# COVID-19: Estimating mean recovery rate

This summer one of my colleagues asked me if I know what is the recovery rate from COVID-19. While I have read a lot of papers, some of them claiming to use "realistic" parameters, I still had no clue (as those "realistic" parameters were in some cases wildly different). Yet as we have enough data available, we can estimate recovery rate ourselves.

Actually we need to define what we mean by "recovery rate", \( \lambda \). Let us say that it is an inverse of a mean time it takes for a newly discovered case to be reported as recovered. If we knew the discovery and recovery dates for each individual, we could simply average the differences. Yet we have only the aggregated time series.

Here in this post we will use CSSE time series for Lithuania. Data was downloaded on August 7th, 2020.

## Manual approach

We can estimate the mean recovery time simply by shifting the recovered cases time series, \( R(t) \), by some \( \Delta t \) in respect to the confirmed cases time series, \( I(t) \). \( \Delta t \) for which both time series look the most alike is our estimate of the mean recovery time.

Though due to deaths we can't make a direct comparison. We have to discount \( I(t) \) by a factor of \( 1 - d \) to account for deaths. Here \( d \) can be seen as an effective death probability, though the true death probability will be slightly smaller.

To automatically estimate likeness between the time series we use root-mean-square error (abbr. RMSE). As of time of writing the post, we find the best result to be \( \Delta t = 28 \) and \( d = 0.11 \). Though these numbers may change as we update the data.

Feel free to check our estimate using the interactive app below. You just need to change the parameter values and observe how RMSE changes (smaller values are better).

In this app we have used red color to plot \( I(t) \times (1-d) \) and blue color to plot \( R(t - \Delta t) \).

## Exponential recovery model

In SIR models all transitions are assumed to Poisson processes. Namely, the transition times are assumed to be distributed exponentially. Lets see what happens!

The simulation result (gray curve) does not look good very good with neither our estimated inverse mean recovery time, \( \lambda^{-1} \approx \Delta t \), nor with other possible values.

Note that in this app we have calculated RMSE between the simulation result and \( R(t) \). RMSE values will slightly vary as you generate new results with the same parameter values.

So, while it seems that our estimate of mean recovery time is decent, the exponential recovery model noticeably fails. We will try a different recovery model in the next post.