# Infection Spreading Under The Sir Model Biology Essay

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In the first chapter, we will discuss the introduction of epidemic. For chapter two, we will introduce the General Epidemic model which had developed by Kermack and McKendrick (1927). In this chapter, we show how to derive the model.

For chapter three, we will discuss the Mathematical Modelling. In chapter four, we will discuss epidemic model modeling in Mathematica program.

In chapter five, we will discuss SIR models simulating in NetLogo program.

In chapter four and five, we will plot the solution for the model.

Last but not least is the chapter six. In this chapter, we will carry out an interpretations and conclusion about the result of epidemic model.

2.0 Mathematical Modelling

2.1 Introduction

A mathematical model is a description of a system by using mathematical language. Mathematical model is useful in natural sciences, engineering disciplines and also in the social sciences. Natural science includes earth science, meteorology, physics and biology. Engineering disciplines include computer science, artificial intelligence. Social sciences include are psychology, political science, economics and sociology. Nowadays, many engineers, physicists, economists, statisticians and operations research analysts use mathematical models to solve the daily problem such as the epidemic problem.

In a mathematical model, mathematical variables represent the explored system and its attributes, functions are represented the activities and equations relationships. Quasi static models and dynamic models represent the two major type of mathematical modelling. Quasi static models show the relationships between the system attributes approximate to equilibrium. The national economy model is one of quasi static models. Dynamic models describe the variation of functions change over the time. The spread of a disease is one of the dynamic models.

Mathematical models are used particularly in the sciences and engineering, such as physics, biology, and electronic engineering but also in the social sciences, such as economics, sociology and political science; physicists, engineers, computer scientists, and economists are the most widely used mathematical model.

There are 6 basic types of mathematical models. There are qualitative versus quantitative, discreet versus continuous, analytical versus numerical, deterministic versus random, microscopic versus macroscopic and principles versus phenomenology. In this project, we are concentrate in microscopic versus macroscopic.

2.2 What is mathematical modelling?

Mathematical models are used to describe our daily life problem. Mathematical modelling is a process to translate our daily life problem into the language of mathematics. General speaking, the mathematical modelling is a process to develop a mathematical model of a specific problem, such as epidemic problem, and using it to analyze and solve the problem.

Building

Studying

Testing

Use

Diagram 2.1: The process of mathematical modelling

We can divide the process of mathematical modelling into four stages, there are building, studying, testing and use. Form the diagram, we found that if any defects or changes appear at the studying stage and testing stage in the process of mathematical modelling, then the studying and testing stages must be repeated and return back to the first stage, building to correct the changes or defects.

2.3 Why we translate our daily life problem into the language of mathematics?

The reasons are as below:

Nowadays, computer is important in our daily life since computer helps human to solve many complications problems such as performing the numerical calculations.

All the results that mathematicians have proved over hundreds of years are at our disposal.

Mathematics is a precise, easy understanding language and also Thus, it is more suitable to formulate ideas and make assumptions. By the way, Mathematics is easy manipulated due to well-defined rules.

2.4 The advantage of mathematical modelling

Mathematical modelling is an interdisciplinary subject. Mathematics and specialists in different fields share their knowledge and experience to improve on extant products, make preferably develop or predict the certain product's behaviour.

The most important of modelling is to gain understanding. If a mathematical model is reflects the essential behaviour of a real-world system of interest, we will easy to gain understanding about the system than using an analysis of the model. In addition, if we want to build a model, we need to find out which factors in the system are most important, and how the different aspect of the relevant system.

We need to predict or simulate in the mathematical modelling. We always want to know what is the real- world system will do in the future, but it is expensive, impractical or unable to experiment directly with the system. Finally, we need to estimate the big values in the mathematical modelling.

2.5 What objectives can mathematical modeling achieve?

The objectives that can be achieved by mathematical modeling depend on:

a) The knowledge about a system

b) How is the mathematical modelling made.

c) The effect of changes in a system;

d) Aid decisions such as tactical decisions and strategic decisions.

3.0 Method

3.1 Methodology of mathematical modelling

Agent Based Modelling (ABM) and Equation Based Modelling (EBM) are one of the basic approaches of mathematical modelling. We solve the epidemic problem with SIR Model (EBM) and agent based modelling (ABM).

Mathematica programme is such as a tool for equation based modelling (EBM) to modelling the epidemic problem. In the other hand, NetLogo program is a tool for agent based modelling (ABM) to simulating the epidemic problem.

