The simplest model of malaria assumes that the mosquito population is at equilibrium and (only) models the infected humans $I$ with the following equation:
$$\frac{dI}{dt} = \frac {}{+}(−)−$$
where is the natural death rate of mosquitoes, $$ is the death rate of humans, $$ is the transmission rate from infected mosquitoes to susceptible humans, and $$ is the transmission rate from humans to mosquitoes.
Find the equilibria of this model and determine stability conditions for the disease free equilibria to be stable or unstable.
So I know that for this DE to be at equilibrium $\frac {dI}{dt}=0$, and the only way I can come up with for this to be at equilibrium is when $I=0$ ie. the disease free equilibria. I am having difficulty finding any other equilibria. Also, how do I go about determining the stability conditions for the disease free equilibrium ($I=0$, I think) to see whether it is stable or unstable.
You need $$ \frac {\alpha\beta I}{\alpha I+Nr}(N-I)-\mu I = 0. $$ If $I\ne 0$ then you can divide both sides by $I$: $$ \frac {\alpha\beta}{\alpha I+Nr}(N-I)-\mu = 0. $$ Can you solve that for $I$?