# New Procedures of a Fractional Order Model of Novel Coronavirus (COVID-19) Outbreak via Wavelets Method

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. A Brief Review of Wavelets

#### 3.1. Sine–Cosine Wavelets and Their Properties

#### 3.1.1. Sine–Cosine Wavelets

#### 3.1.2. Function Approximation

#### 3.1.3. Sine–Cosine Wavelet Operational Matrix of the Fractional Integration

#### Block Pulse Functions (BPFs)

**Definition**

**3.**

**Lemma.**

**Proof.**

#### 3.2. Bernoulli Wavelets and Their Properties

#### 3.2.1. Bernoulli Wavelets

#### 3.2.2. Function Approximation

#### 3.2.3. Bernoulli Wavelet Operational Matrix of the Fractional Integration

## 4. Description of Numerical Method

## 5. Results and Discussion

#### 5.1. Convergence of the Solution

#### 5.2. Verification of the Solution

#### 5.3. Computational Cost

#### 5.4. The Effects of Fractional Orders

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Alluwaimi, A.M.; Alshubaith, I.H.; Al-Ali, A.M.; Abohelaika, S. The Coronaviruses of Animals and Birds: Their Zoonosis, Vaccines, and Models for SARS-CoV and SARS-CoV2. Front. Vet. Sci.
**2020**, 7, 655. [Google Scholar] [CrossRef] [PubMed] - Madjid, M.; Safavi-Naeini, P.; Solomon, S.D.; Vardeny, O. Potential effects of coronaviruses on the cardiovascular system: A review. JAMA Cardiol.
**2020**, 5, 840–931. [Google Scholar] [CrossRef] [PubMed] [Green Version] - WHO. Novel Coronavirus (2019-nCoV): Situation Report 3; WHO: Geneva, Switzerland, 2020. [Google Scholar]
- Chen, T.M.; Rui, J.; Wang, Q.P.; Zhao, Z.Y.; Cui, J.A.; Yin, L. A mathematical model for simulating the phase-based transmissibility of a novel coronavirus. Infect. Dis. Poverty
**2020**, 9, 1–8. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Khan, M.A.; Atangana, A. Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative. Alex. Eng. J.
**2020**, 59, 2379–2389. [Google Scholar] [CrossRef] - Rajagopal, K.; Hasanzadeh, N.; Parastesh, F.; Hamarash, I.I.; Jafari, S.; Hussain, I. A fractional-order model for the novel coronavirus (COVID-19) outbreak. Nonlinear Dyn.
**2020**, 101, 711–718. [Google Scholar] [CrossRef] [PubMed] - Chen, X.; Li, J.; Xiao, C.; Yang, P. Numerical solution and parameter estimation for uncertain SIR model with application to COVID-19. Fuzzy Optim. Decis. Mak.
**2021**, 20, 189–208. [Google Scholar] [CrossRef] - Hashemizadeh, E.; Ebadi, M.A. A numerical solution by alternative Legendre polynomials on a model for novel coronavirus (COVID-19). Adv. Differ. Equ.
**2020**, 2020, 527. [Google Scholar] [CrossRef] - Kisela, T. Fractional Differential Equations and Their Applications; Faculty of Mechanical Engineering Institute of Mathematics: Brno-střed, Czech Republic, 2008. [Google Scholar]
- Solís-Pérez, J.E.; Gómez-Aguilar, J.F.; Atangana, A. A fractional mathematical model of breast cancer competition model. Chaos Solitons Fractals
**2019**, 127, 38–54. [Google Scholar] [CrossRef] - Hussain, A.; Baleanu, D.; Adeel, M. Existence of solution and stability for the fractional order novel coronavirus (nCoV-2019) model. Adv. Differ. Equ.
**2020**, 2020, 1–9. [Google Scholar] [CrossRef] - Atangana, A. Modelling the spread of COVID-19 with new fractal-fractional operators: Can the lockdown save mankind before vaccination? Chaos Solitons Fractals
**2020**, 136, 109860. [Google Scholar] [CrossRef] - Wang, B.H.; Wang, Y.Y.; Dai, C.Q.; Chen, Y.X. Dynamical characteristic of analytical fractional solitons for the space-time fractional Fokas-Lenells equation. Alex. Eng. J.
**2020**, 59, 4699–4707. [Google Scholar] [CrossRef] - Yu, L.J.; Wu, G.Z.; Wang, Y.Y.; Chen, Y.X. Traveling wave solutions constructed by Mittag–Leffler function of a (2 + 1)-dimensional space-time fractional NLS equation. Results Phys.
