1 Introduction

In human history, viruses entered the sources towards the end of the 1800s. In addition to the discovered species, it is known that there are millions of more unidentified viruses. There are different types of viruses that can spread through plants, insects, and animals. Some viruses are known to cause diseases. The worldwide effects of diseases are called epidemics. In 2019, a virus in the coronavirus family affected the whole world, starting from Wuhan, China. The process that started epidemically in 2020 has been raised to a pandemic epidemic level by the World Health Organization (WHO).

Scientific methodologies can help authorities make quick and valuable decisions that reduce the effects of the COVID-19 pandemic. Health authorities from various countries publish data about the spread of COVID-19. While these data are valuable, it is necessary to evaluate whether they provide enough information to manage the pandemic. The WHO publishes various types of data about the COVID-19 pandemic on its website. These data provide different attributes for the spread of COVID-19. However, a combined overall evaluation result is necessary to combine all data specifications and their preferences for the COVID-19 pandemic in a country. The data available from the WHO changes from one day to another. Therefore, sometimes it is better to use criteria that are increasing from one day to another, while on other days, criteria that show decreasing trends may be more appropriate. During peak cases, the “deaths recorded in the last 24 h” is more crucial, while during stable days of the pandemic, "cumulative total deaths" is more important to determine the influence of the COVID-19 pandemic. Additionally, the diversification of these criteria values from one country to another creates a complex decision-making environment for understanding the COVID-19 impacts in different countries. These types of conflicting structures require multi-criteria decision-making (MCDM) methods to model the differentiation of criteria values. MCDM methods are appropriate for incorporating different criteria-related data. MCDM methods can be beneficial in determining the impacts of the COVID-19 pandemic on affected countries because they present a unique combined score that reflects the MCDM problem specification in a better way. There is also the problem of uncertainties in the trend of COVID-19's deployment, which requires the development of fuzzy decision-making models for modeling its impacts on countries. If we analyze published data about the COVID-19 pandemic, we can see that classical fuzzy sets are sufficient for the modeling phase.

The purpose of this study is to use data published by the WHO to develop a multi-criteria decision-making (MCDM) model to monitor the levels of countries affected by the COVID-19 epidemic using the evaluation criteria of cumulative total cases, cases recorded in the last 24 h, cumulative total deaths, and deaths recorded in the last 24 h. The TOPSIS technique, which is one of the MCDM techniques, was chosen for its ease of use and the ease of understanding and interpretation of the results. The evaluation criteria were selected based on criteria in the WHO data. The weights of the evaluation criteria were assigned intuitively. The study was applied as fuzzy TOPSIS (FTOPSIS) by assigning triangular fuzzy number values.

The paper is organized as follows: Sect. 2 presents a literature survey about COVID19. Section three proposes the techniques used in the study. Section four represents the application. Finally, section five presents the conclusions.

2 Literature review

COVID-19 cases were registered in late December 2019 in Wuhan, China. On January 13, 2020, these cases were defined as New Coronavirus Disease (nCov) (Ministry of Health, 2020). On January 20, 2020, a case was detected in the Republic of Korea for the first time outside China (World Health Organization 2020a, b, c, d, e, f, g, h, i, j, k, l, m). The first case was recorded in the United States on January 23, 2020 (World Health Organization 2020a, b, c, d, e, f, g, h, i, j, k, l, m). The number of cases recorded worldwide was 2798, and 37 of these cases were detected in 11 countries outside of China (World Health Organization 2020a, b, c, d, e, f, g, h, i, j, k, l, m). With the total number of cases exceeding six thousand worldwide, the "Pandemic Supply Chain Network" has been established with the cooperation between WHO and the World Economic Forum (WEF). WEF is a platform that provides data sharing, market visibility, operational coordination, and connectivity (World Health Organization 2020a, b, c, d, e, f, g, h, i, j, k, l, m). WHO has worked with technical experts to develop diagnostic kits. Costing, procurement, and assembly issues are identified as priority issues for the future (World Health Organization 2020a, b, c, d, e, f, g, h, i, j, k, l, m).

