Consider a disease that is transmitted both horizontally and vertically. Peoplecan be either susceptible or infected. Infection occurs either from susceptibles being infectedor from infected mothers giving birth to infected children. Susceptible mothers will, of course,give birth to susceptible children. People can also recover from this disease, which does notkill. Write down the differential equations that describe this outbreak. (Let the horizontaltransmission parameter be α, the birth rate be b and the recovery rate be .) Write downall equilibria of this model. Under what conditions on the parameters will this disease beendemic?

I started this problem for my assignment, but i am having trouble with it, this is what i have so far but I am not even sure it is correct, any guidance would be greatly appreciated

S'(t)= -alpha(I)(S) + b
I'(t)= alpha(I)(S)- betaI- mewI
R'(t)= alpha(I)

I'm assuming equilbirum is gotten from isolating the above equations, but I am having trouble doing this, also does one take the limits in order to solve is it is endemic or not? also how is this done?

Thanks for any help and guidance in advance!