# Exploring the Dynamics of COVID-19 with a Novel Family of Models

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. The New Family of Distributions

**Definition 1.**

**Definition 2.**

**Definition 3.**

**Definition 4.**

**Definition 5.**

#### 2.1. Asymptotics

#### 2.2. Useful Representation

#### 2.3. Order Statistics (OS)

#### 2.4. Entropy

#### 2.4.1. Rényi Entropy

#### 2.4.2. Tsallis Entropy

#### 2.4.3. Havrda and Charvat Entropy

#### 2.5. Inference

## 3. Mathematical Characteristics of Sub-Model

#### 3.1. Useful Representation

#### 3.2. Moments and Related Measures

**Theorem 1.**

**Proof.**

**Corollary 1.**

**Theorem 2.**

**Proof.**

**Theorem 3.**

**Proof.**

**Theorem 4.**

**Proof.**

**Theorem 5.**

**Proof.**

**Theorem 6.**

**Proof.**

**Corollary 2.**

**Theorem 7.**

**Proof.**

**Theorem 8.**

**Proof.**

#### 3.3. Residuals and Related Measures

#### 3.3.1. Mean Residual and Mean Inactivity Time

#### 3.4. Entropy

#### 3.4.1. Rényi Entropy

#### 3.4.2. Tsallis Entropy

#### 3.4.3. Havrda and Charvat Entropy

#### 3.5. Order Statistics (OS)

## 4. Inference

#### 4.1. Maximum Likelihood Estimation Method (MLE)

_{1}, x

_{2}, x

_{3}, …, x

_{n}of size n from (6), is given by

#### 4.2. Anderson–Darling Estimation Method (ADE-M)

#### 4.3. Cram’er–Von Mises Estimation Method (CVME-M)

#### 4.4. Least Squares Estimation Method (LSE-M)

#### 4.5. Weighted Least-Squares Estimation Method (WLSE-M)

#### 4.6. Maximum Product of Spacings Estimation Method (MPSE-M)

## 5. Simulation Study

- We set the beginning values for the parameters of our suggested model.
- From our suggested model, we have produced random data sets using the inverse of cdf.
- Use several estimate techniques to find estimators for our proposed model.
- Calculate the bias, MSE, and MRE for each estimator using each estimating technique.
- Repeat steps 1 through 4, 500 times.

## 6. Analysis of COVID-19 Data

## 7. Summary and Conclusions

## 8. Future Recommendations

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Application I: United Kingdom COVID-19 Mortality Rate**

**Application II: Europe COVID-19 Mortality Rate**

**Application III: China COVID-19 Mortality Rate**

**Application IV: United Kingdom COVID-19 Mortality Rate**

**Application V: Nepal COVID-19 Mortality Rate**

**Application VI: Netherland COVID-19 Mortality Rate**

**Application VII: Italy COVID-19 Mortality Rate**

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Model | Base Model | New Model |
---|---|---|

Exponential | $1-{e}^{-\gimel x}$ | ${\left(\right)close="|"\; separators="|">{\left(1+\alpha \right)}^{\left[1-{e}^{-\gimel x}\right]}-{\alpha}^{{\left[1-{e}^{-\gimel x}\right]}^{2}}}_{}\alpha ,\gimel 0$ |

Weibull | $1-{e}^{-\gimel {x}^{\beth}}$ | ${\left(\right)close="|"\; separators="|">{\left(1+\alpha \right)}^{\left[1-{e}^{-\gimel {x}^{\beth}}\right]}-{\alpha}^{{\left[1-{e}^{-\gimel {x}^{\beth}}\right]}^{2}}}_{}\alpha ,\gimel ,\beth 0$ |

Rayleigh | $1-{e}^{-{x}^{2}/2{\gimel}^{2}}$ | ${\left(\right)close="|"\; separators="|">{\left(1+\alpha \right)}^{\left[1-{e}^{-{x}^{2}/2{\gimel}^{2}}\right]}-{\alpha}^{{\left[1-{e}^{-{x}^{2}/2{\gimel}^{2}}\right]}^{2}}}_{}\alpha ,\gimel ,\beth 0$ |

Gompertz | $1-{e}^{-\gimel \left({e}^{\beth x}-1\right)}$ | ${\left(\right)close="|"\; separators="|">{\left(1+\alpha \right)}^{\left[1-{e}^{-\gimel \left({e}^{\beth x}-1\right)}\right]}-{\alpha}^{{\left[1-{e}^{-\gimel \left({e}^{\beth x}-1\right)}\right]}^{2}}}_{}\alpha ,\gimel ,\beth 0$ |

Lomax | $1-{\left[1+\left(x/\gimel \right)\right]}^{-\beth}$ | ${\left(\right)close="|"\; separators="|">{\left(1+\alpha \right)}^{\left[1-{\left[1+\left(x/\gimel \right)\right]}^{-\beth}\right]}-{\alpha}^{{\left[1-{\left[1+\left(x/\gimel \right)\right]}^{-\beth}\right]}^{2}}}_{}\alpha ,\gimel ,\beth 0$ |

