# A New Generalized Modified Weibull Model: Simulating and Modeling the Dynamics of the COVID-19 Pandemic in Saudi Arabia and Egypt

Authors: Walid Emam; Yusra Tashkandy
Journal: Mathematical Problems in Engineering
 (2022)

---

## Abstract

The flare-up of coronavirus disease 2019 (COVID-19) was first identified in China and then spread worldwide. The concerning articles are significant because of their increasing average infection rate. The exact scientific anatomy of this circumstance remains to be discussed. Nevertheless, it is important for governments and all responsible organizations to have sound statistics and analysis to decide on possible necessary actions. This article compares the dynamics of the COVID-19 pandemic between Saudi Arabia and Egypt. We provide an appropriate process of convergence of facts that can be useful for both administrations to implement appropriate quarantine procedures. In addition, the new generalized modified Weibull distribution is provided to provide the preferred description about COVID-19 total cases, daily new cases, total deaths, and daily death statistics of Saudi Arabia and Egypt.


---

## Body

## 1. Introduction

This article has the potential to compare the COVID-19 pandemic outbreak in Saudi Arabia and Egypt. We present a new flexible model to represent the COVID-19 pandemic and closely examine its behavior. The extended model is an improvement of COVID-19 modeling. We refer to some comparisons of epidemic dynamics between different countries in [1–14]. Figure 1 shows the latest information on the total number of COVID-19 cases in Saudi Arabia 1(a) and Egypt 1(b). Figure 2 shows the daily new cases in Saudi Arabia 2(a) and Egypt 2(b).Figure 1
Plots for total COVID-19 cases in Saudi Arabia (a) and Egypt (b).
(a)(b)Figure 2
Plots for daily new cases in Saudi Arabia (a) and Egypt (b).
(a)(b)Figure3 shows the total number of COVID-19 deaths in Saudi Arabia 3(a) and Egypt 3(b). Figure 4 shows daily COVID-19 deaths in Saudi Arabia 4(a) and Egypt 4(b). The gamma, exponential, and Weibull distributions are poor models for fitting the defensible data. Defensible data require the use of a model that is sufficiently fitted within its framework (see Dutta and Perry [15]). Many researchers have adopted new families of distributions (see Ahmad et al. [16], Ahmad et al. [17], Nasir et al. [18], Jamal and Nasir [19], Al-Mofleh [20], Afify et al. [21], Afify and Alizadeh [22], and Cordeiro et al. [23]). In this article, the new method of generalized U-family is used to propose an extended class of statistical models. Suppose Wu;Ξ is the CDF and wu;φ is the PDF of an R.V. The CDF Ϝu;ρ,Ξ of the new generalized-U (NG -U) family of distributions (see Alzaatreh [24]), is given by(1)|u;ρ,Ξ=1−1−Wu;Ξ2ρexpWu;Ξ2,ρ>0,u∈ℝ,where Ξ is the parameter vector. The corresponding PDF, survival function (SF), and hazard rate function (HRF) of R.V.X over the NG-U family are given, respectively, as(2)fu;ρ,Ξ=2e−Wu;Ξ2wu;ΞWu;Ξ1−Wu;Ξ2ρ−11−Wu;Ξ2+ρ,Su;ρ,Ξ=e−Wu;Ξ21−Wu;Ξ2ρ,hu;ρ,Ξ=2wu;ΞWu;Ξ1−Wu;Ξ2−11−Wu;Ξ2+ρ.Figure 3
Plots for total COVID-19 deaths in Saudi Arabia (a) and Egypt (b).
(a)(b)Figure 4
Plots for daily deaths in Saudi Arabia (a) and Egypt (b).
(a)(b)With the new CDF of NG-U, many new flexible models with heavy tails can be obtained. Based on the NG-U approach, some newly added models are presented in Table1. This article is organized as follows: Section 2 defines the new generalized modified Weibull distribution (NG-MW). Section 3 provides the maximum likelihood estimators (MLEs). Section 4 provides the Bayesian estimators. Section 5 performs a simulation discussion. Section 6 analyzes COVID-19 real data to illustrate its potential. Section 7 provides concluding remarks.Table 1
New submodules via the NG-X family.
No.ModelSurvival functionGenerated model1Beta1−Ixa,b2ρ/expIxa,b2a,b,ρ>02Burr1−1−1+xa−2bρ/exp1−1+xa−2ba,b,ρ>03Erlang1−1/a−1!γa,bx2ρ/exp1/a−1!γa,bxa,b,ρ>04Exponential1−1−exp−ax2ρ/exp1−exp−ax2a,ρ>05Frechet1−exp−x/b−a2ρ/expexp−x/b−a2a,b,ρ>06Gamma1−γa,bxγa2ρ/expγa,bxγa2a,b,ρ>07Gumbel1−exp−exp−x−a/b2ρ/expexp−exp−x−a/b2b,ρ>08Half logistic1−1−exp−x/1+exp−x2ρ/exp1−exp−x/1+exp−x2ρ>09Kumaraswamy1−1−1−xa2bρ/exp1−1−xa2ba,b,ρ>010Lindely1−1−exp−ax1+a+ax/1+a2ρ/exp1−exp−ax1+a+ax/1+a2a,ρ>011Linear failure rate1−1−exp−axb−cx2ρ/exp1−exp−axb−cx2a,b,c,ρ>012Log logistics1−1/1+x/b−a2ρ/exp1/1+x/b−aa,b,ρ>013Lomax1−1−1+ax−2bρ/exp1−1+ax−2ba,b,ρ>014Normal1−ϕx−a/b2ρ/expϕx−a/b2b,ρ>015Pareto1−1−xm/x2aρ/exp1−xm/x2aa,xm,ρ>016Power function1−x/b2aρ/expx/b2aa,b,ρ>017Rayleigh1−1−exp−ax22ρ/exp1−exp−ax22a,ρ>018Topp Leone1−xa2−xa2ρ/expxa2−xa2a,b,ρ>019Uniform1−x/a2ρ/expx/a2a,ρ>020Weibull1−1−exp−axb2ρ/exp1−exp−axb2a,b,ρ>0


