The first COVID-19 case was detected in Wuhan city of China, which has a well-known seafood wholesale market. On December 31, 2019, the Wuhan Municipal Health Commission reported a bunch of 27 pneumonia cases of unknown etiology. Initial reports tell us that the people associated with the wholesale business were first infected by zoonotic transmission (transmission of disease from animal to humans) of SARS and MERS. This infection continued on a large scale and covered the whole city. More information about this pandemic can be explored at: https://en.wikipedia.org/wiki/Coronavirus_disease_2019. The comparison of the COVID-19 epidemic dynamics among different countries is of great concern. In this regard, the researchers are producing their best efforts to provide a guideline that can help in controlling this global pandemic. Some previous efforts to compare the epidemic dynamics in Italy and mainland China were introduced; see for detail [1,2]. A comparison of the epidemic dynamics in Ukraine and neighboring countries is provided in [3]. Comparisons of the COVID-19 in Europe, USA, and South Korea have been carried out in [4]. For more detail, we refer the interested readers to [5–12].

In the current situation, it is of great interest to study more about COVID-19 and compare different countries as many as possible. Therefore, in this article, an effort has been made to compare the COVID-19 pandemic outbreak in the two neighboring countries of the Asian region Iran and Pakistan.

This article is structured as follows: Comparing COVID-19 events between Iran and Pakistan is provided in Section 2. The proposed method is discussed in Section 3. A special case of the proposed method is introduced in Section 4. Some properties of the proposed method are discussed in Section 5. Section 6 presents the estimation of the model parameters. Section 7 is devoted to modeling COVID-19 events. Finally, the article is concluded in the last section.

In this section, a very simple and easy method to compare the pandemic dynamics in Iran and Pakistan is presented. The comparison is made by considering the COVID-19 events such as the total number of cases, total recovered, total deaths, and total active cases. The comparison of the COVID-19 daily cases and daily deaths in Iran and Pakistan are presented in Fig. 1. From Fig. 1, one can easily observe that the numbers of daily cases and daily deaths for Iran and Pakistan are increasing over time. Furthermore, a detailed description of the total number of cases, total recovered, total deaths, and total active cases from the COVID-19 epidemic pandemic are presented in Fig. 2. The percentage of the total COVID-19 epidemic pandemic events for Iran and Pakistan are displayed in Fig. 3.

The newly infected and newly recovered cases and the outcome cases (recovery or deaths) in Iran and Pakistan are displayed in Figs. 4 and 5, respectively.

In this section, we propose a new family of statistical models to provide the best description of the COVID-19 daily deaths data. The authors in [13] defined the T–X family method to introduce new families of distributions. Let v(t) be the probability density function (pdf) of a random variable, say T, where T∈[m,n],-∞≤m<n<∞ and let W[F(x)] be a function in F(x) of a random variable, say X, satisfying the conditions given below:

W[F(x)]∈[m,n],

The cumulative distribution function (cdf) of the T–X family has the form

(1) G(x)=∫ mW[F(x)]v(t)dt,x∈ℝ,

where W[F(x)] satisfies the conditions stated above. The probability density function (pdf) corresponding to Eq. (1) is g(x)=∂∂x{W[F(x)]}v{W[F(x)]},x∈ℝ

For the contributed work based on the T–X approach, we refer to [14–19]. Taking inspiration from Eq. (1), we introduce a new class of statistical models suitable for modelling the COVID-19 daily deaths data. Henceforth, we call the proposed method as the new alpha power-X (NAP-X) family of statistical models. Let T~ exp(1), then its cdf is given by

(2) V(t)=1- exp(-t),t≥0.

The pdf corresponding to Eq. (2) reduces to

(3) 0. \label{eqn-3} \end{equation}$$]]> v(t)= exp(-t),t>0.

If v(t) follows Eq. (3) and setting W[F(x)]=-log(α(1-11-log(1-F(x;ξ)))-1α-1) in Eq. (1), we obtain the cdf of the NAP-X family as

(4) 0,\;\alpha \neq 1,\;x,\xi \in \mathbb{R}. \label{eqn-4} \end{equation}$$]]> G(x;α,ξ)=α(1-11-log(1-F(x;ξ)))-1α-1,α>0,α≠1,x,ξ∈ℝ.

The pdf corresponding to Eq. (4) is given by

(5) g(x;α,ξ)=log(α)f(x;ξ)α(1-11-log(1-F(x;ξ)))(α-1)(1-F(x;ξ))(1- log(1-F(x;ξ)))2,x∈ℝ.

Hereafter, a random variable X has the pdf given by Eq. (5) will be denoted by NAP-X (x;α,ξ).

