Standard SIR model with vital dynamics
Holmes 3.11. This problem considers the dynamics of measles, and what strategy to use for vaccinations. Using the same three groups as in the SIR model, the model is
This is known as the standard SIR model with vitaldynamics. Assume that S(0) = S0, I(0) = I0, R(0) = 0, where S0 and I0 are positive.
(a) Show that the above system of equations can be derived from the Law of Mass Action. Do this by finding reactions that give rise to these equations.
Also, assumptions are made about the rate constants to obtain these equations and you should make sure to state what these are. Finally, for each reaction give a sentence or two that explains what assumptions were made about the population to produce the given reaction.
(b) Use a conservation law to reduce the above system to a problem for just S and I.
(c) Nondimensionalize the reduced problem in part (b), using N0 = S0 +I0 to scale both S and I. The final problem should only contain three nondimensional parameters, one of which will be in the initial conditions. Also, use s and i as the nondimensional dependent variables. Explain why the solution must satisfy 0 ? s ? 1 and 0 ? i ? 1 .
(d) What are the steady-states for the problem in (c)? One of them is obtained only if the parameters satisfy a certain inequality. Write this inequality as v > 1 and relate v to the nondimensional parameters in your problem.
(e) One of the steady-states you found has i* = 0 . Under what conditions on the parameters is this steady-state asymptotically stable?
(f) One of the steady-states you found has is i* s 0 , what is called an epidemic equilibrium. Under what conditions on the parameters is this steady-state asymptotically stable?
(g) For measles m = 0.02, þ = 1800, g = 100 (Engbert and Drepper [1994]). Show that in this case the epidemic equilibrium is asymptotically stable.
(h) One strategy for eliminating measles is to vaccinate newborns. If p is the proportion that are vaccinated then the S and R equations need to be modified and they become
S ¢ = (1 - p)m(S + I + R) - (þI + m)S,
R¢ = pm(S + I + R) + gI - mR .
By making the appropriate changes in your earlier calculations determine what p must be so the epidemic equilibrium is unstable. Note you do not need to rewrite everything you did in (b)-(d) to answer this question.
(i) Use the result from (h) to explain why measles is so hard to eliminate.