The current numbers (as of April 5, 2020, 20:15 hrs) pertaining to COVID19 pandemic read as follows. 1,133,758 confirmed cases (of infection), 62,784 confirmed deaths, encompassing over 209 countries/territories. As I jot down this article, no one knows how the numbers would change. No science or mathematics can predict the face of these numbers at the time I am done writing this article. Same goes for you while you read through it. Having said that, there are mathematical models that can project how infectious diseases progress to show the likely outcome of an epidemic like the current COVID-19 pandemic. This article elucidates some popular mathematical models of a typical epidemic and how we can possibly relate it to the present scenario.

Models use basic assumptions or collected statistics along with mathematics to find parameters for various infectious diseases and use those parameters to calculate the effects of different interventions, like mass vaccination programmes. Models are only as good as the assumptions on which they are based. Most models, although not very practical approximations, are based on two main assumptions: (1) Rectangular and stationary age distribution, i.e., everybody in the population lives to age L and then dies, and for each age (up to L) there is the same number of people in the population. (2) Homogeneous mixing of the population, i.e., individuals of the population under scrutiny assort and make contact at random and do not mix mostly in a smaller subgroup.

In any growth model, the pivotal role is played by the basic reproductive number (R0). It is a measure of how transferable a disease is. It is the average number of people that a single infectious person will infect over the course of their infection. This quantity determines whether the infection will spread exponentially, die out, or remain constant:
• If R0 > 0, disease will spread exponentially.
• If R0 < 0, disease will die out.
• If R0 = 0, the disease will become endemic: it will move throughout the population but not increase or decrease.
The basic reproduction number can be computed as a ratio of known rates over time: if an infectious individual contacts 𝛽 other people per unit time, if all of those people are assumed to contract the disease, and if the disease has a mean infectious period of 1/𝛾, then the basic reproduction number is just R0 = 𝛽/𝛾. 𝛽 and 𝛾 signify infection rate and recovery rate respectively.

One of the most basic models used for modelling epidemics is the SIR model. To illustrate this model, let us consider a fixed population with only three compartments: Susceptible, S(t); Infected, I(t); and Removed, R(t).
• S(t) is used to represent the number of individuals not yet infected with the disease at time t, or those susceptible to the disease.
• I(t) denotes the number of individuals who have been infected with the disease and are capable of spreading the disease to those in the susceptible category.
• R(t) is the compartment used for those individuals who have been infected and then removed from the disease, either due to immunization or due to death. Those in this category are not able to be infected again or to transmit the infection to others.

The flow of this model may be considered as the following:
S → I → R

A very critical assumption is that the population is fixed (neither growing nor declining) which seems to be rather impractical. But provided that birth rate and death rate are negligible in comparison to infection rate and recovery rate, this is a fair assumption.

The figure (right) shows the S, I , R curves with respect to time (in days) with 𝛽=0.0004 and 𝛾=0.04.

Now, we move on to assessing the predicament in hand. In most real-world epidemics, there is a fraction (say, equal to x) of the population that is not susceptible to the infection (due to immunity or some other mechanism). Then, in modelling such an epidemic, basic reproductive number, R0 is replaced by effective reproduction number, Re , where Re=R0.x. However, the complicacy associated with an epidemic like COVID-19 is that there is no confirmed method of immunisation as of now. Hence, Re is almost equal to R0. In recent days, Re has been found to be in the range 1.1 to 1.4, with the increasing trend being more prevalent worldwide. In such a precarious situation, it has become indispensable to use all possible methods to flatten the red curve corresponding to the number of infected masses. Flattening this curve, although may not decrease the net number of affected individuals in total, will prevent any situation of medical hospitality saturation. Given that there is no way of immunising the susceptible population at present, the only alternative left is to reduce Re by reducing 𝛽 and increasing 𝛾. Practicing better hygiene, social distancing, better treatment of affected individuals, and other such seemingly trivial practices can actually go a long way in handling this pandemic more effectively.

No one knows with certainty where the current situation is located in the epidemic growth curve, or when the peak will be reached, or when it will come to a standstill. As I had speculated in the beginning, as I get closer to concluding this article, the numbers show a mind-boggling increase: 41,097 confirmed cases worldwide in only 18 hours. With increasing numbers, increases the sense of fear among the survivors. But, isn’t ‘fear’ the need of the hour? For the lack of it (in other words, ‘stupidity’) has not helped the cause so far. All we need to fear now is the lack of fear itself!

REFERENCES

• https://en.wikipedia.org/wiki/ Mathematical_modelling_of_infectious_ disease
• https://www.healthknowledge. org.uk/public-health-textbook/ research-methods/1a-epidemiology/ epidemic-theory
• https://www.youtube.com/ watch?v=gxAaO2rsdIs
• https://www.youtube.com/ watch?v=fgBla7RepXU
• https://www.who.int/emergencies/ diseases/novel-coronavirus-2019