# Sine Frèchet Model: Modeling of COVID-19 Death Cases in Kingdom of Saudi Arabia

Authors: Maha A. Aldahlan
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this research, we discuss a new lifespan model that extends the Frèchet (F) distribution by utilizing the sine-generated family of distributions, known as the sine Frèchet (SF) model. The sine Frèchet distribution, which contains two parameters, scale and shape, aims to give an SF model for data fitting. The sine Frèchet model is more adaptable than well-known models such as the Frèchet and inverse exponential models. The sine Frèchet distribution is used extensively in medicine, physics, and nanophysics. The SF model’s statistical properties were computed, including the quantile function, moments, moment generating function (MGF), and order statistics. To estimate the model parameters for the SF distribution, the maximum likelihood (ML) estimation technique is applied. As a result of the simulation, the performance of the estimations may be compared. We use it to examine a current dataset of interest: COVID-19 death cases observed in the Kingdom of Saudi Arabia (KSA) from 14 April to 22 June 2020. In the future, the SF model could be useful for analyzing data on COVID-19 cases in a variety of nations for possible comparison studies. Finally, the numerical results are examined in order to assess the flexibility of the new model.


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## Body

## 1. Introduction

Diverse academics have recently become interested in the families of distributions that have been produced [1]: the beta-G [2], Kumaraswamy-G [3], the suggested sine-generated (S-G) family [4], modified odd Weibull-G [5], new Kumaraswamy-G [6], xgamma G [7], the truncated Frèchet-G [8], truncated Cauchy power-G and the truncated Weibull-G in [9], odd generalized gamma in [10], odd Weibull-G in [11], exponentiated M-G in [12], odd Chen-Leone-G [13], Topp-Leone odd Fréchet-G [14], odd Burr-G [15], box Cox gamma-G [16], Burr X family [17], Marshall–Olkin exponentiated generalized-G [18], and recently, DUS transformation in [19] and KM-G family [20].As a result, S-G has both a distribution function (CDF) and density function (PDF):(1)Fx;ξ=sinπ2Gx;ξ,x∈R,(2)fx;ξ=π2gx;ξcosπ2Gx;ξ,x∈R,where gx;ξ is considered a PDF of baseline distribution.The Frèchet (F) model is an important model which can be used to analyze the life time data with some monotone failure rates. The PDF and CDF of theF distribution are(3)gx;μ,δ=δμδxδ+1e−μ/xδ,x,μ,δ>0,(4)Gx;α,β=e−μ/xδ,x,μ,δ>0.This work tries to improve the flexibility of the F model by employing the S-G family. The new model has several PDF shapes, including decreasing, right skewness, and unimodal shapes. The hazard rate function (HRF) can also be declining and J-shaped. These findings are depicted in Figures1 and 2. Based on the S-G family of distributions, a unique two-parameter distribution is introduced and analyzed. The model that has been presented is known as the SF model. The SF model is more adaptable and applicable than the F model.Figure 1
The PDF version of the SF model.Figure 2
The HRF version of the SF model.This article is organized as follows: in Section2, a proposal is made for the SF distribution to be built. ML estimators are investigated in Sections 3 and 4, and simulation results of SF parameters are examined in Section 5. Using COVID-19 death data, Section 5 proposes a model that may be applied to real-world data. Section 6 deals with some basic properties of the SF model and the article ends with conclusions.


## 2. Construction of the SF Model

Here, the CDF, PDF, survival function, and HRF of the random variable (rv) X are calculated:(5)Fx=sinπ2e−μ/xδ,x>0,μ,δ>0,fx=πδμδ2xδ+1e−μ/xδcosπ2e−x/μδ,x>0,μ,δ>0,Rx=1−sinπ2e−μ/xδ,(6)hx=πδμδ/2xδ+1e−μ/xδcosπ/2e−μ/xδ1−sinπ/2e−μ/xδ,where μ is a scale parameter and δ is the shape parameter.Figures1 and 2 show the plots of PDF and HRF for various parameter values. This model can have decreasing PDF, right-skewed, and unimodal HRF and decreasing PDF, right-skewed, and unimodal HRF.


