using the standard formulas (2.6) and (2.7). Values c
_ and b
_
can be treated as
statistics-based estimations of parameters c and b from relationships (2.3). Then,
(10.2) yields the estimations for parameters a i .
The reliability of the method can be checked by calculating the correlation
coefficients r i for every epidemic wave with the use of (2.8) and checking how close
is its value to unity. We can use also the F-test for the null hypothesis that says that
the proposed linear relationship (10.1) fits the data set. The experimental values of
the Fisher function can be calculated for every epidemic wave with the use of the
formula:
F i ¼
r
2
i ðn i À mÞ
ð1 À r 2
i Þðm À 1Þ
;
ð10:4Þ
where n i is the number of observations for the ith epidemic wave and m = 2 is the
number of parameters in the regression equation. The corresponding experimental
value F i has to be compared with the critical value F C ðk 1 ; k 2 Þ of the Fisher function
at a desired significance or confidence level (k 1 ¼ m À 1, k 2 ¼ n i À m). When the
values n i and m are fixed, the maximum of the Fisher function coincides with the
maximum of the correlation coefficient. Therefore, to find the optimal values of
parameters N i ; m i ; I i ; R i , we have to find the maximum of the correlation coefficient
for the linear dependence (10.1). To compare the reliability of different predictions
(with different values of n i ), it is useful to apply the ratio F i =F C ð1; n i À 2Þ at a fixed
significance level. We will use the level 0.001; corresponding values of F C ð1; n i À 2Þ
can be taken from [34]. The most reliable prediction yields the highest
F i =F C ð1; n i À 2Þ ratio.
The exact solution (9.13)–(9.14) allows avoiding numerical solutions of differential Eqs. (9.1)–(9.3) and significantly reduces the time spent on calculations. For
large values of V j (e.g., in the case of global dynamics), calculations of integral
(9.14) need more computer time, but even in these cases the calculation of one set
of optimal parameters could be performed on a regular laptop for one day. We have
modified our MATLAB codes (see Chap. 5) in order to perform calculations of the
next pandemic waves.
In the case of sequential calculation of epidemic waves i = 1, 2, 3 …, it is
possible to avoid determining the four optimal unknown parameters N i ; m i ; I i ; R i ,
thereby reducing the amount of calculations and difficulties in isolation of a maximum of correlation coefficient. For parameters I i ; R i , it is possible to use the
numbers of I and R calculated for the previous wave of epidemic at the moment of
time when the following wave began. Then, we need to calculate values
F
Ã
i ðV j ; N i ; m i Þ, linear regression coefficients (10.3), correlation coefficient r i ,
F i =F C ð1; n À 2Þ and to isolate the values of parameters N i and m i corresponding to
the maximum of the correlation cofficient. The MATLAB code mentioned in
Chap. 5 required only minor modifications. Knowing the optimal values of five
parameters N i ; I i ; R i ; m i ; a i , the SIR curves and other characteristics of the corresponding epidemic wave can be calculated with the use of formulas presented in
134
10 Procedures of Parameter Identification …
_ and b
_
can be treated as
statistics-based estimations of parameters c and b from relationships (2.3). Then,
(10.2) yields the estimations for parameters a i .
The reliability of the method can be checked by calculating the correlation
coefficients r i for every epidemic wave with the use of (2.8) and checking how close
is its value to unity. We can use also the F-test for the null hypothesis that says that
the proposed linear relationship (10.1) fits the data set. The experimental values of
the Fisher function can be calculated for every epidemic wave with the use of the
formula:
F i ¼
r
2
i ðn i À mÞ
ð1 À r 2
i Þðm À 1Þ
;
ð10:4Þ
where n i is the number of observations for the ith epidemic wave and m = 2 is the
number of parameters in the regression equation. The corresponding experimental
value F i has to be compared with the critical value F C ðk 1 ; k 2 Þ of the Fisher function
at a desired significance or confidence level (k 1 ¼ m À 1, k 2 ¼ n i À m). When the
values n i and m are fixed, the maximum of the Fisher function coincides with the
maximum of the correlation coefficient. Therefore, to find the optimal values of
parameters N i ; m i ; I i ; R i , we have to find the maximum of the correlation coefficient
for the linear dependence (10.1). To compare the reliability of different predictions
(with different values of n i ), it is useful to apply the ratio F i =F C ð1; n i À 2Þ at a fixed
significance level. We will use the level 0.001; corresponding values of F C ð1; n i À 2Þ
can be taken from [34]. The most reliable prediction yields the highest
F i =F C ð1; n i À 2Þ ratio.
The exact solution (9.13)–(9.14) allows avoiding numerical solutions of differential Eqs. (9.1)–(9.3) and significantly reduces the time spent on calculations. For
large values of V j (e.g., in the case of global dynamics), calculations of integral
(9.14) need more computer time, but even in these cases the calculation of one set
of optimal parameters could be performed on a regular laptop for one day. We have
modified our MATLAB codes (see Chap. 5) in order to perform calculations of the
next pandemic waves.
In the case of sequential calculation of epidemic waves i = 1, 2, 3 …, it is
possible to avoid determining the four optimal unknown parameters N i ; m i ; I i ; R i ,
thereby reducing the amount of calculations and difficulties in isolation of a maximum of correlation coefficient. For parameters I i ; R i , it is possible to use the
numbers of I and R calculated for the previous wave of epidemic at the moment of
time when the following wave began. Then, we need to calculate values
F
Ã
i ðV j ; N i ; m i Þ, linear regression coefficients (10.3), correlation coefficient r i ,
F i =F C ð1; n À 2Þ and to isolate the values of parameters N i and m i corresponding to
the maximum of the correlation cofficient. The MATLAB code mentioned in
Chap. 5 required only minor modifications. Knowing the optimal values of five
parameters N i ; I i ; R i ; m i ; a i , the SIR curves and other characteristics of the corresponding epidemic wave can be calculated with the use of formulas presented in
134
10 Procedures of Parameter Identification …