Does the SEIS model of disease assuming a constant population and homogenous mixing have a disease-free equilibrium? I'm thinking no, because as soon as an individual "recovers" they become susceptible again and are at risk of becoming infected. How can I justify this mathematically?
My equations are: $$ds/dt = -Asi + Ci$$ $$de/dt = -Be + Asi$$ $$di/dt = -Ci + Be$$
Where ds/dt and de/dt can be transformed into these where $R_0 = A/C$ and $s+e+i = 1$ $$ds/dt = -A(1-s-e)(s-1/R_0)$$ $$de/dt = As(1-s-e)-Be$$
When I produce a phase-plane of e vs s, I get. The nullcines don't intersect at (1,0), but they are very close (1,-1.78508E-20). Do I count this as (1,0) and thus a disease-free state?
Starting with the following SEIS model: \begin{align*} {\tfrac {dS}{dT}}&=B-\beta SI-\mu S+\gamma I\\ {\tfrac {dE}{dT}}&=\beta SI-(\epsilon +\mu )E\\ {\tfrac {dI}{dT}}&=\varepsilon E-(\gamma +\mu )I \end{align*} Setting $E=I=0$ and $dS/dT=0$ gives an equilibrium $B=\mu S,$ that is, $S=B/\mu.$ It's the same as if the disease never existed.