Economic, political, and social interactions are embedded in social settings, and the structure of these relations, determines how people behave, and the outcomes of such behavior in society. Such interactions form networks around us, of which we are a part; and these networks come in different shapes and sizes.
Epidemiology in particular, has embraced the potential of network analysis more than any other discipline. Given the interconnectedness of the modern world, infection and disease spread too, are interactions that can give rise to a network. Thus, these phenomena have a high potential for modeling, in order to make out-of-sample predictions for the use of public policy.
Metcalfe’s Law states that the effect of a network is proportional to the square of the number of connected users of the system – that is, the number of unique possible connections in a network of n nodes, is proportional to n2. What this means is that, with just a few nodes, the size of a network grows exponentially over time, thus giving rise to a phenomenon like Facebook growing from 0 users to 800 million users in only seven years.
The idea in itself, though simple, is a powerful one and can be extended to a problem across time. To exemplify, imagine how hard it must have been, to sell the first mobile phone when hardly anyone else was willing to own one, let alone use it. But as more and more people bought and used a mobile phone, thus joining a network, it became increasingly easier for non-users to buy one as well.
In a similar fashion, individuals in a population susceptible to COVID-19, may be represented as nodes in a network, with the links that connect these individuals, representing social contact between them. Simulating the spread of COVID-19 in the network, can help estimate the number of infected cases, over a period of time.
To proceed, we take advantage of the power of exponential growth. For simplicity, we consider the formula for Compound Interest:
A = P (1 + i)n
A is the Final Amount,
P is the Principal Amount,
i is the Interest Rate,
n is the Number of Time Periods elapsed.
In our illustration, we assume that the growth in the number of infected cases, follows the growth of a network. By Metcalfe’s Law, we know that networks grow exponentially, and thus, so does the number of infected cases. Mathematically speaking, compound interest, is nothing but an exponential growth of a principal amount. Therefore, we exploit the formula of Compound Interest, to model the exponential growth of infected cases as follows:
T = N (1 + r)t
T is the Total Number of Infected Cases after time period t,
N is the Number of Infected Cases at t = 0,
r is the Rate of Growth in the Number of Cases,
t is the Number of Time Periods Elapsed.
Now, we take the number of infected cases, N at time period 0, to be 37000 cases. For the simulation, we consider growth over the next t = 50 days, under three different growth rate scenarios, assumed under lockdown conditions:
r1 = 15% per day, which is the worst-case scenario;
r2 = 10% per day, which is the current scenario, based on a moving average estimate, calculated post-lockdown;
r3 = 5% per day, which is the best-case scenario.
The total number of infections, T at the end of 50 days, under the 3 scenarios are:
T1 = 4 crore (40 million) cases approximately;
T2 = 43 lakh (4.3 million) cases approximately;
T3 = 4.2 lakh (420,000) cases approximately.
While there are many problems and uncertainties that this oversimplified exercise suffers from, it cannot be faulted on the grounds of overestimating the situation. Though the figures obtained, are in no way an accurate representation of reality, they serve as lower bound estimates under ideal circumstances of lockdown.
Lockdowns only postpone the exponential growth of the virus, and in no way eliminate it. At the same time, if relaxed, we may face growth rates that are much higher than 10% or 15%, which may even prove to be disruptive. At such growth rates, we will run out of testing kits, and the healthcare system will be brutally overwhelmed. At the most, lockdowns can only help in buying time, during which the state can prepare by ramping up healthcare expenditure, quarantining infected cases, and isolating other potential cases. The best option in this regard, given limited resources, seems to be widespread Testing, contact Tracing, and Treatment of the infected (The T3 Approach), and social distancing.
Lockdowns also come with heavy economic and social costs, especially for the underprivileged. If being tested positive, means being isolated from home and family, as well as losing one’s livelihood, there is strong reason for the poor to evade quarantine, and therefore, even testing itself. Continued cooperation from the public will require economic incentives. Measures of livelihood support such as quarantine allowances, unemployment benefits, Direct Benefit Transfers, and Public Distribution System entitlements, must be undertaken by the state, to ensure the alleviation of further suffering by vulnerable groups.
Consequently, we are faced with two important takeaways to flatten the proverbial curve – We need to strengthen our healthcare infrastructure; and we need to reinforce social security nets.
The data used for the simulation, is based on the Ministry of Health and Family Welfare’s COVID-19 Tracker figures (https://www.mohfw.gov.in/), as on May 2nd, 2020. The total number of cases on the evening of May 2nd, is 37776.
As a robustness check, we can use the power of hindsight, to calculate the rate of growth of the disease spread, over the 30 days before the 2nd of May, 2020. We take the following values:
T = 37776, as on the evening of May 2nd, 2020;
N = 2547, as on the evening of April 3rd, 2020;
t = 30 days.
On solving the equation, we get the rate of growth, r to be 9.4%, which is approximately 10%. 10% also happens to be the rate of growth of the disease spread, post-lockdown.
Thus, the simulation’s conclusions do hold as approximations, when subject to different assumptions and measurements.
Author: Daniel Challam
Daniel Challam is a freelance writer and researcher, currently based in Delhi. He is a Postgrad from the Delhi School of Economics, and an Undergrad from Ramjas College, Delhi. A New Institutionalist by training, he loves taking models to the real world (and secretly enjoys proving them wrong, as long as they’re not his). His current interests lie in Political and Institutional Economics, Public Economics, and the Economics of Discrimination.