SIR Model of Epidemics - Investigation

1.1 Introduction

Epidemic is applied to a disease which, spreading widely, attacks many person at the same time. Epidemic is a widespread outbreak of an infectious disease. When an epidemic exists, it will affect many human populations. There were many factors to simulate the rise of epidemics such as poor population health, immigration and failure of public health programs.

According to the world health organization appeared, disease should have the following conditions: a new pathogens, disease can lead to infection causes a serious complications, pathogen spread easily, especially in interpersonal communication. Generally speaking, this disease is due to some powerful pathogenic microorganisms, and the infections caused by viruses, bacteria.

Historically speaking, terrible epidemics have reoccurs over and over again. Some epidemic diseases, such as the smallpox, plague, and influenza, have been persisted in the history. Smallpox was uprooted worldwide by 1980. In the 18th century, the world's major trade routes, several destructive appear cholera pandemic caused great infectious diseases.

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In the past worldwide, the main cause of many people were killed is infectious diseases and there are more deaths 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 cities and towns respectively, this ill-condition seized the progress of civilization for several generations.

1.2 Objectives

• To study the SIR model (developed by Kermack and McKendrick) in the form of the system of nonlinear ordinary differential equations.
• To solve the SIR model numerically using Mathematica and simulating the Agent Based Modeling using NetLogo.
• To investigate the infection spread under the SIR model
• To interpret the results of epidemic problem based on these two model.

1.3 Research Question

• What is an epidemic?
• Can an epidemic be avoided or control?
• Which is better to solve the SIR model, Equation based Model or agent-based model?
• What are the significant parameters that govern the two models?

1.4 Epidemic Problem modeling in Mathematica Program

Mathematica program is a general computer software system and language in used for Mathematica program and other applications. Mathematica program not just for use in computation, it also use for modeling, simulation, development and deployment, visualizationand 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 modeling and data analysis the rate of epidemic. The question can be answered by creating the model of an epidemic with variables corresponding to the different reaction of a population and the characteristics of a virus.

1.5 Epidemic Problem Simulating in NetLogo

NetLogo is multi-agent programming languages and integrated modeling environment and a platform specifically for agent-based modeling. NetLogo is most suitable for complex system modeling development. Model can guide hundreds or thousands of "agent" all operating independently.

NetLogo also let students can simulate and "play", explore their behavior 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, it 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 efforts to control the spread of the disease, we must choose an optimal solution for the maximum public health benefits. In NetLogo programming, system dynamics can use a unique programming. To determine the influence of various factors have 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.

1.6 Prilimanary Research

The Ebola virus modeling in Mathematica 7

The Ebola virus simulating in NetLogo

1.7 Chapter summary

This dissertation is divided into five chapters. In the first chapter, we discuss the introduction of epidemic. For chapter two, we introduce the General Epidemic model by Kermack and McKendrick (1927). In this chapter, we show how to derive the model. For chapter three, we discuss the Mathematical Modeling. 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. Finally is chapter six. In this chapter, we will do an interpretations and conclusion about the result of epidemic model.

2.0 SIR Model

2.1 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 dependent Variables:

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.

2.2 Assumptions of the SIR model

SIR model is based on some assumptions. Suppose the population quantity is huge and constant. Because we ignore births and immigration, thus nobody is added to the susceptible group. Since the only way to leave the susceptible groups will be infected, we assume the time-rate of change for the number of susceptible depends to the number of people who already susceptible, the number of persons that already infected and the amount of the susceptible persons contact with infected person.

In addition, we are hypothesis each infected people have a fixed value β contact per day, and there are enough sufficient to spread the disease. Not all these contacts are with susceptible people. If we assume that the population is homogeneous mixing, the fraction of these contacts that are with susceptible is S(t). Therefore on an average, each infected person will produce β S(t) of new infected persons per day.

We also assume that a fixed fraction γ 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.

2.3 SIR Formulas

There are three basic dependent differential equations:

S'(t) = - β S(t) I(t)

I'(t) = β S(t) I(t) - γ I(t)

R'(t) = γ I(t)

The model starts with some basic notation. That are S(t) is the number of susceptible persons at time t, I(t) is the number of infected persons at time t, and R(t) is the number of recovered persons at time t

These equations describe the transitions of persons from S to I to R. By adding the three equations, the size of the population is constant and equal to the initial population size, which we denote with the parameter N. Therefore the total population

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

We call the parameter β the infection rate and the parameter γ the recovery rate with β and γ must equal or greater to zero. The term γ 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 β is determined by both the encounter frequency and the efficiency of spreading the diseases per encounter.

2.4 Dynamics

If we imagine the process in a disease that a very suitable for the SIR framework, we will get a flow of people from the susceptible group will move to the infection group, then to the removed group. 

