where
I
I
N pop
; S
S
N pop
;
R
R
N pop
ð7:10Þ
System of differential Eqs. (7.7)–(7.9) contains two unknown parameters a and
q, but it would be wrong to assume that its solution in relative quantities depends
only on these two parameters, since initial conditions (4.6) contain an unknown
ratio
N ¼ N=N pop . Really, Eq. (4.6) can be rewritten as follows:
Iðt
Ã
1 Þ ¼ 1=N pop ;
Rðt
Ã
1 Þ ¼ 0; Sðt
Ã
1 Þ ¼
N À 1=N pop
ð7:11Þ
In particular, this ratio influences the value S 1 S 1 =N pop , which can be defined
form the nonlinear equation
S 1 ¼
N À
1
N pop
e
S1 À
N
m ; m ¼
q
a
ð7:12Þ
following from (4.9). After solving Eq. (7.12), the final size of the epidemic
V 1 can
be calculated from
V 1 ¼
N À S 1
ð7:13Þ
Equation (7.13) follows from (4.5). Nonlinearity of Eq. (7.12) leads to complicated nonlinear approximations (7.5) or (7.6).
Another important epidemic characteristic—its duration must be calculated from
nonlinear relationship:
T d ¼ À
2
a
Z m
NÀ1=N pop
d
U
U m ln
U À
U þ
N À m ln
N À 1=N pop
À
Á
Â
Ã
ð7:14Þ
which follows from (4.25). Equations (7.12)–(7.14) demonstrate that correct estimations of the epidemic parameters can be done only with the use of four unknown
independent parameters N, m, a, t
Ã
1 (for system (4.1)–(4.3)) or
N, m, a, t
Ã
1 (for system
(7.7)–(7.9)). Dependences (7.5) and (7.6) can be treated as some approximations
which allow us to estimate the final sizes of the first epidemic waves without
identification of the optimal values of these parameters as was done in Chaps. 5 and
6. The accuracy of these approximations will be checked later in this chapter.
Figure 7.2 demonstrates the absolute values of parameters taken from Table 7.1
versus values of N. The values a are shown by “stars” and V 1 —by “circles”; data
points for China and South Korea are represented by “triangles”. It is seen that with
increasing N, final size of the first wave V 1 increases and the parameter a
decreases. In the logarithmic scale, both dependences are very close to linear (including the data points for China and South Korea).
7 Comparison of the First Waves of the COVID-19 …
93
I
I
N pop
; S
S
N pop
;
R
R
N pop
ð7:10Þ
System of differential Eqs. (7.7)–(7.9) contains two unknown parameters a and
q, but it would be wrong to assume that its solution in relative quantities depends
only on these two parameters, since initial conditions (4.6) contain an unknown
ratio
N ¼ N=N pop . Really, Eq. (4.6) can be rewritten as follows:
Iðt
Ã
1 Þ ¼ 1=N pop ;
Rðt
Ã
1 Þ ¼ 0; Sðt
Ã
1 Þ ¼
N À 1=N pop
ð7:11Þ
In particular, this ratio influences the value S 1 S 1 =N pop , which can be defined
form the nonlinear equation
S 1 ¼
N À
1
N pop
e
S1 À
N
m ; m ¼
q
a
ð7:12Þ
following from (4.9). After solving Eq. (7.12), the final size of the epidemic
V 1 can
be calculated from
V 1 ¼
N À S 1
ð7:13Þ
Equation (7.13) follows from (4.5). Nonlinearity of Eq. (7.12) leads to complicated nonlinear approximations (7.5) or (7.6).
Another important epidemic characteristic—its duration must be calculated from
nonlinear relationship:
T d ¼ À
2
a
Z m
NÀ1=N pop
d
U
U m ln
U À
U þ
N À m ln
N À 1=N pop
À
Á
Â
Ã
ð7:14Þ
which follows from (4.25). Equations (7.12)–(7.14) demonstrate that correct estimations of the epidemic parameters can be done only with the use of four unknown
independent parameters N, m, a, t
Ã
1 (for system (4.1)–(4.3)) or
N, m, a, t
Ã
1 (for system
(7.7)–(7.9)). Dependences (7.5) and (7.6) can be treated as some approximations
which allow us to estimate the final sizes of the first epidemic waves without
identification of the optimal values of these parameters as was done in Chaps. 5 and
6. The accuracy of these approximations will be checked later in this chapter.
Figure 7.2 demonstrates the absolute values of parameters taken from Table 7.1
versus values of N. The values a are shown by “stars” and V 1 —by “circles”; data
points for China and South Korea are represented by “triangles”. It is seen that with
increasing N, final size of the first wave V 1 increases and the parameter a
decreases. In the logarithmic scale, both dependences are very close to linear (including the data points for China and South Korea).
7 Comparison of the First Waves of the COVID-19 …
93