# Modelling the H1N1 Epidemic in Canada

by ribbonrebel31 (student)

IB Maths SL Internal Assessment | Tanya Waqanika

## Rationale

As I would like to study Medicine in the future, I decided to investigate the mathematics of epidemiology. Epidemiology is also interesting to me because the constant and fast evolution of dangerous pathogens and diseases often threatens the human population. In the last few decades alone, epidemics such as AIDs, Measles, Avian Influenza (H5N1) and Swine Influenza (H1N1 (pdm09)) have caught the attention of the world. 2009 was a particularly significant year, as a certain strain of Swine Influenza was declared by the World Health Organisation to be potentially pandemic due to a series of epidemics all over the world. Even though the world never succumbed to a Swine Influenza pandemic, in December, a published article stated H1N1 was re-emerging in Canada. This caused a stir in the public and encouraged individuals to get themselves vaccinated against H1N1 to prevent another epidemic breaking out. Having lived in Canada for a year, this situation was incredibly interesting and relevant to me as I have a lot of friends and family living in Canada as well. I decided to use the information and statistics on the re-emerging H1N1 virus to try and predict whether the virus had the potential to be epidemic again, how long the epidemic would last if it broke out and how many individuals would need to be vaccinated in order to achieve herd immunity.

## Epidemiology

Epidemiology is defined as a branch of medicine that deals with the study of the incidence, distribution and possible control of diseases and other factors relating to health. Within epidemiology, mathematics is used to produce epidemic models in order to predict the potential spread of a disease throughout a population. The use of mathematical modelling in epidemiology is particularly useful when a new infectious disease is thought to have the potential to cause an epidemic. Models in epidemiology have predicted the spread of diseases such as whooping cough, west Nile virus infections and influenza viruses.

We use mathematical models to predict the spread of a disease because it’s not physically or ethically possible to test it out within lab conditions to get accurate data. To try and get the most accurate representation of the potential spread of a disease through a population, all epidemic models are based on certain assumptions about the population and the disease itself as well as parameters such as how many people an individual comes into contact with on a daily basis, how long an infected individual remains infectious, etc. By using epidemic models, it’s also possible to find and understand the factors that might affect the propagation or hindrance of a disease and devise methods to properly limit or eliminate a disease within a population.

## The SIR Model

One way Epidemiology uses mathematics to predict an epidemic is through the use of compartmental models. The SIR model is one of the simplest epidemiological models in use in epidemiology and was created in 1927 by W.O. Kermack and A.G. Kendrick. It’s the model that subsequent more intricate and detailed models were based on. The SIR model is made up of these compartments;

• S – Susceptibles, the fraction of the population who are capable of being infected by the disease that’s being modelled.
• I – Infecteds, the fraction of the population who are infected by the disease and are able to infect members of the susceptible compartment
• R Recovered/removed, the fraction of the population who have either
1. Recovered from the disease and therefore have been gained
lifelong immunity to the disease
2. Died from the disease.
• N – The total population used in the model, given by the formula S + I + R = N

Figure 1: The SIR model

The SIR model is used to try and predict the course of an epidemic through a population based on the assumptions that:

1. The population is a large and a homogenous mixture
2. Infectious pathogens are spread through the contact or close proximity of susceptible (healthy) and infected individuals, therefore all susceptible individuals have an equal chance of contracting the pathogen and becoming infected at the same rate, β
3. The population is closed and static, which means the SIR model ignores births, deaths and migration over time.
4. Individuals can only be in each compartment once and the model flows in one direction only, from Susceptible, to Infected to Removed
5. All three compartments, S, I and R are dynamic and variable by time, thus, they are all functions of time, t

As the SIR model is compartmental, because of assumption (3), the only way for the number of individuals to change within a compartment is if they move to another compartment and using assumption (4) as a diagram, these changes can be represented as:

Figure 2: The SIR model with rates of compartment change included

Where:
β – The rate at which an infected individual is able to make new infections from the susceptible compartment
γ – The rate at which an infected individual is able to recover and move into the removed compartment.

## The SIR Model’s Differential Equations

As the SIR model is a compartmental model, with the rates depicting the ‘flow’ of individuals into each compartment, these rates can be expressed as differential equations:

Equation 1: The differential equation for S

This is the differential equation that represents the flow of individuals from ...