Epidemic is applied to a disease which, distributing widely, onslaughts many individual at the same clip. Epidemic is a widespread eruption of an infective disease. When an epidemic exists, it will impact many human populations. There were many factors to imitate the rise of epidemics such as hapless population wellness, in-migration and failure of public wellness plans.

Harmonizing to the universe wellness organisation appeared, disease should hold the undermentioned conditions: a new pathogens, disease can take to infection causes a serious complications, pathogen spread easy, particularly in interpersonal communicating. By and large talking, this disease is due to some powerful infective micro-organisms, and the infections caused by viruses, bacteriums.

Historically talking, awful epidemics have reoccurs over and over once more. Some epidemic diseases, such as the variola, pestilence, and grippe, have been persisted in the history. Smallpox was uprooted worldwide by 1980. In the eighteenth century, the universe ‘s major trade paths, several destructive appear cholera pandemic caused great infective diseases.

In the past worldwide, the chief cause of many people were killed is infective diseases and there are more deceases than all the war, such as the Black Death that struck Europe in 1347 had kill between one-third and one-half of the people in many metropoliss and towns severally, this ill-condition seized the advancement of civilisation for several coevalss.

1.2 Aims

To analyze the SIR theoretical account ( developed by Kermack and McKendrick ) in the signifier of the system of nonlinear ordinary differential equations.

To work out the SIR theoretical account numerically utilizing Mathematica and imitating the Agent Based Modeling utilizing NetLogo.

To look into the infection spread under the SIR theoretical account

To construe the consequences of epidemic job based on these two theoretical account.

1.3 Research Question

What is an epidemic?

Can an epidemic be avoided or command?

Which is better to work out the SIR theoretical account, Equation based Model or agent-based theoretical account?

What are the important parametric quantities that govern the two theoretical accounts?

1.4 Epidemic Problem mold in Mathematica Program

Mathematica plan is a general computing machine package system and linguistic communication in used for Mathematica plan and other applications. Mathematica plan non merely for usage in calculation, it besides use for mold, simulation, development and deployment, visualizationand certification. Mathematica calculations can be divided into 3 chief categories which are Numeric, Graphical and Symbolic.

Different occupations covering with different things, but the Mathematica plan is a comprehensive system to supply unprecedented work flow, dependability, sustainability and invention. In this undertaking Mathematica plan is used as a mold and information analysis the rate of epidemic. The inquiry can be answered by making the theoretical account of an epidemic with variables matching to the different reaction of a population and the features of a virus.

1.5 Epidemic Problem Simulating in NetLogo

NetLogo is multi-agent scheduling linguistic communications and incorporate mold environment and a platform specifically for agent-based mold. NetLogo is most suited for complex system patterning development. Model can steer 100s or 1000s of “ agent ” all operating independently.

NetLogo besides let pupils can imitate and “ drama ” , explore their behaviour in different conditions. NetLogo has extended files and tutorial. It besides comes with a theoretical account library, which is a big aggregation of pre-written simulation, it can be used and alteration.

If an epidemic occurs, the variables matching to a population reaction and features of disease will impact its continuance and badness. In attempts to command the spread of the disease, we must take an optimum solution for the maximal public wellness benefits. In NetLogo scheduling, system kineticss can utilize a alone scheduling. To find the influence of assorted factors have on the continuance and serious infective disease, we can alter the variable and expression at the form of the graph differs between tallies in NetLogo.

1.6 Prilimanary Research

The Ebola virus patterning in Mathematica 7

The Ebola virus imitating in NetLogo

1.7Chapter drumhead

This thesis is divided into five chapters. In the first chapter, we discuss the debut of epidemic. For chapter two, we introduce the General Epidemic theoretical account by Kermack and McKendrick ( 1927 ) . In this chapter, we show how to deduce the theoretical account. For chapter three, we discuss the Mathematical Modeling. In chapter four, we will discourse epidemic theoretical account patterning in Mathematica plan. In chapter five, we will discourse SIR theoretical accounts imitating in NetLogo plan. In chapter four and five, we will plot the solution for the theoretical account. Finally is chapter six. In this chapter, we will make an readings and decision about the consequence of epidemic theoretical account.

2.0 SIR Model

2.1 Introduction

In 1927, W. O. Kermack and A. G. McKendrick created a theoretical account of epidemic. The independent variable for this theoretical account is clip ( T ) . Assume the population is a disjoint brotherhood, there are three dependent Variables:

1. S = S ( T ) , which is the figure of susceptible individuals

2. I = I ( T ) , which is the figure of septic individuals

3. R = R ( T ) , which is the figure of cured individuals

The entire population = S ( T ) + I ( T ) + R ( T ) .

