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I'm having a really hard time interpreting and understanding the transmissibility term ($\tau$ in particular,...

It looks to me like $\tau$ is a probability of infection per contact.

The number of transmissions per day per person is then binomially distributed. According to a parameter

$$\text{new infections per time} \sim \text{Binomial}( \underbrace{n=c \cdot I \cdot S}_{\substack{\llap{\text{number of contacts per time }}\rlap{\text{}} \\ \llap{\text{= contacts per infected }}\rlap{\text{per susceptible per time}}\\ \llap{\text{ $\times$ infected}}\rlap{\text{$\times$ susceptible}}}} \, , \, \overbrace{p=\tau}^{\llap{\text{probability of infection}}\rlap{\text{ per contact}}})$$

(or can also be approximated with a Poisson distribution if $c$ is large and $ \tau$ is small)

The standard SIR model is not deterministic and it does not explicitly sample the binomial distribution in order to compute the results.

Instead, it uses the mean or mode of this distribution $\beta = c\tau$. Which is the most likely outcome. (see also https://en.wikipedia.org/wiki/Thermodynamic_limit)

This approach is correct in the limit, when the infected population is very large. E.g. when $n_I$ people will each infect other people according to a Poisson distribution with parameter $\beta$, the total number of infected people will be distributed according to a Poisson distribution with parameter $n_I \beta$ and the ratio of the standard deviation and mean of that distribution will shrink as $n_I$ is large. So the error becomes relatively small for large populations.

Why isn't the equation something more like:

 

[I:infected people] * [c:contacts/day] * [tau:likelihood-of-infection/contact] * [P: proportion of contacts with susceptible people left in the population ]

It is like that. Your parameter P, the proportion of susceptible people left, is accounted for by the term $S$

$$I_{t+1} = I_t - \gamma*I_t + \beta*I_t*S_t$$

you will also often see a division by $N$

$$I_{t+1} = I_t - \gamma*I_t + \beta*I_t*\underbrace{\frac{S_t}{N}}_{\llap{\text{proportion of susceptible}}\rlap{\text{ people left}}}$$

This division with $N$ might make the interpretation of parameters more logical, but it is mathamatically not neccesary (it is effectively just a scaling of the parameters).

It depends a bit on whether you see the $\beta$ as 'infections per infected per susceptible' or as 'infections per infected'.

I'm having a really hard time interpreting and understanding the transmissibility term ($\tau$ in particular,...

It looks to me like $\tau$ is a probability of infection per contact.

The number of transmissions per day per person is then binomially distributed. According to a parameter

$$\text{new infections per time} \sim \text{Binomial}( \underbrace{n=c \cdot I \cdot S}_{\substack{\llap{\text{number of contacts per time }}\rlap{\text{}} \\ \llap{\text{= contacts per infected }}\rlap{\text{per susceptible per time}}\\ \llap{\text{ $\times$ infected}}\rlap{\text{$\times$ susceptible}}}} \, , \, \overbrace{p=\tau}^{\llap{\text{probability of infection}}\rlap{\text{ per contact}}})$$

(or can also be approximated with a Poisson distribution if $c$ is large and $ \tau$ is small)

The standard SIR model is not deterministic and it does not explicitly sample the binomial distribution in order to compute the results.

Instead, it uses the mean or mode of this distribution $\beta = c\tau$. Which is the most likely outcome. (see also https://en.wikipedia.org/wiki/Thermodynamic_limit)

This approach is correct in the limit, when the infected population is very large. E.g. when $n_I$ people will each infect other people according to a Poisson distribution with parameter $\beta$, the total number of infected people will be distributed according to a Poisson distribution with parameter $n_I \beta$ and the ratio of the standard deviation and mean of that distribution will shrink as $n_I$ is large. So the error becomes relatively small for large populations.

Why isn't the equation something more like:

[I:infected people] * [c:contacts/day] * [tau:likelihood-of-infection/contact] * [P: proportion of contacts with susceptible people left in the population ]

It is like that. Your parameter P, the proportion of susceptible people left, is accounted for by the term $S$

$$I_{t+1} = I_t - \gamma*I_t + \beta*I_t*S_t$$

you will also often see a division by $N$

$$I_{t+1} = I_t - \gamma*I_t + \beta*I_t*\underbrace{\frac{S_t}{N}}_{\llap{\text{proportion of susceptible}}\rlap{\text{ people left}}}$$

This division with $N$ might make the interpretation of parameters more logical, but it is mathamatically not neccesary (it is effectively just a scaling of the parameters).

