## Appendix D.5. Application of the model in $m^l$

The model in $m^l$ is now applied, as an identification model, to period 1 which is the beginning of the epidemic in France. This application leads to the following numerical values:

$$A = 107.9402; B = 0.0022803; m = 0.7642119.$$

Whereas period 1 presents a convexity, the value of B belongs to an interval corresponding to an inflexion point, thus expressing that the model in $m^l$ is here inappropriate, as figure D.21 shows it.

<figure><img src="image_22.png" /><figcaption>Figure D.21: Comparative evolution, during period 1, of the contaminated recorded by the Ministry and obtained with the models in $t^m$ and $m^l$: the comparison well illustrates the representativity difference of the models</figcaption></figure>

In conclusion, figure D.21 conveys an eloquent result: the comparative illustration of the representativity of the models in $t^m$ and $m^l$, clearly proves to be in favor of the model in $t^m$. But this drawback of the model in $m^l$ is due to its strategy requirement, in the sense that this model aims, with a unique parameterization, to represent both convex and concave phases of an epidemic growth.

# Appendix E. Comparative tables of the official data and the predictions obtained with the FPM model

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<table><caption>Table E.2: Basic period used for identification of the contaminated in lockdown phase in 2020</caption><thead><tr><th></th><th>29/03</th><th>30/03</th><th>31/03</th><th>01/04</th><th>02/04</th><th>03/04</th><th>04/04</th><th>05/04</th><th>06/04</th></tr></thead><tbody><tr><th>b<sub>k</sub></th><td>39610</td><td>46115</td><td>51404</td><td>56032</td><td>60222</td><td>64094</td><td>67720</td><td>71148</td><td>74412</td></tr><tr><th>a<sub>k</sub></th><td>40174</td><td>44550</td><td>52128</td><td>56989</td><td>59105</td><td>64338</td><td>68605</td><td>70478</td><td>74390</td></tr><tr><th>E<sub>r</sub></th><td>0.0142</td><td>-0.0339</td><td>0.0141</td><td>0.0171</td><td>-0.0185</td><td>0.0038</td><td>0.0131</td><td>-0.0094</td><td>-0.0003</td></tr></tbody></table>

<table><caption>Table E.3: Periods used for prediction of the contaminated in lockdown phase in 2020: mean(E<sub>r</sub>) = 0.03568 for week 1; mean(E<sub>r</sub>) = 0.05525 for week 2; mean(E<sub>r</sub>) = 0.023988 for week 3; mean(E<sub>r</sub>) = -0.023308 for week 4; mean(E<sub>r</sub>) = -0.056094 for week 5</caption><tbody><tr><td rowspan="4">Week 1</td><td>date</td><td>07/04</td><td>08/04</td><td>09/04</td><td>10/04</td><td>11/04</td><td>12/04</td><td>13/04</td></tr><tr><td>b<sub>k</sub></td><td>77536</td><td>80540</td><td>83439</td><td>86244</td><td>88967</td><td>91613</td><td>94192</td></tr><tr><td>a<sub>k</sub></td><td>78167</td><td>82048</td><td>86334</td><td>90676</td><td>93790</td><td>95403</td><td>98076</td></tr><tr><td>E<sub>r</sub></td><td>0.0081</td><td>0.0187</td><td>0.0347</td><td>0.0514</td><td>0.0542</td><td>0.0414</td><td>0.0412</td></tr><tr><td rowspan="4">Week 2</td><td>date</td><td>14/04</td><td>15/04</td><td>16/04</td><td>17/04</td><td>18/04</td><td>19/04</td><td>20/04</td></tr><tr><td>b<sub>k</sub></td><td>96708</td><td>99167</td><td>101573</td><td>103930</td><td>106241</td><td>108509</td><td>110737</td></tr><tr><td>a<sub>k</sub></td><td>103573</td><td>106206</td><td>108847</td><td>109252</td><td>111821</td><td>112206</td><td>114657</td></tr><tr><td>E<sub>r</sub></td><td>0.0710</td><td>0.0710</td><td>0.0716</td><td>0.0512</td><td>0.0525</td><td>0.0341</td><td>0.0354</td></tr><tr><td rowspan="4">Week 3</td><td>date</td><td>21/04</td><td>22/04</td><td>23/04</td><td>24/04</td><td>25/04</td><td>26/04</td><td>27/04</td></tr><tr><td>b<sub>k</sub></td><td>112928</td><td>115083</td><td>117205</td><td>119295</td><td>121354</td><td>123386</td><td>125389</td></tr><tr><td>a<sub>k</sub></td><td>117324</td><td>119151</td><td>120804</td><td>122577</td><td>124114</td><td>124575</td><td>125770</td></tr><tr><td>E<sub>r</sub></td><td>0.0389</td><td>0.0353</td><td>0.0307</td><td>0.0275</td><td>0.0227</td><td>0.0096</td><td>0.0030</td></tr><tr><td rowspan="4">Week 4</td><td>date</td><td>28/04</td><td>29/04</td><td>30/04</td><td>01/05</td><td>02/05</td><td>03/05</td><td>04/05</td></tr><tr><td>b<sub>k</sub></td><td>127367</td><td>129320</td><td>131250</td><td>133156</td><td>135041</td><td>136904</td><td>138748</td></tr><tr><td>a<sub>k</sub></td><td>127290</td><td>128442</td><td>129581</td><td>130185</td><td>130979</td><td>131287</td><td>131863</td></tr><tr><td>E<sub>r</sub></td><td>-0.0006</td><td>-0.0068</td><td>-0.0127</td><td>-0.0223</td><td>-0.0301</td><td>-0.0410</td><td>-0.0496</td></tr><tr><td rowspan="4">Week 5</td><td>date</td><td>05/05</td><td>06/05</td><td>07/05</td><td>08/05</td><td>09/05</td><td>10/05</td><td>11/05</td></tr><tr><td>b<sub>k</sub></td><td>140572</td><td>142377</td><td>144165</td><td>145934</td><td>147687</td><td>149424</td><td>151145</td></tr><tr><td>a<sub>k</sub></td><td>132967</td><td>137150</td><td>137779</td><td>138421</td><td>138854</td><td>139063</td><td>139519</td></tr><tr><td>E<sub>r</sub></td><td>-0.0541</td><td>-0.0367</td><td>-0.0443</td><td>-0.0515</td><td>-0.0598</td><td>-0.0693</td><td>-0.0769</td></tr></tbody></table>

Kermack, W., McKendrick, A., 1927. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond A, 700-721. doi:http://doi.org/10.1098/rspa.1927.0118.

