International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Application of Adomian Decomposition Method in Solving Second Order Nonlinear...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The Journal will bring together leading researchers, engineers and scientists in the domain of interest from around the world. Topics of interest for submission include, but are not limited to:
A Hybrid Immunological Search for theWeighted Feedback Vertex Set ProblemMario Pavone
In this paper we present a hybrid immunological inspired algorithm (HYBRID-IA) for solving the Minimum Weighted Feedback Vertex Set (MWFVS) problem. MWFV S is one of the most interesting and challenging combinatorial optimization problem, which finds application in many fields and in many real life tasks. The proposed algorithm is inspired by the clonal selection principle, and therefore it takes advantage of the main strength characteristics of the operators of (i) cloning; (ii) hypermutation; and (iii) aging. Along with these operators, the algorithm uses a local search procedure, based on a deterministic approach, whose purpose is to refine the solutions found so far. In order to evaluate the efficiency and robustness of HYBRID-IA several experiments were performed on different instances, and for each instance it was compared to three different algorithms: (1) a memetic algorithm based on a genetic algorithm (MA); (2) a tabu search metaheuristic (XTS); and (3) an iterative tabu search (ITS). The obtained results prove the efficiency and reliability of HYBRID-IA on all instances in term of the best solutions found and also similar performances with all compared algorithms, which represent nowadays the state-of-the-art on for MWFV S problem.
Legendre Wavelet for Solving Linear System of Fredholm And Volterra Integral ...IJRES Journal
In this work, we employ Legendre wavelet method to find numerical solution of system of linear Fredholm and Volterra integral equations, which uses zeros of Legendre wavelets for collocation points is introduced and used to reduce this type of system of integral equations to a system of algebraic equations.
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Application of Adomian Decomposition Method in Solving Second Order Nonlinear...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The Journal will bring together leading researchers, engineers and scientists in the domain of interest from around the world. Topics of interest for submission include, but are not limited to:
A Hybrid Immunological Search for theWeighted Feedback Vertex Set ProblemMario Pavone
In this paper we present a hybrid immunological inspired algorithm (HYBRID-IA) for solving the Minimum Weighted Feedback Vertex Set (MWFVS) problem. MWFV S is one of the most interesting and challenging combinatorial optimization problem, which finds application in many fields and in many real life tasks. The proposed algorithm is inspired by the clonal selection principle, and therefore it takes advantage of the main strength characteristics of the operators of (i) cloning; (ii) hypermutation; and (iii) aging. Along with these operators, the algorithm uses a local search procedure, based on a deterministic approach, whose purpose is to refine the solutions found so far. In order to evaluate the efficiency and robustness of HYBRID-IA several experiments were performed on different instances, and for each instance it was compared to three different algorithms: (1) a memetic algorithm based on a genetic algorithm (MA); (2) a tabu search metaheuristic (XTS); and (3) an iterative tabu search (ITS). The obtained results prove the efficiency and reliability of HYBRID-IA on all instances in term of the best solutions found and also similar performances with all compared algorithms, which represent nowadays the state-of-the-art on for MWFV S problem.
Legendre Wavelet for Solving Linear System of Fredholm And Volterra Integral ...IJRES Journal
In this work, we employ Legendre wavelet method to find numerical solution of system of linear Fredholm and Volterra integral equations, which uses zeros of Legendre wavelets for collocation points is introduced and used to reduce this type of system of integral equations to a system of algebraic equations.
AN IMPROVED ITERATIVE METHOD FOR SOLVING GENERAL SYSTEM OF EQUATIONS VIA GENE...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
Lyapunov Functions and Global Properties of SEIR Epidemic ModelAI Publications
The aim of this paper is to analyze an SEIR epidemic model in which prophylactic for the exposed individuals is included. We are interested in finding the basic reproductive number of the model which determines whether the disease dies out or persist in the population. The global attractivity of the disease-free periodic solution is obtained when the basic reproductive number is less than unity and the disease persist in the population whenever the basic reproductive number is greater than unity, i.e. the epidemic will turn out to endemic. The linear and non–linear Lyapunov function of Goh–Volterra type was used to establish the sufficient condition for the global stability of the model.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The concept of an intuitionistic fuzzy number (IFN) is of importance for representing an ill-known quantity. Ranking fuzzy numbers plays a very important role in the decision process, data analysis and applications. The concept of an IFN is of importance for quantifying an ill-known quantity. Ranking of intuitionistic fuzzy numbers plays a vital role in decision making and linear programming problems. Also, ranking of intuitionistic fuzzy numbers is a very difficult problem. In this paper, a new method for ranking intuitionistic fuzzy number is developed by means of magnitude for different forms of intuitionistic fuzzy numbers. In Particular ranking is done for trapezoidal intuitionistic fuzzy numbers, triangular intuitionistic fuzzy numbers, symmetric trapezoidal intuitionistic fuzzy numbers, and symmetric triangular intuitionistic fuzzy numbers. Numerical examples are illustrated for all the defined different forms of intuitionistic fuzzy numbers. Finally some comparative numerical examples are illustrated to express the advantage of the proposed method.
