clarification
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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.)

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, $\beta + \delta$ (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.)

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.)

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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 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.)

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.)

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, $\beta + \delta$ (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.)

added 130 characters in body
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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.)

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, $\beta + \delta$. 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.)

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.)

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fileunderwater
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