MODELLING EBOLA USING AN
SIR MODEL
HL Maths Exploration
NOVEMBER 25, 2014
CANDIATE: METHUSHAA SUTHANTHIRAKUMARAN
School: North London Collegiate School
Teacher: Ms Copin
Modelling Ebola using an SIR model 2014
Contents
1.
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 .................................................................................................................... 17
1
Modelling Ebola using an SIR model 2014
Using the SIR model for Ebola outbreaks
1. Introduction and aim
The 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.
"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
Modelling Ebola using an SIR model 2014
2. Common sense description of the SIR model
The 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 + 𝑅5
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 = time in days. I will set 𝑡 = 0 at the
start of August 2014.
The model uses two parameters which can be used calibrate it, 𝛽 and 𝛾 with 𝛽, 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:
𝑑𝑆
𝑑𝑡
= − 𝛽𝐼𝑆6
𝑑𝑆
In Equation 1, 𝑑𝑡 means the rate of change of the number of people susceptible to the disease over
𝑑𝑆
time. 𝑑𝑡 decreases proportionally to 𝐼 because in order to become infected, you are no longer
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.
6
"Modelling Infectious Diseases." IB Maths Resources from British International School Phuket. N.p., 17 May
2014. Web. 04 Nov. 2014.
3
Modelling Ebola using an SIR model 2014
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
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:
𝑑𝑅
𝑑𝑡
In equation 2,
= 𝛾𝐼7
𝑑𝑅
𝑑𝑡
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:
𝑑𝐼
𝑑𝑡
= 𝛽𝐼𝑆 − 𝛾𝐼8
𝑑𝐼
In equation 3, 𝑑𝑡 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
𝑑𝐼
decreases, therefore the rate at which 𝑑𝑡 increases is inversely proportional to the 𝑆 because 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:
𝑑𝐼
𝑑𝑡
= −
𝑑𝑆
𝑑𝑡
−
𝑑𝑅
𝑑𝑡
into which we can substitute
equation 1 and 2.
Parameterisation of the model
In order to calculate 𝛽 (the rate of infection) and 𝛾 (the rate of recovery), it helps to define two more
parameters.
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.
4
Modelling Ebola using an SIR model 2014
𝐷 = Duration of disease for those recovered
𝑀 = Mortality rate for those who die per day (0.7 for Ebola)
This leads to two further equations.
Equation 4: 𝛾 =
1 9
𝐷
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:
1
10
= 0.1 = 10%
5
Equation 5: 𝛽 =
𝑀 10
𝑆
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=
0.5
= 0.005
100
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.
5
Modelling Ebola using an SIR model 2014
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 = 429400011, and according to data from
WHO12, the number of people infected, I = 84613 and the number of people dead is 48114. 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=
1
= 0.1
10
According to WHO, the mortality rate of Ebola is 0.715 and the number of people susceptible is
4292419.
Therefore from equation 5, 𝛽 (the rate of infection) =
11
0.7
4292419
= 1.63 × 10−7
"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.
12
"Ebola Virus Disease Update - West Africa." WHO. N.p., n.d. Web. 08 Nov. 2014
13
"Ebola Virus Disease Update - West Africa." WHO. N.p., n.d. Web. 24 Nov. 2014.
14
"Ebola Virus Disease Update - West Africa." WHO. N.p., n.d. Web. 24 Nov. 2014.
15
"WHO Finds 70 Percent Ebola Mortality Rate." - Africa. N.p., n.d. Web. 08 Nov. 2014.
6
Modelling Ebola using an SIR model 2014
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 solution16.
Therefore, I will use a numerical approach, as follows. For each day, I will calculate the values of
𝑑𝑆 𝑑𝐼
,
𝑑𝑡 𝑑𝑡
𝑑𝑅
𝑎𝑛𝑑 𝑑𝑡 using equations 1, 2 and 3. Then assume that the 𝑆 value for the following day is the
𝑑𝑆
previous 𝑆 value + 𝑑𝑡 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.
𝑑𝑆
|
𝑑𝑡 𝑡=0
= (− 1.63 × 10−7 ) × 846 × 4292419= -581
𝑑𝐼
|
𝑑𝑡 𝑡=0
= (1.6 × 10−7 ) − (0.1 × 846) = 496
𝑑𝑅
|
𝑑𝑡 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
J
1
T
S
I
R
dS/dt
dI/dt
dR/dt
gamma
2
T
B2
C2
D2
E2
F2
G2
beta
b
-g*I3*B3
B*I3*B3 –
g*I3
g*I3
S+I+R
B2 + C2 +
D2
B3+E3+C2+
F2+D2+G2
3
t+1
B3+E3
C2+F2
D2+G2
g
b
g
This generates the following table:
t
0
1
2
3
4
5
6
7
8
9
10
11
12
Susceptible
S
Infected
I
4292419
4291838
4290916
4289454
4287134
4283457
4277631
4268406
4253821
4230814
4194644
4138095
4050446
16
Recovered
R
846
1342
2130
3379
5361
8502
13478
21355
33804
53432
84258
132381
206792
ds/dt
735
820
954
1167
1505
2041
2891
4239
6374
9755
15098
23524
36762
dI/dt
-581
-922
-1462
-2319
-3677
-5827
-9225
-14585
-23008
-36169
-56549
-87649
-134016
dr/dt
496
788
1249
1981
3141
4977
7877
12449
19627
30826
48123
74411
113337
S+I+R
85
134
213
338
536
850
1348
2136
3380
5343
8426
13238
20679
Matemàtic, Materials. "MAT 2." Publicació Electrònica De Divulgació Del Departament De Matemàtiques
De La Universitat Autònoma De Barcelona (n.d.): n. pag. Www.mat.uab.cat/matmat. 01 July 2013. Web. 12
Nov. 2014.