On The Distribution of Non - Zero Zeros of Generalized Mittag – Leffler Funct...IJERA Editor
In this work, we derive some theorems involving distribution of non – zero zeros of generalized Mittag – Leffler functions of one and two variables. Mathematics Subject Classification 2010: Primary; 33E12. Secondary; 33C65, 26A33, 44A20
AN IMPROVED ITERATIVE METHOD FOR SOLVING GENERAL SYSTEM OF EQUATIONS VIA GENE...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
Lyapunov Functions and Global Properties of SEIR Epidemic ModelAI Publications
The aim of this paper is to analyze an SEIR epidemic model in which prophylactic for the exposed individuals is included. We are interested in finding the basic reproductive number of the model which determines whether the disease dies out or persist in the population. The global attractivity of the disease-free periodic solution is obtained when the basic reproductive number is less than unity and the disease persist in the population whenever the basic reproductive number is greater than unity, i.e. the epidemic will turn out to endemic. The linear and non–linear Lyapunov function of Goh–Volterra type was used to establish the sufficient condition for the global stability of the model.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The concept of an intuitionistic fuzzy number (IFN) is of importance for representing an ill-known quantity. Ranking fuzzy numbers plays a very important role in the decision process, data analysis and applications. The concept of an IFN is of importance for quantifying an ill-known quantity. Ranking of intuitionistic fuzzy numbers plays a vital role in decision making and linear programming problems. Also, ranking of intuitionistic fuzzy numbers is a very difficult problem. In this paper, a new method for ranking intuitionistic fuzzy number is developed by means of magnitude for different forms of intuitionistic fuzzy numbers. In Particular ranking is done for trapezoidal intuitionistic fuzzy numbers, triangular intuitionistic fuzzy numbers, symmetric trapezoidal intuitionistic fuzzy numbers, and symmetric triangular intuitionistic fuzzy numbers. Numerical examples are illustrated for all the defined different forms of intuitionistic fuzzy numbers. Finally some comparative numerical examples are illustrated to express the advantage of the proposed method.
On The Distribution of Non - Zero Zeros of Generalized Mittag – Leffler Funct...IJERA Editor
In this work, we derive some theorems involving distribution of non – zero zeros of generalized Mittag – Leffler functions of one and two variables. Mathematics Subject Classification 2010: Primary; 33E12. Secondary; 33C65, 26A33, 44A20
Abstract: We present a new algorithm, called the soft-tissue filter that can make both soft and bone tissue clearly visible in digital cephalic radiographies under a wide range of exposures. It uses a mixture model made up of two Gaussian distributions and one inverted lognormal distribution to analyze the image histogram. The image is clustered in three parts: background, soft tissue, and bone using this model. Improvement in the visibility of both structures is achieved through a local transformation based on gamma correction, stretching, and saturation, which is applied using different parameters for bone and soft-tissue pixels. A processing time of 1 s for 5 M pixel images allows the filter to operate in real time. Although the default value of the filter parameters is adequate for most images, real-time operation allows adjustment to recover under- and overexposed images or to obtain the best quality subjectively. The filter was extensively clinically tested: quantitative and qualitative results are reported here Index Terms: Digital radiography, histogram-based clustering, image enhancement, local gamma correction
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
We can define heteroscedasticity as the condition in which the variance of the error term or the residual term in a regression model varies. As you can see in the above diagram, in the case of homoscedasticity, the data points are equally scattered while in the case of heteroscedasticity, the data points are not equally scattered.
Two Conditions:
1] Known Variance
2] Unknown Variance
We can define heteroscedasticity as the condition in which the variance of the error term or the residual term in a regression model varies. As you can see in the above diagram, in the case of homoscedasticity, the data points are equally scattered while in the case of heteroscedasticity, the data points are not equally scattered.
Two Conditions:
1] Known Variance
2] Unknown Variance
A GENERALIZED SAMPLING THEOREM OVER GALOIS FIELD DOMAINS FOR EXPERIMENTAL DESIGNcscpconf
In this paper, the sampling theorem for bandlimited functions over
domains is
generalized to one over ∏
domains. The generalized theorem is applicable to the
experimental design model in which each factor has a different number of levels and enables us
to estimate the parameters in the model by using Fourier transforms. Moreover, the relationship
between the proposed sampling theorem and orthogonal arrays is also provided.
A Generalized Sampling Theorem Over Galois Field Domains for Experimental Des...csandit
In this paper, the sampling theorem for bandlimited functions over
domains is
generalized to one over ∏
domains. The generalized theorem is applicable to the
experimental design model in which each factor has a different number of levels and enables us
to estimate the parameters in the model by using Fourier transforms. Moreover, the relationship
between the proposed sampling theorem and orthogonal arrays is also provided.
KEY
Data Science - Part XII - Ridge Regression, LASSO, and Elastic NetsDerek Kane
This lecture provides an overview of some modern regression techniques including a discussion of the bias variance tradeoff for regression errors and the topic of shrinkage estimators. This leads into an overview of ridge regression, LASSO, and elastic nets. These topics will be discussed in detail and we will go through the calibration/diagnostics and then conclude with a practical example highlighting the techniques.
Please Subscribe to this Channel for more solutions and lectures
http://www.youtube.com/onlineteaching
Elementary Statistics Practice Test 5
Module 5
Chapter 10: Correlation and Regression
Chapter 11: Goodness of Fit and Contingency Tables
Chapter 12: Analysis of Variance
Qualitative Analysis of a Discrete SIR Epidemic Modelijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
An Improved Iterative Method for Solving General System of Equations via Gene...Zac Darcy
Various algorithms are known for solving linear system of equations. Iteration methods for solving the
large sparse linear systems are recommended. But in the case of general n× m matrices the classic
iterative algorithms are not applicable except for a few cases. The algorithm presented here is based on the
minimization of residual of solution and has some genetic characteristics which require using Genetic
Algorithms. Therefore, this algorithm is best applicable for construction of parallel algorithms. In this
paper, we describe a sequential version of proposed algorithm and present its theoretical analysis.
Moreover we show some numerical results of the sequential algorithm and supply an improved algorithm
and compare the two algorithms.
