Modelling Ebola using an SIR model - NLCS Maths ... · Web viewThe Ebola virus disease was first...

Modelling Ebola using an SIR model

HL Maths Exploration

NOVEMBER 25, 2014Candiate: Methushaa Suthanthirakumaran

School: North London Collegiate SchoolTeacher: Ms Copin

Modelling Ebola using an SIR model 2014

Contents1. Introduction and aim..........................................................................................................2

2. Common sense description of the SIR model....................................................................3

Parameterisation of the model..........................................................................................4

3. Running the SIR model on the initial figures of the Ebola Outbreak in Liberia in 2014. .6

Graphical interpretation....................................................................................................8

Validity of the model......................................................................................................10

Advantages.....................................................................................................................10

Disadvantages.................................................................................................................11

4. Comparing the model to actual figures and trying to improve it.....................................12

5. Conclusion........................................................................................................................15

6. Bibliography.....................................................................................................................16

1


Using the SIR model for Ebola outbreaks

1.Introduction and aimThe Ebola virus disease was first discovered in 1976 in the present Democratic Republic of Congo1.

Since then, there have been many outbreaks, with the greatest being the current 2014 outbreak2 which

has spread through many countries. Hoping to study medicine in the future and eventually becoming a

doctor, I became fascinated with the repetition of the outbreak of Ebola and the fact that despite the

advancements in technology, little was done in preparation for it. Therefore, by combining my interest

in mathematics which lies in modelling functions along with curiosity for the repetition of the disease,

I decided to model the Ebola Epidemics in Liberia in 2014 and Democratic Republic of Congo 1976

and compare their spread using an SIR3 model. An SIR model is an epidemiological model which

measures the number of people infected with a particular disease over a period of time using three

fundamental equations4.

Therefore in doing so, I aim to develop my understanding on the mathematics of the SIR model and

about its possible limitations for discussing the spread of the disease, in turn, this should shed light on

the spread of Ebola.

In order to do this, I will:

Describe the SIR model

Use the model on the initial data from the outbreak in Liberia 2014

Compare the model with real data Liberia 2014

1 "Ebola Virus Disease." WHO. N.p., n.d. Web. 17 Nov. 2014.2 "2014 Ebola Outbreak in West Africa." Centers for Disease Control and Prevention. Centers for Disease Control and Prevention, 06 Mar. 2015. Web. 07 Nov. 2014.3 "Kermack-McKendrick Model." -- from Wolfram MathWorld. N.p., n.d. Web. 17 Nov. 2014.4 "The Mathematics of Diseases." The Mathematics of Diseases. N.p., n.d. Web. 17 Nov. 2014.

2


2.Common sense description of the SIR modelThe SIR model is used to illustrate the transfer of the epidemic through the interaction of the

following three different variables:

S=number of people that are susceptible to Ebola

I=number of people infected with Ebola

R=number of people recovered from Ebola with total immunity

It makes sense to assume that a fixed population of N people, whereby there are no births and deaths

by natural cause, consists of the number of people susceptible plus the number of people infected plus

number of people resistant:

N=S+ I +R

This is because the population is fixed and therefore, there are only three compartments in which the

population may fit into. Thus, the total of the number of people susceptible infected and recovered in

equivalent to the total population. The assumption that N is fixed, with no births or deaths, makes

sense given 60 days, although it is a simplification.

These variables change over time, so I will define the variable t=timein days. I will set t = 0 at the

start of August 2014.

The model uses two parameters which can be used calibrate it, β andγ withβ , γ>0. Given these

parameters, the model uses 3 differential equations. These will be different numbers for any given

disease and situation, and will depend on things like method of transmission, and the contact rate. I

will calculate those later using actual data for the current Ebola epidemic mathematically, thought of

as contrast of population, but I want to first give an idea of why these equations are true and what

these might mean.

