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# SIR Style of Epidemics - Investigation

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## 1. 1 Introduction

Epidemic is applied to a disease which, spreading greatly, disorders many person at the same time. Epidemic is a common outbreak of infectious disease. When an epidemic is present, it will have an impact on many individual populations. There were many factors to simulate the climb of epidemics such as poor human population health, immigration and inability of open public health programs.

According to the planet health organization appeared, disease should have the following conditions: a new pathogens, disease can result in infection causes a significant complications, pathogen propagate easily, especially in interpersonal communication. In most cases, this disease is due to some powerful pathogenic microorganisms, and the microbe 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 annals. Smallpox was uprooted worldwide by 1980. Inside the 18th century, the world's major trade routes, several destructive appear cholera pandemic triggered great infectious diseases.

In the past worldwide, the main reason behind many individuals were wiped out is infectious diseases and there are more deaths than all the conflict, including the Black Fatality that struck European countries in 1347 acquired remove between one-third and one-half of folks in many metropolitan areas and towns respectively, this ill-condition seized the improvement of civilization for a number of generations.

### 1. 2 Objectives

• To study the SIR model (developed by Kermack and McKendrick) by means of the machine of nonlinear standard differential equations.
• To solve the SIR model numerically using Mathematica and simulating the Agent Based mostly Modeling using NetLogo.
• To investigate the problem pass on under the SIR model
• To interpret the results of epidemic problem based on both of these model.

### 1. 3 Research Question

• What can be 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 guidelines that govern the two models?

### 1. 4 Epidemic Problem modeling in Mathematica Program

Mathematica program is an over-all computer software system and terms 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 records. Mathematica computations can be split into 3 main classes which can be Numeric, Graphical and Symbolic.

Different jobs dealing with different things, however the Mathematica program is a comprehensive system to provide unprecedented workflow, consistency, sustainability and advancement. In this task Mathematica program can be used as a modeling and data analysis the speed of epidemic. The question can be clarified by creating the model of an epidemic with factors corresponding to the various reaction of a human population and the characteristics of your virus.

### 1. 5 Epidemic Problem Simulating in NetLogo

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

NetLogo also let students can simulate and "play", explore their patterns in different conditions. NetLogo has intensive files and article. It also comes with a model library, which is a big collection of pre-written simulation, it could be used and adjustment.

If an epidemic occurs, the variables matching to a population reaction and characteristics of disease will affect its length of time and severeness. In efforts to regulate the spread of the disease, we must choose an optimum solution for the maximum public health advantages. In NetLogo coding, system dynamics can use a unique programming. To look for the influence of various factors have on the length of time and serious infectious disease, we can transform the variable and appearance at the condition of the graph differs between runs in NetLogo.

### 1. 6 Prilimanary Research

The Ebola pathogen modeling in Mathematica 7

The Ebola virus simulating in NetLogo

### 1. 7 Section summary

This dissertation is divided into five chapters. Inside the first chapter, we discuss the launch of epidemic. For section two, we create the overall Epidemic model by Kermack and McKendrick (1927). On this section, we show how to derive the model. For section three, we discuss the Mathematical Modeling. In section four, we will discuss epidemic model modeling in Mathematica program. In section five, we will discuss SIR models simulating in NetLogo program. In section four and five, we will plot the solution for the model. Finally is section six. With this chapter, we can do an interpretations and bottom line 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 indie variable for this model is time (t). Presume the population is a disjoint union, there are three based mostly Variables:

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

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

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

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

SIR model was based on the model in the spread of disease of the populace. SIR model is a straightforward but good style of infectious diseases, such as measles, chicken-pox and rubella, which after the person attacked with, will not be infecting again.

### 2. 2 Assumptions of the SIR model

SIR model is based on some assumptions. Imagine the population quantity is huge and constant. Because we ignore births and immigration, thus no one is added to the prone group. Because the only way to leave the prone teams will be contaminated, we believe the time-rate of change for the number of susceptible depends to the number of people who already prone, the number of people that already afflicted and the amount of the susceptible persons contact with infected person.

In addition, we live hypothesis each afflicted people have a set value contact each day, and there are enough sufficient to disperse the disease. Not absolutely all these connections are with prone people. If we suppose that the populace is homogeneous combining, the fraction of these connections that are with susceptible is S(t). Therefore on the average, each afflicted person will produce S(t) of new contaminated persons per day.

We also expect that a preset portion in the afflicted group will recover gradually in any given day. For example, if the common duration of infection is four times, then typically, one-fourth of the populace under contaminated will recovers every day.

