%% 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 Covid-19 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 Covid-19 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 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 $y_{ikt}\sim\mbox{Poisson}(R_{ikt}-R_{ikt-1})$ \section{Data} \subsection{Age-mixing} 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 $T=C+C^{T}$ 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. \section{Model} \subsection{Connectivity matrix} We assemble a country-wide 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 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 \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 age-LAD 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://fhm-chicas-code.lancs.ac.uk/jewell/covid19uk}. \section{Quick results} See attached SPI-M report. \end{document}