MathematicalModelingModule Department of Mathematics Addis Ababa UniversitySept. 2013

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Mathematical Modelling Module

Department of Mathematics

College of Natural Sciences Addis Ababa University

Sept. 2013

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PREFACE This module is designed for Mathematics undergraduate students taking the course mathematical modelling. The target groups are mainly mathematics students in the undergraduate programme at the Department of Mathematics, Addis Ababa University. This material is prepared so as to address the problem of shortage of reference and/or text books, to some extent. It is believed that this module will be a valuable resource for introducing mathematical modelling and motivating students to do research in applied mathematics in real life problems. The module is organised into five chapters. There are exercises that may help the reader to understand basic techniques in Modeling.

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Content

Introduction

Discrete Models

Continuous models

Methods of Mathematical Modeling

Prototype Models

Exercises

Appendix

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CHAPTER 1

INTRODUCTION Applied Mathematics is concerned with application of Mathematics. And application of Mathematics consists of applying mathematical tools and skills to obtain useful answers to real world problems. In order to apply Mathematics one needs Mathematical skills (Knowledge and Methods) and Technique to apply Mathematical skills.

Basic Definitions A Mathematical Model is a Mathematical construction designed to study a particular real-world system or phenomenon. It includes graphical, symbolic, simulation and experimental constructions. Model: A model is representation or description of a physical phenomena or object of interest. Modeling is a cognitive activity in which one perceives and describes the behavior of an object (a physical phenomenon) of interest. Modeling can be done in several languages such as

- Drawing/Sketches (analogy or emulation ) - Physical models (Statue or Sculpture) - Mathematical Models (formulate or equations)

Mathematical model Vs Scientific Method. Roughly speaking Mathematical modeling and Scientific method are similar procedures. However there are some basic differences. Scientific Method Real world + Conceptual world It is to confirm or deny

Mathematical Mode Mathematical description of real world Not to confirm or deny, but to test reasonableness

Real World: By Real World we mean various phenomenon or behavior to be observed. Conceptual World: By Conceptual world (World of Mind) we mean the understanding about what is going on in the real world. Conceptual world may be viewed as having three stages namely

Observation Modeling Prediction

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Observation: In this stage we measure what is happening and gather empirical evidence on the ground. Observation might be direct such as using our sense organs or indirect such as taking measurement using devices. Modeling: Major steps in this stage are

- Analysis of observed data (behavior). - Description of the behavior observed. - Explain (Reason) why such a behavior occurred in a way it does. - Develop a mechanism that helps predict future behaviors that are as yet unseen. Prediction: Here we build equations (relations) that will be used to identify the computations that are needed to be made and answers that may result. Methods of Mathematical Modeling: are techniques we employ to establish a mathematical model. These methods include

- dimensional homogeneity - abstraction and scaling - conservation and balance laws - consequences of linearly, etc

The process of studying and gaining an understanding of a problem is called modelling. If the understanding is the result of using mathematics, the process is known as mathematical modelling. Thus, mathematical modelling is a way of describing a real problem using mathematics. Modelling can come in a number of different forms:

description in words graphics, such as the graph of a straight line equation, such as the equation of heat flow in conducting materials drawing, such as a scale drawing of a room diagram, such as a flow chart or arrow diagram computer simulation, e.g imitation of a real situation, such as a bending truss

Once a model is built, you can use mathematics to find a solution to the problem. This will be much clearer after you have done it once. For mathematical modelling, you learn by doing. Following is a general summary of the main steps in mathematical modelling.

Steps for building a Mathematical Model

1. Identify the situation. Read and ask questions about the problem. Identify issues you

wish to understand so that your questions are focused on exactly what you want to know.

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2. Simplify the situation. Make assumptions and note the features that you will ignore at first. List key features of the problem. These are the assumptions that you will use to build your model.

3. Build the model and solve the problem. Describe in mathematical terms the relationships among the parts of the problem, and find the answer. The usual way to describe pertinent features mathematically include:

define model variables set up equations draw shapes measure objects calculate values organize values into tables make graphics

4. Evaluate and revise the model. Check whether your solutions make sense, and test your model. If so, use the model until new information becomes available or assumptions change. If not, reconsider the assumptions you made in step 2 and revise them to be more realistic.

Summing up one can say, a model is a representation or an abstraction of a system or a process. We build models because they help us to

1) define our problems, 2) organize our thoughts, 3) understand our data, 4) communicate and test that understanding, and 5) make predictions. A model is therefore an intellectual tool.

One of the most important aims for construction of models is to define the problem such that only important details becomes visible, while irrelevant features are neglected. A mathematical model of a complex phenomenon or situation has many of the advantages and limitations of other types of models. Some factors in the situation will be omitted while others are stressed. When constructing a mathematical system, the modeler must keep in mind the type of information he or she wishes to obtain from it. The interplay between mathematical models and real world scenario can be illustrated by the following schematics.

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The modeler begins with observations of the real world problem. He/she wishes to make some conclusions or predictions about the situation he/she has observed. One way to proceed (E) is to conduct some experiments and record the results. The model builder follows a different path. First, he or she abstracts, or translates, some of the essential features of the real world into a mathematical system. Then by logical argument (L) he or she derives some mathematical conclusions. These conclusions are then interpreted (I) as predictions about the real world. To be useful, the mathematical system should predict conclusions about the real world that are actually observed when appropriate experiments are carried out. If the predictions from the model bear little resemblance to what actually occurs in the real world, then the model is not a good one. The modeler has not isolated the critical features of the situation being studied or the axioms misrepresent the relations among these features. On the other hand, if there is good agreement between what is observed and what the model predicts, then there is some reason to believe that the mathematical system does indeed capture correctly important aspects of the real-world situation. What happens quite frequently is that some of the predictions of a mathematical model agree quite closely with observed events, while other predictions do not agree with the observed events. In such a case, we might hope to modify the model to improve its accuracy. The incorrect predictions may suggest ways of rethinking the assumptions of the mathematical system. One hopes that the revised model will not only preserve the correct predictions of the original one, but that it will also make further correct predictions. The incorrect inferences of the revised model will lead, in turn, to yet another version, more sophisticated more accurate than the previous one. However, it is important to keep in mind, that the goal is not to make the most precise model of the part of the world that is modeled, but that the model (like a road map) includes all the essential features, even if that means that some other features in the model do not present the reality. For example, a model of the flow of blood in human blood vessels, cardiovascular system (the heart, arteries, and veins) could accurately present the systemic arteries and veins and then

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lump the pulmonary circulation into a single compartment. Such a compartment would never represent any of the subsystems correctly. When building mathematical models one should distinguish between the different types of models, some models (deterministic models) can be derived directly from physical laws (e.g. Newtons second law), while other models are based on empirical observations. Both types of models provide insight into the system modeled, but the type of model must be considered carefully. For example, very different types of models are used for predict the weather tomorrow and to determine a rockets trajectory to the moon. Holling (1978) has a diagram that provides a simple and useful classification of problems. The horizontal axis represents how well we understand the problem we are trying to solve; the vertical axis represents the quality and/or quantity of relevant data. Holling divides the quadrant between the two axes into four areas, corresponding to four classes of problems.

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1

2

3

Understanding

Area 1 is a region with good data but little understanding. This is where statistical techniques are useful; they enable one to analyze the data search for patterns or relation, construct and test hypotheses, and so on. Area 3 is a region with good data and good understanding. Many problems in engineering and the physical sciences (for example, the problem of computing a rockets trajectory to the moon) belong to this class of problems. This is the area where models are used routinely and with confidence because their effectiveness has been proved repeatedly. Area 2 has little in the way of supporting data but there is some understanding of the structure of the problem.

Dat

a

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Area 4, in this area there is little knowledge of the structure of the problem and little data to support it. Unfortunately, many problems in sciences (especially in the biological sciences) belong to areas 2 and 4. However, recent explosion in experimental techniques move some of these problems to areas 1 and 3. The main difference from the physical problems is the uncertainty and high levels of noise often found in the data. The modeling challenges for problems in area 2 and 4 are:

- Decisions may have to be made despite the lack of data and understanding. How do we make good, scientific decision under these circumstances?

- How do we go about improving our understanding and suggest new ways for collecting the data necessary to validate the modeling. This is an area where modeling can be used to predict new experimental settings.