Korabel, N., Zaslavsky, G., Tarasov, V., 2007. Coupled oscillators with power-law interaction and their fractional dynamics analogues. Communications in Nonlinear Science and Numerical Simulation 12, 1405-1417. doi:10.1016/j.cnsns.2006.03.015.

Kwuimy, C.A.K., Nazari, F., Jiao, X., Rohani, P., Nataraj, C., 2020. Nonlinear dynamic analysis of an epidemiological model for covid-19 including public behavior and government action. Nonlinear Dynamics 101, 1545-1559. doi:10.1007/s11071-020-05815-z.

Kyurkchiev, N., Iliev, A., Rahnev, A., 2020. On the verhulst growth model with polynomial variable transfer, some applications. International. journal of differential, equations and applications 19, 15-32.

Le Méhauté, A., 2021. About the theoretical roots for managing the third middle illustration using covid-19 dynamics in france. Noworczesne Syslemy Zazadzania. Kwartalny raport naukowy To appear.

Li, W., Zhou, J., Lu, J.a., 2020. The effect of behavior of wearing masks on epidemic dynamics. Nonlinear Dynamics 101, 1995-2001. doi:10.1007/s11071-020-05759-4.

Liang, Z.P., Wang, X.D., Lee, D.J., Peng, X.F., Su, A., 2009. Spreading dynamics of power-law fluid droplets. Journal of Physics: Condensed Matter 21, 464117. doi:10.1088/0953-8984/21/46/464117.

Liouville, J., 1832. Mémoire sur quelques questions de géométrie et de mécanique et sur un nouveau genre de calcul pour résoudre ces équations. Journal de l'Ecole Polytechnique 13, 71-162.

Liu, X., Zheng, X., Balachandran, B., 2020. Covid-19: data-driven dynam-

ics, statistical and distributed delay models, and observations. Nonlinear Dynamics 101, 1527-1543. doi:10.1007/s11071-020-05863-5.

Lu, Z., Yu, Y., Chen, Y., Ren, G., Xu, C., Wang, S., Yin, Z., 2020. A fractional-order seihdr model for covid-19 with inter-city networked coupling effects. Nonlinear Dynamics 101, 1717-1730. doi:10.1007/s11071-020-05848-4.

Mendes, G., da Silva, L., 2009. Generating more realistic complex networks from power-law distribution of fitness generating more realistic complex networks from power-law distribution of fitness. Braz. J. Phys. 39. doi:10.1590/S0103-97332009000400013.

Meyer, S., Held, L., 2014. Power-law models for infectious disease spread. The Annals of Applied Statistics 8, 1612-1639. doi:10.1214/14-AOAS743.

Oustaloup, A., 1995. La dérivation non-entièrė : théorie, synthèse et applications. Hermès, Paris.

Oustaloup, A., 2014. Diversity and Non-integer Differentiation for System Dynamics. Wiley-ISTE.

Oustaloup, A., Levron, F., Mathieu, B., Nanot, F., 2000. Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 47, 25-39. doi:10.1109/81.817385.

Rémond, J., Rémond, Y., 2020. Sur une modélisation simplifiée de l'épidémie du covid-19 de 2020. URL: https://hal.archives-ouvertes.fr/hal-02551464. working paper or preprint.

Richards, F., 1959. A flexible growth function for empirical use. Journal of Experimental Botany 10, 290-300.

Rohith, G., Devika, K.B., 2020. Dynamics and control of covid-19 pandemic with nonlinear incidence rates. Nonlinear Dynamics 101, 2013-2026.
<table><caption>Table E.2: Basic period used for identification of the contaminated in lockdown phase in 2020</caption><thead><tr><th></th><th>29/03</th><th>30/03</th><th>31/03</th><th>01/04</th><th>02/04</th><th>03/04</th><th>04/04</th><th>05/04</th><th>06/04</th></tr></thead><tbody><tr><th>b<sub>k</sub></th><td>39610</td><td>46115</td><td>51404</td><td>56032</td><td>60222</td><td>64094</td><td>67720</td><td>71148</td><td>74412</td></tr><tr><th>a<sub>k</sub></th><td>40174</td><td>44550</td><td>52128</td><td>56989</td><td>59105</td><td>64338</td><td>68605</td><td>70478</td><td>74390</td></tr><tr><th>E<sub>r</sub></th><td>0.0142</td><td>-0.0339</td><td>0.0141</td><td>0.0171</td><td>-0.0185</td><td>0.0038</td><td>0.0131</td><td>-0.0094</td><td>-0.0003</td></tr></tbody></table>

