With a third of the global population in some degree of lockdown, we are still lacking a clear exit strategy. Mathematical models can help us (understand how to) tackle the coronavirus.
Gaute Linga, Kristian Stølevik Olsen, Bjarke Frost Nielsen and Lone Simonsen
“Test, test, test.”
On 16 March, Tedros Adhanom Ghebreyesus, Director-General of the World Health Organization (WHO), had a simple message for countries struggling with the COVID-19 pandemic. He emphasized the importance of contact tracing, a strategy usually applied in the containment stage of emerging infectious diseases. The strategy encouraged by the WHO is the following:
- Test individuals with symptoms,
- identify the positives,
- isolate them,
- find out who they have been in contact with – and continue recursively (that is, go back to step 1).
While several countries have shifted their strategy in dealing with COVID-19 from “containment” to “suppression”, it seems that a reliable exit strategy is currently missing. After an extended period of lockdown, societies must inevitably reopen. Even if the disease may seem under control at that point, the reopening of society is perilous. Being too quick to return to our daily lives is likely to lead to a resurgence of infections. A possible solution is to use a combination of a period of lockdown (strict physical distancing) followed by aggressive contact tracing (CT) to attain low levels of infection. Such a scheme would allow society to partially reopen without overwhelming the healthcare system. This might possibly go on until a vaccine is available, which might take a year and a half. This approach was named “the hammer and the dance” in a recent blogpost – currently with 40 million reads – by non-epidemiologist Thomas Pueyo. Despite heavy scrutiny by epidemiologists, the message seems to align with the measures currently being implemented by several nations. At the time of writing, South Korea appears to have their outbreak under control, partially due to their extensive testing and contact tracing program, which relies heavily upon GPS data. It remains to be seen whether this will persist.
The current global situation is without precedent and as such we cannot rely on experience or anecdotal evidence. At the same time, the stakes are much too high to allow for unaided experimentation. It is in cases like this that computer models become indispensable. In science, the quality of a model is not merely judged by how many digits of existing measurements it can reproduce, but by how well it can capture the essential features of a physical phenomenon with as few rules as possible. Thus, simple models can yield more insight than complicated ones. As statistician George Box put it, “all models are wrong, but some are useful.”
A simplified model shown in a Washington Post article that recently went viral (excuse the pun) has helped increase the public understanding of the qualitative effects of physical distancing and how this will contribute to “flattening the curve”. In that model, people are represented by particles which move in straight lines and infect each other when they collide. While not exactly realistic for human behavior (for example, humans tend not to move only in straight lines, but rather based on their personal preference and social relations), this simple type of dynamics is able to qualitatively reproduce the behavior of the compartmental models that leading epidemiologists and governments base their predictions and decisions on.
In the following, we will consider a model where not only distancing measures are included, but also aggressive contact tracing. The individuals in our simplified society are represented as particles that can only move along horizontal and vertical lines. When two particles come into contact, they reorient themselves and continue in a new, random direction, until they meet the next individual. We assume, like the WHO, that the disease spreading is driven by close contact: It is when two particles collide – that is, when individuals have contact – that the disease can be transmitted. At the start of the outbreak, everyone is susceptible to the disease, except for the few infected individuals (patient zeros). When an infected individual makes contact with a susceptible one, it has a certain probability of transmitting the disease to the latter. An infected individual will then remain sick and contagious for a certain period, after which it recovers and becomes immune to the disease. (For simplicity, we do not model the death of any individuals but this doesn’t make much of a difference, since individuals become immune when recovered and are thus taken mostly out of the equation). Putting these rules into play, we obtain the following typical simulation:
Based on this simulation, we can plot how the fractions of susceptible, infected and recovered (immune) individuals evolve in time:
The few initially infected individuals begin infecting others, and the number of infected individuals grows exponentially in the beginning. Later we see a slowing down of the infection, as some degree of immunity begins to develop in the population. This eventually leads to the well-known bump-shaped curve that we have all been told to help flatten. In this case, almost all of the population become infected eventually and thus develop immunity. Further, the peak load of infection is excessively high: almost half of all individuals become sick simultaneously at one point. This is exactly the kind of unmitigated epidemic that would overload the healthcare system.
