Almost every year, a new book on mathematical modeling is published, so, why another? The answer springs directly from the fact that it is very rare to find a book that covers modeling with all types of differential equations in one volume. Until now. Mathematical Modeling: Models, Analysis and Applications covers modeling with all kinds of differential equations, namely ordinary, partial, delay, and stochastic. The book also contains a chapter on discrete modeling, consisting of differential equations, making it a complete textbook on this important skill needed for the study of science, engineering, and social sciences.

More than just a textbook, this how-to guide presents tools for mathematical modeling and analysis. It offers a wide-ranging overview of mathematical ideas and techniques that provide a number of effective approaches to problem solving. Topics covered include spatial, delayed, and stochastic modeling. The text provides real-life examples of discrete and continuous mathematical modeling scenarios. MATLAB® and Mathematica® are incorporated throughout the text. The examples and exercises in each chapter can be used as problems in a project.

Features • Addresses all aspects of mathematical modeling with mathematical tools used

in subsequent analysis• Incorporates MATLAB and Mathematica• Covers spatial, delayed, and stochastic models• Presents real-life examples of discrete and continuous scenarios • Includes examples and exercises that can be used as problems in a project

Since mathematical modeling involves a diverse range of skills and tools, the author focuses on techniques that will be of particular interest to engineers, scientists, and others who use models of discrete and continuous systems. He gives students a foundation for understanding and using the mathematics that is the basis of com-puters, and therefore a foundation for success in engineering and science streams.

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Engineering Mathematics

MATHEMATICAL MODELINGMATHEMATICAL

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Models, Analysis and ApplicationsModels, Analysis and Applications

Sandip Banerjee

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MATHEMATICAL MODELING

Models, Analysis and Applications

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MATHEMATICAL MODELING

Models, Analysis and Applications

Sandip BanerjeeIndian Institute of Technology

Roorkee

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MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® soft-ware or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.

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i

To my wife Usha

and

son Aditya

who brought

love and joy

to my life

and to Otso

who taught me

how to do research

Contents

1 About Mathematical Modeling 11.1 What is Mathematical Modeling? . . . . . . . . . . . . . . . 11.2 History of Mathematical Modeling . . . . . . . . . . . . . . . 21.3 Importance of Mathematical Modeling . . . . . . . . . . . . 41.4 Latest Developments in Mathematical Modeling . . . . . . . 51.5 Limitations of Mathematical Modeling . . . . . . . . . . . . 6

2 Mathematically Modeling Discrete Processes 92.1 Difference Equations . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Linear Difference Equation with Constant Coefficients 102.1.2 Solution of Homogeneous Equations . . . . . . . . . . 102.1.3 Difference Equation: Equilibria and Stability . . . . . 14

2.1.3.1 Linear Difference Equation . . . . . . . . . . 142.1.3.2 System of Linear Difference Equations . . . . 142.1.3.3 Non-Linear Systems . . . . . . . . . . . . . . 16

2.2 Introduction to Discrete Models . . . . . . . . . . . . . . . . 172.3 Linear Models . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Population Model Involving Growth . . . . . . . . . . 182.3.2 Newton’s Law of Cooling . . . . . . . . . . . . . . . . 192.3.3 Bank Account Problem . . . . . . . . . . . . . . . . . 202.3.4 Drug Delivery Problem . . . . . . . . . . . . . . . . . 222.3.5 Economic Model (Harrod Model) . . . . . . . . . . . . 232.3.6 Arms Race Model . . . . . . . . . . . . . . . . . . . . 242.3.7 Linear Prey-Predator Problem . . . . . . . . . . . . . 24

2.4 Non-Linear Models . . . . . . . . . . . . . . . . . . . . . . . 272.4.1 Density Dependent Growth Models . . . . . . . . . . . 272.4.2 The Learning Model . . . . . . . . . . . . . . . . . . . 27

2.5 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . 282.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3 Continuous Models Using Ordinary Differential Equations 473.1 Introduction to Continuous Models . . . . . . . . . . . . . . 473.2 Formation of Various Continuous Models . . . . . . . . . . . 48

3.2.1 Carbon Dating . . . . . . . . . . . . . . . . . . . . . . 483.2.2 Drug Distribution in the Body . . . . . . . . . . . . . 493.2.3 Growth and Decay of Current in an L-R Circuit . . . 50

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3.2.4 Rectilinear Motion under Variable Force . . . . . . . . 523.2.5 Mechanical Oscillations . . . . . . . . . . . . . . . . . 53

3.2.5.1 Horizontal Oscillations . . . . . . . . . . . . 533.2.5.2 Vertical Oscillations . . . . . . . . . . . . . . 543.2.5.3 Damped Force Oscillation . . . . . . . . . . . 55

3.2.6 Dynamics of Rowing . . . . . . . . . . . . . . . . . . . 573.2.7 Arms Race Models . . . . . . . . . . . . . . . . . . . . 583.2.8 Mathematical Model of Influenza Infection (within

Host) . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2.9 Epidemic Models . . . . . . . . . . . . . . . . . . . . . 61

3.3 Steady State Solutions . . . . . . . . . . . . . . . . . . . . . 653.4 Linearization and Local Stability Analysis . . . . . . . . . . 663.5 Phase Plane Diagrams of Linear Systems . . . . . . . . . . . 683.6 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

