Chapter 1 Introduction

1.1 Goals of this E-book

This e-book provides an introduction to mathematical modeling in epidemiology, focusing on how to develop and analyze models, and how they inform public health decisions. We will learn about compartmental and agent-based models and their extensions that incorporate heterogeneities (age, space, behaviour). We will also explore disease control strategies. Simulation-based demonstrations and coding exercises will help connect theory to real-world insights.

1.2 Recommended Background

Recommended pre-requisites are: introductory calculus, ordinary differential equations, linear algebra, and probability and statistics. For additional review, we suggest the following open-access resources. Topics to be familiar with have also been noted with the chapters they can be found in.

• Calculus: Active Calculus
  1. Active Prelude to Calculus
  2. Active Calculus
  3. Active Calculus Multivariable:
     • 11.9 Change of Variables (for understanding the Jacobian Matrix)
       
• Calculus (alternative): OpenStax Calculus
  1. Calculus Volume 1
  2. Calculus Volume 2
  3. Calculus Volume 3
     • 5.7 Change of Variables in Multiple Integrals (for understanding the Jacobian Matrix)
       
• Linear Algebra: Understanding Linear Algebra
  • Ch 1 Systems of equations
  • Ch 2 Vectors, matrices, and linear combinations
    • Ch 2.1 Vectors and linear combinations
    • Ch 2.2 Matrix multiplication and linear combinations
  • Ch 3 Invertibility, bases, and coordinate systems
    • Ch 3.4 Determinants
  • Ch 4 - Eigenvalues and eigenvectors
    
• Basic Understanding of Complex Numbers
  
• Probability and Statistics: OpenIntro Statistics

1.3 History of Mathematical Epidemiology

Throughout human history, diseases have spread through groups of people.

Definition 1.1 (Epidemic) An Epidemic is a widespread occurrence of an infectious disease at a particular time.

Definition 1.2 (Endemic Disease) An Endemic disease is one that is always present in certain populations.

One of the earliest and deadliest epidemics known is the Black Death, also known as the bubonic plague, which started in Asia in 1346 and spread throughout Europe. Another widely known epidemic is the Spanish Flu epidemic of 1918-19 which caused more than 50 million deaths. Malaria and typhus are examples of endemic diseases (Definition 1.2). Most recently, COVID-19 caused a worldwide lockdown from 2020-2021, infecting 779 million and killing 7 million. The purpose of epidemiology is to understand the causes of a disease, predict its course, and develop ways of controlling its spread.

The first model in mathematical epidemiology was the work of Daniel Bernoulli on the inoculation of smallpox. Bernoulli calculated the increase in life expectancy if smallpox were removed as a cause of death. The beginnings of compartmental modelling started through the work of public health physicians. In 1906, W.H. Hamer proposed that the spread of infection should be observed by looking at individuals in compartments: susceptible and infectious. Dr. Ronald Ross, who worked on malaria, was the first to introduce the concept of:

Definition 1.3 (Basic Reproductive Number) The basic reproductive number (R₀), which quantifies the number of secondary infections caused by one case in a susceptible population. The framework was then expanded and formalized by Kermack and McKendrick in the late 1920s and early 1930s, laying the foundation for modern epidemic modeling. [3]

1.4 Role of Models in Public Health

Due to ethical concerns, it is difficult to design controlled experiments in epidemiology. In addition, epidemiologists may encounter incomplete and inaccurate data from measurement error, which can hinder parameter estimation and model fitting. [26] [4]. This is why mathematical modelling in epidemiology plays a major role in our understanding of mechanisms of disease spread. This understanding can suggest strategies for public health and health care responses regarding diseases and epidemics [25] [4] [22]. Models can also identify variables that data may not show. For instance, some models can identify threshold behaviors of epidemics—an indicator of when a disease will become an epidemic, or dies out [4]. These indicators are based on (R₀) (Definition 1.3) in which, if the variable is above, below, or equal to 1, an epidemic (Definition 1.1) may occur, die out or become endemic (Definition 1.2), respectively. This information could then be used to determine vaccination plans within a population as well as individually [4]. Being able to understand the future spread of disease will give insurmountable advantages in developing a plan of action to combat new infections; additionally, the development of herd immunity (Definition 2.8) can safeguard an entire population through communal protection.

Compartmental models used today (originating from the Kermack-McKendrick model), can show disease transmission while observing time and infection number in relation to an epidemic [25]. These models help understand diseases by dividing a population into compartments, such as the SIR model. These models utilize diverse mathematical equations and relationships to understand the flow and movementts between these models, defining these connections help predict the spread of disease regarding an outbreak. Compartmental models differ with diseases, and their development often relies on various policies. These models must undergo rigorous sensitivity testing to determine ideal parameters that impact model outcomes [22].

In Chapter 3 of this book, we will be exploring the models that are used in modern epidemiology, such as the SIR model [25].