$$\frac{d{s}_j\left(t\right)}{dt}=-{\beta }_j{s}_j\left(t\right)i_j\left(t\right)$$ (2a)

$$\frac{d{i}_j\left(t\right)}{dt}={\beta }_j{s}_j\left(t\right)i_j\left(t\right)-{\gamma }_j{i}_j\left(t\right)$$ (2b)

$$\frac{d{r}_j\left(t\right)}{dt}={\gamma }_j{i}_j\left(t\right)$$ (2c)