OMNI course (Part I) organisation

History of Mathematical Epidemiology

11-14 October 2022

Julien Arino (julien.arino@umanitoba.ca)

Department of Mathematics & Data Science Nexus
University of Manitoba*

Canadian Centre for Disease Modelling
Canadian COVID-19 Mathematical Modelling Task Force
NSERC-PHAC EID Modelling Consortium (CANMOD, MfPH, OMNI/RÉUNIS)

* The University of Manitoba campuses are located on original lands of Anishinaabeg, Cree, Oji-Cree, Dakota and Dene peoples, and on the homeland of the Métis Nation.

Outline

  • About Part I of the course
  • This week's lectures
  • A few remarks about terminology
  • Mathematical epidemiology

About Part I of the course

(One Health Modelling for Emerging Infectious Diseases)

An Introduction and three Themes

  1. Introduction
  2. Coronaviruses
  3. Influenza
  4. Environmental transmission of bacterial and fungal pathogens

Each block is 3 weeks, running from the week of 11 October 2022 to the week of 23 January 2023

Except for the Introduction, each block finishes with a week of group work and presentations of the result of group work

Tutorials, assigned work, etc.

  1. Introduction co-taught by several of us

Each Theme block is taught by one instructor:

  1. Coronaviruses: Marina Freire-Gormaly
  2. Influenza: Assefa Woldegebriel
  3. Environmental transmission: Julien Arino

Each block will have assignments, tutorials, etc. The instructor for that block will give you details

This week's lectures - Outline

  • A (super) brief introduction to mathematical epidemiology (rest of these slides)
  • Introduction to compartmental models in epidemiology
    • The SIR epidemic model of Kermack and McKendrick
    • The SIS endemic model
  • Numerical methods for epidemiological models
    • Solving ODE
    • Simulating continuous time Markov chains

A few remarks about terminology

Incidence versus Prevalence

Incidence: number of new cases in a population generated within a certain time period

Prevalence: number of cases of a disease at a single time point in a population

in an epidemiological model is prevalence, not incidence

(I will come back to that)

Exposition versus Exposed

  • Some bright bulb (not sure who) in days of yore: let's call exposed someone who has contracted the disease but is not yet showing symptoms ( SEIR model)

  • "Real" epidemiologist: let's trace people who were exposed to the virus, i.e., people having come into contact with the virus (whether they have contracted the disease or not)

  • Interestingly, I have embarked on a quixotic quest to make people use instead of , only to be told by real epidemiologists that they don't care :)

Epidemic curves

  • Used to record the occurrence of new cases as a function of time
  • When not too many cases, usually "individualised" (bar plots)
  • When number of cases is large, continuous curve

Some more terminology

  • Epidemic: diseases that are visited upon a population
  • Pandemic: epidemic that has spread across a large region, e.g., multiple continents or worldwide
  • Endemic: diseases that reside within a population
  • We don't say "panendemic"

WHO pandemic (influenza) phases

Period Phase Description
Interpandemic 1 No animal influenza virus circulating among animals has been reported to cause infection in humans
2 Animal influenza virus circulating in domesticated or wild animals known to have caused infection in humans and therefore considered a specific potential pandemic threat
Pandemic alert 3 Animal or human-animal influenza reassortant virus has caused sporadic cases or small clusters of disease in people, but has not resulted in H2H transmission sufficient to sustain community-level outbreaks
4 Human-to-human transmission of an animal or human-animal influenza reassortant virus able to sustain community-level outbreaks has been verified
5 Same identified virus has caused sustained community-level outbreaks in at least 2 countries in 1 WHO region
Pandemic 6 In addition to criteria in Phase 5, same virus has caused sustained community-level outbreaks in at least 1 other country in another WHO region

Mathematical Epidemiology

Domain is quite old ..

.. but has only become a thing in recent years!

Daniel Bernoulli (1760)

  • BNF scan or pdf
  • Probably the first epidemic model
  • About petite vérole (smallpox) inoculation

Ross (early 1900)

  • On 20 August 1897, observed malaria parasites in the gut of a mosquito fed several days earlier on a malaria positive human
  • Nobel Prize for Medicine 1902
  • Started considering malaria eradication using mathematical models; for some history, read this 2012 paper

Kermack and McKendrick (1927+)

  • We spend more time on this later
  • Groundbreaking set of papers starting in 1927
    • We will see one particular case, the most well known, but I point out here and point out in Lecture 02 that this is just the tip of the iceberg of their work

Some activity later, but not much until 1990s

  • In recent years, explosion
  • Since the beginning of COVID-19: just nuts..

Some landmarks in mathematical epidemiology (IMBO)

  • Macdonald. The epidemiology and control of malaria. 1957
  • Baroyan, Rvachev et al. Deterministic epidemic models for a territory with a transport network. Kibernetika, 1967
  • Hethcote & Yorke. Gonorrhea Transmission Dynamics and Control. LNBM 56, 1984
  • Anderson & May. Infectious diseases of humans: dynamics and control. 1991
  • Capasso. Mathematical Structures of Epidemic Systems. LNBM 97, 1993
  • Hethcote. The mathematics of infectious diseases. SIAM Review, 2000
  • van den Driessche & Watmough. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. MBS, 2002

Computational epidemiology - A more recent trend

Network / Agent-based models (ABM)

ABM

  • Early in the life of these models, they were called IBM (individual-based models)
  • Over the years, a "philosophical" distinction has emerged:
    • IBM are mathematical models that consider individuals as the units; e.g., DTMC, CTMC, branching processes, etc.
    • ABM are computational models whose study is, for the most part, only possible numerically

ABM vs Network models

  • Network models endow vertices with simple systems and couple them through graphs
  • Can be ABM, but some networks can also be studied analytically

Use of data in epidemiology - Undergoing a transformation

  • Epidemiology has long relied on data
  • Many developments in statistics originate there
  • Data has traditionally been better for chronic diseases than for infectious ones
  • Near-real-time surveillance of infectious diseases ongoing since the 1980s (e.g., Réseau Sentinelles)
  • SARS-CoV-1 saw the beginning of a move towards real-time emerging infectious disease data
  • With SARS-CoV-2, the system has really progressed a lot, both in terms of "citizen science" and governmental initiatives