My Math Forum Markov Chain Epidemics

March 22nd, 2011, 05:52 AM   #1
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Markov Chain Epidemics

Hello everyone!

For a programming task I have to use a discrete time Markov Model. A population consists of n individuals of which m are initially infected.

The progression of the disease is illustrated in the graph below, where

s = susceptibles
i = new infectives
j remaining infectives of previous peiriod
d = diseased
r = removed

All probabilities are given: mu_t depends on i_t,j_t and d_t

I've already used a multivariate Markov chain with the tupels (s,i,j,d) as state space but the transition matrix becomes huge if
the population size increases.

I've also heard of total size approximation models that use the number of deaths as time scale that simplifies the model as only one infective is considered in one time step. The problem is that I need to know the distribution after a certain time.

I'm looking forward to receive any kind of idea to solve this. It has to be a markov chain - so no SDEs are possible.

Attached Images
 disease.JPG (29.3 KB, 410 views)

 March 22nd, 2011, 06:43 AM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: Markov Chain Epidemics I don't understand the difficulty. You can write this as a 5 X 5 matrix if you're using a language that supports them, or else calculate the new values as indicated one step at a time. The matrix approach is better if you need to look at large isolated time values -- say you want to look at time 1,000,000 but don't need to know the first million values, just that one. Maybe if you'll tell me what you're trying to do and why it doesn't work, or what you don't understand? Also, what language are you using?
March 22nd, 2011, 11:50 AM   #3
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Re: Markov Chain Epidemics

First of all, thank you very much for your response.

Maybe the graph was a bit confusing as it shows only the transitions for ONE individual. But the probability that a susceptible becomes infetected ist ?t = p(it + jt + dt) and thus depends on the number of infectives and diseased of the last period. I want to know how many healthy, diseased and dead individuals there are after x periods.

I attached the model I derived so far - just to illustrate where this is going (details not important)
I cant use it anyway because the transition matrix gets to big as there are too many possible transitions between 4 dimensional states. For a population size of 1000 it has
m x m = 1771129865064208650001 entries !

I have to implement the model in JAVA and I'm looking for a simpler approach - like a bivarate markov chain (as in common SIR-models). The problem is that the infectious period can have a length of 1,2 or 3 which is not really compliant with the markov property.

Any ideas are appreciated!
C.Sager
Attached Images
 model.jpg (117.6 KB, 398 views)

March 22nd, 2011, 02:19 PM   #4
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Re: Markov Chain Epidemics

Quote:
 Originally Posted by CraigSager The problem is that the infectious period can have a length of 1,2 or 3 which is not really compliant with the markov property.
Sure it is. You just split the infected people into "just infected", "infected one period ago", etc. Then whatever probability they would have of staying in the same state, you instead give that probability of moving into the next category.

If the model was

State 1: Noninfected (10% chance per turn to become infected, otherwise stay state 1)
State 2: Infected (takes 3 turns to become zombies; 1% chance per turn to transition to state 4; otherwise stay state 2)
State 3: Zombie (25% chance per turn to be destroyed; otherwise stay state 3)
State 4: Destroyed (stays that way)

Then just rewrite it as

State 1: Noninfected (10% chance per turn to go to state 2a, otherwise stay state 1)
State 2a: Infected (1% chance per turn to transition to state 4; otherwise go to state 2a)
State 2b: Infected (1% chance per turn to transition to state 4; otherwise go to state 2b)
State 2c: Infected (1% chance per turn to transition to state 4; otherwise go to state 3)
State 3: Zombie (25% chance per turn to be destroyed; otherwise stay state 3)
State 4: Destroyed (stays that way)

and it's Markov enough for you, right?

March 23rd, 2011, 01:00 AM   #5
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Re: Markov Chain Epidemics

Quote:
 Originally Posted by CRGreathouse If the model was State 1: Noninfected (10% chance per turn to become infected, otherwise stay state 1) State 2: Infected (takes 3 turns to become zombies; 1% chance per turn to transition to state 4; otherwise stay state 2) State 3: Zombie (25% chance per turn to be destroyed; otherwise stay state 3) State 4: Destroyed (stays that way)
We do not have four states - it is a 4D state space, so (S,I,J,D) is one single state. Therefore I had to use a multinomial distribution to caluclate tranistion probabilities.
The initial state in (N-m, m, 0, 0) so we have m infectives and n-m susceptibles. I want to know how many susceptibles, infectives, diseased there are after 5, 10, 15 periods..

Quote:
 Originally Posted by CRGreathouse 10% chance per turn to become infected
That probability depends on the number of infectives - so its not constant.

March 23rd, 2011, 10:10 AM   #6
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Re: Markov Chain Epidemics

Quote:
 Originally Posted by CraigSager We do not have four states
I know -- thus my use of the subjunctive "If the model was".

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