# Interpreting the reciprocal of $R_0$

$$R_0$$ is the average number of secondary cases arising from a single infectious individual in a fully susceptible population.

In many of the compartmental models for epidemiology, the parameter $$\frac{1}{R_0}$$ appears frequently.

In particular, in an SIS model, the non-trivial steady state is $$(S^*, I^*) = \left(\frac{1}{R_0}, 1 - \frac{1}{R_0} \right)$$.

Is there any intuitive interpretation one can give to the reproductive ratio's reciprocal?

I'm thinking some sort of average rate of secondary cases, but this doesn't feel right since $$S^*$$ is a number.

It's funny: $$R_0$$ doesn't actually follow as nicely as $$1/R_0$$ when focusing on the infected class.

In a SIS model: $$\frac{\mathrm{d}S}{\mathrm{d}t} = -\beta SI + \alpha I \\ \frac{\mathrm{d}I}{\mathrm{d}t} = \beta SI - \alpha I$$

take finding the steady state of $$I$$. Whether a disease is able to invade is dependent on whether or not the steady state is greater than 0, which means that $$I(S\beta - \alpha)$$ must be positive. That is, $$S\beta$$ must be greater than $$\alpha$$, and $$I$$ isn't important as to whether or not the infected class changes (unless infecteds are extinct at 0). This means that $$S$$ must be greater than $$\alpha / \beta$$ for the disease to invade. Since disease prevalence is $$I/(S+I)$$ and there is usually a negligible-sized initial prevalence (i.e., $$S(0) \approx 1$$), we often care about $$\alpha / \beta$$, which is known as the relative removal rate. This is intuitive because $$\alpha / \beta$$ is the ratio of the rate of loss to the rate of gain of the infected class. If the rate of loss is greater than the the rate of gain, $$\alpha / \beta$$ will be greater than 1 and the disease will go extinct.

The reciprocal of the relative removal rate, $$\alpha / \beta$$, is $$\beta / \alpha$$, which is known as $$R_0$$.

1/R0 is the threshold fraction. If fraction of population vulnerable to particular infection is more than 1/R0, only then infection can spread further. And if it is less than 1/R0 then infection can not progress and eventually goes away

And, 1-1/R0 is the fraction of population which requires vaccination so that we can have herd immunity.

• Thanks for the response - however everything you have said is an immediate corollary of the fact that it is the steady state, so I was aware of this already. I was hoping for a more intuitive explanation for how one could "see" that $\frac{1}{R_0}$ is an obvious threshold from its definition? For example, the reciprocal of the rate of recovery is the average period of infection obviously, but I'm not so sure what abut the definition of $R_0$ means that $\frac{1}{R_0}$ is so important or lending to be a threshold? – PhysicsMathsLove May 20 at 20:37