How Network Science is of help on studying Epidemics like Covid-19

During these days we are frequently reading in newspapers about the Logarithmic Curve, the Reproduction Number and the Epidemic Threshold, sometime called R-Zero. But where are all these words coming from? During my studies of Complexity Science with the Santa Fe Institute one of the most interesting topic that caught my attention was, and still is, Network Science. Network Science encompasses a variety of scientific fields from Technology to Biological, Financial and Social Networks, just to mention a few, and its beauty stems in the fact that there is a toolbox of quantitative models that can be applied to all these networks in order to extract some interesting information from them.

Here below the topological representation of a couple of Networks take from Caldarelli (2013)

Today I would like to focus on how scientists have used Network Science and its Quantitative Tools to study Epidemics. Some airborne diseases, like influenza and measles, spread over networks of contact between individuals others like HIV are communicated when people have sex. All these contact structures can be represented and summarized as networks and many academics and scientist have spent quite a bit of time on studying the "Structure" of such networks. I attach here below a very interesting Community Lecture held by the Santa Fe Institute last year on Preventing Pandemics. I encourage the public to watch these two lectures because they are very interesting

It is also interesting to point out that like epidemics, there are many other "spreading" processes like the spread of news, gossip and rumors that can be quantitatively modeled like the spread of a disease but those are out of the scope of my article.

The biological mechanism that characterize what happens when an individual or "host" catches an infection is very complicated but luckily there are simplified models that describe the spread of a disease that can be a good guide to disease behavior.

The SIR Model

This is a simple model of infection that is mostly used at the moment to inform the general public on how to defeat Covid-19. It is a three-state model that stands for Susceptible-Infected-Recovered, i.e SIR Model.

The dynamics of the SIR Model has two stages. In the first stage, susceptible individuals become infected when they have an interaction or contact with infected individuals. Contacts between individuals are assumed to happen at an average rate β per person. In the second stage, infected individuals recover (or die) at some constant average rate γ. The model is governed by three differential equations

Where

• s represents the fraction of individuals who are susceptible (s = S / n)
• x represents the fraction of individuals who are infected (x = X / n)
• r represents the fraction of individuals who have recovered (r = R / n)
• n represents the size of the population

Working on the set of equations above, after some math we end up with the integral below

This integral cannot be evaluated in closed form, but it can be evaluated numerically. Here below an example of the time evolution from the book "Networks" written by professor Paul Newman

From this figure we can note several interesting things.

1. The fraction of susceptible in the population decreases monotonically as susceptible are infected and the fraction of recovered individuals increase monotonically. The fraction of infected on the other side, goes up at first as folks get infected, then down again as they recover, and eventually goes to zero as t goes to infinite.
2. The asymptotic value of r has an important implication; it represent the total number of individuals who ever catch the disease during the entire course of the epidemic, i.e the total size of the outbreak and can be represented by the following equation

This is the most important equation, the one that directly or indirectly we all hear nowadays in the news, from Doctor Fauci, to Ms Birks and others. What is the R-Zero that we want to push below 1 via Social-Distancing? It is part of the Equation above. Recall that

• ? represents the "contact rate" of a population, essentially measuring the number of social contacts/interactions that can transmit the disease per unit of time
• the inverse of γ, i,e (1/γ) represents the Infectious Period

R-Zero is simply (?/γ) and if we are able to push this ratio toward 1 the epidemic goes continuously to zero; if we are even able to push ?/γ below 1, there is no epidemic at all. If ? <= γ then infected individuals recover faster than susceptible individuals become infected, so a disease cannot get a hold in the population. The transition between the regimes where there is and there is NOT an epidemic happens at the point ? = γ = 1 which is called epidemic threshold.

Without a vaccine, we can only try to push R-Zero toward, and eventually below 1 only working on reducing the ? in the equation above, that means aggressive social-distancing.

It seems that in the US, after one month of social-distancing, we are having some success. As you can see from the chart below, the rate of growth of infected people in the US has been somewhat reduced.

Looking at Traffic Data for Washington, DC you can see that folks are doing their best to stay at home. The difference in traffic between this week (red line) and the same week last year (dotted-line) is remarkable.

Conclusion:

Social Distancing is a blunt way to reduce R-Zero but until we don't find a vaccine for this disease, it is probable that we will experience a second wave when the US Government will relax restrictions. I don't see a clear road map from this Administration on how they will reduce to the minimum the emergence of a second wave during the summer/fall.

It is very important to re-start the economic engine, fixing the Supply Side should not be an insurmountable task, but if there will be no Demand, because people will not dare to go to a shopping mall or to a restaurant, the economy will not likely gain much momentum, in my opinion.

References

1. Andrew Stier, Mark Berman, Luis Bettencourt, Covid-19 attack rate increase with city size, (2020)
2. Mark Newman, Networks (2018)
3. Melanie Mitchell, Complexity: A guided tour (2018)
4. Santa Fe Institute, www.santafe.edu
5. Guido Caldarelli, Scale-Free Networks, complex webs in nature and technology (2013)