**2020**, 17, 103156. [Google Scholar] [CrossRef] - Fang, J.J.; Mou, D.S.; Wang, Y.Y.; Zhang, H.C.; Dai, C.Q.; Chen, Y.X. Soliton dynamics based on exact solutions of conformable fractional discrete complex cubic Ginzburg–Landau equation. Results Phys.
**2021**, 20, 103710. [Google Scholar] [CrossRef] - Wang, B.-H.; Wang, Y.-Y. Fractional white noise functional soliton solutions of a wick-type stochastic fractional NLSE. Appl. Math. Lett.
**2020**, 110, 106583. [Google Scholar] [CrossRef] - Wang, B.-H.; Wang, Y.-Y.; Dai, C.-Q. Fractional optical solitons with stochastic properties of a wick-type stochastic fractional NLSE driven by the Brownian motion. Waves Random Complex Media
**2021**, 1–14. [Google Scholar] - Noeiaghdam, S. A novel technique to solve the modified epidemiological model of computer viruses. SEMA J.
**2019**, 76, 97–108. [Google Scholar] [CrossRef] - Noeiaghdam, S.; Micula, S. Dynamical Strategy to Control the Accuracy of the Nonlinear Bio-Mathematical Model of Malaria Infection. Mathematics
**2021**, 9, 1031. [Google Scholar] [CrossRef] - Noeiaghdam, S.; Suleman, M.; Budak, H. Solving a modified nonlinear epidemiological model of computer viruses by homotopy analysis method. Math. Sci.
**2018**, 12, 211–222. [Google Scholar] [CrossRef] [Green Version] - Tuan, N.H.; Mohammadi, H.; Rezapour, S. A mathematical model for COVID-19 transmission by using the Caputo fractional derivative. Chaos Solitons Fractals
**2020**, 140, 110107. [Google Scholar] [CrossRef] [PubMed] - Yi, M.; Huang, J. Wavelet operational matrix method for solving fractional differential equations with variable coefficients. Appl. Math. Comput.
**2014**, 230, 383–394. [Google Scholar] [CrossRef] - Yi, M.; Wang, L.; Huang, J. Legendre wavelets method for the numerical solution of fractional integro-differential equations with weakly singular kernel. Appl. Math. Model.
**2016**, 40, 3422–3437. [Google Scholar] [CrossRef] - Razzaghi, M.; Yousefi, S. Sine-cosine wavelets operational matrix of integration and its applications in the calculus of variations. Int. J. Syst. Sci.
**2002**, 33, 805–810. [Google Scholar] [CrossRef] - Saeed, A.; Saeed, U. Sine-cosine wavelet method for fractional oscillator equations. Math. Methods Appl. Sci.
**2019**, 42, 6960–6971. [Google Scholar] [CrossRef] - Wang, Y.; Yin, T.; Zhu, L. Sine-cosine wavelet operational matrix of fractional order integration and its applications in solving the fractional order Riccati differential equations. Adv. Differ. Equ.
**2017**, 2017, 222. [Google Scholar] [CrossRef] [Green Version] - Kajani, M.T.; Ghasemi, M.; Babolian, E. Numerical solution of linear integro-differential equation by using sine–cosine wavelets. Appl. Math. Comput.
**2006**, 180, 569–574. [Google Scholar] [CrossRef] - Kilicman, A.; Al Zhour, Z.A.A. Kronecker operational matrices for fractional calculus and some applications. Appl. Math. Comput.
**2007**, 187, 250–265. [Google Scholar] [CrossRef] - Keshavarz, E.; Ordokhani, Y. Bernoulli wavelets method for solution of fractional differential equations in a large interval. Math. Res.
**2016**, 2, 17–32. [Google Scholar] [CrossRef] - Rahimkhani, P.; Ordokhani, Y.; Babolian, E. Numerical solution of fractional pantograph differential equations by using generalized fractional-order Bernoulli wavelet. J. Comput. Appl. Math.
**2017**, 309, 493–510. [Google Scholar] [CrossRef] - Tohidi, E.; Bhrawy, A.; Erfani, K. A collocation method based on Bernoulli operational matrix for numerical solution of generalized pantograph equation. Appl. Math. Model.
**2013**, 37, 4283–4294. [Google Scholar] [CrossRef] - Soltanpour Moghadam, A.; Arabameri, M.; Baleanu, D.; Barfeie, M. Numerical solution of variable fractional order advection-dispersion equation using Bernoulli wavelet method and new operational matrix of fractional order derivative. Math. Methods Appl. Sci.
**2020**, 43, 3936–3953. [Google Scholar] [CrossRef]