RESIDUALS AND INFLUENCE IN NONLINEAR REGRESSION FOR REPEATED MEASUREMENT DATAorajjournal
All observations don’t have equal significance in regression analysis. Diagnostics of observations is an important aspect of model building. In this paper, we use diagnostics method to detect residuals and influential points in nonlinear regression for repeated measurement data. Cook distance and Gauss newton method have been proposed to identify the outliers in nonlinear regression analysis and parameter estimation. Most of these techniques based on graphical representations of residuals, hat matrix and case deletion measures. The results
show us detection of single and multiple outliers cases in repeated measurement data. We use these techniques
to explore performance of residuals and influence in nonlinear regression model.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
THE IMPORTANCE OF MARTIAN ATMOSPHERE SAMPLE RETURN.Sérgio Sacani
The return of a sample of near-surface atmosphere from Mars would facilitate answers to several first-order science questions surrounding the formation and evolution of the planet. One of the important aspects of terrestrial planet formation in general is the role that primary atmospheres played in influencing the chemistry and structure of the planets and their antecedents. Studies of the martian atmosphere can be used to investigate the role of a primary atmosphere in its history. Atmosphere samples would also inform our understanding of the near-surface chemistry of the planet, and ultimately the prospects for life. High-precision isotopic analyses of constituent gases are needed to address these questions, requiring that the analyses are made on returned samples rather than in situ.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Multi-source connectivity as the driver of solar wind variability in the heli...Sérgio Sacani
The ambient solar wind that flls the heliosphere originates from multiple
sources in the solar corona and is highly structured. It is often described
as high-speed, relatively homogeneous, plasma streams from coronal
holes and slow-speed, highly variable, streams whose source regions are
under debate. A key goal of ESA/NASA’s Solar Orbiter mission is to identify
solar wind sources and understand what drives the complexity seen in the
heliosphere. By combining magnetic feld modelling and spectroscopic
techniques with high-resolution observations and measurements, we show
that the solar wind variability detected in situ by Solar Orbiter in March
2022 is driven by spatio-temporal changes in the magnetic connectivity to
multiple sources in the solar atmosphere. The magnetic feld footpoints
connected to the spacecraft moved from the boundaries of a coronal hole
to one active region (12961) and then across to another region (12957). This
is refected in the in situ measurements, which show the transition from fast
to highly Alfvénic then to slow solar wind that is disrupted by the arrival of
a coronal mass ejection. Our results describe solar wind variability at 0.5 au
but are applicable to near-Earth observatories.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
(May 29th, 2024) Advancements in Intravital Microscopy- Insights for Preclini...Scintica Instrumentation
Intravital microscopy (IVM) is a powerful tool utilized to study cellular behavior over time and space in vivo. Much of our understanding of cell biology has been accomplished using various in vitro and ex vivo methods; however, these studies do not necessarily reflect the natural dynamics of biological processes. Unlike traditional cell culture or fixed tissue imaging, IVM allows for the ultra-fast high-resolution imaging of cellular processes over time and space and were studied in its natural environment. Real-time visualization of biological processes in the context of an intact organism helps maintain physiological relevance and provide insights into the progression of disease, response to treatments or developmental processes.
In this webinar we give an overview of advanced applications of the IVM system in preclinical research. IVIM technology is a provider of all-in-one intravital microscopy systems and solutions optimized for in vivo imaging of live animal models at sub-micron resolution. The system’s unique features and user-friendly software enables researchers to probe fast dynamic biological processes such as immune cell tracking, cell-cell interaction as well as vascularization and tumor metastasis with exceptional detail. This webinar will also give an overview of IVM being utilized in drug development, offering a view into the intricate interaction between drugs/nanoparticles and tissues in vivo and allows for the evaluation of therapeutic intervention in a variety of tissues and organs. This interdisciplinary collaboration continues to drive the advancements of novel therapeutic strategies.
In silico drugs analogue design: novobiocin analogues.pptx
Eigenvalues for HIV-1 dynamic model with two delays
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. II (Jan - Feb. 2015), PP 34-41
www.iosrjournals.org
DOI: 10.9790/5728-11123441 www.iosrjournals.org 34 | Page
Eigenvalues for HIV-1 dynamic model with two delays
M. C. Maheswari1
, K. Krishnan2
, C. Monica3
1
(Department of Mathematics, V.V.Vanniyaperumal College for Women (Autonomous), Virudhunagar)
2
(Research Department of Mathematics, Cardamom Planters' Association College, Bodinayakanur)
3
(Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chepauk, Chennai)
Abstract : In this paper we provide the asymptotic expansion of the roots of nonlinear dynamic system with
two delays. We develop a series expansion to solve for the roots of the nonlinear characteristic equation
obtained from the HIV-1 dynamical system. Numerical calculation are carried out to explain the mathematical
conclusions.
Keywords: Asymptotic expansion, Delay Differential equations, Eigenvalues, HIV-1.
I. INTRODUCTION
The disease of human immunodeficiency virus (HIV) infection has become a worldwide pandemic [1,2].
Very effective treatments for HIV infected individuals have been developed. The emergence of drug resistance
is one of the most prevalent reasons for treatment failure in HIV therapy. Although the correlates of immune
protection in HIV infection remain largely unknown, our knowledge of viral replication dynamics and virus-
specific immune responses has grown.
A number of mathematical models have been proposed to understand the effect of drug therapy on
viremia [3-6]. Drugs e.g., fusion inhibitors, reverse transcriptase inhibitors and protease inhibitors have been
developed so as to attack on different phases of viral life cycle during infection [7,8]. Combination of these
drugs is also used in some article [9]. The eclipse phase is an important phase of virus life cycle on which the
drug therapy primarily depends. Assuming that the level of target cells is constant and that the protease inhibitor
is 100% effective, they obtained the expression of the viral load and explored the effect of the intracellular delay
on viral load change [10].
The nonlinear characteristic equation corresponding to the HIV-1 model belongs to a well-known class of
equations known as exponential polynomials [11-13] and many researchers in DDEs have (understandably)
chosen to study them [14,15]. In the broader mathematics community, research into this class of algebraic
equations is very active and extends to numerical schemes for computing the roots (see papers by Jarlebring
such as [16]) and applications to fields afar as quantum computing [17].
In this paper we presented a new approach to solve the transcendental characteristic equation of HIV-1
infection dynamical system of DDEs, based upon the series expansion for the roots of the nonlinear eigenvalue
problem. Also we provide the numerical results to support our claims of accuracy.
