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I have the following model for simple endemic with susceptible (S), exposed (E), and infective (I), $$\frac{dS}{dt}=-\beta SI,$$ $$\frac{dE}{dt}=\beta SI-\delta E,$$ $$\frac{dI}{dt}=\delta E.$$

I have already found the steady states I think which are $(0,N)$ and $(N,0)$ for (S,I). I understand that the first one, $(0,N)$ is stable and have proved this but I am struggling to show that the second $(N,0)$ is unstable, so I am looking for pointers for this. I have tried finding the eigenvalues however, I don't know how to proceed from there.

Additionally, I need to identify if an epidemic occurs and if so can the disease become endemic in the population?

user00134857693
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1 Answers1

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If $x = (S,E,I)$, the equations of motion are given by $\dot{x} = F(x)$, where $F(x) = (-\beta SI, \beta SI-\delta E, \delta E)$. Its Jacobian matrix is thus $$ J_F(x) = \begin{pmatrix} -\beta I & 0 & \beta S \\ \beta I & -\delta & \beta S \\ 0 & \delta & 0 \end{pmatrix} \quad\Rightarrow\quad J_F(N,0,0) = \begin{pmatrix} 0 & 0 & \beta N \\ 0 & -\delta & \beta N \\ 0 & \delta & 0 \end{pmatrix} $$ whose eigenvalues are $$ \begin{cases} \lambda_0 = 0 \\ \lambda_\pm = -\frac{1}{2}\left(\delta\pm\sqrt{4\beta\delta N+\delta^2}\right) \end{cases} $$ which cannot be strictly negative at the same time, hence unstability. In conclusion, as the only stable point is $(S,E,I) = (0,0,N)$, all the population will be infected in the long run.

Abezhiko
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