'Small-world' networks

When a pandemic like COVID-19 hits, policymakers must find the most efficient ways to allocate key resources like test kits, medicines, hospital beds, ventilators, etc. They also have to make guesses about the likely speed of propagation of the disease. This leads to decisions such as ordering shutdowns, and possible emergency fiscal and monetary measures.

Obviously, data is crucial and one of the problems is that governments routinely lie, even to themselves, when it comes to the magnitude of bad news. However, even though official data understates levels of infection and mortality rate, epidemiologists use an array of mathematical techniques to model epidemics.

Most epidemics are routinely described as “exponential”. That is, after a certain base level of infections is hit, the number will grow at great speed, until the epidemic is brought under control. The mathematical understanding can be obscured by the casual use of “exponential”. The number of infections can grow at speed without being exponential, and the number of infections may be governed by other mathematical functions.

An exponential curve is created when a base number is multiplied many times by itself. For example, let’s say one infected person returned to Delhi from Wuhan. Then that person infected two persons, who in turn, infected two more persons each, and so on. In each infection cycle, the number of infected increases by two raised to the power of the infection cycle itself.

The first cycle starts with 1, then 2, then 4, 8, etc, until the infected population jumps to 16,384 at the 14th cycle.  The first phase of an epidemic seems to go like this. Once an epidemic comes under control, this curve ceases to be exponential and the number levels off. “Flattening the curve” as the popular phrase goes, involves cutting growth rates, one way or another.

An important mathematical technique involves examining social networks when testing for an infectious disease. First of course, there’s the contact tracing and testing of anybody known to have been in touch with an infected person. This is targeted and designed to pick up people in constant contact with any individual such as family, work colleagues, domestic help, neighbours, friends, etc.

But testing should also involve the random sampling of the population at large, as done during an opinion poll. Public health officials need to pick random individuals, as representative of the population, and test them to see if there’s community spread. 
For example, one patient may have sat on a bus or a plane, or gone to a religious event, and infected a stranger. Contact tracing will not pick up such examples of community spread. Random sampling on a large scale as in South Korea and China may be necessary to find such patients, and to isolate them as soon as possible.

Importantly, most people belong to “small world” networks. That is, we tend to have a cluster of friends, family and professional colleagues, etc., who know each other too. Any Facebook user will understand what it means when you discover you and some FB friend have “55 mutual friends”. You are both part of the same “small world” network.  

Characteristically, you will also have peripheral acquaintances, who belong to other small world networks. For example, your best friend’s second cousin who you’ve met once at your friend’s wedding, has just one mutual friend in common with you. Hence, she has just one link to your small world network. But she has an entire separate small world network of her own.

Small world networks therefore, have multiple internal connections, and much fewer links to other small world networks. This means the chances of catching an infection from somebody outside of your own small world is much lower than the chances of catching it from someone in the same network. If your friend’s cousin is infected, she had just one shot at directly infecting you.

This has an important mathematical implication. If one person is infected, exponential growth is likely, until everybody within that person’s small world network gets infected (or proves immune). But if there are no linkages to other small world networks, the infection also levels off quickly.

Shutdowns of public transport services, flights, malls, movies, religious gatherings, etc., are designed to cut linkages between different small world networks. There are huge implications to shutdowns, so this is not something any government should undertake lightly.

Some non-peer reviewed papers do suggest the small world hypothesis is working with COVID-19. The data is insufficient to prove this yet, one way or another. In India’s case, the lack of random testing means we simply don’t know about extent of community spread. But shutdowns could bring the pandemic under control quicker, even if there’s a massive economic cost.

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