Burr | $1-{\left[1+{x}^{\gimel}\right]}^{-\beth}$ | ${\left(\right)close="|"\; separators="|">{\left(1+\alpha \right)}^{\left[1-{\left[1+{x}^{\gimel}\right]}^{-\beth}\right]}-{\alpha}^{{\left[1-{\left[1+{x}^{\gimel}\right]}^{-\beth}\right]}^{2}}}_{}\alpha ,\gimel ,\beth 0$ |

Pareto | $1-{\left(x/{x}_{min}\right)}^{\beth}$ | ${\left(\right)close="|"\; separators="|">{\left(1+\alpha \right)}^{\left[1-{\left(x/{x}_{min}\right)}^{\beth}\right]}-{\alpha}^{{\left[1-{\left(x/{x}_{min}\right)}^{\beth}\right]}^{2}}}_{}\alpha ,\beth 0$ |

Half Log-Logistic | $\left(1-{e}^{-\gimel x}\right)/\left(1+{e}^{-\gimel x}\right)$ | ${\left(\right)close="|"\; separators="|">\left[{\left(1+\alpha \right)}^{\left[\left(1-{e}^{-\gimel x}\right)/\left(1+{e}^{-\gimel x}\right)\right]}-{\alpha}^{{\left[\left(1-{e}^{-\gimel x}\right)/\left(1+{e}^{-\gimel x}\right)\right]}^{2}}\right]}_{}\alpha ,\gimel 0$ |

Kumaraswamy | $1-{\left(1-{x}^{\gimel}\right)}^{\beth}$ | ${\left(\right)close="|"\; separators="|">{\left(1+\alpha \right)}^{\left[1-{\left(1-{x}^{\gimel}\right)}^{\beth}\right]}-{\alpha}^{{\left[1-{\left(1-{x}^{\gimel}\right)}^{\beth}\right]}^{2}}}_{}\alpha ,\gimel ,\beth 0$ |

Power Function | ${\left[x/M\right]}^{\theta}$ | ${\left(\right)close="|"\; separators="|">{\left(1+\alpha \right)}^{{\left[x/M\right]}^{\theta}}-{\alpha}^{{\left[x/M\right]}^{2\theta}}}_{}\alpha ,\theta 0,M\ge x$ |

Uniform | $x/M$ | ${\left(\right)close="|"\; separators="|">{\left(1+\alpha \right)}^{\left[x/M\right]}-{\alpha}^{{\left[x/M\right]}^{2}}}_{}\alpha 0,M\ge x$ |

$\mathit{n}$ | Est. | Par. | MLE-M | ADE-M | CVME-M | MPSE-M | LSE-M | WLSE-M |
---|---|---|---|---|---|---|---|---|

20 | BIAS | $\widehat{\alpha}$ | 0.3769 | 0.3296 | 0.3288 | 0.3120 | 0.3697 | 0.3932 |

$\widehat{\theta}$ | 0.3723 | 0.3683 | 0.3915 | 0.3293 | 0.3241 | 0.4479 | ||

MSE | $\widehat{\alpha}$ | 0.2035 | 0.1671 | 0.1498 | 0.1352 | 0.1689 | 0.1946 | |

$\widehat{\theta}$ | 0.2903 | 0.2374 | 0.3020 | 0.1866 | 0.1949 | 0.4083 | ||

MRE | $\widehat{\alpha}$ | 0.7538 | 0.6593 | 0.6576 | 0.6240 | 0.7394 | 0.7864 | |

$\widehat{\theta}$ | 0.2482 | 0.2455 | 0.2610 | 0.2196 | 0.2161 | 0.2986 | ||

40 | BIAS | $\widehat{\alpha}$ | 0.3226 | 0.3177 | 0.2933 | 0.2955 | 0.3218 | 0.3115 |

$\widehat{\theta}$ | 0.2652 | 0.2654 | 0.2669 | 0.2128 | 0.2719 | 0.2403 | ||

MSE | $\widehat{\alpha}$ | 0.1698 | 0.1469 | 0.1170 | 0.1294 | 0.1392 | 0.1394 | |

$\widehat{\theta}$ | 0.1128 | 0.1173 | 0.1192 | 0.0843 | 0.1251 | 0.1109 | ||

MRE | $\widehat{\alpha}$ | 0.6451 | 0.6354 | 0.5865 | 0.5910 | 0.6435 | 0.6229 | |

$\widehat{\theta}$ | 0.1768 | 0.1770 | 0.1780 | 0.1419 | 0.1813 | 0.1602 | ||

80 | BIAS | $\widehat{\alpha}$ | 0.2865 | 0.2372 | 0.2258 | 0.2450 | 0.2577 | 0.2499 |

$\widehat{\theta}$ | 0.1836 | 0.1790 | 0.1891 | 0.1689 | 0.1741 | 0.1767 | ||

MSE | $\widehat{\alpha}$ | 0.1457 | 0.0925 | 0.0828 | 0.0998 | 0.1098 | 0.1059 | |

