How do you find these equations that explain a infectious disease? (SIR model)

Question

I have the following model for a infectious disease spread, but can't figure out why they chose the equations they chose. (I know it is closely related to the standard SIR model)

Let $G$ be a network with $N$ nodes.

Let $S_k(t), I_k(t), R_k(t)$ be the proportion of the population with $k$ connections that is respectively susceptible, infected and recovered by the virus at the time t.

Let $\alpha$ be the probability that a susceptible node will be infected per unit time if it is connected to one or more infected nodes.

Let $\beta$ be the probability that we detect the virus and remove it.

Let$\theta (t)$ the proportion of edges in G that connect to infected nodes at time t.

Then we model the problem by

$$\frac{dS_k(t)}{dt} = -\alpha k S_k(t)\theta(t)$$ $$\frac{dI_k(t)}{dt} = -\beta I_k(t) + \alpha k S_k(t)\theta(t) $$ $$\frac{dR_k(t)}{dt}= \beta I_k(t)$$

How do you get this equations?

I mean, why does it make sense to have (for instance) the equation $$\frac{dS_k(t)}{dt} = -\alpha k S_k(t)\theta(t)$$ such that it models the susceptible fraction of the population?

In other words, how can I interpret this equations (geometrically, a priori)?

Answer 1

Actually it's not quite right.

For each connection of a susceptible and an infective, the rate at which infection occurs (on the average) is $\alpha$. When there are $S_k$ susceptibles having $k$ connections, and fraction $\theta$ of all connections of susceptibles are to infectives, that makes $k S_k \theta$ connections from susceptibles to infectives, the total rate is $\alpha k S_k \theta$ infections per unit time. Each of those infections decreases $S_k$ by one and increases $I_k$ by one, so that's where they get the $$ \dfrac{d S_k}{dt} = - \alpha k S_k(t) \theta(t)$$ and also the term $+\alpha k S_k(t) \theta(t)$ in $dI_k/dt$.

The part that's wrong is that $\theta(t)$ is the fraction of all connections in the graph that are to infectives, not the fraction of connections of susceptibles. These are not likely to be the same. For example, we might imagine a scenario in which there are two distinct subpopulations which are not connected to each other at all, and you start out with some infectives in subpopulation 1 but none in subpopulation 2. After a while, nearly everybody in subpopulation 1 is infective, but everyone in subpopulation 2 is still susceptible and will remain so, despite that fact that $\theta$ (which measures connections to susceptibles in the population as a whole) is near $1/2$.

Answer 2

The first one basically says that the number of susceptible people decreases because some of them get sick. The factor $k$ says that the chance somebody gets infected is proportional to the number of connections they have. The factor $\alpha$ is the chance somebody gets infected from one connection. The factor $\theta(t)$ is the proportion of somebody's connections that are sick, so the number of sick people that person is exposed to is $k\theta(t)$

The decrease in susceptible people matches the second term in the equation for infected people. The first term in the equation for infected people is the number we cure. That decrease is matched by the increase in recovered people.