SIR theoretical account was based on the theoretical account in the spread of disease of the population. SIR theoretical account is a simple but good theoretical account of infective diseases, such as rubeolas, chicken-pox and German measles, which one time the individual infected with, will non be infecting once more.

2.2 Premises of the SIR theoretical account

SIR theoretical account is based on some premises. Suppose the population measure is immense and changeless. Because we ignore births and in-migration, therefore cipher is added to the susceptible group. Since the lone manner to go forth the susceptible groups will be infected, we assume the time-rate of alteration forA the figure of susceptible depends to the figure of people who already susceptible, the figure of individuals that already infected and the sum of the susceptible individuals contact with septic individual.

In add-on, we are hypothesis each septic people have a fixed value I? contact per twenty-four hours, and there are adequate sufficient to distribute the disease. Not all these contacts are with susceptible people. If we assume that the population is homogenous commixture, the fraction of these contacts that are with susceptible isA S ( T ) . Therefore on an mean, each infected individual will bring forth I?A S ( T ) A of new septic individuals per twenty-four hours.

We besides assume that a fixed fractionA I? in the septic group will retrieve bit by bit in any given twenty-four hours. For illustration, if the mean continuance of infection is four yearss, so on norm, one-quarter of the population under infected will retrieve each twenty-four hours.

2.3 SIR Formulas

There are three basic dependent differential equations:

S ‘ ( T ) = – I? S ( T ) I ( T )

I ‘ ( T ) = I? S ( T ) I ( T ) – I? I ( T )

R ‘ ( T ) = I? I ( T )

The theoretical account starts with some basic notation. That are S ( T ) is the figure of susceptible individuals at clip T, I ( T ) is the figure of septic individuals at clip T, and R ( T ) is the figure of cured individuals at clip T

These equations describe the passages of individuals from S to I to R. By adding the three equations, the size of the population is changeless and equal to the initial population size, which we denote with the parametric quantity N. Therefore the entire population

N= S ( T ) + I ( T ) + R ( T ) .

We call the parametric quantity I? the infection rate and the parametric quantity I? the recovery rate with I? and I? must be or greater to zero. The term I? is a standard kinetic footings, based on the thought that the figure of unit clip to meet between the susceptible and infective will be relative to the Numberss value. The infection I? is determined by both the brush frequence and the efficiency of distributing the diseases per brush.

2.4 Dynamicss

If we imagine the procedure in a disease that a really suited for the SIR model, we will acquire a flow of people from the susceptible group will travel to the infection group, so to the removed group. i‚?

The diagram of SIR theoretical account

I? S I R I

susceptible

infected

recovered

Diagram 2.4.1

The individual perchance moves from the susceptible to the infected group when person comes in contact with an septic individual. Qualify as a contact in the population, depends on the disease. For HIV virus a contact may be sexual contact or a blood transfusion. For Ebola virus it contact with septic organic structure ‘s funeral, and contact with septic individuals without exercising proper cautious.

2.5 Derivations of the SIR theoretical account

The theoretical account is described by three ordinary differential equations:

For the susceptible differential equation,

When we plotting the graph of S ( T ) versus T with I? and I? is a invariables, that is a negative exponential relationship between S and t. Since S ( 0 ) a‰ 0 when T = 0, , therefore the graph will started with a?z .

The graph of S versus T

Figure 2.5.1

2. For the septic differential equation,

When we plotting the graph of I ( T ) versus T with I? and I? is a invariables, that is a exponential relationship between I and t. Since I ( 0 ) a‰ 0 when T = 0, , therefore the graph will non started with 0.

The graph of I versus t

Figure 2.5.2

3. For the cured differential equation,

When we plotting the graph of R ( T ) versus T with I? and I? is a invariables, really clearly, that is a additive relationship between R and t. Since R ( 0 ) = 0 when T = 0, the graph will started with 0.

The graph of R versus T

Figure 2.5.3

4 Vector Notation

If work outing with numerical values for the invariables a and B, utilizing vector notation can do the system easier to cover with.

Let

so

2.6 A Graphic Solution to the SIR Model

To demo a solution to the SIR theoretical account, we try to plot the differential equations with value a = B = 1 and allow the initial value S ( 0 ) = 5, I ( 0 ) = 0 and R ( 0 ) = 0.

Then

S ‘ ( T ) = – S ( T ) I ( T ) ( Moore, 2000 )

I ‘ ( T ) = S ( T ) I ( T ) – I ( T )

R ‘ ( T ) = I ( T )

Figure 2.6.1

The three populations versus clip give the end product. The septic is relative to the alteration in clip, the figure of septic and the figure of susceptible. The alteration in the septic population addition from the susceptible group and lessening into the cured group.