$\widehat{\theta}$ | 0.0632 | 0.0521 | 0.0550 | 0.0538 | 0.0547 | 0.0535 | ||

MRE | $\widehat{\alpha}$ | 0.5730 | 0.4744 | 0.4516 | 0.4901 | 0.5154 | 0.4999 | |

$\widehat{\theta}$ | 0.1224 | 0.1193 | 0.1261 | 0.1126 | 0.1161 | 0.1178 | ||

100 | BIAS | $\widehat{\alpha}$ | 0.2538 | 0.2231 | 0.2604 | 0.2140 | 0.2631 | 0.2727 |

$\widehat{\theta}$ | 0.1768 | 0.1661 | 0.1741 | 0.1501 | 0.1812 | 0.1577 | ||

MSE | $\widehat{\alpha}$ | 0.1142 | 0.0816 | 0.1062 | 0.0832 | 0.1109 | 0.1151 | |

$\widehat{\theta}$ | 0.0556 | 0.0482 | 0.0519 | 0.0413 | 0.0528 | 0.0449 | ||

MRE | $\widehat{\alpha}$ | 0.5076 | 0.4462 | 0.5207 | 0.4279 | 0.5262 | 0.5455 | |

$\widehat{\theta}$ | 0.1178 | 0.1107 | 0.1161 | 0.1001 | 0.1208 | 0.1051 | ||

150 | BIAS | $\widehat{\alpha}$ | 0.2357 | 0.2039 | 0.2321 | 0.2295 | 0.2092 | 0.2634 |

$\widehat{\theta}$ | 0.1523 | 0.1357 | 0.1600 | 0.1158 | 0.1319 | 0.1414 | ||

MSE | $\widehat{\alpha}$ | 0.1210 | 0.0779 | 0.0974 | 0.1005 | 0.0906 | 0.1184 | |

$\widehat{\theta}$ | 0.0479 | 0.0300 | 0.0528 | 0.0248 | 0.0431 | 0.0302 | ||

MRE | $\widehat{\alpha}$ | 0.4713 | 0.4078 | 0.4641 | 0.4590 | 0.4183 | 0.5269 | |

$\widehat{\theta}$ | 0.1015 | 0.0905 | 0.1067 | 0.0772 | 0.0879 | 0.0942 | ||

200 | BIAS | $\widehat{\alpha}$ | 0.2177 | 0.1687 | 0.2269 | 0.1551 | 0.2126 | 0.2236 |

$\widehat{\theta}$ | 0.1270 | 0.1073 | 0.1115 | 0.0926 | 0.1097 | 0.1222 | ||

MSE | $\widehat{\alpha}$ | 0.1030 | 0.0600 | 0.0978 | 0.0514 | 0.0872 | 0.0980 | |

$\widehat{\theta}$ | 0.0273 | 0.0193 | 0.0217 | 0.0159 | 0.0200 | 0.0248 | ||

MRE | $\widehat{\alpha}$ | 0.4353 | 0.3375 | 0.4538 | 0.3101 | 0.4251 | 0.4472 | |

$\widehat{\theta}$ | 0.0847 | 0.0715 | 0.0743 | 0.0617 | 0.0731 | 0.0815 |

$\mathit{n}$ | Est. | Par. | MLE-M | ADE-M | CVME-M | MPSE-M | LSE-M | WLSE-M |
---|---|---|---|---|---|---|---|---|

20 | BIAS | $\widehat{\alpha}$ | 0.4524 | 0.3208 | 0.3965 | 0.2681 | 0.3433 | 0.3426 |

$\widehat{\theta}$ | 0.1327 | 0.1265 | 0.1600 | 0.1030 | 0.1352 | 0.1349 | ||

MSE | $\widehat{\alpha}$ | 0.2718 | 0.1350 | 0.1910 | 0.1034 | 0.1455 | 0.1650 | |

$\widehat{\theta}$ | 0.0383 | 0.0277 | 0.0426 | 0.0238 | 0.0330 | 0.0374 | ||

MRE | $\widehat{\alpha}$ | 0.6032 | 0.4278 | 0.5287 | 0.3575 | 0.4577 | 0.4568 | |

$\widehat{\theta}$ | 0.2655 | 0.2529 | 0.3199 | 0.2060 | 0.2705 | 0.2698 | ||

40 | BIAS | $\widehat{\alpha}$ | 0.3773 | 0.2905 | 0.3601 | 0.2529 | 0.3060 | 0.3293 |

$\widehat{\theta}$ | 0.0824 | 0.0786 | 0.1078 | 0.0616 | 0.0942 | 0.0795 | ||

MSE | $\widehat{\alpha}$ | 0.2131 | 0.1143 | 0.1548 | 0.1018 | 0.1112 | 0.1374 | |

