 Solution

ID: past-exam-of-the-short-options-course-of-the-university-of-oxford/2017/9/b/i/solution

Assume

S_{0} > 0

and scale by the initial susceptible count:

u = \frac{S}{S _{0}}, v = \frac{I}{S _{0}}, w = \frac{R}{S _{0}}, τ = γ t, v_{0} = \frac{I _{0}}{S _{0}}, R_{0} = \frac{β S _{0}}{γ} .

Substitution into the SIR model equations gives

\frac{d u}{d τ} = - R_{0} uv, \frac{d v}{d τ} = (R_{0} u - 1) v, \frac{d w}{d τ} = v .

The initial data become

u (0) = 1

v (0) = v_{0}

w (0) = 0

. The conserved total is

u + v + w = 1 + v_{0}

, since the scaling is by

S_{0}

, not by

S_{0} + I_{0}

. With this normalization,

R_{0}

is the reproduction ratio at the initial susceptible level.

 New to topics? Read the docs here!