I have been reading a paper about a host-parasites models and for the model:
$$\begin{array}{rll} \displaystyle{\frac{dx}{dt}}&=\lambda -dx -\beta v x & \text{Susceptible host} \\ \displaystyle{\frac{dy}{dt}}&= \beta v x -(a+d)y & \text{Infected host} \\ \displaystyle{\frac{dv}{dt}}&= cy-uv & \text{Free parasites} \\ \end{array}$$
Where $\lambda$ is rate of susceptible host are born, die at rate $dx$, and infected at rate $\beta vx$. Infected host are assumed to die at rate $(a+d)y$. The free parasites are released from infected hosts at rate $cy$ and die at rate $uv$.
For this model, the basic reproduction number according with the paper is: $$R_0=\frac{\lambda \beta c}{d(a+d)u}$$
I read another article about how to find $R_0$ (spectral radius of a special matrix), but does not apply to this case.
I understand that $R_0$ greater than $1$ is required for successful invasion of parasites, but not know how to determine in this case.
Can someone explain in detail how to determine $R_0$ in this case?
Thank you very much.
EDIT: you can find the paper here, in the last two paragraphs of Section 2 explains a little about R, but not on how to determine.
$R_0$ in this context is an average number of cells, which were infected by 1 infectious cells put in a naive (fully susceptible) population.
The rate at which infected cells are produced is given by $$ \frac{\beta c x}{u}. $$ If all the population susceptible then $$ x=\frac{\lambda}{d}. $$ Finally, since the average lifespan of infected cell is $$ \frac{1}{a+d}, $$ we get the desired answer $$ R_0=\frac{\beta c}{u}\cdot \frac{\lambda}{d}\cdot \frac{1}{a+d}. $$