$\widehat{\theta}$ | 0.0141 | 0.0115 | 0.0183 | 0.0083 | 0.0152 | 0.0106 | ||

MRE | $\widehat{\alpha}$ | 0.5031 | 0.3873 | 0.4802 | 0.3372 | 0.4080 | 0.4390 | |

$\widehat{\theta}$ | 0.1649 | 0.1572 | 0.2157 | 0.1231 | 0.1884 | 0.1590 | ||

80 | BIAS | $\widehat{\alpha}$ | 0.2763 | 0.2139 | 0.3093 | 0.2312 | 0.2794 | 0.2613 |

$\widehat{\theta}$ | 0.0497 | 0.0542 | 0.0661 | 0.0419 | 0.0647 | 0.0602 | ||

MSE | $\widehat{\alpha}$ | 0.1272 | 0.0754 | 0.1120 | 0.0938 | 0.0978 | 0.0858 | |

$\widehat{\theta}$ | 0.0053 | 0.0061 | 0.0081 | 0.0035 | 0.0078 | 0.0062 | ||

MRE | $\widehat{\alpha}$ | 0.3684 | 0.2852 | 0.4125 | 0.3083 | 0.3725 | 0.3484 | |

$\widehat{\theta}$ | 0.0993 | 0.1084 | 0.1321 | 0.0838 | 0.1293 | 0.1203 | ||

100 | BIAS | $\widehat{\alpha}$ | 0.2598 | 0.1988 | 0.2691 | 0.1739 | 0.2452 | 0.2783 |

$\widehat{\theta}$ | 0.0428 | 0.0445 | 0.0514 | 0.0311 | 0.0547 | 0.0606 | ||

MSE | $\widehat{\alpha}$ | 0.1216 | 0.0647 | 0.0946 | 0.0633 | 0.0823 | 0.0992 | |

$\widehat{\theta}$ | 0.0045 | 0.0043 | 0.0045 | 0.0024 | 0.0047 | 0.0065 | ||

MRE | $\widehat{\alpha}$ | 0.3464 | 0.2650 | 0.3589 | 0.2318 | 0.3270 | 0.3711 | |

$\widehat{\theta}$ | 0.0856 | 0.0890 | 0.1028 | 0.0623 | 0.1094 | 0.1212 | ||

150 | BIAS | $\widehat{\alpha}$ | 0.2659 | 0.1839 | 0.2481 | 0.1731 | 0.2767 | 0.2420 |

$\widehat{\theta}$ | 0.0446 | 0.0318 | 0.0465 | 0.0334 | 0.0446 | 0.0439 | ||

MSE | $\widehat{\alpha}$ | 0.1286 | 0.0632 | 0.0792 | 0.0642 | 0.0901 | 0.0779 | |

$\widehat{\theta}$ | 0.0046 | 0.0022 | 0.0035 | 0.0027 | 0.0033 | 0.0031 | ||

MRE | $\widehat{\alpha}$ | 0.3546 | 0.2453 | 0.3307 | 0.2308 | 0.3690 | 0.3227 | |

$\widehat{\theta}$ | 0.0893 | 0.0636 | 0.0930 | 0.0668 | 0.0891 | 0.0879 | ||

200 | BIAS | $\widehat{\alpha}$ | 0.2047 | 0.1354 | 0.2469 | 0.1375 | 0.2556 | 0.2536 |

$\widehat{\theta}$ | 0.0317 | 0.0268 | 0.0484 | 0.0223 | 0.0428 | 0.0427 | ||

MSE | $\widehat{\alpha}$ | 0.0950 | 0.0386 | 0.0761 | 0.0481 | 0.0814 | 0.0781 | |

$\widehat{\theta}$ | 0.0030 | 0.0018 | 0.0037 | 0.0013 | 0.0029 | 0.0032 | ||

MRE | $\widehat{\alpha}$ | 0.2730 | 0.1805 | 0.3292 | 0.1833 | 0.3408 | 0.3381 | |

$\widehat{\theta}$ | 0.0633 | 0.0536 | 0.0967 | 0.0446 | 0.0856 | 0.0854 |

$\mathit{n}$ | Est. | Par. | MLE-M | ADE-M | CVME-M | MPSE-M | LSE-M | WLSE-M |
---|---|---|---|---|---|---|---|---|

20 | BIAS | $\widehat{\alpha}$ | 1.1442 | 0.2833 | 0.3719 | 0.3823 | 0.4131 | 0.3160 |

$\widehat{\theta}$ | 0.1602 | 0.1293 | 0.1434 | 0.1475 | 0.1504 | 0.1414 | ||

MSE | $\widehat{\alpha}$ | 1.6432 | 0.1557 | 0.2312 | 0.2903 | 0.2905 | 0.1607 | |

$\widehat{\theta}$ | 0.0408 | 0.0307 | 0.0316 | 0.0331 | 0.0361 | 0.0305 | ||

MRE | $\widehat{\alpha}$ | 0.4577 | 0.1133 | 0.1488 | 0.1529 | 0.1653 | 0.1264 | |

