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Issue #554: Model vignette fixes #555

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Merged
seabbs merged 2 commits into from
Mar 14, 2025
Merged

Issue #554: Model vignette fixes #555

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bdd5090
Add missing reference
athowes Mar 13, 2025
dbb874a
Fixes to maths
athowes Mar 13, 2025
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5 changes: 3 additions & 2 deletions vignettes/model.Rmd
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Original file line number Diff line number Diff line change
Expand Up @@ -132,7 +132,7 @@ $$
&= \int_{P_L}^{P_R} \int_{S_L}^{S_R} g_P(x\,|\,P_L,P_R) f_x(y-x) \,dy\, dx
\end{aligned}
$$
where $ g_P(x\,|\,P_L,P_R)$ represents the conditional distribution of primary event given lower $P_L$ and upper $P_R$ bounds.
where $g_P(x\,|\,P_L,P_R)$ represents the conditional distribution of primary event given lower $P_L$ and upper $P_R$ bounds.

# The naive model

Expand All @@ -159,7 +159,8 @@ Mathematically this model is described as follows.
We look at the conditional probability that the secondary event $S$ falls between $S_L$ and $S_R$, given that the primary event $P$ falls between $P_L$ and $P_R$ and that the secondary event $S$ occurs before the truncation time $T$:
$$
\begin{aligned}
\mathbb{P}(S_L < S < S_R \, | \, P_L < P < P_R, S<T) &= \mathbb{P}(P_L < P < P_R, S_L < S < S_R, S<T \, | \, P_L < P < P_R, S<T)\\
&\mathbb{P}(S_L < S < S_R \, | \, P_L < P < P_R, S<T)\\
&= \mathbb{P}(P_L < P < P_R, S_L < S < S_R, S<T \, | \, P_L < P < P_R, S<T)\\
&= \frac{\mathbb{P}(P_L < P < P_R, S_L < S < S_R, S<T)}{\mathbb{P}(P_L < P < P_R, S<T)}\\
&= \frac{\int_{P_L}^{P_R} \int_{S_L}^{S_R} g_P(x) f_x(y-x) \,dy\, dx}{\int_{P_L}^{P_R} \int_{z}^T g_P(z) f_z(w-z) \, dz \,dw }\\
&= \frac{\int_{P_L}^{P_R} \int_{S_L}^{S_R} g_P(x) f_x(y-x) \,dy\, dx}{\int_{P_L}^{P_R} g_P(z) F_z(T-z) \,dw }\\
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13 changes: 13 additions & 0 deletions vignettes/references.bib
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Expand Up @@ -126,3 +126,16 @@ @article{kallioinen2024detecting
year={2024},
publisher={Springer}
}

@article {ward2022transmission,
author = {Ward, Thomas and Christie, Rachel and Paton, Robert S and Cumming, Fergus and Overton, Christopher E},
title = {Transmission dynamics of monkeypox in the United Kingdom: contact tracing study},
volume = {379},
elocation-id = {e073153},
year = {2022},
doi = {10.1136/bmj-2022-073153},
publisher = {BMJ Publishing Group Ltd},
URL = {https://www.bmj.com/content/379/bmj-2022-073153},
eprint = {https://www.bmj.com/content/379/bmj-2022-073153.full.pdf},
journal = {BMJ}
}
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