As a global strategic preparedness and response plan outlining public health measures, the global pandemic preparedness and response plan was developed by WHO and partners. The plan aims to stop the spread of the disease in all countries (World Health Organization 2020a, b, c, d, e, f, g, h, i, j, k, l, m). WHO has worked with the international network of statistics and mathematical modelers to determine the epidemiological parameters of COVID-19, such as the incubation period (time between infection and symptom), case fatality rate, and serial interval (time between symptom onset of primary and secondary cases) (World Health Organization 2020a, b, c, d, e, f, g, h, i, j, k, l, m). WHO updated its assessment of spreading risk and impact risk on February 28. From this date, regional and global levels have been added to China, which is in the high-risk class (World Health Organization 2020a, b, c, d, e, f, g, h, i, j, k, l, m). While the COVID-19 pandemic is at a high level, World Health Organization has provided online resources in various languages ​​for effective preparedness and response (World Health Organization 2020a, b, c, d, e, f, g, h, i, j, k, l, m). These resources are crucial guides for healthcare professionals, decision-makers, and the public. On March 11, 2020, WHO classified the COVID-19 virus as a pandemic, taking into account past cases and the speed of its spread worldwide (World Health Organization 2020a, b, c, d, e, f, g, h, i, j, k, l, m).

Although the COVID-19 outbreak is a new issue, numerous academic studies have been included in the literature. For example, Majumder, Biswas, and Majumder (2020) presented an approach to examine deaths from COVID-19 and identify important risk factors. They created a model for analyzing deaths from COVID-19 using Technique for Order Preferences by Similarity to Ideal Solution (TOPSIS) techniques and compared it with the existing model. This new model is more accurate and consistent compared to existing regression models. Since there is no weight given to factors in the new model obtained, it has been possible to evaluate the cases originating from COVID-19 (Majumder et al. 2020). Grida et al. (2020) have created a study to prevent COVID-19 impacts under uncertainty in supply, demand, and logistics, which determined to chain supply. They used the Best Worst Method (BWM) and TOPSIS methods based on the pathogenic cluster. While BWM weighs the policies that prevent the spread of the virus, TOPSIS has listed the supply chain aspects for the food, electronics, pharmaceutical, and textile industries (Grida et al. 2020). Alqahtani and Rajkhan (2020) conducted a study on the education sector, one of the sectors affected by the COVID-19 era, to determine the critical success factors of online learning using the Analytic Hierarchy Process (AHP) and TOPSIS methods.

Ic (2021) proposed a forecasting model to predict a potential reactivated COVID19 patient. His paper aims to explore the applicability of a modified TOPSIS integrated Design of Experiment (DoE) method to predict a potential reactivated COVID19 patient in real-time clinical or laboratory applications. Mohammed et al. (2020) presented a benchmarking methodology for the selection of an optimal COVID19 diagnostic model based on entropy and TOPSIS Methods. Their integrated MCDM approach is proposed where TOPSIS applied for the benchmarking and ranking purpose while Entropy is used to calculate the weights of criteria. Hezer et al. (2021) purposed a study to evaluate the safety levels of 100 regions in the world in terms of COVID19 using TOPSIS, VIKOR, and COPRAS methods. Mohammed et al. (2021) proposed a novel convalescent plasma transfusion intelligent framework for rescuing COVID19 patients across centralized/decentralized telemedicine hospitals based on the matching component process to provide an efficient CP from eligible donors to the most critical patients using the TOPSIS method. Gupta et al. (2021) presented a CRITIC-TOPSIS approach to stress intensity in the urban areas of India during the COVID19 pandemic. Ecer and Pamucar (2021) aimed to identify insurance companies’ priority ranking in terms of healthcare services in Turkey during the COVID19 outbreak through an MCDM model. They proposed a Measurement of Alternatives and Ranking according to the Compromise Solution model under an intuitionistic fuzzy environment to rank insurance companies.