$\widehat{\theta}$ | 0.2136 | 0.1724 | 0.1912 | 0.1967 | 0.2005 | 0.1885 | ||

40 | BIAS | $\widehat{\alpha}$ | 0.8187 | 0.1869 | 0.3060 | 0.2338 | 0.3459 | 0.2289 |

$\widehat{\theta}$ | 0.1534 | 0.0950 | 0.1106 | 0.0853 | 0.1104 | 0.1052 | ||

MSE | $\widehat{\alpha}$ | 1.1100 | 0.0648 | 0.1467 | 0.1348 | 0.2198 | 0.0965 | |

$\widehat{\theta}$ | 0.0428 | 0.0135 | 0.0195 | 0.0115 | 0.0204 | 0.0181 | ||

MRE | $\widehat{\alpha}$ | 0.3275 | 0.0747 | 0.1224 | 0.0935 | 0.1384 | 0.0916 | |

$\widehat{\theta}$ | 0.2045 | 0.1267 | 0.1474 | 0.1138 | 0.1473 | 0.1402 | ||

80 | BIAS | $\widehat{\alpha}$ | 0.5424 | 0.1206 | 0.1838 | 0.1014 | 0.1968 | 0.1544 |

$\widehat{\theta}$ | 0.1142 | 0.0673 | 0.0815 | 0.0586 | 0.0770 | 0.0759 | ||

MSE | $\widehat{\alpha}$ | 0.6064 | 0.0286 | 0.0564 | 0.0168 | 0.0653 | 0.0417 | |

$\widehat{\theta}$ | 0.0250 | 0.0072 | 0.0104 | 0.0057 | 0.0090 | 0.0082 | ||

MRE | $\widehat{\alpha}$ | 0.2170 | 0.0483 | 0.0735 | 0.0406 | 0.0787 | 0.0618 | |

$\widehat{\theta}$ | 0.1523 | 0.0897 | 0.1087 | 0.0781 | 0.1027 | 0.1012 | ||

100 | BIAS | $\widehat{\alpha}$ | 0.5402 | 0.0992 | 0.1572 | 0.0818 | 0.1865 | 0.1131 |

$\widehat{\theta}$ | 0.1370 | 0.0632 | 0.0556 | 0.0457 | 0.0674 | 0.0526 | ||

MSE | $\widehat{\alpha}$ | 0.5514 | 0.0184 | 0.0420 | 0.0111 | 0.0506 | 0.0200 | |

$\widehat{\theta}$ | 0.0362 | 0.0064 | 0.0051 | 0.0037 | 0.0075 | 0.0046 | ||

MRE | $\widehat{\alpha}$ | 0.2161 | 0.0397 | 0.0629 | 0.0327 | 0.0746 | 0.0452 | |

$\widehat{\theta}$ | 0.1826 | 0.0843 | 0.0741 | 0.0609 | 0.0898 | 0.0702 | ||

150 | BIAS | $\widehat{\alpha}$ | 0.3914 | 0.0701 | 0.1261 | 0.0587 | 0.1387 | 0.0879 |

$\widehat{\theta}$ | 0.0915 | 0.0393 | 0.0524 | 0.0397 | 0.0517 | 0.0486 | ||

MSE | $\widehat{\alpha}$ | 0.3582 | 0.0099 | 0.0252 | 0.0066 | 0.0318 | 0.0118 | |

$\widehat{\theta}$ | 0.0172 | 0.0025 | 0.0043 | 0.0026 | 0.0040 | 0.0038 | ||

MRE | $\widehat{\alpha}$ | 0.1566 | 0.0281 | 0.0504 | 0.0235 | 0.0555 | 0.0352 | |

$\widehat{\theta}$ | 0.1221 | 0.0523 | 0.0699 | 0.0530 | 0.0690 | 0.0648 | ||

200 | BIAS | $\widehat{\alpha}$ | 0.2187 | 0.0536 | 0.1108 | 0.0354 | 0.1192 | 0.0755 |

$\widehat{\theta}$ | 0.0579 | 0.0378 | 0.0487 | 0.0299 | 0.0453 | 0.0406 | ||

MSE | $\widehat{\alpha}$ | 0.0880 | 0.0064 | 0.0195 | 0.0022 | 0.0225 | 0.0082 | |

$\widehat{\theta}$ | 0.0068 | 0.0025 | 0.0035 | 0.0017 | 0.0031 | 0.0027 | ||

MRE | $\widehat{\alpha}$ | 0.0875 | 0.0215 | 0.0443 | 0.0142 | 0.0477 | 0.0302 | |

$\widehat{\theta}$ | 0.0772 | 0.0504 | 0.0649 | 0.0398 | 0.0604 | 0.0542 |

$\mathit{n}$ | Est. | Par. | MLE-M | ADE-M | CVME-M | MPSE-M | LSE-M | WLSE-M |
---|---|---|---|---|---|---|---|---|

20 | BIAS | $\widehat{\alpha}$ | 0.6551 | 0.5539 | 0.5667 | 0.5194 | 0.5117 | 0.5046 |

