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?