Question

A biologist doing an experiment has a bacteria population cultured in a petri dish. After measuring, she finds that there are 11 million bacteria infected with the zeta-virus and 5.2 million infection-free bacteria. Her theory predicts that 50% of infected bacteria will remain infected over the next hour, while the remaining of the infected manage to fight off the virus in that hour. Similarly, she predicts that 80% of the healthy bacteria will remain healthy over the hour while the remaining of the healthy will succumb to the affliction Modeling this as a Markov chain, use her theory to predict the population of non-infected bacteria after 3 hour(s).

Answer 1

11.4 million

Explanation:

Let's define the variables i and i' to represent the number of infected bacteria initially and after 1 hour, and the variables n and n' to represent the number of non-infected bacteria initially and after 1 hour. The biologist's theory predicts ...

0.50i +0.20n = i'

0.50i +0.80n = n'

In matrix form, the equation looks like ...

`\left[\begin{array}{cc}0.5&0.2\\0.5&0.8\end{array}\right] \left[\begin{array}{c}i&n\end{array}\right]=\left[\begin{array}{c}i'&n'\end{array}\right]`

If i''' and n''' indicate the numbers after 3 hours, then (in millions), the numbers are ...

`\left[\begin{array}{cc}0.5&0.2\\0.5&0.8\end{array}\right]^3 \left[\begin{array}{c}11&5.2\end{array}\right]=\left[\begin{array}{c}i'''&n'''\end{array}\right]`

Carrying out the math, we find i''' = 4.8006 (million) and n''' = 11.3994 (million).

The population of non-infected bacteria is expected to be about 11.4 million after 3 hours.