$\widehat{\theta}$ | 0.5853 | 0.5147 | 0.5672 | 0.5333 | 0.5370 | 0.5565 | ||

MSE | $\widehat{\alpha}$ | 0.8185 | 0.4264 | 0.4388 | 0.3312 | 0.3496 | 0.3353 | |

$\widehat{\theta}$ | 0.5296 | 0.4334 | 0.5226 | 0.3938 | 0.4555 | 0.4805 | ||

MRE | $\widehat{\alpha}$ | 0.4367 | 0.3693 | 0.3778 | 0.3463 | 0.3411 | 0.3364 | |

$\widehat{\theta}$ | 0.2341 | 0.2059 | 0.2269 | 0.2133 | 0.2148 | 0.2226 | ||

40 | BIAS | $\widehat{\alpha}$ | 0.4699 | 0.4312 | 0.4480 | 0.4797 | 0.4561 | 0.4303 |

$\widehat{\theta}$ | 0.4674 | 0.4015 | 0.4695 | 0.4247 | 0.3905 | 0.4035 | ||

MSE | $\widehat{\alpha}$ | 0.2956 | 0.2428 | 0.2840 | 0.3135 | 0.2724 | 0.2252 | |

$\widehat{\theta}$ | 0.3515 | 0.2245 | 0.3717 | 0.2447 | 0.2339 | 0.2738 | ||

MRE | $\widehat{\alpha}$ | 0.3133 | 0.2875 | 0.2987 | 0.3198 | 0.3041 | 0.2869 | |

$\widehat{\theta}$ | 0.1870 | 0.1606 | 0.1878 | 0.1699 | 0.1562 | 0.1614 | ||

80 | BIAS | $\widehat{\alpha}$ | 0.3615 | 0.3707 | 0.3991 | 0.3688 | 0.3920 | 0.3369 |

$\widehat{\theta}$ | 0.3561 | 0.2647 | 0.3746 | 0.3453 | 0.3465 | 0.2991 | ||

MSE | $\widehat{\alpha}$ | 0.1906 | 0.1830 | 0.2170 | 0.1995 | 0.2006 | 0.1527 | |

$\widehat{\theta}$ | 0.1944 | 0.1216 | 0.2243 | 0.1818 | 0.1802 | 0.1374 | ||

MRE | $\widehat{\alpha}$ | 0.2410 | 0.2471 | 0.2661 | 0.2459 | 0.2613 | 0.2246 | |

$\widehat{\theta}$ | 0.1424 | 0.1059 | 0.1498 | 0.1381 | 0.1386 | 0.1197 | ||

100 | BIAS | $\widehat{\alpha}$ | 0.3808 | 0.3453 | 0.3706 | 0.4104 | 0.3278 | 0.3254 |

$\widehat{\theta}$ | 0.3515 | 0.2927 | 0.3085 | 0.3352 | 0.2728 | 0.2765 | ||

MSE | $\widehat{\alpha}$ | 0.1986 | 0.1607 | 0.1639 | 0.2329 | 0.1525 | 0.1499 | |

$\widehat{\theta}$ | 0.1995 | 0.1271 | 0.1579 | 0.1677 | 0.1200 | 0.1195 | ||

MRE | $\widehat{\alpha}$ | 0.2539 | 0.2302 | 0.2470 | 0.2736 | 0.2185 | 0.2170 | |

$\widehat{\theta}$ | 0.1406 | 0.1171 | 0.1234 | 0.1341 | 0.1091 | 0.1106 | ||

150 | BIAS | $\widehat{\alpha}$ | 0.2997 | 0.2534 | 0.3034 | 0.3213 | 0.3070 | 0.3110 |

$\widehat{\theta}$ | 0.2417 | 0.2349 | 0.2779 | 0.2508 | 0.2770 | 0.2720 | ||

MSE | $\widehat{\alpha}$ | 0.1447 | 0.1010 | 0.1283 | 0.1701 | 0.1295 | 0.1376 | |

$\widehat{\theta}$ | 0.0890 | 0.0881 | 0.1206 | 0.0988 | 0.1082 | 0.1020 | ||

MRE | $\widehat{\alpha}$ | 0.1998 | 0.1689 | 0.2023 | 0.2142 | 0.2047 | 0.2073 | |

$\widehat{\theta}$ | 0.0967 | 0.0940 | 0.1112 | 0.1003 | 0.1108 | 0.1088 | ||

200 | BIAS | $\widehat{\alpha}$ | 0.2640 | 0.2652 | 0.3074 | 0.3507 | 0.2594 | 0.2394 |

$\widehat{\theta}$ | 0.2121 | 0.2225 | 0.2266 | 0.2427 | 0.2349 | 0.2063 | ||

MSE | $\widehat{\alpha}$ | 0.1234 | 0.1032 | 0.1193 | 0.1943 | 0.0987 | 0.0902 | |