3.2 Equation Based Modelling (EBM)

EBM is a top-down approach (is also known as step-wise design) which is essentially the breaking down of a system to gain insight into its compositional sub-systems. In a top-down approach an overview of the system is formulated, specifying but not detailing any first-level subsystems. Each subsystem is then refined in yet greater detail, sometimes in many additional subsystem levels, until the entire specification is reduced to base elements. A top-down model is often specified with the assistance of "black boxes", these make it easier to manipulate. However, black boxes may fail to elucidate elementary mechanisms or be detailed enough to realistically validate the model

EBM begins with a set of equations that express relationships among observables. The evaluation of these equations produces the evolution of the observables over time. These equations may be algebraic or they may capture variability over time (ordinary differential equations, as used in system dynamics) or over time and space (partial differential equations). The modeler may recognize that these relationships result from the interlocking behaviours of the individuals but those behaviours have no explicit representation in EBM.

EBM solving the epidemic problem using the SIR model from macroscopic level to microscopic level by using the system of ordinary differential equations (ODE) and partial differential equations (ODE).

3.3 Sir Model

3.3.1 Introduction

In 1927, W. O. Kermack and A. G. McKendrick created a model of epidemic. The independent variable for this model is time (t). Assume the population is a disjoint union, there are three compartments:

1. S = S (t), which is the number of susceptible persons

2. I = I (t), which is the number of infected persons

3. R = R (t), which is the number of recovered persons

The total population = S (t) + I (t) + R (t).

SIR model was based on the model in the spread of disease of the population. SIR model is a simple but good model of infectious diseases, such as measles, chicken-pox and rubella, which once the person infected with, will not be infecting again.

3.3.2 Assumptions of the SIR model

SIR model is based on some assumptions and suppose that the population quantity is huge and constant.

Nobody is added to the susceptible group because we ignore births and immigration. Since the only way to leave the susceptible groups will be infected persons, therefore we assume that the rate of change (time) forÂ the number of susceptible depends to the number of people who already susceptible and the number of persons that already infected and also the amount of the susceptible persons contact with infected person.

In addition, we have make hypothesis that each infected people have a fixed value contact (Î²) per day and there are enough sufficient to spread the disease. If we assume that the population is homogeneous mixing, the fraction of these contacts with susceptible isÂ S (t). Therefore, on an average, each infected person will produce Î²Â S (t)Â of new infected persons per day.

We also assume that a fixed fractionÂ Î³ in the infected group will recover gradually in any given day. For example, if the average duration of infection is four days, then on average, one-fourth of the population under infected will recovers each day.

3.3.3 SIR formulas

There are three basic dependent differential equations:

S'(t) = - Î² S(t) I(t)

I'(t) = Î² S(t) I(t) - Î³ I(t)

R'(t) = Î³ I(t)

The model starts with three basic notations. S (t) represent the number of susceptible persons at time t, I (t) represents the number of infected persons at time t, and R (t) represents the number of recovered persons at time t.

These equations describe the transitions of persons from S to I and then to R. By adding the three equations, we get the total population which we denote it with the parameter N. The size of the population is constant and equals to the initial population size. The total population equation is as below:

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

The parameter Î² is the infection rate and the parameter Î³ is the recovery rate.

The Î² and Î³ must equal or greater than zero. The term Î³ is a standard kinetic terms, based on the idea that the number of unit time to encounter between the susceptible and infectious will be proportional to the numbers value. The infection Î² is determined by both the encounter frequency and the efficiency of spreading the diseases per encounter.

3.3.4 Dynamics

In the SIR framework, we will find that a group of people from the susceptible group will move to the infection group, and also a group of people from the infection group will move to the recovered group. î‚¸

The diagram of SIR model

Î² S(t) I(t) R(t) I(t)

susceptible

infected

recovered

Figure 3.1: The diagram of SIR model

The person possibly moves from the susceptible to the infected group when someone contacts with an infected person. Each epidemic disease has its own infectious way. For example, HIV virus' infectious way is sexual contact or blood transfusion, the infectious way of Ebola virus is infected body's funeral.

3.3.5 Derivations of the SIR model

The model is described by three ordinary differential equations:

For the susceptible differential equation,

When we plot the graph of S (t) versus t with Î² and Î³ are a constants, which is a negative exponential relationship between S and t. Since S (0) â‰ 0, when t = 0, thus the graph will started with the population size.

