One of the most harrowing aspects of diving into uncharted waters, is the uncertainty of what’s to come. And the Coronavirus pandemic is no exception. With each passing day, the numbers of infected individuals rise, and the numbers by which they rise, rise. And this continues until the elusive inflection point is reached, after which the numbers of infected individuals continue to rise, but by lesser each passing day, until they don’t. But how do we come to expect an outcome of this kind? What factors govern the likelihood of such an outcome?
These questions are best answered through the use of “compartment” models. These are mathematical models which segment a population into a number of finite categories, and study the macroscopic evolution of these “compartments” in time.
Epidemiologists work towards modelling disease scenarios in many ways. Obviously, the susceptible sections of the population and the infected ones form a crucial part of this modelling. This should give us the most simple model – the SI model. Here, we consider only two possible states for any randomly selected member of a population – susceptible to infection (S), or infected (I), where they have the disease and are capable of transmitting it to others. In this scenario, the disease runs amok, “consuming” susceptible people until there aren’t any left. The infected eventually die. This takes rather a bleak view on things. But if correct quarantine measures are taken, the damage is limited to a small group of individuals, and without sufficient charge, the “fire” eventually dies down. The next variant is the SIR model, where we consider an additional class of people – the recovered (R). This set of people are unable to transmit the infection any further, having successfully repelled the invaders, and develop lasting immunity from the disease. This model fits Covid-19, in that cases of recurrence of the disease are sparse. The SIS and SIRS models are variants of the aforementioned models, which represent scenarios in which the infection relapses in cured or recovered individuals. These, along with the SEIR and SEIRS models (which incorporate an additional “exposed” section, E) comprise the different types of compartment models used to model epidemics.
The fundamental premise of using mathematical models to inform administrative decision-making stems from their ability to offer a long-term picture of the disease’s progression. Predictions obtained in this manner are sensitive, not only to the initial conditions (say, the initial numbers of susceptible and infected individuals), but also the choice of parameters used by the modeller. Often, the existence of non-trivial equilibrium states (potential long-term scenarios) is conditional upon these parameters, which may be “tuned” to preempt unfavourable outcomes, or steer the system towards more favourable ones. One particularly famous parameter is R0, the basic reproduction number. When R0>1, the forward progression of an epidemic is inevitable, until a stable “endemic” equilibrium (the so-called flat part of the curve we are so anxious to flatten, only we’d like to do so before it consumes the entire susceptible population) is reached wherein the number of infected individuals plateaus, and eventually decays as people recover, or die from the disease. This results because any population is finite, and the disease cannot grow boundlessly. But R0 is not a purely biological constant, and depends on the rate of increase of susceptible individuals, and the vital dynamics (birth, death rates) of the population in elementary models. This is precisely why scientists urge us to maintain social distance, whenever possible. If the value of R0 is driven down to less than 1, the disease automatically dies out, as the number of infected individuals decays to zero.
The ultimate test of model accuracy is how closely it predicts (the trend, at least) witnessed on the ground. As of today, the numbers of infected individuals in South Korea, India, Italy and the US are 10,537, 14,792, 175,925 and 722,182 respectively. These have been presented in the following linear and semi-log plots (fig. 1A, 1B).
fig.1 A Numbers of Infected individuals, fig.1 B Log, both plotted against time since first case
The semi-log plot allows us to fit the exponential progression of the disease in linear terms. Where the trajectory of the plots appears sloped, the disease is in its exponential growth phase. Where it plateaus, it is in its stationary phase. If this seems a bit confusing, you’re not alone. Italy, for its part, recorded its first case in late January, but met with skepticism in its efforts to raise an emergency in the country, as the numbers of cases still appeared to be small, and within control. It was only much later that a full-fledged lockdown was imposed – as late as March – after the numbers of infected individuals had gotten out of hand more rapidly than anticipated. This resulted in part, due to a lack of non-linear thinking and clear foresight. In other words, the kind of mindset you would require to appreciate the exponential progression of the disease. (Think baffled media spokespeople reporting an increase by 1000 cases in a day on your favourite news channel).
The US adopted a somewhat nonchalant stance as well, and has since emerged as a major disease hotspot, quickly surpassing China and Italy in late March. To better visualize these scenarios, we devised “forest-fire plots”, as depicted in fig. 2.
fig.2 Forest-fire plots
If we blindly compare numbers between India and other countries, we will notice that the number of infected individuals in India is less than the deaths in the USA. So this must be a good sign. But is it really? India has had infected patients for a while now, with the patient zero tracing back to a case in Kerala in early January. The country has still managed to keep its numbers low for quite a few months. While this looks good on the outside, most of the testing has been done in metros and larger towns with better medical facilities. A spike is expected in testing soon with the acquisition of more RT PCR kits. If we compare India’s numbers with another country where there was a scare – South Korea, we see a different pattern. While South Korea has infection numbers similar to India, there is a crucial difference, however, in that the rate of increase in the infected has reduced in Korea. More commonly known as “flattening the curve”, South Korea has managed to contain the infection. It would be a bit too premature to determine if India has flattened its curve. This is because there is one statistic where Korea is far better than India – tests performed per unit population. Korea can hence say with more certainty that it has contained the infection. However, it is no small task to test and be sufficiently sure for a country as large as India, but we are steadily approaching a better state. But this state will not come easy. It is bound to take its toll on the medical and economic apparatuses of the country, as well as on those in a state of lockdown, but it is essential that we all comply. The forest-fire plots below demonstrate exactly this.
These plots work on the SEIR model, and can be easily modified to work on other models. But these models are far from perfect. The view they present is somewhat naive, in that it only addresses the time-dynamics of the system, not the complexities which might result from the spatial distribution and motions of the subjects under study, or microscopic interactions on the ground. Hence, their validity is limited to macroscopic or “big-picture” analyses, where such quantities as the fatality rate due to the disease, or the recruitment rate of susceptible individuals are assumed constant (and chosen conservatively). In particular, there may be different numbers of each class of individuals in different regions, causing the model parameters to be different in say, different states in a country, or districts in a state and so on. So, an SIR model applied to 100,000 individuals could not be assumed to hold for a subset of 1000 individuals located in a particular area, where the specific dynamics may vary.
The graphics used in this article were generated in-house. The python scripts used to generate them are completely open source and can be found on our Github.