$\widehat{\theta}$ | 0.0762 | 0.0796 | 0.0746 | 0.0966 | 0.0821 | 0.0654 | ||

MRE | $\widehat{\alpha}$ | 0.1760 | 0.1768 | 0.2050 | 0.2338 | 0.1729 | 0.1596 | |

$\widehat{\theta}$ | 0.0848 | 0.0890 | 0.0906 | 0.0971 | 0.0940 | 0.0825 |

Model | Estimates | Fitted Measures | |||||||
---|---|---|---|---|---|---|---|---|---|

$\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\gamma}}$ | AIC | CVM | AD | KS | p-Value | SS | |

New-PFD | 0.0314 | 0.4012 | - | 283.6512 | 0.1934 | 1.2679 | 0.0777 | 0.7488 | 0.1147 |

Tr-PF | 0.7647 | - | 1.5949 | 291.6001 | 0.3140 | 1.9964 | 0.1329 | 0.1367 | 0.3548 |

Kum-PF | 1.1071 | 0.4754 | 1.3700 | 302.9045 | 0.4286 | 2.6819 | 0.1582 | 0.0446 | 0.5521 |

OGE-PF | 6.4256 | 0.0690 | 2.0309 | 309.0742 | 0.3635 | 2.3533 | 0.1959 | 0.0058 | 0.9499 |

PF-I | 0.4375 | - | - | 302.9309 | 0.4179 | 2.6213 | 0.1976 | 0.0053 | 0.9699 |

NG-PF | 0.6775 | 0.0727 | 5.8366 | 305.8946 | 0.3440 | 2.1834 | 0.0777 | 0.0038 | 1.0395 |

Gen-PF | 2.2790 | - | - | 332.1345 | 0.3500 | 2.1641 | 0.3180 | 0.0000 | 3.0369 |

Model | Estimates | Fitted Measures | |||||||
---|---|---|---|---|---|---|---|---|---|

$\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\gamma}}$ | AIC | CVM | AD | KS | p-Value | SS | |

New-PFD | 0.1921 | 0.3262 | - | 470.1407 | 0.1433 | 0.9490 | 0.1232 | 0.6889 | 0.0760 |

ZTP-PF | 2.8368 | 0.7329 | - | 476.5421 | - | - | 0.1728 | 0.2790 | 0.1924 |

Kum-PF | 1.2485 | 0.2696 | 0.6369 | 467.5365 | 0.1976 | 1.2621 | 0.1797 | 0.2393 | 0.2962 |

OGE-PF | 4.7180 | 0.0646 | 0.0047 | 475.6623 | 0.1195 | 0.8823 | 0.2213 | 0.0818 | 0.4595 |

Gen-PF | 1.0351 | - | - | 494.4784 | 0.1204 | 0.7690 | 0.3991 | 0.0001 | 1.8877 |

Model | Estimates | Fitted Measures | |||||||
---|---|---|---|---|---|---|---|---|---|

$\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\gamma}}$ | AIC | CVM | AD | KS | p-Value | SS | |

New-PFD | 0.1044 | 0.4908 | - | 642.4877 | 0.1217 | 0.8393 | 0.0814 | 0.7740 | 0.0924 |

Tr-PF | 0.5604 | - | 1.7920 | 643.7063 | 0.1588 | 1.0521 | 0.1032 | 0.4836 | 0.1446 |

NG-PF | 1.1644 | 17.2847 | 0.0531 | 648.6244 | 0.1868 | 1.2314 | 0.0814 | 0.2892 | 0.1946 |

Kum-PF | 0.3894 | 1.6505 | 1.0472 | 650.1455 | 0.2125 | 1.3840 | 0.1464 | 0.1182 | 0.2983 |

PF-I | 0.6247 | - | - | 646.2304 | 0.2118 | 1.3797 | 0.1521 | 0.0945 | 0.3314 |

Model | Estimates | Fitted Measures | |||||||
---|---|---|---|---|---|---|---|---|---|

$\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\gamma}}$ | AIC | CVM | AD | KS | p-Value | SS | |

New-PFD | 0.0287 | 1.3984 | - | −41.7665 | 0.0255 | 0.2089 | 0.0835 | 0.9910 | 0.0199 |

NG-PF | 1.5971 | 18.5499 | 0.2342 | −38.5436 | 0.0361 | 0.3030 | 0.0835 | 0.9651 | 0.0265 |

Kum-PF | 0.9756 | 2.1057 | 1.7528 | −36.7425 | 0.0532 | 0.4306 | 0.1276 | 0.7836 | 0.0555 |

Tr-PF | 0.8341 | - | 1.0780 | −39.9779 | 0.0382 | 0.3138 | 0.1277 | 0.7826 | 0.0522 |

PF-I | 1.4845 | - | - | −37.2526 | 0.0498 | 0.4067 | 0.2310 | 0.1311 | 0.2499 |

Estimates | Fitted Measures | ||||||||
---|---|---|---|---|---|---|---|---|---|

Model | $\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\gamma}}$ | AIC | CVM | AD | KS | p-Value | SS |

