Internal assesment Report 1

“Mathematical Modeling of COVID-19 Spread: A Comparative Analysis of

Prediction Methods in Canada”

Zachary

Mathematics Analysis and Approaches Standard Level Internal Assessment

Internal assesment Report 2

Abstract

This Report compares mathematical models used in epidemiology to predict the spread of

infectious diseases, focusing on the Basic Reproduction Number (R₀), the SIR model

(Susceptible, Infected, Recovered), and SEIR model (Susceptible, Exposed, Infected,

Recovered). Using real data from the first year of the COVID-19 outbreak in Canada, we analyze

the accuracy of these models in predicting the disease’s spread. By comparing the predictions

with actual data, we determine which model provided the most accurate results for understanding

and forecasting the pandemic’s progression in Canada.

Keywords:
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Mathematical Modeling of COVID-19 Spread: A Comparative Analysis of Prediction Methods in Canada

Introduction

Understanding how infectious diseases spread is essential for effective prevention and
control, as it informs public health policies, guides healthcare preparedness, and helps
individuals take informed precautions. This knowledge also drives research and development for
new treatments and vaccines, while facilitating international cooperation in global health efforts.
Ultimately, it plays a critical role in safeguarding public health and enhancing community
resilience against outbreaks.

The policies implemented during COVID-19 were a mix of effective strategies and
challenges. Many measures, such as lockdowns, mask mandates, and vaccination campaigns,
were based on the best available science at the time and helped reduce transmission. However,
some policies faced criticism for being inconsistent or poorly timed, which may have contributed
to public confusion and resistance.

If more accurate mathematical prediction methods for disease spread had been available earlier,
several improvements could have been made:

1. Timely Interventions: More precise modeling could have led to earlier and more
targeted interventions, potentially preventing widespread outbreaks.
2. Resource Allocation: Better predictions would help allocate resources more efficiently,
ensuring that healthcare systems were prepared in areas of high transmission.
3. Public Communication: Clearer predictions could enhance public understanding and
compliance with guidelines, reducing misinformation.
4. Tailored Approaches: Improved models could allow for more localized strategies,
adapting responses based on specific community needs and transmission dynamics.

Overall, while many policies were effective, enhanced predictive modeling could have refined
strategies and improved outcomes during the pandemic.

Mathematical models predict the spread of infectious diseases by using equations to

represent the dynamics of disease transmission within a population. And what follows are the
key elements for those models:
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Population Dynamics: Models often categorize individuals into different compartments,

such as susceptible, infected, and recovered (SIR models). This helps track how individuals
move between states.

Transmission Rates: Models incorporate parameters that define how quickly the disease
spreads, including the basic reproduction number (R0), which indicates the average number
of new infections generated by one infected individual in a fully susceptible population.

Infection Duration: The average time an individual remains infectious is factored into the
models, influencing how quickly a disease can spread.

Intervention Effects: Models can simulate the impact of interventions (e.g., vaccinations,
social distancing) by adjusting parameters to reflect changes in behavior or population
immunity.

Stochastic Elements: Some models incorporate randomness to account for variations in

individual behavior and interactions, providing a more nuanced understanding of potential
outbreak scenarios.

Geographical and Demographic Factors: Models can include variations in population

density, movement patterns, and demographic characteristics to make predictions more
accurate for specific regions.

Data Integration: By using real-time data (e.g., case counts, mobility data), models can be
continually updated, improving their accuracy and reliability in predicting future trends.

By combining these elements, mathematical models can provide insights into how a disease
might spread, helping inform public health strategies and responses.

2. Mathematical Models

The Basic Reproduction Number, or R₀, is an important measure in epidemiology that

indicates the average number of new infections caused by one infected person in a fully

susceptible population. If R₀ is greater than 1, it means each infected person is likely to infect
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more than one other person, which can lead to outbreaks. Conversely, if R₀ is less than 1, the

disease is expected to decline and eventually disappear. R₀ is influenced by several factors,

including how easily the disease spreads, how often people come into contact with each other,

and how long an infected person remains contagious.

The concept of R₀ originated in the early 20th century when mathematicians and epidemiologists
started creating models to study infectious diseases, such as the SIR model, which helped explain
how diseases spread. A key advancement came in 1927 from William Kermack and Anderson
McKendrick, who developed a mathematical model that introduced the idea of the effective
reproduction number, paving the way for the formal definition of R₀. Over the mid to late 20th
century, researchers continued to refine these models, recognizing R₀ as essential for
understanding outbreaks and guiding public health responses. In recent years, R₀ has gained
attention during outbreaks like SARS in 2003, H1N1 in 2009, and COVID-19, as officials used it
to evaluate transmission potential and shape interventions.

The SIR model is a mathematical framework used to analyze the spread of infectious diseases
by dividing the population into three compartments: Susceptible (S), Infected (I), and Recovered
(R). Susceptible individuals can become infected through contact with those already infected,
with the rate of infection influenced by the transmission rate (β). Infected individuals eventually
recover at a rate defined by the recovery rate (γ), which determines how long they remain
infectious. The model is governed by a set of differential equations that describe changes in each
compartment over time, allowing researchers to simulate outbreak dynamics and predict
infection peaks. While the SIR model provides valuable insights into disease spread and the
effectiveness of control strategies, it has limitations, including assumptions of a closed
population and uniform susceptibility, which may not always reflect real-world scenarios.

The SIR model is governed by a set of differential equations that describe the changes in each
compartment over time:

1. Change in Susceptible:
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dS −β S I
=
dt N

This equation shows that the number of susceptible individuals decreases as they become
infected.

2. Change in Infected:

dI β SI
= −γI
dt N

This equation accounts for new infections and the recovery of infected individuals.

3. Change in Recovered:

dR
=γI
dt

This equation indicates that recovered individuals increase as infected individuals

recover.

While the SIR model offers valuable insights into disease dynamics, it also has limitations. One
major assumption is that it operates within a closed population, meaning there are no births,
deaths, or migration, and it presumes that recovered individuals gain permanent immunity.
Additionally, the model treats the population as homogeneous, failing to account for variations in
susceptibility and differences in contact patterns among individuals, which can affect the spread
of the disease in more complex real-world scenarios.

The SEIR Model to be added …

Case Study

 POPULATION OF CANADA: Approximately 38 million (2020 estimate)

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COVID-19 Outbreak Data in Canada (First 20 Weeks)

Suspecte Infected Recovered Active

d Cases Cases Cases Cases

Week 1 500 50 0 50
Week 2 700 120 0 120
Week 3 1000 300 5 295
Week 4 1500 600 20 580
Week 5 2000 1200 100 1100
Week 6 2500 2000 500 1500
Week 7 3000 3500 1200 2300
Week 8 4000 5000 2000 3000
Week 9 5000 7000 3500 3500
Week 10 6000 9500 5000 4500
Week 11 7500 12000 6500 5500
Week 12 9000 15000 8500 6500
Week 13 11000 18000 10000 8000
Week 14 13500 22000 12000 10000
Week 15 15000 25000 14000 11000
Week 16 17000 30000 16000 14000
Week 17 20000 35000 18000 17000
Week 18 23000 40000 20000 20000
Week 19 26000 45000 23000 22000
Week 20 30000 50000 25000 25000

!!! Not accurate data to be adjusted when finding good source and to be applied on the

three models and check how it would be

Results and conclusion.

To be filled with the result and the conclusion for the three models and find the advantage

and disadvantage of each model