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Chris Jewell
covid19uk
Commits
a82e7c8f
Commit
a82e7c8f
authored
Mar 16, 2020
by
Chris Jewell
Browse files
Added model description.
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%% LyX 2.2.4 created this file. For more info, see http://www.lyx.org/.
%% Do not edit unless you really know what you are doing.
\documentclass
[english]
{
article
}
\usepackage
[T1]
{
fontenc
}
\usepackage
[latin9]
{
inputenc
}
\usepackage
{
geometry
}
\geometry
{
verbose,tmargin=2.5cm,bmargin=2.5cm,lmargin=2.5cm,rmargin=2.5cm
}
\setlength
{
\parskip
}{
\medskipamount
}
\setlength
{
\parindent
}{
0pt
}
\usepackage
{
url
}
\usepackage
{
bm
}
\usepackage
{
amsmath
}
\usepackage
{
babel
}
\begin{document}
\title
{
UK Age and Space structured Covid19 model
}
\author
{
Chris Jewell, Barry Rowlingson, Jon Read
}
\maketitle
\section
{
Concept
}
We wish to develop a model that will enable us to assess spatial spread
of Covid19 across the UK, respecting the availability of human mobility
data as well as known contact behaviour between individuals of different
ages.
A deterministic SEIR model is posited in which infection rate is written
as a function of known agestructured contact from Polymod, known
human mobility between MSOAs (Middle Super Output Area), and Censusderived
age structured population density in regions across the UK.
Currently, this model is populated with data for England only, though
we are in the process of extending this to Scotland and Wales.
Noise in daily case numbers
$
y
_{
it
}$
for agegroup
$
i
$
in location
$
k
$
on day
$
t
$
is assumed to be Poissondistributed such that
\[
y
_{
ikt
}
\sim\mbox
{
Poisson
}
(
R
_{
ikt
}

R
_{
ikt

1
}
)
\]
\section
{
Data
}
\subsection
{
Agemixing
}
Standard Polymod data for the UK are used, with 17 5year age groups
$
[
0

5
)
,
[
5

10
)
,
\dots
,
[
75

80
)
,
[
80

\infty
)
$
. Estimated contact matrices
for termtime
$
M
_{
tt
}$
and schoolholidays
$
M
_{
hh
}$
were extracted
of dimension
$
n
_{
m
}
\times
n
_{
m
}$
where
$
n
_{
m
}
=
17
$
.
\subsection
{
Human mobility
}
2011 Census data from ONS on daily mean numbers of commuters moving
from each Residential MSOA to Workplace MSOA. MSOAs are aggregated
to Local Authority Districts (LADs) for which we have agestructured
population density. The resulting matrix
$
C
$
is of dimension
$
n
_{
c
}
\times
n
_{
c
}$
where
$
n
_{
c
}
=
152
$
. Since this matrix is for Residence to Workplace
movement only, we assume that the mean number of journeys between
each LAD is given by
\[
T
=
C
+
C
^{
T
}
\]
with 0 diagonal.
\subsection
{
Population size
}
Agestructured population size within each LAD is taken from publicly
available 2019 Local Authority data giving a vector
$
N
$
of length
$
n
_{
m
}
n
_{
c
}$
, i.e. population for each of
$
n
_{
m
}$
age groups and
$
n
_{
c
}$
LADs.
\section
{
Model
}
\subsection
{
Connectivity matrix
}
We assemble a countrywide connectivity matrices as Kronecker products,
such that
\[
K
^{
\star
}
=
I
_{
n
_{
c
}}
\bigotimes
M
\]
and
\[
T
^{
\star
}
=
C
\bigotimes\bm
{
1
}_{
n
_{
m
}
\times
n
_{
c
}}
\]
giving two matrices of dimension
$
n
_{
m
}
n
_{
c
}$
.
$
K
^{
\star
}$
is block
diagonal with Polymod mixing matrices.
$
T
^{
\star
}$
expands the mobility
matrix
$
C
$
such that a block structure of connectivity between LADs
results.
\subsection
{
Disease progression model
}
We assume an SEIR model described as a system of ODEs. We denote the
number of individual in each agegroupLAD combination at time
$
t
$
by the vectors
$
\vec
{
S
}
(
t
)
,
\vec
{
E
}
(
t
)
,
\vec
{
I
}
(
t
)
,
\vec
{
R
}
(
t
)
$
. We
therefore have
\begin{align*}
\frac
{
\mathrm
{
d
\vec
{
S
}
(t)
}}{
dt
}
&
=
\beta
_{
1
}
\left
[K^{\star}\vec{I}(t)+\beta_{2}\bar{K}T^{\star}\frac{{\vec{I}(t)}}{N}\right]
\frac
{
\vec
{
S
}
(t)
}{
N
}
\\
\frac
{
\mathrm
{
d
}
\vec
{
E
}
(t)
}{
dt
}
&
=
\beta
_{
1
}
\left
[K^{\star}\vec{I}(t)+\beta_{2}\bar{K}T^{\star}\frac{{\vec{I}(t)}}{N}\right]
\frac
{
\vec
{
S
}
(t)
}{
N
}

\nu\vec
{
E
}
(t)
\\
\frac
{
\mathrm
{
d
}
\vec
{
I
}
(t)
}{
dt
}
&
=
\nu\vec
{
E
}
(t)
\gamma\vec
{
I
}
(t)
\\
\frac
{
\mathrm
{
d
}
\vec
{
R
}
(t)
}{
dt
}
&
=
\gamma\vec
{
I
}
(t)
\end{align*}
with parameters: baseline infection rate
$
\beta
_{
1
}$
, commuting infection
ratio
$
\beta
_{
2
}$
, latent period
$
\frac
{
1
}{
\nu
}$
, and infectious
period
$
\frac
{
1
}{
\gamma
}$
.
\subsection
{
Noise model
}
Currently, and subject to discussion, we assume that all detected
cases are synonymous with individuals transitioning
$
I
\rightarrow
R
$
.
We assume the number of new cases in each ageLAD combination are
given by
\[
y
_{
ik
}
(
t
)
\sim\mbox
{
Poisson
}
\left
(
R
_{
ik
}
(
t
)
R
_{
ik
}
(
t

1
)
\right
)
\]
This could be relaxed to a Negative Binomial distribution to account
for Poisson overdispersion.
\subsection
{
Implementation
}
The model is currently implemented in Python3, using Tensorflow 2.1
with the RK5 differential equation solver implemented in the
\texttt
{
DormandPrince
}
class provided by Tensorflow Probability 0.9. Code may be found at
\url
{
http://fhmchicascode.lancs.ac.uk/jewell/covid19uk
}
.
\section
{
Quick results
}
See attached SPIM report.
\end{document}
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