Blog: What do modelling earthquakes and epidemics have in common?

By Thomas Le Guenan, Research Engineer, BRGM
 
These days it is easy to think every scientist is working on COVID-19. Of course, those in medicine are focusing on finding a treatment or a vaccine for the virus. But it also seems that, for social scientists, economists, mathematicians, physicists, psychologists and so on, everyone's new priority is the pandemic and its consequences for the world.
 
Which leaves us with the question: what are the geoscientists doing?
 
In SECURe, one of our objectives is to model induced seismicity - the increase in microseismicity frequently observed during subsurface operations. We use the term "micro" seismicity because, in the great majority of cases, the seismic events are undetectable without dedicated, buried sensors.
 
The science of seismicity (and induced seismicity) is complex and non-linear: for a given known cause, such as a change in subsurface pressure, the effect will not necessarily be the same every time. If you clap your hands twice with the same force, for example, you expect it to produce the same sound. This is not true in induced seismicity: for a given injected quantity of a liquid or gas, the observed seismicity may differ.
 
One famous phenomenon is what we call aftershocks: following a (often large) main shock, the base rate of seismicity will remain high for a period of time and produce many (often lower) aftershocks. In fact, this distinction between main and aftershocks is only theoretical. In practice, any event of any size is capable of triggering its own aftershock sequence: one earthquake generates other earthquakes, each of which will itself generate its own sequences.

This is similar, in principle, to an epidemic: an infected person infects the people around them. Once these people are infected, they can each infect new persons and so on. This causes the R, or reproduction, rate to rise above one, leading to the exponential growth we all followed anxiously at the onset of this epidemic. Indeed, one of the best known models for natural and induced earthquakes is called the Epidemic Type Aftershock Sequence model or ETAS.

In Figure 1, you can see a computer simulation of induced seismicity modelled with ETAS. Each event (micro earthquake) is represented by a red dot. The highest magnitude produced here (generated by the computer rather than real life!) is slightly lower than 2, which would not be felt at the surface. The blue line represents the instantaneous rate: the higher this line is, the higher the probability that an event will occur soon.

You will also notice a solid black line, which represents a quantity of fluid that is injected (again, in a computer, not real data) as a function of time. We can see that when the flow rate stops, we expect seismicity to drop “on average”. In this simulation, there is still a large spike (or a “cluster” as an epidemiologist would say) about 20 days after injection stopped! In this case, it is difficult to say if this spike has been caused indirectly by the injection, or if this increase would have happened anyway due to inherent randomness. It is a good illustration of a non-linear process.

As with COVID-19,  we could say that the strategy for geoenergy applications is to "flatten the curve": the release of the accumulated energy should be spread out over time to reduce the risk of inducing a larger event.

Figure 1 : Simulation of induced seismicity using the ETAS model. Source: Thomas Le Guenan

This model, as well as many others, is reviewed and tested in SECURe deliverable D2.3. It comes from what is now called a "Hawkes process" after Professor Alan Hawkes, the British mathematician. The principle is quite simple: "the occurrence of any event increases the probability of further events occurring".

According to Hawkes, the first application of these processes was in the field of earthquakes; subsequent updates led to the ETAS model. However, these types of models have been used in a range of fields: neuroscience, ecology, molecular genetics, crime prediction, terrorism, and… epidemiology. This is what earthquakes and an epidemic have in common.

References
Hawkes, A. G. (2018). Hawkes processes and their applications to finance: a review. Quantitative Finance, 18(2), 193-198.

Ogata, Y. (1988). Statistical models for earthquake occurrences and residual analysis for point processes. Journal of the American Statistical association, 83(401), 9-27.