SIR modeling (need MATLAB code to solve) (a) = the rate of exposure; (a) = the m

Question

SIR modeling (need MATLAB code to solve)

(a) = the rate of exposure; (a) = the mortality rate; s(a) = the survival rate {a is an age}

Bmodel = {u' [a] == -([a]) + [a]), w'[a] == [a] s[a] u[a] - [a] w[a], u[0] == 1., w[0]== 0.}

First equation u' (a) = -((a) + (a)) u(a) says that the rate at which the unexposed changes is proportional to the number of unexposed, and the rate is the sum of the exposure rate and the death rate. Therefore, u(a) is always decreasing. Second equation w' (a) = (a) s(a) u(a) - (a) w(a) says that the rate of survivors is determined by two factors. New survivors are a percentage of those who are exposed to smallpox and that increase is governed by the survival rate, s(a). Furthermore, the survivors are decreased in proportion to the mortality rate (a).

Using Bernoulli’s model, assume that the death rate due to causes other than the disease in question is = 0.01.   Also assume that the rate at which the population is immunized is = 0.3 and assume that survival rate when administering the vaccine is some unknown constant s. Finally, assume that if the vaccine did not exist, the rate at which the population would die is = 0.005.

If everyone survives the vaccination, then it is clear that everyone should get vaccinated. On the other hand, if the survival rate is very low, it is not a good idea to be vaccinated. How high the survival rate must be to make the vaccine effective?

Bernoulli’s epidemiological model for an immunizing infection in a cohort which is in equilibrium with respect to time. s(a) = probability of surviving the infection. (a) = force of infection; (a) = death-rate due to other diseases.

Explanation / Answer

SIR Epidemic Model: S' = -a S I S(0) = So and a = contagious constant I' = a S I – b I I(0) = Io and b = recovery constant R' = b I ,R(0) = R0. Note, (S + I + R)' = S' + I' + R' = 0 so that S + I + R = constant and S(0) + I(0) + R(0) = So + Io + R0. Note, I'(0) = (a S(0) – b) I(0) > 0 if and only if a S(0) – b > 0 and I(0) > 0. The Matlab command ode45 is used to appoximate the solution of our system of differential equation. This command is a robust implementation for systems of differential equations, which uses a variable step size method and the fourth and fifth order Runge-Kutta method. Since there are three unknowns and three differential equations and we wish to use Matlab's ode45 scheme, the file, ypsir.m, must contain the three right sides of the differential equations where y(1) = S, y(2) = I and y(3) = R. The choice of a and b in ypsir.m will determine whether or not the I(t) will increase; if I'(0) = (a S(0) – b) I(0) > 0, then I(t) will increase. The sir.m file contains the call to ode45 and the graphs of the three population groups are generated by the Matlab command plot. The initial populations are S(0) = 99, I(0) = 1 and R(0) = 0.

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