<table><caption>Table E.3: Periods used for prediction of the contaminated in lockdown phase in 2020: mean(E<sub>r</sub>) = 0.03568 for week 1; mean(E<sub>r</sub>) = 0.05525 for week 2; mean(E<sub>r</sub>) = 0.023988 for week 3; mean(E<sub>r</sub>) = -0.023308 for week 4; mean(E<sub>r</sub>) = -0.056094 for week 5</caption><tbody><tr><td rowspan="4">Week 1</td><td>date</td><td>07/04</td><td>08/04</td><td>09/04</td><td>10/04</td><td>11/04</td><td>12/04</td><td>13/04</td></tr><tr><td>b<sub>k</sub></td><td>77536</td><td>80540</td><td>83439</td><td>86244</td><td>88967</td><td>91613</td><td>94192</td></tr><tr><td>a<sub>k</sub></td><td>78167</td><td>82048</td><td>86334</td><td>90676</td><td>93790</td><td>95403</td><td>98076</td></tr><tr><td>E<sub>r</sub></td><td>0.0081</td><td>0.0187</td><td>0.0347</td><td>0.0514</td><td>0.0542</td><td>0.0414</td><td>0.0412</td></tr><tr><td rowspan="4">Week 2</td><td>date</td><td>14/04</td><td>15/04</td><td>16/04</td><td>17/04</td><td>18/04</td><td>19/04</td><td>20/04</td></tr><tr><td>b<sub>k</sub></td><td>96708</td><td>99167</td><td>101573</td><td>103930</td><td>106241</td><td>108509</td><td>110737</td></tr><tr><td>a<sub>k</sub></td><td>103573</td><td>106206</td><td>108847</td><td>109252</td><td>111821</td><td>112206</td><td>114657</td></tr><tr><td>E<sub>r</sub></td><td>0.0710</td><td>0.0710</td><td>0.0716</td><td>0.0512</td><td>0.0525</td><td>0.0341</td><td>0.0354</td></tr><tr><td rowspan="4">Week 3</td><td>date</td><td>21/04</td><td>22/04</td><td>23/04</td><td>24/04</td><td>25/04</td><td>26/04</td><td>27/04</td></tr><tr><td>b<sub>k</sub></td><td>112928</td><td>115083</td><td>117205</td><td>119295</td><td>121354</td><td>123386</td><td>125389</td></tr><tr><td>a<sub>k</sub></td><td>117324</td><td>119151</td><td>120804</td><td>122577</td><td>124114</td><td>124575</td><td>125770</td></tr><tr><td>E<sub>r</sub></td><td>0.0389</td><td>0.0353</td><td>0.0307</td><td>0.0275</td><td>0.0227</td><td>0.0096</td><td>0.0030</td></tr><tr><td rowspan="4">Week 4</td><td>date</td><td>28/04</td><td>29/04</td><td>30/04</td><td>01/05</td><td>02/05</td><td>03/05</td><td>04/05</td></tr><tr><td>b<sub>k</sub></td><td>127367</td><td>129320</td><td>131250</td><td>133156</td><td>135041</td><td>136904</td><td>138748</td></tr><tr><td>a<sub>k</sub></td><td>127290</td><td>128442</td><td>129581</td><td>130185</td><td>130979</td><td>131287</td><td>131863</td></tr><tr><td>E<sub>r</sub></td><td>-0.0006</td><td>-0.0068</td><td>-0.0127</td><td>-0.0223</td><td>-0.0301</td><td>-0.0410</td><td>-0.0496</td></tr><tr><td rowspan="4">Week 5</td><td>date</td><td>05/05</td><td>06/05</td><td>07/05</td><td>08/05</td><td>09/05</td><td>10/05</td><td>11/05</td></tr><tr><td>b<sub>k</sub></td><td>140572</td><td>142377</td><td>144165</td><td>145934</td><td>147687</td><td>149424</td><td>151145</td></tr><tr><td>a<sub>k</sub></td><td>132967</td><td>137150</td><td>137779</td><td>138421</td><td>138854</td><td>139063</td><td>139519</td></tr><tr><td>E<sub>r</sub></td><td>-0.0541</td><td>-0.0367</td><td>-0.0443</td><td>-0.0515</td><td>-0.0598</td><td>-0.0693</td><td>-0.0769</td></tr></tbody></table>

Kermack, W., McKendrick, A., 1927. A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond A, 700-721. doi:http://doi.org/10.1098/rspa.1927.0118.

Korabel, N., Zaslavsky, G., Tarasov, V., 2007. Coupled oscillators with power-law interaction and their fractional dynamics analogues. Communications in Nonlinear Science and Numerical Simulation 12, 1405-1417. doi:10.1016/j.cnsns.2006.03.015.

Kwuimy, C.A.K., Nazari, F., Jiao, X., Rohani, P., Nataraj, C., 2020. Nonlinear dynamic analysis of an epidemiological model for covid-19 including public behavior and government action. Nonlinear Dynamics 101, 1545-1559. doi:10.1007/s11071-020-05815-z.

Kyurkchiev, N., Iliev, A., Rahnev, A., 2020. On the verhulst growth model with polynomial variable transfer. some applications. International. journal of differential, equations and applications 19, 15-32.

Le Méhauté, A., 2021. About the theoretical roots for managing the third middle illustration using covid-19 dynamics in france. Noworczesne Syslemy Zazadzania. Kwartalny raport naukowy To appear.

Li, W., Zhou, J., Lu, J.a., 2020. The effect of behavior of wearing masks on epidemic dynamics. Nonlinear Dynamics 101, 1995-2001. doi:10.1007/s11071-020-05759-4.

Liang, Z.P., Wang, X.D., Lee, D.J., Peng, X.F., Su, A., 2009. Spreading dynamics of power-law fluid droplets. Journal of Physics: Condensed Matter 21, 464117. doi:10.1088/0953-8984/21/46/464117.

Liouville, J., 1832. Mémoire sur quelques questions de géométrie et de mécanique et sur un nouveau genre de calcul pour résoudre ces équations. Journal de l'Ecole Polytechnique 13, 71-162.

Liu, X., Zheng, X., Balachandran, B., 2020. Covid-19: data-driven dynam-

ics, statistical and distributed delay models, and observations. Nonlinear Dynamics 101, 1527-1543. doi:10.1007/s11071-020-05863-5.

Lu, Z., Yu, Y., Chen, Y., Ren, G., Xu, C., Wang, S., Yin, Z., 2020. A fractional-order seihdr model for covid-19 with inter-city networked coupling effects. Nonlinear Dynamics 101, 1717-1730. doi:10.1007/s11071-020-05848-4.

Mendes, G., da Silva, L., 2009. Generating more realistic complex networks from power-law distribution of fitness generating more realistic complex networks from power-law distribution of fitness. Braz. J. Phys. 39. doi:10.1590/S0103-97332009000400013.

Meyer, S., Held, L., 2014. Power-law models for infectious disease spread. The Annals of Applied Statistics 8, 1612-1639. doi:10.1214/14-AOAS743.

Oustaloup, A., 1995. La dérivation non-entière : théorie, synthèse et applications. Hermès, Paris.

Oustaloup, A., 2014. Diversity and Non-integer Differentiation for System Dynamics. Wiley-ISTE.

Oustaloup, A., Levron, F., Mathieu, B., Nanot, F., 2000. Frequency-band complex noninteger differentiator: characterization and synthesis. IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications 47, 25-39. doi:10.1109/81.817385.

Rémond, J., Rémond, Y., 2020. Sur une modélisation simplifiée de l'épidémie du covid-19 de 2020. URL: https://hal.archives-ouvertes.fr/hal-02551464. working paper or preprint.

Richards, F., 1959. A flexible growth function for empirical use. Journal of Experimental Botany 10, 290-300.