Equation 1: dSdt

=−βIS

In Equation 1, dSdt means the rate of change of the number of people susceptible to the disease over

time. dSdt decreases proportionally to I because in order to become infected, you are no longer

susceptible to the diseases any more. Since the only way to leave the set of susceptible people is

through becoming infected with the disease itself, therefore the number of people who are susceptible

3


to the disease is determined by the number of people who are already susceptible, the number of

individuals who are already infected and the amount of contact between the susceptible and infected.

An assumption is made that every individual has the same probability of becoming infected with the

disease. In real life, this is highly improbable and it is a limitation that I discuss later. The equation

also decreases proportionally to S because individuals are repeatedly being removed from the

susceptible section and being transferred into the infectious section.

Equation 2: dRdt

=γI

In equation 2, dRdt means the rate of change of the number of people recovered over time. This

illustrates that the rate of the number of people recovering is dependent upon the number of people

infected as in order to become recovered. This is because, in order to become recovered from a

disease, one must have been infected at some point over a certain period of time and if the duration of

time is shorter, then the rate of becoming infected increases. Therefore, this increases proportionally

with the rate of the disease being infected.

Equation 3: dIdt

=βIS−γI

In equation 3, dIdt means the rate of change of the number of people infected. This is dependent on the

number of people susceptible and the number of people infected as well as the infection rate of the

disease between the two compartments. As the population of I increases, the population of S

decreases, therefore the rate at which dIdt increases is inversely proportional to theSbecause in order

for there to be more infected people, there must be a decrease in the number of susceptible people.

Thus, this equation is a consequence of the fact that: dIdt

=−dSdt

−dRdt into which we can substitute

equation 1 and 2.

Parameterisation of the modelIn order to calculate β(the rate of infection) and γ(the rate of recovery), it helps to define two more

parameters.

D = Duration of disease for those recovered

M=Mortality rate for those who die per day ( 0.7 for Ebola )

4


This leads to two further equations.

Equation 4: γ= 1D 5

In equation 4, the rate at which the disease is spread can be found by dividing 1 by the duration of the

disease. This is because; a certain individual can only experience one recovery in a given period of

time. For example if the duration of the infective period is 10 days, then the rate at which those who

are infected become recovered is:

110

=0.1=10 %

5

Equation 5: β= MS

Equation 5 illustrates that the infection rate of the disease is dependent upon the mortality rate and the

number of people susceptible to the disease. It demonstrates the rate at which the disease passes from

a susceptible individual to an infected individual. The value for β always lies between 0 and 1,

because a value of 1 suggests 100% infection rate and a value of 0 suggests 0% infection rate. For

example, if the mortality rate of the population is 50% and the number of people susceptible is 100,

then the rate in infection will be calculated as follows:

β= 0.5100

=0.005

5 Dolgoarshinnykh, Regina, Columbia University, Steven P. Lalley, and University Of Chicag. "Epidemic Modeling: SIRS Models." Epidemic Modeling: SIRS Models (n.d.): n. pag. Web.

5


3.Running the SIR model on the initial figures of the Ebola Outbreak in Liberia in 2014

If we now take the example of the Ebola outbreak in Liberia 2014, we can assign the parameters with

the following values. The total population of Liberia, N = 42940006, and according to data from

WHO7, the number of people infected, I = 8468 and the number of people dead is 4819. Seeing as R,

includes the number of people who have received permanent immunity, this includes those who have

died as they have permanent immunity, in addition to those who have recovered with permanent

immunity.

Therefore, number of people recovered R=481+(0.3 ×846 )=735

I will now use this data to provide the parameters with the following values.

N=4294000

I=846

R=735

Therefore, S=N−I+R=4294000−(735+846 )=4292419

The duration of the disease ranges from 2 to 18 days, therefore we could roughly estimate the duration

of the disease at the midpoint, i.e. 10 days.

D=10

γ= 110

=0.1

According to WHO, the mortality rate of Ebola is 0.710 and the number of people susceptible is

4292419.