New-PFD | 0.0205 | 0.8481 | - | 993.5991 | 0.0471 | 0.4678 | 0.0519 | 0.8044 | 0.0531 |

W-PF | 4.2216 | 8.4898 | 0.2024 | 996.1838 | 0.0597 | 0.5134 | 0.0548 | 0.7468 | 0.0594 |

ZTP-PF | 4.3987 | 1.7816 | - | 998.4550 | 0.1012 | 0.8065 | 0.0679 | 0.4803 | 0.1028 |

MOE-PF | 0.1416 | 1.9682 | - | 994.8340 | 0.0874 | 0.7155 | 0.0681 | 0.4774 | 0.0815 |

Gen-PF | 1.7756 | - | - | 979.7377 | 0.5117 | 3.7593 | 0.0716 | 0.4132 | 0.1176 |

Kum-PF | 0.4168 | 3.0564 | 1.9823 | 1012.3107 | 0.1206 | 0.9920 | 0.0790 | 0.2948 | 0.1706 |

Tr-PF | 0.9380 | - | 1.2519 | 1002.0116 | 0.0784 | 0.6836 | 0.0819 | 0.2563 | 0.1993 |

OGE-PF | 7.6256 | 0.1239 | 2.8673 | 1035.8485 | 0.2114 | 1.5872 | 0.1621 | 0.0006 | 1.0577 |

PF-I | 0.8615 | - | - | 1039.2363 | 0.1050 | 0.8905 | 0.1992 | 0.0000 | 1.6514 |

Model | Estimates | Fitted Measures | |||||||
---|---|---|---|---|---|---|---|---|---|

$\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\gamma}}$ | AIC | CVM | AD | KS | p-Value | SS | |

New-PFD | 0.0253 | 0.8957 | - | 156.8105 | 0.1062 | 0.6493 | 0.1070 | 0.8467 | 0.0522 |

PF-Poi | −3.5098 | 1.7625 | - | 156.8312 | 0.1076 | 0.6575 | 0.1093 | 0.8283 | 0.0549 |

Tr-PF | 0.8025 | - | 1.3066 | 160.1585 | 0.1701 | 1.0188 | 0.1617 | 0.3725 | 0.1589 |

NG-PF | 1.3880 | 164.1245 | 0.1819 | 1.1150 | 0.1070 | 0.3563 | 0.1619 | ||

Kum-PF | 0.4400 | 2.4843 | 1.2740 | 167.1066 | 0.2316 | 1.4057 | 0.2097 | 0.1232 | 0.2811 |

OGE-PF | 9.8176 | 0.1003 | 2.8782 | 168.6015 | 0.1665 | 0.9571 | 0.2378 | 0.0562 | 0.3771 |

Model | Estimates | Fitted Measures | |||||||
---|---|---|---|---|---|---|---|---|---|

$\widehat{\mathit{\alpha}}$ | $\widehat{\mathit{\theta}}$ | $\widehat{\mathit{\gamma}}$ | AIC | CVM | AD | KS | p-Value | SS | |

New-PFD | 0.0442 | 0.8961 | - | 335.7237 | 0.0979 | 0.7027 | 0.1128 | 0.4103 | 0.1011 |

Tr-PF | 0.8247 | - | 1.4090 | 338.2626 | 0.1176 | 0.8627 | 0.1162 | 0.3742 | 0.1269 |

Kum-PF | 0.5666 | 2.2183 | 1.4513 | 345.9732 | 0.1472 | 1.1037 | 0.1172 | 0.3639 | 0.1927 |

NG-PF | 1.4465 | 27.0812 | 0.0996 | 341.4392 | 0.1092 | 0.8408 | 0.1128 | 0.3496 | 0.1173 |

OGE-PF | 8.7252 | 0.1239 | 2.8958 | 349.0370 | 0.1462 | 1.0418 | 0.1414 | 0.1718 | 0.3554 |

ZTP-PF | 3.8076 | 1.9272 | - | 339.4020 | 0.1315 | 0.7659 | 0.1450 | 0.1514 | 0.1789 |

PF-I | 1.0090 | - | - | 346.1583 | 0.1420 | 1.0706 | 0.1694 | 0.0598 | 0.4853 |

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## Share and Cite

**MDPI and ACS Style**

Alghamdi, A.S.; Abd El-Raouf, M.M.
Exploring the Dynamics of COVID-19 with a Novel Family of Models. *Mathematics* **2023**, *11*, 1641.
https://doi.org/10.3390/math11071641

**AMA Style**

Alghamdi AS, Abd El-Raouf MM.
Exploring the Dynamics of COVID-19 with a Novel Family of Models. *Mathematics*. 2023; 11(7):1641.
https://doi.org/10.3390/math11071641

**Chicago/Turabian Style**

Alghamdi, Abdulaziz S., and M. M. Abd El-Raouf.
2023. "Exploring the Dynamics of COVID-19 with a Novel Family of Models" *Mathematics* 11, no. 7: 1641.
https://doi.org/10.3390/math11071641