S. Lakshmanan

This paper deals with stability and Hopf bifurcation analyses of a mathematical model of HIV infection of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mtext>CD</mml:mtext><mml:msup><mml:mrow><mml:mn mathvariant="normal">4</mml:mn></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math>T-cells. The model is based on a system of delay differential … This paper deals with stability and Hopf bifurcation analyses of a mathematical model of HIV infection of<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mtext>CD</mml:mtext><mml:msup><mml:mrow><mml:mn mathvariant="normal">4</mml:mn></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msup></mml:math>T-cells. The model is based on a system of delay differential equations with logistic growth term and antiretroviral treatment with a discrete time delay, which plays a main role in changing the stability of each steady state. By fixing the time delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. We study the effect of the time delay on the stability of the endemically infected equilibrium. We derive explicit formulae to determine the stability and direction of the limit cycles by using center manifold theory and normal form method. Numerical simulations are presented to illustrate the results.

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On Fractional SIRC Model with<i>Salmonella</i>Bacterial Infection

2014-01-01

Fathalla A. Rihan , Dumitru Băleanu , S. Lakshmanan +1 more

We propose a fractional order SIRC epidemic model to describe the dynamics of Salmonella bacterial infection in animal herds. The infection-free and endemic steady sates, of such model, are asymptotically … We propose a fractional order SIRC epidemic model to describe the dynamics of Salmonella bacterial infection in animal herds. The infection-free and endemic steady sates, of such model, are asymptotically stable under some conditions. The basic reproduction number<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msub><mml:mi>ℛ</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math>is calculated, using next-generation matrix method, in terms of contact rate, recovery rate, and other parameters in the model. The numerical simulations of the fractional order SIRC model are performed by Caputo’s derivative and using unconditionally stable implicit scheme. The obtained results give insight to the modelers and infectious disease specialists.

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Common Coauthors

Coauthor	Papers Together
Fathalla A. Rihan	5
R. Rakkiyappan	2
Prakash Mani	1
Dumitru Băleanu	1
D.H. Abdel Rahman	1
Abdulmajeed Alkhajeh	1
A. H. Hashish	1
E. Ahmed	1
A. G. Sreedevi	1
H. Maurer	1
P. Balasubramaniam	1
K. R. Yogesh kumar	1
M. Venkateshkumar	1

Commonly Cited References

A time delay model of tumour–immune system interactions: Global dynamics, parameter estimation, sensitivity analysis

2014-02-15

Fathalla A. Rihan , D.H. Abdel Rahman , S. Lakshmanan +1 more

Nonstandard finite difference method by nonlocal approximation

2003-01-01

Roumen Anguelov , Jean Lubuma

Nonlinear functional differential equations of arbitrary orders

1998-07-01

A‎. ‎M‎. ‎A‎. El-Sayed

Nonlinear dynamics and chaos in a fractional-order financial system

2006-10-03

Wei-Ching Chen

AN ALGORITHM FOR THE NUMERICAL SOLUTION OF DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

1997-01-01

Kai Diethelm

Differential equations involving derivatives of non-integer order have shown to be adequate models for various physical phenomena in areas like damping laws, diffusion processes, etc. A small number of algorithms … Differential equations involving derivatives of non-integer order have shown to be adequate models for various physical phenomena in areas like damping laws, diffusion processes, etc. A small number of algorithms for the numerical solution of these equations has been suggested, but mainly without any error estimates. In this paper, we propose an implicit algorithm for the approximate solution of an important class of these equations. The algorithm is based on a quadrature formula approach. Error estimates and numerical examples are given.

Analytical approximations for a population growth model with fractional order

2008-07-17

Hang Xu

Modelling aspects of cancer dynamics: a review

2006-04-11

Helen M. Byrne , Tomás Alarcón , Markus R. Owen +2 more

Cancer is a complex disease in which a variety of factors interact over a wide range of spatial and temporal scales with huge datasets relating to the different scales available. … Cancer is a complex disease in which a variety of factors interact over a wide range of spatial and temporal scales with huge datasets relating to the different scales available. However, these data do not always reveal the mechanisms underpinning the observed phenomena. In this paper, we explain why mathematics is a powerful tool for interpreting such data by presenting case studies that illustrate the types of insight that realistic theoretical models of solid tumour growth may yield. These range from discriminating between competing hypotheses for the formation of collagenous capsules associated with benign tumours to predicting the most likely stimulus for protease production in early breast cancer. We will also illustrate the benefits that may result when experimentalists and theoreticians collaborate by considering a novel anti-cancer therapy.

