A New Flexible Statistical Model: Simulating and Modeling the Survival Times of COVID-19 Patients in China

The spread of the COVID-19 epidemic, since December 2019, has caused much damage around the world, disturbed every aspect of daily life, and has become a serious health threat. The COVID-19 epidemic impacted nearly 150 countries around the globe between December 2019 and March 2020. Since December 2019, researchers have been trying to develop new suitable statistical models to adequately describe the behavior of this deadly pandemic. In this paper, a ﬂexible statistical model has been proposed that can be used to model the lifetime events associated with this deadly pandemic. The new distribution is derived from the combination of an extended Weibull distribution and a trigonometric strategy referred to as the arcsine- X approach. Hence, the new model may be referred to as the arcsine new ﬂexible extended Weibull model. The proposed model is capable of capturing ﬁve diﬀerent behaviors of the hazard rate function. The model parameters are estimated via the maximum likelihood approach. Furthermore, a Monte Carlo study is conducted to assess the behavior of the estimators. Finally, the applicability of the new model is demonstrated using the data of ﬁfty-three patients taken from a hospital in China.


Introduction
COVID-19 (coronavirus disease 2019) was first spotted in China in December 2019 and then spread worldwide exponentially. e main reasons for the exponential growth of COVID-19 include unawareness and ignorance of this deadly virus, inadequate tools, low efficiency in detection, and delays in formulating appropriate and effective policies during the initial stage of the epidemic [1]. As  e comparison of the COVID-19 epidemic between different countries is worth studying and is of great concern. In this regard, researchers are devoting serious efforts to make comparisons between different countries. e problems related to the COVID-19 epidemic in Italy have been discussed in [2]. Comparison of COVID-19 in some European countries, South Korea, and the USA has been done in [3].
e COVID-19 pandemic in Australia has been discussed in [4]. e phenomena of the spread of  in Lebanon are provided in [5]. A case study from Spain has been discussed in [6]. Mathematical analysis of COVID- 19 in Mexico is provided in [7]. A case study from Brazil is discussed in [8]. e issues related to the COVID-19 epidemic in Pakistan are provided in [9]. A mathematical model developed for assessing the transmission of the COVID-19 epidemic in India is introduced in [10]. Among the Asian countries, the phenomena of the spread of the COVID-19 epidemic are discussed in [11]. e comparison between Iran and mainland China has appeared in [12]. e comparison between two neighbor countries Iran and Pakistan has appeared in [13]. e problems related to the COVID-19 pandemic in Indonesia have been discussed in [14]. From the literature cited above, we see that there is a great interest to learn and know more about the COVID-19 epidemic. In the field of big data science and other related sectors, providing the best description of the real phenomena is a prominent research topic (refer [15][16][17][18][19][20] for details). Recent studies have pointed out the potentiality of the statistical models in different sectors of applied sciences. e goal of this article is to carry out this research area of distribution theory and propose a new statistical model to provide a better fit to the survival times data.
Recently, Liao et al. [21] introduced a modification of the Weibull distribution called a NFEW (new flexible extended Weibull) and suggested its application in modeling lifetime events. Let X have the NFEW distribution with two shape parameters (η 1 , η 2 > 0) and two scale parameters (κ 1 , κ 2 > 0); then, its DF (distribution function) F(x; Ξ) is as follows: where Ξ � (η 1 , η 2 , κ 1 , κ 2 ). e corresponding PDF (probability density function) f(x; Ξ) is as follows: In this work, we propose a new modification of the NFEW model called ASNFEW (arcsine new flexible extended Weibull) distribution using the arcsine-X strategy [22], which can be obtained as a subcase of [23]. e DF and PDF of the arcsine-X distributions are given by respectively. e DF of the proposed ASNFEW distribution is obtained by inserting equation (1) in (3). In the next section, we introduce the proposed model and sketch some possible behaviors of the PDF and HRF (hazard rate function) of the ASNFEW distribution.

The Arcsine New Flexible Extended Weibull Model
A random variable X has the ASNFEW model if its DF G(x) is given by with SF (survival function) denoted by S(x), and it can be expressed as follows: For η 1 � 0.5, κ 1 � 1.2, η 2 � 0.5, and κ 2 � 0.9, the plots for the DF and SF of the ASNFEW model are sketched in Figure 1.

