Today, I learnt new techniques in the field of statistics that is extremely useful. Previously, when I was working on my HiMCM and IMMC (both modelling competitions), there were several questions that I couldn’t answer—most notably:

1. A lot of the variables are interrelated. To find how a variable would change after, say, 10 years, do I have to calculate how every value would change for every single year?

2. But what if I want to know when the variables will stabilize, or what will happen when the year approaches infinity?

My deepest concerns were entirely addressed by this new mathematical concept: Markov chains and eigenvalue.

First, a Markov chain is a method of approximating the future when one has full knowledge of the present conditions and past history. For instance, consider a SIR model, often used for disease modelling. S stands for susceptible, while I is for infected, and R is for removed. But what makes the calculations complicated in a SIR model is that all three variables are closely interrelated with each other. Markov chains express this entangled nature in a diagram (*fig. 1*) so that it is much easier to formulate.

fig. 1. Markov Chain in SIR model

However, after creating the linear multivariable equations that express each variable at time , a problem arises: because the value at time *t* is dependant on the values of every other variable at time , one has to calculate the value of every single variable until reaching time *t*. That is to say, because of the interlinked nature of such variables, it is hard to predict long-term behavior.

To this, what I did in my previous project was simple (and quite stupid at the same time). I chose an arbitrary value *t*, that was big enough to represent long-term behavior, and I bashed until I got to that value of *t*. Needless to say, this method is likely to have high inaccuracies.

Instead, the use of matrices can largely help us predict their long term behavior. Consider a model for variables x_t, y_t, z_t represented by this system of equations:

Then, this system can be expressed as

Note that multiplying this vector by itself produces a diagonal vector as well!

Generalizing this form, we can see that

Note that in this form, we can obtain a value for each of as tends to infinity. This would allow us to predict long-term behavior. This technique would be extremely useful especially when modelling long-term effects of government policies.