Rohith, G., Devika, K.B., 2020. Dynamics and control of covid-19 pandemic with nonlinear incidence rates. Nonlinear Dynamics 101, 2013-2026.
<table><caption>Table E.4: Basic period used for identification of the contaminated in vaccination phase in 2021</caption><thead><tr><th>date</th><th>26/04</th><th>27/04</th><th>28/04</th><th>29/04</th><th>30/04</th><th>01/05</th><th>02/05</th></tr></thead><tbody><tr><th>b<sub>k</sub></th><td>5502175</td><td>5537601</td><td>5566631</td><td>5592158</td><td>5615361</td><td>5636865</td><td>5657053</td></tr><tr><th>a<sub>k</sub></th><td>5503996</td><td>5534313</td><td>5565852</td><td>5592390</td><td>5616689</td><td>5642359</td><td>5652247</td></tr><tr><th>E<sub>r</sub></th><td>0.00033</td><td>-0.00059</td><td>-0.00014</td><td>0.000041</td><td>0.00024</td><td>0.00097</td><td>-0.00085</td></tr></tbody></table>

<table><caption>Table E.5: Periods used for prediction of the contaminated in vaccination phase in 2021: mean(E<sub>r</sub>) = -0.000893 for week 1; mean(E<sub>r</sub>) = 0.00017 for week 2; mean(E<sub>r</sub>) = 0.0010 for week 3; mean(E<sub>r</sub>) = -0.00131 for week 4; mean(E<sub>r</sub>) = -0.00430 for week 5; mean(E<sub>r</sub>) = -0.00944 for week 6</caption><tbody><tr><td rowspan="4">Week 1</td><td>date</td><td>03/05</td><td>04/05</td><td>05/05</td><td>06/05</td><td>07/05</td><td>08/05</td><td>09/05</td></tr><tr><td>b<sub>k</sub></td><td>5676178</td><td>5694419</td><td>5711910</td><td>5728753</td><td>5745027</td><td>5760798</td><td>5776117</td></tr><tr><td>a<sub>k</sub></td><td>5656007</td><td>5680378</td><td>5706378</td><td>5728090</td><td>5747214</td><td>5767959</td><td>5777087</td></tr><tr><td>E<sub>r</sub></td><td>-0.0036</td><td>-0.0025</td><td>-0.0010</td><td>-0.0001</td><td>0.0004</td><td>0.0012</td><td>0.0002</td></tr><tr><td rowspan="4">Week 2</td><td>date</td><td>10/05</td><td>11/05</td><td>12/05</td><td>13/05</td><td>14/05</td><td>15/05</td><td>16/05</td></tr><tr><td>b<sub>k</sub></td><td>5791028</td><td>5805569</td><td>5819771</td><td>5833661</td><td>5847263</td><td>5860598</td><td>5873684</td></tr><tr><td>a<sub>k</sub></td><td>5780379</td><td>5800170</td><td>5821668</td><td>5841129</td><td>5848154</td><td>5863839</td><td>5877787</td></tr><tr><td>E<sub>r</sub></td><td>-0.0018</td><td>-0.0009</td><td>0.0003</td><td>0.0013</td><td>0.0002</td><td>0.0006</td><td>0.0007</td></tr><tr><td rowspan="4">Week 3</td><td>date</td><td>17/05</td><td>18/05</td><td>19/05</td><td>20/05</td><td>21/05</td><td>22/05</td><td>23/05</td></tr><tr><td>b<sub>k</sub></td><td>5886537</td><td>5899171</td><td>5911600</td><td>5923835</td><td>5935886</td><td>5947764</td><td>5959478</td></tr><tr><td>a<sub>k</sub></td><td>5881137</td><td>5898347</td><td>5917397</td><td>5932812</td><td>5945612</td><td>5958223</td><td>5967927</td></tr><tr><td>E<sub>r</sub></td><td>-0.0009</td><td>-0.0001</td><td>0.0010</td><td>0.0015</td><td>0.0016</td><td>0.0018</td><td>0.0014</td></tr><tr><td rowspan="4">Week 4</td><td>date</td><td>24/05</td><td>25/05</td><td>26/05</td><td>27/05</td><td>28/05</td><td>29/05</td><td>30/05</td></tr><tr><td>b<sub>k</sub></td><td>5971034</td><td>5982442</td><td>5993707</td><td>6004835</td><td>6015833</td><td>6026707</td><td>6037460</td></tr><tr><td>a<sub>k</sub></td><td>5970156</td><td>5973311</td><td>5985957</td><td>5999890</td><td>6011158</td><td>6021833</td><td>6030374</td></tr><tr><td>E<sub>r</sub></td><td>-0.0001</td><td>-0.0015</td><td>-0.0013</td><td>-0.0008</td><td>-0.0008</td><td>-0.0008</td><td>-0.0012</td></tr><tr><td rowspan="4">Week 5</td><td>date</td><td>31/05</td><td>01/06</td><td>02/06</td><td>03/06</td><td>04/06</td><td>05/06</td><td>06/06</td></tr><tr><td>b<sub>k</sub></td><td>6048098</td><td>6058625</td><td>6069046</td><td>6079363</td><td>6089582</td><td>6099704</td><td>6109734</td></tr><tr><td>a<sub>k</sub></td><td>6031585</td><td>6041433</td><td>6050176</td><td>6058340</td><td>6065293</td><td>6071947</td><td>6077017</td></tr><tr><td>E<sub>r</sub></td><td>-0.0027</td><td>-0.0028</td><td>-0.0031</td><td>-0.0035</td><td>-0.0040</td><td>-0.0046</td><td>-0.0054</td></tr><tr><td rowspan="4">Week 6</td><td>date</td><td>07/06</td><td>08/06</td><td>09/06</td><td>10/06</td><td>11/06</td><td>12/06</td><td>13/06</td></tr><tr><td>b<sub>k</sub></td><td>6119674</td><td>6129527</td><td>6139297</td><td>6148984</td><td>6158593</td><td>6168125</td><td>6177582</td></tr><tr><td>a<sub>k</sub></td><td>6078181</td><td>6084199</td><td>6089756</td><td>6094231</td><td>6098102</td><td>6102074</td><td>6104929</td></tr><tr><td>E<sub>r</sub></td><td>-0.0068</td><td>-0.0074</td><td>-0.0081</td><td>-0.0089</td><td>-0.0098</td><td>-0.0107</td><td>-0.0118</td></tr></tbody></table>

doi:10.1007/s11071-020-05774-5.

Roosa, K., Lee, Y., Luo, R., Kirpich, A., Rothenberg, R., Hyman, J.M., Yan, P., Chowell, G., 2020. Short-term forecasts of the covid-19 epidemic in guangdong and zhejiang, china: February 13-23, 2020. Journal of clinical medicine 9, 596. doi:10.3390/jcm9020596.

