Open Access Original Research Article

Theoretical Analysis and Simulation of Acute and Chronic Phase HIV-1 Dynamics

Frank Nani, Mingxian Jin

Journal of Advances in Mathematics and Computer Science, Page 1450-1479
DOI: 10.9734/BJMCS/2014/1815

Aims: The dynamics of HIV-1 induced AIDS is attributed to several biological variables, which characterize the stage, virulence and morbidity of the disease. The aim of this research is to use a necessary and sufficient subset of these immunological variables to construct a clinically plausible mathematical model of the patho-physiological dynamics of HIV-1 induced AIDS during the acute and chronic phases. This model incorporates the interactions between uninfected CD4+ T cells, HIV-1 infected CD4+ T cells, HIV-1 virions in the blood plasma, and specific cytotoxic CD8+ T cells. The major objective is to derive mathematical criteria depicting conditions under which the HIV-1 virions can be maintained definitely at the subclinical viral blood plasma level such that the HIV-1 seropositive person does not develop full-blown AIDS.
Study Design: The model is based on contemporary published patho-physiological data on acute and clinical chronic phase HIV-1 induced AIDS. These data are meticulously condensed into a clinically plausible four-compartmental mathematical model that incorporates the dynamics and interactions between non-HIV-1 infected CD4+ T lymphocytes. HIV-1 infected lymphocytes, free HIV-1 virions in the blood plasma, and HIV-1 specific cytotoxic CD8+ T lymphocytes. The relevant stoichiometric interaction rate constants, apoptotic rate constants, rate constants for viral recruitment from latent reservoirs, and other relevant parameters are clearly exhibited in the mathematical model. The role of CD4+ T cell-induced syncytia is explicitly incorporated into the HIV-1 virion dynamical equation.
Place and Duration of Study: This research was done at Fayetteville State University, North Carolina USA and is sponsored by the FSU Mini-Grant Award and the HBCU Graduate STEM Grant. The research was conducted during the Spring of 2012.
Methodology: The deterministic nonlinear HIV-1 AIDS patho-physio-dynamical equations are analyzed using the techniques of dynamical system theory, principles of linearized stability, Hartman-Grobman theory and other relevant mathematical techniques. The clinically desirable equilibrium states are and their local existence and global stability are analyzed. Investigative computer simulations are performed illustrating some physiological outcomes.
Results: Mathematical criteria are derived under which the clinically desired outcomes can occur. These criteria are presented in terms of theorems. Investigative computer simulations are presented which elucidate a number of physiological scenarios of primary HIV-1 infection, involving the annihilation and persistence of HIV-1 in the absence of AIDS Pharmacotherapy.
Conclusion: This research has demonstrated the existence of plausible criteria under which an HIV-1 sero-positive person can be maintained at an asymptomatic chronic state indefinitely. Some of the criteria are configured in terms of clinically measurable and biological quantifiable parameters which have been verified by the computer simulations.

Open Access Original Research Article

The Roles of Condition Number for the Interval Hull Solution Set in Least Squares Equation

Stephen Ehidiamhen Uwamusi

Journal of Advances in Mathematics and Computer Science, Page 1480-1495
DOI: 10.9734/BJMCS/2014/8110

The paper considers nearness to singularity for which the Co-variance matrix in the least squares equation is well known. A synchronization of the condition number with ill-conditioning is highlighted which relates the quality of approximate solution to the described system. Various theoretical lower and upper bounds to the perturbed least squares problems have been described for which, reach ability theory has strong representation. In particular, a theorem due to Rump as exemplified by Popova was revisited and examined in detail; a slight modification was made to the theorem by neglecting the second term appearing in the equation. This was found to have strong favourable appeals on the interval least squares problem. As a comparison to the computed results, a procedure described in Kramer/Rohn was used to crop the corner points solution of the linear interval system which is obtained from least squares equation based on the appropriate choice of the orthant (where there are possibilities). This leads to solving systems of linear inequalities for the interval Hull of solution set. Furthermore, the Rump/Krawczyk method was used to narrow, the computed corner point solution in order to obtain tighter approximate solution bounds of the interval Hull which may be applicable to both non-parametric and parametric interval linear equations. The loss function for the computed result obtained from Rump method for the set of data points is reported.

Open Access Original Research Article

Statistical Inference for a Simple Step-Stress Model Based on Censored Data from the Kumaraswamy Weibull Distribution

H. R. Rezk, A. A. EL Helbawy, G. R. AL Dayian

Journal of Advances in Mathematics and Computer Science, Page 1496-1514
DOI: 10.9734/BJMCS/2014/7560

The step-stress accelerated life tests allow increasing the stress levels on test units at fixed time during the experiment. In this paper, accelerated life tests are considered when lifetime of a product follows a Kumaraswamy Weibull distribution. The shape parameter is assumed to be a log linear function of the stress and a cumulative exposure model holds. Based on Type II and Type I censoring, the maximum likelihood estimates are obtained for the unknown parameters. The reliability and hazard rate functions are estimated at usual conditions of stress. In addition, confidence intervals of the estimators are constructed. Optimum test plans are obtained to minimize the generalized asymptotic variance of the maximum likelihood estimators. Monte Carlo simulation is carried out to investigate the precision of the maximum likelihood estimates. An application using real data is used to indicate the properties of the maximum likelihood estimators.