Therefore from equation 5, β ( the rate of infection)= 0.74292419

=1.63× 10−7

6 "Modelling Infectious Diseases." IB Maths Resources from British International School Phuket. N.p., 17 May 2014. Web. 04 Nov. 2014.7 "Modelling Infectious Diseases." IB Maths Resources from British International School Phuket. N.p., 17 May 2014. Web. 04 Nov. 2014.8 "Modelling Infectious Diseases." IB Maths Resources from British International School Phuket. N.p., 17 May 2014. Web. 04 Nov. 2014.9 "Modelling Infectious Diseases." IB Maths Resources from British International School Phuket. N.p., 17 May 2014. Web. 04 Nov. 2014.10 "The Spread Of Infectious Diseases." The British Medical Journal 2.1281 (1885): 108. Web.

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In order to use the SIR model to predict the evolution of the disease, it would be helpful if we could

solve the system of differential equations. Unfortunately, we cannot completely solve these equations

with an explicit formula solution11.

Therefore, I will use a numerical approach, as follows. For each day, I will calculate the values of

dSdt

,

dIdt

∧dR

dt using equations 1, 2 and 3. Then assume that the Svalue for the following day is the

previous S value+ dSdt

for that point in time .

I will do this explicitly for the transition from t = 0 to t = 1. Using equations 1, 2 and 3 from earlier,

the following values for the three rates of change of S, I and R can be calculated.

dSdt |t=0

=(−1.63× 10−7)× 846 ×4292419= -581

dIdt |t=0

=(1.6× 10−7)−(0.1 ×846 )= 496

dRdt |t=0

=0.1× 846 = 85

Therefore, at t=1,

S = 4292419−581=4291838

I will now use excel to do this over a two month period, by putting in the formulae in the

following way. I defined γ in as g in cell I1 and β as b in cell J3:

A B C D E F G H I J1 T S I R dS/dt dI/dt dR/dt S+I+R gamma beta

2 T B2 C2 D2 E2 F2 G2 B2 + C2 + D2 g b

3 t+1 B3+E3 C2+F2 D2+G2 -g*I3*B3 B*I3*B3 – g*I3 g*I3 B3+E3+C2+

F2+D2+G2 gb

This generates the following table: Susceptible Infected Recovered

t S I R ds/dt dI/dt dr/dt S+I+R0 4292419 846 735 -581 496 85 42940001 4291838 1342 820 -922 788 134 42940002 4290916 2130 954 -1462 1249 213 42940003 4289454 3379 1167 -2319 1981 338 42940004 4287134 5361 1505 -3677 3141 536 42940005 4283457 8502 2041 -5827 4977 850 42940006 4277631 13478 2891 -9225 7877 1348 42940007 4268406 21355 4239 -14585 12449 2136 42940008 4253821 33804 6374 -23008 19627 3380 4294000

11 "Appendix: Additional Results and Technical Notes for the EbolaResponse Modeling Tool." Centers for Disease Control and Prevention. Centers for Disease Control and Prevention, 23 Sept. 2014. Web. 08 Nov. 2014.