7
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
Modelling Ebola using an SIR model 2014
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
3916430
3715828
3425269
3024975
2513073
1924493
1337601
844874
498167
286544
167290
101437
64431
42895
29843
21608
16213
12556
10000
8164
6812
5796
5016
4409
3927
3541
3228
2971
2757
2579
2429
2302
2193
2100
2019
1950
1889
1837
1791
1750
1714
1683
1655
1631
1609
1589
1572
1557
320129
488718
730405
1057658
1463795
1905995
2302288
2564786
2655015
2601136
2460277
2280102
2089097
1901724
1724604
1560378
1409735
1272418
1147733
1034796
932668
840418
757155
682047
614324
553277
498263
448694
404038
363813
327581
294951
265564
239101
215271
193814
174493
157096
141433
127330
114633
103201
92909
83642
75300
67789
61028
54940
Graphical interpretation
From this table, we can plot S, I, R against t.
8
57441
89454
138326
211366
317132
463512
654111
884340
1140819
1406320
1666434
1912461
2140471
2349381
2539554
2712014
2868052
3009025
3136267
3251040
3354520
3447787
3531828
3607544
3675749
3737181
3792509
3842335
3887205
3927608
3963990
3996748
4026243
4052799
4076709
4098236
4117618
4135067
4150777
4164920
4177653
4189116
4199436
4208727
4217091
4224621
4231400
4237503
-200602
-290559
-400294
-511902
-588580
-586892
-492727
-346707
-211622
-119255
-65853
-37006
-21537
-13052
-8235
-5395
-3657
-2556
-1836
-1352
-1017
-779
-608
-481
-386
-313
-257
-213
-178
-150
-127
-109
-93
-80
-70
-60
-53
-46
-41
-36
-31
-28
-25
-22
-19
-17
-15
-14
168589
241687
327253
406137
442200
396292
262498
90229
-53879
-140859
-180175
-191004
-187373
-177121
-164226
-150643
-137316
-124686
-112937
-102128
-92250
-83262
-75108
-67724
-61046
-55014
-49569
-44656
-40226
-36231
-32631
-29386
-26463
-23830
-21458
-19321
-17397
-15663
-14103
-12697
-11432
-10292
-9266
-8342
-7511
-6762
-6087
-5480
32013
48872
73041
105766
146379
190600
230229
256479
265501
260114
246028
228010
208910
190172
172460
156038
140973
127242
114773
103480
93267
84042
75716
68205
61432
55328
49826
44869
40404
36381
32758
29495
26556
23910
21527
19381
17449
15710
14143
12733
11463
10320
9291
8364
7530
6779
6103
5494
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
4294000
Modelling Ebola using an SIR model 2014
SIR Model
Model for
for the
theEbola
Ebolaoutbreak
outbreakininLiberia,
Liberia,August
August2014
2014
4500000
4500000
Number of
people
ofpeople
Number
4000000
4000000
3500000
3500000
3000000
3000000
2500000
2500000
SS
2000000
2000000
I
1500000
1500000
1000000
1000000
500000
500000
0
R
R
0
N
N
0
0
10
20
30
40
50
10
20
30
40
50
Time
over aa two
twomonth
monthperiod
period
Time(days)
(days) from
from August
August 2014
2014 over
60
60
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.
9
Modelling Ebola using an SIR model 2014
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:
10
Modelling Ebola using an SIR model 2014
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:
Gamma value of 0.1
Gamma value of 0.3
4500000
4500000
4000000
4000000
3500000
3500000
3000000
3000000
Number of 2500000
people 2000000
S
Number of 2500000
people 2000000
S
1500000
R
1500000
1000000
R
N
1000000
N
I
500000
I
500000
0
0
0
10
20
30
40
50
Time (days) from August 2014 over a two month period
60
0
10
20
30
40
50
60
Time (days) from August 2014 over a two month period
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
Modelling Ebola using an SIR model 2014
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 data17 of the number of people infected. From this, I can plot a graph using excel.
Time
I model
I actual
(days)
84258
1378
10
730405
1680
20
2564786
1871
25
2089097
2046
30
1272418
2407
35
757155
3022
40
448694
3280
45
265564
3696
50
157096
3834
55
92909
4076
60
54940
4262
3000000
Number of people infected
15
Comparison of the model data to the actual data for
the number of people infected
2500000
2000000
1500000
1000000
500000
0
0
10
20
I model
30
40
50
60
I actual
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:
Number of people infected
Real data for the number of people infected
1200
1000
800
600
400
I Actual
200
0
0
20
40
60
80
Time (days)
17
"Modeling Ebola in West Africa: Cumulative Cases by Date of Reporting." Contagious Disease Surveillance.
N.p., n.d. Web. 24 Nov. 2014.
12
Modelling Ebola using an SIR model 2014
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.
Number of people infected
Fitting the model with the real data
2000
1800
1600
1400
1200
1000
800
600
400
200
0
0
10
20
30
40
50
60
70
Time (days)
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
Modelling Ebola using an SIR model 2014
 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
Modelling Ebola using an SIR model 2014
5. Conclusion
By 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
Modelling Ebola using an SIR model 2014
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.
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Modelling Ebola using an SIR model 2014
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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