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Coronavirus: Modes of diffusion

Last updated on 2020-03-17

In these times, we are all worried about what to do to stop the virus. Politicians have to think about measures that put people out of anxiety. Experts try to convince politicians about what to do to stop the spreading as quickly as possible. Business people will try to save their companies from bankruptcy.

Maybe, we should try to get to some basics.

In this, and in the next posts I will propose a few ideas and analyze the available data to show what is happening and what might be used to learn some lessons from data.

The social network

The virus is passed from person to person through contact and proximity. This is what we all know. However, what we don’t know is how many people are in the proximity of anyone else.

We should think of every person as a node in a network that we call the social network. In this network, I am a node and my friends and relatives and colleagues are also nodes. They are connected to me through edges (or links). The edge represents the potential to be in close proximity to them. But people that I never meet, are not connected to me in this social network.

In this network, there are two structures. In the local structure, the friend of my friend is also likely to be a friend of mine. As an example, my colleague Markus is also a colleague of my colleague Andrea. In the long distance structure, a friend of mine in one part of the network is most likely not a friend of another friend of mine, who is in a different part of the network. For example, my football friend Jan is not connected to my sister.

Modes of diffusion

On the long distance structure the diffusion is fast. We call this process, social hopping. Someone could first infect some friends in one part of the network. Some of these friends may hop along a long-distance connection somewhere else in the network. What could happen is the following. One person infects two people, who hop somewhere else on the network and each infects other two people. These four people can hop into some other distant part of the network and each infects another two people. On the long time, at each hopping event the number of people infected has duplicated. So, maybe the effective number of hoppers is not two (as we shall see from the data) but the effect is the same.

With social hopping on the long distance structure the number of infected people grows exponentially.

On the local structure, where a friend of my friend is also my friend, the infection propagates more slowly because if someone infects both friends they cannot infect each other again. We call this the small community diffusion. We can think of it as a circle. When someone is infected, he or she is at the center of a small circle of friends. This circle of friends will get infected. Due to the local structure of the network, at the next time step a circle of friends around the first small circle is also infected. At the next time step, the next larger circle is infected. Summing it over the time, the number of infected people is proportional to the area of a circle whose radius grows linearly with time. As the area of a circle is proportional to the square of the radius, the total number of infected grows like a polynomial of degree two of the time.

On the local network the number of infected people grows like a polynomial of time.

Maybe the growth is not like a polynomial of degree two, maybe it grows like a volume or something different. We don’t know how to compute that exactly from first principles, but we know what to look for.

An exponential growth is like a polynomial of very high degree. When growth is exponential it is a very dangerous situation. Nevertheless, such a growth cannot last forever. Indeed, whatever the law of growth is, at a certain point the proportion of people who have been already infected and those who are no longer susceptible becomes so large that propagation slows down and then eventually stops. This is why we need vaccines: to decrease the proportion of susceptible people and slow down or even stop completely the diffusion of a disease over the social network.

What do the numbers say?

Well, I will show some numbers and plots in the next posts. What is clear is that the initial spreading must occur per social hopping and will therefore be of an exponential type. Each social hop will create a new hot spot where the infection propagates through the local small community. On the long term, when social hopping is reduced (because big social hoppers get sick or because of policy measures to prevent contagion), the small community diffusion takes over and the number of infected grows more slowly than exponentially in time.

The difficulty of policy makers

Policy makers have to take decisions to limit the spread, from exponential to polynomial, to linear, to eventually stop the diffusion. The difficult part of the equation is that whatever is decided today, will have its effects in 10 to 14 days. There is no policy that has an immediate effect.


Today (March 17) I have found an article originally published on February 19 in the Washington Post in which the model of diffusion through the small community is shown graphically. It’s a great reading.

Readers from Associazione Culturale Polyhedra have pointed out another article from the Washington Post with an animation explaining various modes of containment of the diffusion.

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