Models that lie in areas 2 and 4 are bound to be speculative. They will never have the respectability of models build for solving problems in area 3 because it is unlikely they will be sufficiently accurate of that they can ever be tested conclusively. We therefore build models to explore the consequences of what we believe to be true. Those who have a lot of data and little understanding of their problem (area 1) gain understanding by living with their data, looking at it in different ways, and searching for patterns and relationships. Because we have so little data in areas 2 and 4, we learn by living with our models, by exercising them, manipulating them, questioning their relevance, and comparing their behavior with what we know (or think we know) about the real world. This process often forces us to review and evaluate our beliefs, and that evaluation in turn leads to new versions off the models. The mere act of assembling the pieces and building a model (however speculative the model might be) usually improves our understanding and enables us to find and/or use data we had not realized were relevant. That in turn leads us to a better model. The process is one of boot-strapping: If we begin with little data and understanding in the bottom left-hand corner of Hollings diagram, models help us to zigzag upwards and to the right. This is a better approach than one of just collecting data because we improve our understanding as we go along. (Those who collect data without building models run the very real risk of discovering, when they eventually analyze their data, that they have collected the wrong data!)

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The process of mathematical modeling as a scientific method might be summarized schematically as in the following diagram:

One worth mentioning feature of mathematical models is their often astounding success. Frequently they work much better than might be expected. This is best illustrated in physics. It is remarkable that a wide range of physical phenomena can be modeled in terms of a very small number of physical principles. For example, general relativity can be used to describe the behavior of objects ranging from billiard balls and bicycles to rockets and planets. Maxwell's equations allow us to describe all electro-magnetic interactions. Quantum mechanics provides the basis for chemistry. Physics has been a highly successful science primarily because the basic physical principles can be readily modeled by precise mathematical equations.

The real world

Part of the real world (the problem)

System

Mathematical system

Model results

Action/Insight

(a) problem formulation

(b) System identification

(c) Mathematical formulation

(d) Mathematical analysis

(e) Interpretation and analysis of results

(f) Analysis of the model validity

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According to Max Born, "all great discoveries in experimental physics have been made due to the intuition of men who made free use of models which for them were not products of the imagination but representations of real things".

Why are mathematical models so successful? In 1960 Eugene Wigner, a Nobel-prize winner in physics, gave a famous lecture on "the unreasonable effectiveness of mathematics in the natural sciences". He concluded that the amazing applicability of mathematics to the physical world is a mysterious, undeserved and inexplicable gift.

Part of the mystery is due to the fact that sometimes mathematics developed for purely mathematical purposes later turns out to have unexpected physical applications. For example, in 1609 Johannes Kepler found that planetary orbits can best be described in terms of ellipses, mathematical curves that had been studied two thousand years earlier by Greek mathematicians. Steven Weinberg, another Nobel-prize winner in physics, remarks

"physicists generally find the ability of mathematicians to anticipate the mathematics needed in the theories of physics quite uncanny. It is as if Neil Armstrong on 1969 when he first set foot on the surface of the moon had found in the lunar dust the footprints of Jules Verne"

Nevertheless, the applicability of mathematics concerns not just a few isolated successes in physics. Rather, it pertains to the much broader applicability of mathematics as a global research strategy. Physicists, from Kepler and Galileo onwards, have been gripped by the conviction that mathematics is the ultimate language of the universe. Physicists probe nature with an eye for mathematical structures and analogies in nature.

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CHAPTER 2

DISCRETE MODELS Dynamics is the study of quantities that change over time. From an applications viewpoint a dynamical system is a system that changes over time. Model variables may change continuously with time or attain values with a leap in time, accordingly the pertinent model is continuous or discrete.

MODELING CHANGE WITH DIFFERENCE EQUATIONS

In discrete models approximation of a change is made using difference equations. We shall demonstrate this situation with the following example. Example : Suppose you take 16ml of cough medicine. Assume that the medicine is immediately absorbed into your blood system. Assume that every 4 hours, you kidneys remove 25% of it from you blood system. The following sequence represents the amount in your blood system at consecutive 4 hour intervals.

A = {16, 12, 9, 6.75, 5.0625, } Definition: Let A = { a0, a1, a2, a3 , . } be a sequence. The first differences are a0 = a1 a0 a1 = a2 a1 a2 = a3 a2 The nth first difference is an = an+1 an Specifically, in our example

a0 =16 , a1 = 12 , a2 = 9 , a3 = 6.75, a4 = 5.0625 Note: an = -0.25an or an+1 = an - 0.25an Recall that a sequence is a function whose domain is the set of nonnegative integers and whose range is a subset of the real numbers. Mathematically, a discrete dynamical system is a

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relationship among terms in a sequence. A numerical solution is a table of values satisfying the dynamical system. We can define the sequence in our example as: a0 = 16 an+1 = 0 .75an Usually, each term of a discrete dynamical system is determined iteratively from the preceding term or terms.

APPROXIMATING CHANGE WITH DIFFERENCE EQUATIONS

In developing a model of the growth of a population, it is often useful to find a function that describes the change in the population. Thus, if pn is the population at time t = n then

pn = pn+1 - pn The most basic model of population growth is based on the assumption that the change is proportional to the size of the population. That is,

pn = pn

where is a parameter that determines the rate of growth. Note that relationship between population and time is not linear. This model actually implies exponential growth of the population! Such growth cannot be sustained indefinitely. It appears to be approaching some limiting value. This value is called the carrying capacity , k. (Why?) A more refined or a modified model is developed by assuming

pn = (k - pn ) pn or ( )m

nnn P

rPPrP2

1 1 +=+

This model states that the population change is proportional to the quantity (k-pn ) pn . The model is non-linear. Spread of a contagious disease: This is an interesting model of a simple epidemic. It forms a starting point for the mathematical analysis of more complex epidemic situations. It is based on the assumption that a disease spreads when infected people come in contact with uninfected or susceptible people. Thus for a population of size 400, In = kIn(400-In) where In(400-In) represents the number of possible interactions between infected and susceptible at time n. k is the fraction of these interactions that lead to a new infection.

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We identify two types of dynamical systems, namely homogeneous and non-homogeneous:

a) an+1 = ran ; r is a constant. b) an+1 = ran + b ; r and b are constants.

We define an equilibrium point as a point/value with the property that, once a system reaches this value, it stays there. In other words the change of the system is zero. If we are interested in a long term behaviour of the system then we have to allow the temporal variable t to grow arbitrarily large. For instance,

a) If an+1 = ran what happens if for various possible values of r ?

i,e. if r = 0 , r = 1 , r < 0 , r1

b) If an+1 = ran + b then there is an equilibrium value if an+1= an. Call the equilibrium value a and then solve a = ra + b.

Clearly, the solution exists if r 1.

What happens if r=1 , r1 ? What does a graph of an vs. n look like?

How can we use the value of r to determine what type of equilibrium we have for a given model?

Exercise: Show that the solution of the dynamical system an+1 = ran+b, r1 is given by

1k

kba r c

r= +

SYSTEMS OF DIFFERENCE EQUATIONS Competitive Model This model basically, describes competition between two species for survival in a habitat, for instance goat and cow or owl and hawk. We assume that each species would exhibit unconstrained growth in the absence of the other species. If On and Hn represent sizes of the owl and hawk populations at the end of day n, then in isolation On = k1On and Hn = k2Hn where k1 and k2 are positive constants representing growth rates.

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Assume the presence of the other species diminishes the growth rate and that this effect is proportional to the frequency of interactions between the two species. With one additional assumption (what is it?), we obtain: On = k1On-k3OnHn Hn = k2Hn-k4OnHn Solving for On+1 and Hn+1 gives us: On+1 = (1+k1) On k3OnHn Hn+1 = (1+k2) Hn k4OnHn Next we choose specific values for k1, k2, k3, k4 and consider the system: On+1 = 1.2 On 0.001 OnHn

Hn+1 = 1.3 Hn 0.002 OnHn What do these values suggest about the hawks and owls? Exercise: a) Find the equilibrium values of the system

b) Can (0,0) and (150,200) be equilibrium values ? c) Is there a possibility of coexistence for this system? d) Predator-Prey Model: Assume this time a habitat with owl and hawk. The

owls primary food source is mice however mice never predate on owls. Develop the pertinent model.

Finite Difference Solution of Discrete Logistic Model. The nonlinear logistic model,

( )m

nnn P

rPPrP2

1 1 +=+ . is not that easy to solve. Thus, but a slight modification (replacing 1

2 by +nnn PPP ) produces an

equation, which is tractable by making the substitutionn

n PV 1= .