3.0 Mathematical Modeling

3.1 Introduction

Mathematical mold is a replacing of an object studied by its image. The mathematical mold is the method of making a mathematical theoretical account of a job, and utilizing it to analyse and work out the job.

In a mathematical theoretical account, mathematical variables represented the explored system and its properties, maps are represented the activities and equations relationships.

Quasistatic theoretical accounts and Dynamic theoretical accounts represent the two major type of mathematical mold. Quasistatic theoretical accounts shows the relationships between the system attributes approximate to equilibrium. The national economic system theoretical accounts is one of quasistatic models.Dynamic theoretical accounts describe the fluctuation of maps change over the clip. The spread of a disease is one of the dynamic theoretical accounts.

Mathematical theoretical accounts are used peculiarly in the scientific disciplines and technology, such as natural philosophies, biological science, and electrinic technology but besides in the societal scientific disciplines, such as economic sciences, sociology and political scientific discipline ; physicists, applied scientists, computing machine scientists, and economic experts are the most widely used mathematical theoretical account.

3.2 Important of Mathematical Modeling

Mathematical mold is an interdisciplinary topic. Mathematicss and specializers in different Fieldss portion their cognition and experience to uninterrupted betterment on extant merchandises, make sooner develop, or predict the certain merchandise ‘s behavior.

The most of import of mold is to derive understanding. If a mathematical theoretical account is reflects the indispensable behaviour of a real-world system of involvement, we will easy to derive understanding about the system than utilizing an analysis of the theoretical account. In add-on, if we want to construct a theoretical account, we need to happen out which factors in the system are most of import, and how the different facet of the relevant system.

We need to foretell or imitate in the mathematical mold. We ever want to cognize what is the real- universe system will make in the hereafter, but it is expensive, impractical or unable to experiment straight with the system. Finally, we need to gauge the large values in the mathematical mold.

3.3 Methodology of Mathematical Modeling

Agent Based Modeling ( ABM ) and Equation Based Modeling ( EMB ) are the attacks of mathematica modeling.

ABM and EBM portion some common concerns, but in two different ways: the basic relation theoretical account between entities, and do them the degree at which they their focal point. These two attacks have recognized that the universe has two sorts of entities: observables and persons.

EBM start with a set of equations that express relationships among observables. The rating of these equations produces the development of the observables over clip. These equations may be algebraic, or they may capture variableness over clip or over clip and infinite. The modeller may acknowledge these relationships result from the meshing behaviours of the persons, but those behaviours have no obvious representation in EBM.

ABM do n’t get down with equations that relate observables to one another, but with behaviours via the interact between persons with one another. These behaviours may affect more personal straight or non straight through sharing environment. The modeller devising much attending to the observation as the theoretical account runs, and may value a inferior history of the dealingss among those observation, but the history is due to the mold and simulation of motion, non its get downing point. The modeller doing start representative of each single behaviour, so turns them over the interaction

In decision, EBM work outing the theoretical account from macroscopic degree to microscopic degree by utilizing the system of ordinary differential equations ( ODE ) and partial differential equations ( ODE ) . Besides that, ABM work outing the theoretical account from microscopic degree to macroscopic degree by utilizing the complex dynamical system ( CDS ) .

4.0 Modeling in Mathematica Programme

4.1 Introduction

Mathematica package consists of wolfram research company. Mathematica 1.0 version released on June 23, 1988. After the release in scientific discipline, engineering, media, and other Fieldss caused a esthesis, considered a radical betterment. Several months subsequently, in all over the universe have 1000s of Mathematica users. Today, in all over the universe have Mathematica 1000000s of loyal clients.

Mathematica 7 usage letters, Numberss and other mathematical symbols or inequality, constitute the equation, images or with diagrams of mathematical logic to depict the features of the system. Mathematica is studied and the motion regulations of system is a powerful tool, it ‘s analysis, design, prediction and anticipation and command existent system.

When we use Mathematica input the epidemic job, it will be use as a numerical and symbolic reckoner and print out the reply.

4.2 Graphic Interface of Mathematica

In most computing machine systems, Mathematica supports a “ notebook ” interface in which we interact with Mathematica by making synergistic paperss.

If use computing machine via a strictly graphical interface, we normally double-click the Mathematica icon to get down with the Mathematica. If use computing machine via a textually based in the operating system, we can normally input the bid mathematica to get down Mathematica.

When Mathematica starts up, it normally gives a clean notebook. When we enter Mathematica input into the notebook, so type Shift-Enter ( keep down the Shift key, so imperativeness Enter. ) to do Mathematica procedure the input.

