Chapter 10 of Dynamical Processes on Complex Networks contains a model of information spreading. This model is strongly based on a SIR model of epidemics. In this model, there were three kinds of people: ignorant, spreaders, and stiflers. $\lambda$ is the rate of change at which ignorant transition to spreaders, and $\alpha$ is the transition rate for the other two categories. The interactions are:
$$I + S \xrightarrow{\lambda} 2S$$ $$S + R \xrightarrow{\alpha} 2R$$ $$S + S \xrightarrow{\alpha} R + S$$
Next, they set up the following evolution equations:
$$\frac{di}{dt} = -\lambda \langle k \rangle i(t) s(t)$$
$$\frac{ds}{dt} = + \lambda \langle k \rangle i(t) s(t) - \alpha \langle k \rangle s(t)[s(t) + r(t)]$$
$$\frac{dr}{dt} = \alpha \langle k \rangle s(t)[s(t) + r(t)]$$
So, I understand that these equations are saying that $s$ is gaining all of the people that are switching from $i$ and losing all of the people that are switching to $r$. I also understand how $I$, $S$, and $R$ relate to each other in the first set of equations.
I'm a little confused about how the second set of equations were derived from the first, though. In particular, all of the terms in the first set of equations involve addition, but there is almost entirely multiplication, and I don't understand where the 2 goes.
Can someone explain what I'm missing?
You need to read the first set of equations like a chemical reaction system. The second set then is the corresponding dynamic of the densities. Thus $i(t)s(t)$ is a measure for the probability of an encounter of individuals of $I$ and $S$, $λ$ provides the rate of conversion. The result is the removal of one unit of $I$ that is converted and added to $S$.
If the reaction were $I+2S\to 3S$, then the reaction term would be $λi(t)s(t)^2$, etc.