Understanding beta/tau in the SIR model of Infectious disease for COVID modeling

Inspired by the SARS-CoV-2 pathogen and COVID-19, I am writing an agent-based simulation to allow the modeling of heterogeneous populations (e.g., some folks abide by quarantine, some resist, or some are more likely to die, some aren't).

Naturally, I want to be able to run a baseline model that approximates the traditional understanding of infection trajectories, and so I built in a function to set initial parameters based on the $$R_0$$ or doubling time in the SIR model. I'm having a really hard time interpreting and understanding the transmissibility term ($$\tau$$ in particular, which is a component of $$\beta$$).

I can set parameters on my current version to approximate the same peak and timeframe of the SIR model, but the actual trajectories of infected/susceptible/recovered during the infection are very different – so I need help better understanding this term.

SIR Equation for New Infections

Terms rearranged from https://code-for-philly.gitbook.io/chime/what-is-chime/sir-modeling

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

Interpretation

(# infections tomorrow) = (#infections today) - (# infections recovered today) + (new infections today)

And New Infections = $$\beta$$ * (# infections today) (# Susceptible today)

And $$\beta = c * \tau$$; or contacts x 'transmissibility'

c is easy enough: number of candidate contacts per day in which the infection could be spread.

Different sources describe tau and beta differently: infectivity, virulence, transmissibility…  $$\tau$$ is a characteristic of the pathogen that appears to be constant in the models I've looked at.  I can use existing formulas to calculate it from the doubling time and I can calculate it from the $$R_0$$ based on existing equations, but being able to calculate a value doesn't really mean anything.

Subterm interpretation

I * c = the total number of contacts-that-can-transmit-the-infection today. Makes sense, and c is a attribute of the individual and can be modified if, say, people socially distance.

tau: I want tau to be "the average likelihood of spread during an unprotected, meaningful social interaction."  That meets the other assumptions: it is a relatively constant attribute of the pathogen, or at least the pathogen given the current timeframe. I was considering building into the agent-based model the ability for tau to decrease during the summer as some estimate due to higher humidity. Regardless, tau changes universally for the entire susceptible population, it's still an attribute of the infection * context, not related to human behavior like c

So then the infection term (beta * I * S) equals (c*I) * tau * S, or [infected contacts today] * [spread per contact] * [raw susceptible population]

That only seems to make sense if there is exactly one infected individual and the entire population is susceptible. Right, because this basically needs to break down to the percentage of the susceptible population that gets newly infected TODAY due to the current infection.

Consider a point on the upswing and downswing of the infection in which the same number of people are infected (I) and all the other assumptions are true:

Infection term during the upswing = c * tau * I * S_up Infection term during the downswing = c * tau * I * S_down

The number of infected contacts (c) for the day are the same because the (unknowingly) infected people are doing all the same things.

Tau is constant because it's a characteristic of the pathogen or infection…

S is the only thing that differs. And it's not just that the susceptible population is different, but the chance that you are interacting with a susceptible individual is much less likely because everyone is already infected or already recovered. And this would sensibly change the trajectory of the infection over time.

It seems like in the (c * I) * tau * S equation, instead, the interpretation of tau is "the likelihood that any susceptible member of the population is infected by any contact that the infected person has," regardless of whether infected people actually are in contact with susceptible people. This seems to have a weird assumption that anyone susceptible is equally likely to become infected, even if other criteria make that unlikely. In practical terms, it includes me kissing the person behind the counter who already recovered and then kisses their partner who was never infected, who then gets it. And the implications of this are significant. Sure, fewer people get infected on the downswing because S_down < S_up, but it should be many fewer people because S_down < S_up AND a larger proportion of the c contacts are not candidate infections because they happen between people who cannot be infected.

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 ]    What am I missing? I can't effectively set parameters in my agent-based simulation to approximate the general SIR equation until I can better understand this parameter.

• "Tau is constant because it is a characteristic of the disease." Leaps out as a simplifying assumption, but not as reality: first, the pathogen causing a given disease varies in its infectiousness based on (1) environmental conditions, (2) host characteristics, (3) ecological relationships, (4) individual virus characteristics (e.g., genetic variation); second "the" disease is a generalization applied to a constellation of signs and symptoms, in the case of COVID-19 none of which neccesarily result from infection with SARS-CoV-2, many of which are common, many of which are not. Commented May 1, 2020 at 15:40
• @Alexis, You're totally right that I'm trying to encapsulate pathogen characteristics not "COVID-19". I'll update the post accordingly. But otherwise, I did suggest in the original post that Tau incorporates the current context/timeframe and likely changes. So, it does account for virus characteristics and at least the macro environmental and ecological relationships (though this may depend on your unit of analysis). For the case at hand, the important separation is that [B] is c * T, where [c] is a personcontext characteristic and tau is a pathogencontext characteristic. Commented May 1, 2020 at 16:58

1 Answer

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'.

• Thank you - this is great and very helpful. And I understand that at the limit, the division of N may be insignificant - but for comparing models of the same N, it does have a very significant effect on the overall characteristics of the result over time.... It's going to significantly "widen" the distribution / slow the acceleration, right? (sorry about the terminology - you might call me an educational sociologist, programmer, or data scientist, but I'm definitely not a mathematician).... Commented May 1, 2020 at 17:19
• And I guess I am thinking of $\tau$ as "infections per infected" - based on my description, $c * \tau * I_t$ is "[raw possible] infections per infected," but doesn't take into account the percent of the susceptible population at all. So it does seem like I need to include $S_t/N$ to correct my equation, right? Perhaps it's more appropriate for me to think of it as $I_t / N * \beta * S_t = I_t / N * \tau * c * S_t$, or [the proportion of the population that is currently infected] * [infectible contacts per day] * [infections / infectible contact] * [remaining susceptible population] Commented May 1, 2020 at 17:21
• You can use a proportion but you don't need to. It depends on how you define $\beta$ as 'new infections per infected per susceptible per day' $$new = \beta \cdot I \cdot S$$ or 'new infections per infected per day' (in fully susceptible population) $$new = \beta \cdot I \cdot \frac{S}{N}$$ or 'new infections per susceptible per day' (in fully infected population) $$new = \beta \cdot S \cdot \frac{I}{N}$$ in the first case the $\beta$ has a different dimension and different value (it is scaled by $N$). The third case is not so common. The first case might be most useful if $N$ isn't constant. Commented May 1, 2020 at 19:06
• Thanks so much again! I've now been able to develop both the SIR model using the traditional equation and adapting the parameters to the agent-based model to arrive at similar outcomes - so I can now add in the additional features that led me down this route to begin with! Thanks again! Commented May 2, 2020 at 1:53