$R_0$ in frequency and density dependent SIR I am reading the book Modelling Infectious Diseases in Humans and Animals by Matt J Keeling and Pejman Rohani 
and density dependent transmission model is give as,  
$\frac{dX}{dt}  =v-\beta{X}Y-\mu X$
$\frac{dY}{dt}= \beta XY - (\gamma+\mu) Y-\frac {\rho}{1-\rho} (\gamma+\mu)  Y$
$\frac{dZ}{dt}= \gamma Y-\mu Z$    
where $\rho$ is the probability that an individual in I class dying from the infection before either recovering or dying from natural causes and X,Y,Z are respectively the number of susceptible, infected and recovered individuals in the population. Here,since disease induced mortality could lead to an ever deceasing population size and in order to keep the population size fixed, a fixed birth rate$(v)$ is incorporated to the susceptible equation. $v$ is independent of population size.  
1) What I don't understand is how they have obtained $R_0.R_0$ is given as $R_0=\frac{\beta(1-\rho)v}{(\mu +\gamma)\mu}$.
Also under frequency dependent transmission  a change is made as
$\frac{dY}{dt}= \frac{\beta XY}{N} - (\gamma+\mu) Y-\frac {\rho}{1-\rho} .   (\gamma+\mu)  Y$
Here $R_0=\frac {(\mu +\gamma)}{\beta (1-\rho)}$.
2) How is this $R_0$ obtained and how does it difffer from previous value.    
Also after solvig the equations the endemic equilibrium is found as
$X*=\frac{v(1-\rho)(\gamma+\mu)}{\mu(\beta(1-\rho)-\mu\rho-\gamma\rho)}$.
But when I set $\frac{dY}{dt}=0$ I get $X*=\frac{(\gamma+\mu)N}{(1-\rho)\beta}$.
 3)How can I obtain the equilibrium points?
 A: SaiRam. That's a very good question. To understand how the expression of Ro arrives, it is important to understand what is $\beta$. Please refer to these links to understand the subtleties in these expressions :
(1) http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.164.3014&rep=rep1&type=pdf
(2) https://parasiteecology.wordpress.com/.../density-dependent-vs-frequency-dependent.
$\beta$ is the product of rate of contacts($k$) between individuals in the population and probability of transmission of the disease from infected to susceptible($v$). In density dependent transmission , $\beta$ depends on the total population.
Intuitively, k = number of contacts in the population divided by total population per unit time. 
Now, we know that susceptible population at disease free equilibrium stateis $\nu/\mu$. Thus rate of contacts gets modified because of the changed susceptible population. Therefore, the term $\nu/\mu$ is multiplied to $k$.  Next, the term $1-\rho$ is the correction term which is multiplied because of infection induced mortality. To get it mathematically, simplify the $dY/dt$ term in the model. We would obtain the expression $-[(\gamma + \mu) /(1-\rho)]*Y$. Thus the term $(\gamma + \mu) /(1-\rho)$ is net capita rate at which infected people go out of the infected class. This rate is nothing but the conglomeration of natural mortality rate, recovery rate and infection induced mortality rate. Thus the reciprocal of this conglomerate rate is $(1-\rho)/(\gamma + \mu)$ which is the average time that an individual spends in the infectious class. Now to get Ro, we multiply this average infectious period by the transmission term which is $\beta$. To account for the population size we multiply $\beta$ by $\nu/\mu$ as explained before. Thereby we obtain the desired expression for Ro.
Next, in case of frequency dependent transmission, rate of contacts is independent of the population size. Therefore the transmission of the disease does not vary as population size changes. So, there is no need to multiply $\nu/\mu$.
Next, regarding the calculation of equilibrium points substitute the Ro expression for frequency dependent transmission and we can obtain the expression for $X^* = N^*/ Ro$. Derivation for other equilibrium points is unknown to me.
