clarification
fileunderwater
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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, $$\beta + \delta$$$$1/(\beta + \delta)$$, 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.)

language + formatting
fileunderwater
• 16.6k
• 3
• 47
• 88

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 defineddescribed as:

$$R_0 = \gamma *c * d$$,

where $$\gamma$$ is the probability of transmitting the disease, $$c$$ is the average contact rate (encounter rate towith 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, $$\beta + \delta$$ (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.)

fileunderwater
• 16.6k
• 3
• 47
• 88

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

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

$$R_0 = \gamma *c * d$$, where $$\gamma$$ is the probability of transmitting the disease, $$c$$ is the average contact rate (encounter rate to 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, $$\beta + \delta$$ (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.)

fileunderwater
• 16.6k
• 3
• 47
• 88