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9 4230814 53432 9755 -36169 30826 5343 429400010 4194644 84258 15098 -56549 48123 8426 429400011 4138095 132381 23524 -87649 74411 13238 429400012 4050446 206792 36762 -134016 113337 20679 429400013 3916430 320129 57441 -200602 168589 32013 429400014 3715828 488718 89454 -290559 241687 48872 429400015 3425269 730405 138326 -400294 327253 73041 429400016 3024975 1057658 211366 -511902 406137 105766 429400017 2513073 1463795 317132 -588580 442200 146379 429400018 1924493 1905995 463512 -586892 396292 190600 429400019 1337601 2302288 654111 -492727 262498 230229 429400020 844874 2564786 884340 -346707 90229 256479 429400021 498167 2655015 1140819 -211622 -53879 265501 429400022 286544 2601136 1406320 -119255 -140859 260114 429400023 167290 2460277 1666434 -65853 -180175 246028 429400024 101437 2280102 1912461 -37006 -191004 228010 429400025 64431 2089097 2140471 -21537 -187373 208910 429400026 42895 1901724 2349381 -13052 -177121 190172 429400027 29843 1724604 2539554 -8235 -164226 172460 429400028 21608 1560378 2712014 -5395 -150643 156038 429400029 16213 1409735 2868052 -3657 -137316 140973 429400030 12556 1272418 3009025 -2556 -124686 127242 429400031 10000 1147733 3136267 -1836 -112937 114773 429400032 8164 1034796 3251040 -1352 -102128 103480 429400033 6812 932668 3354520 -1017 -92250 93267 429400034 5796 840418 3447787 -779 -83262 84042 429400035 5016 757155 3531828 -608 -75108 75716 429400036 4409 682047 3607544 -481 -67724 68205 429400037 3927 614324 3675749 -386 -61046 61432 429400038 3541 553277 3737181 -313 -55014 55328 429400039 3228 498263 3792509 -257 -49569 49826 429400040 2971 448694 3842335 -213 -44656 44869 429400041 2757 404038 3887205 -178 -40226 40404 429400042 2579 363813 3927608 -150 -36231 36381 429400043 2429 327581 3963990 -127 -32631 32758 429400044 2302 294951 3996748 -109 -29386 29495 429400045 2193 265564 4026243 -93 -26463 26556 429400046 2100 239101 4052799 -80 -23830 23910 429400047 2019 215271 4076709 -70 -21458 21527 429400048 1950 193814 4098236 -60 -19321 19381 429400049 1889 174493 4117618 -53 -17397 17449 429400050 1837 157096 4135067 -46 -15663 15710 429400051 1791 141433 4150777 -41 -14103 14143 429400052 1750 127330 4164920 -36 -12697 12733 429400053 1714 114633 4177653 -31 -11432 11463 429400054 1683 103201 4189116 -28 -10292 10320 429400055 1655 92909 4199436 -25 -9266 9291 429400056 1631 83642 4208727 -22 -8342 8364 429400057 1609 75300 4217091 -19 -7511 7530 429400058 1589 67789 4224621 -17 -6762 6779 429400059 1572 61028 4231400 -15 -6087 6103 429400060 1557 54940 4237503 -14 -5480 5494 4294000

Graphical interpretationFrom this table, we can plot S, I, R against t.

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0 10 20 30 40 50 600

50000010000001500000200000025000003000000350000040000004500000

SIR Model for the Ebola outbreak in Liberia, August 2014

SIRN

Time (days) from August 2014 over a two month period

Num

ber o

f peo

ple

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

4500000

0 10 20 30 40 50 60

Num

ber o

f peo

ple


SIR Model for the Ebola outbreak in Liberia, August 2014

S

I

R

N

As expected, this shows that initially, the number of people infected increases steeply, however, over

a longer period of time, the numbers eventually decrease. This happens simultaneously as the number

of people recovered increases because as those infected decreases, they are being transferred into the

recovered category. The main reason is due to the increased awareness of the disease leasing to

further medical support being given in order to help combat the transmission of the disease.

Furthermore, an increased awareness results in more people being aware of methods of protection.

The steep increase in the beginning of the first 15 days is most likely to be due to the great uncertainty

that lied with Ebola allowing a greater rate of transmission. The peak of the graph illustrates the

maximum number of people ever to be infected and after this point, there is a transition whereby the

numbers decrease.

