A numerical approach to solve the model for HIV infection of CD4+T cells

In this study, we will obtain the approximate solutions of the HIV infection model of CD4⁺T by developing the Bessel collocation method. This model corresponds to a class of nonlinear ordinary differential equation systems. Proposed scheme consists of reducing the problem to a nonlinear algebraic equation system by expanding the approximate solutions by means of the Bessel polynomials with unknown coefficients. The unknown coefficients of the Bessel polynomials are computed using the matrix operations of derivatives together with the collocation method. The reliability and efficiency of the proposed approach are demonstrated in the different time intervals by a numerical example. All computations have been made with the aid of a computer code written in Maple 9.     

具有感染年龄结构的CD4+ T-细胞感染HIV病毒模型分析

本文建立和研究一类具有感染年龄结构的CD4+ T-细胞感染HIV病毒的动力学模型.得到决定该模型的未感染平衡点和感染平衡点的存在性和局部渐近稳定性条件,即当一个感染细胞在其整个感染期间产生病毒的总数不超过某-个阈值时,系统总存在局部渐近稳定的未感染平衡点;当-个感染细胞在其整个感染期间产生病毒的总数超过这一阈值时,未感染平衡点不稳定,此时存在局部渐近稳定的感染平衡点.     

A differential equation model of HIV infection of CD4T-cells with cure rate

A differential equation model of HIV infection of CD4⁺T-cells with cure rate is studied. We prove that if the basic reproduction number R₀<1, the HIV infection is cleared from the T-cell population and the disease dies out; if R₀>1, the HIV infection persists in the host. We find that the chronic disease steady state is globally asymptotically stable if R₀>1. Furthermore, we also obtain the conditions for which the system exists an orbitally asymptotically stable periodic solution. Numerical simulations are presented to illustrate the results.     

一类带有治愈率的HIV感染的CD4＋T细胞模型的动力学性质

本文主要介绍一类带有治愈率的HIV感染的CD4 T细胞模型的动力学性质,同时证明了如果基本再生数R0＜1,HIV感染消失;如果R0＞1,HIV感染持续.然后进行数值模拟,给出了地方性平衡点E·全局稳定的参数域,得到了地方性平衡点E·不稳定时周期解存在.     

A fractional-order model of HIV infection with drug therapy effect

In this paper, the fractional-order model that describes HIV infection of CD4⁺ T cells with therapy effect is given. Generalized Euler Method (GEM) is employed to get numerical solution of such problem. The fractional derivatives are described in the Caputo sense.     

基于CD4细胞计数的HIV仓室模型与数据分析

迄今我国每年统计艾滋病的新增人数与死亡人数仍呈上升趋势,由于潜伏期长和初期无明显症状等原因,还存在大量未发现的HIV携带者,这给HIV的防控带来巨大挑战.旨在利用中国疾控中心网报数据和深圳市患者随访数据,结合传染病动力学和统计分析方法与临床知识,建立依微观CD4细胞计数划分的宏观HIV仓室数学模型,通过数值计算方法与MCMC参数估计方法实现稳健的参数拟合,进而利用不确定性和敏感性以及随机森林方法进行灵敏度分析.研究结果表明:2013年广东省未确诊HIV携带者约为13.1061万人,且该地区HIV疫情传播的基本再生数为2.8133.敏感性分析揭示艾滋病疫情防控最优方法是控制患者有效接触人数与沉默系数,由此建议制定针对控制艾滋病传播中一些现象的法律法规,在艾滋病高发地区实施清洁针具交换工作等,对疫情防控提出指导性建议.     

A novel time delayed HIV/AIDS model with vaccination & antiretroviral therapy and its stability analysis

In this paper, we propose a novel time delayed HIV/AIDS mathematical model and further analyze the effect of vaccination and ART (antiretroviral therapy) on this time delayed model, in which the time delay is due to the strong immune response to AIDS for the HIV-infected-aware because of the good physical conditions. We introduce the different stages of the period of AIDS infection having different abilities of transmitting disease, which reflects the developing progress of AIDS infection more realistically. By using suitable Lyapunov functionals and the LaSalle invariant principle, we obtain the basic reproduction number _R0${\mathfrak{R}}_0$ and derive that if _R0<1${\mathfrak{R}}_0<1$ and some parameters satisfy a given condition, the disease-free equilibrium is globally asymptotically stable, while the disease will be died out. Numerical simulations are carried out to verify the obtained stability criteria and demonstrate the effect of the vaccination rate and _R0${\mathfrak{R}}_0$ and the ART on the infective individuals.     

Global dynamics of a mathematical model for HTLV-I infection of CD4 T-cells

In this paper, a mathematical model for HILV-I infection of CD4⁺ T-cells is investigated. The force of infection is assumed be of a function in general form, and the resulting incidence term contains, as special cases, the bilinear and the saturation incidences. The model can be seen as an extension of the model [Wang et al. Mathematical analysis of the global dynamics of a model for HTLV-I infection and ATL progression, Math. Biosci. 179 (2002) 207-217; Song, Li, Global stability and periodic solution of a model for HTLV-I infection and ATL progression, Appl. Math. Comput. 180(1) (2006) 401-410]. Mathematical analysis establishes that the global dynamics of T-cells infection is completely determined by a basic reproduction number _R0${\mathfrak{R}}_0$. If _R0?1${\mathfrak{R}}_0?1$, the infection-free equilibrium is globally stable; if _R0>1${\mathfrak{R}}_0>1$, the unique infected equilibrium is globally stable in the interior of the feasible region.     