Replacing 1

2 by +nnn PPP , leads to

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( )m

nnnn P

PrPPrP 11 1

++ +=

And hence, the expression

( )

+= ++m

nnn P

rPrPP 11 1

Now, substituting

nn P

V 1= , 1

11

++ =

nn P

V and m

m PV 1=

we get

( )

+=++ 11

111

n

m

n VrV

rVnV

Finally, we arrived at ( ) mnn rVVrV += +11

This equation might be solved as a recurrence relation as follows.

Step 1: Find the homogeneous solution to the equation: ( ) 01 1 =+ +nn VrV

If nn qV = ( )0q

( )( ) 011

01 1

=+=+ +

qrqrq nn

rq += 1

1

n

rAVn

+= 11

Step 2: For the particular solution try BVt = ( ) mrVBrB += 1 mVB =

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Therefore the general solution of the equation is:

m

n

n VrAV +

+= 11

Applying the boundary condition 0 when 0 == nVVn gives,

m

m

VVAVAV=

+=0

0

Therefore the required solution of the equation is:

( ) mn

mn VrVVV +

+= 11

0

Now by back substitution, we can rewrite the solution in terms of P.

1

0

11

11

+

+

= mn

mn PrPP

P

If we compare this equation with that of question (2) we find that the solution of this equation behaves differently. Initially the population N rises rapidly, until the population can no longer be sustained. Then the population level falls until it begins to level out at the value of mN the maximum population, which can be sustained. A graph can illustrate this result.

P

mP

t

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Population Growth in a Limited Environment

The study of an insect population in a limited environment revealed that as the population increased the available food per insect declined. In consequence the number of eggs laid

eN decreased in proportion to the population size according to the law te NAN = where tN is the size of the population and A and are constants. None of the adult insects survive the winter but a fraction of the eggs hatch out the following year. Equilibrium size / value is attained, when tt NN =+ 1 for following years, ( )

( )( )tt

tt

tt

te

NANNN

NANNAN

==

==

+

+

1

1

Finding the equilibrium value tNN of

( )NAN

ANANN

t

t

tt

=+==+=+

1

1

Writing tt nNN += , we can obtain and solve a difference equation for tn and hence obtain an expression for tN assuming 0NNt = at t = 0. Also 11 ++ += tt nNN ( )( )

( )tt

t

tt

tt

nnNnA

NnNAn

nNAnN

=+=

=+=+

+

+

+

1

1

1

1

( )( )

( ) ( ) 00

323

02

12

01

1 nn

nnn

nnn

nn

ttt

=====

=

20

and

( ) ( ) ( )( ) ( ) ( )( ) ( )

++

+=+==

==+=

111

1

1

ie

0

0

0

00

AANN

NNNN

NNNN

NNn

NNnnNN

ttt

ttt

ttt

tttt

We can draw a graph showing the behaviour of tN for two typical situations,

(i) when < 1 (ii) when > 1

(i) (ii)

t

N

tN From the relationship above tN tends towards N when < 1

t

N

tN

Over time tN tends towards zero when > 1

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Consider the system of finite difference equations,

tttt

tttt

YXaYaYYXaXaX

22211

12111

+=+=

+

+

If we introduce difference approximation of derivatives, as under,

ttYY

dtdY

ttXX

dtdX

ttt

ttt

+=

+=

+

+

1

1

1

1

then we have

( )

tttt

tttt

ttttt

YbXaXdt

dX

YXaXadt

dX

XYXaXadt

dX

=

+=

+=

1211

1211

1

and similarly we obtain,

( )

tttt

tttt

dYYcXdt

dY

YXaYadt

dY

=

+= 2221 1

Taking 11a =1.5, 025.012 =a , 75.021 =a , 0025.022 =a , and with initial values X=100, Y=20, we can show that equilibrium is maintained. Observe that for equilibrium size the graph is constant.

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By using different initial values, we can show that the system is not stable. Firstly by considering the values X = 90, Y = 20, we can plot the population of the predators and preys.

Population maintained in equilibrium

0

20

40

60

80

100

120

1 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79

Time

Pop

ulat

ion

size

Prey (X)Predator (Y)

Unstable System

0

50

100

150

200

250

300

350

400

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51

Time

Popu

latio

n Si

ze

Prey (X)Predator (Y)

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Secondly we can try another set of initial values for the population of the predators and preys. Therefore taking X = 100, Y = 25, we can plot the resulting population over time.

In the end, we shall see competing species with limited resource. Competing & Co-operating Species The differential equation representing growth of population with a limited food supply requires modification if a second species also competes for the same food resources. In this case the equation becomes:

( )2211111 NcNMNrdtdN =

where 1N is the number of species 1, 1r is the intrinsic growth rate, and 1M is the saturation population for species 1, 2N is the number of species 2 and 2c is the inhibitory factor of species

2 on species 1. The growth rate dt

dN 2 is given by the corresponding equation:

( )1122222 NcNMNrdtdN =

Unstable System

0

100

200

300

400

500

600

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39

Time

Pop

ulat

ion

Siz

e

Prey (X)Predator (Y)

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By taking axes to represent 1N and 2N , we can draw lines representing equilibrium conditions for species 1 and 2. Any point ( 1N , 2N ) represents a possible population configuration. By considering the directions in which this point would tend to move, we can show under what initial conditions:

(a) Species 1 would be eliminated.

(b) Species 2 would be eliminated.

1N

2N

1+2

1

dtdN1

02 =dt

dN

2N

1N

1+2

2 dt

dN 2

01 =dt

dN

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(c) Both species would co-exist in stable equilibrium.

1N

(d) Equilibrium would be unstable.

2N

1+2

1

2

1M

2M

01 =dt

dN

02 =dt

dN

1N

2N

1+2

2

1

02 =dt

dN

01 =dt

dN

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CHAPTER 3

CONTINUOUS MODELS

Here we recall that a continuous model is one in which the model variable temporal variable) takes all numbers in a given interval of real. In what follows we shall present prototypes of these models.

S-I-R Model The SIR Model is used in epidemiology to compute the amount of susceptible, infected, recovered people in a population. This model is an appropriate one to use under the following assumptions:

1) The population is fixed. 2) The only way a person can leave the susceptible group is to become infected. The

only way a person can leave the infected group is to recover from the disease. Once a person has recovered, the person received immunity.

3) Age, sex, and race do not affect the probability of being infected. 4) There is no inherited immunity. 5) The member of the population mix homogeneously (have the same interactions with

one another to the same degree).

The model starts with some basic terminologies: =tS Number of susceptible individuals at time t

=tI Number of infected individuals at time t tR = number of recovered individuals at time t

N = total population size

The assumptions lead us to the following set of differential equations.

tt ISdtdS = (1)

tt IkSdtdI )( = (2)

tkIdtdR = (3)

where =k Fraction of the infected group that recovers = probability of becoming infected = number of people infected person comes in contact with in each period of time on average

= average number of transmissions from an infected person in a time period , and NRIS ttt =++

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From these equations (3) & (4), we can discover how the different groups will act as .t We can see from equation (1), that the susceptible group will decrease over time. How the infected group behaves is more complicated. We can find

NISNRdsIk tttt

s ==0

which means that

+dtdI .

(Optional) By recourse to numerical schemes, namely, applying Eulers method for systems we can solve the differential equations. The solutions to the differential equations are:

tISSS nnnn =+ 1 (4)

tkSII nnn +=+ )1(1 (5) tkIRR nnn +=+1 (6)

where , and are the number of susceptible, infected and recovered people at time (n+1) ( is a small change, and will be equal to one from now on). These equations are primarily used to calculate and k .

These equations can be modified to included people who die from the disease. To do so we eliminate the recovered equation. In its place, we have two equations, one for death ( tD ) and the other for immunity ( tV ) . Specifically

tkIdtdV =

tIkdtdD )1( =

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Using Eulers method for systems, the solutions to the above equations become

tIkDDtkIVV

tnn

nnn

+=+=

+

+)1(1

1

where 1+nV and 1+nD are the number of dead and immune people at time (n+1) . An important part of modeling spread of diseases is the Basic Reproductive Ratio, denoted as RB . The Basic Reproductive Ratio is important since it tells us if a population is at risk from a disease. RB is affected by the infection and removal rates, i.e. k, , and is obtained by

0SkBR

= . When 1>RB , the occurrence of the disease will increase. When 1

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The Control Vaccination Number, denoted VC , is the average number of secondary cases generated by an infectious case during epidemic with control measures, i.e. vaccinations. To calculate this number using following formula

]1[ hfBC RV = (7)

Where h is the vaccine efficacy (the effectiveness of the vaccine) and f is vaccination coverage (the fraction of the population that has been vaccinated). The goal of researchers and health officials to have 1

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In what follows we shall take a closer look at the aforementioned models one after the other.

