I'm currently going through Computational Methods course notes and came across this:
Consider a population of $N$ individuals where individuals are either of type $A$ or $B$ but not both. At any point in time, an individual can only be be of one type. All individuals can be assumed to be in contact with all other individuals. At time $t=0$, there are $b(0)$ type $B$ individuals and all other individuals are type $A$. Type $B$ individuals turn to type $A$ at a fixed rate $\gamma$. Type $A$ individuals turn into type $B$ at a rate $\frac{\beta b}{N}$, i.e., proportional to the number of their contacts that are type $B$.
I want to find the fixed points of this system along with their stability. To this end, I want to express $\dot{b}$ as a function of $b$ and equate it to 0 then solve for the corresponding values of $a$ and $b$. However I'm confused as my model (as I understand it) $$\dot{b}=\frac{\beta}{N}b - \gamma$$ differs from the model given by the notes $$\dot{b}=\beta\frac{b}{N}a - \gamma b.$$
I'm confused as it seems that the dimensions don't agree in the second differential equation, it seems to me that the LHS is $\frac{people}{time}$ but the RHS is $\frac{people^2}{time}$.
For this example, how does one find the fixed points of the system?