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SIS epidemics transition matrix

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  1. May 12, 2017 #1
    1. The problem statement, all variables and given/known data

    The population is 50

    The diseases is known to follow SIS dynamics with the following probabilities

    The number of infected individuals increases with probability 0.1

    and it decreases with probability 0.05

    the probability that nothing happens is 0.85

    a) what is the transition matrix P?


    3. The attempt at a solution

    Would this mean my transition matrix would look like this
    \begin{bmatrix}0.85 & 0.1 & 0.05 \\ 0.05 & 0.85 & 0.1\\ 0.1 & 0.05 & 0.0.85 \end{bmatrix}

    Does this mean my sate space would be (1 to 50)
     
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  3. May 12, 2017 #2

    Ray Vickson

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    The question makes very little sense as written. Do you mean that each non-infected individual becomes infected with probability 0.1 (independent of others), that each infected individual becomes non-infected with probability .05 (independent of others), and that each individual's status remains unchanged with probability 0.85? If so, you need to have state-space 0,1,2,...,50 (the number of infected individuals at any time). Then, if the state at time ##t## is ##X(t) = k## for some ##k \in \{0,1,\ldots, 50\},## how would you compute the probabilities that ##j## individuals are infected at time ##t+1##? Your transition matrix will be a ##51 \times 51## matrix whose entries are those probabilities.
     
  4. May 12, 2017 #3

    StoneTemplePython

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    My guess is that the question is this is a birth-death process, where you get one more sick person with probability ##p##, one less sick person with probability ##q##, and stay in same state with probability ##1 - p - q##. The question... could definitely be tightened up in its wording though.

    The two exceptions for these transition probabilities are corner states --- i.e. when all 50 people are sick OR when 1 (or 0?) people are sick -- then you need to make appropriate tweaks to ##p## and ##q## respectively. (If it's a zoonotic or infected foreign persons come 'pop in' or what have you, you could still have 0 people sick in your population area yet have reoccurrences of the disease.) So the state space is ##\{0, 1, 2, ..., 49, 50\}## or ##\{1, 2, ..., 49, 50\}## depending on whether 0 is allowed.
     
  5. May 12, 2017 #4

    Ray Vickson

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    Yes: that makes a lot of sense, and gives a much more tractable model.
     
  6. May 13, 2017 #5
    when multiplying the state space by the transition matrix

    How would I know if the individual is infected or not?
     
  7. May 13, 2017 #6

    StoneTemplePython

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    I'm not sure I follow. The states refer to total number of infected individuals -- knowing whether "the individual" is infected is unrelated / doesn't make sense to ask with respect to the model I'm suggesting. What course is this for? Are you familiar with a birth-death process? You should be able to work out the steady state by hand....

    Alternatively, type in the exact problem as I'm just guessing this is a birth death process. The question posed in your original post.... leaves a lot to be desired in terms of clarity.
     
  8. May 13, 2017 #7
    A Health company would like to see how a disease will spread if one infected individual was to arrive in a country with a population of 50 people.

    The diseases is known to follow SIS (susceptible-infected-susceptible) dynamics with the following probabilities

    The number of infected individuals increases with probability 0.1

    and it decreases with probability 0.05

    the probability that nothing happens is 0.85

    Would the state space still be 1 to 50 for this? and a 50x50 transition matrix?
     
  9. May 13, 2017 #8

    StoneTemplePython

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    I still don't know what course this is for / whether you are familiar with a birth-death process. I have never heard of an SIS dynamic before -- SIS is not a standard term in probability or linear algebra. Providing definitions to the message board for non-standard terms is... generally beneficial.

    With respect to your question, which I bolded, what I said before still stands

    so its either a 50 x 50 transition matrix or 51 x 51, depending on whether zero is legal.
     
  10. May 13, 2017 #9
    its a probability course and I have heard of the birth-death process

    SIS means going from susceptible to the disease to being infected to returning to being susceptible

    given that 1 individual is already infected the state space must be 1 to 50
     
  11. May 14, 2017 #10

    StoneTemplePython

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    It really depends on what you're trying to model and why. Birth-death could makes sense.

    You could also model it on an individual person level (susceptible, infected, susceptible), figure out on average the percent of time a person is sick (there will be one eigenvalue = 1, and all others have magnitude less than that in your original 3x3 transition matrix), and from there use linearity of expectations to figure out on average the number of people who are sick, and so on.

    The question really could use a re-write for clarity on what exactly it is getting at and perhaps why. (Though that is out of your hands, I know.)
     
  12. May 14, 2017 #11

    Ray Vickson

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    Not necessarily: if eventually the disease can be eradicated, then the state space is {0,1,...,50}. Just because you start at state 1 does not mean that you cannot achieve state 0 in the future.
     
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