Infection delays 3

\[I_{n} = \sum\limits_{{}_{d = 1}}^{{}_{D}}g_{d}I_{n - d}\]

‘Resolution’ refers to the projection onto the characteristic solutions:

\[I_{n} = \sum\limits_{{}_{m = 1}}^{{}_{M}} A_m a_m^n\]

The \(a_m\) are the zeros of the characteristic polynomial (calculated according to the Aberth method), while the \(A_m\) are coefficents that give \(I_0=1\) and zero for the first few negative values. The coefficients were determined using my shallow knowledge of Vandermonde determinants and Cramer’s (and some nice cancellations). [I would also note that \(D = M\).] I have used the resolution to repeat the ‘backward’ projection. The default gives the parameters for the ‘Three-day delay’ example (2) for comparison with the previous graphs. The parameters can be changed (even to the Two-day problem or more complicated ones).

It may be (or not) worth pointing out that the Fibonacci sequence derives from such a structure with \(g_1,g_2\) both equal to 1. If you insert these values in the box and hover a mouse over the first few points one gets 1, 1, 2, 3, 5, 13, and so on. In terms of an infectious disease this would be the result of the infected person birthing a new infected person each day in the first two days of the infection. If you solve the quadratic characteristic equation \(x^2-x-1=0\), you find the solutions \((1\pm\sqrt{5})/2\). The positive solution, \((1+\sqrt{5})/2\approx1.618\), an object also known as the “Golden Ratio”, symbolized as \(\phi\). This value can also be found from the factor button above and hovering over the flat part of the line. As it happens, the negative solution, \((1-\sqrt{5})/2\approx-0.618\), is also related to \(\phi\): in fact, \((1-\sqrt{5})/2=-1/\phi\). Notice that the negative solution has an absolute value less than 1, so it is damped out as the sequence proceeeds forwards in time: it only provides the early oscillations that can be seen in the factor/log graphs. The backward plot shows this factor dominating, giving increasing oscillations as one goes back in time, while the positive solution is damped out. On a more general note, one could envisage these sequences enabling the approximation of roots of a wide range of equations.