1. Rationale:
Over my highschool career my interest in the area of medicine and medical science has deepened. With vaccination being a topic of conversation in the forefront of medical news, I thought it would be valid to explore from a more statistical, and modelled perspective.
After conducting research into various models to present my data, the S-I-R model proved to demonstrate the spread of infectious diseases most effectively. It allowed for the incorporation of the effect of vaccination in its mathematical processes. Upon reflection, I wondered if there was any direct correlation between the model itself and my progression through the standard level mathematics course. I realised that the S-I-R model relates directly to the topic of calculus, with the use of differentiation being present in the mathematical process of constructing the graph. Furthermore, The model itself is frequently referred to as the differential equation model which further consolidated my understanding of the apparent connection. The S-I-R model follows and graphs the spread of disease following the human conditions of susceptible (S), infectious (I) and removed (R). Those three states, as described and diagrammed below (refer to figure 1), follow the common pattern through which a person within a population who is exposed to a particular disease would follow.
Susceptible: This state refers to the initial pre-exposure period wherein a person has developed no immunity to the infectious disease present. This person is at risk of contracting said disease.
Infectious: The second stage refers to the period in which an individual is infected and is able to spread the disease to other susceptible individuals
Removed: Due to the process of established immunity, an individual can be assumed to become immune, or removed, after having contracted and survived an infectious disease. Notably, this state can be achieved without the presence of a vaccine. This person is no longer susceptible to that strain of the disease.
Figure 1. The flow of disease
Contraction and spread following
the parameters of the S-I-R model
2. Introduction
2.1: Aim and applications
My mathematical process will involve modelling the spread of two different infectious diseases, and then further exploring the effectiveness of vaccines on limiting the spread of those diseases. Within this paper I aim to explore the effectiveness of vaccination use on the spread of communicable diseases through the medium of mathematical modeling. In order to determine effectiveness I will be replicating the presence of a disease within a population, with the introduction of vaccine versus no introduced vaccine. I have chosen not to model the spread of smallpox, as it is not considered to be relevant on either a global or local scale due to its eradication. I have decided to apply the S-I-R model to the common infectious disease measles, and the more recently occurring disease, Coronavirus (2020-nCoV).
2.2: Variables used within the model
The S-I-R model used within my exploration requires technology to model its components effectively. The variables used within the model which I will refer to within my exploration include:
Susceptible (S), infectious (I) and removed values (R)
The days over which the model is demonstrating i.e. course of disease.
The population size on which the graph is being modelled on, and the R₀ (R naught) value.
The R₀ value represents the basic reproduction number of an infectious disease. It essentially represents the number of new cases (infected individuals) generated by an individual, on average, over the course of the infectious period.
Constants including:
– the contact rate, how often a contact between a susceptible and an infected individual results in a new infection
The value can be found using the equation:
R₀ =
– the recovery rate, how quickly an individual will recover from the illness and shift into the recovered population
The value can be calculated by the formula
D=1, where D is the duration of infection
R₀ value conditions:
When the R₀ > 1 the spread of the disease is exponentially progressive, with new cases allowing for the disease to increase in rate of spread.
When the R₀ < 1 disease transmission ceases and the disease is removed from the population.
The R₀ values I will apply to my exploration are for infectious diseases measles and 2020-nCoV.
Measles R₀ value = 17
2020-nCoV R₀ value = 2 – 5.47
This value is presented as a range due to the variance globally of its basic reproductive number. However, as a range cannot be applied to the model the average must be found
Calculating the average of the 2020-nCoV R₀ value range:
Adding the lowest and highest values
2+5.47=7.47
Dividing that value by the number of variables added together
7.472=3.735
∴ 2020-nCoV R₀ value = 3.735
The R₀ value can also be used to predict the proportion of a population which requires vaccination to prevent the spread of an infectious disease .
Through vaccination of a population the concept of herd immunity can be established. It is considered to be a more indirect preventative protection of a population from an infectious disease. Through the administering of vaccination both those who have been vaccinated, and those who are unable to receive vaccination (e.g. immunocompromised individuals such as the elderly or children). Vaccination reduces the population within the susceptible bracket of the S-I-R model.
An assumption can be made that:
VT= 1 -1R₀
Where VT represents the proportion of the population which needs to be vaccinated. The percentages solved below represent the proportion of the population which need to be vaccinated in order to achieve herd immunity.