II. EIGENVALUES OF DELAY DIFFERENTIAL EQUATION WITH MULTI DELAY
Consider a delay differential equation with multi delay τi, i = 1,2, …,
,0;)(...)()()()( 23121 ttyatyatyatyaty ii (1)
y(t) = υ(t); t≤0,
where υ(t) is the initial history function. The Laplace transform of (1) is
𝑠 = 𝑎1 + 𝑎2 𝑒−𝑠𝜏1 + ⋯+ 𝑎𝑖 𝑒−𝑠𝜏 𝑖 + ⋯, (2)
a transcendental equation with an infinite number of roots {𝑠𝑗 } in the complex plane. Thus the solutions are
constructed as an infinite series
𝑌 𝑡 = 𝐶𝑗 𝑒 𝑠 𝑗 𝑡∞
𝑗 =−∞ ,
and we note that these basis functions 𝑒 𝑠 𝑗 𝑡
𝑥 form a Schauder basis for 𝐿 𝑝
functions over a finite domain. The
coefficients 𝐶𝑗 can be computed via evaluating the history function υ(t) and the basis functions 𝑒 𝑠 𝑗 𝑡
at 2N+1
time points in [−τ,0] and then linearly solving for 2N+1 of the 𝐶𝑗 ’s.
The exponential term, 𝑒−𝑠𝜏 𝑖 , induced by the time delay term 𝑥(𝑡 − 𝜏𝑖), makes the characteristic equation
transcendental (i.e., infinite dimensional and nonlinear). Thus, it is not feasible to find roots of equation (1),
which has an infinite number of roots. We develop a series expansion for the roots to the nonlinear eigenvalue
2. Eigenvalues for HIV-1 dynamic model with two delays
DOI: 10.9790/5728-11123441 www.iosrjournals.org 35 | Page
problem corresponding to delay differential equation with multi delay. Our goal is to make the truncated sums to
be more computationally efficient to evaluate.
III. HIV-1 INFECTION MODEL WITH TWO DELAY
The general model that have been used to study [3,8,18-20], HIV-1 infection dynamics have a form
similar to
𝑑𝑇
𝑑𝑡
= 𝑠 − 𝑑 𝑇 𝑇 − 𝑘𝑉𝑇,
𝑑𝑇∗
𝑑𝑡
= 𝑘𝑉𝑇 − 𝛿𝑇∗
, (3)
𝑑𝑉
𝑑𝑡
= 𝑁𝛿𝑇∗
− 𝑐𝑉.
where the notations are described below in Table-I.
TABLE - I: In vitro model parameters and its values which are taken from [21Error! Reference source not
found.] and [22]
Notation Description Values
𝑇[cells mm3
] Uninfected target cells 180
𝑇∗
[cells mm3
] Infected cells that are producing virus 3.6
𝑉 [virions mm3
] Virus 134
(Both infectious and non-infectious virus)
s [cells mm3
/day] Rate at which new target cells are generated 1−10
𝑑 𝑇 [ day] Specific death rate of target cells 0.01−0.03
k[mm3
virions/day] Constant rate that characterizing 3.5 × 10−5
− 4.2 × 10−5
target cell infection
δ [ day] Total death rate of target cells 0.26−0.68
N [virions cells] New virus particles 326−503
Models of this standard form have been adequate to summarize the effects of drug therapy on the virus
concentration. Perelson [21] used this model to analyse the response to the protease inhibitor therapy. We
assumed that the target cell density remained unchanged during the observed period. Also we introduced two
delays 𝜏1 as the total intracellular delay, considered in Dixit and Perelson [23], and 𝜏2 as the delay representing
the time necessary for a newly infected virus to become mature and then infectious.
Thus the standard model reduced to a model with delay kernal as
𝑑𝑇∗
𝑑𝑡
= 𝑘𝑇0 𝑉𝐼(𝑡 − 𝜏1) − 𝛿𝑇∗
(𝑡),
𝑑𝑉𝐼
𝑑𝑡
= (1 − 𝑛 𝑝)𝑁𝛿𝑇∗
(𝑡 − 𝜏2) − 𝑐𝑉𝐼(𝑡), (4)
𝑑𝑉𝑁𝐼
𝑑𝑡
= 𝑛 𝑝 𝑁𝛿𝑇∗
(𝑡 − 𝜏2) − 𝑐𝑉𝑁𝐼(𝑡) .
where 𝑇0 is the constant level of target cells and 𝑛 𝑝 is the efficacy of the protease inhibitor scaled such that
𝑛 𝑝 = 1 corresponds to a completely effective drug that results only in the production of non infectious virions,
𝑉𝑁𝐼 .
IV. ASYMPTOTIC EXPANSION FOR THE EIGENVALUES
In this section, we develop our series expansion for the roots to the nonlinear characteristic equation
arising from the two delay DDE. Note that for the multi delay case, the conventional approach focuses on
developing bounds for the roots, whereas here, we are explicitly computing the roots.
The linearized HIV-1 equation (4) can be expressed in state space as,
𝑥 𝑡 = 𝑎𝑥 𝑡 + 𝑏𝑥 𝑡 − 𝜏1 + c𝑥(𝑡 − 𝜏2) (5)
where, 𝑎 =
−𝛿 0 0
0 −𝑐 0
0 0 −𝑐
, 𝑏 =
0 𝑘𝑇0 0
0 0 0
0 0 0
and 𝑐 =
0 0 0
1 − 𝑛 𝑝 𝑁𝛿 0 0
𝑛 𝑝 𝑁𝛿 0 0
.