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Recent applications of fractional calculus to science and engineering

2003-01-01

Lokenath Debnath

This paper deals with recent applications of fractional calculus to dynamical systems in control theory, electrical circuits with fractance, generalized voltage divider, viscoelasticity, fractional‐order multipoles in electromagnetism, electrochemistry, tracer in … This paper deals with recent applications of fractional calculus to dynamical systems in control theory, electrical circuits with fractance, generalized voltage divider, viscoelasticity, fractional‐order multipoles in electromagnetism, electrochemistry, tracer in fluid flows, and model of neurons in biology. Special attention is given to numerical computation of fractional derivatives and integrals.

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Numerical Treatments for Volterra Delay Integro-differential Equations

2009-01-01

Fathalla A. Rihan , E. H. Doha , Mohamed I. Hassan +1 more

Abstract This paper presents a new technique for numerical treatments of Volterra delay integro-differential equations that have many applications in biological and physical sciences. The technique is based on the … Abstract This paper presents a new technique for numerical treatments of Volterra delay integro-differential equations that have many applications in biological and physical sciences. The technique is based on the mono-implicit Runge — Kutta method for treating the differential part and the collocation method (using Boole’s quadrature rule) for treating the integral part. The efficiency and stability properties of this technique have been studied. Numerical results are presented to demonstrate the effectiveness of the methodology.

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Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis

1994-03-01

Vladimir A. Kuznetsov , Iliya A. Makalkin , Mark A. Taylor +1 more

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Global existence theory and chaos control of fractional differential equations

2006-11-29

Wei Lin

A Mathematical Tumor Model with Immune Resistance and Drug Therapy: An Optimal Control Approach

2000-04-18

L. G. de Pillis , Ami Radunskaya

We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four‐population model that includes tumor cells, host cells, … We present a competition model of cancer tumor growth that includes both the immune system response and drug therapy. This is a four‐population model that includes tumor cells, host cells, immune cells, and drug interaction. We analyze the stability of the drug‐free equilibria with respect to the immune response in order to look for target basins of attraction. One of our goals was to simulate qualitatively the asynchronous tumor‐drug interaction known as “Jeffs phenomenon.” The model we develop is successful in generating this asynchronous response behavior. Our other goal was to identify treatment protocols that could improve standard pulsed chemotherapy regimens. Using optimal control theory with constraints and numerical simulations, we obtain new therapy protocols that we then compare with traditional pulsed periodic treatment. The optimal control generated therapies produce larger oscillations in the tumor population over time. However, by the end of the treatment period, total tumor size is smaller than that achieved through traditional pulsed therapy, and the normal cell population suffers nearly no oscillations.

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On the fractional-order logistic equation

