exponential Times
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Accumulator

In the Covid-19 reports there are usually figures for “new” and “total” cases. Up to here, we have focused on the “new” aspect. At least in the distinct eigenvalue case, the derivation of the “total” result is relatively straightforward. It depends on knowledge of the behaviour of “geometric series”.*
We want to sum such series from “Day Zero", when “Patient Zero” was infected. The sum formula of geometric series is well known.*
We have the daily infections, “new cases”, in the form of a sum of characteristic solutions, \(I_n=\sum\limits_{m=0}^{M}A_m a_m^n\). We note that each component of the sum is separately geometric.
The total infections formula, “total cases”, is \(T_n=\sum\limits_{k=0}^n I_k=\sum\limits_{k=0}^n \sum\limits_{m=0}^{M}A_m a_m^k\). We can reverse the order of summation and apply the geometric series sum formula: \[T_n= \sum\limits_{m=0}^{M}\left[\sum\limits_{k=0}^nA_m a_m^k\right]=\sum\limits_{m=0}^{M}A_m \frac{a_m^{n+1}-1}{a_m-1}\]
If the maximum absolute value of some of the eigenvalues is greater than 1, the behaviour will eventually be dominated by the largest of these, \(a_{m^*}\). \[T_n \sim A_{m^*} \frac{a_{m^*}^{n+1}-1}{a_{m^*}-1} \sim \frac{A_{m^*}a_{m^*}}{a_{m^*}-1}a_{m^*}^n\]
At the same time, we have \(I_n \sim A_{m^*}a_{m^*}^n\). These asymptotic relations hold as \(n\rightarrow\infty\). The asymptotic regime kicks in when \(a_{m^*}^{n+1}\gg1\).
Also as \(n\rightarrow\infty\), \[T_n\rightarrow\frac{a_{m^*}}{a_{m^*}-1}I_n\]
I indicated that such a behaviour is not unexpected when I looked at Covid-19 reports in my articles on the SIR model.
The case where all the eigenvalues are different is somewhat special. Then all the \(r^{n+1}\rightarrow0\) as \(n\rightarrow\infty\). We are then left with a constant: \[I_n\rightarrow\sum\limits_{m=0}^{M} \frac{-A_m}{a_m-1}=\sum\limits_{m=0}^{M} \frac{A_m}{1-a_m}\]
This constant is positive, if all the \(A_m\) are positive.