Eliminate $R=N-S-I$ to obtain the <autonomous differential equation>[two-dimensional autonomous differential equation]
$$
\dot S=\alpha(N-S-I)-\beta SI,
\qquad
\dot I=\beta SI-\gamma I.
$$
Its <Jacobian matrix> at the endemic equilibrium is
$$
J_*=
\begin{pmatrix}
-\alpha-\beta I^*&-\alpha-\gamma\\
\beta I^*&0
\end{pmatrix}.
$$
Since $I^*>0$,
$$
\operatorname{tr}J_*=-(\alpha+\beta I^*)<0,
\qquad
\det J_*=\beta I^*(\alpha+\gamma)>0.
$$
The <trace-determinant stability criterion> shows that both <eigenvalue>[eigenvalues] have negative real part. The <Endemic equilibrium of the SIR model with waning immunity> is therefore locally <asymptotic stability>[asymptotically stable].

Solution

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

{wiki=Autonomous_system_(mathematics)}

An autonomous differential equation has the form $\dot x=f(x)$, with no explicit dependence of the vector field $f$ on time.


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