Chapter 10
Procedures of Parameter Identification
for the Waves of Epidemics
In this chapter, we will discuss the methods of SIR parameter identification for
second and next waves of epidemics. Some examples will be presented.
In the case of a new epidemic, the values of its parameters are unknown and
must be identified with the use of limited data sets. For the first wave of an epidemic
starting with one infected person, the number of unknown parameters is only four,
since I 1 = 1 and R 1 = 0. The corresponding statistical approach was proposed in
Chaps. 5 and 6 to estimate the values of four unknown parameters for the first
waves of the COVID-19 pandemic in different countries and regions. Some results
were published in [35, 38, 66, 67, 69–77].
For the next epidemic waves (i > 1), the moments of time t
Ã
i corresponding to
their beginning are known. Therefore, the exact solution (9.13)–(9.14) depends only
on five parameters—N i ; I i ; R i ; m i ; a i . Then, the registered number of victims V j
corresponding to the moments of time t j can be used in Eq. (9.14) in order to
calculate F i;j ¼ F
Ã
i ðV j ; N i ; m i ; I i ; R i Þ for every fixed values of N i ; m i ; I i ; R i and then to
check how the registered points fit the straight line (9.13).
Equation (9.13) can be rewritten as follows:
y F
Ã
i ðV; N i ; I i ; R i ; m i Þ ¼ a i t À a i t
Ã
i ;
ð10:1Þ
Equation (10.1) coincides with (2.2), if we assume
c ¼ a i ; b ¼ Àa i t
Ã
i
ð10:2Þ
As in Chap. 2, we can estimate the values of parameters c and b, by treating the
values y j F
Ã
i ðV j ; N i ; I i ; R i ; m i Þ and corresponding time moments t j as random
variables, and use the observations of the accumulated number of cases and the
linear regression in order to calculate the coefficients c
_ and b
_
of the regression line
y
_ ¼ c
_ t þ b
_
ð10:3Þ
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
I. Nesteruk, COVID-19 Pandemic Dynamics,
https://doi.org/10.1007/978-981-33-6416-5_10
133
Procedures of Parameter Identification
for the Waves of Epidemics
In this chapter, we will discuss the methods of SIR parameter identification for
second and next waves of epidemics. Some examples will be presented.
In the case of a new epidemic, the values of its parameters are unknown and
must be identified with the use of limited data sets. For the first wave of an epidemic
starting with one infected person, the number of unknown parameters is only four,
since I 1 = 1 and R 1 = 0. The corresponding statistical approach was proposed in
Chaps. 5 and 6 to estimate the values of four unknown parameters for the first
waves of the COVID-19 pandemic in different countries and regions. Some results
were published in [35, 38, 66, 67, 69–77].
For the next epidemic waves (i > 1), the moments of time t
Ã
i corresponding to
their beginning are known. Therefore, the exact solution (9.13)–(9.14) depends only
on five parameters—N i ; I i ; R i ; m i ; a i . Then, the registered number of victims V j
corresponding to the moments of time t j can be used in Eq. (9.14) in order to
calculate F i;j ¼ F
Ã
i ðV j ; N i ; m i ; I i ; R i Þ for every fixed values of N i ; m i ; I i ; R i and then to
check how the registered points fit the straight line (9.13).
Equation (9.13) can be rewritten as follows:
y F
Ã
i ðV; N i ; I i ; R i ; m i Þ ¼ a i t À a i t
Ã
i ;
ð10:1Þ
Equation (10.1) coincides with (2.2), if we assume
c ¼ a i ; b ¼ Àa i t
Ã
i
ð10:2Þ
As in Chap. 2, we can estimate the values of parameters c and b, by treating the
values y j F
Ã
i ðV j ; N i ; I i ; R i ; m i Þ and corresponding time moments t j as random
variables, and use the observations of the accumulated number of cases and the
linear regression in order to calculate the coefficients c
_ and b
_
of the regression line
y
_ ¼ c
_ t þ b
_
ð10:3Þ
© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021
I. Nesteruk, COVID-19 Pandemic Dynamics,
https://doi.org/10.1007/978-981-33-6416-5_10
133