A locally sequential refinement of the growth dynamics identification

Figures & data

Figure 1. High-frequency oscillations during the reconstruction without a general regularization, the example from [13].

Figure 2. Low-frequency oscillations during the reconstruction with a general regularization, the example from [13].

Table 1. The determination of the parent element of the invariant family: item 10 – the parent element $\text{Å}$, item 16 – the element $\check{a}$ with the minimum shift measure ${\mathcal{M}}_{min}$ = 0.005315, $\mathrm{\wp }$ = −0.10056.

#	λ	$\mathcal{M}$	$a_1^{\mathrm{\prime }\mathrm{\prime }}$	$a_2^{\mathrm{\prime }\mathrm{\prime }}{|}_{t=0}$	$\int a_2^{2}dt$	$\int {\left({\displaystyle \frac{d{a}_2}{dt}}\right)}^{2}dt$	$\int {\left({\displaystyle \frac{d^2{a}_2}{d{t}^2}}\right)}^{2}dt$	$\mathrm{\Lambda }$	${\tilde{a}}_2{|}_{t=0}=-\mathrm{\Lambda }/y_0-{\displaystyle \frac{y_0}{2y{|}_{t\to \mathrm{\infty }}}}$
1	20	157521.8	20.2	396.8889	8053.236	79.43446	0.7887467	−20.10056	402.01
2	10	39806.34	10.2	199.5140	2033.389	20.05461	0.1991119	−10.10056	202.01
3	5	10169.76	5.2	100.8439	518.5167	5.113462	0.0507692	−5.100563	102.01
4	1	475.4257	1.2	21.80286	24.14696	0.2380336	0.0023648	−1.100562	22.01
5	0.2	35.66094	0.3999984	5.969981	1.802236	0.0177391	0.0001765	−0.300560	6.009968
6	0.1	15.93609	0.3	3.990102	0.802846	0.0078935	0.786686510^−4	−0.200562	4.01
7	0.017814	5.593619	0.2178144	2.362757	0.279923	0.0027456	0.274284810^−4	−0.118376	2.366288
8	0.00046581	4.087380	0.2004658	2.019224	0.203960	0.0019984	0.199843410^−4	−0.101028	2.019316
9	0.00001	4.050987	0.20001	2.010198	0.202126	0.0019804	0.198046010^−4	−0.100572	2.0102
10	0.0	4.050191	0.2	2.01	0.202086	0.0019800	0.198006710^−4	−0.100562	2.01
11	−0.8153710^−5	4.049541	0.1999918	2.009839	0.202053	0.0019797	0.197974610^−4	−0.100554	2.009837
12	−0.00001	4.049394	0.19999	2.009802	0.202046	0.0019796	0.197967410^−4	−0.100552	2.0098
13	−0.001	3.970928	0.199	1.990198	0.198092	0.0019407	0.194092410^−4	−0.099562	1.99
14	−0.01	3.292850	0.19	1.811979	0.163937	0.0016050	0.160621410^−4	−0.090562	1.81
15	−0.1	0.005882	0.1	0.029703	0.000027	0.2333710^−7	0.933528710^−9	−0.000562	0.01
16	−0.10056	0.005315	0.0994375	0.018563	0.000019	0.1388910^−7	0.123153610^−9	0.0	−0.00125

Figure 3. The observations [35] and the model-calculated population with constant parameters: 1 – the sample $\{y_i^{(\delta )}{\}}_{\overline{i=1,11}}$, 2 – the direct problem solution $y{|}_{a_{0,1,2}^{(1)}=const}$, 3 – the inflection area of the sample, 4 – the inflection point of the solution $y{|}_{a_{0,1,2}^{(1)}}$.

Figure 4. Processing of the observations by smoothing spline: 1 – the population, 2 – the growth rate of the population, 3 – the inflection points of the population.

Figure 5. The reconstruction: 1– $a_2^{(3)}$ by the locally sequential refinement, ${\rho }_{rms}^{}{|}_{a_{1,2}^{(3)}}$ = 0.004469; 2– ${\tilde{a}}_2$ by the formula (18),${\rho }_{rms}^{}{|}_{a_1^{(3)},{\tilde{a}}_2}$ = 0.00937; $1/a_1^{(3)}$ = 33.0.

Table 2. The solutions to the inverse problems (13) – (16) for the different α₁ and the allocation of the parent element: item 7 – the minimum of the stabilizer, item 10 – the parent element.

#	$a_1$	Stabilizer (13)	$a_2{|}_{t=0}$	$a_2{|}_{t=160}$	$\mathrm{\Lambda }$	Shooting $a_2{|}_{t=0}$
1	0.03300	0.28686580	0.0161405	0.0285415	0.0024	−0.12009
2	0.03200	0.22001042	0.00105456	0.0274765	0.0014	−0.07463
3	0.03100	0.19202371	0.00009819	0.0271296	0.0004	−0.02918
4	0.03090	0.19035260	0.0000856769	0.0270345	0.0003	−0.02463
5	0.03080	0.18937900	0.0000811075	0.0269352	0.0002	−0.02009
6	0.03070	0.18744988	0.0000705780	0.0268116	0.0001	−0.01554
7	0.03060	0.18635412	0.0000495330	0.0267949	0	−0.01100
8	0.03050	0.18641411	0.0000024956	0.0260564	−0.00001	−0.00645
9	0.03040	0.18928670	0.0000013815	0.0259858	−0.00020	−0.00191
10	0.0303596	0.19071940	−0.0000786108	0.0259731	−0.000241	−0.000072727
11	0.03030	0.19314427	−0.0000854877	0.0257162	−0.0003	0.00263
12	0.03020	0.20192082	−0.000186326	0.0256957	−0.0004	0.00718
13	0.03010	0.20901307	−0.000220475	0.0252427	−0.0005	0.01172
14	0.03000	0.22177508	−0.00692703	0.0249941	−0.0006	0.01627
15	0.02900	0.31305743	−0.02434342	0.0242376	−0.0016	0.06172

Figure 6. The total coronavirus cases in Germany, the source [59]: 1 – the sample $y^{(\delta )}$, 2 – the spline approximation, $\tilde{y}$.

Figure 7. The spline approximation of the growth rate, $d\tilde{y}/dt$, the period of the renewal and decline cycle, $1/a_1$ = 6.98 days.

Figure 8. The parameter ${\tilde{a}}_2^{(6)}$ determined by the formula (18): (a) the initial period, (b) the attenuation period .

Figure 9. The parameter reconstruction: (a) the general regularization (5) without the locally sequential refinement, (b) the general regularization (5) together with the locally sequential refinement, (c) the used approximation scheme.

Figure 10. The patterns of the COVID-19 spread in Germany (the symbols $◯1\text{--}◯7$ mark the moments of the dynamics changes).

Figure 11. The reconstruction into the large consistency corridor, the worst residuals with observations: ${\rho }_{max}^{}{|}_{i=7}=1.4$, ${\rho }_{max}^{}{|}_{i=49}=167$, ${\rho }_{max}^{}{|}_{i=105}=1261$, ${\rho }_{max}^{}{|}_{i=424}=1855$.

Figure 12. The parameters reconstruction for the different countries, the source of the COVID-19 cases [59]: (a) Sweden; (b) Switzerland; (c) Brazil.

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