Malthusian Model

Thomas R. Malthus in 1798 the proposed a mathematical model of population growth.

>=>=

)2.1.3(0)(

)1.1.3(0,

00 PtP

PdtdP

Where in equation (3.1.1) expresses population growth rate and it reflects a strong impact on how fast the population will grow.

Equation (3.1.1) indicatesdtdP increases to infinity as t increases, i.e. the model population

increases to infinity as time goes to infinity , while equation (3.1.2) expresses the population size when 0tt = . Malthuss model voices such principles: (1) Food is necessary for human existence; (2) Human population grows faster than the power in the earth to produce food (3) The effects of these two unequal powers must be kept equal. Analytical Solution of Malthusian Model In the next few lines, we give the analytical solution of equations (3.1.1) - (3.1.2). From equation (3.1.1), that is

PdtdP = , ( Separable ODE )

We have dtdPP =1 (3.1.3) = tttt dtdPP 00 1 (integrating for two sides of (3.1.3)) (3.1.4)

ttt ceeePP == )( 00 (3.1.5) Where 00

tePc = is a constant for some fixed .

This solution (3.1.5) of equations (3.1.1) - (3.1.2) clearly indicates P is an exponential function of base e. Its physical meaning is that population size grow exponentially as a function of time t .

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Lets investigate how good this model is. To do so, we need to estimate the constant . This constant is the continuous relative growth rate,

i.e. dtdP

P1= .

Lets look at some actual data. This chart shows the China population in the years 1949 and 1959, measured in millions.

Using this information with 0=t corresponding to the year 1949, we have 67.5410 =P . We can solve for using the fact that 07.672=P when 10=t . By using (3.1.5), that is

0215.067.54107.672ln

10167.54107.672 10

== e (to 3 s.f.) (3.1.6)

This leads to the general solution

tetP 0215.067.541)( = (3.1.7)

Lets compute the population at later years and compare it with the actual data, see the table.

Year Population 1949 54 1. 67 1959 672. 07

Year Actual Predicted1960 662. 07 686. 19 1961 65 8. 59 701. 10 1962 672. 95 716. 34 1963 661. 72 731. 90 1964 704. 99 747. 81 1965 725. 38 764. 06 1966 745. 72 780. 67 1967 763. 68 797. 63 1968 785. 34 814. 97 1969 806. 71 832. 86 1970 829. 92 850. 78 1971 852. 29 869. 27 1972 871. 77 888. 16

32

Fig. 2-2 Population vs time

500

550

600

650

700

750

800

850

900

950

1960 1962 1964 1966 1968 1970 1972Year

Pop

ulat

ion

size

s(m

illion

)

Predicted Actual

From this graph, one can see when 0215.0 , how approximate of the prediction value and actual value of population sizes as time increases except of 1963. It showed the efficiency of the model of Malthus under supposes of the Malthuss model. If we used the above model to predict the China population in 2000 and 2049, we may get,

58.1621)2000( P million, 16.4650)2049( P million.

Malthuss model is unconstrained growth, i.e. model in which the population increases in size without bound. It is an exponential growth model governed by a differential equation of the form

==dtdP

PP

dtdP 1 (Constant)

As we have seen, the equation solution is equation (3.1.5).

ttt ceeePP == )( 00

Where 00tePc = is a constant for the constant . Therefore, the population number P increases

to infinity as time t goes to infinity.

Nevertheless, most populations are constrained by limitations on resources, even in the short run and none is unconstrained forever. The following figure 2-3 shows three possible courses for growth of the population. The red curve expresses super-exponential growth and approaching a vertical asymptote (the dashed line), the green curve follows an exponential growth pattern, and the blue curve is constrained so that the population is always less than some numberK.

33

When the population is small related toK, the patterns are virtually identical -- in particular, the constraint doesn't make much difference. But as P becomes a significant fraction ofK, the curves begin to diverge, and, in the constrained case, as P gets close toK, the growth rate drops to 0. (Here =K )

Fig. 3-3 Population growths as time

Logistic ( P. Verhulst ) Model In 1840 Piere Verhulst modified Malthuss Model. He thought population growth not only depends on the population size but also on how far this size is from its upper limit. He proposed a new model which is,

)1(MPP

dtdP = (3.2.1)

Where 0> expresses population growth rate, and 0>M is called the carrying capacity or the maximum supportable population. This equation is also known as a logistic equation.

Analytical Solution of the Logistic Model

We may account for the growth rate declining to 0 by including in the exponential model a factor of - P -- which is close to 1 (i.e., has no effect) when P is much smaller than , and which is close to 0 when P is close to . The constant solutions are P =0 and P =M. The non-constant solutions may obtained by separating the variables as follows.

34

dt

MPdP

dP = )1(

(3.2.2)

Taking indefinite integration for the two sides of equation (3.2.2)

= dtMPP

dP )1(

(3.2.3)

The partial fraction techniques give

+= dPMP

MP

MPP

dP )1

11(

)1( (3.2.4)

Which gives

ctMPP += 1lnln (3.2.5)

Easy algebraic manipulations give

tCe

MP

P =1

(3.2.6)

where C is constant. Finally, solving for P we get,

t

t

CeMMCeP

+= (3.2.7)

If we consider the initial condition 0)0( PP = (assuming that 0P is not equal to 0 or M), we get

0

0

PMMP

C = (3.2.8)

Which, once substituted into the expression for )(tP and simplified, renders

tePMPMP

tP += )()( 000 (3.2.9)

Prediction of long term behavior can be conduct as based on this equation in (3.2.9)

35

Evidently,

MtPt

=+ )(lim (3.2.10)

And hence, one can plot the population P against time t to get the following figure.

Fig. 3-4 Population sizes vs. time ( P0 < M)

The above curve produced by the logistic equation resembles an S. That is why it is called an S-shaped curve or a Sigmoid. As you can see, when the population starts to grow, it does go through an exponential growth phase, but as it gets closer to the carrying capacity, the growth slows down and it reaches a stable level. This slow down to a carrying capacity is perhaps the result of war, pestilence, and starvation as more and more people contend for the resources that are now at their upper bound. There are many examples in nature that show that when the environment is stable the maximum number of individuals in a population fluctuates near the carrying capacity of the environment. However, if the environment becomes unstable, the population size can have dramatic changes.

Exercise: Produce a logistic curve in the event that P0 > M.

36

Practical Data Test of the Logistic Model

If all we know about P , its values at certain times t, then we have to approximate the rate of change,

dtdP

by the fraction

tP

.

The relative rate of growth is

dtdP

P1

.

Using population data from 1842~1970 we can estimate the relative growth rate at the years 1895, 1900, 1905, etc, with the formula

PtPtP

tP

PdtdP

P 10)5()5(11 +=

(3.2.11)

This table has a few of these estimates

Year Population P

tPtP10

)5()5( + 1895 495. 50 0.0018083

1910 509. 12 0.0018070 1925 523. 11 0.0018084 1940 537. 48 0.0018066 1955 614. 48 0.020409 1970 825. 81 0.018625

Table 2.3

We can find a better model by using this data to estimate how the relative growth rate is changing with respect to population. That is, instead of assuming that the relative growth rate is constant, lets try to find a function which estimates this rate as a function of the population P .

37

In the next figure, we show plot the estimates fordtdP

P1 versus P along with the line which best

fits the points.

Fig. 3-5 Population Growth rate vs. population The equation of this line is

PdtdP

P+= 51087612.502662.01 (3.2.12)

If the carrying capacity M is 14.8 billion then, the growth rate is

14801087612.502662.01480

1 5 += dtdP (3.2.13)

060347.01480

1 =dtdP (3.2.14)

Clearly, one can see that the growth rate increases as population size increases.

38

Coalition Model (H. Forster et. al) Coalition growth model is proposed by a team of people, Heinz von Foerster, Patricia Mora, and Larry Amiot in 1960. These modelers argued that the growth pattern in the historical data can be explained by improvements in technology and communication that have molded the human population into an effective coalition in a vast game against Nature reducing the effect of environmental hazards, improving living conditions, and extending the average life span. They proposed a coalition growth model for which the productivity rate is not constant, but rather is an increasing function of the population P, namely, a function of the form P , where the power is positive and presumably small. (If = 0, this would reduce to the natural model , which we now know it does not fit.) The mathematical description i.e. differential equation for this model is

0,,1 >= + PdtdP (3.3.1)

Or PdtdP

P=1

which is a separable ODE with growth rate of population, and productivity rate.

