# SIR model in terms of fluids

Let $$s_i (t)$$ be the fraction of people who are susceptible of getting the disease in community (node) $$i$$ at time $$t$$, $$x_i (t)$$ is the fraction of infected people in community $$i$$ at time $$t$$, and $$r_i (t)$$ is the rate of recovered people (which we do not model). These parameters verify $$x_i (t)+s_i (t)+r_i (t)=1$$, $$t$$ is a discrete time parameter. The system of ODE is the following: $$\frac{d s_{i}}{d t}=-\beta s_{i} x_{i}+\alpha \sum_{j=0}^{N-1} \frac{A_{i j}}{q_{i}}\left(s_{j}-s_{i}\right)$$

$$\frac{d x_{i}}{d t}=\beta s_{i} x_{i}-\mu x_{i}+\alpha \sum_{j=0}^{N-1} \frac{A_{i j}}{q_{i}}\left(x_{j}-x_{i}\right)$$

I understand that the term $$\mu x_{i}$$ correspond to a flux of particles that leaves the state $$x_i$$. What could $$\beta s_{i} x_{i}$$ mean in term of fluid dynamics?. Matrix $$A$$ is the adjacency matrix and $$q_i$$ is the degree of node $$i$$, is it possible to interpret $$\frac{A_{i j}}{q_{i}}\left(s_{j}-s_{i}\right)$$ in terms of fluid dynamics?

Here we have similarly let the implicit third state, "recovered", correspond to the "rest" or empty background. However the first two states are as if we have two different species of particles, $$s$$ and $$x$$. It's like having a mixture of two fluids, with the total density additionally allowed to varying.
Worse, in the epidemiological model individuals convert between the different states. This is the meaning of the terms $$\beta s_i x_i$$: we have individuals converting between states $$x$$ and $$s$$ at some rate $$\beta$$. On the whole the model is more like a model with multiple species of fluid and a chemical reaction.
The layout of the space you are working with here is a graph, which is a more general case than is usually seen for fluid models. Fluid dynamics takes place in 3-dimensional space; if we discretize a model into a cubic lattice, as is often done in simulations, each cell exchanges particles or fluid density with its 6 (for example, one could choose to add diagonal exchanges) spatial neighbors. We can view setup a specific graph where each location is connected to exactly 6 other locations in a regular pattern. You wouldn't write a factor like $$q$$, because it is the same for each location. You could write a matrix like $$A$$ which keeps track of which cells are spatial neighbors and thus interact, but more usually we just write a note that the sum is over nearest-neighbor pairs only.
I regard this more as a combined flow and chemical reaction analogy. $$\beta s_ix_i$$ is analogous to the rate of a 2nd order chemical reaction, representing the rate at which susceptible people become infected. The summation terms represent the flow (interchange) of $$s_i$$ susceptible people between nodes i and j, and the flow (interchange) of infected people between nodes i and j, respectively. The term $$-\mu x_i$$ is also like the rate of a chemical reaction, representing the rate at which infected people are cured and no longer infected (or die).