Quantifying COVID mitigation efforts by countries

Coronavirus is a raging pandemic of this generation that has infected over a million lives so far. Several countries across the world have swept in with different mitigation efforts at different points in time during the crisis. Some started late, some started early, some were strict about it and some were lenient about it. Here, we try to quantify the COVID mitigation efforts by different countries around the world using Bass Model Diffusion Curves. Live statistics of COVID cases can be found here:

Coronavirus Cases:

The Bass Model

The Bass Model is a mathematical representation of innovators and imitators stemming from the word-of-mouth strategy affecting the sales of a product in a market. It goes by:

“The probability of adopting by those who have not yet adopted is a linear function of those who had previously adopted.” — Prof. Bass, An Imitation Model, 1963.

Unfortunately, in our case, the product that is getting adopted in the market is coronavirus.

The equation is:

Equation 1

f(t) — the portion of M that adopts at time t
F(t) — the portion of M that have adopted by time t
A(t) — cumulative number of adopters of the product (or adoptions) at t
t — the time passed from product launch
M — the potential market (the ultimate number of adopters)
p — coefficient of innovation
q — coefficient of imitation

Equation 2

The derivation of Equation 2 from Equation 1 can be found is in this link.

You can see that Equation 2 looks complex enough. So, to make it simpler, we express F (t) as a linear combination of the Equation 2 in its limiting cases at p=0, where it reduces to a logistic function and q=0, where it reduces to an exponential function.

We use a parameter λ to tune between fast uptake curve i.e. exponential curve and slow uptake curve i.e. sigmoid curve.

Equation 1

Variation of F(t) w.r.t. lambda in Eq 3

Goals

In this blog, we use the Bass Model to assess:

1. How mitigation efforts diffuse through a population to curb the spread of coronavirus?
2. Forecast the rate of drop in COVID cases with the adoption of new mitigation techniques
3. Compare between different countries to assess who did it better?

Python Implementation

I sourced the data tracked by Johns Hopkins.

ncov - CSSE

The case when p=0; define a sigmoid curve a. k. a. S_shape_curve.

The case where q=0; exponential curve a. k. a. rapid_curve

The case where we have a linear combination of sigmoid and exponential i.e. transition_curve

For this blog post, I have chosen 2 countries: China and Korea to explore and compare the impact of mitigation efforts made by them. Since this is a postmortem analysis, I chose countries that have reached their steady-state.

Curve fit for COVID-19 confirmed cases — China

Curve fit for COVID-19 confirmed cases — Korea

You can see from the above two plots, how well Korea has been successful in its mitigation measures compared to China. Though being the first hotspot for COVID-19, China could contain it in the later stages of the spread. Whereas South Korea took preemptive mitigation measures, which helped contain the virus. Now, let's quantify the three graphs with the help of Bass Model Diffusion curve parameters.

Note: A fast curve has a Curve Type = 10; slow curve had Curve Type = 0

Mitigation efforts helped flatten the curve much faster.

Key Takeaways

We must take the projected peak affected population without mitigation with a grain of salt because of the absence of enough data. But we can observe the following trends:

1. The projected peak number of confirmed cases with mitigation is always lower than the ones without mitigation
2. Curve type shows how fast we progressed towards the peak, with China and Korea, where we see that the curves with mitigation have a higher value for curve type, this means they were progressing towards the peak at a faster pace. Hence their mitigation efforts helped to flatten the curve much faster.
3. We can also see that with the help of mitigation, time to peak seems to have increased with a lower number of peak infections as compared to the curves without mitigation. This is also good news because this gives the countries more time to deal with infected patients.

The code for this project can be found here [Github]

References

[1]: Bass, Frank M. 1963. “A Dynamic Model of Market Share and Sales Behavior,” Frank M. Bass, Proceedings, Winter Conference American Marketing Association, Chicago, IL, (Bass Model section starts on page 269).

[2]: Srinivasan, V. Seenu and Charlotte Mason. 1986. Nonlinear least squares estimation of new product diffusion models. Marketing Science, 5 (2), 169–178.

Future Work

1. Deaths can be modeled in a similar fashion.
2. Other parametrized curves can be explored instead of Bass Model Diffusion curves.