Splitting the Curve : An alternative approach

In my last medium post, I discussed the basics of SIR Modelling of disease spread with major emphasis on the real meaning of the term Ro and Rt. We also discussed how containment measures (reducing Ro) can effect S∞ in addition to ‘flattening the curve’. If you haven’t read it and are unfamiliar with basics of SIR modelling, read it here:

QUICK RECAP (Abbreviations) :

S∞ = Fraction of population that remained susceptible after the pandemic.

β = Index measuring infection spread

v = Index measuring recovery

Ro = β/v

Rt = Ro*S

I_max = Peak value of infected individuals at an instant

AUC = Area under the SIR-Curve

‘Flattening the Curve’ : The Math behind it.

Towards the end I referenced a few studies and through a few simulations I had done by then, concluded that even transient containments have effects on S∞. But an intuitive explanation to this was missing. This article tries to intuitively explain a solution ‘Splitting the Curve’ for placing a transient fixed- period lockdown in absence of other social measures.

How effective must the lockdown be?

It all started when we wondered what (if anything) remains constant when we ‘flatten the curve’ transiently— some people suggested Area under Curve (AUC) while some suggested total infections.

To test that out, I plotted Figure 1 through a MATLAB simulation wherein I plotted the AUC and (1- S∞) due to a lockdown of varying intensity (change in Ro) fixed during day 10–40 with initial and post-lockdown Ro fixed at 2.2

Figure 1 : X axis : Ro during 30 days (10–40) lockdown and Y axis : AUC and (1- S∞)

What we observed was not only that these two things varied, they showed a similar trend but also that they were lowest at a point between (0 and 2.2), which is a pretty counterintuitive result at first. How can a more effective lockdown cause a net increased total infections?

Approach to Intuition

So as I like to think about this, when one imposed a very strict lockdown for a fixed time, it is not reducing the pandemic but rather delaying it. A small initial peak (during lockdown) doesn’t decrease the S enough to prevent a much larger peak later.

In the limit of a lockdown of Ro=0 (during the lockdown), the pandemic is just out of phase with a natural pandemic by 30 days i.e the natural pandemic starts (with S=1) 30 days after expected(just after the lockdown is over).This is best shown in Figure 2 where we plotted the SIR curve for 3 random values of Ro during lockdown. Observe the distribution of peaks in the three cases.

Figure 2

Essentially if we have to split the pandemic into a during and an after(also before) lockdown phase with a fixed number of days (amounting to the economic hit a country can take) — its better split evenly in terms of number of infections. (Why number of infections? — read on to find out)

When should the lockdown start?

Now the next step obviously was changing what we kept constant in the last plot, the day at which this 30-day lockdown started. So a similar simulation had us plotting Figure 3.

Figure 3: X axis — Day at which 30 days lockdown started with Y axis : S∞

So keeping the Ro during lockdown fixed at 1.3 (note we got the minima around that in Figure 1), this graph varied the start day of lockdown. What we found is that there exists a special day (around day 15 here) where it’s most effective starting. Note how the graph tapers both sides.

Approach to Intuition

To make sense of this, first we must realise that after a set point (around day 20 here), the pandemic is almost done with its destruction, so initiating lockdown post that should not have any effect.

The interesting part is how even too early a lockdown can result in an insignificant decrease in total infections (Disclaimer : This model looks over early testing/contact tracing and quarantine’s effect) by a similar small peak and phase shift analogy. Figure 4 shows this in a more intuitive way by plotting the SIR curves for start of lockdown on Days 5, 10 and 15. Again, observe the peaks during the different cases.

Figure 4

Again by a similar analogy, a sweet spot in terms of the start day of a fixed period lockdown decreases S in two almost equal pieces (‘Splitting the Curve’), and both extreme values result in the decrease in S happening in a short time frame (therefore a larger I peak) which is not ideal.

“The most effective interval for imposing a limited-duration lockdown is one which straddles the peak of the spread of the disease in society in the absence of the lockdown” [1]

Figure adapted from [1] Shayak et al. 25th March 2020

Which variable is important?

In this article and the last we have gone through many things, notably — I_max (maximum infected at a time), S∞ (total susceptible left by the end of the pandemic) and AUC (Area under the SIR curve). But the question remains — what is more important practically.

Figure 5: SIR graph with days of X-axis and fraction of population in S (Red) , I (Blue) and R (Yellow) stated on Y-axis.

We saw already that (1-S∞) and AUC almost parallel each other for the most part(Figure 1). Figure 5 shows us something to think about. Notice how we get a peak near Day 20 and a tiny peak near Day 70. Practically the second peak is just a few patients walking in the hospital for a long time, this wouldn’t be a disaster as such (as herd immunity is already reached by then). But also notice what a major change it resulted in the S∞ dropping it by about 15% of the population.

Thus we conclude that I_max is a better judge of practical hazard this pandemic brings as compared to S∞.

Getting 3-Dimensional

We just saw how two different variables if applied in just the right way can give one the best outcome. Analysing both the variables simultaneously was essential to getting a feel of what was happening down here.

Figure 6: X axis — Ro during 30 day lockdown , Y axis : Day at which lockdown starts , Z axis : corresponding I_max

So at the end we plotted Figure 6 to encompass our earlier two observations in a compact manner. Figure 1 & 2 were mere slices of it at different levels.

Again it is quite apparent that an ideal day of start and intensity of lockdown do exist given a fixed length of lockdown. Quantitative evaluation of the same still stands in our way. Though practical application of this little exercise of ours is debatable.

Disclaimer

The observations reported above are based on a personal simulation exercise and is in no way supposed to predict and/or advice strategies to combat the pandemic underway. We have looked over many variables, an inherent limitation of the SIR model namely :

1. Testing lapses evident in most countries
2. Incubation period (Symptomatic & Infectious have been taken to be synonymous)
3. Age wise difference in morbidity and mortality

Also note that no fitting of data to any region has been attempted during this study. Some assumptions like Initial and Final Ro = 2.2 and v = 0.3 (i.e average infectious period = 3.3 days) were taken as an estimate of what might be possible and again are not meant to be ideal. Though we have worked over a range of possible values of initial Ro and v, this concept is seen to stand.

You can try creating your own situations :

https://cmmid.github.io/visualisations/covid-transmission-model

Conclusions

Containment measures like lockdowns should be thought of as a source of ‘preparation time’ to handle a pandemic. For a fixed period (as allowed by a region’s economic stability), there exists an alternate (and complementary) solution to flattening the curve — i.e Splitting the curve through timely placement of this lockdown.

Additional References

1. Transmission Dynamics of COVID-19 and Impact on Public Health Policy : B. Shayak et al. 25th March 2020 : https://www.medrxiv.org/content/10.1101/2020.03.29.20047035v1.full.pdf