Early CoVID-19 growth often obeys: N{t}=N/I\exp[+K/o\t], with K/o\=[(ln2)/(t/dbl\)], where t/dbl\ is the pandemic doubling time, prior to society-wide Social Distancing. Previously, we modeled Social Distancing with t/dbl\ as a linear function of time, where N[t]=1exp[+K/A\ t/(1+ gamma/o\ t)] is used here. Additional parameters besides {K/o\,gamma/o\} are needed to better model different rho[t]=dN[t]/dt shapes. Thus, a new Orthogonal Function Model [OFM] is developed here using these orthogonal function series:
N(Z) = sum[m=0,M/F\] g/m\ L/m\(Z) exp[-Z] ,
R(Z) = sum[m=0,M/F\] c/m\ L/m\(Z) exp[-Z] ,
where N(Z) and Z[t] form an implicit N[t]=N(Z[t]) function, giving:
G/o\ = [K/A\ / gamma/o\ ] ,
Z[t] = +[ G/o\ / (1+ gamma/o\t) ] ,
rho[t] = [ gamma/o\ / G/o\ ] (Z^2) R(Z) ,
with L/m\(Z) being the Laguerre Polynomials. At large M/F\ values, nearly arbitrary functions for N[t] and rho[t]=dN[t]/dt can be accommodated. How to determine {K/A\, gamma/o\} and the {g/m\; m=(0,+M/F\)} constants from any given N(Z) dataset is derived, with rho[t] set by:
c/(M/F\ - k)\ = sum[m=0,k] g/m\ .
The bing.com USA CoVID-19 data was analyzed using M/F\=(0,1,2) in the OFM. All results agreed to within about 10 percent, showing model robustness. Averaging over all these predictions gives the following overall estimates for the number of USA CoVID-19 cases at the pandemic end:
<N/max\> = 5,009,677 (+/-) 269,450 (data to 5/3/20), and
<N/max\> = 4,422,803 (+/-) 162,580 (data to 6/7/20),
which compares the pre- and post-early May bing.com revisions. The CoVID-19 pandemic in Italy was examined next. The M/F\=2 limit was inadequate to model the Italy rho[t] pandemic tail. Thus, regions with a quick CoVID-19 pandemic shutoff may have additional Social Distancing factors operating, beyond what can be easily modeled by just progressively lengthening pandemic doubling times (with 13 Figures).
Orthogonal Functions for Evaluating Social Distancing Impact on CoVID-19 Spread
