391% A simplified model for the spread of a disease goes like this. The total po

Question

391% A simplified model for the spread of a disease goes like this. The total population size is N of which some are diseased and the remainder healthy. Assume that all encounters between people occur between pairs of individuals. Take the basic time interval so small that in an interval there is a probability P 1 of one pair of individuals encountering, And the probability of more than one encounter can be neglected. Also assume that an encounter between any pair of individuals is just as likely as between any other pair Finally, assume that if a diseased and non diseased individual encounter one another, there is a probability P1 that the non diseased individual will become diseased. Write down the transition probabilities for the number of diseased individuals.

Explanation / Answer

At any time T suppose there are n diseased individuals

Now at time T+1 there can be n+1 diseased individuals if there is interaction of only one pair at time T+1 and the pair interacted should be of a diseased and non-diseased and the non-diseased should be diseased. So, probability of only one interaction pair = P

Probability of diseased and non-diseased interaction = 1/2 ( DD,ND,DN,NN - 4 possibilities out of 2 are for Diseased and Non-Diseased)

Probability of teansmitting disease = P

So, P(N(T+1) = n+1 | N(T) = n) = P*1/2*P = P^2 / 2

So, P(N(T+1) = n | N(T) = n) = 1-P^2 /2

So, TPM is :

and so on.

....n n+1 n+2 n+3... ....n 1 - P^2 / 2 P^2/2 0 0 n+1 0 1 - P^2 / 2 P^2 / 2 0 n+2 0 0 1 - P^2 /2 P^2 /2 n+3 .... 0 0 0 1 - P^2 /2

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