Does Lockdown Help Control a Pandemic? – A Probabilistic Model

Let me clarify at the outset – I am neither a biologist, nor an epidemiologist or a doctor! I am daring to take up this burning topic to discuss some interesting facts. Having tried my hands with the structure and behaviour of social networks, I am trying to understand the spreading rate of current pandemic, Covid – 19. Being in lockdown for around a month, we all may be wondering, does this really help? Yes, it does. We can prove this mathematically. 

What is a Social Network?

The connectedness found among groups of people, internet, research articles etc. is known as a social network. Generally, a social network is represented as a graph consisting of nodes and edges (either directed or undirected). Few examples:

• A network of people on Facebook/Twitter: Here, people are represented by nodes and friendships are denoted by edges. The Facebook network is an undirected graph, as friendship is mutual (Refer Figure 1). The Twitter network is directed graph, as person A is following person B doesn’t mean that B also follows A. 
• A web graph: A webpage is treated as a node and hyperlink from one page to other page is an edge. Obviously, this is a directed graph. 
• Citation Network: A research article may cite many other sources. Here, each source and the article itself are nodes. The referencing links are edges. Again, this is a directed graph.

In the study of any contagious disease, a network can be a group of people who came in contact with, either directly or indirectly. 

Influence in Social Networks:

“Tell me who your friends are, I will tell you who you are” is an age-old quote. Researchers have shown that social networks play a key role in cascading an idea/behaviour/habits etc. If your friend is obese, eventually you also may end up being obese! If your friend is happy, you will also be happy! But, spreading of contagious diseases are different from spreading of idea/habits in following ways:

• People involved in the networks do not have a choice when getting infected by a disease. Whereas, they have a choice while adopting any new habit from their friends.
• Spreading of disease is an invisible process. That is, you will not come to know through whom you got a disease, when you were in contact with multiple people.

Thus, it is interesting to know how any disease spreads across the networks. The following are the key factors required to model any pandemic:

• The degree of contagiousness of a pathogen: How quickly a virus/pathogen spreads in the network and by which mode are important factors to be understood. For example, diseases like Measles and Flu spread quite quickly, compared to Ebola and HIV. Moreover, Flu may spread to any person who is in physical proximity, whereas HIV is spread mostly via sexual contact and blood transfusion.
• Type of network: If the network (or graph) is dense, then more people get affected very quickly. Less people get affected if the network is sparse. A sparse graph is the one where there are very few edges connecting the nodes.

A Probabilistic Model to demonstrate spreading of disease in a network:

Consider a simple network shown in Figure 2. In reality, a network of people will not be a tree-structure as shown, rather it will be complicated with multiple cross-connections. But, to understand how a disease spreads, let us assume that Person A has four friends B, C, D and E. Getting infected by a pathogen also depends on immunity of the person, and hence some people may get away without being infected, even if they come in contact with an infected person. So, assume that A is infected with a disease, and the probability that he can infect anybody else with whom he comes in contact with is 0.5. Note that, each of B, C, D and E getting infected from A are independent of each other. 

Given this scenario, what is the expected (average) number of people A can infect? As the probability of getting infected is given as 0.5, we can treat the situation as a random experiment of tossing a coin. If the head turns out, the person is affected, otherwise not. So, to calculate the expected number of affected people, let us take a random variable X denoting total number of people getting infected. Let, Xi denote the ith person getting infected. Then, 

Thus, person A can infect two people on an average, if he is in contact with four people and if the probability of infecting is 0.5.

Now, let us generalize this model. Assume, the person A is in contact with k different people and let the probability of infecting any person be p. Then, expected number of people who can be infected by A is –

This number pk is known as the basic reproductive number denoted by R0. 

Now, generalize the model to multiple levels as shown in Figure 3. Let there be multiple levels of people who may come in contact with. 

Figure 3. Generalized model for spreading of pathogen

Note that, the probability of getting infected will reduce over several levels. For example, 

Let us assume that each person at Level 1 will come in contact with exactly k people. So, there will be k2 number of people in Level 2. Thus, the expected number of people getting infected at second level would be p2k2. Continuing this assumption, we can say that the expected number of people who are getting affected by the pathogen at Level i would be (pk)i. 

If a disease can spread to multiple levels in the network, then such disease is known as epidemic. And, the value of the basic reproductive number (R0 – the number of secondary infections) indicates whether the disease is going to be an epidemic or not. Let us illustrate the role of R0 with the help of examples as shown:

It is clear from the above illustrations that even if R0 is 1.1, the disease may turn out to be pandemic and if R0 is just smaller than 1 (like 0.9), the disease gradually dies away.

What to do to stop a disease from becoming pandemic?

Answer is simple: Reduce the value of the basic reproductive number! We know that R0 is a product of p and k. We can reduce either of these. The probability of getting infected can be reduced by being more hygiene and improving the immune system. But, when we are not sure how to improve the immune system (like in the case of novel Covid -19), we can think of reducing k – which is the number of people who comes in contact with an infected person. Reducing k will make sure that after a few levels, R0 is dragged down. Thus, the lockdown and/or social distancing can help to reduce the impact of pandemic.

Remember that, for illustration purposes, we have considered a network in the form of a tree. The real-world network of individuals will be much more complicated. However, the model discussed here can be applied there too.

References:

1. Networks, Crowds and Markets by David Easley and Jon Kleinberg, Cambridge University Press, 2010 
2. Social and Economic Networks by Matthew O. Jackson, Princeton University Press, 2010.
3. Obesity is Contagious – The Harvard Gazette (https://news.harvard.edu/gazette/story/2007/07/obesity-is-contagious/)
4. Social Networks: A Course by Prof. Sudarshan S Iyengar, IIT Ropar (https://nptel.ac.in/courses/106/106/106106169/)