**Figure 1.**Comparison of the numerical solutions of S

_{p}(t) by using the sine–cosine wavelets method, Bernoulli wavelets method, ALPs and RK4 methods for $\upsilon =1$.

**Figure 2.**Comparison of the numerical solutions of ${e}_{p}(t)$ by using the sine–cosine wavelets method, Bernoulli wavelets method, ALPs and RK4 methods for $\upsilon =1$.

**Figure 3.**Comparison of the numerical solutions of ${i}_{p}(t)$ by using the sine–cosine wavelets method, Bernoulli wavelets method, ALPs and RK4 methods for $\upsilon =1$.

**Figure 4.**Comparison of the numerical solutions of ${a}_{p}(t)$ by using the sine–cosine wavelets method, Bernoulli wavelets method, ALPs and RK4 methods for $\upsilon =1$.

**Figure 5.**Comparison of the numerical solutions of ${r}_{p}(t)$ by using the sine–cosine wavelets method, Bernoulli wavelets method, ALPs and RK4 methods for $\upsilon =1$.

**Figure 6.**Comparison of the numerical solutions of $w(t)$ by using the sine–cosine wavelets method, Bernoulli wavelets method, ALPs and RK4 methods for $\upsilon =1$.

**Figure 7.**The effect of $\upsilon $ on the distribution of ${s}_{p}\left(t\right)$. Black: $\upsilon =0.2$, Blue: $\upsilon =0.4$, Red: $\upsilon =0.6$, Green: $\upsilon =0.8$.

**Figure 8.**The effect of $\upsilon $ on the distribution of ${e}_{p}\left(t\right)$. Black: $\upsilon =0.2$, Blue: $\upsilon =0.4$, Red: $\upsilon =0.6$, Green: $\upsilon =0.8$.

**Figure 9.**The effect of $\upsilon $ on the distribution of ${i}_{p}\left(t\right)$. Black: $\upsilon =0.2$, Blue: $\upsilon =0.4$, Red: $\upsilon =0.6$, Green: $\upsilon =0.8$.

**Figure 10.**The effect of $\upsilon $ on the distribution of ${a}_{p}\left(t\right)$. Black: $\upsilon =0.2$, Blue: $\upsilon =0.4$, Red: $\upsilon =0.6$, Green: $\upsilon =0.8$.

**Figure 11.**The effect of $\upsilon $ on the distribution of ${r}_{p}\left(t\right)$. Black: $\upsilon =0.2$, Blue: $\upsilon =0.4$, Red: $\upsilon =0.6$, Green: $\upsilon =0.8$.

**Figure 12.**The effect of $\upsilon $ on the distribution of ${w}_{p}\left(t\right)$. Black: $\upsilon =0.2$, Blue: $\upsilon =0.4$, Red: $\upsilon =0.6$, Green: $\upsilon =0.8$.

Variables and Parameters | Definition | Variables and Parameters | Definition |
---|---|---|---|

n_{p} | The birth rate of people. | µ’_{p} | The shedding coefficients from A_{p} to W. |

m_{p} | The death rate of people. | δ_{p} | The proportion of asymptomatic infection rate of people. |

$\frac{1}{{\omega}_{p}}$ | The incubation period of people. | β_{p} | The transmission rate from I_{p} to S_{p}. |

$\frac{1}{{{\omega}^{\prime}}_{p}}$ | The latent period of people. | β_{W} | The transmission rate from W to S_{p}. |

$\frac{1}{{\gamma}_{p}}$ | The infectious period of symptomatic infection in people. | k | The multiple of the transmissibility of A_{p} to that of I_{p}. |

$\frac{1}{{{\gamma}^{\prime}}_{p}}$ | The infectious period of asymptomatic infection in people. | $\frac{1}{\epsilon}$ | The lifetime of the virus in W. |