## 2. Submodel Description

The modified Weibull distribution is one of the most memorable modifications of the Weibull distribution, developed to provide adaptability to the exponential, Rayleigh, linear failure rate, and Weibull distributions (see [25]). We further develop this area of distribution theory and provide a new memorable version of the modified Weibull model to improve its adaptability. The modified Weibull distribution with three parameters Ξ=δ,η,ζ has CDF, PDF, SF, and HRF and is given by(3)Wx;δ,η,ζ=1−e−δx−ηxζ,x>0,ζ>0,δ,η≥0,ξ+η>0,wx;δ,η,ζ=e−δx−ηxζδ+xζ−1ηζ,x>0,ζ>0,δ,η≥0,δ+η>0,Sx;δ,η,ζ=e−δx−ηxζ,x>0,ζ>0,δ,η≥0,δ+η>0,hx;δ,η,ζ=δ+ηζxζ−1,x>0,ζ>0,δ,η≥0,δ+η>0.We introduce the NG-MW distribution and examine the setting of its PDF, SF, and HRF. An R.V.X follows the NG-MW distribution if its CDF takes the form(4)Fx;δ,η,ζ,ρ=1−e−1−e−xδ−xζη21−1−e−xδ−xζη2ρ.where x>0,ζ,ρ>0,δ,η≥0, and δ+η>0. The PDF, SF, and HRF of the NG-MW distribution are, respectively, given by(5)fx;δ,η,ζ,ρ=2e−δx−ηxζ−−1+e−δx−ηxζ2δ+ηζxζ−11−e−δx−ηxζ×1−−1+e−xδ−xζη2ρ−11−−1+e−xδ−xζη2+ρ,Sx;δ,η,ζ,ρ=e−1−e−xδ−xζη21−1−e−xδ−xζη2ρ,(6)hx;δ,η,ζ,ρ=2e−xδ−xζηδ+x−1+ζηζ1−e−xδ−xζη1−1−e−xδ−xζη2+ρ1−1−e−xδ−xζη2.The model NG-MW reduces to the NG-Weibull distribution whenδ=1, and reduces to the modified Weibull distribution when ρ=1.