2006-11-29

A‎. ‎M‎. ‎A‎. El-Sayed , A.E.M. El-Mesiry , H. A. A. El-Saka

Numerical Modeling of Fractional-Order Biological Systems

2013-01-01

Fathalla A. Rihan

We provide a class of fractional-order differential models of biological systems with memory, such as dynamics of tumor-immune system and dynamics of HIV infection of CD4<sup >+</sup> T cells. Stability … We provide a class of fractional-order differential models of biological systems with memory, such as dynamics of tumor-immune system and dynamics of HIV infection of CD4<sup >+</sup> T cells. Stability and nonstability conditions for disease-free equilibrium and positive equilibria are obtained in terms of a threshold parameter <svg style="vertical-align:-3.25793pt;width:22.5px;" id="M1" height="16.025" version="1.1" viewBox="0 0 22.5 16.025" width="22.5" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,11.737)"><path id="x211B" d="M852 669l7 -19q-33 -13 -66 -38q83 -53 83 -139q0 -65 -49 -111q-49 -47 -131 -50v-2q54 -40 54 -84q0 -36 -38 -97t-38 -77q0 -34 32 -34q52 0 145 101l17 -16q-100 -115 -176 -115q-38 0 -65.5 23.5t-27.5 58.5q-1 28 20.5 61t43.5 63t20 51q0 48 -33 48q-30 0 -47 11
q-81 -166 -177 -242.5t-219 -76.5q-77 0 -125 36t-48 87q0 31 17 51t42 20q22 0 36.5 -13.5t14.5 -34.5t-11.5 -34.5t-31.5 -13.5q-23 0 -30 12h-2q0 -28 35 -54q34 -25 85 -25q94 0 152 52q60 53 145 203q75 133 131 214q54 75 121 124q-76 48 -192 48q-150 0 -250 -75
q-101 -75 -101 -174q0 -41 30 -68.5t73 -27.5q158 0 205 245h20q-8 -107 -69 -191q-62 -84 -158 -84q-63 0 -102.5 35.5t-39.5 94.5q0 107 111 191t282 84q124 0 221 -59q38 24 84 41zM814 486q0 64 -48 102q-7 -6 -14 -14t-14.5 -18l-10.5 -14q-4 -5 -35 -55l-6 -11
q-6 -11 -10 -20t-10 -20l-6 -11l-17 -36q-4 -7 -7.5 -14.5l-6 -13t-3.5 -7.5q34 0 60 -17q57 6 92 50q36 43 36 99z" /></g> <g transform="matrix(.012,-0,0,-.012,16.112,15.825)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g> </svg> (minimum infection parameter) for each model. We provide unconditionally stable method, using the Caputo fractional derivative of order <svg style="vertical-align:-0.1254pt;width:9.7749996px;" id="M2" height="7.9499998" version="1.1" viewBox="0 0 9.7749996 7.9499998" width="9.7749996" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D6FC" d="M545 106q-67 -118 -134 -118q-24 0 -40 37.5t-30 129.5h-2q-47 -72 -103 -119.5t-108 -47.5q-47 0 -76 45.5t-29 119.5q0 113 85 204t174 91q47 0 70 -33.5t43 -119.5h3q32 47 80 140l55 13l10 -9q-47 -80 -138 -201q17 -99 27.5 -136t22.5 -37q23 0 69 61zM333 204
q-14 98 -31 149.5t-50 51.5q-49 0 -94 -70t-45 -164q0 -55 15.5 -86t40.5 -31q70 0 164 150z" /></g> </svg> and implicit Euler’s approximation, to find a numerical solution of the resulting systems. The numerical simulations confirm the advantages of the numerical technique and using fractional-order differential models in biological systems over the differential equations with integer order. The results may give insight to infectious disease specialists.

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A generalized Banach contraction principle that characterizes metric completeness

2007-12-06

Tomonari Suzuki

We prove a fixed point theorem that is a very simple generalization of the Banach contraction principle and characterizes the metric completeness of the underlying space. We also discuss the … We prove a fixed point theorem that is a very simple generalization of the Banach contraction principle and characterizes the metric completeness of the underlying space. We also discuss the Meir-Keeler fixed point theorem.

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Multiscale models for the growth of avascular tumors

2007-05-09

Marcelo L. Martins , Silvio C. Ferreira , M.J. Vilela

ON THE FOUNDATIONS OF CANCER MODELLING: SELECTED TOPICS, SPECULATIONS, AND PERSPECTIVES

2008-04-01

Nicola Bellomo , N. K. LI , Philip K. Maini

This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, … This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution.

Mathematical Models of Avascular Tumor Growth

2007-01-01

Tiina Roose , S. Jonathan Chapman , Philip K. Maini

This review will outline a number of illustrative mathematical models describing the growth of avascular tumors. The aim of the review is to provide a relatively comprehensive list of existing … This review will outline a number of illustrative mathematical models describing the growth of avascular tumors. The aim of the review is to provide a relatively comprehensive list of existing models in this area and discuss several representative models in greater detail. In the latter part of the review, some possible future avenues of mathematical modeling of avascular tumor development are outlined together with a list of key questions.

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Dynamics Analysis and Limit Cycle in a Delayed Model for Tumor Growth with Quiescence

2006-02-27

Radouane Yafia

Modeling immunotherapy of the tumor - immune interaction

1998-09-03

Denise E. Kirschner , John C. Panetta

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Computational methods for delay parabolic and time‐fractional partial differential equations

2009-08-11

Fathalla A. Rihan

Abstract This article is concerned with ϑ ‐methods for delay parabolic partial differential equations. The methodology is extended to time‐fractional‐order parabolic partial differential equations in the sense of Caputo. The … Abstract This article is concerned with ϑ ‐methods for delay parabolic partial differential equations. The methodology is extended to time‐fractional‐order parabolic partial differential equations in the sense of Caputo. The fully implicit scheme preserves delay‐independent asymptotic stability and the solution continuously depends on the time‐fractional order. Several numerical examples of interest are included to demonstrate the effectiveness of the method. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Nonstandard numerical methods for a mathematical model for influenza disease

2008-04-17

L. Jódar , R.-J. Villanueva , Abraham J. Arenas +1 more

A delay differential equation model for tumor growth

2003-08-01

Minaya Villasana , Ami Radunskaya

Macrophage T lymphocyte interactions in the anti-tumor immune response: a mathematical model.

