The document describes the Reed-Frost epidemic model, which was developed in the 1920s to mathematically model how diseases spread through populations. The model uses a binomial distribution to calculate the probability of infection rates at different time periods based on variables like population size, number of initial infections, and the probability (p) of transmission. The document provides an example application, modeling a hypothetical disease spreading through a friend group at a high school under different risk levels for transmission probability (p).

1. REED-FROST EPIDEMIC MODEL
IB Candidate Number: 002904-0032
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Mathematically Modeling Epidemics Through the Use of the Reed-Frost Equation
Alexander Kaunzinger
IB Candidate Number: 002904-0032
Biotechnology High School
May, 2015 Exam Session
Exploration: Mathematics SL
Instructor: Laura Widmer

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The study of infectious diseases, originally started out merely as reactions to events that
had already occurred. The focus had been on the avoidance/elimination of active cases (people
that have the disease), followed by treatment of the active cases, and finally prophylactic
treatment with the advent of vaccines in the late 18th century. However, during the early 20th
century, a paradigm shift occurred in the study of infectious disease. Scientists realized that in
order to better prevent and treat infectious diseases, they needed to more completely understand
how the diseases spread, specifically how they spread within a population. These scientists
called the field “Epidemiology”, meaning the study of epidemics and their patterns.
In order to fully access the benefits that this area of study could provide, one needs a way
to accurately determine and model how the disease will spread through a population. An
equation, now known as the Reed-Frost Epidemic Model, was developed by two researchers in
the 1920s working at Johns Hopkins University. These two researchers were named Lowell Reed
and Wade Frost, hence the name of the equation.
The aim of this exploration is to examine the Reed-Frost Epidemic Model, with
discussions of the many variables it contains, the assumptions and limitations of the model, and
its uses. The model will then be used to simulate a hypothetical outbreak using the total
population, number of susceptibles, and the number of infectious people to to calculate in order
to determine the number of infected individuals at a given point in time. This number will be
compared with records from the outbreak to evaluate the accuracy of the Reed-Frost Epidemic
Model. Furthermore, the model will be used to simulate past outbreak using the total population,
number of susceptibles, and the number of infectious people to ultimately determine the number
of infected individuals at a given point in time as an example of the Reed-Frost Epidemic
Model’s practical application in the modern world.
The full equation of the Reed-Frost Epidemic Model is as follows:
r= 1-(1-p)𝐶 𝐶
.
A description of variables is as follows:
● Si denotes the number of susceptible individuals (people that can contract the disease) in
the population at time period i.
● Ci denotes the number of infected individuals (people that have the disease) in the
population at time period i.
● p is the probability of an infected individual infecting a susceptible individual.
● i is the generation of infection.
The Reed-Frost Epidemic Model makes the following assumptions:
● The latent period (time between infection and being infective) and incubation period
(time between infection and the appearance of the first symptoms) are constant. This
essentially means that, once infected, every person will always become infective in the
same amount of time and will always develop symptoms in the same amount of time.
● The period of infectiousness is reduced to a single point.

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● A single attack of the epidemic gives immunity. This means that once an infected person
stops being infectious, they will become immune and will not continue to transmit the
disease.
● The latent period is taken to be the unit of time. Essentially the latent period is equal to i.
The applications of these assumptions is ultimately that any epidemic that starts within a
group of susceptible individuals starts with one infected person, Ci= 1. The epidemic will
continue in stages that are separated by time intervals, each the duration of a latent period, i, until
all individuals are infected or there are no infected individuals left in the population.
There is an initial population of x susceptibles and y infected people with the
understanding that there is spatial homogeneity, which is the random mixing of people within the
population causing each person an equal probability to mix with all the other people in the
population. At any given time, each susceptible has a probability of p of having contact with an
infected person and being infected. Infections are independent events and so have probabilities of
1-p of avoiding infection at time i. A person has a chance of (1-p)𝐶 𝐶 to not be infected at all.
When individuals are infected, they can infect other individuals. Therefore the probability of a
susceptible becoming infected is 1-(1-p)𝐶 𝐶.
When Ci+1 people are infected, there are Si+1 people who will not be infected due to
insufficient contact with an infected person in i and so will continue to be susceptible. Therefore,
there are N possible outcomes of infection that will form the requisite numbers of susceptible
and infected people for time i so that . This all results in a binomial distribution with Si trials,
with each trial having a success probability (which is an infection) of 1-(1-p)𝐶 𝐶.
We can use this binomial distribution to determine the number of infected individuals at
any given time period in an epidemic that satisfies the assumptions discussed earlier. It can be
applied to many hypothetical cases, including the one given below.
Let’s consider some things that are fairly commonplace throughout the entire world:
children and schools. When a child gets sick, it is recommended in many nations that the child
should stay home from school in order to not infect the rest of the school’s population. This style
of dealing with disease is a throwback to the original idea that if you avoid someone that has a
disease, you will not be infected with that disease. Many times this course of action will work.
However, what if there is a scenario in which the infected child or adult for that matter does not
stay home from school? Let’s say that a student had a really important test that they did not want
to make up and so they went to school despite the supposed cold they had or the rash that was on
their arm. An alternative to that is that the student might not even know they have a disease.
There are many infections such as the common cold that can be easily mistaken for simple
environmentally-related allergies. One sneezes both when one has a cold and when one has their
pollen-allergy acting up. The point is, can we model an infection sweeping through the certain
population if a student shows up to school sick?
The answer to that is yes, using the Reed-Frost Epidemic Model. Since this example is a
hypothetical example, let’s say at Regular High School in Regulartown, U.S.A., we can set a
total population size. For simplicity’s sake, let us take a look at a single friend group to

