SIR infection induced mortality I am reading the book Modelling Infectious Diseases in Humans and Animals by Matt J Keeling and Pejman Rohani.   
In it the SIR model is given as,   
$\frac{dS}{dt}  =\mu-\beta{S}I-\mu S$
$\frac{dI}{dt}= \beta SI - \gamma I-\mu I$
$\frac{dR}{dt}= \gamma I-\mu R$     
where the rate at which individuals in any epidemiological class suffer natural mortality is given by $\mu$ and S,I,R are respectively the proportion of susceptible, infected and recovered individuals in the population.  
For this model when the infection induced mortality is introduced the following change is made. (page 34)
$\frac{dI}{dt}= \beta SI - (\gamma+\mu) I-\frac {\rho}{1-\rho} (\gamma+\mu)  I$  where $\rho $ is the probability that an individual in I class dying from the infection before either recovering or dying from natural causes.     
In this I don't understand
1) How does $\frac {\rho}{1-\rho} (\gamma+\mu)  I$ capture the infection induced mortality.
2) Also it says rather than having the per capita disease-induced mortality rate for infected individuals it is preferable to think of $\rho$, probability that an individual in I class dying from the infection before either recovering or dying from natural causes. Why is this probability easy than mortality rate?

Thank you for your explanation. It is really useful to understand the relationship between  (probability of dying due to infection) and m (mortality rate).
I translated your language into a equation to show what you had clarified.
$$\frac{I}{I(1-)} = \frac{\int_t^∞{mIdt}}{\int_t^∞{(+)Idt}}$$
Then,
$$\frac{I}{I} = \frac{(1-)\int_t^∞{mIdt}}{\int_t^∞{(+)Idt}}$$
$$\frac{I}{I} = \frac{\int_t^∞{(1-)mIdt}}{\int_t^∞{(+)Idt}}$$
As a result,
$$m = \frac{}{1-}(+)$$