## 3. Maximum Likelihood Estimation

Let the observed valuesx1,x2,…,xn of X1,X2,…,Xn be a random sample from the model NG-X. The NG-X probability is(7)l=2ne−∑i=1nWxi;Ξ2∏i=1nwxi;ΞWxi;Ξ1−Wxi;Ξ2ρ−11−Wxi;Ξ2+ρ.The corresponding NG-X log-likelihood is given by(8)L=nLog2−∑i=1nWxi;Ξ2+∑i=1nLogwxi;Ξ+∑i=1nLogWxi;Ξ+ρ−1∑i=1nLog1−Wxi;Ξ2+∑i=1nLog1+ρ−Wxi;Ξ2.MLE can be derived by maximizing equation (8) as follows:(9)∂L∂ρ=∑i=1nLog1−Wxi;Ξ2+∑i=1n11+ρ−Wxi;Ξ2.Then the NG-MW log-likelihood function has the form(10)L=nLog2−∑i=1n1−e−δxi−ηxiζ2+∑i=1nLoge−xiδ−xiζηδ+xiζ−1ηζ+∑i=1nLog1−e−δxi−ηxiζ+ρ−1∑i=1nLog1−1−e−δxi−ηxiζ2+∑i=1nLog1+ρ−1−e−δxi−ηxiζ2.Maximizing equation (10) to obtain the MLE estimates as follows:(11)∂L∂δ=∑i=1nxieδxi+ηxiζ−1+2ρ−1∑i=1nxie−δxi−ηxiζe−δxi−ηxiζ−11−1−e−δxi−ηxiζ2+2∑i=1n11+ρ−1−e−δxi−ηxiζ2+1xie−δxi−ηxiζe−δxi−ηxiζ−1+∑i=1n1−xiδ+ηζxiζ−1δ+ηζxiζ−1,(12)∂L∂η=∑i=1nxiζeδxi+ηxiζ−1+2ρ−1∑i=1nxiζe−δxi−ηxiζe−δxi−ηxiζ−11−1−e−δxi−ηxiζ2+2∑i=1n11+ρ−1−e−δxi−ηxiζ2+1xiζe−δxi−ηxiζe−δxi−ηxiζ−1+∑i=1nxiζ−1ζ−xiδ+ηζxiζ−1δ+ηζxiζ−1,(13)∂L∂ζ=∑i=1nηxiζLogxieδxi+ηxiζ−1+2ηρ−1∑i=1ne−δxi−ηxiζ−1xiζe−δxi−ηxiζLogxi1−1−e−δxi−ηxiζ2+2η∑i=1ne−δxi−ηxiζ−1xiζe−δxi−ηxiζLogxi1+ρ−1−e−δxi−ηxiζ2+2η∑i=1ne−δxi−ηxiζ−1xiζe−δxi−ηxiζLogxi+∑i=1n−ηLogxixiζδ+ηζxiζ−1+ηxiζ−1+ηζLogxixiζ−1δ+ηζxiζ−1,(14)∂L∂ρ=∑i=1n11+ρ−1−e−δxi−ηxiζ2+∑i=1nLog1−1−e−δxi−ηxiζ2.The maximum likelihood estimators of the NG-MW distribution parametersδ,η,ζ, and ρ are the solutions of the above nonlinear equations. Since the equations expressed in (11)–(14) cannot be solved analytically, one must use a numerical procedure to solve them. The asymptotic C.Is of δ,η,ζ, and ρ can be calculated. The variance-covariance matrix Vκ^,η^,ζ^,ρ^ was given by (15)Vκ^,η^,ζ^,ρ^=−∂2L∂δ2∂2L∂δ∂η∂2L∂δ∂ζ∂2L∂δ∂ρ∂2L∂η∂δ∂2L∂η2∂2L∂η∂ζ∂2L∂η∂ρ∂2L∂ζ∂δ∂2L∂ζ∂η∂2L∂ζ2∂2L∂ζ∂ρ∂2L∂ρ∂δ∂2L∂ρ∂η∂2L∂ρ∂ζ∂2L∂ρ2−1.A1001−ε% two-sided approximate C.Is for δ,η,ζ, and ρ are, respectively, given by(16)δ^±zδ/2Vδ^,η^±zδ/2Vη^,ζ^±zδ/2Vζ^,ρ^±zδ/2Vρ^,where Vδ^, Vη^, Vζ^, and Vρ^ are given by the diagonal elements of Vδ^,η^,ζ^,ρ^, and zε/2 is the upper ε/2 percentile of the standard normal distribution.