Santamaria, J., 2017. De la simplicité à la complexité. Le Monde, Voyage dans le cosmos.

Satsuma, J., Willox, R., Ramani, A., Grammaticos, B., Carstea, A., 2004. Extending the sir epidemic model. Physica A: Statistical Mechanics and its Applications 336, 369-375. doi:https://doi.org/10.1016/j.physa.2003.12.035.

Savageau, M.A., 1970. Biochemical systems analysis: Iii. dynamic solutions using a power-law approximation. Journal of Theoretical Biology 26, 215-226. doi:10.1016/S0022-5193(70)80013-3.

Scharbarg, E., Moog, C.H., Mauduit, N., Califano, C., 2020. From the hospital scale to nationwide: observability and identification of models for the covid-19 epidemic waves. Annual Reviews in Control 50, 409-416. doi:10.1016/j.arcontrol.2020.09.007.

Sher, P.H., Smith, J.M., Dalgarno, P.A., Warburton, R.J., Chen, X., Dobson, P.J., Daniels, S.M., Pickett, N.L., O'Brien, P., 2008. Power law carrier dynamics in semiconductor nanocrystals at nanosecond timescales. Applied Physics Letters 92, 101111. doi:10.1063/1.2894193.

Tanner, L.H., 1979. The spreading of silicone oil drops on horizontal surfaces.

Journal of Physics D: Applied Physics 12, 1473-1484. doi:10.1088/0022-3727/12/9/009.

Tolle, J., 2003. Can growth be faster than exponential, and just how slow is the logarithm? The Mathematical Gazette 87, 522-525. doi:10.1017/S0025557200173802.

Trigeassou, J., Maamri, N., 2019. The Distributed Model of the Fractional Integrator. John Wiley & Sons, Ltd. chapter 6. pp. 127-158. doi:10.1002/9781119476917.ch6.

Tuan, N.H., Mohammadi, H., Rezapour, S., 2020. A mathematical model for covid-19 transmission by using the caputo fractional derivative. Chaos, Solitons & Fractals 140, 110107. doi:https://doi.org/10.1016/j.chaos.2020.110107.

Verhulst, P.F., 1838. Notice sur la loi que la population poursuit dans son accroissement. Correspondance mathématique et physique, 113-121.

Verhulst, P.F., 1845. Recherches mathématiques sur la loi d'accroissement de la population. Nouveaux mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles 18, 14-54.

Viboud, C., Simonsen, L., Chowell, G., 2016. A generalized-growth model to characterize the early ascending phase of infectious disease outbreaks. Epidemiology 15, 27-37. doi:https://doi.org/10.1016/j.epidem.2016.01.002.

Wang, X.S., Wu, J., Yang, Y., 2012. Richards model revisited: Validation by and application to infection dynamics. Journal of Theoretical Biology 313,
<table><caption>Table E.4: Basic period used for identification of the contaminated in vaccination phase in 2021</caption><thead><tr><th>date</th><th>26/04</th><th>27/04</th><th>28/04</th><th>29/04</th><th>30/04</th><th>01/05</th><th>02/05</th></tr></thead><tbody><tr><th>b<sub>k</sub></th><td>5502175</td><td>5537601</td><td>5566631</td><td>5592158</td><td>5615361</td><td>5636865</td><td>5657053</td></tr><tr><th>a<sub>k</sub></th><td>5503996</td><td>5534313</td><td>5565852</td><td>5592390</td><td>5616689</td><td>5642359</td><td>5652247</td></tr><tr><th>E<sub>r</sub></th><td>0.00033</td><td>-0.00059</td><td>-0.00014</td><td>0.000041</td><td>0.00024</td><td>0.00097</td><td>-0.00085</td></tr></tbody></table>

<table><caption>Table E.5: Periods used for prediction of the contaminated in vaccination phase in 2021: mean(E<sub>r</sub>) = -0.000893 for week 1; mean(E<sub>r</sub>) = 0.00017 for week 2; mean(E<sub>r</sub>) = 0.0010 for week 3; mean(E<sub>r</sub>) = -0.00131 for week 4; mean(E<sub>r</sub>) = -0.00430 for week 5; mean(E<sub>r</sub>) = -0.00944 for week 6</caption><tbody><tr><th rowspan="4">Week 1</th><th>date</th><td>03/05</td><td>04/05</td><td>05/05</td><td>06/05</td><td>07/05</td><td>08/05</td><td>09/05</td></tr><tr><th>b<sub>k</sub></th><td>5676178</td><td>5694419</td><td>5711910</td><td>5728753</td><td>5745027</td><td>5760798</td><td>5776117</td></tr><tr><th>a<sub>k</sub></th><td>5656007</td><td>5680378</td><td>5706378</td><td>5728090</td><td>5747214</td><td>5767959</td><td>5777087</td></tr><tr><th>E<sub>r</sub></th><td>-0.0036</td><td>-0.0025</td><td>-0.0010</td><td>-0.0001</td><td>0.0004</td><td>0.0012</td><td>0.0002</td></tr><tr><th rowspan="4">Week 2</th><th>date</th><td>10/05</td><td>11/05</td><td>12/05</td><td>13/05</td><td>14/05</td><td>15/05</td><td>16/05</td></tr><tr><th>b<sub>k</sub></th><td>5791028</td><td>5805569</td><td>5819771</td><td>5833661</td><td>5847263</td><td>5860598</td><td>5873684</td></tr><tr><th>a<sub>k</sub></th><td>5780379</td><td>5800170</td><td>5821668</td><td>5841129</td><td>5848154</td><td>5863839</td><td>5877787</td></tr><tr><th>E<sub>r</sub></th><td>-0.0018</td><td>-0.0009</td><td>0.0003</td><td>0.0013</td><td>0.0002</td><td>0.0006</td><td>0.0007</td></tr><tr><th rowspan="4">Week 3</th><th>date</th><td>17/05</td><td>18/05</td><td>19/05</td><td>20/05</td><td>21/05</td><td>22/05</td><td>23/05</td></tr><tr><th>b<sub>k</sub></th><td>5886537</td><td>5899171</td><td>5911600</td><td>5923835</td><td>5935886</td><td>5947764</td><td>5959478</td></tr><tr><th>a<sub>k</sub></th><td>5881137</td><td>5898347</td><td>5917397</td><td>5932812</td><td>5945612</td><td>5958223</td><td>5967927</td></tr><tr><th>E<sub>r</sub></th><td>-0.0009</td><td>-0.0001</td><td>0.0010</td><td>0.0015</td><td>0.0016</td><td>0.0018</td><td>0.0014</td></tr><tr><th rowspan="4">Week 4</th><th>date</th><td>24/05</td><td>25/05</td><td>26/05</td><td>27/05</td><td>28/05</td><td>29/05</td><td>30/05</td></tr><tr><th>b<sub>k</sub></th><td>5971034</td><td>5982442</td><td>5993707</td><td>6004835</td><td>6015833</td><td>6026707</td><td>6037460</td></tr><tr><th>a<sub>k</sub></th><td>5970156</td><td>5973311</td><td>5985957</td><td>5999890</td><td>6011158</td><td>6021833</td><td>6030374</td></tr><tr><th>E<sub>r</sub></th><td>-0.0001</td><td>-0.0015</td><td>-0.0013</td><td>-0.0008</td><td>-0.0008</td><td>-0.0008</td><td>-0.0012</td></tr><tr><th rowspan="4">Week 5</th><th>date</th><td>31/05</td><td>01/06</td><td>02/06</td><td>03/06</td><td>04/06</td><td>05/06</td><td>06/06</td></tr><tr><th>b<sub>k</sub></th><td>6048098</td><td>6058625</td><td>6069046</td><td>6079363</td><td>6089582</td><td>6099704</td><td>6109734</td></tr><tr><th>a<sub>k</sub></th><td>6031585</td><td>6041433</td><td>6050176</td><td>6058340</td><td>6065293</td><td>6071947</td><td>6077017</td></tr><tr><th>E<sub>r</sub></th><td>-0.0027</td><td>-0.0028</td><td>-0.0031</td><td>-0.0035</td><td>-0.0040</td><td>-0.0046</td><td>-0.0054</td></tr><tr><th rowspan="4">Week 6</th><th>date</th><td>07/06</td><td>08/06</td><td>09/06</td><td>10/06</td><td>11/06</td><td>12/06</td><td>13/06</td></tr><tr><th>b<sub>k</sub></th><td>6119674</td><td>6129527</td><td>6139297</td><td>6148984</td><td>6158593</td><td>6168125</td><td>6177582</td></tr><tr><th>a<sub>k</sub></th><td>6078181</td><td>6084199</td><td>6089756</td><td>6094231</td><td>6098102</td><td>6102074</td><td>6104929</td></tr><tr><th>E<sub>r</sub></th><td>-0.0068</td><td>-0.0074</td><td>-0.0081</td><td>-0.0089</td><td>-0.0098</td><td>-0.0107</td><td>-0.0118</td></tr></tbody></table>