3.6.1 Saddle-Node Bifurcation . . . . . . . . . . . . . . . . . 733.6.2 Transcritical Bifurcation . . . . . . . . . . . . . . . . . 753.6.3 Pitchfork Bifurcation . . . . . . . . . . . . . . . . . . . 773.6.4 Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . 79

3.7 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . 803.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4 Spatial Models Using Partial Differential Equations 1114.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1114.2 Different Mathematical Models Using Diffusion . . . . . . . . 112

4.2.1 Fluid Flow through a Porous Medium . . . . . . . . . 1124.2.2 Heat Flow through a Small Thin Rod (One Dimen-

sional) . . . . . . . . . . . . . . . . . . . . . . . . . . . 1134.2.3 Wave Equation . . . . . . . . . . . . . . . . . . . . . . 1154.2.4 Vibrating String . . . . . . . . . . . . . . . . . . . . . 1174.2.5 Traffic Flow . . . . . . . . . . . . . . . . . . . . . . . . 1194.2.6 Theory of Car-Following . . . . . . . . . . . . . . . . 1244.2.7 Crimes Model . . . . . . . . . . . . . . . . . . . . . . . 126

4.3 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . 1274.3.1 One Species with Diffusion . . . . . . . . . . . . . . . 1274.3.2 Two Species with Diffusion . . . . . . . . . . . . . . . 128

4.4 A Research Problem: Spatiotemporal Aspect of a MathematicalModel of Cancer Immune Interaction Considering the Role ofAntibodies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1324.4.1 Background of the Problem . . . . . . . . . . . . . . . 1324.4.2 Spatiotemporal Model Formulation . . . . . . . . . . . 1324.4.3 Qualitative Analysis . . . . . . . . . . . . . . . . . . . 1334.4.4 Numerical Results . . . . . . . . . . . . . . . . . . . . 1374.4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 138

4.5 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . 1394.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

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5 Modeling with Delay Differential Equations 1535.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535.2 Different Models Using Delay Differential Equations . . . . . 154

5.2.1 Delayed Protein Degradation . . . . . . . . . . . . . . 1545.2.2 Football Team Performance Model . . . . . . . . . . . 1555.2.3 Breathing Model . . . . . . . . . . . . . . . . . . . . . 1565.2.4 Housefly Model . . . . . . . . . . . . . . . . . . . . . . 1575.2.5 Shower Problem . . . . . . . . . . . . . . . . . . . . . 1585.2.6 Two-Neuron System . . . . . . . . . . . . . . . . . . . 159

5.3 Linear Stability Analysis . . . . . . . . . . . . . . . . . . . . 1605.3.1 Linear Stability Criteria . . . . . . . . . . . . . . . . . 161

5.4 Miscellaneous Examples . . . . . . . . . . . . . . . . . . . . . 1635.4.1 A Research Problem: Immunotherapy with Interleukin-

2, a Study Based on Mathematical Modeling [8] . . . . 1715.4.1.1 Background of the Problem . . . . . . . . . . 1715.4.1.2 The Model . . . . . . . . . . . . . . . . . . . 1735.4.1.3 Positivity of the Solution . . . . . . . . . . . 1755.4.1.4 Linear Stability Analysis with Delay . . . . . 1755.4.1.5 Estimation of the Length of Delay to Preserve

Stability . . . . . . . . . . . . . . . . . . . . 1785.4.1.6 Numerical Results . . . . . . . . . . . . . . . 1815.4.1.7 Conclusion . . . . . . . . . . . . . . . . . . . 182

5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

6 Modeling with Stochastic Differential Equations 1916.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

6.1.1 Probability Space . . . . . . . . . . . . . . . . . . . . . 1926.1.2 Stochastic Process . . . . . . . . . . . . . . . . . . . . 193

6.1.2.1 Wiener Process (Brownian Motion) . . . . . 1946.1.3 Stochastic Differential Equation (SDE) . . . . . . . . . 1956.1.4 Gaussian White Noise . . . . . . . . . . . . . . . . . . 1956.1.5 Stochastic Stability . . . . . . . . . . . . . . . . . . . . 195

6.2 Some Stochastic Models . . . . . . . . . . . . . . . . . . . . . 1966.2.1 Stochastic Logistic Growth . . . . . . . . . . . . . . . 1966.2.2 Heston Model . . . . . . . . . . . . . . . . . . . . . . . 1976.2.3 Resistor-Inductor-Capacitor(RLC)Electric Circuit with

Randomness . . . . . . . . . . . . . . . . . . . . . . . 1976.2.4 Two Species Competition Model . . . . . . . . . . . . 199

6.3 A Research Problem: Cancer Self-Remission and Tumor Stabil-ity - A Stochastic Approach [116] . . . . . . . . . . . . . . . 2006.3.1 Background of the Problem . . . . . . . . . . . . . . . 2006.3.2 The Deterministic Model . . . . . . . . . . . . . . . . 2026.3.3 Equilibria and Local Stability Analysis . . . . . . . . . 2036.3.4 Biological Implications . . . . . . . . . . . . . . . . . . 2056.3.5 The Stochastic Model . . . . . . . . . . . . . . . . . . 206

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6.3.6 Stochastic Stability of the Positive Equilibrium . . . . 2076.3.7 Numerical Results and Explanations . . . . . . . . . . 2116.3.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . 211