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effectively model the possibilities of the spread of the infection. Our total number of people in
this scenario is set to eight, which is an average size for a high school friend group. This means
N is equal to eight. Infections like the common cold, which is caused by a virus, tend to spread
rather quickly, especially if they are airborne.
We start with a single infected person, C0 = 1, meaning that at the onset of the infection,
which would be time period zero, there would be one infected case. This means that there are
seven susceptible people as, N = Ci + Si, at any given point, therefore 8 = 1 + S0 with S0 = 7.
Since this is a binomial distribution equation, there are only two possibilities for any person in
the friend group, that they remain susceptible, meaning they have not been infected, or they
become infected and so can infect others in the next time period. Therefore, the risk of a
susceptible person becoming infected during the first time period (i=0) is 1-(1-p)𝐶0which is then
equal to 1-(1-p)1 which is equal to p, as shown below:
1-(1-p)𝐶0and C0 = 1
1-(1-p)1
1-(1-p)
1-1+p
p
Now in the second time period, (i=1), there are N possible outcomes, since there are N
people in the school, that means there are 8 possible outcomes of infection, ranging from 0 new
infections to 7 infections in the first period. The binomial distribution of this shows the chances
of N infections is as follows:
P(C1 = 0) = 8C0(p0)(1-p)8
P(C1 = 1) = 8C1(p1)(1-p)7
P(C1 = 2) = 8C2(p2)(1-p)6
P(C1 = 3) = 8C3(p3)(1-p)5
P(C1 = 4) = 8C4(p4)(1-p)4
P(C1 = 5) = 8C5(p5)(1-p)3
P(C1 = 6) = 8C6(p6)(1-p)2
P(C1 = 7) = 8C7(p7)(1-p)1
There is not an equation for the last portion of a standard binomial model (P(C1 = 8) =
8C8(p8)(1-p)0) because there cannot be 8 infected individuals during the second time period. In
accordance with the assumption that a single attack of the epidemic gives immunity, the original
infected person in time period one becomes immune and therefore cannot become infected again,
causing the maximum number of infected individuals to be 7, rather than 8.
In order to further calculate P(C1 = x) for any of the above binomial distributions, we
need to know the value of p. p is the probability of an infected individual infecting a susceptible
individual, which will actually change for each individual depending on a number of factors such
as the proximity of the infected person to the susceptible person, what kind of interactions the
susceptible person had with the infected person, and the susceptible person’s natural immunities
to whatever disease the infected had.