Figure 3.2: The graph of Susceptible Population versus Time

Figure 3.2 show that the susceptible population start from very large number which is almost equal to the total population. Since the number of infected people increases, and the disease spreads at quickly rate; therefore the number people of susceptible group decrease. At last, the number of peoples remaining in the susceptible population stop until the susceptible close to the equilibrium. The equilibrium is the lowest value of susceptible which is no more infectious diseases.

For the infected differential equation,

When we plot the graph of I (t) versus t with Î² and Î³ is a constants, which is an exponential relationship between I and t. Since I (0) = 0 when t = 0, thus the graph will start with value 0.

Figure 3.3: The graph of Infected Population versus Time

Figure 3.3 shows the inflected population increase rapidly until the susceptible close to the equilibrium. It will proceed to decrease to a bit less than the recovery.

The recovered differential equation,

When we plot the graph of R (t) versus t with Î² and Î³ are constants, very clearly, that is a linear relationship between R and t. Since R (0) = 0 when t = 0, the graph will start with value 0.

Figure 3.4: The graph of Recovered Population versus Time

Figure 3.4 shows the recovered population versus time. Initially, the size of the recovered population growth at a constantly decreasing rate until the entire inflected population had been recovered. The figure show, initially, the rate of recovery is high, but as time goes on, the rate slow down until it is almost constant.

iv. Vector notations

If solving with numerical values for the constants a andb, using vector notation can make the system easier to deal with.

Let

Then,

3.4 How to use Mathematica programme

3.4.1 Introduction

Wolfram Research Company develops and design Mathematica programme. Mathematica 1.0 version was released on June 23, 1988. After it has been released in science, technology, media, and other fields, it caused a sensation and is considered a revolutionary improvement. Several months later, all over the world have thousands of Mathematica users. Today, all over the world have Mathematica millions of loyal customers.

Mathematica start with a set of equations that express relationships among observables. The evaluation of these equations produces the evolution of the observables over time. These equations can algebraic or they may capture variability over time or over time and space.

The simplest way to use Mathematica is like a calculator. The range of calculations with Mathematica is however far greater than with a traditional electronic calculator. A traditional system might support perhaps 30 mathematical operations; Mathematica has over 750 built in. In addition, while traditional systems handle only numerical computations, Mathematica is not only handle numerical computations, it also handles symbolic and graphical computations.

When we use Mathematica and input the epidemic problem, it will be use as a numerical and symbolic calculator and print out the answer.

3.4.2 Graphical interface of Mathematica programme

Mathematica systems are divided into two parts: the kernel and the front end. The kernel is actually performs computations and run the Mathematica. The front end is handles interaction with the user.

The front end is optimized for particular computers and graphical user interfaces. On the other hand, the front end for Mathematica supports a "notebook" interface in which we interact with Mathematica by creating interactive documents. If use computer via a purely graphical interface, we usually double-click the Mathematica icon to start with the Mathematica. If use computer via a textually based in the operating system, we can usually input the command Mathematica to start Mathematica.

When Mathematica starts up, it usually gives a blank notebook. When we enter Mathematica input into the notebook, then type Shift-Enter (hold down the Shift key, then press Enter.) to make Mathematica process the input.

In addition, we also can prepare the input by using the standard editing functions of graphical interface, which may go on for several lines. After send Mathematica input from the notebook, Mathematica will label the input with In [n]:=. It labels the corresponding output Out [n] =.

When we type 2 + 2, then end the input with Shift-Enter. Mathematica will process the input, and then adds the input label In [1]:=, later gives the output is as follow:

Figure 3.5: The diagram of Mathematica's notebook

Mathematica use as numerical computations

Example: Find the numerical value of log (8Ï€)

The N tells Mathematica that you want a numerical result.

Here is log (8Ï€) to 40 decimal places.

Figure 3.6: The diagram of Mathematica's Numerical computations notebook

Mathematica can deal with numerical data, allowing us to do statistical and other analysis, and perform operations such as Fourier transforms interpolation and least-square fitting.

Mathematica can do numerical computations, it not only with individual numbers, but also owns items such as matrices. It supports linear algebra operations such as matrix inversion method and characteristics of system computation.

