We are working on the SIRS model in my physical analysis course, and I want to prove that all solutions in the triangular region $\Delta$ tend to the equilibrium point $(\tau,0)$ when the total population doesn't exceed threshold level for the disease. I know that when $\tau = \frac{v}{\beta},$ the two equilibria coalesce at $(\tau,0).$ That being said, I am having some issues connecting this idea to the behavior of $\Delta$ in the model. I think that I may need some assistance on this problem. For reference, $\Delta$ is the region where $I,S \geq 0 \wedge S+I \leq \tau.$
2026-03-28 08:47:08.1774687628
Ideas Surrounding the SIRS model
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First notice that the $\Delta$ region is a positively invariant set. You can verify that analyzing the slope field at the boundary of $\Delta$. Expect to find no vector pointing outwards $\Delta$. So once a solution curve gets in $\Delta$ it never gets out.
You can better understand the behavior of the system near $(\tau, 0)$ by studying the linearized system around it. Therefore I suggest to linearize the system around $(\tau, 0)$ and rewrite it as a matrix differential equation.
Let A be the matrix of coefficients.
Consider the case when $\tau \lt \frac{v}{\beta}$.
Verify that A has two negative real eigenvalues. This means that the equilibrium point $(\tau, 0)$ for some neighborhood of itself behaves like a spiral sink (on a twisted fashion) and so it is asymptotically stable.
Now suppose that $\tau = \frac{v}{\beta}$.
Hence A no longer have a pair of negative real eigenvalues. Take a look at the direction field around the point $(\tau, 0)$. It no longer resembles any linear pattern we know. We would need a more refined technique solely to analyze this singularity.
Thus I will leave you with this question to reason:
What are the chances of finding an instance on the nature of the modeled phenomenon such that $\tau = \frac{v}{\beta}$ for all time?
(Bibliography: Hirsch, Smale, and Devaney, Differential Equations, Dynamical Systems, and an Introduction to Chaos (Edition: 2), Elsevier Science & Technology Books, 2004)