6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

7 Hints and Solutions 219

Bibliography 243

Index 253

Foreword

When we talk about mathematical modeling it is worthwhile to discuss aconcept of a model that is considerably wider than just the concept of amathematical model. It is very difficult, if not impossible, to give a precisedefinition of the notion of a model covering the wide spectrum of possibilities.We shall be satisfied with the following statement: A model is an image ofa sector of reality which is created in order to accomplish a giventask. Consequently the starting point of a modeling process has to be theidentification of the purpose to be satisfied by the model and the task to beaccomplished by the modeling process. This is essential, because it determinesboth the sector of reality to be considered and the type of model to be created.It is important that the model preserves those structures of the real world thatare essential for the goals of the modeling process. It is equally important thatthe model does not include unnecessary features. An obvious reason for thisis that otherwise the model may become intractable numerically (if it is amathematical model). However, because of the enormous progress made incomputing technology this reason nowadays is in many cases of little impor-tance. The real, but less obvious, reason is that unnecessary components of amodel may obscure the structure which is important in view of the problemto solve or they may hide the mechanisms that are essential for the aims ofthe modeling process. To determine what is important and has to be includedin a model and what is not important and therefore should be neglected isthe essence of modeling.

Types of Models. The “definition” of a model given above includes a ratherwide range of different types of models including, of course, mathematicalmodels. Even if we are mainly interested in mathematical models it is usefulto also consider other types of models in order to demonstrate that certainguidelines should be observed in a modeling process. Below we list a few typesof models:

• Verbal descriptions: Here the sector of reality could be a political event,like a session of parliament or a demonstration, and the image, i.e., themodel, text in a newspaper. Ideally one would assume that the goalfor writing the article is to give an unbiased report of the event whichcaptures the politically relevant facts correctly. However, the purpose ofthe model (i.e., of the article) could also be different. For example thetext should create the impression in the reader that the event demon-

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strates the superiority of the program of a certain political party overthe program of another party. Other examples of models in the form ofa verbal description are, for instance, witness reports in the case of acar accident, or minutes of a meeting.

• Structured lists of measurements: Such lists of measurements or attributescan also be considered as models, the purpose being to classify certainobjects with sufficient precision for a given purpose. As an example wemention the numbers arranged in a size chart for blue jeans which allowclothing manufacturers to produce jeans such that a high percentage ofcustomers can find jeans that fit.

• Maps of various kinds: Here the sector of reality usually is a part of thesurface of the earth and the image is a geometric two-dimensional struc-ture. The appearance of a map is clearly dependent on the purpose itwas designed for. A road map looks very different than a physical or apolitical map. If one tries to navigate through a city with public busesusing an aerial photograph (which also can be considered as a modelof the city) instead of a map for the bus lines one will realize that thephotograph, though it contains much more detail, is totally unsuited forthe purpose of using buses in the city.

• Real models: These are replicas of real objects which preserve certain ge-ometric properties or some of the functionality of the real object. Asan example we mention models of aircraft for wind tunnel experiments,which are down-scaled replicas of real aircraft preserving the shape ofthe real aircraft as much detail as possible in order to study the airflowaround the airplane. Another example is provided by dummies as modelsfor the human body in crash tests. Here the shape and internal structureof a human body are imitated only to the extent that damages to thedummy during a crash test allow one to conclude what kind of injuriesa person would have suffered in a car accident of the kind simulated bythe crash test.

• Mathematical models: In this case the image of a part of reality is a math-ematical structure given, for instance, by a set of linear or nonlinearequations, by a system of ordinary or partial differential equations, by astochastic process etc. Of course, this is the kind of model we are mainlyinterested in these notes, and which are the topic of this book.

We can distinguish different types of mathematical models. Examples are:

• Dynamical models: Here, the time behavior of the real system is of primeinterest. Consequently the main components of the model are ordinarydifferential equations, delay equations, stochastic differential equationsand/or partial differential equations of parabolic or hyperbolic type, forinstance.

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• Geometric models: In this case the model should represent the geometry ofthe real system. One example is given by models which can be used inorder to optimize the shape of the real system according to given criteria(minimal weight and still resist forces of a given magnitude withoutdeformation). Another example is provided by computer tomography ormagnetic resonance imaging in medicine.

• Classifying models: These are models which allow classifying members of aset of objects or of a population on the basis of available data. Manystatistical models fall into this class.

Goals of a Modeling Process: It was stressed above that the first stepin a modeling process is to give a clear definition of the goal to be achievedby this process. In the following we provide a list of possible goals, which isfar from being complete:

• Simulations instead of experiments: Compared to numerical simulations ex-periments are in general much more expensive. Therefore a very commongoal of a modeling process is to replace experiments by numerical sim-ulations on the basis of a mathematical model. However, that does notmean that experiments eventually will be replaced completely by numer-ical simulations. We always shall need experiments in order to validatea model which has been developed. Besides the cost advantage of nu-merical simulations with a model another reason for doing simulationscan be that it may be impossible to conduct an experiment because itis too dangerous or it is prohibited by ethical reasons. Of course in sucha case simulation results have to be considered with great care, becausevalidation of the model may also be limited.