3. Eigenvalues for HIV-1 dynamic model with two delays
DOI: 10.9790/5728-11123441 www.iosrjournals.org 36 | Page
Defining 𝑥 = {𝑇∗
, 𝑉𝐼, 𝑉𝑁𝐼} 𝑇
, where 𝑇
indicates transpose. 𝑎, 𝑏 and c are the linearized coefficient matrix of the
HIV-1 infection model and are functions of in vitro model parameters.We will rescale and rearrange terms,
defining several new variables and parameters in order to reduce the equation to its essential mathematical
features. We rescale time by the first delay 𝑡 =
𝑡
𝜏1
, the parameters so that 𝑎1 = 𝑎𝜏1, 𝑏1 = 𝑏𝜏1, 𝑐1 = 𝑐𝜏1, 𝜏 =
𝜏2
𝜏1
,
and denote 𝑥 𝑡 =
𝑑
𝑑𝑡
𝑥 𝑡 = 𝜏1
𝑑
𝑑𝑡
𝑥 𝑡 to obtain the equation
𝑥 𝑡 = 𝑎1 𝑥 𝑡 + 𝑏1 𝑥 𝑡 − 𝜏1 + c1 𝑥(𝑡 − 𝜏2)
After a Laplace transform, the corresponding nonlinear characteristic equation for 𝑠 ∈ 𝐶 is, thus
𝑠
= 𝑎1 + 𝑏1 𝑒−𝑠
+ 𝑐1 𝑒−𝑠𝜏
. 6
We then let 𝜆 = 𝑠 − 𝑎1, 𝑏2 = 𝑏1 𝑒−𝑎1 , 𝑐2 = 𝑐1 𝑒−𝑎1 𝜏
, and multiply through by 𝑒 𝜆𝜏
, equation (6) becomes much
cleaner equation as,
𝜆𝑒 𝜆𝜏
=
𝑏2 𝑒(𝜏−1)𝜆
+ 𝑐2. (7)
Inspired by the original expansion in [24Error! Reference source not found.], we note that for |𝑐2| near to or
far from zero, we can write λ as a small perturbation u of ln 𝑐2,
𝜆 = (ln 𝑐2 + 𝑢)/𝜏.
and then safely assume that
|𝑢| ≪ |ln 𝑐2 |. (8)
Substituting this form of λ into equation (7Error! Reference source not found.) we find that
1
τ
1 +
𝑢
ln c2
𝑐2 𝑒 𝑢
=
𝑏2𝑐2𝜏−1𝜏ln 𝑐2 (𝑒𝑢)𝜏−1𝜏+ 𝑐2ln 𝑐2. (9)
To proceed, we will also assume that
𝑏2
𝑐2
1/𝜏 ln 𝑐2
≪ 1, (10)
in addition to (8). This restriction on the coefficients is a result of the chosen rearrangement of(6) into (7). There
is nothing particularly special about (10Error! Reference source not found.) and choosing to recast (6)
differently would simply necessitate different assumptions on the parameters in order for an expansion to
converge. Moving forward, we can solve (9) for u as
𝑢 ≈ ln 𝜏 + ln(1/ ln 𝑐2),
which means that
𝑢
= ln 𝜏 + ln 1 ln 𝑐2 + 𝑣, (11)
where our attention now focuses on solving for a new variable v. As discussed by Corless et al. [25Error!
Reference source not found.], the nested logarithms in (11) need not select the same branch. As Corless et al.
[25Error! Reference source not found.] do, we will let the outer logarithm be the principal branch and denote
Ln as the inner logarithm for which we have not yet chosen a branch, i.e.,
𝑢 = ln 𝜏 + ln 1 ln 𝑐2 + 𝑣.
By substituting u back into (9), we can cancel several terms and simplify our problem to one of finding the roots
v to the equation
1 +
ln 𝜏
𝐿𝑛 𝑐2
+
ln 1 Ln 𝑐2
𝐿𝑛 𝑐2
+
1
𝐿𝑛 𝑐2
=
𝑏2
𝑐2
𝑐2 𝜏
𝐿𝑛 𝑐2
𝜏−1
𝜏
𝑒−𝑣 𝜏
+ 𝑒−𝑣
.
If we let
𝜎 =
1
𝐿𝑛 𝑐2
, 𝛾 =
𝑏2
𝑐2
𝑐2 𝜏
𝐿𝑛 𝑐2
𝜏−1
𝜏
, 𝜇 =
ln 𝜏 ln(1 𝐿𝑛 𝑐2)
𝐿𝑛 𝑐2
− 𝛾.
then our efforts will focus on identifying the roots of
4. Eigenvalues for HIV-1 dynamic model with two delays
DOI: 10.9790/5728-11123441 www.iosrjournals.org 37 | Page
𝑓 𝑣 ≡ 𝑒−𝑣
+ 𝛾𝑒−𝑣 𝜏
− 𝜎𝑣 − 1 − 𝛾 − 𝜇. (12)
Note that the motivation for adding and subtracting γ will become self-evident in the proof for the following
lemma.
Lemma 1 If there exists positive numbers α and β such that |σ|<α and |μ|<α then (12Error!
Reference source not found.) has a single root inside the region 𝑣 ≤ 𝛽.
Proof. Let δ denote the lower bound of |e−v
− 1| on |v| = β and δτ denote the lower bound of |γe−v/ τ
- γ| on |v| =
β. We can arbitrarily choose β = π min{1, τ} since e−v
−1 has only one root at zero for β ∈ (0,2πτ) and γ(e−v/ τ
−1)
also has only one root at v = 0 for β ∈ (0,2πτ). Thus there is only one root at v = 0 for e−v
−1+γ(e−v/ τ
−1) in the
region |v| ≤ π min{1, τ} and that on |v| = π min{1, τ} we have a lower bound of min{δ,δτ}.
Let α =
min {δ,δτ}
2(π+1)
and then for |σ| < α, |µ| < α, and |v| = π min{1, τ}, we have that
|σv + µ| ≤ |σ||v| + |µ| ≤
min {δ,δτ}
2(π+1)
π +
min {δ,δτ}
2(π+1)
=
min {δ,δτ}
2(π+1)
< min{δ,δτ}.
