covid19uk issues https://fhm-chicas-code.lancs.ac.uk/jewell/covid19uk/-/issues 2020-12-09T11:01:30Z https://fhm-chicas-code.lancs.ac.uk/jewell/covid19uk/-/issues/13 Branching process proposal 2020-12-09T11:01:30Z Chris Jewell Branching process proposal Chris Jewell Chris Jewell https://fhm-chicas-code.lancs.ac.uk/jewell/covid19uk/-/issues/12 Poisson observation model 2020-12-09T11:00:53Z Chris Jewell Poisson observation model Chris Jewell Chris Jewell https://fhm-chicas-code.lancs.ac.uk/jewell/covid19uk/-/issues/10 Continuous time likelihood calculation 2020-10-21T11:01:10Z Chris Jewell Continuous time likelihood calculation # Purpose This document outlines a new form of the likelihood for an epidemic model, using a continuous-time approximation to a discrete Markov process. # State transition model Epidemic models are represented as State Transition Models, where at any time in an epidemic individuals are assumed to exist in one of a number of states $s = 1, \dots, S$ connected by directed edges, or transitions, $r=1,\dots,R$. For example: mermaid graph LR; s1 -- r1 --> s2; s2 -- r2 --> s3; s2 -- r3 --> s4;  # Notation Let us assume a $y_{tmr}$ represents the number of events occurring at timepoint $t$ in epidemiological unit $m=1,\dots,M$ of type $r=1,\dots,R$, where $r$ represents a transition (or edge) in a State Transition Model graph. For convenience, we refer to the __event tensor__ Y with shape [T, M, R]. We represent the number of individuals in each epidemiological unit $m$ at time $t$ in state $s$ as $x_{tms}$, and refer to a __state tensor__ X with shape [T, M, S]. # Likelihood computation Consider a timepoint $t$ and state $s$. Under the (approximate) continuous-time model, the probability of $y_{tmr}$ events transitioning _out_ of $s$ via transition $r$ at the end of a time interval $\delta t$ is math f(y_{tmr} | \lambda_{mr}(t), x_{tms}) = (\lambda_{mr}(t))^{y_{tmr}} \exp \left( -x_{tms}\sum_{r \in o(s)} \lambda_{mr}(t) \delta t \right)  where $\lambda_{mr}(t)$ is the hazard rate for the $r$th transition, and $o(s)$ is the set of _outgoing_ transitions from state $s$. __Remarks__: * This is essentially survival analysis, where the lifespan function is a product of a hazard rate ($\lambda_{mr}(t)$) and survivor function math S(T > t) = 1 - F(T \leq t) = \exp (-\lambda_{mr}(t) \delta t  under the exponential distribution. # Purpose This document outlines a new form of the likelihood for an epidemic model, using a continuous-time approximation to a discrete Markov process. # State transition model Epidemic models are represented as State Transition Models, where at any time in an epidemic individuals are assumed to exist in one of a number of states $s = 1, \dots, S$ connected by directed edges, or transitions, $r=1,\dots,R$. For example: mermaid graph LR; s1 -- r1 --> s2; s2 -- r2 --> s3; s2 -- r3 --> s4;  # Notation Let us assume a $y_{tmr}$ represents the number of events occurring at timepoint $t$ in epidemiological unit $m=1,\dots,M$ of type $r=1,\dots,R$, where $r$ represents a transition (or edge) in a State Transition Model graph. For convenience, we refer to the __event tensor__ Y with shape [T, M, R]. We represent the number of individuals in each epidemiological unit $m$ at time $t$ in state $s$ as $x_{tms}$, and refer to a __state tensor__ X with shape [T, M, S]. # Likelihood computation Consider a timepoint $t$ and state $s$. Under the (approximate) continuous-time model, the probability of $y_{tmr}$ events transitioning _out_ of $s$ via transition $r$ at the end of a time interval $\delta t$ is math f(y_{tmr} | \lambda_{mr}(t), x_{tms}) = (\lambda_{mr}(t))^{y_{tmr}} \exp \left( -x_{tms}\sum_{r \in o(s)} \lambda_{mr}(t) \delta t \right)  where $\lambda_{mr}(t)$ is the hazard rate for the $r$th transition, and $o(s)$ is the set of _outgoing_ transitions from state $s$. __Remarks__: * This is essentially survival analysis, where the lifespan function is a product of a hazard rate ($\lambda_{mr}(t)$) and survivor function math S(T > t) = 1 - F(T \leq t) = \exp (-\lambda_{mr}(t) \delta t  under the exponential distribution. Alison Hale Alison Hale