Differential Equation: Find the instability criterion

Question

My attempt:

Question (d)

I took the derivative of the original differential equation,

$$dI/dt = BI(N-I) -uI = g(I)$$

$$g'(I) = BN - 2BI - u$$

Set $$g'(I) = 0$$

Isolate $$I = Ro$$

$$I = Ro = N/2 - u/2B$$

Why is this wrong?

Answer 1

We are given that $$\frac{dI}{dt} = \beta I(N-I)-\mu I$$ To find $R_0$ consider the case where $\frac{dI}{dt} > 0:$

\begin{align*}\frac{dI}{dt} > 0 &\implies \beta I(N-I)-\mu I > 0 \\ & \implies\beta I(N-I) > \mu I \\ &\implies \beta(N-I) > \mu \\ &\implies \frac{\beta(N-I)}{\mu} > 1\end{align*}

But we have that $I =0$, thus we have that $$R_0= \frac{\beta N}{\mu} $$