The impact of zidovudine (AZT) therapy on the survivability of those with the progressive HIV infection

An analysis of the data regarding the impact of zidovudine therapy on the Survivability of those with progressive HIV disease demonstrates that this therapy extends longevity for perhaps 5.5 months on the average but does not prevent the disease from eventually being fatal. All of the benefits of zidovudine therapy in extending survivability appear to accrue within a relatively short treatment period, perhaps within a few months, but the effectiveness of this drug wanes in time, suggesting that zidovudine therapy could eventually be stopped without influencing survivability. Since the efficacy of zidovudine therapy in extending survivability appears to be independent of the stage of the HIV infection in infecteds whose CD4 T-cell densities fall below 200 cells/mm³, zidovudine therapy should be initiated when the patient's infection reaches a potentially fatal stage. Zidovudine therapy may be viewed as causing the HIV infection to regress to a previous stage of the disease, but the infection's progression promptly resumes and follows a course similar to one uninfluenced by zidovudine.     

The Modeling Dynamics of HIV and CD4$$^{}$$ T-cells During Primary Infection in Fractional Order: Numerical Simulation

This article is devoted to introduce a numerical treatment using Adams–Bashforth–Moulton method of the fractional 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. The fractional derivative is described in Caputo sense. Special attention is given to present the local stability of the proposed model using fractional Routh–Hurwitz stability criterion. Qualitative results show that the model has two equilibria: the disease-free equilibrium and the endemic equilibrium points. We compare our numerical solutions with those numerical solutions using fourth-order Runge–Kutta method (RK4). The obtained numerical results of the proposed model show the simplicity and the efficiency of the proposed method.     

Global dynamics of a mathematical model for HTLV-I infection of CD4 T cells with delayed CTL response

Human T-cell leukaemia virus type I (HTLV-I) preferentially infects the CD4⁺ T cells. The HTLV-I infection causes a strong HTLV-I specific immune response from CD8⁺ cytotoxic T cells (CTLs). The persistent cytotoxicity of the CTL is believed to contribute to the development of a progressive neurologic disease, HTLV-I associated myelopathy/tropical spastic paraparesis (HAM/TSP). We investigate the global dynamics of a mathematical model for the CTL response to HTLV-I infection in vivo. To account for a series of immunological events leading to the CTL response, we incorporate a time delay in the response term. Our mathematical analysis establishes that the global dynamics are determined by two threshold parameters R₀ and R₁, basic reproduction numbers for viral infection and for CTL response, respectively. If R₀≤1, the infection-free equilibrium P₀ is globally asymptotically stable, and the HTLV-I viruses are cleared. If R₁≤1<R₀, the asymptomatic-carrier equilibrium P₁ is globally asymptotically stable, and the HTLV-I infection becomes chronic but with no persistent CTL response. If R₁>1, a unique HAM/TSP equilibrium P₂ exists, at which the HTLV-I infection is chronic with a persistent CTL response. We show that the time delay can destabilize the HAM/TSP equilibrium, leading to Hopf bifurcations and stable periodic oscillations. Implications of our results to the pathogenesis of HTLV-I infection and HAM/TSP development are discussed.     

Small world network models of the dynamics of HIV infection

It has long been recognised that the structure of social networks plays an important role in the dynamics of disease propagation. The spread of HIV results from a complex network of social interactions and other factors related to culture, sexual behaviour, demography, geography and disease characteristics, as well as the availability, accessibility and delivery of healthcare. The small world phenomenon has recently been used for representing social network interactions. It states that, given some random connections, the degrees of separation between any two individuals within a population can be very small. In this paper we present a discrete event simulation model which uses a variant of the small world network model to represent social interactions and the sexual transmission of HIV within a population. We use the model to demonstrate the importance of the choice of topology and initial distribution of infection, and capture the direct and non-linear relationship between the probability of a casual partnership (small world randomness parameter) and the spread of HIV. Finally, we illustrate the use of our model for the evaluation of interventions such as the promotion of safer sex and introduction of a vaccine.     

The equation $$x^4+2^ny^4=z^4$$ x 4 + 2 n y 4 = z 4 in algebraic number fields

Acta Mathematica Hungarica - Let n be a positive integer. We show that if the equation $$(1) \qquad \qquad \qquad x^4+2^ny^4=z^4$$ has a solution (x,y,z) in a cubic number field K with $$xyz \neq...     

Analysis of tumor growth under direct effect of inhibitors with time delays in proliferation

In this paper, a free boundary problem modeling tumor growth under the direct effect of an inhibitor with time delays is studied. The delays represent the time taken for cells to undergo mitosis. Nonnegativity of solutions, the existence of the stationary solutions and their asymptomatic behavior are studied. The results show that when the inhibitor is large, and the initial tumor is not too large, the tumor will disappear. If however, the initial tumor is large enough, then it will grow. When the inhibitor is not as large, the growth of the tumor is determined by the size of the nutrients and whether the initial tumor is large or not. When the inhibitor is smaller, the tumor will grow no matter if the initial tumor is large or not.     

Electron spin resonance of Cr5+ in CaWO4

Electron Spin Resonance of CaWO₄ with 0·1% of Cr has been investigated at liquid nitrogen and liquid helium temperatures. The observed ESR spectrum is attributed to Cr⁵⁺ ion in the substitutional site of W which has a compressed tetrahedral surroundings. A simple point charge calculation based on this geometry explains the observedg anisotropy and hyperfine anisotropy and places the magnetic electron in a predominantly \(3d_{z^2 } \) orbital. A comparison of these results with those obtained on other isoelectronic systems in similar and different co-ordinations justifies our assignment.