# Markov chain with external input

by Miss Q   Last Updated May 22, 2020 23:20 PM

Could anyone explain to me this Markov chain model? $$S_{k+1}= P(S_k+S_k^0).$$

Please allow me to give a link from the paper I was read this equation $$(6)$$ here https://drive.google.com/file/d/132FbOj-up5J4VO8ujj0wBI03aSQ28KJy/view?usp=sharing

I was actually reading this paper https://www.medrxiv.org/content/10.1101/2020.04.21.20073668v1.full.pdf but they refer to the above model which I didn't understand. Thanks for a bit of discussion and over-all idea about the model. Thanks a lot.

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A simple Markov chain model multiplies an $$N\times N$$ square matrix of transition probabilities ($$P$$ in your question) by a state vector of height $$N$$ ($$S_k$$) in your question. (I'm sure you know this, but sometimes $$P$$ is on the right or left, which affects the structure of that matrix--i.e. whether it's the rows or columns that must each sum to 1 because they represent probabilities. This difference is illustrated in the two papers.)
So in a simple Markov chain model, the only thing that produces a new state $$S_{k+1}$$ is the transition probability matrix $$P$$ operating on the old state $$S_k$$. We multiply the the square matrix $$P$$ and the column vector $$S_k$$ to produce the new state vector $$S_{k+1}$$.
In the model you are asking about, there is also an external source of change at time $$k$$, a vector of the same size as $$S_k$$, named $$S^0_k$$. This represents changes to the values represented in $$S_k$$ from some other source. It looks like this represents newly infected people to be added in to the vector of states of previously infected people, but I have not read the article closely.
Once the vectors $$S_k$$ and $$S^0_k$$ are added together, the next step proceeds just as in the simple Markov model: $$P$$ is multiplied by the resulting column vector.