Searching for a Lyapunov function for a SIRS model

Question

I am trying to prove the global stability of the endemic equilibrium $$\left(\frac{\nu}{\beta},\gamma\frac{N-(\nu/\beta)}{\nu+\gamma}\right)$$ of a SIRS model $$S'=-\beta SI+\gamma(N-S-I) \qquad I'=\beta SI- \nu I$$

I've been looking for a Lyapunov function that can help me, but I haven't found one that works with this specific system. Do you know one? Or, can you suggest another way to prove this?

Answer 1

As you have a two-dimensional system you could use the Markus-Yamabe Theorem.

First, shift your equation such that the equilibrium point is in $(0,0).$. Let $\dot{x}=f(x)$, in which $x\in \mathbb{R}^2$. Then determine the Jacobian of $f(x)$. If the eigenvalues of the Jacobian have a strictly negative real part for every $x\in\mathbb{R}^2$, then $(0,0)$ is an globally asymptotic equilibrium point.

Answer 2

Keeping in mind some aspects of the concrete situations the SIRS is modelling might help. First, the functions $S$, $I$ and $R$ measure sizes of populations such that $S+I+R=N$ hence, if the model would make sense, the domain $$D=\{(S,I)\mid S>0,I>0,S+I<N\}\subseteq\mathbb R^2$$ should be stable. Unsurprisingly, $D$ is indeed stable by the dynamics, as one sees by looking at the vector field giving $(S',I')$ on the boundary of $D$. To wit, this vector field is pointing to $D$ on the lines $[S>0,I>0,S+I=N]$ and $[S=0,0<I<N]$, and the line $[I=0,0<S<N]$, which completes the boundary of $D$, is made of solutions.

The non trivial equilibrium point is $$S^*=\frac\nu\beta\qquad I^*=\frac\gamma{\nu+\gamma}(N-S^*)$$ and, provided the condition $$\nu<N\beta$$ holds, $(S^*,I^*)$ is in $D$. Furthermore, the dynamics can be rewritten as $$S'=-(\beta I+\gamma) (S-S^*)-(\nu+\gamma)(I- I^*)\qquad I'=\beta I(S-S^*)$$ Starting from these identities, one can prove without too much effort that

$$V=S-(S^*+S_*)\ln(S+S_*)+I-I^*\ln I\qquad\text{with}\qquad S_*=\frac\gamma\beta$$

yields $$V'=-(\beta I^*+\gamma)\frac{(S-S^*)^2}{S+S_*}\leqslant0$$ hence indeed, $(S^*,I^*)$ is the limit of every solution starting in $D$.

Note that we carefully excluded the boundary of $D$ from the set of admissible initial conditions, to avoid "seeing" the other fixed point $(S,I)=(N,0)$, which is repelling except along the line $I=0$. But one could also use from the start the domain $$\bar D=\{(S,I)\mid S\geqslant0,I\geqslant0,S+I\leqslant N\}\subseteq\mathbb R^2$$ then the conclusions are that $(S,I)\to(S^*,I^*)$ for every $(S_0,I_0)$ in $\bar D$ such that $I_0\ne0$ and that $(S,I)\to(N,0)$ for every $(S_0,0)$ in $\bar D$.