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However, in order for p to be calculated without detailed medical reports on the natural
immunities of the susceptible, and accurate records of the interactions and proximity, if any, of
the infected and susceptible persons. we must assume a value p based on the severity of the
potential risk of the infected person infecting the susceptible person. For example, actions such
as direct contact between the infected person and the susceptible person would carry a high risk
of infection for the susceptible person, a risk we could assume as p = 0.75. These assumptions
should be graduated depending on the level of risk, therefore a high level of risk is p = 0.75, a
moderate level of risk is p = 0.50, and a low level risk is p = 0.25. There is no level of risk for p
= 0 for as long the infected person and the susceptible people are in the same vicinity, there is
always a risk of infection.
Therefore, for the second time period (i=1), we can now calculate the probability of each
scenario occurring at each level of risk, which is as follows:
High Risk (p = 0.75):
P(C1 = 0) = 8C0(p0)(1-p)8
P(C1 = 0) = 8C0(0.750)(1-0.75)8
P(C1 = 0) = 0.00001525878906
P(C1 = 1) = 8C1(p1)(1-p)7
P(C1 = 1) = 8C1(0.751)(1-0.75)7
P(C1 = 1) = 0.0003662109375
P(C1 = 2) = 8C2(p2)(1-p)6
P(C1 = 2) = 8C2(0.752)(1-0.75)6
P(C1 = 2) = 0.0038452148
P(C1 = 3) = 8C3(p3)(1-p)5
P(C1 = 3) = 8C3(0.753)(1-0.75)5
P(C1 = 3) = 0.0230712891
P(C1 = 4) = 8C4(p4)(1-p)4
P(C1 = 4) = 8C4(0.754)(1-0.75)4
P(C1 = 4) = 0.086517334
P(C1 = 5) = 8C5(p5)(1-p)3
P(C1 = 5) = 8C5(0.755)(1-0.75)3
P(C1 = 5) = 0.2076416016
P(C1 = 6) = 8C6(p6)(1-p)2
P(C1 = 6) = 8C6(0.756)(1-0.75)2
P(C1 = 6) = 0.3114624023
P(C1 = 7) = 8C7(p7)(1-p)1
P(C1 = 7) = 8C7(0.757)(1-0.75)1
P(C1 = 7) = 0.2669677734

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Moderate Risk (p = 0.50):
P(C1 = 0) = 8C0(p0)(1-p)8
P(C1 = 0) = 8C0(0.500)(1-0.50)8
P(C1 = 0) = 0.00390625
P(C1 = 1) = 8C1(p1)(1-p)7
P(C1 = 1) = 8C1(0.501)(1-0.50)7
P(C1 = 1) = 0.03125
P(C1 = 2) = 8C2(p2)(1-p)6
P(C1 = 2) = 8C2(0.502)(1-0.50)6
P(C1 = 2) = 0.109375
P(C1 = 3) = 8C3(p3)(1-p)5
P(C1 = 3) = 8C3(0.503)(1-0.50)5
P(C1 = 3) = 0.21875
P(C1 = 4) = 8C4(p4)(1-p)4
P(C1 = 4) = 8C4(0.504)(1-0.50)4
P(C1 = 4) = 0.2734375
P(C1 = 5) = 8C5(p5)(1-p)3
P(C1 = 5) = 8C5(0.505)(1-0.50)3
P(C1 = 5) = 0.21875
P(C1 = 6) = 8C6(p6)(1-p)2
P(C1 = 6) = 8C6(0.506)(1-0.50)2
P(C1 = 6) = 0.109375
.
P(C1 = 7) = 8C7(p7)(1-p)1
P(C1 = 7) = 8C7(0.507)(1-0.50)1
P(C1 = 7) = 0.03125
Low Risk (p = 0.25):
P(C1 = 0) = 8C0(p0)(1-p)8
P(C1 = 0) = 8C0(0.250)(1-0.25)8
P(C1 = 0) = 0.100112915
P(C1 = 1) = 8C1(p1)(1-p)7
P(C1 = 1) = 8C1(0.251)(1-0.25)7
P(C1 = 1) = 0.2669677734
P(C1 = 2) = 8C2(p2)(1-p)6
P(C1 = 2) = 8C2(0.252)(1-0.25)6
P(C1 = 2) = 0.3114624023
P(C1 = 3) = 8C3(p3)(1-p)5

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P(C1 = 3) = 8C3(0.253)(1-0.25)5
P(C1 = 3) = 0.2076416016
P(C1 = 4) = 8C4(p4)(1-p)4
P(C1 = 4) = 8C4(0.254)(1-0.25)4
P(C1 = 4) = 0.086517334
P(C1 = 5) = 8C5(p5)(1-p)3
P(C1 = 5) = 8C5(0.255)(1-0.25)3
P(C1 = 5) = 0.0230712891
P(C1 = 6) = 8C6(p6)(1-p)2
P(C1 = 6) = 8C6(0.256)(1-0.25)2
P(C1 = 6) = 0.0038452148
.
P(C1 = 7) = 8C7(p7)(1-p)1
P(C1 = 7) = 8C7(0.257)(1-0.25)1
P(C1 = 7) = 0.0003662109375
From these calculated probabilities, we can see that the original infected person has a
much higher chance of infecting more individuals during the high risk scenarios, in more than
50% of the scenarios infecting six or all seven susceptible individuals. During the moderate risk
scenarios, the original infected person most often infected between three and five of the

