A mathematical approach to fight the pandemic

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While attending a thesis defence of a fellow researcher everybody was bewildered by the answer of the speaker who explained the singularities of a complex valued function in his presentation. The question asked by someone from the audience was about the concrete or non-abstract applications of the singularities to which he answered, “supposing a nuclear bomb is thrown at some place, a person well versed in finding the singularities can save himself by locating the singularities of the wave/destruction function created by the bomb’. Everybody present in the audience giggled as an act to snubbing the scholar with the presumption that it is an absolute stupidity to look for a pencil, paper and to calculate the singularities in such a situation when the bomb is thrown! May be at that point of time we were not smart enough to simulate the situation. But today I realise the essence of his example and the singularities, the only thing we need to do is to superimpose the nuclear bomb situation with the present epidemic we are going through, his argument stands quite relevent and valid. Recalling here that a singularity of a function is a point where the function behaves in contrast to its normal behaviour. Now replicating the notion to the present situation the corona infected persons are the singularities of the normal ‘life function’ who can be dealt with by using this approach. So the notion from pure/abstract Mathematics stands as a  claimant to tackle a concrete problem. Anyway, this is just an example to claim and give a reason that Mathematics can be related and can give solutions (though have to encounter with a greater degree of difficulty) to problems which don’t look even remotely connected with this divine subject commonly known as ‘the queen of all sciences’. Here I would like to share a different Mathematical mechanism which directly relates to the problem and can be used to propound solution to deal with the prevailing pandemic.

Understanding the possibilities of an infectious disease to get augmented into an outbreak or a pandemic, what we need to know is ‘Basic Reproduction Number’ which is universally denoted by R0 (pronounced as R naught) which indicates how contagious an infectious disease is. It is determined by several factors, biological and non-biological, and is not an intrinsic property of an infectious disease caused by a virus or bacteria. Putting the definition in the simplest form, we can say R0 is directly proportional to the quotient of number of infected people, their contacts in a given period of time to the contacts in a given period of time of the infected people.

More specifically,

R0  = τ.c.d

where τ is transmissibility i.e., probability of infection given contact between a susceptible and infected individual, c the average rate of contact between susceptible and infected individuals and d the duration of infectiousness.

Now here I would not like to go into the nitty-gritty of the model and symbols which certainly needs a good deal of a branch of Mathematics known as differential equation that is what Mathematicians at Institute of Mathematical Sciences, Chennai (IMSC) are working on. But! Of course I will be discussing about the implications of the (numerical) value of    R0 . Let us fix the reference point to be the 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 be one (R0 =1) each existing infection causes one new infection. In this case the disease 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 the disease dies down its own death.

If it is greater than one (R0>1) , each existing infection will cause more than one new infection. In this case disease will spread and will result in an outbreak or an epidemic. Its values (greater than 1) indicates the number of susceptible persons who can get infected while coming in contact with an infected person.

To interpret the vitality of case (iii) we treat the process with two particular values of R0 if for instance, the value of R0 is 2 the disease 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 up the process and things can get out of our hands. During the initial stages of the present pandemic the highest value of R0 was attained in the city Wuhan which was calculated to be 5.7. Here in India the value calculated on a weekly basis 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. Then on 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 value came 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 from the values of R0. At this point of time we need to make use of our resources and strategies optimally in synchronicity with the values of R0. Undeniably lockdown has proved to be instrumental in containing the virus but it is unwise to continue with unprecedented lockdowns, for which a bigger price is being paid. The lockdown strategy can be used judiciously in a sense that we can calculate the value of R0 regionally which can guide the authorities in imposition of the region-wise lockdowns. I would like to conclude with an audacious statement that theories like ‘herd immunity’ don’t make sense as long as the value of R0 remains below five.