In add-on, we besides can fix the input by utilizing the criterion redacting maps of graphical interface, which may travel on for several lines. After send Mathematica input from the notebook, Mathematica will label the input with In [ n ] : = . It labels the corresponding end product Out [ n ] = .

When type 2 + 2, so stop the input with Shift-Enter. Mathematica will treat the input, and so adds the input label In [ 1 ] : = , subsequently gives the end product.

Throughout this book, “ duologues ” with Mathematica are shown in the undermentioned manner:

With a notebook interface, we merely type in 2 + 2 and so type Shift-Enter. Mathematica so adds the label In [ 1 ] : = , and print out the consequence.

In [ 1 ] : = 2 + 2

Out [ 1 ] =

5.0 Imitating in NetLogo Programme

5.1 Introduction

NetLogo is a programmable mold environment for imitating complex scientific phenomena, both natural and societal. It is one of the most widely used multi-agent modeling tools today, with a community of 1000s of users worldwide. Its “ low-threshold, noceiling ” design doctrine is inherited from Logo. NetLogo is simple plenty that pupils and instructors can easy plan and run simulations, and advanced plenty to function as a powerful tool for research workers in many subjects. Novitiates will happen an easy-to-learn, intuitive, and well-documented scheduling linguistic communication with an elegant graphical interface.

Experts and research workers can utilize NetLogo ‘s advanced characteristics, such as automatic running experiments, 3-D support, and user expansibility. NetLogo besides includes HubNet, which prepare a web of scholars to collaboratively, explore and command a simulation. NetLogo connects NetLogo Lab by external physical devices utilizing the consecutive port, and a System Dynamics Modeler do assorted agent-based and polymerisation representations.

NetLogo has extended certification, including a library with more than 150 sample theoretical accounts in a series of sphere, tutorials, a simple vocabulary, and sample codification illustrations. This package is free and works on all major calculating platforms. Fabrication, make the system kineticss assorted agent-based

5.2 Graphic interface of NetLogo

This theoretical account simulated the transmittal and saving of all people are infected with the virus. Ecological life scientists suggested several influence factors within a population infected straight. This theoretical account is initialized with 150 people, including 10 are infected.

Peoples of the universe indiscriminately move in one of the three provinces below:

healthy but susceptible to infection ( green ) ,

sick and infective ( ruddy ) ,

healthy and immune ( Grey ) . Peoples may decease of infection or a natural decease.

The factors in this theoretical account are summarized below with an account

Controls ( BLUE ) – allow to run and command the flow of executing

1. SETUP button

resets the artworks and secret plans

distributes with 140 green susceptible people and 10 ruddy septic people

2. Travel button

get down the simulation.

Settings ( GREEN ) – allow to modify parametric quantities

3. Peoples slider

Density of the population

Population denseness frequently affect infection, immune and susceptible personal contact each other.

4.INFECTIOUSNESS skidder

Some familiar virus easy spread.

Some viruses spread from every smallest contact

Others ( illustration: the HIV virus ) require important contact before the virus transmitted.

5.CHANCE-RECOVER skidder

Population turnover.

Classify the people that had into group of susceptible, septic and immune.

Determined the opportunities of people dices of the virus or a natural decease.

All of the new born people replace those who decease.

6.DURATION skidder

Duration of infectiousness

Time of the virus infected wellness people.

Duration of a people infected before they recover or decease.

7.TICKS

Number of hebdomad in the clip graduated table.

Positions ( BEIGE ) – allow to expose information

8.OUTPUT

3 end product show show the per centum of population is infected and immune, and the figure of old ages have already passed.

I ) proctors – expose the current value of variables

two ) plots – show the history episode of a variable ‘s value

three ) artworks window – the chief thought of the NetLogo universe

The secret plan shows ( in their several colourss ) the figure of people which is susceptible, infected, and immune. It besides shows the entire figure of people in the population.

5.3 The epidemic simulated by NetLogo Programme

1. The HIV virus simulated by NetLogo Programme

Let the theoretical account is initialized with 150 people, of which 10 are infected and 1 old ages that have passed.

Figure 5.2.1

The HIV virus has a really long continuance, a really low recovery rate, but a really low infectiousness value.

2.The Ebola virus simulated by NetLogo Programme

Let the theoretical account is initialized with 150 people, of which 10 are infected and 1 old ages that have passed.

Figure 5.2.2

The celebrated Ebola virus in cardinal Africa has a really short continuance, a really high infectiousness value, and a really low recovery rate.

The celebrated Ebola hemorrhagic fever in cardinal Africa a really short clip, really high infectiousness value, an highly low recovery rate.