1985-04-01

Rob J. de Boer , Paulien Hogeweg , Hub F. J. Dullens +2 more

Abstract In this paper we present a model of the macrophage T lymphocyte interactions that generate an anti-tumor immune response. The model specifies i) induction of cytotoxic T lymphocytes, ii) … Abstract In this paper we present a model of the macrophage T lymphocyte interactions that generate an anti-tumor immune response. The model specifies i) induction of cytotoxic T lymphocytes, ii) antigen presentation by macrophages, which leads to iii) activation of helper T cells, and iv) production of lymphoid factors, which induce a) cytotoxic macrophages, b) T lymphocyte proliferation, and c) an inflammation reaction. Tumor escape mechanisms (suppression, antigenic heterogeneity) have been deliberately omitted from the model. This research combines hitherto unrelated or even contradictory data within the range of behavior of one model. In the model behavior, helper T cells play a crucial role: Tumors that differ minimally in antigenicity (i.e., helper reactivity) can differ markedly in rejectability. Immunization yields protection against tumor doses that would otherwise be lethal, because it increases the number of helper T cells. The magnitude of the cytotoxic effector cell response depends on the time at which helper T cells become activated: early helper activity steeply increases the magnitude of the immune response. The type of cytotoxic effector cells that eradicates the tumor depends on tumor antigenicity: lowly antigenic tumors are attacked mainly by macrophages, whereas large highly antigenic tumors can be eradicated by cytotoxic T lymphocytes only.

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Dynamic analysis of a fractional order prey–predator interaction with harvesting

2013-05-02

Mohammad Masoud Javidi , Nemat Nyamoradi

Global stability analysis for delayed virus infection model with general incidence rate and humoral immunity

2013-03-01

Tianlei Wang , Zhixing Hu , Fucheng Liao +1 more

OPTIMAL CONTROL OF MIXED IMMUNOTHERAPY AND CHEMOTHERAPY OF TUMORS

2008-03-01

L. G. de Pillis , K. Renee Fister , Wenling Gu +5 more

We investigate a mathematical population model of tumor-immune interactions. The populations involved are tumor cells, specific and non-specific immune cells, and concentrations of therapeutic treatments. We establish the existence of … We investigate a mathematical population model of tumor-immune interactions. The populations involved are tumor cells, specific and non-specific immune cells, and concentrations of therapeutic treatments. We establish the existence of an optimal control for this model and provide necessary conditions for the optimal control triple for simultaneous application of chemotherapy, tumor infiltrating lymphocyte (TIL) therapy, and interleukin-2 (IL-2) treatment. We discuss numerical results for the combination of the chemo-immunotherapy regimens. We find that the qualitative nature of our results indicates that chemotherapy is the dominant intervention with TIL interacting in a complementary fashion with the chemotherapy. However, within the optimal control context, the interleukin-2 treatment does not become activated for the estimated parameter ranges.

Modelling of the hamstring muscle group by use of fractional derivatives

2009-09-10

Nenad Grahovac , Miodrag Žigić

Nonlinear fractional dynamics on a lattice with long range interactions

2006-03-21

Nick Laskin , G. M. Zaslavsky

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GLOBAL DYNAMICS OF AN IN-HOST HIV-1 INFECTION MODEL WITH THE LONG-LIVED INFECTED CELLS AND FOUR INTRACELLULAR DELAYS

2012-04-30

Shifei Wang , Yicang Zhou

In this paper, we investigate global dynamics for an in-host HIV-1 infection model with the long-lived infected cells and four intracellular delays. Our model admits two possible equilibria, an uninfected … In this paper, we investigate global dynamics for an in-host HIV-1 infection model with the long-lived infected cells and four intracellular delays. Our model admits two possible equilibria, an uninfected equilibrium and infected equilibrium depending on the basic reproduction number. We derive that the global dynamics are completely determined by the values of the basic reproduction number: if the basic reproduction number is less than one, the uninfected equilibrium is globally asymptotically stable, and the virus is cleared; if the basic reproduction number is larger than one, then the infection persists, and the infected equilibrium is globally asymptotically stable.

