Relationship between Reproductive Ratio and Jacobian in Population Model In class we defined the Reproductive Ratio, $R_0$ of a population modelled by SIR, SEIR,... as the average number of secondary infections caused by an average infected individual in an average infectious period in a totally susceptible population. Later on we were given a different formulation,
$$ R_0 = \frac{\lambda }{\gamma} + 1 $$
Where $\gamma$ is the average rate of leaving the infected class (so that $1/\gamma$ is the average infectious period), and $\lambda$ is the dominant eigenvalue of the Jacobian of the SIR system at time $0$
\begin{align}
(\dot{S},\dot{I},\dot{R}) = f(S,I,R)
\end{align}
Or the appropriate system for a different model e.g. SEIR.
How do these two formulations relate? Even some intuition for the simplest case, SIR or SIS would help.
In addition, how does the WAIFW (Who Acquires Infection From Whom) matrix $B = \{\beta_{ij} S_i\}$ in a risk-structured SIR(say) model fit in to this framework? There should be a relationship as we were told that in this case,
$$R_0 = \frac{\lambda}{\gamma} $$
where $\lambda$ is the dominant eigenvalue of $B$ at $t=0$ now, and $1/\gamma$ is the average infectious period shared by the risk classes (I am aware of the more complicated model which has different infectious periods for different classes but I wish to understand this simpler case first).
We have attempted to contact our lecturer regarding this but it isn't term time and he seems to be quite busy.
 A: There are a number of way you can solve for $R_0$. Here is an example taken from (Otto and Day). Consider the system below for a population of susceptible $S$ and infected $I$ individuals.  
\begin{align}
\frac{dS}{dt}&=\theta-dS-\beta SI+\gamma I\\
\frac{dI}{dt}&=\beta S I -(d+v+\gamma)I
\end{align}
$\beta$ is the transmission rate, $d$ death rate, $v$ death rate due to infection and $\gamma$ is the clearance rate. 
Case1: (This approach may not always be possible.)
Consider the second DE above. For there to be an epidemic, the rate $dI/dt$ must greater than zero. 
$$\beta S I -(d+v+\gamma)I>0$$
If you rearrange the above expression you will get:
$$\frac{\beta S}{d+v+\gamma}>1$$
The expression on the left is $R_0$. Numerator is successful transmissions, denominator is duration of infection.  
Case 2: (Using the Jacobian)
The Jacobian allows us to determine the stability of the disease free system (locally). First we need the disease free equilibrium.
$$(S^*,I^*)=\Big(\frac{\theta}{d},0\Big)$$ 
The Jacobian (J) is a linearization of the system of equations about the equilibrium point. 
$$J=\begin{bmatrix}
-d&-\beta\frac{\theta}{d}+\gamma\\
0 & \beta\frac{\theta}{d}-(d+v+\gamma)\\
\end{bmatrix}$$
Essentially what you are doing is studying a small perturbation of this equilibrium state. If the perturbation grows, the equilibrium is not stable, if the perturbation shrinks, it is stable. The dominant eigenvalue of this linear system determines whether it grows or shrinks. 
$$\lambda=\beta\frac{\theta}{d}-(d+v+\gamma)$$
If there is to be an epidemic, we need the disease free state to be unstable, and this perturbation to grow. Therefore, we want the dominant eigenvalue to be greater than zero, giving us the same $R_0$ from before: 
$$\frac{\beta S^*}{d+v+\gamma}>1$$
where $S^*=\theta/d$
