Mathematics of corona-virus

I hope to cultivate the mathematical principles that governthe transmission of the deadly virus within a community. Firstly, I shall laydown the assumptions — not too remote from the actual case. Then I’llformulate certain mathematical laws that will model the phenomenon. Finally, Iwill describe a possible,though not the optimal, way to contain the epidemic.No attempt at all will be made at mathematical rigour. In fact, little morethan elementary mathematics is used.

At the heart of every large-scale transmission–whether ofinformation, rumour, microbial growth or infection– lies the concept of theexponential function. Exponential function underlies why infections, in certaincases that I will discuss presently, spread like fire. To understand it better,let’s say someone is infected and each day the infection is transmitted to anew person. After 30 days, say, a total of 31 people will be infected. In thespirit of mathematical abstraction, let’s stretch reality a bit — instead ofinfecting one person let’s suppose each infected person transmits the virus toHALF a person. After a day, there will be one and a half new infected peopleand these will infect another one and a half plus a half and a quarter peopleon the third day. Proceeding this way there will about 2 lakh infected peopleon the 30th day — an very large number. You see, even though the rate oftransmission per day is smaller in this case by half an individual, still thescale of transmission completely dwarfs that in the first case where the rateis linear. There’s a contagion boom! Such is the power of the exponent (punintended!)

   

Back here in the real world, we have to look out for a numberof demographic and environmental factors that govern the transmission. It woulddo us good to introduce a single number, T, to denote the collective impact ofthese factors. T is the number of people that can be infected by each alreadyinfected person. Let’s assume that an infected person encounters U new peoplewhile infected. On an average, T out of these U people will be infected. So,each has a T/U chance of getting infected. But if W of these U people aresomehow resistant to the virus, the number of infected people will be T/U *(U-W).  For the infection to becontrolled,  this number must not exceed1 for otherwise we’d have an exponential growth. Setting the last expressionequal to 1 we can easily solve for W/U which is the fraction of population thatare resistant to the virus, to control the outbreak.

Thus, we have the golden rule: If we could break thismenacing pattern of exponential growth, we can check the rate at which thevirus transmits. To take a concrete example, if the transmission rate of thevirus is 1.5 that is, each infected person transmits the virus to 1.5 new peopleon average,  (1 – 1/1.5) or, 33 % of thepopulation has to be resistant to the virus to prevent the epidemic– forexample by being super-vigilant, and maintaining good hygiene–that way notonly do they protect themselves from the virus but also,  and more importantly,  they check the rate of transmission withinthe community.

While the preceding discussion captures the essence of thesituation, there are a few crucial points that I would like to address here.The first point is that it is not completely true that the rate at which theepidemic grows is always proportional to the extent to which it has currentlygrown. This is true because, loosely speaking, a time reaches when the number of people who are infected far outweighsthose who aren’t. At that critical point the number of new cases isn’t simplyproportional to the number of existing cases but the rate itself gets smallerthan would be warranted by an exponential curve. This marks a point ofinflection meaning that after this point the rate of of change continues todecrease upto a point where the outbreak plateaus. This gives us the logisticcurve. Another important point that must be made here is that the basicexponential equation doesn’t take into account the deaths and recoveries; forexample if someone dies or completely recovers they cease to spread the virusand therefore don’t factor into the growth equation. However, this doesn’tmatter so long as the growth is fast enough because in that case most peoplewould have been infected only recently, but if the growth rate is slower thisbecomes significant.

Even with these limitations this model works fairly well tomake an informed assessment of the situation. Plus it involves a delightfulcalculus exercise!

The writer is an alumnus of NIT Srinagar

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