Morbid forecasting

It’s important, if very distressing, to look at the daily numbers released for Covid-19 cases, hospitalisations and deaths by the UK government. It’s important since we really do need to know what the situation is with the virus. It’s an infectious disease, such that if we’re out and about, there’s a solid chance we will pick it up, and pass it to others, often without realising. And that’s happening a lot, evidently.

If you go for cases on the government’s dashboard, you get given first off the positive tests by specimen date. That isn’t the same as the headline number released each day, but rather it puts all of those positive tests announced each day into the day in which people were tested positive. You can download the plots below really easily and they show the difference:

Cases by specimen date are updating for a good few days, and hence the number for January 7 of 28,141 will increase for the next few days (grey bars on the plot on the left). On January 4, there were 75,191 positive tests, and that number isn’t likely to jump too much more.

But what do these numbers of infections mean for hospitalisations? If people have Covid-19, there’s a good (well, bad) chance their condition deteriorates to the extent they need to go into hospital. And if things continue to go badly, there’s a chance they will die. Both hospitalisations and deaths are also recorded on the government dashboard.

We can look at how things have been over the last six months in terms of whether the number of cases on a given day can predict hospitalisations, and deaths, in the next few days or weeks.

The problem, of course, is that the testing regime has changed much over the course of the last year. It’s not right to look at the first wave, since we didn’t have mass testing at that point. That seems to have been always coming in, but from this BBC graphic from a few weeks ago, it appears to be some time before June 1:

So I’ve taken June 1 as the date that I’m looking at hospitalisations, and deaths, as predicted by cases.

I’m using conventional time series methods to model the data. That is, we’re looking at highly non-stationary data series, and hence while it may make sense to run a linear regression of hospitalisations on some lag of cases, that may lead to spurious significance. The way to get around this is to add in a lag of hospitalisations too.

This then yields an autoregressive distributed lag model to estimate. The elegance of these models is that so-called long-term effects can be estimated. That is, the total change in X for a change in Y, where X is hospitalisations (or deaths), and Y is cases. This is helpful since it allows us to include a lot of lagged information, in order to effectively pick up all of the systematic relationship between cases and hospitalisations or deaths through time.

I’ve included a lot of lags of cases — 14 for hospitalisations, and 28 for deaths, and a lag of the dependent variable in each case (hospitalisations or deaths), and then calculated what that means. I’ve put the regression table at the bottom for the interested. Don’t focus too much on the coefficients of the individual lags, since collinearity makes them less important to focus on in relation to the F-test of overall significance (right at the bottom), and they are less informative than the long-run solution.

The full (long term) effect of an increase in cases is, formally, the sum of all the coefficients on the lags of cases divided by 1 minus the coefficient on the lag of hospitalisations or deaths. That reduces those huge regressions down to something simple, and statistically well grounded.

For hospitalisations, it looks like about 8.6% of an increase in cases turn into hospitalisations. For deaths it looks like about 2.3% of an increase in cases turn into deaths.

Remember, this is based on information from over six months of data on cases, hospitalisations and deaths.

It means that each day, when 60,000 new cases are announced, that means that over 5,000 people will go into hospital, in all likelihood, in the next two weeks. Hospitalisations are already way above what they were in the first wave. Can our hospitals cope with all of these extra Covid patients? It also means that about 1,378 people will die in the next 28 days.

Observing lockdown rules in the next 28 days will do nothing about this, as these are people that already have Covid-19. The only thing might be improvements in treatment. But if our hospitals are nearing breaking point, can they realistically offer that?

But observing lockdown rules can do something about how many cases there will be tomorrow, the day after, the day after that, and so on. Please, please, observe the lockdown.