## 4. Bayesian Estimation

Assume thatδ,η,ζ, and ρ are R.Vs corresponding to the prior PDFs Gamma δ;a1,b1, Gamma η;a2,b2, Gamma ζ;a3,b3, respectively. Gamma ρ;a4,b4 follows, where ai and bi are positive constants and i=1,2,3,4. The posterior DF of δ,η,ζ,ρ and the data under the gamma priors can take the following forms:(17)π1δ=b1a1ζa1δa1−1exp−b1δ,δ,a1,b1>0,π2η=b2a2ζa2ηa2−1exp−b2η,η,a2,b2>0,π3ζ=b3a3ζa3ζa3−1exp−b3ζ,ζ,a3,b3>0,π4ρ=b4a4ζa4ρa4−1exp−b4ρ,ρ,a4,b4>0.Then, the posterior density ofδ,η,ζ,ρ and the data can be extracted as(18)π∗δ,η,ζ,ρ|x∝πδ,η,ζ,ρ∏i=1nfxi;δ,η,ζ,ρ=J−1δa1−1ηa2−1ζ1a3−1ρa4−1e−b1δ+b2η+b3ζ+b4ρ−∑i=1nδxi+ηxiζ+1−e−δxi−ηxiζ2∏i=1nδ+ηζxiζ−11−e−δxi−ηxiζ1−1−e−δxi−ηxiζ2ρ−1∏i=1n1+ρ−1−e−δxi−ηxiζ2,where xi≥0,δ,η,ζ,ρ,ai,bi>0,i=1,2,3,4. And J is the normalizing constant.


## 5. MCMC Method

We use the Metropolis–Hastings (M-H) procedure as follows:(1)
Set initial valuesδ0,η0,ζ0, and ρ0. Then, simulate a sample of size n from NG-MW, next set l=1.(2)
Simulateδ∗,η∗,ζ∗, and ρ∗ using the proposal distributions Nδl−1,Vδ^, Nηl−1,Vη^, Nζl−1,Vζ^, and Nρl−1,Vρ^.(3)
Obtain the probabilityr=minπ∗δ∗,η∗,ζ∗,ρ∗/π∗δl−1,ηl−1,ζl−1,ρl−1,1.(4)
SimulateU from Uniform (0, 1).(5)
IfU<r, then δl,ηl,ζl,ρl=δ∗,η∗,ζ∗,ρ∗. If U≥r, then δl−1,ηl−1,ζl−1,ρl−1=δ∗,η∗,ζ∗,ρ∗.(6)
Setl=l+1.(7)
Iterate steps 2–6,M repetitions, and get δl,ηl,ζl and ρl for l=1,…,M.Now, if we use the squared error loss function, given byLSEϑ,ϑ^=ϑ−ϑ^2, where ϑ^ is an estimate of the unknown parameter ϑ=δ,η,ζ, and ρ against the loss function SE and is the posterior mean. Using the generated random samples from the Gibbs sampling procedure above and for N is the burn, then the Bayesian estimator of ϑ, say ϑ^SE, can be obtained as(19)ϑ^SE=Eϑϑ|x=1M−N∑l=N+1Mϑl.The second loss function is the LINEX loss function, given by(20)LLEϑ,ϑ^=expϖϑ−ϑ^−ϖϑ−ϑ^−1,ρ≠0.The approximate Bayes estimate ofϑ=σ,δ,ζ, and ρ under LE loss function based on the Gibbs sampling technique, becomes(21)ϑ^LE=−1ϖlogEϑexp−ϖϑ|x=−1ϖlog∑l=N+1Mexp−ϖϑlM−N.Finally, the general entropy (GE) loss function, given by(22)LGEϑ,ϑ^=ϑ^ϑε−εlogϑ^ϑ−1.The approximate Bayes estimate of the parameters, given by(23)ϑ^GE=Eϑϑ−ε|x−1/ε=1M−N∑l=N+1Mϑl−ε−1/ε.The MCMC HPD credible interval algorithm procedure is as follows:(1)
Sortδ∗,η∗,ζ∗, and ρ∗ in rising values.(2)
The lower bounds ofδ,η,ζ, and ρ in the rank M−N∗ε/2.(3)
The lower bounds ofδ,η,ζ, and ρ in the rank M−N∗1−ε/2.(4)
Iterate the previous stepsM times. Get the average value of the lower and upper bounds of δ,η,ζ, and ρ.The first and second lines show the credible HPD interval and the corresponding width of the parameter, respectively.


