differential equations in SIR epidemic model and obtain Ro I need to know why the differential equation system that expresses epidemic's model SIR in some texts appears:
$$\frac{dS}{dt}  =-\beta\frac{S}{N}I$$
 $$\frac{dI}{dt}= \beta \frac{S}{N}I - \gamma I$$
$$\frac{dR}{dt}=\gamma I$$
and in other is expressed in this way:
$$\frac{dS}{dt}=-\beta S I$$
$$dI = \beta S I - \gamma I$$
$$\frac{dR}{dt}= \gamma I$$
Another question that I have is the origin of $R_o$ number, how can I obtain this number from this system
Please could you in addition recommend me some bibliography about this topic specially considering vaccination
I really appreciate your help
 A: The difference is if $S,\,I,\,R$ are numbers (not frequencies) of susceptibles, infected and recovered, the transmission function of the form
$$
-\beta SI \tag{1}
$$
describes "mass-action" kinetics. I.e., it means that in the considered population the average number of contacts per one individual is proportional to the total population size.
If you have
$$
-\frac{\beta}{N}SI, \tag{2}
$$
this means that the average number of contacts per one individual is fixed and does not depend on $N$ (think about sexually transmitted disease). This does not lead to different results qualitatively in your case, but may have important implications in more realistic models. 
$R_0$ is defined as the average number of individuals infected by one person, who is put in totally susceptible population. It is a threshold parameter, if $R_0>1$ then epidemics occur, if $R_0\leq 1$ then there is no epidemic. All this can be made rigorous within the theory of branching processes (see the links below). For deterministic models $R_0$ is defined as the spectral radius of next generation operator. 
In your case you can find it as
$$
R_0=\frac{\beta N}{\gamma}
$$
if $(1)$ is used, and
$$
R_0=\frac{\beta}{\gamma}
$$
if $(2)$ is used. To find it consider the condition $\frac{dI}{dt}\geq 0$ for small $t$ assuming that $S(0)\to N$.
(Here how you can see it:
$$
\frac{dI}{dt}\geq 0\implies \beta S(0)I(0)-\gamma I(0)\geq 0\implies \frac{\beta S(0)}{\gamma}\geq 1\implies R_0=\frac{\beta N}{\gamma}
$$)
Actually, the question in which form to use the transmission function was quite a heated one, you can read it a little more about it, for example, here.
To calculate $R_0$ in more general situation (for deterministic models), you might want to study this paper.
Much more details are given in Mathematical Tools for Understanding Infectious Disease Dynamics.
A: Let $s = \frac{S}{N}$. Then $\frac{ds}{dt} = \frac{1}{N}\frac{dS}{dT}$, so
$$\frac{ds}{dt} = \frac{1}{N}\frac{dS}{dt} = \frac{1}{N}\left(-\beta SI\right) = -\beta\frac{S}{N}I.$$
So in some texts, $S$ is presented as is, and in others, it is scaled by $N$. It's just a simple variable re-scaling.
You have not defined $R_0$, so it is impossible to say what it's derivation is.
As far as references, I believe that this is covered in the Lin and Segel text Mathematics Applied to Deterministic Problems in the Natural Sciences in very good detail.
