Actuarial Techniques for COVID-19 in Malta: Markov Chains

A version of this article has been published in the Times of Malta and can be viewed here.

On 7th March 2020, the first three cases of coronavirus were reported in Malta. As at the time of writing (16th March 2020) the number of total cases to date are 30, two of which recovered. The aim of this series of articles is to apply actuarial techniques in a simple manner to explain the onset of this virus in Malta.

Before we start, let’s set up a few, uh, provisos, a, a couple of quid pro – any prediction here is bound to be incorrect for a number of reasons:

• Limited amount of time is being dedicated to this prediction (actually I am making sure to dedicate a maximum of one hour per session not to make the math unapproachable),
• Only limited public data is being used (knowing the number of swabs and visiting individuals would be great additional parameters),
• I am not a healthcare expert,
• Statistics seem to vary by country - so it is hard to latch to some benchmark data,
• We are still at early stages so extrapolation is quite unreliable, and
• Any model is bound to be imperfect anyway.

Building the Markov Chain

This first article will link what is happening to Markov Chains. Let me start by explaining the key concepts.

The base example of a Markov Chain would include three possible states: healthy, sick and dead. The arrows in the state-space diagram would show the possible movements over a period of discrete time. Once dead, you are bound to be dead the next day/month/year (only one known case to the contrary about two thousand years ago and a cat in Sliema in 2015). If you are healthy, in most likelihood you would be healthy the next period, so that arrow should have a high percentage. 

In this case we are identifying coronavirus cases and hence we should consider death by coronavirus as a separate outcome. We also add the state representing persons who were sick from coronavirus and recovered. We consider these as immune to COVID-19. I found a research study based on 1099 cases to be the best approximation for the number of deaths (1.4%) but this ranged up to 4% in other reported statistics (Prof Grech uses 2% in his excellent analysis for Malta.). The rate for the Markov model would be smaller as it would evaluate the death over a period (for example if the death rate is 2% and people spend twenty days sick, it could be 0.1% per day if uniform [it is not]). At the time of writing, we have 28 sick and 2 immune that we aware of in Malta.

The next stage would be to subdivide the sickness by coronavirus in two stages – early-onset and late-stage in the sickness. For simplification, I assume that early-on always lead to late stage. Finally, we should consider that not all healthy individuals are at the same risk. Some of them are already exposed to the virus and possibly a carrier. This group puts the others at risk – up to two weeks following being exposed. A study published last week in Germany showed that the transmission is much higher during the first few days of exposure and sickness.

This leads us to a final Markov Chain with 7 states. Ideally this has more states and varies by age group as we know that the elderly and weak are at higher risks. I am also assuming that healthy recovered immune individuals are not carriers. Notwithstanding this helps us build our model. We know that we have 28 sick, about half in each stage, and two that are immune. The key concern is how many are carriers?

The UK has followed an approach of having carriers/exposed that are low-risk groups in order to create a herd immunity, basically creating a link between Healthy (exposed) to Healthy (immune) or assuming low-risk go through the cycles of sickness quickly with little to no deaths. On the other hand, most other countries are worried about flattening the curve – reducing the number of unexposed to exposed and hence sick. The rationale is that many will get sick anyway and the problem is the healthcare system cannot cope with all getting sick at the same time. This goes hand-in-hand with the hope that the virus might calm down once the weather improves and a vaccine might be eventually created.

Initial Estimates

As for the time period, I am working on two, four and seven day intervals. The rationale is that it seems that half of the cases have that gap from exposure to virus to the manifestation of the sickness.

The level of uncertainty for the estimates is quite high so any estimate is bound to have some errors. I would expect in total 41 to 62 cases (with 50 being the median) to be reported by Wednesday (from 25 right now) and 88 to 365 cases to be reported by next Monday 23rd March (median 180). On the plus side, we should be having more recoveries too – one to four more by Wednesday. As for the number of deaths – hopefully none by end of week.

These predictions are best-estimate predictions not a range of possible outcomes. Indeed the wider range for the 7 day period is due to the inherent uncertainty in the early stages of the model.