It depends a bit on whether you see the $\beta$ as 'infections per infected per susceptible' or as 'infections per infected'.

I'm having a really hard time interpreting and understanding the transmissibility term ($\tau$ in particular,...

It looks to me like $\tau$ is a probability of infection per contact.

The number of infections per day per person is then binomially distributed.

$$\text{Infections per time per person} \sim \text{Binomial}( \underbrace{n=c}_{\llap{\text{contacts per person }}\rlap{\text{per time}}} \, , \, \overbrace{p=\tau}^{\llap{\text{probability of infection}}\rlap{\text{ per contact}}})$$

(or can also be approximated with a Poisson distribution if $c$ is large and $ \tau$ is small)

The standard SIR model is not deterministic and it does not explicitly sample the binomial distribution in order to compute the results.

Instead, it uses the mean or mode of this distribution $\beta = c\tau$. Which is the most likely outcome. (see also https://en.wikipedia.org/wiki/Thermodynamic_limit)

This approach is correct in the limit, when the infected population is very large. E.g. when $n_I$ people will each infect other people according to a Poisson distribution with parameter $\beta$, the total number of infected people will be distributed according to a Poisson distribution with parameter $n_I \beta$ and the ratio of the standard deviation and mean of that distribution will shrink as $n_I$ is large. So the error becomes relatively small for large populations.

Why isn't the equation something more like:

[I:infected people] * [c:contacts/day] * [tau:likelihood-of-infection/contact] * [P: proportion of contacts with susceptible people left in the population ]

It is like that. Your parameter P, the proportion of susceptible people left, is accounted for by the term $S$

$$I_{t+1} = I_t - \gamma*I_t + \beta*I_t*S_t$$

you will also often see a division by $N$

$$I_{t+1} = I_t - \gamma*I_t + \beta*I_t*\underbrace{\frac{S_t}{N}}_{\llap{\text{proportion of susceptible}}\rlap{\text{ people left}}}$$

This division with $N$ might make the interpretation of parameters more logical, but it is mathamatically not neccesary (it is effectively just a scaling of the parameters).

I'm having a really hard time interpreting and understanding the transmissibility term ($\tau$ in particular,...

It looks to me like $\tau$ is a probability of infection per contact.

The number of transmissions per day per person is then binomially distributed. According to a parameter

$$\text{new infections per time} \sim \text{Binomial}( \underbrace{n=c \cdot I \cdot S}_{\substack{\llap{\text{number of contacts per time }}\rlap{\text{}} \\ \llap{\text{= contacts per infected }}\rlap{\text{per susceptible per time}}\\ \llap{\text{ $\times$ infected}}\rlap{\text{$\times$ susceptible}}}} \, , \, \overbrace{p=\tau}^{\llap{\text{probability of infection}}\rlap{\text{ per contact}}})$$

(or can also be approximated with a Poisson distribution if $c$ is large and $ \tau$ is small)

The standard SIR model is not deterministic and it does not explicitly sample the binomial distribution in order to compute the results.

Instead, it uses the mean or mode of this distribution $\beta = c\tau$. Which is the most likely outcome. (see also https://en.wikipedia.org/wiki/Thermodynamic_limit)

This approach is correct in the limit, when the infected population is very large. E.g. when $n_I$ people will each infect other people according to a Poisson distribution with parameter $\beta$, the total number of infected people will be distributed according to a Poisson distribution with parameter $n_I \beta$ and the ratio of the standard deviation and mean of that distribution will shrink as $n_I$ is large. So the error becomes relatively small for large populations.

Why isn't the equation something more like:

[I:infected people] * [c:contacts/day] * [tau:likelihood-of-infection/contact] * [P: proportion of contacts with susceptible people left in the population ]

It is like that. Your parameter P, the proportion of susceptible people left, is accounted for by the term $S$

$$I_{t+1} = I_t - \gamma*I_t + \beta*I_t*S_t$$

you will also often see a division by $N$

$$I_{t+1} = I_t - \gamma*I_t + \beta*I_t*\underbrace{\frac{S_t}{N}}_{\llap{\text{proportion of susceptible}}\rlap{\text{ people left}}}$$

This division with $N$ might make the interpretation of parameters more logical, but it is mathamatically not neccesary (it is effectively just a scaling of the parameters).