Since |e−v
− 1 + γ e−v/ τ
− γ| > min{δ, δτ} on the curve defined by |v| = π min{1, τ}, we have satisfied the
hypotheses of Rouche` ’s Theorem [26] and can therefore conclude that there is only one root to (12) inside |v| =
π min{1, τ}.
By Cauchy’s theorem, we can then compute v using the well-known formula from complex analysis
v=
1
2πi
∮
|δ|=β
f'(δ)δ
f(δ)
dδ=
1
2πi
∮
|δ|=β
−e
−δ
−γ
e
−δ/τ
τ
−σ δ
e
−δ
+γe
−δ/τ
−σδ−1−γ−μ
dδ.
(13)
for β=πmin{1, τ}. The 1/f(δ) can be expanded into the following (absolutely and uniformly convergent) power
series
1
𝑓(𝜁)
= (𝑒−𝜁
+ 𝛾𝑒−
𝜁
τ − 1 − 𝛾)−𝑘−𝑚−1
𝜁 𝑘
𝜎 𝑘
𝜇 𝑚
𝛼 𝑚
𝑚+𝑘
,∞
𝑚=0
∞
𝑘=0
(14)
with some coefficients 𝛼 𝑚 . The substitution of (14) into (13) and evaluation of the contour integral generates a
double series in m and k. Note, however, that the m=0 term is zero because the integrand has the same number
of roots (at δ=0) in the numerator and the denominator. The doubly infinite and absolutely convergent power
series for v in σ and μ is, thus
𝑣 = 𝛼 𝑘𝑚 𝜎 𝑘
𝜇 𝑚
∞
𝑚=0
∞
𝑘=0
, (15)
where the 𝛼 𝑘𝑚 are coefficients independent of σ and μ. The coefficients 𝛼 𝑘𝑚 are computed by first computing a
power series expansion in σ and then applying the Lagrange Inversion Theorem [27Error! Reference source
not found.] to obtain v as a function of μ. To facilitate rearrangement of this series into the form of (15Error!
Reference source not found.), we denote
𝑓2 𝑤 ≡ 𝑓 𝑤 − 𝑓 0 ,
and note that the derivative of f(w) with respect to σ is −w. Equation (15) can thus be written (after the power
series expansion in σ around zero)
𝑣 =
𝑚 𝑘∞
𝑚=0
∞
𝑘=0 ℎ 𝑚,𝑘
𝜇 𝑚
𝑚!
𝜎 𝑘
𝑘!
, (16)
where 𝑚 𝑘
is a rising factorial, (We note that in the combinatorics literature, m
(k)
is more commonly used as a
rising factorial, also known as the Pochhammer symbol with 𝑘=(m+k−1)!−(m−1)!) and ℎ 𝑚,𝑘 = limw→0 hm,k(w)
with
hm,k w =
dm−1
dwm−1
wm+k
f2(w)−(m+k)
.
5. Eigenvalues for HIV-1 dynamic model with two delays
DOI: 10.9790/5728-11123441 www.iosrjournals.org 38 | Page
Straightforward application of the product rule yields that
hm,k w =
m − 1
l
m−1
l=0
dm−1−l
dwm−1−l
wm+k
dl
dwl
f2(w)−(m+k)
.
The lth derivative of f2(w)−(m+k)
is computed using Faǎ Di Bruno’s formula [28Error! Reference source not
found.] for higher derivatives of composite functions and thus
hm,k w = aklm wk+l+1
m−1
l=0
bkmp f2(w)−(m+k+p)
l
p=0
𝔹l,p f2
′
w , … , f2
(l−p+1)
w ,
where,
aklm =
m − 1
l
m + k !
k + l + 1 !
, bkmp = (−1)p
(m + k)p
,
and 𝔹𝑙,𝑝 are the partial Bell polynomials [11,29Error! Reference source not found.]. Recall that for an
arbitrary sequence {x
i
} and l,p∈N, the partial Bell polynomials are
𝔹l,p x1, … , xl−p+1 =
l!
j1! j2! … jl−p+1!
x1
1!
j1 x2
2!
j2
…
xl−p+1
(l − p + 1)!
jl−p+1
,
with the sum taken over all sequences {j1, … , jn−k+1} ∈ N such that ji
n−k+1
i=1 = m and iji
n−k+1
i=1 = k (a simple
Diophantine linear system). The sets of indices {ji} satisfying these sums are just a way of describing all
possible partitions of a set of size k [30Error! Reference source not found.].
In what follows, the argument for the Bell polynomial frequently takes the form {f2
1+i
(0)}i=1
n
. To improve the
clarity of the formulas, we denote
Fn = {f2
1+i
(0)}i=1
n
; 𝑛 ∈ 𝑁,
to be a finite sequence of higher derivatives of f
2
. Formally taking the limit for limw→0 hm,k(w) yields
𝛼 𝑘𝑙𝑚 𝛽𝑘𝑙𝑚𝑝𝑞 𝔹2𝑚−2−𝑙−𝑞 ,𝑚−1−𝑝
2𝑚−2−𝑙
𝑞=0
𝑙
𝑝=0
𝑚−1
𝑙=0 𝐹 𝑚−𝑙−𝑞+𝑝 𝔹𝑙,𝑝
(𝑞)
(𝐹𝑙−𝑝+1)
𝑓"2 (0)2𝑚+𝑘−1
where
αklm = αklm
2m + k − 1
2m − l − 2
k + l + 1 !, βklmpq = βkmp
2m − 2 − l
q
m − 1 − p !
2m + k − 1 !
.
The evaluation of most terms in (17) is computationally straightforward and efficient, except the term involving
𝔹𝑙,𝑝
(𝑞)
. For example, the sum in (16) for m = 10 and k = 100 can take around one minute for computation when
using Mathematica to analytically find the qth derivative of 𝔹𝑙,𝑝 and then sum the terms. In [31Error!
