Effect of movement on the stochastic phase of an epidemic

Heidelberg 23 September 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.

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 [..] 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

Thank you to

Collaborators (stochastic phase)

  • Jason Rose (MSc Thesis was much earlier DTMC version)
  • Evan Milliken (U of Louisville)

Collaborators (nursing homes)

  • Tristan Delory (Centre Hospitalier Annecy)
  • Pierre-Yves Boëlle (IPLESP)
  • Paul-Emile Haÿ (Groupe Colisee)
  • Vincent Klotz (Groupe Colisee)
  • Arino & Milliken. Effect of movement on the early phase of an epidemic. To appear in Bulletin of Mathematical Biology
  • Delory, Arino, Haÿ, Klotz & Boëlle. SARS-CoV-2 in nursing homes: analysis of routine surveillance data in four European countries. To appear in Aging and Infection

Outline

  1. Heterogeneity of spread and the stochastic phase of an epidemic
  2. The model in one spatial location
  3. The model with movement between two locations
  4. Observations

Heterogeneity of spread and the stochastic phase of an epidemic

Arino. Spatio-temporal spread of infectious pathogens of humans. Infectious Disease Modelling 2 (2017)

Mobility is complicated but drives disease spatialisation

  • 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:

Transmission within national jurisdictions was heterogeneous

Moving from ISO-3166-3 (nation or territory) level to smaller sub-national jurisdictions, the picture is more contrasted

Next slide: Example of activation of North American health regions/municipios/counties

Why do we observe such disparity?

  • Some jurisdictions (health regions/municipios/counties) had cases throughout
  • Others either had silent transmission chains for more than 3 weeks or had reimportation of cases
  • Critical phase is importation/establishment in locations: sometimes things pick up, other times they don't...
Orange County NY
Jefferson County KY
Winnipeg MB
Campbell County WY
Campbell County WY

We've "all" been there

(Well, all of those who tried to fit some COVID data)

Suppose you want to fit an SIR-ish model to the previous data

  • Campbell County is a sure fail
  • The buildup phase is waaaaay too long, even accounting for some silent transmission chains
  • If you throw in as a free parameter, there is some hope, but then is close to the start of the exponential phase

(Re-)Introductions happen because of movement

JA, Bajeux, Portet & Watmough. Quarantine and the risk of COVID-19 importation. Epidemiology & Infection 148 (2020)

JA, Boëlle, Milliken & Portet. Risk of COVID-19 variant importation - How useful are travel control measures?. Infectious Disease Modelling 6 (2021)

Influenza in Nouvelle-Aquitaine (2018-2019)

left:45% right:45%

Influenza in Nouvelle-Aquitaine & Occitanie (2018-2019)

left:45% right:45%

Zoom on pre-deterministic phase

Research questions

  • How to model the stochastic phase of an epidemic?
  • Does mobility have an effect on the stochastic phase?

The model in one spatial location

A super simple CTMC SIS

CTMC characterized by the transitions

with rates given by

Description Transition Rate
Infection in A
Recovery in A

A super simple CTMC SIS with a twist

Regular chain of this type has as sole absorbing state

We add another absorbing state: if , then the chain has left the stochastic phase and is in a quasi-deterministic phase with exponential growth

Doing this, time to absorption measures become usable additionally to first passage time ones

And the question becomes: how long does the chain "linger on" before it is absorbed? We define the inter-absorption trajectory as the stochastic phase

One obvious problem: what should be ?

  • Choose too small and the stochastic phase will not last long
  • Choose too large and absorption will only be at the DFE
  • So, how does one choose ?
    • A formula of Whittle
    • Multitype branching process (MTBP); explained later in two-patch case

A formula of Whittle (1955)

Start with infectious cases introduced in population of susceptible individuals

: proba. that at time there are still uninfected and infectious not removed

: proba. removal of one infectious in ; : proba. new infection

Probability of an epidemic when :

The model with movement between two locations

The threshold

We could have , and but for simplicity, we use a single , which is triggered when

A little trick for numerics

We set (rate of movement) and (proportion of the population in ); then

,

,

MTBP approximation to choose the threshold

the MTBP approximation of CTMC with infected types and . Define type and offspring probability-generating functions

For any one choice of parameters:

  • Approximate probability of extinction using a MTBP
  • Run an ensemble of simulations for the probability absorption in
  • Repeat for progressively larger values of until the desired precision is reached

Observations

  • On average, sample paths leading to outbreak are longer than those leading to extinction
  • As increases, paths leading to outbreak become more likely, but all paths become more direct and duration decreases
  • Even in a single isolated population, the mean duration of the stochastic phase peaks for to the right of the classic threshold ; the mean duration of paths that lead to extinction (also known as minor epidemics) peaks at
  • Mean duration of the stochastic phase generally decreases with the rate of movement when the patch of disease (re-)emergence is coupled to a patch more favorable to the spread of the disease
  • This relationship is not monotone, however
  • E.g., when , increasing even larger increase in system resembles the patch of disease (re-)emergence being coupled to a patch less favorable to the spread of disease mean duration increases with increasing
  • In contrast, mean duration of the stochastic phase does not seem to depend strongly on . However, when approaches 1, even for fixed , increases to infinity as increases to 1

CMPD 6 - Winnipeg 22 to 26 May 2023

Computational and Mathematical Population Dynamics

(Fort Lauderdale 2019, Taiyuan 2013, Bordeaux 2010, Campinas 2007, Trento 2004)

Follows MPD: University of Mississippi 1986, Rutgers University 1989, Pau 1992, Rice University 1995, Zakopane 1998, Trento 2004

And DeStoBio: Sofia 1997, Purdue 2000, Trento 2004

See information on (embryonic so far) https://cmpd6.github.io/

Merci / Miigwech / Thank you

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