## 3. Statistical Characteristics

Here, we will look at some of the statistical characteristics of the SF distribution.


### 3.1. Moments

Theorem 1.
LetX can be a rv. When using the SF model, its rth moment(7)μr′=∑i=0∞μr−δϖΓ1−r/δδ2i+11−r/δ.Proof.
Let X be a rv with PDF (6). Therth moment of the SF distribution is calculated as(8)μr′=∫0∞xrfxdx=πδμδ2∫0∞xr−δ−1e−μ/xδcosπ2e−μ/xδdx.
By inserting the expansioncosGx=∑i=0∞−1i/2i!Gx2i,n to the previous equation, (9)μr′=∑i=0∞δμδ−1i2i!π22i+1∫0∞xr−δ−1e−2i+1μ/xδdx.
The last equation can be rewritten as(10)μr′=∑i=0∞ϖ∫0∞xr−δ−1e−2i+1μ/xδdx,where ϖ=∑i=0∞δμδ−1i/2i!π/22i+1.
Lety=μ/xδ, then(11)μr′=∑i=0∞μr−δδϖ∫0∞y−r/δe−2i+1ydy.
Then,(12)μr′=∑i=0∞μr−δϖΓ1−r/δδ2i+11−r/δ.
The MGF of X is(13)MXt=∑r=0∞trr!μr′=∑r,i=0∞trμr−δϖΓ1−r/δr!δ2i+11−r/δ.


### 3.2. Quantile Function

If X ∼ SF, then the quantile function of SF is(14)Qu=μlnπ2Arcsinu−1δ,and by substituing u = 0.5, we obtain the median (M) as(15)M=μln3−1/δ.


### 3.3. Order Statistics

LetX1,X2,…,Xn be r sample from the SF model with order statistics X1,X2,…,Xn. The PDF of X(k) of order statistics is(16)fXkx=n!k−1!n−k!Fk−1xfx1−Fxn−k.The PDF ofXk can be expressed as(17)fXkx=n!δμδΞxδ+1k−1!n−k!sinΞk−1cosΞ1−sinΞn−k,where π/2e−μ/xδ=Ξ. We can obtain the PDF of the lowest and largest order statistics at k=1 and k=n, respectively,(18)fX1x=nδμδΞxδ+1cosΞ1−sinΞn−1,(19)fXnx=nδμδΞxδ+1sinΞn−1cosΞ.


## 3.1. Moments

Theorem 1.
LetX can be a rv. When using the SF model, its rth moment(7)μr′=∑i=0∞μr−δϖΓ1−r/δδ2i+11−r/δ.Proof.
Let X be a rv with PDF (6). Therth moment of the SF distribution is calculated as(8)μr′=∫0∞xrfxdx=πδμδ2∫0∞xr−δ−1e−μ/xδcosπ2e−μ/xδdx.
By inserting the expansioncosGx=∑i=0∞−1i/2i!Gx2i,n to the previous equation, (9)μr′=∑i=0∞δμδ−1i2i!π22i+1∫0∞xr−δ−1e−2i+1μ/xδdx.
The last equation can be rewritten as(10)μr′=∑i=0∞ϖ∫0∞xr−δ−1e−2i+1μ/xδdx,where ϖ=∑i=0∞δμδ−1i/2i!π/22i+1.
Lety=μ/xδ, then(11)μr′=∑i=0∞μr−δδϖ∫0∞y−r/δe−2i+1ydy.
Then,(12)μr′=∑i=0∞μr−δϖΓ1−r/δδ2i+11−r/δ.
The MGF of X is(13)MXt=∑r=0∞trr!μr′=∑r,i=0∞trμr−δϖΓ1−r/δr!δ2i+11−r/δ.


## 3.2. Quantile Function

If X ∼ SF, then the quantile function of SF is(14)Qu=μlnπ2Arcsinu−1δ,and by substituing u = 0.5, we obtain the median (M) as(15)M=μln3−1/δ.


