= Solution
An illness episode ends at total rate $\gamma+\delta$, so its mean duration is
$$
\frac1{\gamma+\delta}=10\text{ days}.
$$
The probability that its terminating transition is fatal is $\delta/(\gamma+\delta)=0.1$. Therefore, in day units,
$$
\widehat\delta=0.01,
\qquad
\widehat\gamma=0.09.
$$
Starting healthy, the first infection time is exponential with rate $\lambda_z$. Hence
$$
1-e^{-30\lambda_1}=0.06,
\qquad
1-e^{-30\lambda_0}=0.01,
$$
and
$$
\widehat\lambda_1=-\frac{\log0.94}{30},
\qquad
\widehat\lambda_0=-\frac{\log0.99}{30},
\qquad
\widehat\beta=\log\frac{\widehat\lambda_1}{\widehat\lambda_0}.
$$
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