The number of people susceptible to the disease remains constant for the first 10 days, and then it

steeply decreases to create a negative sigmoidal curve. This means that it is shaped like the letter S,

but in reverse. It must be noted that the number of people never reaches 0, and only tends towards it

allowing the epidemic to reoccur in the future. The only way for the number of susceptible to reach 0

is through the vaccination as this acts as a vehicle to remove the disease from the population. The

number of people susceptible remains constant at the beginning, which is similar to the small increase

in the number of people infected. However, as there are more people infected, there is a steep decline

in the number of people susceptible to the disease. This is because being the number of people

infected comes from the number of people susceptible and they are connected. Therefore, as the

number of people infected begins to decline, the number of people susceptible begins to level off.

This is due to the fact that everyone infected is eventually becoming recovered, thus reducing the

numbers of those who are infected. Therefore, there is very little change in the number of people

susceptible to the disease towards the end of the two month period.

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The number of people recovered from the disease, slowly increase at the beginning with the slow rate

of infection. However, as the number of people infected increases dramatically, this leads to a

consequent steep increase with the number of people recovered, until eventually levelling off

simultaneously to the number of people susceptible. The line illustrating the number of people

recovering increases concurrently as the number of people susceptible decreases. This is because

susceptibility and recovery are inversely proportional to one another. However, the number of people

recovered from the disease, never reaches the total population, and only tends towards it.

Furthermore, this graph illustrates cumulative distributions through the positive sigmoidal curve on

the graph.

The graph also shows that the total population remains constant throughout the two month period via

a linear correlation. This is because, as established earlier, N=S+ I +R and in order to detect a

change in something we need to differentiate it. In this case, the graph suggests no change, so the

differentiation must be equivalent to 0.

Therefore, (S+ I +R)'=S '+ I '+ R' and by substituting the differential equations 1, 2 and 3, we get the

following:

(S+ I +R)'=−βIS+γI+βIS−γI=0

Thus, there is no change in the population and it will remain constant in a given period of time.

The graph is useful because it allows me to see the interaction between the different variables and it is

interesting to relate it to differentiation to determine the changes over time.

Validity of the model

However, in order for the model to be valid and allow to inform government policy, it obviously

needs to correspond fairly close to reality. Before checking against the graph, there are already clear

advantages and disadvantages to this model:

Advantages

The advantages to the model include:

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1. It is very quick to model the data having found the values for the respective parameters and transition

probabilities to allow immediate assessment of the condition that is present. This results in instant

evaluation of the situation as well as valid prediction of the spread of disease in the future.

2. The model is widely used and also widely understood by the medical community making it easier to

explain the effects of the epidemic

3. This model is clear and easy to understand in order to distinguish between the number of people

susceptible, infected and recovered

4. The mechanism to create the data is flexible, allowing it to be easily altered if certain values are

incorrect or have changed

5. It is computationally cheap and there are other software available with very small time intervals

allowing it to be more accurate which I could not complete in excel.

Disadvantages

The disadvantages to the model include:

1. The calculation of the beta values and gamma values are often inaccurate because small deviation

from the ‘correct value’ can result in great changes in the overall model. For example, changing the

gamma value from 0.1 to 0.3 can lead to the following changes:

2. The data itself can be fairly unreliable especially, in countries where death counts are difficult to

manage. (For the Ebola case, although the numbers seem fairly high, there is a high probability that

there were far more people infected)

3. In order for the model to be calculated correctly, you need the right form of data including the number

of people infected, recovered and susceptible. This data can be very particular and calculating the

number of people susceptible as the number of people left over from taking away the number of

people infected and recovered from the total population, is not always the most reliable method.

4. This model is only effective for small environments with heterogeneous population density

distribution

11

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

4500000

0 10 20 30 40 50 60

Number of people


Gamma value of 0.3

S

I

R

N

0

500000

1000000

1500000

2000000

2500000

3000000

3500000

4000000

4500000

0 10 20 30 40 50 60

Number of people


Gamma value of 0.1

S

I

R

N


4.Comparing the model to actual figures and trying to improve it

The table below compares the data collected from the model for the number of people infected and the

real life data12 of the number of people infected. From this, I can plot a graph using excel.

