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.
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).
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Figure 3 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 is the CDF and is the PDF of an R.V. The CDF of the new generalized-U (NG -U) family of distributions (see Alzaatreh [24]), is given bywhere 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
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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 Table 1. 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.
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 has CDF, PDF, SF, and HRF and is given by
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 formwhere , and . The PDF, SF, and HRF of the NG-MW distribution are, respectively, given by
The model NG-MW reduces to the NG-Weibull distribution when , and reduces to the modified Weibull distribution when .
3. Maximum Likelihood Estimation
Let the observed values of be a random sample from the model NG-X. The NG-X probability is
The corresponding NG-X log-likelihood is given by
MLE can be derived by maximizing equation (8) as follows:
Then the NG-MW log-likelihood function has the form
Maximizing equation (10) to obtain the MLE estimates as follows:
The maximum likelihood estimators of the NG-MW distribution parameters , and 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 , and can be calculated. The variance-covariance matrix was given by
A two-sided approximate C.Is for , and are, respectively, given bywhere , , , and are given by the diagonal elements of , and is the upper percentile of the standard normal distribution.
4. Bayesian Estimation
Assume that , and are R.Vs corresponding to the prior PDFs Gamma , Gamma , Gamma , respectively. Gamma follows, where and are positive constants and . The posterior DF of and the data under the gamma priors can take the following forms:
Then, the posterior density of and the data can be extracted aswhere . And is the normalizing constant.
5. MCMC Method
We use the Metropolis–Hastings (M-H) procedure as follows:(1)Set initial values , and . Then, simulate a sample of size from NG-MW, next set .(2)Simulate , and using the proposal distributions , , , and .(3)Obtain the probability .(4)Simulate from Uniform (0, 1).(5)If , then . If , then .(6)Set .(7)Iterate steps 2–6, M repetitions, and get and for .
Now, if we use the squared error loss function, given by , where is an estimate of the unknown parameter , and against the loss function SE and is the posterior mean. Using the generated random samples from the Gibbs sampling procedure above and for is the burn, then the Bayesian estimator of , say , can be obtained as
The second loss function is the LINEX loss function, given by
The approximate Bayes estimate of , and under LE loss function based on the Gibbs sampling technique, becomes
Finally, the general entropy (GE) loss function, given by
The approximate Bayes estimate of the parameters, given by
The MCMC HPD credible interval algorithm procedure is as follows:(1)Sort , and in rising values.(2)The lower bounds of , and in the rank .(3)The lower bounds of , and in the rank .(4)Iterate the previous steps times. Get the average value of the lower and upper bounds of , and .
The first and second lines show the credible HPD interval and the corresponding width of the parameter, respectively.
6. Numerical Calculations
First, we simulate M = 1000 samples of size n = 50 from the model NG-MW over initial parameter values , and . 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: when and when . The biases and ERs are given respectively by
Coverage probabilities (CPs) are also calculated for the 95% and 90% HPD credibility intervals. Simulation results are shown in Table 2 for parameters , and .
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 , and .
Table 4 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.
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 Table 4. 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.
Data Availability
All the datasets used in this paper are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This study was funded by Researchers Supporting Project number (RSP2022R488), King Saud University, Riyadh, Saudi Arabia.