The diagram of SIR model

β S I R I

susceptible

infected

recovered

Diagram 2.4.1

The person possibly moves from the susceptible to the infected group when somebody comes in contact with an infected person. 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 infected body's funeral, and contact with infected persons without exercise proper cautious.

2.5 Derivations of the SIR model

The model is described by three ordinary differential equations:

For the susceptible differential equation,

When we plotting the graph of S(t) versus t with β and γ is a constants, that is a negative exponential relationship between S and t. Since S(0) ≠ 0 when t = 0, , thus the graph will started with ∞.

The graph of S versus t

2. For the infected differential equation,

When we plotting the graph of I(t) versus t with β and γ is a constants, that is a exponential relationship between I and t. Since I(0) ≠ 0 when t = 0, , thus the graph will not started with 0.

The graph of I versus t

Figure 2.5.2

3. For the recovered differential equation,

When we plotting the graph of R(t) versus t with β and γ is a constants, very clearly , that is a linear 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 solving with numerical values for the constants a and b, using vector notation can make the system easier to deal with.

Let

then

2.6 A Graphical Solution to the SIR Model

To show a solution to the SIR model, we try to plot the differential equations with value a = b = 1 and let 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 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 increase from the susceptible group and decrease into the recovered group.

3.0 Mathematical Modeling

3.1 Introduction

Mathematical modeling is a replacement of an object studied by its image. The mathematical modeling is the method of creating a mathematical model of a problem, and using it to analyze and solve the problem.

In a mathematical model, mathematical variables represented the explored system and its attributes, functions are represented the activities and equations relationships.

Quasistatic models and Dynamic models represent the two major type of mathematical modeling. Quasistatic models shows the relationships between the system attributes approximate to equilibrium. The national economy models is one of quasistatic 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 electrinic 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.

3.2 Important of Mathematical Modeling

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

The most important of modeling is to gain understanding. If a mathematical model is reflects the essential behavior 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 modeling. 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 modeling.

3.3 Methodology of Mathematical Modeling

Agent Based Modeling (ABM) and Equation Based Modeling (EMB) are the approaches of mathematica modelling.

ABM and EBM share some common concerns, but in two different ways: the basic relation model between entities, and make them the level at which they their focus. These two approaches have recognized that the world has two kinds of entities: observables and individuals.

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 modeler may recognize these relationships result from the interlocking behaviors of the individuals, but those behaviors have no obvious representation in EBM.

ABM don't start with equations that relate observables to one another, but with behaviors via the interact between individuals with one another. These behaviors may involve more personal directly or not directly through sharing environment. The modeler 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 modeling and simulation of movement, not its starting point. The modeler making start representative of each individual behavior, then turns them over the interaction

In conclusion, EBM solving the 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 model from microscopic level to macroscopic level by using the complex dynamical system (CDS).

4.0 Modeling in Mathematica Programme

4.1 Introduction

Mathematica software consists of wolfram research company. Mathematica 1.0 version released on June 23, 1988. After the release in science, technology, media, and other fields caused a sensation, considered a revolutionary improvement. Several months later, in all over the world have thousands of Mathematica users. Today, in all over the world have Mathematica millions of loyal customers.

Mathematica 7 use letters, numbers and other mathematical symbols or inequality, constitute the equation, images or with diagrams of mathematical logic to describe the characteristics of the system. Mathematica is studied and the movement rules of system is a powerful tool, it's analysis, design, forecasting and prediction and control actual system.

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

4.2 Graphical Interface of Mathematica

In most computer systems, 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 type 2 + 2, then end the input with Shift-Enter. Mathematica will processes the input, and then adds the input label In[1]:=, later gives the output.

Throughout this book, "dialogs" with Mathematica are shown in the following way:

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

In[1]:= 2 + 2

Out[1]=

5.0 Simulating in NetLogo Programme

5.1 Introduction

NetLogo is a programmable modeling 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, noceiling" 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 HubNet, 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 Modeler 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. Manufacturing, make the system dynamics mixed agent-based

5.2 Graphical interface of NetLogo

This model simulated the transmission and preservation of all people are 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

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.

5.3 The epidemic simulated by NetLogo Programme

1. The HIV virus simulated by NetLogo Programme

Let the model is initialized with 150 people, of which 10 are infected and 1 years that have passed.

The HIV virus has a very long duration, a very low recovery rate, but a very low infectiousness value.

2. The Ebola virus simulated by NetLogo Programme

Let the model is initialized with 150 people, of which 10 are infected and 1 years that have passed.

The famous Ebola virus in central Africa has a very short duration, a very high infectiousness value, and a very low recovery rate.

The famous ebola in central Africa a very short time, very high infectiousness value, an extremely low recovery rate.