I have the following infection system:
$ \frac{dx}{dt}=\lambda-d\cdot x(t)-\beta\cdot v(t)\cdot x(t) $ $\rightarrow$ Susceptible cells
$ \frac{dy}{dt}=\beta\cdot v(t)\cdot x(t)-(a+d)\cdot y(t) $ $\rightarrow$ Infected cells
$ \frac{dv}{dt}=c\cdot y(t)-u\cdot v(t) $ $\rightarrow$ Free parasites
Q1: How exactly do I calculate the possible equilibria? I know that the infection-free equilibrium is calculated by setting $\frac{dx}{dt}=0$, $y(t)=0$ and $v(t)=0$, but how exactly do I calculate other possible equilibria?
Q2: In many articles, they mention that the rate at which infected cells are produced in this particular system in $\frac{\beta\cdot c\cdot x(t)}{u}$, but how exactly do I get to that expression? In general, what's the logical thinking that is used to calculate get to this expression?
Q3: How exactly do I get to the expression that represents the average lifespan of an infected cell? Many articles mention that for this system, it's $\frac{1}{a+d}$.
Q4: And finally, based on all the questions above, how do I calculate the basice reproduction number for the system?
Note: I would really appreaciate it if you could provide me with an introductory reference where these calculations are detailed and well-explained.
Partial answer
To answer Q1, equilibria of a dynamical system are generally found by setting all derivatives to $0$, so you get the system
$$0=\lambda-dx-\beta vx$$ $$0=\beta vx-(a+d)y$$ $$0=cy-uv$$
You can solve this system to find $v=y=0,\ x=\lambda/d$, or $x=\frac{u(a+d)}{c\beta},\ v=\frac{\lambda c}{u(a+d)}-\frac d\beta,\ y=\frac{\lambda}{a+d}-\frac{du}{c\beta}$. No other equilibria exist as far as I can tell.
You might then be interested in the stability of these equilibria, which can be calculated by linearising the system around this point. See wikipedia