We referred to a flow chart to select the most appropriate MCDM methodology to apply in our study. Figure 1 shows a flow chart to determine the suitable method for our problem. According to Fig. 1, the green illustration proposes the TOPSIS method as the most appropriate MCDM method for this study. As a result, the TOPSIS method is selected as a multi-criteria decision-making method to rate the COVID-19 pandemic-influenced countries. Also, we provide a sub-flow chart to illustrate the proposed methodology that incorporates the Value at Risk (VaR), fuzzy TOPSIS, and IF–Then rules.

Fig. 1
figure 1

MCDM model selection flow chart (adapted from Koçak et al. 2021; Ic and Simsek 2019; Sen and Yang 1998; Yorulmaz and Ic 2022)

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In this paper, instead of using a traditional crisp multi-criteria decision-making model, we calculate an overall performance score for a selected country in the world using the fuzzy TOPSIS method. We use various criteria to determine which countries are most negatively affected by the COVID-19 pandemic. We also demonstrate the applicability of the developed decision support framework in this paper.

3 Methodology

3.1 TOPSIS method

In Multi-Criteria Decision Making (MCDM) problems, the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method is commonly used as a guide for decision-makers as it deals with the distances to the ideal solution. It is widely applied in various studies in literature due to its practical interpretability of the results. For example, İç and Yurdakul (2020) presented a paper that examined the development of manufacturing companies and provided sectoral rankings using the integrated TOPSIS and fuzzy AHP methods. In another study, Zhang, Zhang, Sun, Zou, and Chen (2018) presented a TOPSIS approach that evaluated public transport performance levels in Wuhan.

3.2 Fuzzy logic and fuzzy set theory

Fuzzy logic has been widely used in various research areas such as renewable energy, computer security, and robotic systems (Cobos-Guzman et al. 2020; Barai and Nonami 2008; Seraji and Howard 2002; Fateh 2010). The popularity of fuzzy logic is due to its ability to approximate reasoning under uncertainty simply and suitably for modeling purposes. When building a fuzzy system, the information from real-world systems takes precedence over decision-making approaches. Since obtaining accurate information from different criteria of COVID-19 spread considering the data structure is not easy and there are no clear categories of the involved criteria, we have developed a fuzzy decision-making model to rate the countries. The design of the fuzzy decision-making system includes the following tasks:

• Selection of the membership functions for the data and criteria weights.

• Establishing the fuzzy rules base.

• Development of the MCDM model’s structure.

- Selection of the defuzzification approach that converts the fuzzy output into a crisp one.

A fuzzy set is a class of objects expressed in degrees of membership with continuity. The values ​​of the elements in the membership function take membership values ​​between 0 and 1 (Zadeh 1965). Fuzzy numbers are shown as a triangle, and trapezoids are widely used in research and practice. Triangular fuzzy numbers are represented as (a, m, b). Figure 2 illustrates that a is the smallest value on the left, m is the most appropriate value, and b is the largest value (Cheng 2004). Functions of triangle fuzzy numbers;

Fig. 2
figure 2

Membership function of triangular fuzzy numbers (a, m, b)

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$$t(x;a,m, b)= \left\{\begin{array}{l}1-\frac{m-x}{m-a}, \quad a\le x\le m,\\ 1-\frac{x-m}{b-m}, \quad m\le x\le b, \\ 0, \quad elsewhere.\end{array}\right.$$
(1)

As with normal clusters, operations can be performed on fuzzy clusters as well. For example, let F1 = (a1, m1, b2) and F2 = (a2, m2, b2) be two triangular fuzzy numbers:

$${F}_{1}+{F}_{2}=({a}_{1}+{a}_{2}, {m}_{1}+{m}_{2}, {b}_{1}+{b}_{2})$$
(2)
$$k{F}_{1}=\left(k{a}_{1}, k{m}_{1}, k{b}_{2}\right) \quad k\,\ge\, 0$$
(3)

