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?
1/ To answer your first question: "infection induced mortality" implies "dying because of infection".
Consider the first term. Using the provided definition of $\rho$, then $(1-\rho)$ is the probability of an individual in class $I$ dying from natural causes. This means that $\frac{\rho}{1-\rho}$ can be thought of as the chance/proportion of infected individual at time $t$ who will die from disease complication.
On to the second term. Since the removal rates are $\gamma + \mu$, then the total number of individual being removed at time $t$ from the infected population $I$ is $(\gamma + \mu)I$.
Combining the two term, you have the proportion of individual dying from disease complication is $\frac{\rho}{1-\rho}(\gamma + \mu)I$.
2/ To answer your second question. The mortality rate is not specific enough for practical purposes. For example, if you want to measure how dangerous a disease is, the mortality rate is not enough to give you this information. To do this, you need the disease induced mortality and infection rate per contact and the contact rate at the minimum.