In preceding posts we dealt with the susceptible-infectious-susceptible (SIS) model of disease transmission. Under the assumption of a closed population of hosts, the time evolution of infectious hosts is the solution to the differential equation:
\[\begin{equation}\frac{dx}{dt}=(\beta - \gamma) x - \tfrac{\beta}{N}x^2 \end{equation}\]Seen through the lense of neural networks, the right-hand side of this model can be written as: \(\mathfrak{f}_w: x \mapsto w_1 x \oplus w_2 x \oplus (w_3 x\otimes x)\)
where $\mathfrak{f}_w$ is a real-valued function given unknown weights $w=(w_1, w_2, w_3)$.
The operators used in the definition of $\mathfrak{f}_w$ are reminiscent of tensor calculus: the direct product $\oplus$ and the tensor product $\otimes$ allow to do arithmetics on mathematical objects which are sort of high-dimensional generalizations of matrices.
The function \(\mathfrak{f}_w\) should look very familiar. If we set $w_1=\beta, w_2=-\gamma, w_3=-\tfrac{\beta}{N}$, it is exactly our vector field \(f(x)=(\beta - \gamma) x - \tfrac{\beta}{N}x^2\).
But what do we gain by rewriting the vector field in this form? Well, the goal is to build an anolog inference machine.
that measures input voltage $x$ and output voltage $\mathfrak{f}_w(x)$ and where we can modulate the weights $w_1, w_2, w_3$ such that for a certain target voltage can be met.