• Control of processes in the real world: Many processes in the real world haveto be controlled in order to work in a satisfactory way. For instance, thegoal can be to keep the dynamics of a system in a prescribed status de-spite perturbations which may act on the system. A very simple exampleis the temperature control of a room which should keep the temperaturein the room at a given value. Perturbations in this case are caused byoutside temperature changes, changes in the number of people in theroom, opening and closing doors and windows etc. A considerably morecomplicated example is provided by an autopilot which should keep anairplane on a prescribed course. The design of controllers with predefinedproperties has to be based on a mathematical model for the process andthe application of results from control theory to that model. Thereforemethods of optimal control and optimization are integral components ofmany modeling processes. Of course, control problems arise not only fortechnical systems. For instance in disease control it may be necessary todevelop a model in order to design a vaccination strategy to keep thenumber of infectious individuals below a given threshold.

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• Models for measurements: Almost all standard measuring devices use somephysical mechanism which relates the quantity to be measured to aquantity which can be observed directly. For instance, a classical ther-mometer shows the length of the column of mercury enclosed in a glasspipe, which is related to the room temperature by two physical pro-cesses: one is the heat exchange between the air in a room and themercury in the thermometer and the other is the fact that the volumeof mercury increases as temperature increases. From this it is also clearthat a classical thermometer cannot follow rapid temperature changes.From this simple example we should learn that the physical processes onwhich the measuring device is based in order to interpret the quality ofthe measurements correctly. With the advance of computer technology,more complex measuring devices frequently implement the numerical al-gorithms simulating the physical mechanisms on which the functioningof the device is based. This is already a simple application of mathe-matical modeling in order to obtain measurements. In more complexsituations models are used in order to ‘measure’ (i.e., to identify) aquantity which cannot be measured directly by measuring another moreaccessible quantity. Such procedures are in particular used in order to‘measure’ physiological quantities.

• Gain of understanding through mathematical modeling: An important fac-tor in this context is the fact that mathematical modeling requires a sys-tematic approach concentrating on the main structural and functionalfeatures of the real system under consideration. This inevitably leads toa better understanding of the real process.

• Models for educational purposes: Real models have been used for a longtime as teaching devices. With the availability of affordable computingpower in combination with visualization techniques, complex models canbe used to demonstrate the functioning of complex real systems undervarying conditions. In many cases the requirement that the model hasto be validated is relaxed in favor of a more complete representation ofthe real system which is at least qualitatively correct.

Validation of a Model: An important concept in mathematical model-ing is given by the domain of validity of a model. This can be defined as thatsector (time-wise, spatial or functional) of reality that is represented by themodel with sufficient accuracy. Validation of a model requires that simulationresults of the model have to be compared with data from experiments thatmay have been performed for the sole purpose of validating the model. Thismeans that parameters of the model have to be adjusted in order to get anoptimal agreement between simulations and data; i.e., we choose the parame-ters such that some functional measuring the difference between model outputand data is minimized. The validation step should also check that the origi-nally defined goals for the modeling process are reached; i.e., if the domain of

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validity of the model included the sector of reality for which the model wasdeveloped.

Simulations with a model involving substantial changes in the model pa-rameters in order to predict the behavior of the real system in more extremesituations has to be done with great care, because the real system correspond-ing to the changed parameters may not hold true in the domain of validity ofthe model.

Guidelines: When one is involved in a modeling process it is importantto observe guidelines which have proven to be helpful for accomplishing thetask. A few of these guidelines are:

• The first step of a modeling process is to identify the sector of reality to beconsidered and to establish the goals of the modeling process.

• Next, it is important to determine the mechanisms to be considered, andthe details that should be neglected, using empirical observations andavailable knowledge of the real system.

• Variables and parameters of the model should have interpretations in real-ity. For models that reflect causal mechanisms acting in the real system,this guideline must be followed. Parameters of model components whichreflect physical, chemical or biological mechanisms need to have an in-terpretation in the relevant mechanism. This allows interpreting changesin the parameters as changes in real system and vice versa.

• If a model does not achieve the given goals, then it has to be modified orabandoned and replaced by a new one. This guideline seems to be ob-vious. However, mathematicians involved in the modeling process tendto like a mathematical model that provides an interesting mathematicalstructure and the possibility to prove nice theorems. But it may be thatthe mathematically interesting model has a rather poor performancewith respect to the goals of the modeling process.

Important Factors in Mathematical Modeling: We want to drawattention to some important factors for the rapid development and successof mathematical modeling in various applied sciences. One is the requirementfor precision in control or simulation of more complicated real processes. Thisand other needs require mathematical modeling of real processes of increas-ing complexity also in fields where in the past this was not necessary or notpossible. This has accelerated the interaction between applied sciences andmathematics and is leading to new mathematical methodologies. Throughoutthe history of mathematics, interaction with applications has been an impor-tant factor for the further development of mathematics per se.

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Perhaps the most important factor for the rapid development of mathemat-ical modeling in various applications is the availability of enormous computingpower, which allows considering models of a complexity that was unthinkablebefore we had programmable computers. In this context one has to point outthe possibilities for visualizing simulation results provided by modern com-puters.

An important factor that is quite often underestimated in the context ofmathematical modeling is the introduction of new technologies for measure-ments and data collection. This is not surprising in view of the needs for dataon a real system for the validation phase of a modeling process.