The model asserts that the derivative of P should be proportional to a power of P, that is, the rate of change should be a power function of P. If that is the case, then the logarithm of the derivative should be a linear function of the logarithm of the population, i.e.

PdtdP ln)1(lnln ++= (3.3.2)

Where +1 is the power.

Analytical Solution of Coalition Model

We now use the separation of variables technique to obtain a symbolic representation of the solutions of this differential equation. Then we will consider the implications of faster-than-exponential growth. Separate the variables in the differential equation

+= 1PdtdP (3.3.3)

And write it in the form

39

dtdPP =+ )1( (3.3.4) Taking indefinite integration for the two sides of equation (3.3.4), there is the following

=== +++++ dtdPdtdPdtdPP 1)1(1)1()1( 1 (3.3.5) That is

ctP += i.e. (3.3.6)

1

][

1

ct

P

+= , (3.3.7)

Where c is constant. Finally

[ ] 1)(1

tTP

= (3.3.8)

This model only makes sense if t is less than T (why?). This calculation shows that there is a finite time T at which the population P becomes infinite or would if the growth pattern continues to follow the coalition model.

Remark on the models

Notice that Malthuss model works fairly well for a short period that is the predicted population sizes are quite similar to those of the actual sizes. It starts to fall apart after that. However, this model assumes that the relative growth rate is constant. In fact, even if we ignore natural disasters, wars, and changes in social behavior, the growth rate would change as the population increased due to crowding, disease, and lack of natural resources. The model predicts that the population would grow without bound, but this cannot possibly happen indefinitely. One of the biggest failures in the Malthusian theory was that Malthus failed to foresee the immense technological innovation that was to occur, which increased crop yields and discovered new resources. Malthus interprets his mathematical conclusions in terms of the real world and compares the real world to the model. Well, ideally, he would do that. But, he doesn't live in an information-rich age, and he's dealing with lengthy time spans, so he can't make such comparisons very easily.

40

The logistic growth equation is a useful model for demonstrating the effects of density-dependent mechanisms: discrete-time model, in population growth. Under such model, it is possible for the population to overshoot its carrying capacity. There is no instantaneous adjustment of the population growth rate. The discrete-time model tells us something about what happens when the effects of density-dependence aren't instantaneous, but lag behind the population's growth in time. However, the logistic growth equation utility in real populations is limited because the dynamics of populations are complex and because it is difficult to come up with the real value for M in a given habitat. In addition, M is not a fixed number over time; it is always changing depending on many conditions. It is often limited by the current level of technology, which is subject to change. More generally, species can sometimes alter and expand their niche. If the carrying capacity of a system changes during a period of logistic growth, a second period of logistic growth with a different carrying capacity can superimpose on the first growth pulse. For example, cars first replaced the population of horses but then took on a further growth trajectory of their own. Comparatively speaking, the coalition growth model is the most comprehensive among the models. However, this model only makes sense if t is less than T. The calculation shows that there is a finite time T at which the population P becomes infinite or would if the growth pattern continues to follow the coalition model. Summing up, there are quite a lot of potentially valuable factors which affect population growth have been left out in all three models. Therefore, we cannot say which model is the better one. Modified Logistic Model There are many shortcomings in the three models. However, here, we only use the logistic model as a prototype to do the modification. To this effect we try to give attention to some factors/ variables that were ignored in the model.

A) Introducing health-condition factor

It is important to include the effect of health conditions (not sure about the term) in the model. Having a model of this sort can help population researchers (same as the above term) set scientifically sound population dying limits? If H is the number of people taken by, for example, disease each time period (usually years) then the logistic model can be modified to account for this by subtracting H from the population each time period. The resulting model is:

HMPPtP = )1()(

41

B) Introducing minimum viable population factor

There is a second modification to the logistic model that further increases the model's realism and usefulness. This is the addition of the ecological idea of a minimum viable population. The idea here is that a population of any species has a minimum level at which the population can thrive. If the population drops below this minimum level various environmental and genetic factors lead to the elimination of the population. The relevant factors might include inability to find mates, loss of genetic diversity and increased vulnerability to short and long term environmental changes and disease events. The logistic model can be modified to account for the existence of a minimum viable population. One way of doing this is to use the difference equation:

HMP

MinPPtP = )1)(1

.()(

Remark

Thus far, we examined Malthuss model, logistic model and coalition model. Using mathematical techniques of differentiation and integration, we exactly reach the explicit solutions for each model. They are greatly clear and simple for tests of practical data and analysis for each model. Furthermore, with chart, tables and figures, we compare predicted data and actual data of population growth for the models. They prove the efficiency of the models. Then the advantages and shortcomings of these models are summarized. Finally two new probably improved mathematical models for the predication of population growth are proposed. Using mathematics techniques, Malthuss population model, the logistic model, as well as the coalition model, we predicted the virtual population size without family planning policies. We didnt try to analyze the influences of family planning polices, and hence did not evaluate its merits and shortcomings. Otherwise, our study, though very limited and sallow should have showed that family planning programs have benefited the whole country drastically and even avoided some terrible social or environmental disasters. The success of family planning programs in some countries deserves attention from other developing countries that also face the problem of massive population and its rapid growth and the negative impacts have already affected the whole world. In this regard, one may suggest a global population program to be planned and launched as an important part of sustainable development which emphasis an ideal relationship among population growth, economic development and environmental protection.

42

Finally, we close this chapter by presenting a couple of worked exercises that are intended to stabilize, i.e. help internalize the main points in the chapter, Worked Problem 1: Fish Reproduction Model

A fish hatchery is stocked with a population of 0N juvenile fish, which is subject to a depletion rate r. After a time T (to allow the fish to grow) fishing commences and causes an additional depletion rate q. We can obtain an expression for the population at time t (t >T) and determine the proportion of the original 0N fish which will be recovered by fishing.

rT

rt

rt

eNN

eNNAeN

rNdtdN

Tt

==

=====

=

0

0

00

T when tNow

Tfor t

NA NN 0when t

( )( )

( )

( ) Tfor t

NN Twhen tTfor t

0

0

0

0

===

====

+=

+

+

+

tqrqT

qT

qTrTrT

tqrrT

tqr

eeNNeN

eeeNBBee

BeN

Nqrdt

dNTt

43

The proportion of the original 0N fish, which will be recovered by fishing.

( )

( )( )

( )

qrqe

qree

Tt

Aqr

eqe

dteqe

dtN

eeqN

dtNqN

NqNP

rt

qTrT

tqrqT

tqrqT

tqrqT

t

t

+=+===++=

=

=

=

=

+

+

+

qT

0

0

0

0

qeA

0P and

( )( )

( )( )tqrrTrTTqrrt

eqr

qeqr

qeqr

eqeP

+

+

+=+++=

1

Worked Problem 2: Fish Growth Model

Experiment and observation indicate that the surface area of a fish is proportional the square of its length and that its weight is proportional to the cube of its length. The energy used in seeking food is proportional to its weight, while the energy provided by the food is proportional to its surface area. The rate of increase in weight is proportional to the difference between these two energies. We can obtain an equation showing how the weight of the fish varies with time.

44

( )DwGaMdtdw

GaClwDwkla

EEdtdw

=

====

23

12

12

E

E

=====

==

=

=

wwwdtdww

dtdv

wwv

wwdtdw

DwNwMdtdw

klalw

lw

32

32

32

31

321

32

32

2

232

3

31

31

MD and MN call

since

33

333

1

=+

=

vdtdv

wdtdv

Integrating factor dte = 3

Teve

evedtdv

e

tt

tt

t

+=

=

=

33

33

3

33

3

45

( )

( )( )3

3

3

3

331

after tfish of weight Total

1

(4)question in recoveredfish theof weight totalThe

+==

+=

+=

+==

+

t

tqrrT

t

t

Tep

eqr

qep

Tetw

Tevw

Worked Problem 3: Predator-Prey Model

A predator-prey situation occurs with two populations, when encounters between members of the populations benefit the members of one population (predators) but adversely affect the members of the other population (prey). For a simple model assume that the prey has an adequate food supply and that in the absence of interaction there will be a positive rate of increase proportional to the population size. Likewise, assume that in the absence of the prey, the predators food supply will be inadequate and that there will be a negative rate of increase proportional to the population size. The incidence of interaction will depend on the size of both populations and may be taken to be proportional to their product, and the effect will increase the rate of growth of the predator population and decrease the rate of growth of the prey population.

