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I have been looking at Markov matrices and have been having some difficulty getting my head around them. I was wondering if someone could explain to me in somewhat simple manners how they work.

Also, how would one go about using them to model migration? I have been thinking of doing an investigation on this for a project, and read some papers on modelling migration or disease spread.

Thanks!

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Markov matrices $A$ are matrices with nonnegative entries whose rows sum to $1$*. We say a probability vector $p$ is a row vector with nonnegative entries which sums to $1$. When you multiply a probability vector on the left with a Markov matrix (as $pA$), you get another probability vector $q$. In this situation $p$ describes the initial distribution on the state space and $q$ describes the distribution after one step.

In migration, the state space is some collection of locations, and the entry $a_{ij}$, for two locations $i,j$, is the probability that a migrant goes from location $i$ to location $j$ (it could be that $i=j$ in which case the "migrant" does not migrate). Implicit in this description is that the probability that the migrant goes to $j$ from $i$ does not depend on their previous history. This is called the Markov property, and it is not particularly realistic for migration, especially of humans.

From here one could use the machinery of Markov chains to determine how people will be distributed around the region of interest in the long run, among other things.

*Some authors instead decide that the columns sum to $1$ and that probability vectors are column vectors, but the row convention is more common.

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    Ahh ok. I've read one paper that models disease spread in an animal population, though entailed in this is differential equations (https://www.ncbi.nlm.nih.gov/pubmed/11895380). Why would they need to use differentials in here, and also, what difference would there be in modelling migration in comparison to migration? – 2017-01-24
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    @ChangingTime I think you meant "disease in comparison to migration". In my view the big difference there is that in disease there is no way to neglect the increase and decrease in the *total population*, so that there is both (in mathematical parlance) "diffusion" and "reaction". (However I am not an expert on disease or migration modeling.) – 2017-01-24
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    @ChangingTime Where differential equations come into play here is when you look at a *continuous time* Markov chain. This is given by a generator matrix $G$ whose off-diagonal entries are nonnegative and whose rows sum to zero (so the diagonal entries are nonpositive, usually strictly negative). Then you get the so-called Forward Kolmogorov equation or Fokker-Planck equation for the evolution of the distribution: $\frac{dp}{dt} = pG$. – 2017-01-24