doi:10.1007/s11071-020-05774-5.

Roosa, K., Lee, Y., Luo, R., Kirpich, A., Rothenberg, R., Hyman, J.M., Yan, P., Chowell, G., 2020. Short-term forecasts of the covid-19 epidemic in guangdong and zhejiang, china: February 13-23, 2020. Journal of clinical medicine 9, 596. doi:10.3390/jcm9020596.

Santamaria, J., 2017. De la simplicité à la complexité. Le Monde, Voyage dans le cosmos.

Satsuma, J., Willox, R., Ramani, A., Grammaticos, B., Carstea, A., 2004. Extending the sir epidemic model. Physica A: Statistical Mechanics and its Applications 336, 369-375. doi:https://doi.org/10.1016/j.physa.2003.12.035.

Savageau, M.A., 1970. Biochemical systems analysis: Iii. dynamic solutions using a power-law approximation. Journal of Theoretical Biology 26, 215-226. doi:10.1016/S0022-5193(70)80013-3.

Scharbarg, E., Moog, C.H., Mauduit, N., Califano, C., 2020. From the hospital scale to nationwide: observability and identification of models for the covid-19 epidemic waves. Annual Reviews in Control 50, 409-416. doi:10.1016/j.arcontrol.2020.09.007.

Sher, P.H., Smith, J.M., Dalgarno, P.A., Warburton, R.J., Chen, X., Dobson, P.J., Daniels, S.M., Pickett, N.L., O'Brien, P., 2008. Power law carrier dynamics in semiconductor nanocrystals at nanosecond timescales. Applied Physics Letters 92, 101111. doi:10.1063/1.2894193.

Tanner, L.H., 1979. The spreading of silicone oil drops on horizontal surfaces.

Journal of Physics D: Applied Physics 12, 1473-1484. doi:10.1088/0022-3727/12/9/009.

Tolle, J., 2003. Can growth be faster than exponential, and just how slow is the logarithm? The Mathematical Gazette 87, 522-525. doi:10.1017/S0025557200173802.

Trigeassou, J., Maamri, N., 2019. The Distributed Model of the Fractional Integrator. John Wiley & Sons, Ltd. chapter 6. pp. 127-158. doi:10.1002/9781119476917.ch6.

Tuan, N.H., Mohammadi, H., Rezapour, S., 2020. A mathematical model for covid-19 transmission by using the caputo fractional derivative. Chaos, Solitons & Fractals 140, 110107. doi:https://doi.org/10.1016/j.chaos.2020.110107.

Verhulst, P.F., 1838. Notice sur la loi que la population poursuit dans son accroissement. Correspondance mathématique et physique, 113-121.

Verhulst, P.F., 1845. Recherches mathématiques sur la loi d'accroissement de la population. Nouveaux mémoires de l'Académie Royale des Sciences et Belles-Lettres de Bruxelles 18, 14-54.

Viboud, C., Simonsen, L., Chowell, G., 2016. A generalized-growth model to characterize the early ascending phase of infectious disease outbreaks. Epidemics 15, 27-37. doi:https://doi.org/10.1016/j.epidem.2016.01.002.