It depends a bit on whether you see the $\beta$ as 'infections per infected per susceptible' or as 'infections per infected'.

I'm having a really hard time interpreting and understanding the transmissibility term ($\tau$ in particular,...

It looks to me like $\tau$ is a probability of infection per contact.

The number of infections per day per person is then binomially distributed.

$$\text{Infections per time per person} \sim \text{Binomial}( \underbrace{n=c}_{\llap{\text{contacts per person }}\rlap{\text{per time}}} \, , \, \overbrace{p=\tau}^{\llap{\text{probability of infection}}\rlap{\text{ per time}}})$$

(or can also be approximated with a Poisson distribution if $c$ is large and $ \tau$ is small)

The standard SIR model is not deterministic and it does not explicitly sample the binomial distribution in order to compute the results.

Instead, it uses the mean or mode of this distribution $\beta = c\tau$. Which is the most likely outcome. (see also https://en.wikipedia.org/wiki/Thermodynamic_limit)

This approach is correct in the limit, when the infected population is very large. E.g. when $n_I$ people will each infect other people according to a Poisson distribution with parameter $\beta$, the total number of infected people will be distributed according to a Poisson distribution with parameter $n_I \beta$ and the ratio of the standard deviation and mean of that distribution will shrink as $n_I$ is large. So the error becomes relatively small for large populations.

Why isn't the equation something more like:

[I:infected people] * [c:contacts/day] * [tau:likelihood-of-infection/contact] * [P: proportion of contacts with susceptible people left in the population ]

It is like that. Your parameter P, the proportion of susceptible people left, is accounted for by the term $S$

$$I_{t+1} = I_t - \gamma*I_t + \beta*I_t*S_t$$

you will also often see a division by $N$

$$I_{t+1} = I_t - \gamma*I_t + \beta*I_t*\underbrace{\frac{S_t}{N}}_{\llap{\text{proportion of susceptible}}\rlap{\text{ people left}}}$$

This division with $N$ might make the interpretation of parameters more logical, but it is mathamatically not neccesary (it is effectively just a scaling of the parameters).

I'm having a really hard time interpreting and understanding the transmissibility term ($\tau$ in particular,...

It looks to me like $\tau$ is a probability of infection per contact.

The number of infections per day per person is then binomially distributed.

$$\text{Infections per time per person} \sim \text{Binomial}( \underbrace{n=c}_{\llap{\text{contacts per person }}\rlap{\text{per time}}} \, , \, \overbrace{p=\tau}^{\llap{\text{probability of infection}}\rlap{\text{ per contact}}})$$

(or can also be approximated with a Poisson distribution if $c$ is large and $ \tau$ is small)

The standard SIR model is not deterministic and it does not explicitly sample the binomial distribution in order to compute the results.

Instead, it uses the mean or mode of this distribution $\beta = c\tau$. Which is the most likely outcome. (see also https://en.wikipedia.org/wiki/Thermodynamic_limit)

This approach is correct in the limit, when the infected population is very large. E.g. when $n_I$ people will each infect other people according to a Poisson distribution with parameter $\beta$, the total number of infected people will be distributed according to a Poisson distribution with parameter $n_I \beta$ and the ratio of the standard deviation and mean of that distribution will shrink as $n_I$ is large. So the error becomes relatively small for large populations.

Why isn't the equation something more like:

[I:infected people] * [c:contacts/day] * [tau:likelihood-of-infection/contact] * [P: proportion of contacts with susceptible people left in the population ]

It is like that. Your parameter P, the proportion of susceptible people left, is accounted for by the term $S$

$$I_{t+1} = I_t - \gamma*I_t + \beta*I_t*S_t$$

you will also often see a division by $N$

$$I_{t+1} = I_t - \gamma*I_t + \beta*I_t*\underbrace{\frac{S_t}{N}}_{\llap{\text{proportion of susceptible}}\rlap{\text{ people left}}}$$

This division with $N$ might make the interpretation of parameters more logical, but it is mathamatically not neccesary (it is effectively just a scaling of the parameters).