Momentum managing epidemic spread and Bessel functions

Abstract

Starting with the power law for the total number of detected infections, we propose differential equations describing the effect of momentum epidemic management. Our 2-phase formula matches very well the curves of the total numbers of the Covid-19 infection in many countries; the first phase is described by Bessel functions. It provides projections for the saturation, assuming that the management is steady. We discuss Austria, Brazil, Germany, Japan, India, Israel, Italy, the Netherlands, Sweden, Switzerland, UK, and the USA, including some analysis of the second waves.

Keywords: 33C10; 92B05; Bessel functions; Epidemic psychology; Epidemic spread.

Fig. 1

$u\left(t\right)=1.7t^{(c+1)/2}{J}_{(c-1)/2}\left(\sqrt{a}t\right)$for $c=2.2,a=0.2$.

Fig. 2

Covid-19 in the United States.

Fig. 3

$u\left(t\right)=2.2t^{(c+1)/2}{J}_{(c-1)/2}\left(\sqrt{a}t\right)$for $c=2.4,a=0.2$.

Fig. 4

$u\left(t\right)=3t^{(c+1)/2}{J}_{(c-1)/2}\left(\sqrt{a}t\right)$for $c=2.4,a=1/3$.

Fig. 5

Covid-19 in Switzerland.

Fig. 6

Two types of momentum management.

Fig. 7

USA: $c=2.2,a=0.2,C=1.7;b=0.35,D=1.9$.

Fig. 8

UK: $c=2.4,a=0.2,C=2.2;b=0.35,D=2.5$.

Fig. 9

$w\left(t\right)=2.1t^{(c+1)/2}{J}_{c-1}\left(2\sqrt{bt}\right);b=0.3,c=2.4$.

Fig. 10

$u\left(t\right)=2.2t^{(c+1)/2}{J}_{(c-1)/2}\left(\sqrt{a}t\right),$$c=2.6,a=0.3$.

Fig. 11

Brazil: $c=4.47,a=0.04,C=0.65$.

Fig. 12

$India:3/20-8/23,c=5.2,a=0.035,C=0.1$.

Fig. 13

Israel: $c=2.6,a=0.3,d=0.6$.

Fig. 14

Italy: $c=2.6,a=0.2,d=0.5$.

Fig. 15

Germany: $c=2.6,a=0.35,d=0.56$.

Fig. 16

Germany: $u=u_1=1.7t^{1.8}{J}_{0.8}\left(t\sqrt{0.2}\right)$.

Fig. 17

Japan: $u=1.5t^{1.8}(J_{0.8}-0.4J_{-0.8})\left(t\sqrt{0.3}\right)$.

Fig. 18

The Netherlands: $u=0.5t^{1.7}{J}_{0.7}\left(t\sqrt{0.2}\right)$.

Fig. 19

USA, the sum of the curves for individual states.

Fig. 20

The forecast for the USA as of 6/21.

Fig. 21

The forecast for Europe as of 7/14.

Fig. 22

The forecast for Europe as of 8/02.

Fig. 23

Second wave in Israel: 6/13-8/23.

Fig. 24

Second wave in the USA: 6/15-8/23.