Reference source not found.], Mishkov derived a formula for 𝔹𝑙,𝑝
(𝑞)
, but to implement it would consists of
solving a cascade of linear Diophantine equations to obtain the correct indices for the sums. We will use a
recursive property of 𝔹𝑙,𝑝
(𝑘)
to avoid having to solve systems of Diophantine equations. Note that for an arbitrary
sequence X={x
j
}
𝜕
𝜕𝑥𝑗
𝔹𝑙,𝑝 𝑋 =
𝑙
𝑗
𝔹𝑙−𝑗,𝑝−1 𝑋 ,
and thus in the context of (17) we have that
𝔹𝑙,𝑝
𝑞
𝐹𝑙−𝑝+1 =
𝑙
𝑟1
𝑙−𝑝+1
𝑟1=1
𝑞 − 1
𝑟2
𝑞−1
𝑟2=0 𝔹𝑙−𝑟1,𝑝−1
𝑞−𝑟2−1
𝐹𝑙−𝑝+1 (18)
recursively allows (eventual) computation of the derivatives by evaluation of the partial Bell polynomials
themselves The use of Mathematica’s memoization features for recursive sums substantially sped up the
computation as well, naturally at the expense of increased memory usage. We now have that the principal
branch of the roots of (7) is
𝜆 =
ln 𝑐2 + ln 𝜏 + ln 1 ln 𝑐2 + 𝑣
𝜏
, (19)
where
6. Eigenvalues for HIV-1 dynamic model with two delays
DOI: 10.9790/5728-11123441 www.iosrjournals.org 39 | Page
𝑣 = 𝑚 𝑘
∞
𝑚=0
∞
𝑘=0
ℎ 𝑚,𝑘
𝜇 𝑚
𝑚!
𝜎 𝑘
𝑘!
, (20)
We note that for i≥1
f2
1+i
0 = −1 i
1 + cτ − i .
and thus h
m,k
is defined as
𝛼 𝑘𝑙𝑚𝑝𝑞 𝔹2𝑚 −2−𝑙−𝑞 ,𝑚 −1−𝑝
2𝑚 −2−𝑙
𝑞=0
𝑙
𝑝=0
𝑚 −1
𝑙=0 𝐹 𝑚 −𝑙−𝑞+𝑝 𝔹𝑙,𝑝
(𝑞)
(𝐹 𝑙−𝑝+1)
−(1+𝜂τ−1)2𝑚 +𝑘−1
(21)
where
Fn = { −1 n
1 + ητ−n
} i=1
n
,
αklmpq =
−1 p
k + m Γ(m)Γ(m − p)Γ(k + m + p)
q! l! Γ(2 + k + l)Γ(−l + m)Γ(−1 − l + 2m − q)
,
𝜎 =
1
𝐿𝑛 𝑐2
, 𝛾 =
𝑏2
𝑐2
𝑐2 𝜏
𝐿𝑛 𝑐2
𝜏−1
𝜏
, 𝜇 =
ln 𝜏 ln(1 𝐿𝑛 𝑐2)
𝐿𝑛 𝑐2
− 𝛾.
and recalling that 𝔹𝑙,𝑝
(𝑞)
is defined recursively in (18).
The full description of the jth branch root for the characteristic equation in (6Error! Reference source
not found.) is therefore
sj
=
1
𝜏
(lnj 𝑐2 + ln 𝜏 + ln 1 lnj 𝑐2 + 𝑣𝑗 ) + 𝑎1,
Where lnj(𝑧) = 𝑧 + arg 𝑧 + 2𝜋𝑖𝑗, v is defined in (20Error! Reference source not found.), and the
subscript on the v indicates the chosen branch for the logarithms in σ,ε, and μ.
V. NUMERICAL CALCULATION
In this section, we provide numerical results for the estimation of the eigenvalues. Consider the vector
form of the delay differential equation (5) with two delay.
We note that the accuracy of our asymptotic expansion rests on the assumption that (10) is small and so
we choose
𝑏2 =
0 5.382 × 10−4
0
0 0 0
0 0 0
, 𝑐2 =
1.0832 0 0
0 0.0068 0
0 0 0.0068
, and 𝑎1 =
−0.96 0 0
0 −9 0
0 0 −9
.
a=
−0.32 0 0
0 −3 0
0 0 −3
, b=
0 0.0058 0
0 0 0
0 0 0
, and c=
0 0 0
15.36 0 0
138.24 0 0
,
and thus
𝑏2
𝑐2
1/𝜏 ln 𝑐2
≈
0 0.0782 0
0 0 0
0 0 0
.
Numerical investigations suggested that the series converges in m about a factor of four times faster than it
converges in k. Accordingly, Figure - 1 depicts the convergence in m with 𝑘 = 𝑚4
. We note that up to 𝑚 = 8,
the series converges nicely. For 𝑚 > 8, however, the series exhibits a reduction of accuracy, likely due to the
numerical issues inherent in evaluating such large series. Lastly, in subsequent computations of 𝑠𝑗 , we choose to
truncate the series at 𝑚 = 8 and 𝑘 = 1000 to maintain accuracy.
7. Eigenvalues for HIV-1 dynamic model with two delays
DOI: 10.9790/5728-11123441 www.iosrjournals.org 40 | Page
Figure - 1
Figure - 1: Eigenvalues s
j
computed using the expansion in Section 4 with the parameter values in
Table-I.
VI. CONCLUSION
In this paper, the new approach for the eigenvalue of the HIV-1 infected dynamical problem, have been
presented using the series expansion of the roots of the nonlinear characteristic equation. Since Asl and Ulsoy’s
[32Error! Reference source not found.] original article, the connection between DDE eigenvalues and
solutions to Lambert W has been substantially extended and expanded [16,25,28,29,31Error! Reference source
not found.]. These results, however, do not extend to multi delay DDEs as (to the best of our knowledge) the
nonlinear eigenvalue problem can be cast as a Lambert W equation only when the DDE has a single delay. We
have developed an asymptotic expansion for the roots to a nonlinear eigenvalue problem associated with two
delay model of HIV-1 infection dynamics. We have provided numerical evidence supporting the accuracy of our
expansion.
