analysis of SIR model

Question

Currently doing an analysis of the standard SIR model with vital dynamics and trying to re-express the equations for proportions in dimensionless form using dimensionless time coordinate and the dimensionless parameters, however, do not quite understand what biological meanings dimensionless time coordinate τ = (γ + µ)t , and dimensionless parameter ε = µ/(γ + µ) have in relation to the standards SIR model with vital dynamics? Any help would be highly appreciated! Thanks

Answer 1

I'm using the info from your recently "on hold" post to answer this question because it is missing some context.

By setting $\tau = (\gamma + \mu)t$, where $\gamma$ and $\mu$ are the recover and the death rates in your SIR equation respectively, you are scaling the time to a biological time that focuses on the recovering effect of the SIR model with respect to demography (birth/death).

I know you also asked for $R_0 = \frac{\beta}{\gamma + \mu}$. This is your basic reproduction number. You can obtain this via many ways - one of which is non-dimensionalization according to the time scale given.

Finally $\epsilon = \frac{\mu}{\gamma + \mu}$. This is the trickiest, because it doesn't have a clear biological meaning. Instead, it is useful because you are trying to scale the demographic term ($\mu$) with respect to the total recovery and demographic term. In the context of diseases that are short term, then $\mu$ is very small - because the disease is short-term so not many people are born or die during this period due to natural death. This means $\epsilon$ would be very small when the disease is short-term. On the other hand, if the disease is long term, then $\epsilon$ would be larger, however, still smaller than $1$. It is a useful scaling to show the effect of short- vs. long- terms diseases.