The product formula suggests appropriate discrete models describing the dynamics of host pathogen interactions with HIV1 and is applied to perform numerical simulations based on the model of the HIV infection process and disease progression. Finally, the results of our numerical simulations are visualized and it is observed that our results agree with medical and physiological aspects.     

A fractional order HIV-TB coinfection model with nonsingular Mittag-Leffler Law

The biological models for the study of human immunodeficiency virus (HIV) and its advanced stage acquired immune deficiency syndrome (AIDS) have been widely studied in last two decades. HIV virus can be transmitted by different means including blood, semen, preseminal fluid, rectal fluid, breast milk, and many more. Therefore, initiating HIV treatment with the TB treatment development has some advantages including less HIV-related losses and an inferior risk of HIV spread also having difficulties including incidence of immune reconstitution inflammatory syndrome (IRIS) because of a large pill encumbrance. It has been analyzed that patients with HIV have more weaker immune system and are susceptible to infections, for example, tuberculosis (TB). Keeping the importance of the HIV models, we are interested to consider an analysis of HIV-TB coinfected model in the Atangana-Baleanu fractional differential form. The model is studied for the existence, uniqueness of solution, Hyers-Ulam (HU) stability and numerical simulations with assumption of specific parameters.     

Modelling and analysis of the spread of AIDS epidemic with immigration of HIV infectives

We propose and analyze, a nonlinear mathematical model of the spread of HIV/AIDS in a population of varying size with immigration of infectives. It is assumed that susceptibles become infected via sexual contacts with infectives (also assumed to be infectious) and all infectives ultimately develop AIDS. The model is studied using stability theory of differential equations and computer simulation. Model dynamics is also discussed under two particular cases when there is no direct inflow of infectives. On analyzing these situations, it is found that the disease is always persistent if the direct immigration of infectives is allowed in the community. Further, in the absence of inflow of infectives, the endemicity of the disease is found to be higher if pre-AIDS individuals also interact sexually in comparison to the case when they do not take part in sexual interactions. Thus, if the direct immigration of infectives is restricted, the spread of infection can be slowed down. A numerical study of the model is also carried out to investigate the influence of certain key parameters on the spread of the disease.     

Modelling the effect of tuberculosis on the spread of HIV infection in a population with density-dependent birth and death rate

A nonlinear mathematical model is proposed to study the effect of tuberculosis on the spread of HIV infection in a logistically growing human population. The host population is divided into four sub classes of susceptibles, TB infectives, HIV infectives (with or without TB) and that of AIDS patients. The model exhibits four equilibria namely, a disease free, HIV free, TB free and an endemic equilibrium. The model has been studied qualitatively using stability theory of nonlinear differential equations and computer simulation. We have found a threshold parameter R₀ which is if less than one, the disease free equilibrium is locally asymptotically stable otherwise for R₀>1, at least one of the infections will be present in the population. It is shown that the positive endemic equilibrium is always locally stable but it may become globally stable under certain conditions showing that the disease becomes endemic. It is found that as the number of TB infectives decreases due to recovery, the number of HIV infectives also decreases and endemic equilibrium tends to TB free equilibrium. It is also observed that number of AIDS individuals decreases if TB is not associated with HIV infection. A numerical study of the model is also performed to investigate the influence of certain key parameters on the spread of the disease.