3.3 Fuzzy TOPSIS

\(\widetilde{D}\) Represents the fuzzy decision matrix and criteria weights can assign with fuzzy weight vector (İç et al. 2017):

Step 1: Decision matrix is constructed:

$$\widetilde{D}=\left[\begin{array}{ccc}\widetilde{{x}_{11}}& \cdots & \widetilde{{x}_{1n}}\\ \vdots & \ddots & \vdots \\ \widetilde{{x}_{m1}}& \cdots & \widetilde{{x}_{mn}}\end{array}\right]$$
(4)
$$where, \quad \widetilde{{x}_{ij}};\,i\,=\,1, 2, \dots m alternatives, and \quad j=1, 2, \dots n, criteria$$

Step 2: Determination of the normalized decision matrix:

A normalized decision matrix is obtained using the Eqs. (5) and (6) for B and C being high-affected and low-affected clusters.

$${\widetilde{r}}_{ij}={\widetilde{x}}_{ij}\left(\div \right){\widetilde{x}}_{j}^{*}=\left(\frac{{a}_{ij}}{{c}_{j}^{*}},\frac{{m}_{ij}}{{b}_{j}^{*}},\frac{{b}_{ij}}{{a}_{j}^{*}}\right),\quad j\epsilon B$$
(5)
$${\widetilde{r}}_{ij}={\widetilde{x}}_{j}^{-}\left(\div \right){\widetilde{x}}_{ij}=\left(\frac{{a}_{j}^{-}}{{b}_{ij}},\frac{{b}_{j}^{-}}{{m}_{ij}},\frac{{c}_{j}^{-}}{{a}_{ij}}\right),\quad j\epsilon\, C$$
(6)
$$\left({a}_{j}^{*}, {b}_{j}^{*},{c}_{j}^{*}\right)={max}_{i}\left({a}_{ij}, {m}_{ij}, {b}_{ij}\right)\, if \,j\,\epsilon B$$
(7)
$$\left({a}_{j}^{-}, {b}_{j}^{-},{c}_{j}^{-}\right)={min}_{i}\left({a}_{ij}, {m}_{ij}, {b}_{ij}\right)\, if\, j\,\epsilon C$$
(8)

\(where, {\widetilde{x}}_{ij}=({a}_{ij}, {m}_{ij}, {b}_{ij})\) and \({\widetilde{x}}_{j}^{*}=({a}_{j}^{*}, {b}_{j}^{*},{c}_{j}^{*})\).

Step 3: Computation of the weighted normalized matrix in which fuzzy weights are calculated as \({\widetilde{w}}_{j}=({\alpha }_{j}, {\beta }_{j}, {\gamma }_{j})\):

$$\widetilde{v}={\left[{\widetilde{v}}_{ij}\right]}_{mxn} i=1, 2, 3, \dots , m, \,j\,=\,1, 2, 3, ..., n$$
(9)
$${\widetilde{v}}_{ij}={\widetilde{r}}_{ij}\otimes {\widetilde{w}}_{j}.$$
(10)

where the weight \({\alpha }_{j}\) is the smallest value on the left, \({\beta }_{j}\) is the most appropriate value, and \({\gamma }_{j}\) is the largest value of the triangular fuzzy weight number of \({\widetilde{w}}_{j}.\)

Step 4: Defuzzification: Fuzzy performance values are converted to crisp values by calculating the means of the fuzzy numbers (Ic et al. 2013; Chen and Hwang 1992):

$$\mathop {x_{{}} }\limits^{\_} (\mathop {A_{{}} }\limits^{\sim } ) = \frac{{\int {x\mu_{{\mathop {A_{{}} }\limits^{\sim } }} (x)dx} }}{{\int {\mu_{{\mathop A\limits^{\sim } }} (x)dx} }}$$
(11)

