16 September 2021
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 Emerging Infectious Disease Modelling Consortium
for receiving me in your homes/offices and to ..
Funding and logistical support: Public Health Agency of Canada (PHAC)
History of the Peloponnesian War
∂∂tS(x,t)=−λ(x,t)S(x,t)−μS(x,t)+μN(x)+γ1I(x,t)∂∂tI(x,t)=λ(x,t)S(x,t)−(μ+γ1+γ2)I(x,t)∂∂tR(x,t)=γ2I(x,t)−μR(x,t)\begin{aligned} \frac{\partial}{\partial t}S(x,t) &= -\lambda(x,t)S(x,t) - \mu S(x,t) + \mu N(x) + \gamma_1 I(x,t) \\ \frac{\partial}{\partial t}I(x,t) &= \lambda(x,t)S(x,t)-(\mu+\gamma_1+\gamma_2) I(x,t) \\ \frac{\partial}{\partial t}R(x,t) &= \gamma_2I(x,t)-\mu R(x,t) \end{aligned} ∂t∂S(x,t)∂t∂I(x,t)∂t∂R(x,t)=−λ(x,t)S(x,t)−μS(x,t)+μN(x)+γ1I(x,t)=λ(x,t)S(x,t)−(μ+γ1+γ2)I(x,t)=γ2I(x,t)−μR(x,t)
with force of infection
λ(x,t)=1N∫0Ldx′β(x,x′)I(x′,t)\lambda(x,t) = \frac 1N\int_0^L dx'\beta(x,x')I(x',t) λ(x,t)=N1∫0Ldx′β(x,x′)I(x′,t)
and total population along the road
N=∫0Ldx′N(x′)N = \int_0^L dx'N(x') N=∫0Ldx′N(x′)
= various levels or scales of the functional or spatial aspects of a diffusion process. Scale (cone of resolution) takes on two dimensions: functional (decisions made by different groups of individuals) and spatial (manifestations of these decisions as observed in a spatial context)
Nc1′=∑p∈L∖{1}mc1pNcp−Nc1∑p∈L∖{1}mcp1N_{c1}' = \textcolor{red}{\sum_{\mathclap{p\in\mathcal{L}\setminus\{1\}}}m_{c1p}N_{cp}} \textcolor{blue}{-N_{c1}\sum_{\mathclap{p\in\mathcal{L}\setminus\{1\}}}m_{cp1}} Nc1′=p∈L∖{1}∑mc1pNcp−Nc1p∈L∖{1}∑mcp1
or
Nc1′=∑p∈Lmc1pNcp assuming mc11=−∑p∈L∖{1}mcp1N_{c1}' = \sum_{p\in\mathcal{L}} m_{c1p}N_{cp} \textrm{ assuming } m_{c11}=-\sum_{\mathclap{p\in\mathcal{L}\setminus\{1\}}} m_{cp1} Nc1′=p∈L∑mc1pNcp assuming mc11=−p∈L∖{1}∑mcp1
In each patch, put a system describing the evolution of the number of individuals in each compartment present
\gdef\I{\mathcal{I}} \gdef\U{\mathcal{U}} Assume uninfected (sss) and infected (iii) compartments U\UU and I\II. For all j∈Uj\in\Uj∈U, k∈Ik\in\Ik∈I and ℓ∈L\ell\in\mathcal{L}ℓ∈L
sjℓ′=fjℓ(Sℓ,Iℓ) + ∑q∈Lmjℓqsjqikℓ′=gkℓ(Sℓ,Iℓ) + ∑q∈Lmkℓqikq\begin{align*} s_{j\ell}' &= f_{j\ell}(S_\ell,I_\ell) \textcolor{red}{\;+\;\sum_{\mathclap{q\in\mathcal{L}}} m_{j\ell q}s_{jq}} \\ i_{k \ell}' &= g_{k\ell}(S_\ell,I_\ell) \textcolor{red}{\;+\;\sum_{\mathclap{q\in\mathcal{L}}} m_{k\ell q}i_{kq}} \end{align*} sjℓ′ikℓ′=fjℓ(Sℓ,Iℓ)+q∈L∑mjℓqsjq=gkℓ(Sℓ,Iℓ)+q∈L∑mkℓqikq
fff and ggg describe interactions between compartments in a given location. Might involve more than Sℓ,IℓS_\ell,I_\ellSℓ,Iℓ, but always local (ℓ\ellℓ)
Sums describe movement of (individuals from) compartments between locations
Movement from location q∈Lq\in\mathcal{L}q∈L to location p∈Lp\in\mathcal{L}p∈L occurs at rate mXpqm_{Xpq}mXpq for individuals in compartment XXX
\gdef\M{\mathcal{M}} Movement matrix for compartment XXX:
MX=(−∑q∈L,q≠pmXq1mX12⋯mX1∣L∣mX21−∑q∈L,q≠pmXq2mX2∣L∣mX∣L∣1mX∣L∣2−∑q∈L,q≠pmXq∣L∣)\M^X = \begin{pmatrix} -\sum\limits_{\mathclap{q\in\mathcal{L},q\neq p}} m_{Xq1} & m_{X12} & \cdots & m_{X1|\mathcal{L}|} \\ m_{X21} & -\sum\limits_{\mathclap{q\in\mathcal{L},q\neq p}} m_{Xq2} & & m_{X2|\mathcal{L}|} \\ & & & \\ m_{X|\mathcal{L}|1} & m_{X|\mathcal{L}|2} & & -\sum\limits_{\mathclap{q\in\mathcal{L},q\neq p}} m_{Xq|\mathcal{L}|} \end{pmatrix} MX=⎝⎛−q∈L,q=p∑mXq1mX21mX∣L∣1mX12−q∈L,q=p∑mXq2mX∣L∣2⋯mX1∣L∣mX2∣L∣−q∈L,q=p∑mXq∣L∣⎠⎞
J. Arino, N. Bajeux, S. Portet & J. Watmough. Quarantine and the risk of COVID-19 importation. Epidemiology & Infection, 2020
In Ecology, importations are called introductions and have been studied for a while, because they are one of the drivers of evolution and, more recently, because of invasive species
An importation occurs when an individual who acquired the infection in a jurisdiction makes their way to another jurisdiction while still infected with the disease
Geographies greatly influence reasoning
Modify the usual SSS (susceptible), LLL (latent), III (infectious with symptoms), AAA (infectious without symptoms) and RRR (recovered)
ppp fraction of cases detected a posteriori (stricto sensu)
Q:
S.P Otto, T. Day, J. Arino, C. Colijn et al. The origins and potential future of SARS-CoV-2 variants of concern in the evolving COVID-19 pandemic. Current Biology, 2021
1/λ1/\lambda1/λ the mean time between case importations, 1/λq1/\lambda_q1/λq the mean quarantine-regulated time between case importations, ccc the efficacy of quarantine (in %). Then
λq=(100−c)×λ\lambda_q = (100-c)\times \lambda λq=(100−c)×λ
Suppose 1/λ=1/\lambda=1/λ= 5 days and efficacy of quarantine is 90% at 7 days and 98% at 14 days, respectively
Then 1/λq=1/\lambda_q=1/λq= 50 and 250 days, respectively
Left hand side
Right hand side