## 3.3. Order Statistics

LetX1,X2,…,Xn be r sample from the SF model with order statistics X1,X2,…,Xn. The PDF of X(k) of order statistics is(16)fXkx=n!k−1!n−k!Fk−1xfx1−Fxn−k.The PDF ofXk can be expressed as(17)fXkx=n!δμδΞxδ+1k−1!n−k!sinΞk−1cosΞ1−sinΞn−k,where π/2e−μ/xδ=Ξ. We can obtain the PDF of the lowest and largest order statistics at k=1 and k=n, respectively,(18)fX1x=nδμδΞxδ+1cosΞ1−sinΞn−1,(19)fXnx=nδμδΞxδ+1sinΞn−1cosΞ.


## 4. ML Estimation

Letx1,x2,…,xn be the observed values from the SF model with parameters ε=μ,δ. The total likelihood function corresponding to (6) is(20)lnL=nlnπδ2+δnlnμ−δ+1∑i=1nlnxi−∑i=1nμxiδ+∑i=1nlncosπ2e−μ/xiδ.The ML equations of the SF model are(21)∂lnL∂δ=nδ+nlnμ−∑i=1nμxiδ−∑i=1nμxiδlnμxi+π2∑i=1nμxiδlnμxie−μ/xiδtanπ2e−μ/xiδ,(22)∂lnL∂μ=nδμ−δμ∑i=1nμxiδ+δπ2μ∑i=1nμxiδe−μ/xiδtanπ2e−μ/xiδ.Then, the ML estimators of the parametersε are calculated by substituting Uε=0 and solving it.


## 5. Numerical Results

The ML technique for estimating parameters is evaluated, and the modest numerical results are obtained. Mathematica 9 is used to do Monte Carlo simulations on an SF. The simulation points are arranged as follows:(i)
We generate random samples from SF distribution by using(23)xq=μlnπ2Arcsinu−1/δ.(ii)
Each sample size ofn = 100, 200, 300, and 500 was replicated 3000 times in order to obtain the data.(iii)
According to Tables1–5, a variety of values are picked for the parameters.(iv)
Formulas used for investigating root mean square error (RMSE), lower bound (Z1), upper bound (Z2), and average length (Z3) of 90% and 95% are calculated.Table 1
Numerical results for the SF model atδ=0.5,μ=0.5.
nMLERMSE90%95%Z1Z2Z3Z1Z2Z31000.74530.25970.04361.44701.4034−0.09081.58141.67221.00471.0359−0.40772.41712.8248−0.67822.68753.36572000.53440.02170.28990.77900.48910.24310.82580.58280.59210.09940.12741.05680.92950.03841.14581.10743000.49650.01110.32650.66640.33990.29400.69900.40500.51810.03280.20020.83600.63570.13940.89680.75745000.53640.00780.38890.68400.29510.36060.71220.35160.57270.03160.29620.84930.55310.24320.90230.6591Table 2
Numerical results for the SF model atδ=0.7,μ=0.5.
nMLERMSE90%95%Z1Z2Z3Z1Z2Z31000.82460.17000.15711.49221.33510.02921.62001.59080.70970.3171−0.18661.60611.7927−0.35831.77772.13602000.75120.04330.36341.13910.77570.28921.21330.92420.52750.04520.06770.98730.9196−0.02031.07541.09573000.75240.04320.43061.07420.64360.36901.13590.76690.52720.04510.15410.94040.78620.07891.01560.93685000.71090.02480.48640.93540.44900.44340.97840.53500.52320.04170.23980.80650.56670.18560.86080.6752Table 3
Numerical results for the SF model atδ=1.2,μ=0.5.
nMLERMSE90%95%Z1Z2Z3Z1Z2Z31001.29900.06370.92831.66980.74150.85731.74080.88350.52110.00830.37420.66800.29380.34610.69610.35012001.24990.02411.00261.49730.49470.95521.54470.58940.51450.00460.41140.61760.20620.39170.63730.24563001.22200.01161.05331.39070.33741.02101.42300.40200.50460.00150.43290.57630.14340.41920.59010.17095001.20120.00341.09681.30560.20881.07681.32560.24880.49990.00060.45470.54500.09020.44610.55360.1075Table 4
Numerical results for the SF model atδ=1.8,μ=0.5.
nMLERMSE90%95%Z1Z2Z3Z1Z2Z31001.93170.14521.30352.55981.25631.18332.68011.49690.52830.00770.39030.66630.27600.36380.69270.32892001.86330.11831.44202.28460.84261.36132.36521.00390.51350.00620.41840.60870.19030.40010.62690.22683001.78280.03131.50262.06290.56031.44902.11650.66750.49920.00180.43320.56510.13190.42060.57780.15725001.80490.00731.62591.98380.35781.59172.01810.42640.50030.00040.45870.54190.08320.45070.54980.0991Table 5
Numerical results for the SF model atδ=2.5,μ=0.5.
nMLERMSE90%95%Z1Z2Z3Z1Z2Z31002.78380.53111.75723.81042.05331.56064.00702.44650.53010.00690.39880.66140.26260.37370.68660.31292002.66030.30591.99123.32931.33821.86313.45751.59440.51270.00430.42340.60210.17870.40630.61920.21293002.56960.07952.11733.02200.90472.03063.10861.07800.50920.00160.44600.57250.12650.43390.58460.15075002.47600.03012.20342.74860.54512.15132.80080.64950.49160.00050.45280.53040.07770.44530.53790.0925


