What does $d \tau s(\tau)$ refer to in this equation and how do I integrate it?

Question

Chapter 10 of Dynamical Processes on a Complex Network discusses various aspects of rumor spreading (based on the SIR epidemic model). It contains the following equation for the density of those who remain ignorant of a rumor:

$$i(t) = i(0) exp [- \lambda \langle k \rangle \int ^t_0 d \tau s(\tau )]$$

By combining this with several other equations, they end up with

$$\int_0^t dt \frac{dr}{dt} = \alpha \langle k \rangle \int_0^t d \tau s(\tau) + \frac{\alpha}{\lambda} \int_0^t dt \frac{di}{dt}$$

I suspect I'm missing something obvious, but I'm confused about what the $d \tau s(\tau)$ and $\int_0^t dt \frac{dr}{dt}$ terms refers to and how I can integrate them.

Answer 1

$\int_0^t\mathrm{d}\tau\,s(\tau)$ is just an alternative notation for $\int_0^t s(\tau)\,\mathrm{d}\tau$. There are mainly two reasons why you see this notation from applied mathematics:

1. Usually in applied mathematics the integrand is a complicated function, which doesn't fit in a single line (especially nowadays with two-column layout of journals meaning even less space for displayed equations). So you usually need to split the integral in multiple lines and it looks weird to have $\int$ in one line and $\mathrm{d}x$ a few lines later.
2. With multiple integrals, it is easier to tell the integration range corresponding to which variables, e.g., $$ \int_0^{\pi/2}\int_0^\pi\int_0^1 f(r,\theta,\phi)\,\mathrm{d}r\,\mathrm{d}\phi\,\mathrm{d}\theta $$ versus $$ \int_0^{\pi/2}\mathrm{d}\theta\int_0^\pi\mathrm{d}\phi\int_0^1\mathrm{d}r\, f(r,\theta,\phi) $$