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susceptible individuals. During the low risk scenarios, the original infected person most often
infected between one and three susceptible individuals. Again, no scenario was calculated with
zero or 100% risk as all of those scenarios would have come to the same answer for each trial.
In order to continue in this hypothetical example, it becomes necessary to be able to
calculate all the iterations of each time period. For example, if zero people were infected in the
second time period (i=1), then the epidemic is over as the original infected person did not infect
anyone while also becoming immune after the time period ended. Likewise, if all seven
susceptible people were infected, then the epidemic is also over as all the susceptible people are
infected and so will become immune after the next time period is over. However, if one to six
people were infected in the second time period, then the epidemic continues onto a third time
period, with one immune individual, one to six infected, and the remaining of the eight total
people being susceptible. After that time period, the infected people move to the immune
category and the susceptibles that were infected during the third time period move to the infected
category. Those not infected stay in the susceptible. The epidemic will continue until all the
susceptible people have been infected and have moved from the infected category to the immune
category or there are no more infected people.
However, if we assume that there is a high risk of infection (p=0.75) as these people are
all good friends and so would have a large amount of contact with an infected individual in their
midst, we can further determine the most likely course of the epidemic during the third time
period (i=2). If we look back to the results of the high risk model during the second time period
(i=1), we see that the most likely number of susceptible people that were infected was six. The
individual initially infected in the first time period (i=0) has now become immune, leaving us
with a total population size of eight, a susceptible population size of one, an infected population
of six, and an immune population of one. Now the immune population is essentially irrelevant as
that individual is no longer infectious nor can that individual be infected.
Now in the third time period, (i=2), there are only two possible outcomes, since there is
only one susceptible person. Either that susceptible person becomes infected or they do not. The
binomial distribution of this shows the chances of infection is as follows:
P(C2 = 0) = 2C0(p0)(1-p)2
P(C2 = 1) = 2C1(p1)(1-p)1
Again, we have to make assumptions of risk, which are again graduated depending on the
level of risk, therefore a high level of risk is p = 0.75, a moderate level of risk is p = 0.50, and a
low level risk is p = 0.25. Again, there is no level of risk for p = 0 for as long the infected people
and the susceptible person are in the same vicinity, there is always a risk of infection.
Therefore, for the second time period (i=2), we can now calculate the probability of each
scenario occurring at each level of risk, which is as follows:
High Risk (p = 0.75):
P(C2 = 0) = 2C0(p0)(1-p)2

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P(C2 = 0) = 2C0(0.750)(1-0.75)2
P(C2 = 0) = 0.0625
P(C2 = 1) = 2C1(p1)(1-p)1
P(C2 = 1) = 2C1(0.751)(1-0.75)1
P(C2 = 1) = 0.375
Moderate Risk (p = 0.50):
P(C2 = 0) = 2C0(p0)(1-p)2
P(C2 = 0) = 2C0(0.500)(1-0.50)2
P(C2 = 0) = 0.25
P(C2 = 1) = 2C1(p1)(1-p)1
P(C2 = 1) = 2C1(0.501)(1-0.50)1
P(C2 = 1) = 0.50
Low Risk (p = 0.25):
P(C2 = 0) = 2C0(p0)(1-p)2
P(C2 = 0) = 2C0(0.250)(1-0.25)2
P(C2 = 0) = 0.5625
P(C2 = 1) = 2C1(p1)(1-p)1
P(C2 = 1) = 2C1(0.251)(1-0.25)1
P(C2 =
1) =
0.375

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From these calculated probabilities, we can see that the six infected people have a much
higher chance of infecting the susceptible individual during the moderate scenarios, but at this
point with only one susceptible person against six infected people, the probability that the
susceptible person would not be infected is small.
At this point, in both the scenario where the last susceptible person is infected or the
scenario in which the last susceptible person remains susceptible, the epidemic is over. If the last
susceptible person is infected, then there are no more susceptible people to infect, so after that
person becomes immune, there is no remaining disease. Likewise, in the case of the susceptible
person not being infected, the six infectious people will become immune after this time period
and can no longer transmit the disease, therefore the disease has died out and the susceptible
person will not contract the disease.

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Ennis, D. (n.d.). Reed-Frost Epidemic Model. Retrieved December 16, 2014, from
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Riedel, S. (2005). Edward Jenner and the History of Smallpox and Vaccination. Retrieved
December 21, 2014, from http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1200696/
Using Probabilistic Models to Infer Infection Rates in Viral Outbreaks. (2007). Retrieved
December 19, 2014, from http://www.stats.ox.ac.uk/__data/assets/file/0013/3352/
infection_rates.pdf
Chain binomial epidemic models. Retrieved December 21, 2014, from
http://lalashan.mcmaster.ca/theobio/mmed/images/8/8f/Reed-frost-5jun2013.pdf