Let $\Omega\subset \mathbb{R}^{d}$ ($d\geq 1$) be a bounded domain with a smooth boundary $\partial\Omega$. Let $S, I$ be dependent variables and $x, t$ their independent variables. Additionally, $q:\mathbb{R}\rightarrow \mathbb{R}$ a function taking positive bounded values.
What could be a mathematical interpretation of the following Partial Differential Equation (which actually is part of a particular SIR model)?
$\dfrac{\partial S}{\partial t}-\triangle S = aq(t)-aS+bI-\gamma\dfrac{SI}{N} \text{ in } \Omega\times(t_{0}, +\infty)$
I hold an interpretation for all the mathematical factors (symbols) that make up the equation, but I cannot interpret holistically (and mathematically) the equation (which may imply that I understand it). And it would be awesome if someone could recommend me a book about interpretations of equations.
P.S.: This is what I've got so far:
$S(x,t):$ number of susceptible (not infected but able to be it) individuals at time $t$ and position $x$.
$\dfrac{\partial S}{\partial t}:$ represents the growth rate of S over time.
$\triangle S:$ measure of the state of being concave of the graph of S(x,t). Informally, difference between the average value of $S$ in the neighborhood of a point, and its value at that point.
$a:$ the average number of deaths caused by the disease per person per unit of time.
$b:$ the average excess deaths caused by the disease per infected person per unit of time.
$\gamma:$ average number of contacts between susceptible and infected people.
$q(t):$ seasonal variation of transmission.
Thank for your time.