so I have a scenario whereby a disease is modeled using SEIS (Susceptible, Exposed, Infected, Susceptible) where:
- Susceptible (S) individuals become exposed (E) at a rate $A$ individuals per day
- Exposed (E) individuals become infected (I) at a rate $B$ individuals per day
- Infected (I) individuals do not gain immunity and become susceptible again at a rate $C$ per day.
- $s$, $e$ and $i$ are the frequencies of S, E and I of the total population, $N$.
- $R_0$ is $A$/$C$
- $s$ + $e$ + $i$ = 1
I have derived a series of differential equations for this model: $$ds/dt = -As + Ci$$ $$de/dt = -Be + As$$ $$di/dt = -Ci + Be$$
I am now required to show that:
$$ds/dt = -A(1-s-e)(s-1/R_0)$$ $$de/dt = As(1-s-e)-Be$$
I assume I need to use the fact that $s$ + $e$ + $i$ = 1 to do so but I've tried but am stuck. How do I go about getting these two 'new' equations?