Textbook: Mathematical Models in Population Biology and Epidemiology

Course: Self-study

I have this model with the hypothesis that $r > \mu$

$$\begin{align}

S' &= r(S + I) - \beta S I - \mu S \\

I' &= \beta S I - (\mu + \alpha) I

\end{align}$$

From the second equation the equilibria are given by either $I = 0$ or $S = \frac{\mu + \alpha}{\beta}$. For the first case, $I = 0$, the first equilibrium equation gives us $r S = \mu S$ which implies that $S = 0$ given our hypothesis. We then have an equilibrium $(S,I) = (0,0)$ and through linearizing the system I was able to show that this equilibrium is unstable.

9

For the second case, $S = \frac{\mu + \alpha}{\beta}$, the first equation gives us $I = \frac{(r - \mu) (\mu + \alpha)}{\beta (\alpha + \mu - r)}$. We then have an equilibrium $(S,I) = \left(\frac{\mu + \alpha}{\beta}, \frac{(r - \mu) (\mu + \alpha)}{\beta (\alpha + \mu - r)}\right)$. For simplicity, let $\left(\frac{\mu + \alpha}{\beta}, \frac{(r - \mu) (\mu + \alpha)}{\beta (\alpha + \mu - r)}\right) = (S_\infty, I_\infty)$. Linearizing about this equilibrium we get this two-dimensional system

$$\begin{align}

u' &= (r - \beta I_\infty - \mu) u + (r - \beta S_\infty) v \\

v' &= \beta I_\infty u + (\beta S_\infty - (\mu + \alpha)) v

\end{align}$$

Finding the eigenvalues of the coefficient matrix we get $$(r - \beta I_\infty - \mu - \lambda)(\beta S_\infty - (\mu + \alpha) - \lambda) - \beta S_\infty (r - \beta S_\infty) = 0 $$

This quadratic is really hard to solve by hand, as I am not familiar with any CAS. The authors of my textbook state that this equilibrium is stable, but they omit the verification. Should I take some other approach to determining the stability of this equilibrium?

EDIT:Another hypothesis is that $r < \mu + \alpha$