While attending a thesis defence of a fellow researchereverybody was bewildered by the answer of the speaker who explained thesingularities of a complex valued function in his presentation. The questionasked by someone from the audience was about the concrete or non-abstractapplications of the singularities to which he answered, “supposing a nuclearbomb is thrown at some place, a person well versed in finding the singularitiescan save himself by locating the singularities of the wave/destruction functioncreated by the bomb’. Everybody present in the audience giggled as an act tosnubbing the scholar with the presumption that it is an absolute stupidity tolook for a pencil, paper and to calculate the singularities in such a situationwhen the bomb is thrown! May be at that point of time we were not smart enoughto simulate the situation. But today I realise the essence of his example andthe singularities, the only thing we need to do is to superimpose the nuclearbomb situation with the present epidemic we are going through, his argumentstands quite relevent and valid. Recalling here that a singularity of afunction is a point where the function behaves in contrast to its normalbehaviour. Now replicating the notion to the present situation the coronainfected persons are the singularities of the normal ‘life function’ who can bedealt with by using this approach. So the notion from pure/abstract Mathematicsstands as a claimant to tackle aconcrete problem. Anyway, this is just an example to claim and give a reasonthat Mathematics can be related and can give solutions (though have toencounter with a greater degree of difficulty) to problems which don’t lookeven remotely connected with this divine subject commonly known as ‘the queenof all sciences’. Here I would like to share a different Mathematical mechanismwhich directly relates to the problem and can be used to propound solution todeal with the prevailing pandemic.
Understanding the possibilities of an infectious disease toget augmented into an outbreak or a pandemic, what we need to know is ‘BasicReproduction Number’ which is universally denoted by R0 (pronounced as Rnaught) which indicates how contagious an infectious disease is. It isdetermined by several factors, biological and non-biological, and is not anintrinsic property of an infectious disease caused by a virus or bacteria.Putting the definition in the simplest form, we can say R0 is directlyproportional to the quotient of number of infected people, their contacts in agiven period of time to the contacts in a given period of time of the infectedpeople.
More specifically,
R0 = τ.c.d
where τ is transmissibility i.e., probability of infectiongiven contact between a susceptible and infected individual, c the average rateof contact between susceptible and infected individuals and d the duration ofinfectiousness.
Now here I would not like to go into the nitty-gritty of themodel and symbols which certainly needs a good deal of a branch of Mathematicsknown as differential equation that is what Mathematicians at Institute ofMathematical Sciences, Chennai (IMSC) are working on. But! Of course I will bediscussing about the implications of the (numerical) value of R0 . Let us fix the reference point to bethe numeral one and we know as per the law of trichotomy three cases arise (i)R0 =1 (ii) R0<1 (iii) R0>1 .
If the value of R0 of an infectious disease comes out to beone (R0 =1) each existing infection causes one new infection. In this case thedisease will stay alive and stable, but won’t result in an outbreak or an epidemic.
If the value of R0 comes out to be less than one (R0<1),then an infected person causes less than one new infection. In this case thedisease dies down its own death.
If it is greater than one (R0>1) , each existinginfection will cause more than one new infection. In this case disease willspread and will result in an outbreak or an epidemic. Its values (greater than1) indicates the number of susceptible persons who can get infected whilecoming in contact with an infected person.
To interpret the vitality of case (iii) we treat the processwith two particular values of R0 if for instance, the value of R0 is 2 thedisease with follow a progression which can be put numerical as 2, 4, 8, 16,32, 64,. . . and if the value comes out to be 3 the pattern will be like 3 ,9,27, 71, 213, 639,. . . So, just a small increase in the value of R0 can blow upthe process and things can get out of our hands. During the initial stages ofthe present pandemic the highest value of R0 was attained in the city Wuhanwhich was calculated to be 5.7. Here in India the value calculated on a weeklybasis by the Mathematicians at IMSC Chennai kept fluctuating during the start,but now a downward trend is observed after a country-wide lockdown was imposed.After three weeks the first positive case was reported, its value was 1.7. Thenon 23rd of March (one day before lockdown) the value of R0 came out to be 4,which certainly depicted the situation as alarming. But on April 6, the valuecame down to 1.83 and on April 11, its value has been calculated to be 1.55.
So far so good, the curve is flattening as can be seen fromthe values of R0. At this point of time we need to make use of our resourcesand strategies optimally in synchronicity with the values of R0. Undeniablylockdown has proved to be instrumental in containing the virus but it is unwiseto continue with unprecedented lockdowns, for which a bigger price is beingpaid. The lockdown strategy can be used judiciously in a sense that we cancalculate the value of R0 regionally which can guide the authorities inimposition of the region-wise lockdowns. I would like to conclude with anaudacious statement that theories like ‘herd immunity’ don’t make sense as longas the value of R0 remains below five.