Open Access Original Research Article

Optimal Design and Control for Minimizing the Dynamic Response of an Anisotropic Plate with Variable Thickness

M. A. Hafiz, A. E. Alamir

Journal of Advances in Mathematics and Computer Science, Page 1515-1533
DOI: 10.9734/BJMCS/2014/7515

The problem of minimizing the dynamic response of an anisotropic rectangular plate of variable thickness with minimum possible expenditure of force is presented for various cases of boundary conditions. The plate has a principal direction of anisotropy rotated at an arbitrary angle relative to the coordinate axes. The orientation angle and thickness parameter have been taken as optimization design parameters. The control problem is formulated as an optimization problem by using a performance index, which comprises a weight sum of the control objective and penalty function of the control force. Explicit solutions for the surface shape, the total elastic energy of the plate and the closed-loop distributed control force are obtained by means of Liapunov-Bellman theory. To assess the present solutions, numerical results are presented to illustrate the effect of various thickness parameters, orientation angles, aspect ratios and boundary conditions on the control process.

Open Access Original Research Article

Sine-cosine Method for a Class of Nonlinear Fourth Order Variant of a generalized Camassa-holm equation

Haixia Zhao, Sijia Geng, Shengqiang Tang

Journal of Advances in Mathematics and Computer Science, Page 1534-1541
DOI: 10.9734/BJMCS/2014/7892

In this paper we study a class of nonlinear fourth order analogue of a generalized Camassa-Holm equation by using sine-cosine method. The compactons, solitary wave, solitary patterns, periodic wave and solitary patterns solutions of a class of nonlinear fourth order analogue of a generalized Camassa-Holm equation are successfully obtained. It is shown that the sine-cosine provides a powerful mathematical tool for solving a great many nonlinear partial differential equations in mathematical physics.

Open Access Original Research Article

Some New Results and Generalizations in G-cone Metric Fixed Point Theory

Mohammad Al-Khaleel, Sharifa Al-Sharif, Mona Khandaqji

Journal of Advances in Mathematics and Computer Science, Page 1542-1550
DOI: 10.9734/BJMCS/2014/6840

In this paper, we present sufficient generalized contractive conditions for the existence of fixed points in what so-called G-cone metric spaces. Importantly, we have obtained our results using contractive conditions stated in terms of variable coefficients and with no use of the normality property of cone.

Open Access Original Research Article

Analytical Solutions to the Fractional Fisher Equation by Applying the Fractional Sub-equation Method

H. Yépez-Martínez, J. M. Reyes, I. O. Sosa

Journal of Advances in Mathematics and Computer Science, Page 1551-1566
DOI: 10.9734/BJMCS/2014/9248

The fractional sub-equation method is proposed to construct analytical solutions of nonlinear fractional partial differential equations (FPDEs), involving Jumarie’s modified Riemann-Liouville derivative. The fractional sub-equation method is applied to the fractional Fisher equation. The analytical solutions show that the fractional sub-equation method is very effective for the analytical solutions of the Fisher equation. The fractional sub-equation method introduces a promising tool for solving many fractional partial differential equations.

Open Access Original Research Article

2-tuple Linguistic Bonferroni Mean Operators and Their Application to Multiple Attribute Group Decision Making

Zhiming Zhang, Chong Wu

Journal of Advances in Mathematics and Computer Science, Page 1567-1614
DOI: 10.9734/BJMCS/2014/9590

Aims: The aim of this paper is to develop the 2-tuple linguistic Bonferroni mean and the weighted 2-tuple linguistic Bonferroni mean.
Study Design: Some desirable properties and special cases of the developed operators are discussed. The geometric Bonferroni mean (GBM) is a generalization of the Bonferroni mean and geometric mean. In this paper, we also investigate the GBM under 2-tuple linguistic environments. We develop the 2-tuple linguistic geometric Bonferroni mean and the weighted 2-tuple linguistic geometric Bonferroni mean. We investigate some fundamental properties and special cases of them.
Place and Duration of Study: The Bonferroni Mean (BM) operator is a traditional mean type aggregation operator, which can capture the expressed interrelationship of the individual arguments and which is only suitable to aggregate crisp data.
Methodology: This paper extends the BM operator to 2-tuple linguistic environments.
Results: Based on these operators, we develop two approaches for multiple attribute group decision making with 2-tuple linguistic information.
Conclusion: Two numerical examples are provided to illustrate the effectiveness and practicality of the proposed approaches.

Open Access Original Research Article

On Hyers-Ulam-Rassias Stability for Bernoulli's and First Order Linear and Nonlinear Differential Equations

Maher Nazmi Qarawani

Journal of Advances in Mathematics and Computer Science, Page 1615-1628
DOI: 10.9734/BJMCS/2014/9071

This paper considers the stability of linear and nonlinear differential equations of first order in the sense of Hyers-Ulam-Rassias. It also considers the Hyers-Ulam-Rassias stability for Bernoulli's differential equation. Some illustrative examples are given.

Open Access Original Research Article

A Casson Fluid Model for the Steady Flow through a Stenosed Blood Vessel

Manish Gaur, Manoj Kumar Gupta

Journal of Advances in Mathematics and Computer Science, Page 1629-1641
DOI: 10.9734/BJMCS/2014/9563

Laminar flow of blood considering the blood as a Casson fluid has been studied. It is observed that the axial velocity, volumetric flow rate and pressure gradient increase with the increase in slip velocity and decrease with growth in yield stress. The results derived have been presented both analytically and graphically for a better understanding by choosing the appropriate parameters.