Consider a population of size $N$ with per capita birth rate $b(N)$ and death rate $d(N)$. Assume that it reaches stable steady state $N^*$ in absence of disease, with $b(N^*)=d(N^*)$. Find $R_0$ when disease is introduced into population at disease-free steady state
\begin{align} \frac{dS}{d\tau}&=b(N)N-\beta IS-d(N)S\\ \frac{dI}{d\tau}&=\beta IS-\gamma I-cI-d(N)I\\ \frac{dR}{d\tau}&=\gamma I-d(N)R \end{align}
So the back of the book says that $R_0=\frac{\beta N^*}{\gamma+c+d(N^*)}$. I can't figure out how they deduced it, any hints or clues on how to compute this?
Just look at the derivative of $I$, which can be rewritten as: $$\frac{dI}{dt} = (\beta S)I - (\gamma + c + d(N))I.$$
Note that $\beta S$ is the average rate of infection and $\gamma + c + d(N)$ is the average rate of removal from the infected class (e.g., exponentially distributed). The reciprocal of $\gamma + c + d(N)$ is the average duration an infectious person stays infectious. With that, then we have: \begin{align} R_0(N) & = \text{(average infection rate)} \times \text{(average duration of an infected person stay infected)}\\ & (\beta S) \times \left(\frac{1}{\gamma + c + d(N)}\right). \end{align}
Now, note that we want $R_0$ when you introduce a single infection into an otherwise susceptible community, so you must evaluate $R_0$ at $N^*$, giving you the answer.