Mathematica can do numerical operations on functions, such as numerical integration, numerical minimization, and linear programming. It also can produce numerical solutions to both algebraic equations.

ii. Mathematica use as symbolic computation

Example: Find a formula for the integral

Here is the expression x^3/(x-1) in Mathematica.

This tells Mathematica to integrate the previous expression. Mathematica finds an explicit formula for the integral.

Figure 3.7: The diagram of Mathematica's Symbolic computations notebook

One of the main classes of calculations made possible by Mathematica's symbolic computation capabilities is those involving the manipulation of algebraic formulas. Mathematica can solve many types of algebraic operations. It can expand, factor and simplify polynomials and rational expressions. It also can solve the solutions of algebraic to polynomial equations and systems of equations.

Mathematica can solve calculus. It can evaluate derivatives and integrals symbolically and solve ordinary differential equations. It can derive and manipulate power series approximations, and find the limits. Standard Mathematica also cover areas such as vector analysis and Laplace transforms.

iii. Mathematica use as graphical computations.

Example: Plot the function Cos (x y) for x and y between 0 and Ï€.

This generates a three-dimensional plot of Cos (x y) as a function of x and y. There are many options for controlling graphics in Mathematica.

Figure 3.8: The diagram of Mathematica's 3D graph notebook

Mathematica can plot 2 and 3 dimensional graphics, as well as contour and density of the plot.

Mathematica provides many options for controlling the details of graphics output. In three dimensions, we can control material qualitative, colour, lighting, surface shininess and other parameters. Many versions of Mathematica also support the animation graphics.

3.4.3 The epidemic problem modelling by using Mathematica programme

Euler Method

The numerical solution method such as Euler's Method can be used to solve the epidemic problem. Given the initial value problem

Euler's Method with step size consists of applying the iterative formula

to calculate successive approximations

Let the initial value is starting with the initial point,Â and computing each rise as slopeÂ timesÂ run. Thus,

whereÂ is slope and is a suitably small step size in the time domain.

The SIR model, we want the dependent variable names to beÂ S,Â I,Â and R.Â Thus we have three Euler formulas of the form

Given that the SIR equations,

Thus, the Euler formulas become

To calculate something from these formulas, we must have explicit values for Â ,Â ,Â S(0),Â I(0),Â R(0),Â and Â .

The Graphical Solution to the SIR Model

To show a solution to the SIR model, we try to plot the differential equations with value b = Î³ = 1 and let the initial value S (0) = 759, I (0) = 1 and R (0) = 0.

Then

And the Euler equation will become

When we use Mathematica programme and input the equation, it will be print out the answer as follow.

Figure 3.9: The graph of SIR Population versus Time

The SIR populations versus time give the output. The infected is proportional to the change in time, the number of infected and the number of susceptible. The change in the infected population will increase from the susceptible group and decrease into the recovered group.

3.5 Agent Based Modelling (ABM)

ABM is a bottom-up approach which is the piecing together of systems to give rise to grander systems, thus making the original systems sub-systems of the emergent system. In a bottom-up approach the individual base elements of the system are first specified in great detail. These elements are then linked together to form larger subsystems, which then in turn are linked, sometimes in many levels, until a complete top-level system is formed. This strategy often resembles a "seed" model, whereby the beginnings are small but eventually grow in complexity and completeness. However, "organic strategies" may result in a tangle of elements and subsystems, developed in isolation and subject to local optimization as opposed to meeting a global purpose.

ABM not begins with equations that relate observables to one another, but with behaviours through which individuals interact with one another. These behaviours may involve multiple individuals directly (foxes eating rabbits) or indirectly through a shared environment (horses and cows competing for grass). The modeler begins by representing the behaviours of each individual, and then turns them loose to interact.

ABM solving the epidemic problem using the SIR model from microscopic level to macroscopic level by using the complex dynamical system (CDS).

3.6 How to use NetLogo program

3.6.1 Introduction

NetLogo start with the behaviours via the interaction between individuals with another one. These behaviours may involve more personal directly or not directly through sharing environment.

NetLogo is a programmable modelling environment for simulating complex scientific phenomena, both natural and social. It is one of the most widely used multi-agent modelling tools today, with a community of thousands of users worldwide. Its "low-threshold, no ceiling" design philosophy is inherited from Logo. NetLogo is simple enough that students and teachers can easily design and run simulations, and advanced enough to serve as a powerful tool for researchers in many disciplines. Novices will find an easy-to-learn, intuitive, and well-documented programming language with an elegant graphical interface.

