# COVID-19: Recovery model with convolution

In the previous post we have shown that Weibull recovery model works well when trying to reconstruct the recovered cases time series from the confirmed cases time series. In that post we have used random simulation to generate fake recovered cases time series. In this post we will use convolution to get the expected recovered cases time series.

Suppose we want to obtain expected number of recovered individuals at moment $$t$$ given only the confirmed cases time series, $$I(t)$$. Let us assume that we know recovery time distribution whose probability density function $$p( \tau )$$ is known. At some later point we will assume that the recovery time distribution is Weibull distribution, but for now let us keep things general.

At time $$t$$ we will have to consider all individuals who were confirmed on all moments prior and including $$t$$. Individuals who were confirmed on $$t$$ will recover with probability $$p(0)$$, because they had $$tau = 0$$ days to recover. While individuals confirmed on $$t-1$$ (yesterday) will recover with probability $$p( \tau = 1 )$$. The same logic applies for all previous days in the time frame of epidemic. Adding all these probabilities together we get:

\begin{equation} E[r(t)] = i(t) p(0) + i(t-1) p(1) + \ldots + i(0) p(t) = \sum_{\tau=0}^t i(t-\tau) p(\tau) . \end{equation}

The above is discrete convolution formula. For continuous observations one could replace it with an integral expression. Note that here $$i(t)$$, $$r(t)$$ and $$p(\tau)$$ are discrete. In the app below we discretize $$p(\tau)$$ from Weibull probability density function, which is not precise (it would be better to use cumulative distribution function instead), but works as an easy approximation.

Also note that in the convolution formula we have used lowercase letters to denote time series. This is to indicate that this formula use the daily new confirmed/recovered cases time series. But the formula applies to the cumulative time series (for which we use capital letters) as well:

\begin{equation} E[R(t)] = \sum_{\tau=0}^t I(t-\tau) p(\tau) . \end{equation}

Now after doing this foundation we can do a comparison between the expected recovered cases time series and the empirical recovered cases time series. We have found an optimal RMSE between the series with $$k = 2.67$$ and $$\lambda = 1/32.2$$. Mean recovery time in this case is:

\begin{equation} \langle \tau \rangle = \frac{\Gamma(1+1/k)}{\lambda} \approx 28.62 . \end{equation}

Which is close to our manual estimate from a previous post.

You may find a better set of parameters using the interactive app below. Feel free to try.