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Maths IA GRADE 6
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IA for IB curriculum, grade 6 awarded, in depth analysis, marked against 2021 IB rubric, structured
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SIR Model for DiseaseUsing the SIR Model to model the spread of Ebola in Liberia to calculate the
herd immunity threshold and the effect it would have had on the population
Rationale
The Ebola virus disease was first discovered near the Ebola River in Democratic republic of Congo.1
The virus is highly contagious and is transmitted through contact with bodily fluids such as blood.
The 2014-2016 outbreak in West Africa was Ebola’s largest outbreak since it was first discovered in
1976. Currently living through a global pandemic of COVID-19, the use of maths to predict and
monitor the spread of the virus has been interesting to follow. Personally, the most interesting
aspect of maths is its application to real life situations, which is why I enjoy differential calculus. In
class we had looked at the application of differential calculus to real life examples, however not in
the complex context of epidemics, where there are a system of differential equations that are all
interlinked. Hoping to pursue a career in applied mathematics, doing an investigation on how it is
used to predict the evolution of a disease captivated me. I started researching more about
mathematical modelling for epidemics and found that the most common is the SIR epidemiology
model. Upon further research, I discovered that there were various ways the spread of a disease can
be slowed down. Thoroughly engaged with the news during this global pandemic of COVID-19, I was
exposed to the term herd immunity. Herd immunity is the resistance to the spread of an infectious
diseases when a high proportion of the population has been infected or vaccinated, providing
immunity within the population. 2 It is one of the preventative methods for the spread of an
infectious disease which really caught my interest, as much of the current news is about the effect a
vaccination would have on the spread of the COVID-19 virus. Although there was no vaccination
created at the time of the Ebola outbreak in Liberia, I was intrigued to see what effect it would have
had on the population if they had access to a vaccine. Doing an investigation on how mathematical
modelling of a disease can be used to predict and mitigate a virus was captivating. It is important to
note that all values are represented to full decimal places as the change is very slight, so to create a
1
“Ebola Virus Disease.” World Health Organization. Accessed October 9, 2020. https://www.who.int/news-
room/fact-sheets/detail/ebola-virus-disease.
2
Iftikhar, Noreen. “Herd Immunity: What It Means for COVID-19.” Healthline. Healthline Media, April 2, 2020.
https://www.healthline.com/health/herd-immunity.
1
,graph that is as close to the actual graph, all significant figures are necessary. All secondary data is
collected from the World Health Organization, starting from the day the first Ebola cases were
identified in Liberia, April 8th 2014, up until the latest recorded case, January 19th 2015.
Aim
My aim is to learn more about the SIR model for epidemics and understand how it can be used to
predict the evolution of a disease. Firstly, I will describe and interpret the SIR model used for
infectious diseases and then apply the model to the Ebola outbreak in Liberia using secondary data,
utilizing Euler’s method to solve for the system of differential equations. After modelling the initial
data of the outbreak in Liberia, I will calculate the herd immunity threshold and create a second SIR
model, factoring in the vaccinations to analyze the effects it would have had on the spread of Ebola
through the Liberian population.
The SIR Model
The SIR model was first developed in the early 20th century and has subsequently been applied to
various infectious diseases. The model divides a population into three compartments: S, I and R
representing the susceptible, infectious and recovered.3 The model is used to understand how an
epidemic spreads through a population over a period of time, therefore, our independent variable is
time and will be referred to as t (time in days) in the investigation. Hence, our dependent variables,
S, I and R are functions of time that change according to a set of differential equations.4 An Ordinary
Differential Equation (ODE) is an equation that involves one or more dependent variables ( in this
case
𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛) 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑎 𝑠𝑖𝑛𝑔𝑙𝑒 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 (𝑖𝑛 𝑡ℎ𝑖𝑠 𝑐𝑎𝑠𝑒 𝑡𝑖𝑚𝑒) . 𝑆𝑜 𝑤𝑒 𝑐𝑎𝑛
define these notations as:
St= number of people susceptible to Ebola on day t
It= number of people infected with Ebola on day t
Rt= number of people recovered from Ebola day t
3
“SIR and SIRS Models.” SIR and SIRS models - Generic Model documentation. Accessed August 4, 2020.
https://idmod.org/docs/general/model-sir.html.
4
“The Mathematics of Diseases.” plus.maths.org, April 27, 2020.
https://plus.maths.org/content/mathematics-diseases.
2
, Given that St, It and Rt represent the three possible segments of a population 𝑁, we have
𝑆: + 𝐼: + 𝑅: = 𝑁. We can express each variable as a fraction of the total population 𝑁, and assume
the sum of the fractions equals 1, supposing the population doesn’t change. Making the fraction
sum up to 1 makes it easier to carry out calculations because it avoids using the large numbers for
the population. This simplification also makes it easier for epidemiologists to compare findings with
various populations.5
𝑆: 𝐼: 𝑅:
+ + =1
𝑁 𝑁 𝑁
Assumptions about SIR Model
When modelling real-life situations, predictions and assumptions must be accurate to obtain reliable
results. However, it is important to note that all models are simplifications and extract
approximations of the real world. In the SIR model for epidemics, there are some assumptions that
must be acknowledged.
1. Well-mixed population: the model assumes that each individual has the same probability of
coming into contact with another individual, and ignores the fact the different individuals,
geographically, can be closer.
2. Homogenous mixing: it assumes that everyone is equally susceptible, it does not take into
account that some individuals are more vulnerable than others.
3. Constant population size: we assume there is a fixed population of size N, ignoring births,
deaths and migrants so it is a simplification.
4. No latency: the model assumes that a person is immediately infectious after being infected.
Parameters and Equations
Before deriving the differential equations for the SIR model, it is important to explain the
parameters which affect the rate of change of each of the model compartments: 𝑆: , 𝐼: and 𝑅: .
Firstly, the spread of the epidemic depends on the recovery rate of an individual (𝛾) and the rate of
transmission per contact (𝛽). The rate at which people move from the Infected to the Recovered
5
Abou-Ismail, Anas. “Compartmental Models of the COVID-19 Pandemic for Physicians and Physician-
Scientists.” Sn Comprehensive Clinical Medicine. Springer International Publishing, June 4, 2020.
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7270519/.
3
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