Hi,
I can't seem to resolve a discrepancy when deriving a likelihood for an
SI epidemic model and was hoping someone might be able to clarify my
thinking. I have written the problem below in LaTeX for quick compiling
but hopefully it should be readable in e-mail form (apologies if not).
\documentclass{article}
\begin{document}
The problem is:
If we assume that $n$ individuals in a homogeneous closed population are
either susceptible to infection (S) or infected and infectious (I), then
at time $t$ we can model the rate of transition for a susceptible
individual between the states $S \rightarrow I$ as $\beta I_{t-1}$. This
gives exponentially-distributed inter-event times. So if we assume that
the epidemic saturates the population, and we have $n$ completely
observed ordered event times, then the likelihood is:
\[
L(beta) = \prod_{i=1}^n \beta S_{i-1} I_{i-1} \exp(-\beta S_{i-1}
I_{i-1} [t_i-t_{i-1}])
\]
To derive this heuristically, my thinking is that the newly infected
individual contributes:
\[
\beta I_{i-1} \exp(-\beta I_{i-1}[t_i-t_{i-1}])
\]
to the likelihood, and then the $S_{i-1}-1$ individuals that must have
survived the period $(t_{i-1},t_i)$ contribute
\[
\exp(-\beta I_{i-1} [S_{i-1}-1] [t_i-t_{i-1}]).
\]
Since we do not condition on {\it specific} individuals there are
$S_{i-1}$ ways in which the susceptible individuals can become infected,
and hence we have:
\[
S_{i-1} \times \beta I_{i-1} \exp(-\beta I_{i-1} S_{i-1} [t_i-t_{i-1}]).
\]
So far so good. The discrepancy arises when you consider that you have
{\it individual-level} data. In this case let $\lambda_{it}$ denote the
rate that an individual $i$ is subject to infection at time $t$. Given
event times $t_1<\dots<t_n$ as before, the likelihood can be written:
\[
L(\beta) = \prod_{i=1}^n \lambda_{it_{i-1}} \exp(-\sum_{j=i}^n
\lambda_{jt_{i-1}} [t_i-t_{i-1}]),
\]
in the same way as above, only this time we know explicitly {\it which}
susceptible individual became infected. If we now assume that the rate
of infection is the same for all individuals in the population, e.g.
$\lambda_{1t}=\lambda_{2t} = \dots = \lambda_{nt} = \beta I_{t-1}$, then
we recover the likelihood above save for a factor of $S_{i-1}$ at the
front, i.e.
\[
L(\beta) = \prod_{i=1}^n \beta I_{i-1} \exp(-\beta S_{i-1} I_{i-1}
[t_i-t_{i-1}])
\]
Naively I would expect to recover the same likelihood function, though
maybe this is incorrect. Any pointers in the right direction would be
most appreciated.
Many thanks in advance,
TJ McKinley
\end{document}
--
Dr. TJ McKinley
CIDC - Disease Dynamics Unit
Dept. of Veterinary Medicine
University of Cambridge
Madingley Road
Cambridge
CB3 0ES
UK
Tel: +44 (0)1223 337685
Fax: +44 (0)1223 764667
http://www.vet.cam.ac.uk/cidc/
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