In an alternate scenario, the government in this model system becomes aware of the disease some time into the exponential phase of spreading. This situation would be similar to that in Wuhan, Lombardy and more recently New York, as well as many cities during the 1918 influenza epidemic. At this point, a lockdown is imposed. Citizens are told to practice social distancing (SD) and to stay at home if at all possible. Some individuals will have to move around for society to function, so we assume that a certain fraction remains mobile. A temporary strategy of this sort leads to the following evolution of the epidemic:
The initial effect is obvious. The number of simultaneously infected individuals is curbed by the strict measures. However, when the government removes restrictions, thinking the epidemic has been sufficiently weakened, we see a resurgence of infections – a second peak. In the end, almost the entire population will go through the infection, just as in the initial scenario. The peak load is high as well – the short lockdown only had the effect of delaying the peak.
Repeated lockdown has been proposed as a strategy by the Neil Ferguson group at Imperial College – the one that made Boris Johnson change strategy. Every time there is a tendency towards an increase in the number of cases, it is brought to rest by lockdown. As can be seen in the simulations below, this can lead to a substantial flattening of the curve – the peak load is much lower than before. Hospitals will perhaps be able to cope, but in practice the impact of a periodic lockdown on individuals and the economy may be lasting. Further, strategy is risky, as the outcome may be quite sensitive to the exact timing of these lockdowns.
We now move on to aggressive contact tracing. While this measure is in force, the infected individuals of the model are subjected to testing. If an individual tests positive, it is placed in isolation, and its recent contacts are tracked down based on their location history. We assume that not all contacts can be located, but even with imperfect tracking (75% success rate in the figure below), the initiative turns out to have a marked effect. The tracing scheme is recursive, meaning that every contact that tests positive will also be placed in isolation and have its contacts put under scrutiny.
While the model individuals are in isolation, they are immobilized and cannot infect others. When their isolation period is over, they are immune and free to roam. In a situation where contact tracing is the only intervention, a typical scenario will evolve like this – some time into the exponential phase the government discovers the disease and the contact tracing procedure is set in action:
The random testing and tracing program, that is, the number of tests per time step, has a noisy signal. Once in a while, a new infection chain is discovered, leading to sudden bursts in the number of tests per time step. Each discovery leads to individuals being put in isolation, leading to a reduction in the peak load and final number of individuals that have gone through the disease. Clearly for such a scenario to be possible, an extensive testing program is necessary: during the course of the outbreak more tests are performed than there are individuals in the system.
Finally, we try what Pueyo called the hammer and the dance. A period of lockdown is followed by a period of random testing and contact tracing.
Again, the participation rate in our tracking program is only 75%, and yet the strategy works; the continued monitoring and testing is able to suppress the resurgence of the epidemic. It is likely that an increased participation rate will make the tracking programme more effective. The kind of extensive contact tracing modelled here can only be made feasible by leveraging technology such as GPS and Bluetooth data from smartphones. This will raise important debates over privacy and handling of personal data, but several countries are already developing apps which can notify users when they have been in the proximity of an infected individual – without requiring detailed tracking of the user’s every move.
The most important takeaway from what we have shown is this: although simplified models will never capture the true complexity of an epidemic, it can give valuable insights on the relative importance of various measures. The somewhat uncomfortable truth is that even once we seem to have passed a peak in the number of infected individuals and are slowly starting to go back to our normal lives, a resurgence – that is, a secondary peak – is very likely. Long term measures, like random testing and aggressive contact tracing, may be vital in keeping the infection curve below the health system’s peak capacity.
Moving forward there are several questions that will have to be discussed by scientists, politicians and the public. In society, as opposed to in the mathematical model, the implementation of a contact tracing program comes with its own set of ethical challenges. It is a delicate situation where one tries to balance privacy and medical confidentiality on the one hand, and protection of public health on the other. These are not questions that a simple mechanistic model can answer. As a society we will have to decide which measures we are willing to take; our decisions will be based on experience from around the world, but models can help us understand the mechanisms of what is going on. While our highly connected modern world makes it easy for viruses to spread, the very same connections makes it easier than ever for information and ideas to be shared.
