‘Flattening the Curve’ : The Math behind it.

We have heard enough about ‘the curve’ being flattened by the better social strategies adopted at the international level including lockdowns of entire nations. Some reliable institutions have also questioned the effect these mitigation strategies have on the pandemic we are in.[1]

Which ‘Curve’ is this?

Figure 1 : Infographic taken from The Centre for Disease Control (CDC)

Let us first understand the “Curve” we are talking about. Figure 1 shows the curve for number of infected people at a given time (Y-axis) plotted against progress of the pandemic (X-axis) according to the SIR Model which is a prime choice for epidemiologists to predict progression of disease spread in an isolated population. [2]

So, we have SIR which stands for Susceptible, Infected and Removed respectively and consequently represents the fraction of population in those states.(Side note : Removal includes either recovery or death — in either case the person is removed from the model as he/she is assumed to not be able to get infected again.)

Figure 2 : Adapted from J.H Jones’s Notes on Ro 2007 [3]

The whole curve boils down to three differential equations which seem quite obvious with essentially only two terms predicting the rise and fall of our 3 parameters (S,I,R). We assume that at the start of a pandemic, only a few (ideally one person) people are infected (known commonly as ‘Patient Zero’) and consequently the entire population is susceptible (S = 1).

Why flatten the curve?

We assume a straight line parallel to the X-axis that represents the capacity of our healthcare system to handle extra patients, that is, in addition to those requiring emergency care for other conditions like stroke or heart attack. Most fatalities in many countries (especially of the young and immunocompetent) have been traced to lack of medical attention. So flattening the curve makes the pandemic progress slower (and for longer) so as to not overwhelm the medical systems (beds, equipments and service providers) in place.[4]

The feel of the Differential Equations

The rate of susceptible people becoming infected should depend on the number of infected and the number of susceptible which explains the equation (2) of Figure 2. The rate of recovery however depends on the healthcare services and the natural history of the disease itself, along with being proportional to the number infected people which explains equation (4) of Figure 2. Equation 3 remains self explanatory. See the video below for a thorough explanation to this.

‘The Coronavirus Curve’ by Numberphile

When you try to plot all three of the variables (SIR) on a single plot, the curve looks something like Figure 3.

Figure 3 : SIR Curve for an arbitrary value of β and v. Note that at the end of the curve S asymptotes to a value between 1 and 0 (which is a variable of Ro as we will later see)

Remember there were these two parameters β and v in Figure 2, they are essentially the constants that dictate these differential equations. When we take their ratio (i.e β/v) we can derive a meaningful variable that represents how fast the spread is going to happen. It is interesting to note how these variables ( β and v) change with lockdowns and super-awesome hospitals. This derived variable is what people technically refer to as Ro i.e Basic Reproduction Number (some call it the ‘Basic Reproduction Rate’ but that is technically flawed according to its dimensional unit).

Ro = β/v

Now we can rearrange (3) from Figure 2 in terms of Ro to get :

dI/dt = (Ro*S -1) * I * v

Easily notable from the above is that anytime Ro is less than 1, the number of infections do not rise initially and the pandemic would not happen in the first place. Also notable is, as the pandemic progresses and S decreases until it compensates for the Ro > 1 (at the peak) — signalling the ‘end of the pandemic’. Seeing this beautiful behaviour of Ro*s, we started calling it ‘Rt’, rightly so because of how it varies with time.

Rt = Ro*S

How are we to modify this Ro and/or Rt?

As we saw Ro is a function of β and v. So essentially all we are doing to curb the pandemic is reducing Ro, so as to reach the peak at higher S (and consequently less number of people infected and/or removed).

Modifying Ro depends on 3 essential factors on which it depends :

1. Interaction : More the interaction between infected and susceptible persons, more are the chances of transmission. This is kept in check through lockdowns.
2. Chance of Spread : The probability of transmission to the susceptible person given the above interaction occurs. This is modulated through better hygienic practices.
3. Duration of Illness : Longer a person remains infected, longer will he/she be capable of spreading it to other susceptible people. This is reduced by better medical care.

Modifying Rt (exclusive of Ro) depends on reducing ‘S’ which is essentially done through 2 ways.

1. Vaccination : Artificially ‘removing’ sections of our population bypassing the infected stage.
2. Natural Way : Wait for long and the virus does it for you.

