Effect of movement on the stochastic phase of an epidemic

Fields Institute 15 August 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)

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.

Thank you to

Collaborators:

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

"Enablers":

• Nick Ogden & others (Public Health Agency of Canada Public Health Risk Science Division)
• Funding and logistical support: Public Health Agency of Canada (PHAC)
• Funding NSERC and CIHR

Paper (Effect of movement on the early phase of an epidemic, JA & E Milliken) will appear in BMB soon

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

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:

• Black Death 1347-1353 arrived in Europe and spread following trade routes
• SARS-CoV-1 spread out of HKG following the GATN
• Khan, Arino, Hu et al, Spread of a novel influenza A (H1N1) virus via global airline transportation, New England Journal of Medicine (2009)

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...

(Re-)Introductions happen because of movement

Influenza in Nouvelle-Aquitaine (2018-2019)

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

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

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

Let ,

By threshold theorem of [AllenPvdD2013] and criticality theorem of [Harris1963], if , s.t. . For and sufficiently small, the proba. of extinction from the initial state is

The probability of extinction if there is initially a single infected individual of (resp. ) is (resp. )

In the case of stopping due to reaching , deterministic behavior after that point means that the time series of the process should go to the endemic quasi-stationary distribution with probability 1

The probability of a minor epidemic is denoted and the probability of a major epidemic

By the Markov property, when is sufficiently large the probability of reaching number of infected individuals before reaching zero is

For any one choice of parameters:

• Approximate 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 . This is an important distinction, since 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

Merci / Miigwech / Thank you