### 2. 3 SIR Formulas

There are three basic dependent differential equations:

S'(t) = - S(t) I(t)

I'(t) = S(t) I(t) - I(t)

R'(t) = I(t)

The model starts with some basic notation. That happen to be S(t) is the amount of susceptible folks at time t, I(t) is the number of infected persons at time t, and R(t) is the number of recovered individuals at time t

These equations explain the transitions of persons from S to I to R. With the addition of the three equations, how big is the populace is continuous and equal to the initial inhabitants size, which we denote with the parameter N. Which means total population

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

We call the parameter the infection rate and the parameter the restoration rate with and must equal or better to zero. The word is a typical kinetic terms, based on the theory that the number of unit time for you to encounter between the susceptible and infectious will be proportional to the amounts value. The infection depends upon both the come across consistency and the efficiency of dispersing the diseases per encounter.

### 2. 4 Dynamics

If we envision the procedure in an illness that a very suited to the SIR construction, we will get a flow of individuals from the susceptible group will proceed to the infection group, then to the removed group. ё

The diagram of SIR model

S I R I

susceptible

infected

recovered

Diagram 2. 4. 1

The person possibly goes from the susceptible to the afflicted group when somebody touches an contaminated person. Qualify as a contact in the population, depends on the disease. For HIV disease a contact may be sexual contact or a bloodstream transfusion. For Ebola virus it connection with infected body's funeral, and connection with infected people without exercise proper careful.

### 2. 5 Derivations of the SIR model

The model is described by three ordinary differential equations:

For the prone differential equation,

When we plotting the graph of S(t) versus t with and 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 out with Л†.

The graph of S versus t

2. For the infected differential equation,

When we plotting the graph of I(t) versus t with and is a constants, that is a exponential romance between I and t. Since I(0) 0 when t = 0, , thus the graph will not began 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 and is a constants, very plainly, that is clearly a linear marriage 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 handling with numerical prices for the constants a and b, using vector notation can make the machine easier to offer with.

Let

then

### 2. 6 A Graphical Means to fix 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 original 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 supply the output. The infected is proportional to the change in time, the amount of infected and the amount of vulnerable. The change in the contaminated society increase from the vulnerable group and decrease in to the recovered group.

## 3. 0 Mathematical Modeling

### 3. 1 Introduction

Mathematical modeling is an upgraded of an object examined by its image. The mathematical modeling is the technique of fabricating a mathematical style of problems, and using it to investigate and solve the trouble.

In a mathematical model, mathematical factors displayed the explored system and its qualities, functions are displayed the actions and equations human relationships.

Quasistatic models and Active models represent both major type of mathematical modeling. Quasistatic models shows the romantic relationships between the system attributes approximate to equilibrium. The national current economic climate models is one of quasistatic models. Active models summarize the variation of functions change over the time. The pass on of a disease is one of the strong models.

Mathematical models are used specifically in the sciences and anatomist, such as physics, biology, and electrinic engineering but also in the communal sciences, such as economics, sociology and politics science; physicists, engineers, computer scientists, and economists will be the hottest mathematical model.

### 3. 2 Important of Mathematical Modeling

Mathematical modeling can be an interdisciplinary subject. Mathematics and specialists in several fields share their knowledge and experience to continuous improvement on extant products, make preferably develop, or forecast the certain product's behavior.

The most significant of modeling is to gain understanding. If a mathematical model is shows the essential tendencies of a real-world system of interest, we will easy to get understanding about the machine than using an analysis of the model. Furthermore, if you want to build a model, we need to find out which factors in the system are most significant, and how the different facet of the relevant system.

We need to forecast or simulate in the mathematical modeling. We always want to really know what is the real- world system will do in the future, but it is expensive, impractical or struggling to experiment straight with the machine. Finally, we have to estimate the big ideals in the mathematical modeling.

### 3. 3 Strategy of Mathematical Modeling

Agent Based Modeling (ABM) and Equation Structured Modeling (EMB) will be the techniques of mathematica modelling.

ABM and EBM reveal some typically common concerns, but in two various ways: the basic relationship model between entities, and make them the level of which they their target. These two solutions have accepted that the entire world has two types of entities: observables and individuals.

EBM start with a couple of equations that express connections among observables. The evaluation of these equations produces the progression of the observables as time passes. These equations may be algebraic, or they may capture variability over time or higher time and space. The modeler may identify these relationships result from the interlocking manners of the individuals, but those actions have no obvious representation in EBM.

ABM don't focus on equations that connect observables one to the other, but with habits via the interact between people with each other. These conducts may entail more personal immediately or in a roundabout way through showing environment. The modeler making much attention to the observation as the model runs, and may value a substandard accounts of the relationships among those observation, but the account is due to the modeling and simulation of movements, not its starting place. The modeler making start representative of each individual behavior, then becomes them over the interaction

In realization, EBM dealing with the model from macroscopic level to microscopic level by using the system of common differential equations (ODE) and incomplete differential equations (ODE). Besides that, ABM solving the model from microscopic level to macroscopic level utilizing the complicated dynamical system (CDS).

