Hi,
Thank you, well done so far, if I may, writing as a vaccinologist and
epidemiologist, I have 2 questions re your logic:
1) "...homogeneous closed population...": how realistic is such a situ in
practice even within controlled lab or animal house settings?
and
2) "...If we now assume that the rate of infection is the same for all
individuals in the population...": how likely will this be in practice given
the heterogeneity of immunological competencies for individuals in any given
population?
If these 2 assumptions are modified I realise that the model's complexity
could rise; if so, perhaps these 2 questions could be addressed at a later
time to "fine-tune" the "gross" "first draft" model.
Best wishes,
William Stanbury.
2009/7/30 TJ McKinley <[log in to unmask]>
> 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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