Because of this fact, we can write the fraction of susceptible individuals in terms of the state variable $I$. If the fraction of infected individual is $I$, then the fraction of susceptible individuals is .
The term $\alpha I (1-I)$ is the rate at which infected susceptible recovered individuals get infected die become susceptible again recover. Since a infected susceptible recovered person needs to encounter an infected recovered susceptible person to get the infection, the rate is proportional both to the fraction of infected individuals and the fraction of susceptible individuals. (Write an answer in terms of the state variable in the last two blanks.) The parameter $\alpha$ captures how likely it is for a infected susceptible recovered person to encounter an susceptible infected recovered person and then subsequently become susceptible again get infected die recover.
The term $-\mu I$ is the rate at which recovered infected susceptible individuals recover get infected die and become infected susceptible recovered again. This rate is just proportional to the number of recovered infected susceptible individuals. The parameter $\mu$ captures how quickly susceptible recovered infected individuals get infected recover die.
The zero equilibrium is biologically plausible for small no large any values of the parameters $\alpha$ and $\mu$.
The other equilibrium is biologically plausible if the following condition on the parameters is satisfied: . (This condition should be an inequality involving $\alpha$ and $\mu$).
Online, enter $\alpha$ as alpha and $\mu$ as mu. You can enter $\le$ as <= and $\ge$ as >=.
The zero equilibrium is stable if . The zero equilibrium is unstable if .
The other equilibrium is stable if . The other equilibrium is unstable if .
There is one case where the stability theorem does not tell us if the equilibria are stable or unstable. This happens when $\alpha = $ .
Online, enter $\alpha$ as alpha and $\mu$ as mu.
For the first case, let $\alpha=0.5$ and $\mu=2$.
On this plot of $I$ versus $t$, in addition to the equilibria, show solutions for the initial conditions $I(0)=0.1, 0.5,$ and $0.9$.
For the second case, let $\alpha=2$ and $\mu=0.5$.
A bifurcation point is a value of $\mu$ where the number or stability of the equilibria changes. The above bifurcation diagram should have a bifurcation point, which we'll label $\mu^*$. $\mu^*=$ .