HOW VACCINATION HELPS IN LIMITING THE SPREAD OF THE COVID-19 VIRUS AND THE UNDERLYING MATH BEHIND IT.
-By Akash Kumar
-Batch(2k20), Deptt. of
Chemical Engg.
-BIT Sindri, Dhanbad
It will be two years since we experienced the first lockdown due to COVID 19. Since then it seems that COVID 19 won’t ever be over but the scientists predict some interesting theories explaining how COVID 19 spread can be limited to such low, negligible rates that it might seem to end.
Types of transmission of infection – Linear vs Exponential
For understanding the nature of spreading of infectious diseases let us take a simple example of how rumours spread. Infections spread much like rumours, someone picks it up and passes it on to others. So we can create a simple mathematical model of spreading rumours to try to understand the analogous spread of the virus.
Let's say you hear a juicy rumour and you can't keep it to yourself so you compromise by sharing it with one person and keeping the same trend that is others also share it with only one person. In this way, the gossip won't spread too far. In fact, if only one new person hear the rumour only 30 people would hear it in 1 month.
How fast can it spread if the rumour is to be told to 2 new people every single day? Shockingly the results are very bad and it spreads way too faster than expected. What makes this difference? The rate of change of spreading is the answer. In the first case, the nature of the spread is linear, the rumour spreads to the same number of people every single day, that is to 1 person. This means the number of new people hearing the rumour each day is constant. Whereas in the second case the rumour is transmitted each day to twice as many people as yesterday. The number of people here the rumour isn’t constant. It grows exponentially. 2 on the first day 4 on the second day 8 on the third day and so on, and ultimately 230 in 1 month. This exponential growth makes a drastic difference in the second case whereas in linear growth in the first case the spread is very limited.
Linear functions are characterised by a constant rate of change. Linear growth is slow and steady as in the first case: one new person hears the rumour each day. Whereas in the case of exponential growth that is in the second case, the rate of change increases at each step: two new people hear the rumour next day for new people than eight and so on. Unlike linear growth, exponential growth accelerates- the amount of increase itself continues to increase. Following is a graph to explain the difference between exponential and linear growth.
How community spread of the virus can be limited
To understand this let’s jump into some technical definitions and mathematical calculations. In the above example, we saw how infections spread by an elementary mathematical model of the spread of rumours. We can see how a seemingly small difference in transmission rate can make a huge difference between a few isolated cases and a catastrophic epidemic. In technical language, this transmission rate of any infection is called the basic reproduction number represented by R0.
Basic reproduction number R0 is the average number of new infections each infected person is expected to produce. In the above example of the spread of rumours, the value of R0 in the first case which is linear is 1 and in the second case the rate of increase is exponential R0= 2, the infectious period was one day. In order to limit the spread of any infection, the attempt is to make the nature of the spread of infection linear that is R0 =1, to avoid a severe devastating impact on the population. For practical reasons making R0 = 0 is not possible in a very short frame of time. It may eventually be that the spread of the virus dies out but it will be in the long run of time.
The role of vaccination in making the nature of spread linear
vaccination is a process by which one’s immune system becomes fortified against the infectious agent like a virus, bacteria. It helps is the immune system of the body to develop immunity against a certain disease.
When vaccinated an individual develops resistance to the disease. Success rates vary but for simplicity, we assume the vaccine is 100% effective.
If enough individuals are vaccinated the spread can be made linear that is R0 = 1. So how many people need to be vaccinated for R0 = 1. Let’s jump into the mathematics part of our discussion.
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