# Modelling the H1N1 Epidemic in Canada

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Introduction

IB Maths SL Internal Assessment | Tanya Waqanika

CITY AND ISLINGTON SIXTH FORM COLLEGE |

IB Maths SL Internal Assessment |

Modelling the re-emerging H1N1 virus in Canada |

Tanya Waqanika |

Table of Contents

Rationale

Background

Epidemiology

The SIR Model

The SIR Model’s Differential Equations

The S – Susceptible equation

An Alternative Method for Plotting the SIR model on a Graph

Applying the SIR Model to the Re-Emerging H1N1 virus in Canada

The Basic Reproductive Numberand β

Applying the Calculated Values from the data on the SIR Model

Vaccination and Herd Immunity with the Basic Reproduction Number

Conclusion and Reflection

Appendices

Appendix 1 – Excel data used to produce Graph 2

Appendix 2 – Model, Graph and Calculations for Graph 3

Bibliography

## 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.^{[1]} 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.

Middle

^{[5]}to devise a simpler way of producing a line that is approximate to the line produced by the rate of change in the S compartment as the epidemic progresses. Remembering that S in the SIR model is a function of time, I can use this formula to produce a similar equation for S.

Formula taken from [http://en.wikipedia.org/wiki/Euler_method]

If we then assume that dS/dt can be written as:

Where:

dS/dt --βSI

(n + 1) – Any time in the future

n – The present time

I can then rearrange this in terms S, which would give an approximate formula for the integration of dS/dt:

The same method applied to dI/dt devises the suitable formula for its approximate integration as well:

To keep the model simple, we can rely on the previous assumptions made about the SIR model. Since S + I + R = N, we can rearrange this to make R = N – S – I. When this is written in a similar form to the formulas above, we get:

## Applying the SIR Model to the Re-Emerging H1N1 virus in Canada

The population of Canada is approximately 35,295,770^{[6]}. Within the population 8,292,255^{[7]} is the estimated number of Canadians vaccinated against H1N1 since 2012. The Canadian government has a website called FluWatch, which collects data from all over Canada concerning the number of cases that crop up from each type of influenza strain. The very first case of H1N1 was detected on August 25th 2013 and FluWatch carefully monitors the number of H1N1 cases that are detected and publishes these numbers every week. I have been collecting these cases and have presented them in the following table:

Week | Number of Cases |

35 | 4 |

36 | 1 |

37 | 0 |

38 | 1 |

39 | 1 |

40 | 2 |

41 | 2 |

42 | 1 |

43 | 1 |

44 | 14 |

45 | 19 |

46 | 38 |

47 | 38 |

48 | 73 |

49 | 145 |

50 | 281 |

51 | 503 |

52 | 808 |

53 | 1378 |

Conclusion

Understand Seasonal Flu, Human Swine Flu and Hand-Foot-Mouth Diseases. 2014. [e-book] Hong Kong: Centre for Health Protection. p. 19. Available through: Centre for Health Protection http://www.chp.gov.hk/files/pdf/understand_seasonal_flu,_human_swine_flu_and_hand-foot-mouth_diseases_eng.pdf [Accessed: 13 Feb 2014].

Using Calculus to Model Epidemics. 2014. [e-book] Available through: homepage.math.uiowa.edu http://homepage.math.uiowa.edu/~stroyan/CTLC3rdEd/3rdCTLCText/Chapters/Ch2.pdf [Accessed: 13 Feb 2014].

Wikipedia. 2014. Epidemic model. [online] Available at: http://en.wikipedia.org/wiki/Epidemic_model [Accessed: 13 Feb 2014].

Wikipedia. 2014. Euler method. [online] Available at: http://en.wikipedia.org/wiki/Euler_method [Accessed: 13 Feb 2014].

Wikipedia. 2014. Basic reproduction number. [online] Available at: http://en.wikipedia.org/wiki/Basic_reproduction_number [Accessed: 13 Feb 2014].

Wikipedia. 2014. List of countries by life expectancy. [online] Available at: http://en.wikipedia.org/wiki/List_of_countries_by_life_expectancy#List_by_the_World_Health_Organization_.282013.29 [Accessed: 13 Feb 2014].

Wikipedia. 2014. Herd immunity. [online] Available at: http://en.wikipedia.org/wiki/Herd_immunity [Accessed: 13 Feb 2014].

Wikipedia. 2014. Mathematical modelling of infectious disease. [online] Available at: http://en.wikipedia.org/wiki/Mathematical_modelling_of_infectious_disease [Accessed: 13 Feb 2014].

| Page

[1] http://www.ctvnews.ca/health/h2n1-virus-what-you-need-to-know-1.1609554

[2] http://www.oxforddictionaries.com/definition/english/epidemiology

[3] http://homepage.math.uiowa.edu/~stroyan/CTLC3rdEd/3rdCTLCText/Chapters/Ch2.pdf

[4] http://en.wikipedia.org/wiki/Epidemic_model

[5] http://en.wikipedia.org/wiki/Euler_method

[6] http://www.statcan.gc.ca/start-debut-eng.html

[7] http://statcan.gc.ca/tables-tableaux/sum-som/l01/cst01/health202a-eng.htm

[8] http://en.wikipedia.org/wiki/Basic_reproduction_number

[9] http://en.wikipedia.org/wiki/Mathematical_modelling_of_infectious_disease

[10] http://en.wikipedia.org/wiki/List_of_countries_by_life_expectancy#List_by_the_World_Health_Organization_.282013.29

[11]http://en.wikipedia.org/wiki/Basic_reproduction_number; http://www.chp.gov.hk/files/pdf/understand_seasonal_flu,_human_swine_flu_and_hand-foot-mouth_diseases_eng.pdf

[12] http://motivate.maths.org/content/DiseaseDynamics/Activities/Modelling

[13] http://en.wikipedia.org/wiki/Herd_immunity

[14] http://en.wikipedia.org/wiki/Basic_reproduction_number

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