Letting X, Y denote the size of the prey and predator populations respectively, we can write down two simultaneous differential equations to represent the situation.

( )

( ) 01

01

==+=

===

=

=

XlY

XYlYdtdY

YkX

XYkXdtdX

lYdtdY

kXdtdX

In terms of the constants in these equations, we can obtain the equilibrium values YX and .

46

XXYY

XY

YX

======

==

1 1

01 01

0 0

Let , YYyXXx == , denote the deviations of the population sizes from their equilibrium values. Assuming these deviations are small compared to the size of the populations, we can rewrite the differential equations in terms of x and y and solve the resulting equations, neglecting the second order terms. Hence we can obtain approximate solutions for X and Y.

( )

xldtdy

yk

ykkxy

yxk

yxkdtdx

yYyY

xXxX

=

=

=

+=

+

+=

+=+=

+=+=

1

111

1

1

Renaming

wtqwtpx

xwdt

xdxwdt

xdwAB

ABxdtdyA

dtxd

BxdtdyAy

dtdx

sincos

0let 222

22

22

2

2

+==+==

==

==

47

{ }( )wtqwtp

Aw

wtwqwtwpA

y

dtdx

Ay

cossiny

cossin1

1

=

+=

=

Taking axes to represent X and Y we can sketch a graph to show the variation in populations.

X, Y

t

48

CHAPTER 4

METHODS OF MATHEMATICAL MODELLING DIMENSIONAL ANALYSIS

Dimensional Analysis is a method which helps to determine how the selected variables in a mathematical model are related. It helps to reduce significantly the amount of experimental data that must be collected. Dimensional analysis helps to design scientific experiments. It is also a process by which we ensure dimensional consistency. It does give a better understanding of what to expect experimentally. Indeed it is a method by which we deduce information about a phenomenon from the single premise that the phenomenon can be described by a dimensionally correct equation among certain variables.

UNITS and DIMENSIONS

The physical quantities we use to model objects or systems represent concepts, such as time, length, and mass, to which we also attach numerical values or measurements. Thus, we could describe the width of a soccer field by saying that it is 60 meters wide. The concept or abstraction invoked is length or distance, and the numerical measure is 60 meters. The numerical measure implies a comparison with a standard that enables both communication about and comparison of objects or phenomena without their being in the same place. In other words, common measures provide a frame of reference for making comparisons. Thus, soccer fields are wider than American football fields since the latter are only 49 meters wide. The physical quantities used to describe or model a problem come in two varieties. They are either fundamental or primary quantities, or they are derived quantities. Taking a quantity as fundamental means only that we can assign it a standard of measurement independent of that chosen for the other fundamental quantities. In mechanical problems, for example, mass, length, and time are generally taken as the fundamental mechanical variables, while force is derived from Newtons law of motion. It is equally correct to take force, length, and time as fundamental, and to derive mass from Newtons law. For any given problem, of course, we need enough fundamental quantities to express each derived quantity in terms of these primary quantities. While we relate primary quantities to standards, we also note that they are chosen arbitrarily, while derived quantities are chosen to satisfy physical laws or relevant definitions. For example, length and time are fundamental quantities in mechanics problems, and speed is a derived quantity expressed as length per unit time. If we chose time and speed as primary quantities, the derived quantity of length would be (speed time), and the derived quantity of area would be (speed time)

2.

The word dimension is used to relate a derived quantity to the fundamental quantities selected for a particular model. If mass, length, and time are chosen as primary quantities, then the dimensions of area are (length)

2, of mass density are mass/(length)

3, and of force are (mass

49

length)/(time)2. We also introduce the notation of brackets [] to read as the dimensions of. If

M, L, and T stand for mass, length, and time, respectively, then:

[A =area]=(L)2, (2.1a)

[ =density]=M/(L)3, (2.1b)

[F =force]=(M L)/(T)2. (2.1c)

The units of a quantity are the numerical aspects of a quantitys dimensions expressed in terms of a given physical standard. By definition, then, a unit is an arbitrary multiple or fraction of that standard. The most widely accepted international standard for measuring length is the meter (m), but it can also be measured in units of centimeters (1 cm = 0.01 m) or of feet (0.3048 m). The magnitude or size of the attached number obviously depends on the unit chosen, and this dependence often suggests a choice of units to facilitate calculation or communication. The soccer field width can be said to be 60 m, 6000 cm, or (approximately) 197 feet. Dimensions and units are related by the fact that identifying a quantitys dimensions allows us to compute its numerical measures in different sets of units, as we just did for the soccer field width. Since the physical dimensions of a quantity are the same, there must exist numerical relationships between the different systems of units used to measure the amounts of that quantity. For example,

1 foot (ft) = 30.48 centimeters (cm),

1 centimeter (cm) = 0.000006214 miles (mi),

1 hour (hr) = 60 minutes (min) = 3600 seconds (sec or s).

UNITS The basic accepted SI (System international) units are as follows.

Quantity Unit Symbol Length meter m Mass Kilogram kg Time seconds s

Electric current ampere A Temperature Kelvin k

Luminous intensity Candela cd Amount of substance Mole mol

50

Other commonly used units Non SI units commonly used in science and engineering

DIMENSIONS The Fundamental Dimensions are,

Mass denoted by M Length denoted by L Time denoted by T

In mechanics, all quantities can be expressed in terms of fundamental quantities. For example:

the dimension of area is [area] = L2 the dimension of speed is [speed] = LT-1 the dimension of density is [density] = ML-3

Quantity Unit Symbol Force Newton N (kgm/s2)

Energy Joule J (kgm2/s2) Power Watt W (J/s or kgm2/s3)

Frequency Hertz Hz (1/s)

Pressure Pascal Pa (N/m2 or kg/ms2)

Quantity Unit Symbol Area hectare Ha ( =104 m2)

Energy liter l (=10-3 m3)

Power milliliter ml (=10-6 m3)

Frequency degree celsius 0c ( = 273 k)

Mass gram g (=10-3 kg)

Mass tone t ( = 103 kg)

Energy Kilowatt hour kWh ( =3.6 x 106 J)

Energy Electron volt ev ( = 4.1868 J)

Pressure bar bar ( = 105 Pa)

Pressure atmosphere atm ( 1.013 x105 Pa)

51

If an equation evolves a derivative, the dimensions of derivative are given by the ratio of the dimensions. For example: Consider the pressure gradient

Dimensionless Quantities

Some quantities are dimensionless. These quantities are pure numbers. For example: [angle] = LL-1 = L0 = 1

Recall: Dimensional analysis is based on the premise that physical quantities have dimensions. Furthermore, physical laws are not altered when we change the units used to measure various dimensions. Consider Newtons Law i.e the equation: F = ma What are the units of measurement? Two possibilities are:

1. Newtons, kilograms, meters per second2. 2. Pounds, slugs, feet per second2.

In general, since the three fundamental physical quantities and their associated dimensions are Quantity Dimension mass M length L time T We have velocity = distance Time = (distance) (time) -1

Hence, the dimension of velocity is: [ V ] = LT -1 Similarly, the dimension of

1) area is L2 2) acceleration is LT -2 3) force is MLT -2 ( F = ma) 4) work is ML2 T -2 (Work = Force * distance)

Remark

1. Dimension is based on 3 fundamental physical quantities: mass m, length s, and time t. These quantities are measured in some appropriate system of units whose choice does not affect the assignment of dimension.

dpdz

[ ][ ]pz

[ ] = = ML-2 T-2

52

2. Other physical quantities can be defined as products involving mass, length, and time.

3. To each product, we assign a dimension, which is an expression of the form:

M n Lp Tq where n, p, q R

4. If a basic dimension is missing from a product (e.g. Mo LT -1 = LT -1), the corresponding exponent is zero.

5. If n, p, and q are all zero, the quantity or product is said to be dimensionless.

6. We cannot add products that are dimensionally incompatible or inconsistent. For

example, the equation: B = mv + v2

is not be correct. (Why?)

7. An equation is dimensionally homogeneous if it is true regardless of the system of units in which it is measured.

Example(Free fall): Assume the only force on a falling object is gravity g. Then the time, t, to fall a distance, s, is: a) t s g= 2 / b) t = s / 16.1 dimensionally homogeneous not dimensionally homogeneous. Example (Simple Pendulum): Using dimensional analysis determine the period t of a pendulum

Variables of the model are: r the length of the pendulum, m its mass,

g the gravitational acceleration, the angular displacement.

