According to (4.18), epidemics start exponentially and only two parameters
(c ¼ aN and t
Ã
1 ) describe the process. We have already used Eq. (4.18) in Chap. 2
(Eq. (4.18) coincides with (2.1) if we put b ¼ ct
Ã
1 ).
To follow next epidemic stages, let us use a more exact approximation:
ln N À U
ð
ÞÀln N À 1
ð
Þ% 1 À U
ð
Þ=N:
Then the integral (4.14) can be expressed as follows:
F
Ã
1 ¼
N
m þ ðN À mÞN
ln
NV þ mð1 À VÞ
N
À ln
N À V
N À 1
!
and the solution has the form
V ¼
EN À m
E þ N À m
ð4:19Þ
E ¼
N
N À 1
exp
a t À t
Ã
1
À
Á m þ N À m
ð
ÞN
½
N
The second approximation (4.19) yields limited value of victims, since
V = I + R tends to N at infinity, but the differences between the exact solution
(4.13)–(4.14) and approximate solutions (4.18) and (4.19) can be very large (see
[35]). Further, we will use only the exact solution (4.13)–(4.14) (similar to papers
[38, 50, 66–78]).
The constant parameters of the model N, m, a, t
Ã
1 and SIR curves can be calculated with the use of limited number of observations V j ; j ¼ 1; 2; 3; . . .; n during
some limited period of time T c (a corresponding algorithm will be presented in the
next chapter). The obtained SIR curves allow predicting the epidemic behavior
(after T c ) and restoring the epidemic history (before T c ). When applying the above
formulas, it should be remembered that the model assumes a closed population with
stable conditions of the epidemic (because all four parameters N, m, a, t
Ã
1 are
assumed to be constant from the time moment t
Ã
1 to the time moment t final ). If new
infected people arrive in the region (so known imported cases) and/or there are
changes in quarantine conditions, testing algorithms or social behavior, the accuracy of both predictions and analysis of the past may deteriorate. Some modified
approach will be developed later in order to take into account changes in epidemic
characteristics.
Actually, the presented approach is suitable only for the first waves of an epidemic, since the initial condition (4.6) assumes that the epidemic began with one
person. In August and September 2020, we saw a sharp increase in the daily
number of new COVID-19 cases in many countries, which may indicate a change in
the pandemic characteristics. These new waves cannot be adequately described by
28
4 Classical SIR Model and the Exact …
(c ¼ aN and t
Ã
1 ) describe the process. We have already used Eq. (4.18) in Chap. 2
(Eq. (4.18) coincides with (2.1) if we put b ¼ ct
Ã
1 ).
To follow next epidemic stages, let us use a more exact approximation:
ln N À U
ð
ÞÀln N À 1
ð
Þ% 1 À U
ð
Þ=N:
Then the integral (4.14) can be expressed as follows:
F
Ã
1 ¼
N
m þ ðN À mÞN
ln
NV þ mð1 À VÞ
N
À ln
N À V
N À 1
!
and the solution has the form
V ¼
EN À m
E þ N À m
ð4:19Þ
E ¼
N
N À 1
exp
a t À t
Ã
1
À
Á m þ N À m
ð
ÞN
½
N
The second approximation (4.19) yields limited value of victims, since
V = I + R tends to N at infinity, but the differences between the exact solution
(4.13)–(4.14) and approximate solutions (4.18) and (4.19) can be very large (see
[35]). Further, we will use only the exact solution (4.13)–(4.14) (similar to papers
[38, 50, 66–78]).
The constant parameters of the model N, m, a, t
Ã
1 and SIR curves can be calculated with the use of limited number of observations V j ; j ¼ 1; 2; 3; . . .; n during
some limited period of time T c (a corresponding algorithm will be presented in the
next chapter). The obtained SIR curves allow predicting the epidemic behavior
(after T c ) and restoring the epidemic history (before T c ). When applying the above
formulas, it should be remembered that the model assumes a closed population with
stable conditions of the epidemic (because all four parameters N, m, a, t
Ã
1 are
assumed to be constant from the time moment t
Ã
1 to the time moment t final ). If new
infected people arrive in the region (so known imported cases) and/or there are
changes in quarantine conditions, testing algorithms or social behavior, the accuracy of both predictions and analysis of the past may deteriorate. Some modified
approach will be developed later in order to take into account changes in epidemic
characteristics.
Actually, the presented approach is suitable only for the first waves of an epidemic, since the initial condition (4.6) assumes that the epidemic began with one
person. In August and September 2020, we saw a sharp increase in the daily
number of new COVID-19 cases in many countries, which may indicate a change in
the pandemic characteristics. These new waves cannot be adequately described by
28
4 Classical SIR Model and the Exact …