A few more thoughts about parameter estimation and uncertainty in epidemic modelling

Last night's post was a bit rushed and devoid of analysis. I'm not surprised to see there have have been some other recent model-fitting investigations by recognised researchers in the field, including this one. The model they used is in fact marginally simpler than the one I was considering in that it does not have a latent period, but on the other hand they explicitly model death and present their equation. Which is quite handy as I think I should be able to add this to the SEIR model I'm using :-)

Something I should have pointed out yesterday. If all you have is data from the initial exponential part of an epidemic curve, then it's a straight line (in log space) and only constrains two degrees of freedom - a growth rate, and a magnitude. I had uncertain 4 parameters, 3 biological (which jointly determine the rate) and a starting value (ie magnitude). Clearly, my problem is hugely underconstrained by this data and so my priors will have played a large role in determining the results.

Worse, if the data are likely to be under-reported by a large but unknown factor, then we can't constrain the magnitude at all. This is quite likely the case when we look at reported cases of illness, as the most mild cases may never be discovered. In the UK recently, many very likely cases are not tested at all unless the patient becomes seriously ill. If we include in our estimation process an under-reporting factor which is itself unknown, then the magnitude of the epidemic will also be unknown (ok, pedants will note it is bounded below, but probably by an unreasonably low value). This latest paper above and the Ferguson et al research both used deaths to calibrate their models. Death data are probably much more secure than cases of illness, but the relationship between infection and death is again highly uncertain. By assuming a very low death rate, we can estimate the current infection level to be high (so the epidemic is widespread but probably not so harmful), and conversely a high death rate means that current infection level is low and we are in for a bumpy future.

As far as I can see, the Oxford group has basically played this game by introducing a parameter to represent the proportion of the population which is susceptible to a severe form of the disease. This parameter acts as a simple scaling factor in the death equation for their model. When this is set very small, it means the epidemic is very large, which also means it started earlier than thought (though given the nature of exponential growth, not necessarily by an unbelievable margin). When the parameter is large, the epidemic is currently small but will get much more serious.

These seem to be fairly fundamental road-blocks to doing any more detailed parameter estimation, so all results will necessarily be highly dependent on prior assumptions. Even the most elementary model has more degrees of freedom than can be constrained by time series data in the initial phase. It gets better once we are past the peak, as we know the proportion infected is then a substantial part of the population, and this will also help to get a handle on the fatality rate (though the lag means that could take a little longer). But by then it's too late to do anything about it.

Studying sub-populations that have already experienced the disease is one thing that may help. If the death rate is as low as one end of the Oxford paper suggests, how did 7 people die on that cruise ship (out of 700 cases, where there was regular testing and the total number of people was about 3500)? Bad luck, or were they just a particularly unhealthy bunch? And does their fit to Italian data (which implies that the epidemic is basically over) work for Lombardy as opposed to the whole country? Enquiring minds etc...

Edit: OK according to Wikipedia, Lombardy has 60% of the deaths but only 10% of the population. If the Oxford model (with low-death parameter) was fitted to this, I'm sure that it would put them right at the end of their epidemic with people no longer falling ill at much of a rate. However, despite the lockdown having been in effect for a while, there are still a lot of cases being reported. I say that refutes their idea, though it would take a proper calculation to be sure.