discrepancies between diff.equ. for SIR model of infection

I'm reading this book as background for thesis research, and I'm confused by the explanation of the SIR model given in the first chapter.

On p. 7, I'm given $$\frac{dS}{dt} = −\beta SI$$ and $$\frac{dI}{dt} = \beta SI - \gamma I$$

where S is the number of susceptibles and I is the number of infectious people. $S + I + R = N$ is the total population size. It says "Here, the transmission rate (per capita) is β and the recovery rate is γ (so the mean infectious period is 1/γ)."

Taking the ratio of the above two, they give $$\frac{dI}{dS} = -1 + \frac{1}{\mathcal{R}_0S}$$ where $\mathcal{R}_0S = \frac{\beta N}{\gamma}$, but I get $$\frac{dI}{dS} = -1 + \frac{N}{\mathcal{R}_0S}$$ which is what is in a later chapter (by a different author). Any ideas what's going on?

• In this model, S+I is not constant hence I hope they did not write S+I = N with N constant. – Did Mar 31 '15 at 16:35
• Sorry, that should have been S+I+R=N. My typo! I've corrected it now. Thank you! – MissMonicaE Apr 2 '15 at 14:56

I think it's actually a matter of how the author defines $\mathcal R_0$. I have always seen it defined as a dimensionless quantity (does not depend on population size), given as the product of the transmission rate, $\beta$, and the mean infectious period, $1/\gamma$. This is so that the basic reproduction number can be compared between epidemics having different population sizes (that, and dimensionless quantities are nice to work with). So, taking $\mathcal R_0 = \beta/\gamma$, I get
$$\frac{\mathrm{d}I}{\mathrm{d}S} = \frac{\frac{\mathrm{d}I}{\mathrm{d}t}}{\frac{\mathrm{d}S}{\mathrm{d}t}} = \frac{\beta S I - \gamma I}{-\beta SI} = -1 + \frac{\gamma}{\beta S} = -1 + \frac{1}{\mathcal R_0 S}$$
Note: I have typically used $\mathcal{R}_0$ instead of $\mathbb R_0$ to represent the basic reproduction number to avoid any possible confusion with the common notation $\mathbb R$ representing the real numbers.
• But isn't $\mathcal{R}_0$ supposed to be the number of secondary cases produced by an index case in an otherwise completely susceptible population? So it would have to depend on the population size, right? – MissMonicaE Apr 2 '15 at 17:37