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# ws24 - d r and γ Group all terms by variable 2 Simplify...

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Math 464, Worksheet 24 This is the classical SIR model. A population is divided into three groups: S (susceptible), R (recovered/resistant), and I (infectious). Assumptions are that Everyone is in exactly one group. If you are susceptible there is a daily chance, r , that you will get sick. If your are sick there is a daily chance, γ , that you will get well, and thereafter be resistant. All groups have a daily death rate, d , assumed constant. (We assume the disease does not kill you.) All groups can give birth, also assumed to be a constant rate b . Every newborn is automatically in the susceptible group. Schematically, it looks like this: b S d b r b I d γ R d 1. Develop a difference equation model. You should have three equations with three variables : S , I and R . You also have four unknown parameters : b

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Unformatted text preview: d , r and γ . Group all terms by variable . 2. Simplify the model by assuming constant population N . Note that this allows you to eliminate variables as follows: (a) Since births must match deaths, d = b . (b) Since total population is constant, R = N-S-I . You should end up with two equations in the variables S and I . You also have a new parameter, N . Group by variable . 1 3. Assume constant values b = 0 . 02, r = 0 . 01, γ = 0 . 1 and N = 100. Perform the usual analysis. That is: (a) Think about what happens if S = 0. Sketch solutions if you can. (b) Think about what happens if I = 0. Sketch solutions if you can. (c) Find all equilibria. (d) For each equilibrium, ±nd eigenvalues, ±nd eigenvectors, and sketch solutions. 2...
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ws24 - d r and γ Group all terms by variable 2 Simplify...

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