The Role of Mathematicians:Mathematical modeling not only requiresknowledge of existing mathematical theories or methodologies, but in view ofthe rapidly increasing requirements of mathematical models also the devel-opment of new mathematical methodologies. This is certainly a task wheremathematicians are challenged to contribute their competence and experi-ence. However, the role of mathematicians in mathematical modeling is notonly to provide the necessary mathematics once the model has been devel-oped by the applied scientists. In order to be really helpful in a modelingprocess the mathematician has to be an integral part of this process. To doso the mathematician must be able to communicate with the applied scien-tists on their terms, which means he has to absorb enough of the applied fieldto understand the problems and peculiarities of this field. Applied scientistscannot be expected to enter the relevant mathematical fields to the same de-gree. Moreover, for cooperation of mathematicians and applied scientists in amodeling process to be successful mathematicians have to adopt the goals ofthe modeling process which is a goal in the applied area. Finally it should bestressed that mathematicians can add important competencies as partners ina modeling process because they are trained to attack problems in a formaland structured approach, which makes it easier for them to neglect unimpor-tant details and to recognize the underlying structure of a process.

This book reflects most of the points presented here in concrete situationswhich in my opinion makes it a valuable contribution to the literature onmathematical modeling for the target audience of undergraduate students, aswell as for anyone interested in mathematical modeling.

Graz, September 2013 Franz Kappel

Preface

Almost every year, a new book on mathematical modeling is published, hencethe inevitable question is: Why another? The answer springs from the factthat it is rare to find a book that covers modeling with all types of differentialequations in one volume. In my study of books on mathematical modelingwith differential equations, none of them covered all the aspects of differentialequations. It is this that motivated me to write a book on mathematical mod-eling that will cover modeling with all kinds of differential equations, namely,ordinary, partial, delay and stochastic. The book also contains a chapter ondiscrete modeling, consisting of difference equations. It is this aspect of thebook that makes it unique and complete and is one of its salient features.

Mathematical modeling is an important aspect of the study of science andengineering. Even social sciences involve mathematical modeling. The primaryaim of the book is to work towards creating in the reader an awareness of theconcepts involved in mathematical modeling so as to build a solid foundationin the subject. Since mathematical modeling involves a diverse range of skillsand tools, I have focused on techniques that would be of particular interestto engineers, scientists and others who use models of discrete and continuoussystems. In an age when computers rule, students of engineering and sciencewill be better placed if they understand and use the mathematics that is thebasis of computers. Mathematics is necessary for engineering and science. Thiswill remain unaltered in spite of all changes that may come about in computa-tion. I never tire of telling my students that it is a risky proposition to acceptcomputer calculations without having recourse to some parallel closed-formmodeling to verify the veracity of the results. Finally, it is fun and satisfyingto do mathematical manipulations that explain how and why things happen.

My target audience is undergraduates who are keen to grasp the basicsof mathematical modeling. Graduate students who want to use mathematicalmodeling in their research will also find the book useful. My experience withbooks on mathematical modeling has been that they are effective in some ar-eas of mathematical modeling using a particular kind of differential equationneeded for the model (either ODE/PDE/DDE or even stochastic) but neverconsider the complete picture as far as modeling with differential equation isconcerned. It is important for students to recognize the relevance and applica-tion of their knowledge of mathematics to practical applications. This book, I

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hope, bridges the gap between the student’s abilities and modeling and wouldprepare students to venture forth on their own to solve problems on mathe-matical modeling using differential equations.

It has taken me almost 3 years to complete this book and this experi-ence has taught me that writing a book is not an easy task. However, theend result justifies the hard work that goes into it and leaves one with asense of satisfaction once it is ready. The amount of knowledge that I havegained while working on this book is immense and that is reward in itself. Thebook is divided into six chapters. The first chapter introduces the subject ofmathematical modeling, its history, possibilities and its limitations. Chapter2 deals with discrete models using difference equations. Chapter 3, the largestchapter, discusses modeling with ordinary differential equations and incor-porates innumerous examples from physics, chemistry, biology, ecology andengineering. Chapter 4 is concerned with modeling dealing with partial differ-ential equations including examples on fluid flow, heat flow, traffic flow andpattern formation. Chapter 5 focuses on delay differential equations, namely,solving techniques of simple DDE’s and its stability. Examples from physics,biology, ecology, finance and its numerical solutions have been discussed inthis chapter; a research problem has been included for those interested inadvance mathematical theory of DDE. The effect of stochasticity on popu-lation dynamics, epidemiology, finance models, tumor-immune interaction isthe highlight of Chapter 6.

Years of teaching the subject have honed my skills in the area of mathe-matical modeling; however, perfection is something that one ought to strivefor. I am sure I have not yet achieved it, and hence would welcome any sug-gestion for improvements, comments and criticisms (directly to sandipbaner-jea@gmail.com) for later editions of the book.