Basic Mathematical Properties
In this section, we will establish some statistical properties of the ASNFEW distribution.
3.1. Quantile Function. Let X denote the ASNFEW random variable with DF given by equation (5); then, the QF (quantile function) of X, denoted by Q(u), is given by e QF (also called inverse DF) can be used to generate random numbers. Later, in Section 5, we will use the inverse DF method to carry out the simulation study.

Moments.
is section deals with the derivation of the r th moment of the ASNFEW distribution that can be further used to obtain important characteristics. e r th moment of the ASNFEW distribution can be obtained as follows: Using binomial series that is convergent when |t| < 1 (see https://socratic.org/questions/how-do-you-use-the-binom ial-series-to-expand-f-x-1-sqrt-1-x-2), we have Using equation (11), we have Using expression (12) in (10), we have

Moment-Generating Function.
is section offers the MGF (moment generating function) of the ASNFEW model. Let X has the ASNFEW model; then, the MGF of X is derived as Using equations (16) in (17), we get the MGF of the proposed model.

Skewness and Kurtosis.
Using r � 1, 2, 3, 4 in equation (16), we get the first four moments of the ASNFEW distributions. ese moments can be used to derive the following important characteristics of the ASNFEW distribution: (iii) e CK (coefficient of kurtosis): For κ 1 � κ 2 � 1 and different values of η 1 and η 2 , plots of the important characteristics of the ASN-FEW distribution are sketched in Figure 4.

An Application to the Survival Times of COVID-19 Patients in China
A clear motivation behind proposing new statistical distributions is to increase the level of flexibility and applicability of the existing distributions. e main objective of proposing the ASNFEW distribution is its utilization in describing different types of time-to-event data. In this section, the respective fact is demonstrated using the data of fifty-three patients taken from a hospital in China. e data set is given as follows:  Table 3. e histogram and box plot of COVID-19 data along with the total time test (TTT) plot are sketched in Figure 6.
e PDFs of the competitive models are as follows: (i) MOW distribution: (ii) Ku-W distribution: (iii) OLL-MW distribution: (iv) T-MW distribution: (v) FrW distribution: (vi) B-MW distribution: e decision about the best fitting of the competing distributions is made by considering certain criteria selected for comparison. ese criteria consist of some discrimination and goodness-of-fit measures. e expressions of the discrimination measures (DM) are given as follows: (i) e AIC (Akaike information criterion): 8 Complexity (ii) e BIC (Bayesian information criterion): (iii) e HQIC (Hannan-Quinn information criterion): (iv) e CAIC (corrected Akaike information): e expressions of the goodness-of-fit measures are given as follows: (i) e AD (Anderson-Darling) test statistic: (ii) e CM (Cramer-von Mises) test statistic: (iii) e KS (Kolmogorov-Smirnov) test statistic:    Tables 4  and 5, respectively, whereas the goodness-of-fit measures are provided in Table 6.              Tables 5 and 6, a graphical display of these results is provided in Figures 7 and  8.
Based on the numerical results in Tables 5 and 6 as well as the graphical display of these results in Figures 7 and 8, we can conclude that the ASNFEW is a useful and suitable candidate distribution for modeling COVID-19 and other related data sets. For further confirmation of the best fitting capability of the ASNFEW distribution, the plots of the estimated PDF, DF, SF, and PP (probability-probability) are sketched in Figure 9. e plots sketched in Figure 9 show a closer fit of the ASNFEW distribution to the data under consideration.

Concluding Remarks
e statistical distributions have proven to be of great importance and attracted the attention of researchers to use them for modeling data, particularly, in the fields related to lifetime events. In this work, a generalization of the new flexible extended Weibull model is proposed. An example of real-life data related to the survival times of COVID-19 patients in China is used to demonstrate the applicability of the ASNFEW distribution. Comparisons of the proposed model with the other competitors, including four-parameter and five-parameter models, are provided. Certain analytical tools, including four discrimination measures and three goodness-of-fit measures, are considered to compare the fitted distributions. e numerical results of these analytical measures showed that the ASNFEW model provides a better fit, supported by the graphical sketching and numerical tools. In summary, the proposed model has a wide range of applications because of its flexibility in modeling different types of hazard functions. A simulation study also reveals that the proposed model can be valuable in adequately describing different types of time-to-event data. We hope that beyond the scope of this paper, the ASNFEW distribution can be applied to analyze other forms of the data related to COVID-19 events.

Data Availability
e data set used to support this study is included within the article. However, additional data (if required) will be provided upon request to the corresponding author.