In this section, we define a special sub-model of the NAP-X family, called the new alpha power-Weibull (NAP-W) distribution. Let F(x;ξ) be the cdf of the Weibull distribution given by F(x;ξ)=1-e-γxθ,x≥0 where ξ=(θ,γ). Then, the cdf of the NAP-W distribution is specified by

(6) 0,~\alpha \neq 1. \label{eqn-6} \end{equation}$$]]> G(x;α;ξ)=α(γxθ1+γxθ)-1α-1,x≥0,α,θ,γ>0,α≠1.

The pdf corresponding to Eq. (6) reduces to

(7) 0. \label{eqn-7} \end{equation}$$]]> g(x;α;ξ)=log(α)θγxθ-1(α-1)(1+γxθ)2α(γxθ1+γxθ),x>0.

Some possible shapes for the pdf of the NAP-W distribution for selected values of the model parameters are sketched in Figs. 6 and 7.

In this section, we use the maximum likelihood method to estimate the NAP-W parameters and provide a Monte Carlo simulation study to assess the behavior of these estimators.

As we stated above that comprehensive statistical analyses for events related to COVID-19 pandemic are still to come. Now, it is urgently needed to introduce a suitable model to provide a better description of COVID-19 daily deaths data to get a reliable estimate of the required number of places in hospitals, level of individual protection, and rate of isolation of infected persons, among others.

In this section, we model COVID-19 daily deaths that have been occurred in Iran and Pakistan. The NAP-W distribution is adopted to model the daily deaths data in Iran and Pakistan. Since, the daily deaths data which are analyzed in this paper is discrete by nature. Therefore, before analyzing, we have to transform the discrete data to be continuous. The data were collected in a raw form; therefore, they can be transformed into any type of data. Some available techniques can be adopted to transform discrete data into continuous such as re-sampling and continuity correction factor and as.numeric(), among others. In this paper, the continuity correction factor is adopted to transform the data into continuous form. We found that the proposed NAP-Weibull model can be applied quite effectively to provide the best description of the COVID-19 daily deaths data of the two neighbors countries.

It is well known that selecting the appropriate approximation model is desirable to assign some preference to the alternatives. Note that the distributional parameters of a parametric model play an important role in determining the adequacy of the overall fit. For COVID-19 daily deaths data of Iran, the values of the model parameters are α=1, γ=0.415 and θ=0.487. For the daily deaths data in Pakistan due to COVID-19, the values of the model parameters are α=1.001, γ=0.066 and θ=1.352.

Thus, after computing the values of the model parameters, now we can show how the proposed model provides the best description of COVID-19 daily deaths data. For such purpose, we considered the graphical sketching of the estimated cdf, Kaplan-Meier (KM) survival function, and the probability-probability (PP) plot. The fitted cdf and KM survival plots of the NAP-W model for COVID-19 daily deaths data in Iran are presented in Fig. 10. Further, the fitted cdf and KM survival plots of the NAP-W distribution for COVID-19 daily deaths data in Pakistan are provided in Fig. 11. The PP plots for the COVID-19 daily deaths data in Iran and Pakistan are displayed in Fig. 12. The black lines in Figs. 10 and 11 shows the real behaviors of COVID-19 daily deaths data in Iran and Pakistan, whereas the green and red line show the performance of the proposed NAP-W model.

Figs. 10–12 illustrate that the proposed NAP-W model fits COVID-19 total death data in Iran and Pakistan very closely, leading to the conclusion that the proposed model performs well. Speaking compactly, the Monte Carlo simulation study and the real data examples reveal that the NAP-W model can be a very useful candidate for analyzing COVID-19 data with ranging characteristics.

The COVID-19 pandemic is becoming one of the deadliest viruses, greatly affecting daily life affairs. The situation with the coronavirus pandemic in the Asian region, particularly in Iran and Pakistan, is very threatening. Henceforth, Iran and Pakistan are in urgent need of assistance in speeding up the isolation of infected and contact persons to prevent the new daily infections. The government and other responsible organizations might be interested in providing bases for comparison and providing a better description of the data under consideration to get reliable estimates of the parameters of interest. In the present article, a brief comparison of the COVID-19 events such as total cases, total deaths, totally recovered, and active cases in Iran and Pakistan are provided. Such a brief and clear-cut comparison of the COVID-19 events might be helpful to accommodate the affected peoples.

Furthermore, we propose a new family of statistical distributions that can be used to model COVID-19 pandemic dynamics events. We just provided a platform for the researchers to utilize our proposed method in modeling COVID-19 events. A special sub-model of the proposed family called a new alpha-power Weibull distribution is considered in detail. The model parameters are estimated via the maximum likelihood estimation method. The flexibility of the proposed model is shown by fitting the daily deaths data in Iran and Pakistan due to the COVID-19, and it is observed that the proposed model may provide a close fit to the COVID-19 events. We do expect that the researchers will use our model to utilize it to formulate regression models and tailored NAP-Weibull distribution towards prediction as well as evaluating the link between various risk factors for such and other related purposes.