A book is always the product not only of its author, but also various peoplewho have in their way contributed towards it. Colleagues, research scholars,the work environment where I teach, the discussions, engagements, discoursesthat I have gathered during years of teaching and research, the many confer-ences, referee reports, seminars, university lectures, training courses, summerand winter schools, e-mail correspondence, and most importantly, the patienceand constant support of my family. While it would not be possible to nameall those who have helped me on this journey, their contribution has beenimmense. I take this opportunity to thank Prof. Hien Tran, North CarolinaState University. During my visit to NCSU, I had full access to their fabu-lous library which provided me materials so important for research that wentinto writing the book. I also thank three anonymous referees who has gonethrough my book proposal. Their valuable suggestions have helped me in prop-erly shaping up the book. I gratefully acknowledge the Quality ImprovementProgram (QIP) of IIT Roorkee for providing me financial support for writ-

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ing the book. My sincere thanks to my research scholars, Ram Keval, SubhasKhajanchi, Teekam Singh and Sumana Ghosh, who helped me with typingand proofreading. Special mention is due to Nishi Pulugurtha, who helped mein the composition of the history of mathematical modeling. Finally, a wordof thanks to my publisher for their active effort in publishing this book.

Sandip BanerjeeDepartment of MathematicsIndian Institute of Technology RoorkeeIndia.

Additional material is available from the CRC web sitehttp://www.crcpress.com /product/isbn/9781439854518

MATLAB� is a registered trademark of The MathWorks, Inc.For product information, please contact:

The MathWorks, Inc.3 Apple Hill DriveNatick, MA 01760-2098 USATel: 508-647-7000Fax: 508-647-7001E-mail: info@mathworks.comWeb: www.mathworks.com

Biography

Dr. Sandip Banerjee is Associate Professor in the Department of Math-ematics, Indian Institute of Technology Roorkee (IITR), India. His areas ofinterest are mathematical modeling of interactions between tumors and theimmune system (the spatial aspect and its 3D visualization and validation),mathematical modeling of hepatitis C, application of stochastic delay differ-ential equations to population dynamics and tumor modeling, ecological dy-namics of interacting populations (mainly effects of space and stochasticity),and stage structured prey-predator models with multiple delays. Mathemat-ical modeling is his passion. Dr. Banerjee was the recipient of the Indo-USFellowship in 2009 and he was awarded IUSSTF Research Fellow medal bythe Indo-US Technology Forum for his contribution to the Indo-US Fellow-ship 2009 program at the First Indo-US Research Fellowship Conclave, Pune,India, 15-17 March 2013. In addition to several national and internationalprojects, Dr. Banerjee is involved in the Virtual Network in MathematicalBiology project, which promotes mathematical biology in India.

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Chapter 1

About Mathematical Modeling

1.1 What is Mathematical Modeling? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 History of Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Importance of Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Latest Developments in Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Limitations of Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.1 What is Mathematical Modeling?

Mathematical modeling is the application of mathematics to describe real-world problems and investigating important questions that arise from it. Usingmathematical tools, the real-world problem is translated to a mathematicalproblem, which mimics the real-world problem. A solution to the mathemat-ical problem is obtained, which is interpreted in the language of real-worldproblem to make predictions about the real world.

By real-world problems, I mean problems from biology, chemistry, engi-neering, ecology, environment, physics, social sciences, statistics, wildlife man-agement and so on. Mathematical modeling can be described as an activitywhich allows a mathematician to be biologist, chemist, ecologist, economistdepending on the problem that he/she is tackling. The primary aim of amodeler is to undertake experiments on the mathematical representation of areal-world problem, instead of undertaking experiments in the real world.

Challenges in mathematical modeling........not to produce the most comprehensive descriptive model but toproduce the simplest possible model that incorporates the major fea-tures of the phenomenon of interest.

Howard Emmons

1

2 Mathematical Modeling: Models, Analysis and Applications

1.2 History of Mathematical Modeling

Modeling (from Latin modellus) is a way of handling reality. It is the abilityto create models that distinguish human beings from other animals. Modelsof real objects and things have been in use by human beings since the StoneAge as is evident in cave paintings. Modeling became important in the AncientNear East and Ancient Greek civilizations. It is writing and counting numberswhich were the first models. Two other areas where modeling was used in itspreliminary forms are astronomy and architecture. By the year 2000 BC, thethree ancient civilizations of Babylon, Egypt and India had a good knowledgeof mathematics and used mathematical models in various spheres of life [117].

In Ancient Greek civilization, the development of philosophy and its closerelation to mathematics contributed to a deductive method, which led toprobably the first instance of mathematical theory. From about 600 BC, ge-ometry became a useful tool in analyzing reality. Thales predicted the solareclipse of 585 BC and devised a method for measuring heights by measuringthe lengths of shadows using geometry. Pythagoras developed the theory ofnumbers, and most importantly initiated the use of proofs to gain new re-sults from already known theorems. Greek philosophers Aristotle, Eudoxus,and others contributed further and in the next 300 years after Thales, geom-etry and other branches of mathematics developed further. The zenith wasreached by Euclid of Alexandria who in Circa 300 BC wrote The Elements , averitable collection of almost all branches of mathematics known at the time.This work included, among others, the first precise description of geometryand a treatise on number theory. It is for this that Euclid’s books becameimportant for the teaching of mathematics for many hundreds of years, andaround 250 BC Eratosthenes of Cyrene used this knowledge to calculate thedistances between the Earth and the Sun, the Earth and the Moon and thecircumference of the Earth using a geometric model.

