Compartmental models, networks and the coronavirus

It’s March 25, 2020, and the whole world is (or should be) battling the worst disease outbreak in anyone’s memory. It is definitely a unique situation in that it is the first emerging pandemics in the era of social media, so we get the full spectrum of information—from hard facts to nonsense—all filtered through the worries of the writer. 

In my research, we use compartmental models to understand disease spreading. These models divide people into classes with respect to the disease—like infectious (people who can spread the virus to others), susceptible (those who can get it) or removed (those who had the disease and are immune, or dead, and thus out of the epidemics). The models have transition rules between the classes—when a susceptible meets an infectious, the disease spreads to the susceptible with some probability (1), or an infectious can become removed (2). These two rules define the SIR model. Then we couple these ingredients with a model for how people meet—typically a network of social contacts—that gives us a simulated outbreak.

In almost every aspect, the models are extreme simplifications of reality: In reality, different people can be more or less infectious; in models, they are usually the same. In practice, how infectious you are varies throughout the infection; in models, infectiousness is constant. In reality, the environment (ventilation, temperature, etc.) plays a role, etc., in models, it usually doesn’t matter. One can try to improve models by adding such features, but they will never be as complicated as reality.

So we can’t accurately forecast the outbreak by compartmental models. Still, we can understand some features of how epidemics behave and how we can expect the epidemic to change depending on what we do. Thus this will give us some ideas of how to slow it down (“flatten the curve”) and eventually stop it. I will list six features that we can see both in the models and in the statistics coming in every day. I will also mention some features of models that are not directly showing in the statistics but could be essential to keep in mind when thinking and discussing epidemic outbreaks since they are lurking behind the numbers.

Without a structured population model. Let’s start with some observations in the simplest possible type of compartmental models. These are making the (“well-mixed”) assumption—that anyone can meet anyone with the same chance at every time. This is, of course, absurd, but enables one to solve the problem with simple differential equation models. Here is an excellent online simulator for such models.

Feature 1. There is a threshold behavior for the severity of the outbreak. If infectious people, on average, infect more than one other, the epidemic can spread to everyone. Otherwise, it will die out by itself. This means that if a population (say people in a country) is close to the threshold, making it just a little easier for the disease to spread can make a massive difference in the spreading speed. Here in Japan, we have had interventions (such as school closures) effective for about a month and seemingly been just slightly under the threshold. Now we are about to lift these interventions, and people (such as the brilliant epidemiologist Toshio Nishiura) worry that it will bring us over the threshold, to a situation of uncontrolled spread.

Feature 2. Once the disease gets a hold in the population, it propagates relatively predictably. This is a well-known feature of the SIR model and the explanation of why we can take the curve of the total number of infected (I+R) for any given country, extend it into the future, and get an excellent guess of the number of infected the next few days.

Feature 3. Not everyone will be infected, even if all are susceptible at the beginning. As more people recover or die, the fraction of people who could be infected decreases. Eventually, they will be so diluted that disease goes under the threshold and dies out. The incidence (number of new cases per time) will be a hump-shaped function of time that will reach zero before everyone is infected. When this mechanism has brought the population under the threshold, the population has reached herd immunity. One can estimate the fraction needed to give herd immunity by differential equation models. (Note that this statement assumes that people who have had the disease are permanently immune to it, and could fail if immunity decays sufficiently fast.)

With a structured population. As alluded to above, differential equation models, in their purest form, will give very rough answers. One can improve them in many ways, most importantly (I think) by making the contact patterns more realistic. I, and many others, have been doing this by assuming people are connected by a social network over which the disease can spread.

Feature 4. The people with most contacts will be infected first. Assuming celebrities have more connections than most, they will be infected before others. Furthermore, because they can’t hide their health status, we should be able to spot the approach of the disease by reading gossip media. Even if this is a tongue-in-cheek observation (by network economist Carolina Mattsson) and carries little medical relevance, it gives an intuition on how the disease propagation works. The slightly good news here is that these active people are also the ones who would infect most others, and since they will be the first to be immune (or die), the herd immunity will happen earlier than anticipated from well-mixed models.

Feature 5. Outbreaks happen less likely in a structured population, but when they happen, they spread more violently. The “well-mixed” assumption that the unstructured models rest upon is an extreme situation. In reality, people have very different risk behavior—some meet many others every day; some meet very few. This means that the chance a disease is introduced to a person with few contacts is high, and thus that the chance it will die early is high. Probably viruses similar to SARS-CoV-2 have entered the human population many times without anyone noticing. Conversely, when it hits highly connected people, it will be harder to eradicate.

Feature 6. For long-distance travel restrictions to be efficient, they have to happen early and unfailingly throughout the outbreak. Otherwise, their effect will at best be to delay the spreading (which could still be relevant, to buy time for the health care system to prepare). Once the epidemics start an uncontrolled spread in the community, it does not matter much if new infectious people are introduced from outside. Note that this statement does not mean social distancing is futile—social distancing is (in the absence of vaccine or prophylaxis) the best way of pushing society under the epidemic threshold—it is comparing two scenarios that both have the same rate of contacts, but with or without long-distance travel.

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