Dimensional analysis:

Variable m g t r Dimension M LT-2 T L M0L0T0

Dimensionless product of the variables: magbtcrde (*)

53

i.e : (M)a(LT-2)b(T)c(L)d(M0L0T0)e = MaLb+dTc-2b

A product of the form (*) is dimensionless iff MaLb+dTc-2b = M0L0T0 Solving the linear system:

M: a +0e = 0 L: b +d +0e = 0 T: -2b +c +0e = 0

There are two linearly independent solutions: b = 0 & e =1 a = c =d =0 b = 1 & e = 0 a = 0, c = 2, d = -1 These solutions render the dimensionless products: 1 = and 2 = m0g1t2r-10 = gt2/r That is, we obtain 2 products: 1. 1 = when b = 0, e = 1 2. 2 = gt2/r when b = 1, e = 0 Example (Wind & Speed of a car):How does the speed of a car is affected by the wind force on the car. Observation: The force, F the wind exerts on the wind shield of a car is proportional to the car velocity, v and the surface area, A of the wind shield.

i.e F v A F = k va Ab , k = constant

Dimensional analysis:

Since [ F ] = MLT-2 & [ vA ] = M0L0T0 (LT-1)a(L2)b Dimensional consistency yields:

MLT-2 = M0L0T0 (LT-1)a(L2)b

Variable F K v A Dimension MLT-2 M0L0T0 LT-1 L2

54

Questions: Is this possible?

What is the exponent of M on each side? Revised the model. The force, F the wind exerts on the wind shield of a car depends not only on the speed of the car, v and the surface area, A of the wind shield but also on the density, of the air column. Its dimension is ML-3. (Why?) And hence, our model is of the for F = kvaAb c for suitable values of the scalars a, b & c.

Dimensional analysis: Dimensional consistency renders: MLT-2 = M0L0T0 (LT-1)a(L2)b(ML-3)c This leads to the following linear systems: c =1 a+2b-3c = 1 -a = -2 a=2, b = 1 & c = 1.

Consequently our model becomes: F = k v2A Question : Is this equation dimensionally homogeneous ? Finally we emphasize that a rational equation as an equation in which each independent term has the same net dimensions. Then, taken in its entirety, the equation is dimensionally homogeneous. Simply put, we cannot add length to area in the same equation, mass to time and charge to stiffness. Although we can add quantities having the same dimensions but expressed in different units, e.g., length in meters and length in feet. The fact that equations must be rational in terms of their dimensions is central to modeling because it is one of the best and easiest checks as to whether a model makes sense or not, i.e has been correctly derived! We should remember that a dimensionally homogeneous equation is independent of the units of measurement being used. However, we can create unit-dependent versions of such equations because they may be more convenient for doing repeated calculations or as a memory aid.

Variable F K v A Dimension MLT-2 M0L0T0 LT-1 L2 ML-3

55

Buckinghams Pi Theorem The procedure most commonly used to identify both the number and form of the appropriate non-dimensional parameters is referred to as the Buckingham Pi Theorem. Buckinghams Pi theorem is fundamental to dimensional analysis and it can be stated as follows: A dimensionally homogeneous equation involving n variables in m primary or fundamental dimensions can be reduced to a single relationship among n m independent dimensionless products. A dimensionally homogeneous (or rational) equation is one in which every independent, additive term in the equation has the same dimensions. This means that we can solve for any one term as a function of all the others. If we introduce Buckinghams notation to represent a dimensionless term, his famous Pi theorem can be written as:

1 = (2, 3 ... nm) . (a) or, equivalently,

(1, 2, 3 ... nm) = 0 . (b)

Equations (b) state that a problem with n derived variables and m primary dimensions or variables requires n m dimensionless groups to correlate all of its variables. We apply the Pi theorem by first identifying the n derived variables in a problem: A1, A2, ...An. We choose m of these derived variables such that they contain all of the m primary dimensions, say, A1, A2, A3 for m = 3. Dimensionless groups are then formed by permuting each of the remaining n m variables (A4, A5, ... An for m = 3) in turn with those ms already chosen:

1 = A1 a1

A2 b1

A3 c1

A4,

2 = A1 a2

A2 b2

A3 c2

A5, . . .

n-m = A1 an-m

A2 bn-m

A3 cn-m

An

The ai , bi , and ci are chosen to make each of the permuted groups dimensionless.

One can also write the combination of dimensionless pi terms in functional form as:

k = f (1, 2, i)

Example, modeling the small angle, free vibration of an ideal pendulum. There are six variables to consider in this problem, and they are listed along with their fundamental dimensions. In this case we have m = 6 and n = 3, so that we can expect three dimensionless groups. We will choose l, g , and m as the variables around which we will permute the remaining three variables (T0, , T ) to obtain the three groups. Thus,

56

1 = la1

gb1

mc1

T0,

2 = la2

gb2

mc2 ,

3 = la3

gb3

mc3

T .

Derived quantities Dimensions

Length (l) L Gravitational accel. (g ) L/T2 Mass (m) M Period (T0) T Angle ( ) 1 String tension (T ) (M L)/T2

The Pi theorem applied here then yields three dimensionless groups:

mgT

glT

===

3

2

01 /

These groups show how the period depends on the pendulum length l and the gravitational constant g, and the string tension T on the mass m and g . The second group also shows that the (dimensionless) angle of rotation stands alone, that is, it is apparently not related to any of the other variables. This follows from the assumption of small angles, which makes the problem linear, and makes the magnitude of the angle of free vibration a quantity that cannot be determined.

One of the rules of applying the Pi theorem is that the m chosen variables include all n of the fundamental dimensions, but no other restrictions are given. So, it is natural to ask how this analysis would change if we start with three different variables. For example, suppose we choose T0, g , and m as the variables around which to permute the remaining three variables (l, , T ) to obtain the three groups. In this case we would write:

1 = T0 a1 g

b1 mc1 l,

2 = T0 a2 g

b2 mc2 ,

3 = T0 a3 g

b3 mc3 T .

57

Geometric Similarity and Modeling The basic requirement in this process is to achieve 'similarity' between the 'experimental model and its test conditions' and the 'prototype and its test conditions' in the experiment. In this context, therefore, similarity is defined in simple terms as all relevant dimensionless parameters have the same values for the model and the prototype.

Geometric Similarity

Geometric similarity might be stated as saying, All linear dimensions of the model are related to the corresponding dimensions of the prototype by a constant scale factor, SFG . We shall demonstrate this concept with the following diagram of airfoil (laminar flow),

For this case, geometric similarity requires the following:

SF G = rmr p =

L mL p

= W mW p

=

Remark In geometric similarity:

All angles are preserved. All flow directions are preserved. Orientation with respect to the surroundings must be same for the model and the

prototype

58

Consider two cubes, one of which has sides of unit length in any system of units we care to choose, that it, the cubes volume could be 1in

3 or1m

3 or1km

3. The other cube has sides of length L in the same system

of units, so its volume is either L3

in3

or L3

m3

or L3

km3. Thus, for comparisons sake, we can ignore the

units in which the two cubes sides are actually measured.

L

Two geometrically similar cubes

The total area and volume of the first cube are, respectively, 6 and 1, while the corresponding values for the second cube are 6L

2 and L

3. We see immediately an instance of geometric scaling, that is, the area of

the second cube changes as does L2

and its volume scales as L3. Thus, doubling the side of a cube

increases its surface area by a factor of four and its volume by a factor of eight. Exercices

1. Which of the following is not true about dimensional analysis? a) It helps to decide if an equation is correctly derived b) It helps reduce relevant variables c) It can be used to check if an equation is right or wrong. d) It helps predict possible form of relationship between variables of the system e) None

2. Which of the following is a geometric variable? a) velocity c) viscosity c) length d) pressure e) none

3. Which of the following equations is dimensionally homogeneous?

a) atuts 21+= b) gtuh2

2=

c) 0)()()( =+

++

zu

yu

xu

t d) ghP = e) none

4. Which of the following in not dimensionless?

a) lu b)

lgdt

c) gl

u d) Fum e) none

59

CHAPTER 5

PROTOTYPE MODELS

A) Traffic Flow Models In order to design the roads and the cars that enable and facilitate transportation, we model both the behavior of individual cars with their drivers in a (single) line of cars, and that of groups of cars in one or more lanes of traffic. However, our concern is not with modeling the ergonomics

of operating a car. Rather, we focus on the interactions of vehicles on single highway lanes, both individually and in dense lines.