A further step in the development of mathematical models was taken upby Diophantus of Alexandria in about 250 AD, who, in his book Arithmetica,developed the beginnings of algebra based on symbolism and the idea of avariable. In the field of astronomy, Ptolemy, influenced by Pythagoras’ idea ofdescribing celestial mechanics by circles, developed a mathematical model ofthe solar system using circles to predict the movement of the sun, the moon,and the planets. The model was so accurate that it was used until the earlyseventeenth century when Johannes Kepler discovered a much more simpleand superior model for planetary motion in 1619. This model, with later re-finements done by Newton and Einstein, is in use even today.

Mathematical models are used for real-world problems and are hence im-

About Mathematical Modeling 3

portant for human development. Mathematical models were developed inChina, India and Persia as well as in the Western world. One of the mostfamous Arabian mathematicians was Abu Abd-Allah ibn Musa Al-Hwarizmi,who lived in the late eighth century [117]. Interestingly, his name still survivesin the word algorithm. His well known books are de numero Indorum (aboutthe Indian numbers - today called Arabic numbers) and Al-kitab al-muhtasarfi hisab al-g abr wa’lmuqabala (a book about the procedures of calculationbalancing) [117]. Both these books contain mathematical models and prob-lem solving algorithms for use in commerce, survey, and irrigation. The termalgebra was derived from the title of his second book.

In the Western world, it was only in the sixteenth century that mathe-matics and mathematical models developed. The greatest mathematician inthe Western world after the decline of the Greek civilization was Fibonacci,Leonardo da Pisa (ca. 1170-ca. 1240). The son of a merchant, Fibonacci mademany journeys to the Orient, and familiarized himself with mathematics as ithad been practiced in the Eastern world. He used algebraic methods recordedin Al-Hwarizmi’s books to improve his trade as a merchant. He first realizedthe great practical advantage of using the Indian numbers over the Romannumbers which were still in use in Europe at that time. His book Liber Abaci,first published in 1202, began with a reference to the ten ‘Indian figures’ (0,1, 2,..., 9), as he called them. 1202 is an important year since this is the yearthat saw the number zero being introduced to Europe. The book itself wasmeant to be a manual of algebra for commercial use. It dealt in detail witharithmetical rules using numerical examples which were derived from practicaluse, such as their applications in measure and currency conversion.

The painter Giotto (1267-1336) and the Renaissance architect and sculp-tor Filippo Brunelleschi (1377-1446) are responsible for the development ofgeometric principles. In the later centuries many more and varied mathemati-cal principles were discovered and the intricacy and complexity of the modelsincreased. It is important to note that despite the achievements of Diophantand Al-Hwarizmi, the systematic use of variables was invented by Vieta (1540-1603) [117]. In spite of all these developments it took many years to realizethe true role of variables in the formulation of mathematical theory. It alsotook time for the importance of mathematical modeling to be completely un-derstood. Physics and its application to nature and natural phenomena is amajor force in mathematical modeling and its further development. Later eco-nomics became another area of study where mathematical modeling began toplay a major role.

4 Mathematical Modeling: Models, Analysis and Applications

1.3 Importance of Mathematical Modeling

A mathematical model, as stated, is a mathematical description of a reallife situation. So, if a mathematical model can reflect or mimic the behaviorof a real life situation, then we can get a better understanding of the systemthrough proper analysis of the model using appropriate mathematical tools.Moreover, in the process of building the model, we discover various factorswhich govern the system, factors which are most important to the system andthat reveal how different aspects of the system are related.

The importance of mathematical modeling in physics, chemistry, biology,economics and even industry cannot be ignored. Mathematical modeling inbasic sciences is gaining popularity, mainly in biological sciences, economicsand industrial problems. For example, if we consider mathematical model-ing in the steel industry, many aspects of steel manufacture, from mining todistribution, are susceptible to mathematical modeling. In fact, steel com-panies have participated in several mathematics-industry workshops, wherethey discussed various problems and obtained solution through mathematicalmodeling - problems involving control of ingot cooling, heat and mass transferin blast furnaces, hot rolling mechanics, friction welding, spray cooling andshrinkage in ingot solidification, to mention a few [91]. Similarly, mathemati-cal modeling can be used(i) to study the growth of plant crops in a stressed environment,(ii) to study mRNA transport and its role in learning and memory,(iii) to model and predict climate change,(iv) to study the interface dynamics for two liquid films in the context of or-ganic solar cells,(v) to develop multi-scale modeling in liquid crystal science and many more.

For gaining physical insight, analytical techniques are used. However, todeal with more complex problems, numerical approaches are quite handy. Itis always advisable and useful to formulate a complex system with a simplemodel whose equation yields an analytical solution. Then the model can bemodified to a more realistic one that can be solved numerically. Together withthe analytical results for simpler models and the numerical solution from morerealistic models, one can gain maximum insight into the problem.

About Mathematical Modeling 5

1.4 Latest Developments in Mathematical Modeling

Mathematical modeling is an area of great development and research. In re-cent years, mathematical models have been used to validate hypotheses madefrom experimental data, and at the same time the designing and testing ofthese models has led to testable experimental predictions. There are impres-sive cases in which mathematical models have provided fresh insight into bi-ological systems, physical systems, decision making problems, space models,industrial problems, economical problems and so forth. The development ofmathematical modeling is closely related to significant achievements in thefield of computational mathematics.

