For the past few years I have been using the Next Generation Matrix to calculate the basic reproductive number $R_0$ in many epidemiological applications. Lately, I've tried to understand the proof of the Theorem from the paper "Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission" by P. van den Driessche and James Watmough. The authors rewrite a system of ODEs $x_i' = f_i(x)$ as $$x_i' = f_i(x) = \cal{F}_i(x)-\cal{V}_i(x),$$ for $i = 1, \ldots, n$. The function $\cal{F}_i$ denotes the rate of appearance of new infections in compartment $i$. Writing $\cal{V}_i(x) = \cal{V}_i^-(x) - \cal{V}_i^+(x)$, then $\cal{V}_i^+$ denotes the rate of transfer into of compartment $i$ by all other means (disease progression, for example), whereas $\cal{V}_-$ denotes the rate of transfer out of that compartment.
They label the compartments in such a way that the first $m$ compartments are the disease ones. For example, in the usual SIR model, the state vector could be written as $(I,S,R)$. They also make the following assumptions:
- If $x \geq 0$, then $\cal{F}_i, \cal{V}_i^-, \cal{V}_i^+ \geq 0$ for $i = 1, \ldots, n$.
- If $x_i = 0$ then $\cal{V}_i^- =0.$
- $\cal{F}_i = 0$ if $i > m$.
- If $x$ is a disease free state (i.e., $x_i = 0$ for $i = 1, \ldots, m$), then $\cal{F}_i(x) = \cal{V}_i^+(x)$ for $i = 1, \ldots, m$.
- If $\cal{F}(x)$ is set to zero, then all eigenvalues of $Df(x_0)$ have negative real parts.
The last assumption tells us that any disease free equilibrium would be locally assymptotically stable. They then prove the following lemma: 
I've been able to check everything with the exception of the claim that the last assumption implies that the eigenvalues of $V$ have positive real part. I can see that the eigenvalues of $F-V$ have negative real parts, but how does assumption 5 relate to the eigenvalues of $F$? Thanks in advance!