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| 1 | +\documentclass[english]{article} |
| 2 | +\usepackage[T1]{fontenc} |
| 3 | +\usepackage[latin9]{inputenc} |
| 4 | +\usepackage{geometry} |
| 5 | +\geometry{verbose,tmargin=2cm,bmargin=2cm,lmargin=2cm,rmargin=2cm} |
| 6 | +\setlength{\parskip}{\medskipamount} |
| 7 | +\setlength{\parindent}{0pt} |
| 8 | +\usepackage{bm} |
| 9 | +\usepackage{amsmath} |
| 10 | +\usepackage{babel} |
| 11 | +\begin{document} |
| 12 | + |
| 13 | +\title{Australian Equine Influenza model} |
| 14 | + |
| 15 | +\maketitle |
| 16 | + |
| 17 | +\section{infer-ausei version 4.0} |
| 18 | + |
| 19 | +The following documentation describes the model used in infer-ausei |
| 20 | +version 4.0. Please refer to the \texttt{README} file for details |
| 21 | +of how to make inference on this model. |
| 22 | + |
| 23 | + |
| 24 | +\subsection{Between-property model} |
| 25 | + |
| 26 | +The state of properties during the epidemic is modelled as a continuous |
| 27 | +time SIO (Susceptible, Infected, Onset) process, with pairwise transmission |
| 28 | +rate |
| 29 | +\[ |
| 30 | +\beta_{ij}(t)=\mu\cdot h_{i}(t-I_{i})n_{i}a_{i}^{\xi}\cdot n_{j}a_{j}^{\zeta}\theta^{\bm{1}[j\in\mathcal{V}(t)]}\frac{\delta}{\delta^{2}+\rho_{ij}^{2}}\hspace{1em}i\in\mathcal{I},j\in\mathcal{S} |
| 31 | +\] |
| 32 | +and assumes a fixed 2 day time from infection to onset \emph{at the |
| 33 | +between property level}. $\mu$ is the baseline transmission rate, |
| 34 | +$n_{k}$ is the number of horses on property $k$, $a_{k}$ is the |
| 35 | +area of property $k$ with non-linear parameters $\xi$ and $\zeta$ |
| 36 | +giving the effect (note that if these non-linear parameters were 0, |
| 37 | +this would imply no effect of area). $\theta$ is the effect of vaccination |
| 38 | +on the susceptible properties with $\mathcal{V}(t)$ the set of vaccinated |
| 39 | +properties at time $t$. $\rho_{ij}$ is the Euclidean distances (map-units/1000) |
| 40 | +between centroids(?) of the properties $i$ and $j$, and $\delta$ |
| 41 | +is the decay parameter for our Cauchy distance kernel. The function |
| 42 | +$h_{i}(t-I_{i})$ is the infectivity of property $i$, taken to be |
| 43 | +proportional to the number of animals infected according to the within-property |
| 44 | +model described below. |
| 45 | + |
| 46 | +At time $t$, the \emph{infectious pressure} $\lambda_{j}(t)$ on |
| 47 | +property $j$ is |
| 48 | +\[ |
| 49 | +\lambda_{j}(t)=\epsilon_{t}+\sum_{i\in}\beta_{ij}(t) |
| 50 | +\] |
| 51 | +where |
| 52 | +\[ |
| 53 | +\epsilon_{t}=\begin{cases} |
| 54 | +\epsilon_{0} & \mbox{ if }t<10\\ |
| 55 | +\epsilon_{10} & \mbox{if }10\leq t<44\\ |
| 56 | +\epsilon_{44} & \mbox{if }44\leq t |
| 57 | +\end{cases} |
| 58 | +\] |
| 59 | +corresponding to control events at $t=10$ and $t=44$ (remind me |
| 60 | +what these events were?). |
| 61 | + |
| 62 | + |
| 63 | +\subsection{Within-property model} |
| 64 | + |
| 65 | +At the \emph{within property level}, $h_{i}(s)=I_{i}(s)/n_{i}$ function |
| 66 | +returns the proportion of horses infected on property $i$ at time |
| 67 | +$s$ after $i$'s infection time, where $I_{i}(s)$ is the solution |
| 68 | +to a deterministic SEIR model |
| 69 | + |
| 70 | +\begin{eqnarray*} |
| 71 | +\frac{dS_{i}(t)}{dt} & = & -\beta\frac{S_{i}(t)I_{i}(t)}{n_{i}}\\ |
| 72 | +\frac{dE_{i}(t)}{dt} & = & \beta\frac{S_{i}(t)I_{i}(t)}{n_{i}}-\omega E_{i}(t)\\ |
| 73 | +\frac{dI_{i}(t)}{dt} & = & \omega E_{i}(t)-\nu I_{i}(t)\\ |
| 74 | +\frac{dR_{i}(t)}{dt} & = & \nu I_{i}(t) |
| 75 | +\end{eqnarray*} |
| 76 | + |
| 77 | + |
| 78 | +where $\omega=$1 and $\nu=1/6$. We set $h(s)$=0 for $I_{i}(s)<0.5$ |
| 79 | +horses. |
| 80 | + |
| 81 | +Note that a property ``runs out'' of infectiousness as all the horses |
| 82 | +recover. This is why the SIO model above does not have an ``R'' |
| 83 | +compartment -- ``O'' stage properties simply stop being infectious |
| 84 | +after a while. |
| 85 | +\end{document} |
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