## 6. Numerical Calculations

First, we simulateM = 1000 samples of size n = 50 from the model NG-MW over initial parameter values δ0=2.6,η0=3.2,ζ0=5.01, and ρ0=3.7. In this simulation study, we empirically determine the biases and expected risks (ER) of the MLEs and Bayesian methods for the different parameter combinations of each sample. The point estimates of the parameters using the 200 burns MCMC methods are obtained. Two LINEX loss functions are used: LE1 when ϖ=−0.3 and LE2 when ϖ=0.7. The biases and ERs are given respectively by(24)Biasϑ^=11000∑i=11000ϑ^i−ϑ,ERϑ^=11000∑i=11000ϑ^i−ϑ2.Coverage probabilities (CPs) are also calculated for the 95% and 90% HPD credibility intervals. Simulation results are shown in Table2 for parameters δ,η,ζ, and ρ.Table 2
Point and interval estimation of the parametersδ,η,ζ,ρ.
ParameterInit.aPointIntervalBMLSELE1LE2GEMLHPDSHPDLE1HPDLE2HPDGEδ2.60.990.732.90782.69082.69362.68412.68530.209 5.60662.594 2.9452.5992 2.94632.5854 2.94262.587 2.94260.30780.09080.09360.08410.08535.39750.3510.34710.35710.35561.89590.03010.03040.02950.02960.6359 5.17972.631 2.9252.6332 2.92552.6194 2.92462.6214 2.9254.54380.2940.29230.30530.3035η3.20.010.472.90783.18753.19083.17983.18230.426 5.38962.853 3.2832.8526 3.28692.8525 3.27182.8526 3.2764−0.2922−0.0125−0.0092−0.0202−0.01774.96360.430.43440.41930.42381.60330.0290.02890.02920.02910.8185 4.99712.881 3.2452.8826 3.24982.8788 3.23312.879 3.23644.17850.3640.36720.35430.3574ζ5.010.640.292.44635.02185.02545.01355.01820.0001 5.32994.753 5.2634.7584 5.26574.7445 5.25824.7455 5.2611−2.56360.01180.01540.00350.00825.32980.510.50720.51370.51562.16450.02960.02960.02970.02960.0188 4.87384.788 5.2234.7911 5.22644.7821 5.21384.7857 5.21924.85510.4350.43530.43170.4335ρ3.70.260.523.25583.42743.42833.42553.42620.2639 6.24773.308 3.393.3081 3.39023.3081 3.38873.3081 3.3891−0.4442−0.2726−0.2717−0.2745−0.27385.98380.0820.08220.08070.0812.33010.07980.07940.08070.08030.7371 5.77453.312 3.3843.3125 3.38463.3124 3.38363.3124 3.38415.03730.0720.07210.07120.0717Point estimate: the first line represents estimate, the second line represents bias, and the third line represents ER. Interval estimate:95% and 90% interval estimate, respectively. The first and second lines show the credible HPD interval and the corresponding width of the parameter, respectively.Next, the COVID-19 dataset (see, [4]) is considered as an application of the NG-MW model and is given by 0.054, 0.064, 0.704, 0.816, 0.235, 0.976, 0.865, 0.364, 0.479, 0.568, 0.352, 0.978, 0.787, 0.976, 0.087, 0.548, 0.796, 0.458, 0.087, 0.437, 0.421, 1.978, 1.756, 2.089, 2.643, 2.869, 3.867, 3.890, 3.543, 3.079, 3.646, 3.348, 4.093, 4.092, 4.190, 4.237, 5.028, 5.083, 6.174, 6.743, 7.274, 7.058, 8.273, 9.324, 10.827, 11.282, 13.324, 14.278, 15.287, 16.978, 17.209, 19.092, 20.083. The expected real results are shown in Table 3 for the parameters δ,η,ζ, and ρ.Table 3