µ_{p} | The shedding coefficients from I_{p} to W. | c | The relative shedding rate of A_{p} compared to I_{p}. |

**Table 2.**The values of 10

^{10}S

_{p}and rate of convergence (ROC) at different points for k = 2 and various values of M by Bernoulli wavelets.

t | 10^{10} S_{p}M = 4 | 10^{10} S_{p}M = 5 | 10^{10} S_{p}M = 6 | 10^{10} S_{p}M = 7 | ROC M = 7 | 10^{10} S_{p}M = 8 | ROC M = 8 | 10^{10} S_{p}M = 9 | ROC M = 9 | 10^{10} S_{p}M = 10 | ROC M = 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 1.08 | 1.77 | 2.57 | 2.62 | −18.95 | 2.24 | −0.73 | 2.03 | −0.29 | 2.13 | 1.16 |

0.4 | 2.28 | 0.804 | 2.09 | 1.42 | 4.69 | 1.47 | 3.83 | 1.81 | −0.73 | 1.35 | 0.16 |

0.6 | 1.30 | 1.29 | 1.29 | 1.29 | −0.01 | 1.29 | −234 | 1.29 | 0.02 | 1.29 | −20 |

0.8 | 1.12 | 1.12 | 1.12 | 1.12 | 3.30 | 1.12 | 1.10 | 1.12 | −0.86 | 1.12 | 1.30 |

**Table 3.**The values of e

_{p}and rate of convergence (ROC) at different points for k = 2 and various values of M by Bernoulli wavelets.

t | e_{p}M = 4 | e_{p}M = 5 | e_{p}M = 6 | e_{p}M = 7 | ROC M = 7 | e_{p}M = 8 | ROC M = 8 | e_{p}M = 9 | ROC M = 9 | e_{p}M = 10 | ROC M = 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 5.4570 | 5.4560 | 5.4565 | 5.4564 | 3.17 | 5.4562 | −0.77 | 5.4560 | −0.17 | 5.4559 | 2.09 |

0.4 | 5.2560 | 5.2519 | 5.2535 | 5.2529 | 0.94 | 5.2529 | 3.79 | 5.2529 | −0.25 | 5.2528 | 0.94 |

0.6 | 5.1056 | 5.1054 | 5.1053 | 5.1053 | 0.72 | 5.1052 | 0.79 | 5.1052 | 0.85 | 5.1052 | 0.88 |

0.8 | 4.9861 | 4.9860 | 4.9859 | 4.9859 | 1.13 | 4.9859 | 0.79 | 4.9859 | 0.87 | 4.9859 | 0.88 |

**Table 4.**The values of i

_{p}and rate of convergence (ROC) at different points for k = 2 and various values of M by Bernoulli wavelets.

t | i_{p}M = 4 | i_{p}M = 5 | i_{p}M = 6 | i_{p}M = 7 | ROC M = 7 | i_{p}M = 8 | ROC M = 8 | i_{p}M = 9 | ROC M = 9 | i_{p}M = 10 | ROC M = 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 3.0058 | 3.0058 | 3.0058 | 3.0057 | −0.88 | 3.0058 | 0.39 | 3.0058 | 0.45 | 3.0058 | 5.47 |

0.4 | 2.9974 | 2.9976 | 2.9975 | 2.9975 | 1.79 | 2.9975 | 1.59 | 2.9975 | −0.025 | 2.9975 | −0.8 |

0.6 | 2.9878 | 2.9878 | 2.9878 | 2.9878 | 0.95 | 2.9877 | 0.90 | 2.9877 | 0.88 | 2.9877 | 0.89 |

0.8 | 2.9775 | 2.9775 | 2.9775 | 2.9775 | 0.74 | 2.9775 | 0.88 | 2.9775 | 0.87 | 2.9775 | 0.88 |

**Table 5.**The values of a

_{p}and rate of convergence (ROC) at different points for k = 2 and various values of M by Bernoulli wavelets.

t | a_{p}M = 4 | a_{p}M = 5 | a_{p}M = 6 | a_{p}M = 7 | ROC M = 7 | a_{p}M = 8 | ROC M = 8 | a_{p}M = 9 | ROC M = 9 | a_{p}M = 10 | ROC M = 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 3.0058 | 3.0058 | 3.0058 | 3.0057 | −0.88 | 3.0058 | 0.39 | 3.0058 | 0.45 | 3.0058 | 5.47 |