Experts and researchers can use NetLogo's advanced features, such as automatic running experiments, 3-D support, and user expansibility. NetLogo also includes Hub Net, which prepare a network of learners to collaboratively, explore and control a simulation. NetLogo connects NetLogo Lab by external physical devices using the serial port, and a System Dynamics Modeller make mixed agent-based and polymerization representations.

NetLogo has extensive documentation, including a library with more than 150 sample models in a series of domain, tutorials, a simple vocabulary, and sample code examples. This software is free and works on all major computing platforms.

3.6.2 Graphical interface of NetLogo

This model simulated the transmission and preservation of all people was infected with the virus. Ecological biologists suggested several influence factors within a population infected directly. This model is initialized with 150 people, including 10 are infected.

People of the world randomly move in one of the three states below:

healthy but susceptible to infection (green),

sick and infectious (red),

Healthy and immune (grey). People may die of infection or a natural death.

The factors in this model are summarized below with an explanation

Figure 3.10: The Graphical Interface of NetLogo

Controls (BLUE) - allow to run and control the flow of execution

1. SETUP button

resets the graphics and plots

distributes with 140 green susceptible people and 10 red infected people

2. GO button

Start the simulation.

Settings (GREEN) - allow to modify parameters

3. PEOPLE slider

Density of the population

Population density often affect infection, immune and susceptible personal contact each other.

4. INFECTIOUSNESS slider

Some familiar virus easily spread.

Some viruses spread from every smallest contact

Others (example: the HIV virus) require significant contact before the virus transmitted.

5. CHANCE-RECOVER slider

Population turnover.

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

Determined the chances of people die of the virus or a natural death.

All of the new born people replace those who death.

6. DURATION slider

Duration of infectiousness

Time of the virus infected health people.

Duration of a people infected before they recover or death.

7. TICKS

Number of week in the time scale.

Views (BEIGE) - allow to display information

8. OUTPUT

3 output display show the percent of population is infected and immune, and the number of years have already passed.

i) Monitors - display the current value of variables

ii) Plots - show the history episode of a variable's value

iii) Graphics window - the main idea of the NetLogo world

The plot shows (in their respective colors) the number of people which is susceptible, infected, and immune. It also shows the total number of people in the population.

4.0 Result

4.1 Mathematica result

The H1N1 virus modelling in Mathematica

Figure 4.1: The graph of H1N1 virus via Mathematica

Analysis

The population is N = 760. The susceptible group is decreased when the infected group is increased in first 6 days. The seventh day, since the medicine of H1N1 is found, the infected group is decreased gradually and the recovered group increases rapidly. In first 6 days, the recovered group increases gradually since some infected people recover themselves.

4.2 NetLogo result

Simulating the H1N1 virus in NetLogo programme

Let the model is initialized with 760 people, of which 10 are infected and 52 weeks that have passed.

Figure 4.2: The graph of H1N1 virus via NetLogo

Analysis

The H1N1 virus has a very short duration, a very high infectiousness value, and a very low recovery rate.

The population is N = 760. The susceptible group is decreased when the infected group is increased in first 6 days. After a few days, since the medicine of H1N1 is found, the infected group is decreased gradually and the recovered group increases rapidly. At the beginning, the recovered group increases gradually since some infected people recover themselves.

5.0 Conclusion

5.1 Discussion

Research question

What is an epidemic?

Epidemic is a widespread outbreak of an infectious disease and it attacks many people at the same time in an area or community. Many cases of a common disease, such as the common cold would not classified as an epidemic. Only a few cases of a very rare disease are classified as an epidemic.

According to the World Health Organization (WHO), epidemic disease is a pathogen which can spread easily, and it can lead to a serious complications. Generally speaking, the epidemic disease is stimulated by some powerful pathogenic microorganisms.

Historically speaking, terrible epidemics had reoccurs over and over again. In the past, there are many people were killed by infectious diseases and the amount of death people is more the war, for example, the Black Death that occurred in Europe in year 1347, there were one-third of the people in cities and one-half of the people in towns were killed by the Black Death.

Luckily, some epidemic diseases, such as the smallpox, plague, and influenza have been persisted in the history. Smallpox was uprooted worldwide by 1980. Nowadays, we have many ways to avoid and control the epidemic disease to prevent the epidemics reoccurs again.

Can an epidemic be avoided or control?

