Question

One of the classical models of epidemics is due to Kermack and McKendrick (1927). The model considers three classes of individuals: at time t, I(t) is the number of infected people in the population, S(t) is the number of non-infected susceptible people, and R(t) is the number of “removed” people (either cured or deceased). This model is often referred to as the “SIR” model (where each letter is pronounced, or pronounced as the word “sir”). The equations of the model are

S’(t) = −r*S*I

I’(t) = r*S*I − γ*I

R’(t) = γ*I

5. (8 pts) Eliminate t from the S and I equations to give a single ODE relating S and I by dividing one equation by the other (and justifying this process by the chain rule). Solve this ODE and sketch the solution curves. Compare these solution curves to your direction field picture

Answer 1

Given;

S’(t) = − r S I

I’(t) = r S I − γ I

R’(t) = γ I

where;

S’(t) = dS / dt

I’(t) = dI / dt

R’(t) = dR / dt

Now;

Dividing equation 2 by equation 1 :

dI / dS = ( r S I − γ I ) / ( - r S I )

Verification by chain rule:

dI / dS = (dI / dt) (dt / dS)

dI / dS = ( r S I − γ I ) / ( - r S I )

Hence we are getting the same result.

We have;

dI / dS = - 1 + (γ / r) (1 / S)

Let (γ / r) = k ; where k is a constant

Therefore;

dI / dS = - 1 + k / S

d I = ( - 1 + k / S ) dS

Integrating the above expression;

I = - S + k ln (S) + C

where c is the constant of integration.

For simplicity of plotting; let k = 1. This won't change the nature of the solutions.

Then; we have :

dI / dS = - 1 + 1 / S

I = - S + ln (S) + C

The solution curves for different values of C are given below :

The direction field picture is a plot between dI / dS vs S. The plot is given below:

The solution curves are in accordance with the direction field picture.

At S = 0 ; the solution curves are perpendicular to x-axis. This means they have a slope of 90 °. The same can be observed in the direction field picture.

At S = 1; the solution curves are parallel to x axis. This means they have a slope of 0 °. The same can be observed in the direction field picture.

For S > 1; the solution curves decrease in value and are at obtuse angles with the x-axis. The same trend of decreasing slope can be observed in the direction field picture.