0.4 | 2.9974 | 2.9976 | 2.9975 | 2.9975 | 1.79 | 2.9975 | 1.59 | 2.9975 | −0.025 | 2.9975 | −0.8 |

0.6 | 2.9878 | 2.9878 | 2.9878 | 2.9878 | 0.95 | 2.9877 | 0.90 | 2.9877 | 0.88 | 2.9877 | 0.88 |

0.8 | 2.9775 | 2.9775 | 2.9775 | 2.9775 | 0.74 | 2.9775 | 0.88 | 2.9775 | 0.87 | 2.9775 | 0.89 |

**Table 6.**The values of r

_{p}and rate of convergence (ROC) at different points for k = 2 and various values of M by Bernoulli wavelets.

t | r_{p}M = 4 | r_{p}M = 5 | r_{p}M = 6 | r_{p}M = 7 | ROC M = 7 | r_{p}M = 8 | ROC M = 8 | r_{p}M = 9 | ROC M = 9 | r_{p}M = 10 | ROC M = 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 3.1403 | 3.1422 | 3.1412 | 3.1412 | 6.15 | 3.1417 | −0.80 | 3.1421 | −0.11 | 3.1423 | 2.20 |

0.4 | 3.5476 | 3.5563 | 3.5529 | 3.5542 | 1.00 | 3.5543 | 4.18 | 3.5542 | −0.38 | 3.5544 | 0.47 |

0.6 | 3.8494 | 3.8497 | 3.8499 | 3.8500 | 0.72 | 3.8501 | 0.78 | 3.8501 | 0.85 | 3.8501 | 0.88 |

0.8 | 4.0869 | 4.0870 | 4.0871 | 4.0871 | 1.21 | 4.0872 | 0.78 | 4.0872 | −0.87 | 4.0872 | 0.89 |

**Table 7.**The values of w

_{p}and rate of convergence (ROC) at different points for k = 2 and various values of M by Bernoulli wavelets.

t | w_{p}M = 4 | w_{p}M = 5 | w_{p}M = 6 | w_{p}M = 7 | ROC M = 7 | w_{p}M = 8 | ROC M = 8 | w_{p}M = 9 | ROC M = 9 | w_{p}M = 10 | ROC M = 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 3.5608 | 3.5609 | 3.5609 | 3.5609 | 9.39 | 3.5609 | −0.77 | 3.5609 | −0.10 | 3.5609 | 2.31 |

0.4 | 3.5783 | 3.5789 | 3.5786 | 3.5787 | 1.06 | 3.5787 | 5.1 | 3.5787 | −0.53 | 3.5787 | 0.21 |

0.6 | 3.5898 | 3.5898 | 3.5898 | 3.5898 | 0.71 | 3.5898 | 0.78 | 3.5898 | 0.85 | 3.5898 | 0.88 |

0.8 | 3.5975 | 3.5976 | 3.5976 | 3.5976 | 1.36 | 3.5976 | 0.76 | 3.5976 | −0.87 | 3.5976 | 0.89 |

**Table 8.**The values of 10

^{10}S

_{p}and rate of convergence (ROC) at different points for k = 0 and various values of L by sine–cosine wavelets.

t | 10^{10} S_{p}L = 4 | 10^{10} S_{p}L = 5 | 10^{10} S_{p}L = 6 | 10^{10} S_{p}L = 7 | ROC L = 7 | 10^{10} S_{p}L = 8 | ROC L = 8 | 10^{10} S_{p}L = 9 | ROC L = 9 | 10^{10} S_{p}L = 10 | ROC L = 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 0.368 | 1.26 | 3.06 | 3.62 | −1.68 | 2.44 | −0.63 | 1.17 | 0.10 | 1.37 | −24.1 |

0.4 | 1.16 | 1.36 | 2.20 | 0.874 | 0.31 | 2.11 | −0.13 | 1.39 | 8.58 | 1.40 | 7.31 |

0.6 | 1.53 | 1.52 | 0.753 | 0.92 | 0.08 | 0.821 | −0.14 | 1.47 | 8.87 | 1.46 | 8.38 |

0.8 | 1.96 | 1.91 | 0.840 | 0.246 | −0.19 | 0.865 | −0.07 | 0.78 | 9.24 | 1.75 | −9.38 |

