The dynamics of epidemics can be likened to basic principles of physics, because the host-virus interaction has its physical attributes. In physics, momentum is defined as mass X velocity. In an epidemic, the number of active cases at a given point in time is like mass. The velocity is the speed of spread. The momentum of the epidemic is therefore the number of active cases X speed of spread—momentum may accelerate or decelerate.
When the driver of a speeding vehicle applies the brake, it slows down (decelerates) before coming to a stop. The lag from the time point when the brake was applied to the time the vehicle stops is determined by the momentum before the brake was applied.
In epidemics, the number of new infections generated by previous infections depends on the ‘effective reproduction number’, ‘Re.’ In simple terms, it is the average number of people a single afflicted individual infects at a given point in time. Acceleration and deceleration are indicative of the changes of Re over time. Re > 1 means the epidemic is growing and Re < 1 shows it has started declining. At the transition, which is the peak of the epidemic curve, Re may be ~1 for a few days—so the numerical peak may be blunt and not needle-sharp.
In this analogy, a decrease in Re is the brake that slows down epidemic spread. The ‘resistance’ to spread is the prevalence of immune subjects who resist disease; if the prevalence of immune subjects increases, concurrently the prevalence of susceptible subjects will decrease. As more subjects in a given population get infected, the ‘susceptible pool’ becomes smaller and the speed of spread is impeded. This is why the ascending limb of the epidemic curve has a phase of steep climb, deceleration and slow climb, a short-lived plateau, followed by the descending limb.
What we describe is the fact that the epidemic has a virtual peak when the maximum acceleration (Re for the second wave probably > 3) decelerates to Re = 1 corresponding to a visible numerical peak, the highest number so far, after which the reported daily numbers show a steady decline. We have drawn two graphs using the daily acceleration or deceleration applied to the first and second waves. This is an innovative way of understanding the dynamics of the epidemic; the numerical peak shows the date of maximum case numbers like where the vehicle stopped, but the virtual peak shown on the acceleration-deceleration graph shows where the rate of infection spread rapidly declined like where the brake was applied.
If the epidemic is allowed to run its natural course without interference, the epidemic curve will be roughly symmetrical—bell-shaped. This was clear in the first wave, which took six months to reach the peak (on 16 September 2020) and declined to low numbers in five-and-a-half months, conforming to the bell-shaped symmetry.
Graphs showing daily numbers of new Covid-19 cases in India, prepared by different agencies, are available in the public domain. We used data from the ‘Worldometer Coronavirus India’ website for our analysis. The daily testing is mostly on self-reporting symptomatic individuals and contacts screened for early detection of infection for quarantine purposes.
The daily numbers of new cases shown on the Worldometer bar graph are determined by daily numbers tested. As the number tested is fewer on Sundays, Monday’s numbers are always low. In order to remove the weekend artefact and smoothen the graph, Worldometer India uses a daily rolling average of the preceding seven days on a linear graph, a better reflection of the time trend of spread of infection.
From the seven-day rolling averages available in Worldometer, we computed the daily change in the number of new cases—this increases with acceleration of epidemic spread and decreases with deceleration. The graph of acceleration and deceleration is a novel way of understanding the dynamics of the raw numbers presented in the graph in the Worldometer website.
We can conceive of two peaks in the epidemic, first a peak of acceleration that denotes the time point of the most rapid spread (Re is maximal) and second, the peak in the number of new cases, commonly considered to be ‘the numerical peak of the epidemic’.
When we applied this method to the first wave, as shown in Graph 1, the deceleration started on 11 September 2020 and steadily progressed to about September 22. Therefore, the virtual peak was on September 9, when Re started declining, but the visible numerical peak was on September 16. The lag was seven days, from September 9 to September 16.
Re declined steadily from September 10 to September 22 and the peak was in its middle on September 16; the peak of acceleration was on September 9 but the peak in the number of new cases was seven days later; the ‘momentum’ of the epidemic delayed the numerical peak.
The momentum of the first wave was much smaller than that of the second wave because the number of active cases were fewer and the speed of spread only one-fourth of the speed of spread of the second wave. We can therefore expect a higher momentum, and longer lag time from virtual to numerical peak of the second wave.
The shape of the epidemic curve of the second wave (in Worldometer India) is strikingly different from the first, with a very steep rise, racing towards a peak. The second graph represents acceleration-deceleration of the second wave. We pinpoint the peak of the acceleration-deceleration curve of the second wave as April 21, after which deceleration has progressed steadily to date. On May 6, the day’s caseload was the highest of this season, 4,14,433, marking the visible numerical peak.
The most important lesson from this analysis is that the second wave of epidemic is declining. Now is not the time for a national lockdown to reduce the speed of spread. The appropriate time, in retrospect, was during the last two weeks in March or the first two weeks in April.
The second wave is not uniform in cities, districts and villages of each state. State governments can analyse their new cases using this technique and decide on the need for and timing of regional lockdowns.
Dr M S Seshadri
Medical Director, Thirumalai Mission Hospital, Ranipet and former Professor of Medicine and Clinical Endocrinology, CMC, Vellore
Dr T Jacob John
Former professor of Clinical Virology, CMC, Vellore