No matter how we refer to traffic arteries, the flow of traffic on them is modeled, analyzed, and predicted with traffic flow theory, which we now detail at two levels. The macroscopic modeling of traffic assumes a sufficiently large number of cars in a lane or on a road such that each stream of cars can be treated as we would treat fluid flowing in a tube or stream. Thus, to maintain the biological metaphor, traffic flow is treated as a flow of a fluid field in an artery. Macroscopic models are expressed in terms of three gross or average variables for a whole line of traffic: the number of cars passing a fixed point per unit of time, called the rate of flow; the distance covered per unit time, the speed of the traffic flow; and the number of cars in a traffic line or column of given length, which we identify as the traffic density. The relationship between the speed and the density is embodied by macroscopic modelers in a plot of these two variables called the fundamental diagram. We may also invoke the continuum hypothesis to confirm that it is appropriate to (mathematically) treat the traffic as a field.

The second level of traffic modeling, microscopic modeling, addresses the interaction of individual cars in a line of traffic. Microscopic models describe how an individual follower car responds to an individual leader car by modeling its acceleration as a function of various perceived stimuli, which might be the distance between the leader and follower cars, the relative speeds of the two cars, or the reaction time of the operator of the follower car. Car-following models come in several varieties, and they can be used to construct the speed-density curves that are the underpinning of macroscopic modeling. Such speed-density plots, supported by data taken from real traffic arteries, enable traffic experts to model and understand road or freeway capacity as a function of traffic speed and density even if everyday drivers feel they do not fully understand what is happening around them. (The microscopic models are also used to support the modeling of vehicular control, that is, to implement control strategies that enable lines of traffic to maintain high flow rates at high speeds.

Macroscopic Traffic Flow Models We start by asserting the validity of an analogy, namely, that the flow of a stream of cars can be modeled as a field, much as we would model the flow of a fluid. Thus, the collection of cars taking the 10 east out of Los Angeles on any given evening is mathematically similar to the flow of blood in an artery or water in a home piping system. We want to relate the speed of Find? a

60

line of traffic to the amount of traffic in that line (or lane). We use three variables to describe such traffic flows: the rate of flow, q(x, t ), measured in the number of cars per unit time; the density of the flow, (x, t ), which is the number of vehicles per unit length of road; and the speed of the flow, v(x, t ). How are these three variables related? Conservation of Cars

Suppose we have a highway of infinite length where the velocity and density are known; can we predict the pattern of traffic? First we consider (x, t) and u(x, t) to be the two fundamental traffic variables. We have (x, t) = traffic density, which is the number of cars at time t at position x, and u(x, t) = car velocity at position x and time t, traffic flow = q(x, t), which is the number of cars per hour passing position x at time t, and thus

),(),(),( txutxtxq = . Then the initial traffic density is (x, 0), which is the traffic density at position x and time 0, and the traffic velocity field for all time remains the same, u(x, t). The motion of each car is determined by taking the derivative of position x with respect to t, which satisfies the following first order differential equation:

),( txudtdx = with 0)0( xx = (5.1)

Solving equation (5.1) determines the position of each car at future times. To be able to calculate the traffic density at future times we would need to know the traffic velocity and the initial density. We want to be able to calculate the density easily if we know the velocity. We choose an interval on any particular roadway between say x = a & x = b, as illustrated below:

Figure 5.1 cars entering and leaving a segment of roadway

On this interval [a, b], the number of cars, denoted N, is the traffic density integrated:

( ) === bx ax dxtxtN ,)( (5.2)

61

Even with no exits or entrances on this roadway the number of cars on the interval between x = a and x = b could still change in time. As cars enter at x = a, the number of cars increases and as cars leave at x = b, the number of cars decreases, therefore the traffic flow q (a, t) and q (b, t) is not constant in time.

The rate of change of the number of cars with respect to time,

dtdN

,

is equal to the number of cars per unit time entering the interval [a, b] at x = a minus the number of cars per unit time exiting the interval [a, b] at x = b, where the cars are always moving to the right, as illustrated in the equation below since the rate of change of the number of cars per unit time is the traffic flow at position a minus position b both at time t:

),(),( tbqtaqdtdN = (5.3)

Taking the derivative of both sides of equation (5.2) with respect to time gives the following:

( ) === bx ax dxtxdtddtdN , (5.4) By combining equation (5.3) and equation (5.4), you get the result:

( ) ),(),(, tbqtaqdxtxdtd bx

ax= == (5.5)

And q(a, t) q(b, t) can be rewritten by taking the partial derivative of the right hand side of equation (5.5) with respect to x, and then taking the integral from x = b to x = a gives the following equation:

( ) dxx

txqdxtxdtd ax

bx

bx

ax ==== = ),(, (5.6) To have the integral with the same interval, we need to use an integral property, which is to take the negative of the right hand side of equation (5.6):

( ) dxx

txqdxtxdtd bx

ax

bx

ax ==== = ),(, (5.7)

62

Moving the negative sign inside of the integral gives:

( ) dxx

txqdxtxdtd bx

ax

bx

ax ==== = ),(, (5.8) We can now move the dtd inside of the integral to get the following equation; we can do this because derivatives and integrals are interchangeable. If you move the derivative inside the integral and it has a function of two variables, then the derivative becomes a partial derivative:

( ) dxx

txqdxtxt

bx

ax

bx

ax ==== = ),(, (5.9) Equation (5.9) implies:

dxx

txqtxt

bx

ax == +

),(),( = 0 (5.10) Equation (5.10) implies:

0),(),( =+

x

txqtxt (5.11)

And from equation (5.11) we get

0),(),( =+

xtxq

dttxd

(5.12)

Suppressing equation (5.12), which is just not including the variables of the function, gives us:

0=+

xq

dtd

(5.13)

This is the equation of conservation of cars. We know from above that uq = , and so therefore we can rewrite the xq as the following:

),( uqxx

q =

. (5.14)

63

Which implies

xu

uq

xq

xq

+

=

(5.15)

Now combining equation (5.13) with equation (5.15) we get the following:

0=

+

+

xu

uq

xq

t

(5.16)

Which is still the conservation of cars since equation (5.16) is the same as equation (5.13). Now, assume that )(uu = .Taking the derivative to the velocity with respect to the position, x gives:

0=

xu

(5.17)

Combining equation (5.16) and equation (5.17) we get the following result:

0=

+

x

qt

(5.18)

A Velocity-Density Relationship

There are many factors that have an affect on the speed at which a car can go since it is operated by an individual. The person operating one car may want to drive faster than another person in a different vehicle. Once the traffic becomes a lot heavier, however, lane changing and speed are at a minimum for every driver on the road since it is difficult to change lanes when there are more vehicles on the road and it is not always possible to go the speed you want when there are more vehicles on the road. A lot of times you get stuck going the same speed at which the flow of traffic is moving. With all of these types of observations, we can make a simplifying assumption that at any point along the road the velocity of a car only depends on the density of cars. This is illustrated in the equation below, which was mentioned above in the explanation of the conservation of cars:

)(uu = (5.19)

64

Traffic density

Figure 5.1 Traffic velocity V/s density

Apart from the fact that traffic velocity as a function of traffic density is decreasing, the exact relationship between then is not known. Thus another closer look at such a relationship between the two can be described in the next graphics;

Traffic density

Figure =5.2 Another view of the Traffic velocity V/s density

As mentioned above, cars velocity can be at a maximum when there are very little to no cars at all on the road with them. So when there are no other cars at all on the road, this means that the density is at zero, and therefore the velocity will be at a maximum as illustrated with equation (5.17) with the density at zero below:

max)0( uu = (5.20) As more and more cars per mile that join the road way their presence will slow down the car, and as the density increases more, the velocity of the cars would continue to decrease. Thus the rate of change, which is the derivative of the velocity with respect to density, is defined as below:

0)(' uddu

(5.21)

Once density is at a maximum, then cars will move at zero velocity, or stand still:

65

0)( max =u (5.22) Therefore the car velocity vs. the traffic density is steady decreasing.

Elementary Traffic Model

As shown above, in general the car velocity is a decreasing function of density. At zero density, cars move the fastest which was denoted maxu and the maximum density was denoted

max where the car velocity is zero. The simplest relationship to satisfy these properties is let:

=

maxmax 1)(

uu (5.23)

in which from the fact uq = the flow is given by

( )

=

=

max

2

maxmax

max 1

uuq (5.24)

And the density wave velocity satisfies

( ) ( )

==

maxmax

21' uqc (5.25)

Red Light Turning Green

Now we assume the elementary model of traffic flow so that the traffic density satisfies

021max

max =

+

x

ut

(5.26)

Behind a red light position is at zero,