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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?

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

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  • $\begingroup$ 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? $\endgroup$ – MissMonicaE Apr 2 '15 at 17:37

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