On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations

1990-06-01

Odo Diekmann , Hans Heesterbeek , J.A.J. Metz

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Multiple periodic solutions of a delayed stage-structured predator–prey model with non-monotone functional responses

2006-11-15

Yonghui Xia , Jinde Cao , Sui Sun Cheng

A STUDY OF DIFFERENTIAL EQUATION MODELING MALIGNANT TUMOR CELLS IN COMPETITION WITH IMMUNE SYSTEM

2011-06-01

Radouane Yafia

In this paper, we present a competition model of malignant tumor growth that includes the immune system response. The model considers two populations: immune system (effector cells) and population of … In this paper, we present a competition model of malignant tumor growth that includes the immune system response. The model considers two populations: immune system (effector cells) and population of tumor (tumor cells). Ordinary differential equations are used to model the system to take into account the delay of the immune response. The existence of positive solutions of the model (with/without delay) is showed. We analyze the stability of the possible steady states with respect to time delay and the existence of positive solutions of the model (with and without delay). We show theoretically and through numerical simulations that periodic oscillations may arise through Hopf bifurcation. An algorithm for determining the stability of bifurcating periodic solutions is proved.

Theory and Applications of Fractional Differential Equations

2006-01-01

Anatoly A. Kilbas , H. M. Srivastava , Juan J. Trujillo

Modelling and mathematical problems related to tumor evolution and its interaction with the immune system

2000-08-01

Nicola Bellomo , Luigi Preziosi

Two-Dimensional Chemotherapy Simulations Demonstrate Fundamental Transport and Tumor Response Limitations Involving Nanoparticles

2004-11-18

John P. Sinek , Hermann B. Frieboes , Xiaoming Zheng +1 more

Dynamics of a viral infection model with delayed CTL response and immune circadian rhythm

2012-07-20

Zhenguo Bai , Yicang Zhou

Reaction front in an<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>A</mml:mi><mml:mo>+</mml:mo><mml:mrow><mml:mrow><mml:mover><mml:mrow><mml:mi>B</mml:mi></mml:mrow><mml:mrow><mml:mo>→</mml:mo></mml:mrow></mml:mover></mml:mrow></mml:mrow><mml:mi>C</mml:mi></mml:math>reaction-subdiffusion process

2004-03-31

S. B. Yuste , L. Acedo , Katja Lindenberg

We study the reaction front for the process A+B-->C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion … We study the reaction front for the process A+B-->C in which the reagents move subdiffusively. Our theoretical description is based on a fractional reaction-subdiffusion equation in which both the motion and the reaction terms are affected by the subdiffusive character of the process. We design numerical simulations to check our theoretical results, describing the simulations in some detail because the rules necessarily differ in important respects from those used in diffusive processes. Comparisons between theory and simulations are on the whole favorable, with the most difficult quantities to capture being those that involve very small numbers of particles. In particular, we analyze the total number of product particles, the width of the depletion zone, the production profile of product and its width, as well as the reactant concentrations at the center of the reaction zone, all as a function of time. We also analyze the shape of the product profile as a function of time, in particular, its unusual behavior at the center of the reaction zone.

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The dynamics of an optimally controlled tumor model: A case study

2003-06-01

L. G. de Pillis , Ami Radunskaya

Mathematical analysis of an SIS model with imperfect vaccination and backward bifurcation

2011-08-18

Muntaser Safan , Fathalla A. Rihan

Time delays in proliferation process for solid avascular tumour

2003-06-01

Urszula Foryś , Marek Bodnar

Sensitivity analysis for dynamic systems with time-lags

2003-02-01

Fathalla A. Rihan

On some Routh–Hurwitz conditions for fractional order differential equations and their applications in Lorenz, Rössler, Chua and Chen systems

2006-05-06

E. Ahmed , A‎. ‎M‎. ‎A‎. El-Sayed , H. A. A. El-Saka

Qualitative and Computational Analysis of a Mathematical Model for Tumor‐Immune Interactions

2012-01-01

Fathalla A. Rihan , Muntaser Safan , Mohamed Abdeen +1 more

We provide a family of ordinary and delay differential equations to model the dynamics of tumor‐growth and immunotherapy interactions. We explore the effects of adoptive cellular immunotherapy on the model … We provide a family of ordinary and delay differential equations to model the dynamics of tumor‐growth and immunotherapy interactions. We explore the effects of adoptive cellular immunotherapy on the model and describe under what circumstances the tumor can be eliminated. The possibility of clearing the tumor, with a strategy, is based on two parameters in the model: the rate of influx of the effector cells and the rate of influx of IL‐2. The critical tumor‐growth rate, below which endemic tumor does not exist, has been found. One can use the model to make predictions about tumor dormancy.

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