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:

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

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 rth 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

  S(T > t) = 1 - F(T \leq t) = \exp (-\lambda_{mr}(t) \delta t

  under the exponential distribution.