Point and interval estimation of the parametersδ,η,ζ,ρ via the COVID-19 dataset.
ParameterabPointIntervalMLSELE1LE2GEMLHPDSδ0.270.230.27380.23170.23160.23160.23160.0167 0.52330.1212 0.26390.01671.30E − 099.90E − 313.70E − 321.20E − 320.50660.14270.0568 0.48320.4999 0.69330.42650.1934η0.550.640.63870.24970.24970.24970.24960.1438 1.13620.2247 0.25430.06411.00E − 045.10E − 055.10E − 050.00010.99250.02960.2223 1.05770.2295 0.25220.83550.0227ζ0.50.520.52430.44570.44570.44550.44490.2046 0.83540.3705 0.45420.02592.80E − 320.00040.00040.00040.63090.08370.2545 0.78550.389 0.45420.53110.0652ρ0.180.520.51940.49990.49990.49990.49990.0001 1.13910.4999 0.75980.09982.80E − 322.70E − 321.20E − 322.70E − 321.1390.25990.0001 1.04120.4999 0.69331.04110.1934Point estimate: the first line represents estimate and the second line represents ER. Interval estimate:95% and 90% interval estimate, respectively.Table4 compares the NG-MW distribution based on some recognition criteria, such as the Akaike information criterion (AIC), Bayesian information criterion (BIC), Hannan–Quinn information criterion (HQIC), and the consistent Akaike information criterion (CAIC). The goodness-of-fit results of the NG-MW model are compared with some other models, including Weibull, NG-Weibull (NG-W), and NG-exponential (NG-Exp) distributions.Table 4
Relative quality of the NG-MW vs competing models.
ModelAICCAICBICHQICNG-W273.0032273.493278.9141275.2762NG-MW274.5728275.4061282.454277.6035NG-Exp282.32282.56286.2606283.8354Weibull321.4568321.9466327.3677323.7298


## 7. Conclusion

In this paper, a four-parameter extension of the Weibull distribution, the new generalized modified Weibull distribution, is discussed in detail. Maximum likelihood and Bayesian estimation are used to estimate the model parameters. A simulation study was conducted to evaluate the performance of the new generalized modified Weibull distribution. The flexibility and usefulness of the proposed new generalized modified Weibull distribution is shown in the analysis of COVID-19 data. The new generalized modified Weibull model outperformed many other old models, as shown in the results in Table4. The performance of the Bayesian estimates is more excellent than that of the corresponding ML estimators. The results indicate that the generalized Weibull model, as a special case of the new generalized modified Weibull model, provides a better fit than other competing models. We found that the proposed model provides a better characterization of COVID-19 events by analyzing daily death data. Therefore, it can be a perfect model for predicting future cases.



---

*Source: 1947098-2022-04-21.xml*