Wang, X.S., Wu, J., Yang, Y., 2012. Richards model revisited: Validation by and application to infection dynamics. Journal of Theoretical Biology 313,
<table><caption>Table E.6: Basic period used for identification of the vaccinated from May 8 to May 14 2021</caption><thead><tr><th>date</th><th>08/05</th><th>09/05</th><th>10/05</th><th>11/05</th><th>12/05</th><th>13/05</th><th>14/05</th></tr></thead><tbody><tr><th>b<sub>k</sub></th><td>8257962</td><td>8464151</td><td>8670293</td><td>8876403</td><td>9082490</td><td>9288558</td><td>9494611</td></tr><tr><th>a<sub>k</sub></th><td>8316714</td><td>8371235</td><td>8623700</td><td>8917609</td><td>9183962</td><td>9260435</td><td>9460813</td></tr><tr><th>E<sub>r</sub></th><td>0.0071</td><td>-0.0110</td><td>-0.0054</td><td>0.0046</td><td>0.0112</td><td>-0.0030</td><td>-0.0036</td></tr></tbody></table>

<table><caption>Table E.7: Periods used for prediction of the vaccinated from May 15 to June 11 2021: mean(E<sub>r</sub>) = -0.02898 for week 1; mean(E<sub>r</sub>) = -0.05833 for week 2; mean(E<sub>r</sub>) = -0.04167 for week 3; mean(E<sub>r</sub>) = -0.01112 for week 4</caption><tbody><tr><th rowspan="4">Week 1</th><th>date</th><td>15/05</td><td>16/05</td><td>17/05</td><td>18/05</td><td>19/05</td><td>20/05</td><td>21/05</td></tr><tr><th>b<sub>k</sub></th><td>9700651</td><td>9906680</td><td>10112698</td><td>10318707</td><td>10524708</td><td>10524708</td><td>10936687</td></tr><tr><th>a<sub>k</sub></th><td>9579057</td><td>9612256</td><td>9789129</td><td>9978383</td><td>10158012</td><td>10359164</td><td>10577373</td></tr><tr><th>E<sub>r</sub></th><td>-0.0125</td><td>-0.0297</td><td>-0.0320</td><td>-0.0330</td><td>-0.0348</td><td>-0.0157</td><td>-0.0329</td></tr><tr><th rowspan="4">Week 2</th><th>date</th><td>22/05</td><td>23/05</td><td>24/05</td><td>25/05</td><td>26/05</td><td>27/05</td><td>28/05</td></tr><tr><th>b<sub>k</sub></th><td>11142666</td><td>11348640</td><td>11554607</td><td>11760569</td><td>11966527</td><td>12172479</td><td>12378426</td></tr><tr><th>a<sub>k</sub></th><td>10683925</td><td>10709922</td><td>10761760</td><td>10985380</td><td>11208870</td><td>11472324</td><td>11787631</td></tr><tr><th>E<sub>r</sub></th><td>-0.0412</td><td>-0.0563</td><td>-0.0686</td><td>-0.0659</td><td>-0.0633</td><td>-0.0575</td><td>-0.0477</td></tr><tr><th rowspan="4">Week 3</th><th>date</th><td>29/05</td><td>30/05</td><td>31/05</td><td>01/06</td><td>02/06</td><td>03/06</td><td>04/06</td></tr><tr><th>b<sub>k</sub></th><td>12584370</td><td>12790309</td><td>12996244</td><td>13202175</td><td>13408103</td><td>13614027</td><td>13819948</td></tr><tr><th>a<sub>k</sub></th><td>11968407</td><td>12009982</td><td>12249961</td><td>12550529</td><td>12825866</td><td>13136728</td><td>13496370</td></tr><tr><th>E<sub>r</sub></th><td>-0.0489</td><td>-0.0610</td><td>-0.0574</td><td>-0.0494</td><td>-0.0434</td><td>-0.0351</td><td>-0.0234</td></tr><tr><th rowspan="4">Week 4</th><th>date</th><td>05/06</td><td>06/06</td><td>07/06</td><td>08/06</td><td>09/06</td><td>10/06</td><td>11/06</td></tr><tr><th>b<sub>k</sub></th><td>14025865</td><td>14231780</td><td>14437691</td><td>14643599</td><td>14849505</td><td>15055407</td><td>15261307</td></tr><tr><th>a<sub>k</sub></th><td>13717098</td><td>13777590</td><td>14075211</td><td>14411546</td><td>14728908</td><td>15064328</td><td>15469804</td></tr><tr><th>E<sub>r</sub></th><td>-0.0220</td><td>-0.0319</td><td>-0.0251</td><td>-0.0158</td><td>-0.0081</td><td>0.0006</td><td>0.0137</td></tr></tbody></table>

12 – 19. doi:https://doi.org/10.1016/j.jtbi.2012.07.024.

Wu, K., Darcet, D., Wang, Q., Sornette, D., 2020. Generalized logistic growth modeling of the covid-19 outbreak: comparing the dynamics in the 29 provinces in china and in the rest of the world. Nonlinear Dynamics 101, 1561–1581. doi:10.1007/s11071-020-05862-6.

Xu, C., Yu, Y., Chen, Y., Lu, Z., 2020. Forecast analysis of the epidemics trend of covid-19 in the usa by a generalized fractional-order seir model. Nonlinear Dynamics 101, 1621–1634. doi:10.1007/s11071-020-05946-3.

Young, P., Chen, F., 2021. Monitoring and forecasting the covid-19 epidemic in the uk. Annual Reviews in Control doi:10.1016/j.arcontrol.2021.01.004.

Zhao, S., Lin, Q., Ran, J., Musa, S.S., Yang, G., Wang, W., Lou, Y., Gao, D., Yang, L., He, D., Wang, M.H., 2020a. Preliminary estimation of the basic reproduction number of novel coronavirus (2019-ncov) in china, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak. International Journal of Infectious Diseases 92, 214 – 217. doi:https://doi.org/10.1016/j.ijid.2020.01.050.

Zhao, Y.F., Shou, M.H., Wang, Z.X., 2020b. Prediction of the number of patients infected with covid-19 based on rolling grey verhulst models. International journal of environmental research and public health 17, 4582. doi:10.3390/ijerph17124582.

Zhou, X., Ma, X., Hong, N., Su, L., Ma, Y., He, J., Jiang, H., Liu, C., Shan, G., Zhu, W., Zhang, S., Long, Y., 2020. Forecasting the worldwide spread of covid-19 based on logistic model and seir model. medRxiv doi:10.1101/2020.03.26.20044289.

