Commit a82e7c8f authored by Chris Jewell's avatar Chris Jewell
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Added model description.

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\title{UK Age and Space structured Covid-19 model}
\author{Chris Jewell, Barry Rowlingson, Jon Read}
We wish to develop a model that will enable us to assess spatial spread
of Covid-19 across the UK, respecting the availability of human mobility
data as well as known contact behaviour between individuals of different
A deterministic SEIR model is posited in which infection rate is written
as a function of known age-structured contact from Polymod, known
human mobility between MSOAs (Middle Super Output Area), and Census-derived
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 age-group $i$ in location
$k$ on day $t$ is assumed to be Poisson-distributed such that
Standard Polymod data for the UK are used, with 17 5-year age groups
$[0-5),[5-10),\dots,[75-80),[80-\infty)$. Estimated contact matrices
for term-time $M_{tt}$ and school-holidays $M_{hh}$ 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 age-structured
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
with 0 diagonal.
\subsection{Population size}
Age-structured 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.
\subsection{Connectivity matrix}
We assemble a country-wide connectivity matrices as Kronecker products,
such that
K^{\star}=I_{n_{c}}\bigotimes M
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
\subsection{Disease progression model}
We assume an SEIR model described as a system of ODEs. We denote the
number of individual in each age-group-LAD combination at time $t$
by the vectors $\vec{S}(t),\vec{E}(t),\vec{I}(t),\vec{R}(t)$. We
therefore have
\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)
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 age-LAD combination are
given by
This could be relaxed to a Negative Binomial distribution to account
for Poisson overdispersion.
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
\section{Quick results}
See attached SPI-M report.
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