Reproduction number of a SIR model with mortality

Question

We know that reproduction number $\mathcal{R}_0$ is $\frac{\alpha}{\beta}$ for the following system, such that if $\mathcal{R}_0>1$, there is an epidemic in the population.

Now, assume the system below with mortality rate of $\delta$:

I'm wondering which one of options below represents ${R}_0$?

1. ${R}_0 = \frac{\alpha}{\beta + \delta}$
2. ${R}_0 = \frac{\alpha}{\beta \delta}$
3. ${R}_0 = \frac{\alpha}{\beta}+\frac{\alpha}{\delta}$

Could you mark the right option with its justification?

Thanks

Answer 1

The only answer that makes numerical sense is 1. The product of two rates beta and delta (recovery * death) doesn't mean anything in SIR. And in answer three you're doubling the rate of infection (alpha). Looking at the other way, for R_0 it doesn't matter how people leave the Infected class, once you're either dead or recovered you no longer are transmitting the disease. Therefore you can say that some value Zeta is the exit from Infected class and Zeta would be the sum of all rates which remove a person from being Infectious

Answer 2

Granted, I haven't worked with SIR-models, but to me the answer is definitely nr. 1.

Fundamentally, $R_0$ is defined as the number of secondary infections from a single individual in an uninfected population. It is sometimes described as:

$ R_0 = \gamma *c * d $,

where $\gamma$ is the probability of transmitting the disease, $c$ is the average contact rate (encounter rate with other individuals), and $d$ the average length being infectious (which is $1/\beta$, there $\beta$ is the recovery rate). In your expression above $\alpha = \gamma*c$.

In this simple model of $R_0$ adding a mortality rate basically equates to adding another way of being removed from being infectious (you can either recover or die). Therefore, the expected time being infectious (previously $1/\beta$) is now the sum of two rates, $1/(\beta + \delta)$, where $\delta$ is the death rate (multiplying the two rates would be similar to estimating the probability that someone both recovers and dies from the infection). This means that $R_0$ with a death rate is:

$$ R_0 = \frac {\gamma c}{\beta+\delta} = \frac {\alpha}{\beta+\delta}$$

However, there are also more complicated (and realistic) ways of modelling situations with a death rate.

(I began my answer before @Artems nice answer, which is why I'm posting this as a complementary answer.)