How to prevent epidemic?

Specific for example H1N1, HIV, Ebola

Some epidemic has been control? Small pox, Black Death, lepra, lung (pulmonary) tuberculosis.

How control Small pox, Black Death, lepra, lung (pulmonary) tuberculosis.

What is the equation-based model and agent-based model?

Equation based modelling (EBM)

EBM is a top-down approach which is essentially the breaking down of a system to gain insight into its compositional sub-systems. In a top-down approach an overview of the system is formulated, specifying but not detailing any first-level subsystems. Each subsystem is then refined in yet greater detail, sometimes in many additional subsystem levels, until the entire specification is reduced to base elements. A top-down model is often specified with the assistance of "black boxes", these make it easier to manipulate. However, black boxes may fail to elucidate elementary mechanisms or be detailed enough to realistically validate the model.

EBM begins with a set of equations that express relationships among observables. The evaluation of these equations produces the evolution of the observables over time. These equations may be algebraic or they may capture variability over time (ordinary differential equations, as used in system dynamics) or over time and space (partial differential equations). The modeler may recognize that these relationships result from the interlocking behaviours of the individuals but those behaviours have no explicit representation in EBM.

EBM solving the epidemic problem using the SIR model from macroscopic level to microscopic level by using the system of ordinary differential equations (ODE) and partial differential equations (ODE).

Agent based modelling (ABM)

ABM is a bottom-up approach which is the piecing together of systems to give rise to grander systems, thus making the original systems sub-systems of the emergent system. In a bottom-up approach the individual base elements of the system are first specified in great detail. These elements are then linked together to form larger subsystems, which then in turn are linked, sometimes in many levels, until a complete top-level system is formed. This strategy often resembles a "seed" model, whereby the beginnings are small but eventually grow in complexity and completeness. However, "organic strategies" may result in a tangle of elements and subsystems, developed in isolation and subject to local optimization as opposed to meeting a global purpose.

ABM begins, not with equations that relate observables to one another, but with behaviours through which individuals interact with one another. These behaviours may involve multiple individuals directly (foxes eating rabbits) or indirectly through a shared environment (horses and cows competing for grass). The modeler begins by representing the behaviours of each individual, then turns them loose to interact.

ABM solving the epidemic problem using the SIR model from microscopic level to macroscopic level by using the complex dynamical system (CDS).

Which is more suitable to solve the epidemic problem, equation-based model or agent-based model?

Both approaches simulate the system by constructing a model and executing it on a computer. The differences are in the form of the model and how it is executed.

In agent-based modeling (ABM), the model consists of a set of agents that encapsulate the behaviours of the various individuals that make up the system and execution consists of emulating these behaviors, which is essentially dynamic. In equation-based modeling (EBM), the model is a set of equations (pre-determined static) and execution consists of evaluating them. Thus "simulation" is a general (umbrella) term that applies to both methods, which are distinguished as Agent-based emulation and equation-based evaluation.

In my opinion, agent-based model is more suitable to solve the epidemic problem due to those reasons:

Agent-based model allows the user to define ways of interacting with the simulation.

The ABM programming language is intuitive and very well documented, with a great deal of built-in functionality. The interface is easily customizable and allows for adjustment of the program by people who are not familiar with the language.

The ABM =allows for recording videos of the simulation and exporting plots.

The ABM is easier to construct and distinguish physical space from interaction space.

The ABM offers an additional level of validation.

The ABM support more direct experimentation.

The ABM is easier to translate back into practice.

The ABM gives more realistic results than EBM in many domains.

In conclusion, agent-based model is suitable to solve the epidemic problem.

What are the significant parameters that govern the two models?

SIR model

Agent Based Modelling

t - time

Duration slider

S (t) - susceptible persons at time t

I (t) - infected persons at time t

Infectiousness slider

R (t) - recovered persons at time t.

Chance-recover slider

Î²is the infectionrate

Î³is the recovery rate

N - population

People slider - density of the population

Figure 4.3: The table of parameter

These two approaches in order to show the parameters that govern the two models.

These approaches share the following parameters: time or duration, - infected persons at time, recovered persons at time and the density of the population.These approaches differ in the following parameters: SIR model have susceptible persons at time, the infection rate and recovery rate but agent based model did not have.

These similarities and differences relate to the essential meaning of the approaches because SIR model begins with a set of equations that express relationships among observables but ABM begins with behaviors through which individuals interact with one another.

