Are the parameters $\beta$ and $\gamma$ in (Susceptible, Infected, Recovered) SIR model probability number? Can they larger than 1.0?

Question

I am learning SIR model from this blog post. We also had a very good discussion in CV post

The key parameters of the model are $\beta$ and $\gamma$, people usually describe them as the "infection rate" and "recover rate".

When I do some agent based simulation, I also used some parameter to describe the probability of one person being infected or recovered after unit time.

My question is, are they the same thing? i.e., $\beta$, and $\gamma$ needs to be in the range of 0 to 1.0?,

If yes, why I am seeing some one (fitting with covid19 data), and got two parameters ~20? what does that mean?

If no, why the two links above use BFGS with bounds 0 to 1?

Answer 1

The parameters $\beta$ and $\gamma$ of the standard SIR model in the blog post \begin{align} {\mathrm d S \over \mathrm d t} &= -\beta {S I }\\[1.5ex] {\mathrm d I \over \mathrm d t} &= \beta {S I} - \gamma I \\[1.5ex] {\mathrm d R \over \mathrm d t} &= \gamma I \\ \end{align} are rates in a continuous time model, which means that they can take on any positive value, including values greater than 1. For example $\beta SI$ is the rate at which susceptibles are converted to infecteds i.e. the number of 'conversions' per unit time. The parameter $\beta$ is multiplied by $S$ and $I$, both of which could be quite large, so it's not surprising that the $\beta$ estimates are very small in the situation of the blog post. So I'm guessing that having bounds of 0 and 1 is merely a convenient default setting when you're expecting small but positive parameter estimates.

For discrete time models you can re-interpret these rates as probabilities. Suppose your discrete time model has time step $\delta t$. Then, at each time step, and for each susceptible, convert it to an infected with probability $1-e^{-\beta I \delta t}$, which is approximately $\beta I \delta t$ for small enough $\delta t$. For small enough $\delta t$ this would give a close approximation to the stochastic SIR model with rates as in the deterministic model specified above.

Answer 2

About the blog

The linked blog uses the equation

$$I^\prime = \beta SI - \gamma I$$

Instead of

$$I^\prime = \beta \frac{S}{N} I - \gamma I$$

That is why their results are so strange. Their values are off by a factor $N$.

But other things may be causes as well since compartment models can not be well used for covid-19 or at least the interpretation of the parameters won't make sense. For the $\gamma$ value they reach the lower limit which means that they actually did not reach convergence to the optimal solution.

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About the limits

The beta and gamma are not non-dimensional parameters. They will depend on the time scale. So you can have values above 1.

This limitation between 0 and 1 is technically not necessary.

But, possibly this limitation stems from a model which is taking discrete time steps and then a value $\gamma >1$ could mean that more than 100% of infected people recover, which makes physically no sense.

Also for a differential equation, with infinitesimally small time steps a value $\gamma > 1$ is often strange. This is when time is measured in days because then values $>1$ mean that people are on average cured within 1 day. The value for beta can still be easily above 1 though. Especially when you consider an agent based model where each agent has a different effective beta, superspreaders may cause rates of infection above 1 per day.

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About the agent based model

Now you are not explicitly modelling beta and gamma, but instead transmission probability per contact or per agent. The average rate of infections and recovery can be related to the beta and gamma values.

Obviously parameters can be above 1. E.g. you could model the number of contacts an agent has each time unit as a Poisson distributed variable and the rate parameter could be above 1.