**Table 9.**The values of e

_{p}and rate of convergence (ROC) at different points for k = 0 and various values of L by sine–cosine wavelets.

t | e_{p}L = 4 | e_{p}L= 5 | e_{p}L = 6 | e_{p}L = 7 | ROC L = 7 | e_{p}L = 8 | ROC L = 8 | e_{p}L = 9 | ROC L = 9 | e_{p}L = 10 | ROC L = 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 5.3833 | 5.3951 | 5.4766 | 5.5134 | −0.41 | 5.4717 | 15 | 5.4189 | 1.91 | 5.4222 | −11.6 |

0.4 | 5.2357 | 5.2385 | 5.2857 | 5.2175 | 0.12 | 5.2783 | −0.31 | 5.2441 | 4.96 | 5.2449 | 6.59 |

0.6 | 5.1227 | 5.1198 | 5.0729 | 5.1403 | 0.13 | 5.0800 | −0.30 | 5.1139 | 5.18 | 5.1131 | 6.50 |

0.8 | 5.0548 | 5.0443 | 4.9666 | 4.9308 | −0.38 | 4.9707 | −0.13 | 5.0219 | 2.35 | 8.0187 | −11.0 |

**Table 10.**The values of i

_{p}and rate of convergence (ROC) at different points for k = 0 and various values of L by sine–cosine wavelets.

t | i_{p}L = 4 | i_{p}L = 5 | i_{p}L = 6 | i_{p}L = 7 | ROC L = 7 | i_{p}L = 8 | ROC L = 8 | i_{p}L = 9 | ROC L = 9 | i_{p}L = 10 | ROC L = 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 3.0032 | 3.0036 | 3.0065 | 3.0077 | −0.40 | 3.0063 | −0.14 | 3.0045 | 2.09 | 3.0046 | −11.5 |

0.4 | 2.9969 | 2.9970 | 2.9987 | 2.9962 | 0.13 | 2.9984 | −0.34 | 2.9972 | 4.69 | 2.9972 | 6.39 |

0.6 | 2.9884 | 2.9883 | 2.9865 | 2.9890 | 0.13 | 2.9868 | −0.35 | 2.9810 | 4.78 | 2.9880 | 6.22 |

0.8 | 2.9803 | 2.9798 | 2.9768 | 2.9754 | −0.45 | 2.9769 | −0.14 | 2.9788 | 1.81 | 2.9787 | −12.1 |

**Table 11.**The values of a

_{p}and rate of convergence (ROC) at different points for k = 0 and various values of L by sine–cosine wavelets.

t | a_{p}L = 4 | a_{p}L = 5 | a_{p}L = 6 | a_{p}L = 7 | ROC L = 7 | a_{p}L = 8 | ROC L = 8 | a_{p}L = 9 | ROC L = 9 | a_{p}L = 10 | ROC L = 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 3.080 | 3.3203 | 3.4046 | 3.4426 | −0.41 | 3.3996 | −0.15 | 3.3449 | 1.91 | 3.3483 | −11.6 |

0.4 | 3.1595 | 3.1624 | 3.2111 | 3.1408 | 0.12 | 3.2035 | −0.31 | 3.1682 | 5.00 | 3.1690 | 6.60 |

0.6 | 3.0527 | 3.0497 | 3.0015 | 3.0709 | 0.13 | 3.0088 | −0.30 | 3.0437 | 5.20 | 3.0429 | 6.53 |

0.8 | 2.9942 | 3.9835 | 2.9040 | 2.8672 | −0.38 | 2.9082 | −0.13 | 2.9609 | 2.39 | 2.9577 | −11.0 |

**Table 12.**The values of r

_{p}and rate of convergence (ROC) at different points for k = 0 and various values of L by sine–cosine wavelets.

t | r_{p}L = 4 | r_{p}L = 5 | r_{p}L = 6 | r_{p}L = 7 | ROC L = 7 | r_{p}L = 8 | ROC L = 8 | r_{p}L = 9 | ROC L = 9 | r_{p}L = 10 | ROC L = 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 3.2907 | 3.2666 | 3.1000 | 3.0248 | −0.41 | 3.1099 | −0.15 | 3.2178 | 1.91 | 3.2111 | −11.6 |