## 6. Applications to Real Data

In this section, an actual data set is studied to demonstrate the benefits of the SF model over other known models such as the F and inverse exponential (IE) models.We propose several information criteria (IC) to compare the competitive models, such as the minus of log-likelihood function (L1), Akaike IC (L2), the right Akaike IC (L3), Bayesian IC (L4), Hannan-Quinn IC (L5), and Kolmogorov–Smirnov IC (L6).The data set proposes a specific application using a real-world data set to gauge interest in the SF model. The data evaluated were the daily fatality confirmed cases of COVID-19 in Saudi Arabia from 14 April to 22 June 2020. The data set was collected electronically from the following website:https://covid19.moh.gov.sa (6, 4, 4, 5, 5, 6, 6, 5, 7, 6, 9, 3, 5, 8, 5, 5, 7, 8, 7, 9, 9, 10, 10, 10, 7, 9, 9, 9, 10, 9, 10, 10, 8, 9, 10, 12, 13, 15, 11, 9, 12, 14, 16, 17, 22, 23, 22, 24, 30, 32, 31, 34, 36, 34, 37, 36, 38, 36, 39, 40, 39, 41, 39, 48, 45, 46, 37, 40, 39). Many studies investigated biomedical data sets, including those in [21–24].Some descriptive statistics of the data are provided in Table6. The ML estimates of all competitive models and their SEs are mentioned in Table 7. Values of measures of goodness of fit are provided in Table 8.Table 6
Some descriptive statistics for the data set.
nMinMaxMeanMedianModeVarianceSkewnessKurtosis9619723.6671411583.6391.7772.372Table 7
The ML estimates and SEs for the data set.
ModelML estimates and SEsSFδ,μ0.915 (0.075)13.452 (1.473)Fδ,μ1.168 (0.0948.679 (0.942IEμ9.247 (1.105)Table 8
The values of L1, L2, L3, L4, L5, and L6 the data set.
ModelGoodness-of-fit criteriaL1L2L4L5L3L6SF534.841538.841538.531540.627539.020.132F543.894547.894547.584549.68548.0730.1369IE547.221549.221549.066550.114549.2790.1752We find that the SF model provides a better fit than the other competitive models. It has the lowest value of L1, L2, L3, L4, L5, and L6 values among those considered here. Moreover, the plots of the estimated PDF of the data set for all competitive models are shown in Figure3.Figure 3
Estimated PDF for the data set.


## 7. Conclusion

In this study, we introduced the SF distribution, a unique two-parameter model. In medicine, physics, and nanophysics, the sine Frèchet distribution is widely employed. The statistical features of the SF model, such as the quantile function, moments, moment generating function (MGF), and order statistics, were computed. The ML estimation approach is used to evaluate the estimate of the SF parameters. The simulation results are calculated to show the accuracy of the estimates. The modeling for COVID-19 in KSA from 14 April to 22 June 2020 real data set is used to explain the significance of the SF model in comparison to the other competitive models. Some basic SF model properties are proposed.



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