When using the example of measles where the R₀ value =17
VT= 1 -117
VT= 1617 ≈ 0.94
∴ 94% of the population needs to be vaccinated to ensure the establishment of herd immunity
When using the example of 2020-nCoV where the R₀ value = 3.735
VT= 1 -13.735
VT= 547747 ≈ 0.73
∴ 73% of the population needs to be vaccinated to ensure the establishment of herd immunity
There is a difference in percentage between the two different infectious diseases due to the variance in their contagiousness as seen through their R₀ value.
As the R₀ value is greater for that of measles it can be considered to be more contagious and therefore the percentage of the population which requires vaccination is greater.
S-I-R values:
Certain mathematical expressions have been established to represent the number of the people within the populations of susceptible, infectious and removed.
These expressions are used to graph the S-I-R model.
They are seen below:
a.) dSdt=-IS
b.) dIdt=IS-I
c.) dRdt=I
a.) dSdt symbolises the rate of change of those who are within the susceptible population to
the illness with respect to time.
b.) dIdtsymbolises the rate of change of those who are within the infected population with
respect to time. If this value is high the number of people who are moving into the infected group is increasing, which denotes that the rate of spread of infection is great. If this value is zero however then there is no change in the number of people shifting from the susceptible group to the infected group, therefore the rate of infection remains steady. When the dIdtvalue is negative, the number of people becoming infected is decreasing.
c.)dRdtsymbolises the rate of change of those who have recovered with respect to time.
3. Modelling case studies:
3.1: Sydney measles outbreaks 1867
I chose to model this particular year of disease due to its relevance in relation to more current measles outbreaks. I live in Sydney, Australia, where a notable decline in measles vaccinations (MMR vaccine) has resulted in more frequent outbreaks. The 1867 outbreak occurred during a period pre-vaccine, which allowed me to infer that none of the population could have established a recovered status. This assumption has faults, as certain individuals could have established immunity from prior exposure to the disease, however as no valid data was recorded on this matter, I have chosen to set my initial recovered population size at 0.
At the time of the 1867 measles outbreak, the population of Sydney was resting at 1 million.
Due to the lack of interference of vaccination, the initial susceptible population can be assumed to be 1 million.
The initial infectious population can be assumed to be one, as there is no definitive data of a collective group which initiated the spread of the disease.
The outbreak was seen to be most active over the year of 1867, which therefore resulted in me setting the course of the disease at 365 days.
Solving for the value:
As stated before, the equation to solve for the value is
D=1
The duration of infection is 10 days,
∴ D=10
Sub in value for D,
10=1
Solve for ,
=110=0.1
Solving for the value:
As stated before, the equation to solve for the value is
R₀ =
In the case of measles:
R₀=17
Sub in value for R₀ into first equation,
17=
Sub in value for =110,
17=(110)
Solve for value:
=1.7
Initial values for input:
S(0)=999,999
I(0)=1
R(0)=0
Course of disease =365 days
=1.7
=0.1
Those values were then substituted into the computer function which then results in the graph seen in figure 2.
The graph was scaled to only show 60 days of progression as beyond that point all values showed no further movement
This graph displays the movement of people, without the interference of vaccination.
Figure 2. S-I-R model displaying the flow of people during the 1867 measles outbreak
1867 outbreak of measles with hypothetical vaccination introduction:
In the most ideal case vaccination would be
VT= 1 -117
VT= 1617 ≈ 0.94
∴ 94% of the population needs to be vaccinated to ensure the establishment of herd immunity
Solving for new removed (R)population
0.94 1,000,000 =940,000
∴ Nu = 0.94
S(0)=999,999
I(0)=1
R(0)=0
Course of disease =365 days
=1.7
=0.1
Nu=0.94
There is also the further implementation of the value Nu, this value represents the rate, or percentage of which the susceptible population becomes vaccinated.
Those values were then substituted into the computer function which then results in the graph seen in figure 3.
The graph was scaled to only show 10 days of progression as beyond that point all values showed no further movement
This graph displays the movement of people, with the hypothetical interference of vaccination.
Figure 3. S-I-R model displaying the flow of people during the 1867 measles outbreak with hypothetical vaccination intervention