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colour corrections in text
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davidearn committed Mar 16, 2024
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27 changes: 14 additions & 13 deletions vignettes/brauer-ms.Rnw
Expand Up @@ -1512,12 +1512,12 @@ particular, $\R_0\approx \Sexpr{signif(R0.phila.fitode,2)}$ and $\Tg\approx
\Sexpr{signif(Tg.phila.fitode*365,2)}$ days.

If we convert the \code{fitode} estimates of the SIR parameters to the
parameters of \KM's approximation, we obtain the orange curve in \cref{fig:phila}, which grossly
underestimates the magnitude of the epidemic (the epidemic peak occurs much too
soon). The \KM approximation \eqref{eq:sech} is good initially, but
becomes poorer and poorer over time as the underlying assumption
on which it is based \eqref{eq:KMassumption} becomes less and less
valid.
parameters of \KM's approximation, we obtain the dotted yellow curve
in \cref{fig:phila}, which grossly underestimates the magnitude of the
epidemic (the epidemic peak occurs much too soon). The \KM
approximation \eqref{eq:sech} is good initially, but becomes poorer
and poorer over time as the underlying assumption on which it is based
\eqref{eq:KMassumption} becomes less and less valid.

%%\section{Example for which \KM approximation is poor (via stochastic SIR)}
\section{Fitting the deterministic SIR model to stochastic simulations}\label{sec:stoch}
Expand Down Expand Up @@ -1597,14 +1597,15 @@ stochastic SIR model with initial state $(S_0,I_0,R_0)=(\Sexpr{iii})$,
basic reproduction number $\R_0=\Sexpr{R0}$, and mean generation
interval $\Tg=\gamma^{-1}=\Sexpr{1/mm[["gamma"]]}$ week. In the top
panel, $\dbydt{R}$ \eqref{eq:SIR;R} with the correct initial
conditions and parameter values is shown in green, and the \KM
conditions and parameter values is shown with solid green, and the \KM
approximation \eqref{eq:sech} based on those parameter values is shown
in orange. The \code{fitode} fit (based on $\int (\dbydt{R}) \dt$) and
confidence band are shown in yellow. The time shift between the
deterministic solution and the stochastic realization arises because
the stochastic model captures the demographic noise (which causes a
randomly distributed delay until the tipping point is reached, i.e.,
until the epidemic takes off in a roughly deterministic fashion).
with dotted green. The \code{fitode} fit (based on $\int (\dbydt{R})
\dt$) and confidence band are shown in yellow. The time shift between
the deterministic solution and the stochastic realization arises
because the stochastic model captures the demographic noise (which
causes a randomly distributed delay until the tipping point is
reached, i.e., until the epidemic takes off in a roughly deterministic
fashion).

As expected, with the correct parameter values, \KM's approximation
\eqref{eq:sech} fails once the requirement \eqref{eq:KMassumption}
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