REFERENCES
[1] S. Bonhoeffer, R.M. May, G.M Shaw, M.A. Nowak, Virus dynamics and drug therapy, Proc. Natl. Acad. Sci. USA, 94, 1997, 6971
- 6976.
[2] X.Y. Song, A.U. Neumann, Global stability and periodic solution of the viral dynamics, J. Math. Anal. Appl., 329, 2007, 281 - 297.
[3] M.A. Nowak, S. Bonhoeffer, G.M. Shaw, R.M. May, Anti-viral drug treatment: dynamics of resistance in free virus and infected
cell populations, J. Theor. Biol., 184, 1997, 203 - 217.
[4] T.B. Kepler, A.S. Perelson, Drug concentration heterogeneity facilitates the evolution of drug resistance, Proc. Natl. Acad. Sci.
USA, 95, 1998, 11514 - 11519.
[5] P.K. Roya, A.N. Chatterjee, D. Greenhalgh, J.A. Khanc, Long term dynamics in a mathematical model of HIV-1 infection with
delay in different variants of the basic drug therapy model, Nonlinear Analysis: Real World Applications, 14, 2013, 1621 - 1633.
[6] R.J. Smith, L.M. Wahl, Distinct effects of protease and reverse transcriptase inhibition in an immunological model of HIV-1
infection with impulsive drug effects, Bull. Math. Biol., 66, 2004, 1259 - 1283.
[7] M. Pitchaimani, C. Monica, M. Divya, Stability analysis for HIV infection delay model with protease inhibitor, Biosystems, 114,
2013, 118 - 124.
[8] D. Kirschner, Using mathematics to understand HIV immune dynamics, Notices Am. Math. Soc., 43, 1996, 191 - 202.
[9] P.W. Nelson, A.S. Perelson, Mathematical Analysis of delay differential equation models of HIV-1 infection, Math. Biosci., 179,
2002, 73 - 94.
[10] V. Herz, S. Bonhoeffer, R. Anderson, R. May, M. Nowak, Viral dynamics in vivo: limitations on estimations on intracellular delay
and virus decay, Proc. Nat. Acad. Sci., 93, 1996, 7247.
[11] E.T. Bell, Exponential Polynomials, The Analysis of Mathematics, 35(2), 1934, 258 - 277.
[12] C.J. Moreno, The zeros of exponential polynomials (I), Compositio Mathematica, 26(1), 1973, 69 - 78.
[13] J.F. Ritt, On the Zeros of Exponential Poylnomials, Transactions of the American Mathematical Society, 31(4), 1929, 680 - 686.
[14] C.E. Avellar, J.K. Hale, On the Zeros of Exponential Polynomials, Journal of Mathematical Analysis and Applications, 73, 1980,
434 - 452.
[15] K.L. Cooke, P. Van Den Driessche, On zeros of some transcendental functions, Eunkcialaj Ekvacioj, 29, 1986, 77 - 90.
[16] E. Jarlebring, Critical delays and polynomial eigenvalue problems, Journal of Computational and Applied Mathematics, 224(1),
2009, 296- 306.
[17] Y. Sasaki, On zeros of exponential polynomials and quantum algorithms, Quantum Information Processing, 9(3), 2009, 419 - 427.
[18] A. Perelson, D. Kirschner, R. De Boer, Dynamics of HIV infection of CD4+ T cells, Math. Biosci., 114, 1993, 81 - 90.
[19] A. McLean, S. Frost, Ziduvidine and HIV: mathematical models of within-host population dynamics, Rev. Med. Virol., 5, 1995,
141 - 145.
8. Eigenvalues for HIV-1 dynamic model with two delays
DOI: 10.9790/5728-11123441 www.iosrjournals.org 41 | Page
[20] M. Nowak, R. Anderson, M. Boerlijst, S. Bonhoeffer, R. May, A. McMichael, HIV-1 evolution and disease progression, Science,
274 1996, 1008 - 1011.
[21] A. Perelson, A. Neumann, M. Markowitz, J. Leonard, D. Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span,
and viral generation time, Science, 271, 1996, 1582 - 1585.
[22] P.K. Srivastava, M. Banerjee, P. Chandra, Modeling the drug therapy for HIV infection, J. Biol. systems, 17, 2009, 213 - 223.
[23] N.M. Dixit, A.S. Perelson, Complex patterns of viral load decay under antiretroviral therapy: in
uence of pharmacokinetics and intracellular delay, J. Theor. Biol., 226m 2004, 95 - 109.
[24] N.D. De Bruijn, Asymptotic Methods in Analysis, North Holland Publishing Company, Amsterdam, The Netherlands, Dover
Edition, 1958.
[25] R.M. Corless, G.H. Gonnet, D.E.G. Hare, D.J. Jerey, and D.E. Knuth, On Lambert's W function, Advances in Computational
Mathematics, 5, 1996, 329 - 359.
[26] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1977.
[27] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, 1972.
[28] W.P. Johnson, The Curious History of Fa di Bruno's Formula, The American Mathematical Monthly, 109(3), 2002, 217 - 222.
[29] Wheeler, F. S., 1987. Bell polynomials, ACM SIGSAM Bulletin, 21(3): 44 - 53.
[30] L.H. Encinas, A.M. Del Rey, J.M. Masque, Faa di Bruno's formula, lattices, and partitions, Discrete Applied Mathematics, 148(3),
2005, 246 - 255.
[31] R.L. Mishkov, Generalization of the formula of Fa di Bruno for acomposite function with a vector argument, International Journal
of Mathematics and Mathematical Sciences, 24(7), 2000, 481 - 491.
[32] F.M. Asl, A.G. Ulsoy, Analysis of a system of linear delay differential equations, ASME. J. Dyn. Syst. Meas. Cont., 125, 2003, 215
- 223.