How to determine $R_0$, the basic reproduction number, for following system?

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?

• $R_0$ is given here by $\beta S^*/(\gamma+c+d(N^*))$. Since, we are in the disease free state, then $S^*=N^*$. The result follows.
– KBS
Commented Apr 24, 2022 at 19:33

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.