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.
Let us think of our differential equation model as a simple, serial electrical circuit.
You can think of combining resistors, capacitators and switches in your electric circuit board to form a electric network. In the same way as resistors in your electric circuit control the electric current, every operation between two elementary functions carries a weight. One can even switch off entire parts the network, or keep the weights at a very low level. Like capacitators that can store and release electric energy more or less quickly, the elements of neural networks can be tuned to smooth the outgoing signal. In machine learning this is called hyperparameter tuning. The machine learning task is to infer the weights such that the electric energy that you put into your circuit reproduces the flinkering LED as the output of your particular circuit. 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.