Although SIR model and agent based model don't seem to have anything in common, in actuality, they both will produce a similarities result.

5.2 Conclusion

In efforts to control the spread of the disease, we must select the optimal solution for the maximum public health benefits. Mathematical models can help us to better understand the spread of an infectious disease and to test the control strategies.

In this project, the epidemic problem can be solved by using SIR model and through Mathematica program and simulating the epidemic problem by using agent based modelling through NetLogo program.

EBM start with a set of equations that express relationships among observables. The evaluation of these equations produces the evolution of the observables over time. These equations may be algebraic or they may capture variability over time or over time and space. The modeller may recognize these relationships result from the interlocking behaviours of the individuals, but those behaviours have no obvious representation in EBM.

EBM is a top-down approach. A top-down approach (is also known as step-wise design) is essentially the breaking down of a system to gain insight into its compositional sub-systems. In a top-down approach an overview of the system is formulated, specifying but not detailing any first-level subsystems. Each subsystem is then refined in yet greater detail, sometimes in many additional subsystem levels, until the entire specification is reduced to base elements. A top-down model is often specified with the assistance of "black boxes", these make it easier to manipulate. However, black boxes may fail to elucidate elementary mechanisms or be detailed enough to realistically validate the model.

ABM did not start with equations that relate observables to one another, but with behaviours via the interaction between individuals with another one. These behaviours may involve more personal directly or not directly through sharing environment. The modeller making much attention to the observation as the model runs and may value a inferior account of the relations among those observation, but the account is due to the modelling and simulation of movement, not its starting point. The modeller making start representative of each individual behavior then turns them over the interaction.

ABM is a bottom-up approach. A bottom-up approach is the piecing together of systems to give rise to grander systems, thus making the original systems sub-systems of the emergent system. In a bottom-up approach the individual base elements of the system are first specified in great detail. These elements are then linked together to form larger subsystems, which then in turn are linked, sometimes in many levels, until a complete top-level system is formed. This strategy often resembles a "seed" model, whereby the beginnings are small but eventually grow in complexity and completeness. However, "organic strategies" may result in a tangle of elements and subsystems, developed in isolation and subject to local optimization as opposed to meeting a global purpose.

In conclusion, EBM solving the epidemic problem using the SIR model from macroscopic level to microscopic level by using the system of ordinary differential equations (ODE) and partial differential equations (ODE). Besides that, ABM solving the epidemic problem using the SIR model from microscopic level to macroscopic level by using the complex dynamical system (CDS).

Mathematica program is a programming languages and a platform for equation based modelling (EMB). Mathematica program is a general computer software system and language that is used in Mathematic and other applications. Mathematica 7 use letters, numbers and other mathematical symbols or inequality and constitute the equation, images or with diagrams of mathematical logic to describe the characteristics of the system. Mathematica is studied and the movement rule of system is a powerful tool, it can analysis, design, forecasting and prediction and control the actual system.

Mathematica program is not just use for computation, it also can use for modelling, simulation, development and deployment, visualization and documentation. Mathematica computations can be divided into 3 main classes which are Numeric, Graphical and Symbolic.

Different jobs dealing with different things, but the Mathematica program is a comprehensive system to provide unprecedented workflow, reliability, sustainability and innovation. In this project, Mathematica program is used as a modelling and used for analysis the rate of epidemic. The question can be answered by creating the model of an epidemic with variables which are corresponding to the different reaction of a population and the characteristics of a virus.

NetLogo is a multi-agent programming language and is integrated with modelling environment and a platform for agent based modelling. NetLogo is the most suitable programme for complex system to modelling development. Model can guide hundreds or thousands of "agent" all operating independently.

NetLogo also lets students to simulate and "play", students can explore their behaviors in different conditions. NetLogo has extensive files and tutorial. It also comes with a model library, which is a large collection of pre-written simulation which can be used and modification.

If an epidemic occurs, the variables corresponding to a population reaction and characteristics of disease will affect its duration and severity. In NetLogo programming, system dynamics can use a unique programming. To determine the influence of various factors on the duration and serious infectious disease, we can change the variable and look at the shape of the graph differs between runs in NetLogo.

In conclusion, The NetLogo is more suitable to solve the epidemic problem because NetLogo allows the user to define ways of interacting with the simulation, and NetLogo also led to the discovery of a programming environment.