Epidemics spreading in space and time

24 August 2023

Julien Arino

Department of Mathematics & Data Science Nexus
University of Manitoba*

Canadian Centre for Disease Modelling
Canadian COVID-19 Mathematical Modelling Task Force

* 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.

Pathogens have been mobile for a while

It first began, it is said, in the parts of Ethiopia above Egypt, and thence descended into Egypt and Libya and into most of the King's country [Persia]. Suddenly falling upon Athens, it first attacked the population in Piraeus—which was the occasion of their saying that the Peloponnesians had poisoned the reservoirs, there being as yet no wells there—and afterwards appeared in the upper city, when the deaths became much more frequent.

Thucydides (c. 460 BCE - c. 395 BCE)

History of the Peloponnesian War


  • Mobility and the spread of infectious diseases
  • Formulating metapopulation models
  • Basic mathematical analysis
  • is not the panacea - An urban centre and satellite cities
  • Problems specific to metapopulations
  • Numerical investigations of large-scale systems

Mobility and the spread of infectious diseases

Mobility is complicated and drives disease spatialisation

Mobility is complicated:

  • Multiple modalities: foot, bicycle, personal vehicle, bus, train, boat, airplane
  • Various durations: trip to the corner shop commuting multi-day trip for work or leisure relocation, immigration or refuge seeking
  • Volumes are hard to fathom

And yet mobility drives spatio-temporal spread:

The Black Death: quick facts

  • First of the middle ages plagues to hit Europe
  • Affected Afro-Eurasia from 1346 to 1353
  • Europe 1347-1351
  • Killed 75–200M in Eurasia & North Africa
  • Killed 30-60% of European population

Plague control measures

  • Lazzarettos of Dubrovnik 1377 (30 days)
  • Quarantena of Venice 1448 (40 days)
  • Isolation of known or suspected cases as well as persons who had been in contact with them, at first for 14 days and gradually increased to 40 days
  • Improvement of sanitation: development of pure water supplies, garbage and sewage disposal, food inspection
  • .. Find and kill a snake, chop it into pieces and rub the various parts over swollen buboes. (Snake, synonymous with Satan, was thought to draw the disease out of the body as evil would be drawn to evil)

Pathogen spread has evolved with mobility

  • Pathogens travel along trade routes

  • In ancient times, trade routes were relatively easy to comprehend

  • With acceleration and globalization of mobility, things change

Fragmented jurisdictional landscape

  • Political divisions (jurisdictions): nation groups (e.g., EU), nations, provinces/states, regions, counties, cities..
  • Travel between jurisdictions can be complicated or impossible
  • Data is integrated at the jurisdicional level
  • Policy is decided at the jurisdictional level
  • Long range mobility is a bottom top top bottom process

Why mobility is important in the context of health

All migrants/travellers carry with them their "health history"
  • latent and/or active infections (TB, H1N1, polio)
  • immunizations (schedules vary by country)
  • health/nutrition practices (KJv)
  • treatment methods (antivirals)
Pathogens ignore borders and politics
  • antiviral treatment policies for Canada and USA
  • SARS-CoV-2 anyone?

SARS-CoV-1 (2002-2003)

Overall impact

  • Index case for international spread arrives HKG 21 February 2003

  • Last country with local transmission (Taiwan) removed from list 5 July 2003

  • 8273 cases in 28 countries

  • (Of these cases, 1706 were HCW)

  • 775 deaths (CFR 9.4%)

Polio spread 2002-2006. Pallansch & Sandhu, N Engl J Med 2006; 355:2508-2511

Formulating metapopulation models

General principles (1)

  • geographical locations (patches) in a set (city, region, country..)
  • Patches are vertices in a graph
  • Each patch contains compartments
    • individuals susceptible to the disease
      • individuals infected by the disease
      • different species affected by the disease
      • etc.

General principles (2)

  • Compartments may move between patches, with rate of movement of individuals from compartment from patch to patch
  • Movement instantaneous and no death during movement
  • , defines a digraph with arcs
  • Arc from to if , absent otherwise
  • compartments, so each can have at most arrows multi-digraph

The underlying mobility model

population of compartment in patch

Assume no birth or death. Balance inflow and outflow

when we write

The toy SLIRS model in patches


is the birth rate (typically or )

= latently infected ( exposed, although the latter term is ambiguous)

-SLIRS model

with incidence

-SLIRS (multiple species)

a set of species

with incidence

Oh, what a tangled web we weave

(Walter Scott, not Shakespeare)

Consider, e.g., with ,

  • describes intra- and inter-species possibilities of transmission in a location
  • If the encounter of a susceptible from species and an infectious from species in location can lead to infection of the susceptible then otherwise

Malaria and many vector-borne diseases


Plague (bubonic, not pneumonic)



-SLIRS (residency patch/movers-stayers)

with incidence

General metapopulation epidemic models

uninfected and infected compartments, and

For , and ,

where and

Basic mathematical analysis

Analysis - Toy system

For simplicity, consider -SLIRS with

with incidence

System of equations

Size is not that bad..

System of equations !!!

However, a lot of structure:

  • copies of individual units, each comprising 4 equations
  • Dynamics of individual units well understood
  • Coupling is linear

Good case of large-scale system (matrix analysis is your friend)

Notation in what follows

  • a square matrix with entries denoted

  • if for all (could be the zero matrix); if and with ; if . Same notation for vectors

  • spectrum of

  • spectral radius

  • spectral abscissa (or stability modulus)

  • is an M-matrix if it is a Z-matrix ( for ) and , with and

Behaviour of the total population

Consider behaviour of . We have


Vector / matrix form of the equation

We have

Write this in vector form

where </