# COVID-19: Weibull recovery model

In the previous post we have estimated mean recovery time in COVID-19 Lithuanian data set. We have tried to generate fake recovery time series assuming that recovery times are exponentially distributed. We have failed. This time we will assume that recovery times are Weibull distributed.

Assuming exponential distribution to describe the recovery times has a specific meaning. It means that the recovery rate is constant in time. Namely, it does not matter how long some one is sick, the time one will continue being sick will be exponentially distributed with the same recovery rate.

Weibull distribution takes into account ageing effects. Namely, the recovery rate varies in time:

- With \( k < 1 \) it decreases in time. While larger number of quick recoveries is observed. Though note that extremely long recoveries are also more probable in comparison to simple exponential distribution.
- With \( k > 1 \) the recovery rate increases in time. There is smaller number of quick recoveries, but extremely long recoveries are also less probable in comparison to simple exponential distribution.
- In special case of \( k = 1 \) Weibull distribution is identical to exponential distribution.

We parametrize Weibull distribution as follows (assuming \( \tau > 0 \)):

\begin{equation} p(\tau) = k \lambda \left(\lambda \tau \right)^{k-1} \exp\left[ - \left(\lambda \tau \right)^k \right] . \end{equation}

For the data available we have observed that \( k = 2.5 \) and \( \lambda^{-1} = 32 \) generate quite good simulation results. This parameter set implies average recovery time of:

\begin{equation} \langle \tau \rangle = \frac{\Gamma(1 + 1/k)}{\lambda} \approx 28.4 . \end{equation}

Which is really close to the value we have estimated manually in the previous post.

Likely, a better set of parameters could be found by conducting multiple simulations with the same parameters (estimating confidence intervals for RMSE) or by using convolution (the topic of our next post). At this point we have to satisfy ourselves with a simple example providing an interesting point - recovery times are more likely to be Weibull distributed than to be exponentially distributed.