0.4 | 3.5893 | 3.5836 | 3.4872 | 3.6263 | 0.12 | 3.5023 | −0.31 | 3.5721 | 4.97 | 3.5706 | 6.59 |

0.6 | 3.8144 | 3.8204 | 3.9160 | 3.7785 | 0.13 | 3.9016 | −0.30 | 3.8324 | 5.18 | 3.8340 | 6.51 |

0.8 | 4.9469 | 4.9683 | 4.1265 | 4.1995 | −0.38 | 4.1182 | −0.13 | 4.0137 | 2.36 | 4.0201 | −11.0 |

**Table 13.**The values of w

_{p}and rate of convergence (ROC) at different points for k = 0 and various values of L by sine–cosine wavelets.

t | w_{p}L = 4 | w_{p}L = 5 | w_{p}L = 6 | w_{p}L = 7 | ROC L = 7 | w_{p}L = 8 | ROC L = 8 | w_{p}L = 9 | ROC L = 9 | w_{p}L = 10 | ROC L = 10 |
---|---|---|---|---|---|---|---|---|---|---|---|

0.2 | 3.5675 | 3.5664 | 3.5591 | 3.5557 | −0.41 | 3.5595 | −0.15 | 3.5643 | 1.88 | 3.5640 | −11.6 |

0.4 | 3.5802 | 3.5800 | 3.5758 | 3.5819 | 0.12 | 3.5764 | −0.30 | 3.5795 | 5.01 | 3.5794 | 6.64 |

0.6 | 3.5883 | 3.5885 | 3.5927 | 3.5867 | 0.13 | 3.5921 | −0.29 | 3.5890 | 5.28 | 3.5891 | 6.55 |

0.8 | 3.5915 | 3.5924 | 3.5993 | 3.9024 | −0.37 | 3.5989 | −0.13 | 3.5943 | 2.448 | 3.5946 | −10.8 |

**Table 14.**Comparison of CPU running time in seconds, used in Bernoulli for k = 2 and sine–cosine wavelets for k = 0.

Bernoulli Wavelets | M = 4 $\left(\hat{\mathit{m}}=8\right)$ | M = 5 $\left(\hat{\mathit{m}}=10\right)$ | M = 6 $\left(\hat{\mathit{m}}=12\right)$ | L = 7 $\left(\hat{\mathit{m}}=14\right)$ | M = 8 $\left(\hat{\mathit{m}}=16\right)$ | M = 9 $\left(\hat{\mathit{m}}=18\right)$ | M = 10 $\left(\hat{\mathit{m}}=20\right)$ |
---|---|---|---|---|---|---|---|

CPU running time | 0.28 | 0.67 | 1.71 | 2.55 | 3.91 | 9.00 | 67.9 |

Sine-cosine wavelets | L = 4$\left(\hat{\mathit{m}}=9\right)$ | L = 5$\left(\hat{\mathit{m}}=11\right)$ | L = 6$\left(\hat{\mathit{m}}=13\right)$ | L = 7$\left(\hat{\mathit{m}}=15\right)$ | L = 8$\left(\hat{\mathit{m}}=17\right)$ | L = 9$\left(\hat{\mathit{m}}=19\right)$ | L = 10$\left(\hat{\mathit{m}}=21\right)$ |

CPU running time | 0.82 | 1.73 | 3.08 | 5.35 | 8.86 | 9.67 | 13.9 |

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## Share and Cite

**MDPI and ACS Style**

Hedayati, M.; Ezzati, R.; Noeiaghdam, S.
New Procedures of a Fractional Order Model of Novel Coronavirus (COVID-19) Outbreak via Wavelets Method. *Axioms* **2021**, *10*, 122.
https://doi.org/10.3390/axioms10020122

**AMA Style**

Hedayati M, Ezzati R, Noeiaghdam S.
New Procedures of a Fractional Order Model of Novel Coronavirus (COVID-19) Outbreak via Wavelets Method. *Axioms*. 2021; 10(2):122.
https://doi.org/10.3390/axioms10020122

**Chicago/Turabian Style**

Hedayati, Maryamsadat, Reza Ezzati, and Samad Noeiaghdam.
2021. "New Procedures of a Fractional Order Model of Novel Coronavirus (COVID-19) Outbreak via Wavelets Method" *Axioms* 10, no. 2: 122.
https://doi.org/10.3390/axioms10020122