# The Mathematics of How Virus Can Grow

## Beautiful Mathematics Seminar

We have a beautiful mathematics seminar every week on Thursday at 8 p.m. IST. Today’s topic is “The Mathematics of How Virus Can Grow” at Cheenta Ganit Kendra.

## Prerequisites

• Basic Probability
• Probabilistic Versions of Addition Principle and Multiplication Principle
• The Basic Idea of Limit

## Theory

We will discuss some basic questions. We will develop the theory while seeking answers to these questions only. This kind of pedagogy has always been our approach to mathematics. So, let’s start to explore coronavirus by asking some questions.

Question 1.1 Suppose you are trying to find some vaccine of coronavirus. You know that every single second, a single cell divides into two. Some coronavirus cells were kept in a jar. You go out of the room and came back after 1 minute to find out the whole jar is full of the virus. When was the jar half full of virus?

Question 1.2 What would have been the answer, if the virus divides at a faster rate, say a single cell divides into 5 cells in a second?

Question 2.1 Consider some coronavirus cells in a lab jar. A single virus either survives all along or it gives rise to another virus when it dies. Will a colony of the virus ever become extinct?

Question 2.2 Consider a single coronavirus cell. Now, in each second it dies and gives rise to another child virus with probability p < 1. Will the virus colony become extinct?

Probability of Extinction Let $$X_n$$ is the number of virus cells after $$n^{th}$$ second. The extinction probability at $$n^{th}$$ second, $$u_{n} = P({X_{n}=0})$$ for $$n \geq 1.$$ What is $$u_{\infty}$$, the probability of ultimate extinction?

Question 2.3 Consider a bunch of coronavirus cells. Now, in each second a single cell dies and gives rise to another child virus with probability p < 1. Will the virus colony become extinct?

Research Question 3 Consider a bunch of coronavirus cells. Now, in each second a single cell dies or gives rise to two child virus cells with probability p < 1. Will the virus colony become extinct? Does it depend on the values of p? Explore.

## Reference

Refer to the following document.

## Different Mathematics

Note: The models described above may or may not truly describe the growth of the corona virus. Not sufficient research has been there on how coronavirus grows.

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