Consider a new product being launched by a company. In the developmentprocess, there are critical decisions involved in its launch such as timing, de-termining price, launch sequence, etc. Experts use and develop mathematicalmodels to facilitate such decision making. Similarly, in order to survive marketcompetition, cost reduction is one of the main strategies for a manufacturingplant, where a large amount of production operation costs are involved. Properlayout of equipment can result in a huge reduction in such costs. This leads todynamic facility layout problem for finding equipment sites in manufacturingenvironments, which is one of the developing areas in the field of mathematicalmodeling [122].

Mathematical modeling also intensifies the study of potentially deadly fluviruses from mother nature and bio-terrorists. Mathematical models are alsobeing developed in optical sciences [6], namely, diffractive optics, photonicband gap structures and wave guides, nutrient modeling, studying the dynam-ics of blast furnaces, studying erosion, and prediction of surface subsidence.

In geosciences, mathematical models have been developed for talus. Talusis defined by Rapp and Fairbridge [103] as an accumulation of rock debrisformed close to mountain walls, mainly through many small rockfalls. Hiroyukiand Yukinori [90] have constructed a new mathematical model for talus devel-opment and retreat of cliffs behind the talus, which was later applied to theresult of a field experiment for talus development at a cliff composed of chalk.They developed the model which was in agreement with the field observations.

There has been tremendous development in the interdisciplinary field ofapplied mathematics in human physiology in the last decade, and developmentcontinues. One of the main reasons for this development is the researcher’s im-proved ability to gather data, whose visualization have much better resolutionin time and space than just a few years ago. At the same time, this devel-opment also constitutes a giant collection of data as obtained from advanced

6 Mathematical Modeling: Models, Analysis and Applications

measurement techniques. Through statistical analysis, it is possible to findcorrelations, but such analysis fails to provide insight into the mechanismsresponsible for these correlations. However, when it is combined with mathe-matical modeling, new insights into the physiological mechanisms are revealed.

Mathematical models are being developed in the field of cloud computingto facilitate the infrastructure of computing resources in which large pools ofsystems (or clouds) are linked together via the internet to provide IT services(for example, providing secure management of billions of online transactions)[25]. Development of mathematical models are also noticed(i) in the study of variation of shielding gas in GTA welding,(ii) for prediction of aging behavior for Al-Cu-Mg/Bagasse Ash particularcomposites,(iii) for public health decision making and estimations,(iv) for developing of cerebral cortical folding patterns which have fascinatedscientists with their beauty and complexity for centuries,(v) to predict sunflower oil expression,(vi) in the development of a new three dimensional mathematical ionospheremodel at the European Space Agency/European Space Operators Centre,(vii) in battery modeling or mathematical description of batteries, which playsan important role in the design and use of batteries, estimation of battery pro-cesses and battery design. These are some areas where mathematical modelingplays an important role. However, there are many more areas of application.

1.5 Limitations of Mathematical Modeling

Sometimes although the mathematical model used is well adapted to thesituation at hand, it may give unexpected results or simply fail. This maybe an indication that we have reached the limit of the present mathematicalmodel and must look for a new refinement of the real-world or a new the-oretical breakthrough [10]. A similar type of problem was addressed in [6],which deals with Moire theory, involving the mathematical modeling of thephenomena that occur in the superposition of two or more structures (linegratings, dot screens, etc.), either periodic or not.

In mathematical modeling, more assumptions must be made, as infor-mation about real-world systems become less precise or harder to measure.Modeling becomes a less precise endeavor as it moves away from physicalsystems towards social systems. For example, modeling an electrical circuitis much more straightforward than modeling human decision making or theenvironment. Since physical systems usually do not change, reasonable past

About Mathematical Modeling 7

information about a physical system is quite valuable in modeling future per-formance. However, both social systems and environments often change inways that are not of the past, and even correct information may be of lessvalue in forming assumptions. Thus, to understand a model’s limitations, itis important to understand the basic assumptions that were used to create it.

Real-world systems are complex and a number of interrelated componentsare involved. Since models are abstractions of reality, a good model must try toincorporate all critical elements and interrelated components of the real-worldsystem. This is not always possible. Thus, an important inherent limitationof a model is created by what is left out. Problems arise when key aspects ofthe real-world system are inadequately treated in a model or are ignored toavoid complications, which may lead to incomplete models. Other limitationsof a mathematical model are that they may assume the future will be like thepast, input data may be uncertain or the usefulness of a model may be limitedby its original purpose.

However, despite all these limitations and pitfalls, a good model can beformulated, if a modeler asks himself/herself the following questions about themodel:(i) Does the structure of the model resemble the system being modeled?(ii) Why is the selected model appropriate to use in a given application?(iii) How well does the model perform?(iv) Has the model been analyzed by someone other than the model authors?(v) Is adequate documentation of the model available for all who wish to studyit?(vi) What assumptions and data were used in producing model output for thespecific application?(vii) What is the accuracy of the model output?

One should not extrapolate the model beyond the region of fit. A modelshould not be applied unless one understands the simplifying assumptionson which it is based and can test their applicability. It is also important tounderstand that the model is not the reality and one should not distort realityto fit the model. A discredited model should not be retained and one shouldnot limit himself to a single model, as more than one model may be useful forunderstanding different aspects of the same phenomenon. It is imperative tobe aware of the limitations inherent in models. There is no best model, onlybetter models.