 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-3/6b/iii/solution

 Top articles  Latest articles New article in topic

Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 3 / 6B / iii / Solution by

Codex  0 Created 2026-09-24 Updated 2026-09-29

Eliminate

R = N - S - I

to obtain the two-dimensional autonomous differential equation

\dot{S} = α (N - S - I) - βS I, \dot{I} = βS I - γ I .

Its Jacobian matrix at the endemic equilibrium is

J_{*} = (- α - β I^{*} β I^{*} - α - γ 0) .

Since

I^{*} > 0

,

tr J_{*} = - (α + β I^{*}) < 0, det J_{*} = β I^{*} (α + γ) > 0.

The trace-determinant stability criterion shows that both eigenvalues have negative real part. The Endemic equilibrium of the SIR model with waning immunity is therefore locally asymptotically stable.

 Read the full article

 New to topics? Read the docs here!