## 4. 0 Modeling in Mathematica Programme

### 4. 1 Introduction

Mathematica software involves wolfram research company. Mathematica 1. 0 version released on June 23, 1988. After the release in knowledge, technology, marketing, and other fields caused a feeling, considered a ground-breaking improvement. Almost a year later, in around the globe have thousands of Mathematica users. Today, in worldwide have Mathematica millions of faithful customers.

Mathematica 7 use words, quantities and other mathematical icons or inequality, constitute the equation, images or with diagrams of mathematical logic to spell it out the characteristics of the machine. Mathematica is researched and the movement rules of system is a robust tool, it's examination, design, forecasting and prediction and control genuine system.

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

### 4. 2 Graphical Interface of Mathematica

In most computer systems, Mathematica facilitates a "notebook" software where we interact with Mathematica by creating interactive documents.

If use computer via a purely graphical software, we usually double-click the Mathematica icon to start with the Mathematica. If use computer with a textually located in the operating-system, we can usually input the control mathematica to start Mathematica.

When Mathematica begins, it usually gives a blank notebook. Whenever we enter Mathematica source into the notebook, then type Shift-Enter (carry down the Transfer key, then press Enter. ) to make Mathematica process the source.

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

When type 2 + 2, then end the insight with Shift-Enter. Mathematica will processes the type, and then offers the type label In[1]:=, later gives the output.

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

With a notebook user interface, we just enter 2 + 2 and then type Shift-Enter. Mathematica then offers 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 sophisticated technological phenomena, both natural and sociable. It is one of the most widely used multi-agent modelling tools today, with a community of a large number of users worldwide. Its "low-threshold, noceiling" design beliefs is inherited from Company logo. NetLogo is simple enough that students and professors can certainly design and run simulations, and advanced enough to serve as a robust tool for research workers in many disciplines. Novices will see an easy-to-learn, intuitive, and well-documented programming language with an elegant graphical software.

Experts and research workers may use NetLogo's advanced functions, such as automated running tests, 3-D support, and customer expansibility. NetLogo also includes HubNet, which prepare a network of learners to collaboratively, explore and control a simulation. NetLogo attaches NetLogo Lab by external physical devices using the serial slot, and a System Dynamics Modeler make combined agent-based and polymerization representations.

NetLogo has considerable documents, including a collection with an increase of than 150 sample models in a series of domain, tutorials, a simple vocabulary, and sample code cases. This software is free and works on all major computing platforms. Manufacturing, make the system dynamics merged agent-based

### 5. 2 Graphical software of NetLogo

This model simulated the transmitting and preservation of most people are afflicted with the disease. Ecological biologists suggested several impact factors inside a population infected directly. This model is initialized with 150 people, including 10 are contaminated.

People of the world arbitrarily move around in one of the three claims below:

healthy but susceptible to infection (renewable),

sick and infectious (red),

healthy and immune system (grey). People may die of disease or an all natural death.

The factors in this model are summarized below with an explanation

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

1. Installation button

resets the graphics and plots

distributes with 140 inexperienced vulnerable 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 infections, immune and vulnerable personal contact each other.

4. INFECTIOUSNESS slider

Some familiar trojan easily distributed.

Some viruses propagate from every smallest contact

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

5. CHANCE-RECOVER slider

Population turnover.

Classify the individuals who had into group of susceptible, afflicted and immune system.

Determined the chances of people perish of the pathogen or a natural death.

All of the new created people replace those who death.

6. DURATION slider

Duration of infectiousness

Time of the pathogen afflicted health people.

Duration of any people contaminated before they retrieve or fatality.

7. TICKS

Number of week in enough time scale.

Views (BEIGE) - allow to show information

8. OUTPUT

3 output screen show the percent of populace is afflicted and immune system, and the amount of years have previously passed.

i) monitors - display the current value of variables

ii) plots - show the annals episode of a variable's value

iii) graphics windowpane - the key notion of the NetLogo world

The storyline shows (in their respected colors) the quantity of folks which is prone, infected, and immune system. In addition, it shows the full total number of people in the populace.

### 5. 3 The epidemic simulated by NetLogo Programme

1. The HIV pathogen simulated by NetLogo Programme

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

The HIV trojan has a very long duration, an extremely low recovery rate, but a very low infectiousness value.

2. The Ebola pathogen simulated by NetLogo Programme

Let the model is initialized with 150 people, which 10 are afflicted and 1 years which may have passed.

The famous Ebola trojan in central Africa has an extremely short duration, an extremely high infectiousness value, and a very low restoration rate.

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

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