Time (days)

I model I actual

10 84258 137815 730405 168020 2564786 187125 2089097 204630 1272418 240735 757155 302240 448694 328045 265564 369650 157096 383455 92909 407660 54940 4262

As the graph demonstrates, the real data does not correspond very well to the data received from the

model. Although the actual data may seem to follow a straight like graph, this is untrue as it is only

depicted in this manner due to the limitations on the axis of the graph. The difference between the real

life data and the data from the model is so vast that the straight line looks like a graph of y=0.

Therefore, I decided to plot is separately:

12 "Ebola Virus Disease Update - West Africa." WHO. N.p., n.d. Web. 08 Nov. 2014

12

0 10 20 30 40 50 600

500000

1000000

1500000

2000000

2500000

3000000

Comparison of the model data to the actual data for the number of people infected

I model I actual

Num

ber o

f peo

ple

infe

cted

0 10 20 30 40 50 60 700

200400600800

10001200

Real data for the number of people infected

I Actual

Time (days)

Num

ber o

f peo

ple

infe

cted


Therefore, this shows that the model has significantly overestimated the number of people who will

become infected with Ebola. This is because of the several limitations which the model presents. One

of the main limitations includes the inaccurate beta and gamma values which were calculated. After

altering the beta and gamma values, I was able to find another gamma value which resulted in similar

values to the real data. Here is the graph to show this, with the appropriate gamma value of

0.679995559.

Although, this does not fit the graph exactly, it shows a better positive correlation of the number of

people infected. Therefore, in order to improve the model, several changes must be done, including

altering the gamma value. The value which I eventually used to alter the model, led to being in several

decimal places. This goes to illustrate the necessary precision needed as little deviance can lead to

large changes. This is because; the gamma is calculated through extreme simplification, leaving great

possibilities for further room for errors.

Furthermore, there are many assumptions that are made with creating an SIR model:

Any individual in the population has an equal probability of receiving this disease

The number of people leaving a certain category is equivalent to the number of people joining

a new category. (i.e. the number of people leaving the susceptibility category, is equivalent to

the number of people joining the infected category)

Rate of recovery is faster that the time scale of birth and death

There is a homogenous mixing of the population whereby each individual encounters contact

with similar people in ratio to each category.

13

0 10 20 30 40 50 60 700

200400600800

100012001400160018002000

Fitting the model with the real data

Time (days)

Num

ber o

f peo

ple

infe

cted


Individuals that recovery, automatically recover with permanent immunity

The number of people recovered includes those who have died as well as those alive with

permanent immunity, making it difficult to differentiate one from the other

These assumptions do not always comply with reality as often there is no homogenous mixing within

the population and each individual does not have the same probability of being a victim of the disease

as others. These limitations borne out of the assumption increase the subjectivity of the results,

creating results which often may not correlate to real life data. Furthermore, it is difficult to

differentiate between the number of people who have died and the numbers of people who have

survived with permanent immunity as they both fall under the same category of being ‘recovered’.

The model is used to estimate future predictions of the disease and consequently, it will help to

determine practical elements such as the number of beds needed in the hospitable, leaving these

limitations of little importance.

14


5.ConclusionBy using an SIR model, I was able to see the importance of modelling data, especially in the field of

medicine. This is because, in order to cope with the rapid changes in the medical sector, many

governments must find methods to sustain and maximise the efficiency of the available health care

systems. One of these methods includes mathematical modelling which is becoming increasingly

important in helping identify the future of certain diseases. The application of mathematical models

on diseases can be extended to include the effects of vaccination and impacts of herd immunity on an

outbreak as well. This can help to determine different factors which can help reduce the mortality

rate.