Zilany, M., Bruce, I., Nelson, P., Carney, L., 2009. A phenomenological model of the synapse between the inner hair cell and auditory nerve: long-term adaptation with power-law dynamics. Journal of the Acoustical Society of America 126, 2390–2412. doi:10.1121/1.3238250.
<table><caption>Table E.6: Basic period used for identification of the vaccinated from May 8 to May 14 2021</caption><thead><tr><th>date</th><th>08/05</th><th>09/05</th><th>10/05</th><th>11/05</th><th>12/05</th><th>13/05</th><th>14/05</th></tr></thead><tbody><tr><th>b<sub>k</sub></th><td>8257962</td><td>8464151</td><td>8670293</td><td>8876403</td><td>9082490</td><td>9288558</td><td>9494611</td></tr><tr><th>a<sub>k</sub></th><td>8316714</td><td>8371235</td><td>8623700</td><td>8917609</td><td>9183962</td><td>9260435</td><td>9460813</td></tr><tr><th>E<sub>r</sub></th><td>0.0071</td><td>-0.0110</td><td>-0.0054</td><td>0.0046</td><td>0.0112</td><td>-0.0030</td><td>-0.0036</td></tr></tbody></table>

<table><caption>Table E.7: Periods used for prediction of the vaccinated from May 15 to June 11 2021: mean(E<sub>r</sub>) = -0.02898 for week 1; mean(E<sub>r</sub>) = -0.05833 for week 2; mean(E<sub>r</sub>) = -0.04167 for week 3; mean(E<sub>r</sub>) = -0.01112 for week 4</caption><tbody><tr><th rowspan="4">Week 1</th><th>date</th><td>15/05</td><td>16/05</td><td>17/05</td><td>18/05</td><td>19/05</td><td>20/05</td><td>21/05</td></tr><tr><th>b<sub>k</sub></th><td>9700651</td><td>9906680</td><td>10112698</td><td>10318707</td><td>10524708</td><td>10524708</td><td>10936687</td></tr><tr><th>a<sub>k</sub></th><td>9579057</td><td>9612256</td><td>9789129</td><td>9978383</td><td>10158012</td><td>10359164</td><td>10577373</td></tr><tr><th>E<sub>r</sub></th><td>-0.0125</td><td>-0.0297</td><td>-0.0320</td><td>-0.0330</td><td>-0.0348</td><td>-0.0157</td><td>-0.0329</td></tr><tr><th rowspan="4">Week 2</th><th>date</th><td>22/05</td><td>23/05</td><td>24/05</td><td>25/05</td><td>26/05</td><td>27/05</td><td>28/05</td></tr><tr><th>b<sub>k</sub></th><td>11142666</td><td>11348640</td><td>11554607</td><td>11760569</td><td>11966527</td><td>12172479</td><td>12378426</td></tr><tr><th>a<sub>k</sub></th><td>10683925</td><td>10709922</td><td>10761760</td><td>10985380</td><td>11208870</td><td>11472324</td><td>11787631</td></tr><tr><th>E<sub>r</sub></th><td>-0.0412</td><td>-0.0563</td><td>-0.0686</td><td>-0.0659</td><td>-0.0633</td><td>-0.0575</td><td>-0.0477</td></tr><tr><th rowspan="4">Week 3</th><th>date</th><td>29/05</td><td>30/05</td><td>31/05</td><td>01/06</td><td>02/06</td><td>03/06</td><td>04/06</td></tr><tr><th>b<sub>k</sub></th><td>12584370</td><td>12790309</td><td>12996244</td><td>13202175</td><td>13408103</td><td>13614027</td><td>13819948</td></tr><tr><th>a<sub>k</sub></th><td>11968407</td><td>12009982</td><td>12249961</td><td>12550529</td><td>12825866</td><td>13136728</td><td>13496370</td></tr><tr><th>E<sub>r</sub></th><td>-0.0489</td><td>-0.0610</td><td>-0.0574</td><td>-0.0494</td><td>-0.0434</td><td>-0.0351</td><td>-0.0234</td></tr><tr><th rowspan="4">Week 4</th><th>date</th><td>05/06</td><td>06/06</td><td>07/06</td><td>08/06</td><td>09/06</td><td>10/06</td><td>11/06</td></tr><tr><th>b<sub>k</sub></th><td>14025865</td><td>14231780</td><td>14437691</td><td>14643599</td><td>14849505</td><td>15055407</td><td>15261307</td></tr><tr><th>a<sub>k</sub></th><td>13717098</td><td>13777590</td><td>14075211</td><td>14411546</td><td>14728908</td><td>15064328</td><td>15469804</td></tr><tr><th>E<sub>r</sub></th><td>-0.0220</td><td>-0.0319</td><td>-0.0251</td><td>-0.0158</td><td>-0.0081</td><td>0.0006</td><td>0.0137</td></tr></tbody></table>

12 – 19. doi:https://doi.org/10.1016/j.jtbi.2012.07.024.

Wu, K., Darcet, D., Wang, Q., Sornette, D., 2020. Generalized logistic growth modeling of the covid-19 outbreak: comparing the dynamics in the 29 provinces in china and in the rest of the world. Nonlinear Dynamics 101, 1561–1581. doi:10.1007/s11071-020-05862-6.

Xu, C., Yu, Y., Chen, Y., Lu, Z., 2020. Forecast analysis of the epidemics trend of covid-19 in the usa by a generalized fractional-order seir model. Nonlinear Dynamics 101, 1621–1634. doi:10.1007/s11071-020-05946-3.

Young, P., Chen, F., 2021. Monitoring and forecasting the covid-19 epidemic in the uk. Annual Reviews in Control doi:10.1016/j.arcontrol.2021.01.004.

Zhao, S., Lin, Q., Ran, J., Musa, S.S., Yang, G., Wang, W., Lou, Y., Gao, D., Yang, L., He, D., Wang, M.H., 2020a. Preliminary estimation of the basic reproduction number of novel coronavirus (2019-ncov) in china, from 2019 to 2020: A data-driven analysis in the early phase of the outbreak. International Journal of Infectious Diseases 92, 214 – 217. doi:https://doi.org/10.1016/j.ijid.2020.01.050.

Zhao, Y.F., Shou, M.H., Wang, Z.X., 2020b. Prediction of the number of patients infected with covid-19 based on rolling grey verhulst models. International journal of environmental research and public health 17, 4582. doi:10.3390/ijerph17124582.

Zhou, X., Ma, X., Hong, N., Su, L., Ma, Y., He, J., Jiang, H., Liu, C., Shan, G., Zhu, W., Zhang, S., Long, Y., 2020. Forecasting the worldwide spread of covid-19 based on logistic model and seir model. medRxiv doi:10.1101/2020.03.26.20044289.

Zilany, M., Bruce, I., Nelson, P., Carney, L., 2009. A phenomenological model of the synapse between the inner hair cell and auditory nerve: long-term adaptation with power-law dynamics. Journal of the Acoustical Society of America 126, 2390–2412. doi:10.1121/1.3238250.