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Commits (2)
 ... ... @@ -25,29 +25,28 @@ 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 A deterministic SEIR state transition model is posited in which individuals transition from Susceptible to Exposed (i.e. infected but not yet infectious) to Infectious to Removed (i.e. quarantined, got better, or died). We model the infection rate (rate of S$\rightarrow$E transition) 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 $y_{ikt}\sim\mbox{Poisson}(R_{ikt}-R_{ikt-1})$ we are in the process of extending this to Scotland, Wales, and Northern Ireland. \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 $n_{m}=17$. Standard Polymod social mixing 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} ... ... @@ -67,8 +66,8 @@ with 0 diagonal. 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. $n_{m}n_{c}=2584$, i.e. population for each of $n_{m}$ age groups and $n_{c}$ LADs. \section{Model} ... ... @@ -77,16 +76,16 @@ $n_{c}$ LADs. We assemble a country-wide connectivity matrices as Kronecker products, such that $K^{\star}=I_{n_{c}}\bigotimes M M^{\star}=I_{n_{c}}\bigotimes M$ and $T^{\star}=C\bigotimes\bm{1}_{n_{m}\times n_{c}} C^{\star}=C\bigotimes\bm{1}_{n_{m}\times n_{c}}$ 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. giving two matrices of dimension $n_{m}n_{c}\times n_{m}n_{c}$. $M^{\star}$ is block diagonal with Polymod mixing matrices. $C^{\star}$ expands the mobility matrix $C$ such that a block structure of connectivity between LADs results. \subsection{Disease progression model} ... ... @@ -95,16 +94,19 @@ 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{S}(t)}}{dt} & =-\beta_{1}\left[M^{\star}\vec{I}(t)+\beta_{2}\bar{M}C^{\star}\frac{{\vec{I}(t)}}{N}\right]\frac{\vec{S}(t)}{N}\\ \frac{\mathrm{d}\vec{E}(t)}{dt} & =\beta_{1}\left[M^{\star}\vec{I}(t)+\beta_{2}\bar{M}C^{\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 $\frac{1}{\gamma}$. where $\bar{M}$ is the global mean person-person contact rate. Parameters are: baseline infection rate $\beta_{1}$, commuting infection ratio $\beta_{2}$, latent period $\frac{1}{\nu}$, and infectious period $\frac{1}{\gamma}$. Typically, we assume that contact with commuters is $\beta_{2}=\frac{1}{3}$ of that between members of the same age-LAD combination assuming an 8 hour working day. \subsection{Noise model} \subsection{Noise model (under construction)} Currently, and subject to discussion, we assume that all detected cases are synonymous with individuals transitioning $I\rightarrow R$. ... ... @@ -122,9 +124,7 @@ 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 \url{http://fhm-chicas-code.lancs.ac.uk/jewell/covid19uk}. \section{Quick results} \url{http://fhm-chicas-code.lancs.ac.uk/jewell/covid19uk}. See attached SPI-M report. The code is a work in progress \textendash{} see README. \end{document}