Infection delays
Introduction
In the current covid-19 pandemic, people are interested in projecting forward in time to assess the efficacy of various techniques to control, or even better eliminate, the infection. The simplest models tend to be dominated by exponential behaviour, where the infected population tomorrow is related to today's infection by (roughly) a constant factor. If the factor is greater than 1, the infection grows; less than 1, the infection is fading, and happiness breaks out across the land. We come out of our prison blocks, blinking and shading our eyes from the sun.
Of course, the infection numbers tomorrow aren't simply related to what is happening today: in the case of covid-19 the infection takes up to two weeks to manifest, if at all in some cases. How come a simple constant factor works?
A reasonably simple model for infection delay (i.e. what has crossed what passes for my mind) would have today's newly infected population to be a function of the newly infected populations on previous days. To make life extra simple, we also use a linear function — a sum of the previous populations multiplied by various constant factors:
\[I_{0} = g_{1}I_{- 1} + g_{2}I_{- 2} + g_{3}I_{- 3} + \ldots\]
Using fancy maths notation, with the Greek capital sigma character (Σ) indicating a sum between limits, with D being the length of time the infection lasts in the average person (or at least stops creating new infections):
\[I_{0} = \sum\limits_{{}_{d = 1}}^{{}_{D}}g_{d}I_{- d}\]
Tomorrow's new infections would then be expected to be:
\[I_{1} = g_{1}I_{0} + g_{2}I_{- 1} + g_{3}I_{- 2} + \ldots = \sum\limits_{{}_{d = 1}}^{{}_{D}}g_{d}I_{- d + 1}\]
And on day n :
\[I_{n} = g_{1}I_{n - 1} + g_{2}I_{n - 2} + g_{3}I_{n - 3} + \ldots = \sum\limits_{{}_{d = 1}}^{{}_{D}}g_{d}I_{n - d}\]
There are known to be a number of (characteristic) solutions of these equations that have a simple power form:
\[I_{n} = a^{n}\]
This only works for specific values of a . To find these values we pop the proposed solution into one of the equations:
\[a^{n} = g_{1}a^{n - 1} + g_{2}a^{n - 2} + g_{3}a^{n - 3} + \ldots + g_{D}a^{n - D}\]
If we divide through by \(a^{n - D}\):
\[a^{D} = g_{1}a^{D - 1} + g_{2}a^{D - 2} + g_{3}a^{D - 3} + \ldots + g_{D}\]
This is a D -order polynomial. Note that n has dropped out, since it is not really relevant (and shouldn't be, that's the point) to the value of a . In theory, this equation can have up to D solutions. Allowing complex number solutions, there are exactly D solutions, possibly including multiple roots.