From doing my exploration, I gained further insight in the ways in which modelling can be used to

predict the apparent spread of diseases in order to inform health care superficial of the necessary

precautions that must be in place. Nonetheless, similar to most models, the SIR is also subject to

limitations as often a model is a simplified representation of the real situation and often this can lead

to over simplification, creating conflicts between simplicity and complexity. Ultimately, the aim of

modelling is to clarify certain concepts, but models often attempt to mimic a real life situation through

introducing many variables and can lead to further confusion.

However, the results obtained from modelling data can lead to differing perspectives and

interpretations. This is due to the unequal distribution of data across the world whereby in countries

such as Liberia, there is very little access to the statistics which makes it difficult to make constructive

predictions concerning the outbreak. However, in countries such as UK, the data is more widely

available making developing countries and their governments dependent on them. This caused an

exaggerated media coverage leading to the development of irrational fears which promoted the

prevalence of more resilient and contagious diseases such as tuberculosis. The deaths that arose from

Ebola only account for a tiny fraction in comparison to other causes of deaths such as malaria and

HIV/AIDS. Nonetheless, due to the inflation of the situation, much research has been conducted in

order to create a potential vaccine against it.

Through completing this exploration, I am able to see the impact of mathematical modelling and the

influences it has in helping scientists to analyse epidemics and help prevent further disruption. The

SIR model which I used showed the general trend of the epidemic, however due to its limitations

which eventually outweighed the advantages, the model did not precisely correspond to the real life

data, although they mostly illustrated similar correlation.

15


Therefore, through my exploration, I have gained further insight into the uses of mathematical

modelling in order to determine the spread of diseases as well as evaluating its flaws. Having chosen

Ebola as the disease of concentration, as it is very relevant to the current situation in Africa, it has

enabled a realistic understanding of its rate of transmission. Moreover, this task had allowed me to

combine my interests in maths alongside a disease with which I have great interest in, in order to

simulate an analytical study and gain further understanding of the ways in which health care

professions rely on mathematical studies to help them make important decisions in improving the

healthcare of the population.

16


6.Bibliography"2014 Ebola Outbreak in West Africa." Centers for Disease Control and Prevention. Centers

for Disease Control and Prevention, 06 Mar. 2015. Web. 07 Nov. 2014.

Dolgoarshinnykh, Regina, Columbia University, Steven P. Lalley, and University Of Chicag.

"Epidemic Modeling: SIRS Models." Epidemic Modeling: SIRS Models (n.d.): n.

pag. Web.

"Ebola Virus Disease." WHO. N.p., n.d. Web. 17 Nov. 2014.

"Ebola Virus Disease." WHO. N.p., n.d. Web. 18 Nov. 2014.

"Estimating the Future Number of Cases in the Ebola Epidemic — Liberia and Sierra Leone,

2014–2015." Centers for Disease Control and Prevention. Centers for Disease

Control and Prevention, 07 Oct. 2014. Web. 18 Nov. 2014.

"Kermack-McKendrick Model." -- from Wolfram MathWorld. N.p., n.d. Web. 05 Nov. 2014.

Matemàtic, Materials. "MAT 2." (n.d.): n. pag. Www.mat.uab.cat/matmat. Publicació

Electrònica De Divulgació Del Departament De Matemàtiques De La Universitat

Autònoma De Barcelona. Web. 18 Nov. 2014.

"Modeling Ebola in West Africa: Cumulative Cases by Date of Reporting." Contagious

Disease Surveillance. N.p., n.d. Web. 24 Nov. 2014.

"Modelling Infectious Diseases." IB Maths Resources from British International School

Phuket. N.p., 17 May 2014. Web. 17 Nov . 2014.

"The Mathematics of Diseases." The Mathematics of Diseases. N.p., n.d. Web. 14 Nov. 2014.

"The Spread Of Infectious Diseases." The British Medical Journal 2.1281 (1885): 108. Web.

"WHO Finds 70 Percent Ebola Mortality Rate." - Africa. N.p., n.d. Web. 18 Nov. 2014.

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