End of the Pandemic

However you treat the course, it does come to an end. What differs is the number of people that come out alive from it. The hospital overload is the primary determinant of this and has been talked about extensively in literature. However what this article will focus on is how changes in Ro (measures we take) have an effect on the total number of people infected throughout the course of the pandemic irrespective of how the fate of the infected turn out (death/infected ratio).

Figure 4 : SIR Model graph with arbitrary β and v to fix Ro = 2. Note the asymptote of S.

Figure 5 : SIR Model graph with arbitrary β and v to fix Ro = 3. Note the S∞.

It can be demonstrated through a computer simulation (Figure 4 and Figure 5) that with decreasing Ro, more fraction of the population remains susceptible at infinite time to the start, and assuming no vaccines were involved, this means less people suffered the infection.

This can be theoretically proved by integrating dS and dI from Figure 2, with limits of I (from 0 -> 0) and of S (from 1 -> S∞) to arrive at [3] :

log(S∞) = Ro(S∞ − 1)

Figure 6 : Solutions for equation y = log(S∞)-Ro(S∞ − 1). Note that 2 solutions exist for all Ro > 1 (one being always S =1 and the other being between 0 and 1). Adapted from J.H Jones’s Notes on Ro 2007 [3]

What should be the plan ahead?

It has been shown earlier by Rajesh Singh et al. 2020 that unplanned lifting of measures like lockdown in the Indian setting can result in a second peak (as in Figure 7) after the pandemic has supposedly ended. [5]

Figure 7 : Prediction of CoVID — 19 pandemic changes post lockdown lifting in India. Taken from Rajesh Singh et al. 2020

Such scenarios can also be simulated given enough data on changes in Ro post changes in pandemic management is available.

These results can be theoretically explained on the basis of our model as a spontaneous change in Ro midway in the SIR curve. Though the ‘S’ at that point was compensating for the Ro (with lockdown), the sudden change (increase) in Ro may cause the net Rt to cross the threshold of 1, converting the pandemic from a sub-critical one (technically not a pandemic) to critical.

Therefore, we predict that before increasing R0 (post the peak) through any measures (like lifting the lockdown), all sense of false security must be wiped and the Ro change must be predicted. The drop in ‘S’ (while in lockdown) should be able to compensate for the new Ro (not only the old Ro) so as to avoid a resurgence (and another peak).

It can be also be shown (by simulations — though the mathematics behind it is a bit tricky) that even a temporary containment (lowered Ro) would decrease (to varying degrees) the S∞.

READ MORE :

https://medium.com/@bhavikbansal/splitting-the-curve-an-alternative-solution-4ca2bd1f23db?sk=3ed77d53244be25d968f0a5550c32152

Conclusions

Containment of the pandemic through social measures by ‘flattening the curve’ even transiently not only decreases the workload on our health services, but also prevents a greater fraction of the population from contracting the infection in the first place (throughout the course of the pandemic)

Assumptions & Limitations

The whole article was formulated as if every step is quantifiable and every person classifiable. The problem of lack of data due to inefficient testing was not addressed. The period between exposure and infection was neglected (though some studies predict a better suited SEIR Model). [6]

References

1. https://www.imperial.ac.uk/media/imperial-college/medicine/sph/ide/gida-fellowships/Imperial-College-COVID19-NPI-modelling-16-03-2020.pdf
2. A contribution to the mathematical theory of epidemics : William Ogilvy Kermack and A. G. McKendrick. Published : 01 August 1927 https://doi.org/10.1098/rspa.1927.0118
3. Notes on Ro : J.H Jones. http://web.stanford.edu/class/earthsys214/notes/Jones_R0_notes2019.pdf
4. How will country-based mitigation measures influence the course of the COVID-19 epidemic? : RM Anderson et al. https://doi.org/10.1016/S0140-6736(20)30567-5
5. Age-structured impact of social distancing on the COVID-19 epidemic in India : Rajesh Singh
   arXiv:2003.12055
6. A note on the stationary distribution of stochastic SEIR epidemic model with saturated incidence rate : Qixing Han et al. 10.1038/s41598–017–03858–8

Additional Content

1. Epidemic Calculator : https://gabgoh.github.io/COVID/index.html
2. The MATH of Epidemics — https://www.youtube.com/watch?v=Qrp40ck3WpI
3. Simulating an epidemic (3b1b) — https://www.youtube.com/watch?v=gxAaO2rsdIs
4. Hans Nesse — Global Health — SEIR Model http://www.public.asu.edu/~hnesse/classes/seir.html