Since the criteria values used in this study are crisp values, they will be expressed as (m; m; m) using the center of the fuzzy number to perform mathematical operations. For this reason, crisp values are identified in triangular fuzzy number format to multiplying by the fuzzy triangular numbers. Within the scope of the study, the averages of the crisp values multiplied by the weight values without dividing by the value of 3 and, the clarified values will be calculated as in the following Eq. (12):

$$\mathop x\limits^{\_} (\mathop A\limits^{\sim } ) = \frac{(m + m + m)}{3} = (m + m + m)$$
(12)

The fuzzy weighted normalized values (\(\mathop {\tilde{v}}\nolimits_{ij}\)) provided in Eq. (10) is defuzzified using Eq. (12) and the crisp weighted normalized values (vij) is obtained.

Step 5: Identify positive ideal and negative ideal solutions: The A* and A − are defined in terms of the weighted normalized values, as shown in Eqs. (13) and (14):

$$A^{*} = \left\{ {\mathop {(\max }\limits_{i} v_{ij} \left| {i \in I),(\mathop {\min }\limits_{i} v_{ij} \left| {i \in I^{\prime} } \right.} \right.} \right\}$$
(13)
$$A^{ - } = \left\{ {\mathop {(\min }\limits_{i} v_{ij} \left| {i \in I),(\mathop {\max }\limits_{i} v_{ij} \left| {i \in I^{\prime} } \right.} \right.} \right\}$$
(14)

where I is the set of benefit type criteria and I′ is the set of cost type criteria.

Step 6: Calculation of separation measures:

$$d_{i}^{ + } = \sqrt {\sum\limits_{j = 1}^{{\mathop n\nolimits_{{}} }} {(v_{ij} - v_{j}^{*} )^{2} } }$$
(15)
$$d_{i}^{ - } = \sqrt {\sum\limits_{j = 1}^{{\mathop n\nolimits_{{}} }} {(v_{ij} - v_{j}^{ - } )^{2} } }$$
(16)

where, vi* is the positive-ideal value for ith criteria, and vi- is the negative-ideal value for the ith criteria.

Step 7: Calculation of ranking scores:

$$C_{i}^{*} = \frac{{d_{i}^{ - } }}{{d_{i}^{ - } + d_{i}^{ + } }}\quad i = 1, \ldots m.$$
(17)

3.4 Value at risk (VaR)

We apply the value at risk (VaR) approach for historical data and applied the “historical simulation” methodology for the problem. First of all, we calculated the “return values” indicated that the daily log-return, Rt + 1, using the daily value Ki, t + 1:

$${R}_{t+1}\equiv \mathrm{ln}\left(\frac{{K}_{i,t+1}}{{K}_{i,t}}\right)$$
(18)

In the financial engineering literature, the term “return” indicates the revenue. However, criteria using the COVID19 pandemics reflects the “minimum value is better” type (cost type) evaluation. So, we used the calculated VaR value within a contrary perspective that “maximum VaR is better” perspectives. We can now directly model the volatility of the COVID19 portfolio return, RPF,t+1, call it KPF,t+1, and then calculate the VaR for the portfolio.

Given a confidence interval of p ∈ [0,1] and assumed the time index of t and t + 1, we can find the volatility of the portfolio of the ΔV(α) over the period. Let Fα(x) be the cumulative distiribution function of ΔV(α). Considering the hold of a short position with a period α with probability p, and the position ΔV(α) ≥ 0, the VaR is defined as

$$p={\mathbb{P}}\left[\mathrm{\Delta V}\left(\mathrm{\alpha }\right)\ge \mathrm{VaR}\right]=1-{\mathbb{P}}\left[\mathrm{\Delta V}\left(\mathrm{\alpha }\right)\le \mathrm{VaR}\right]=1-\mathrm{F\alpha }(\mathrm{VaR})$$
(19)

We can define the p-quantile of Fα(x) and the given confidence level of p ∈ [0,1] (0.05 for 95% confidence level) is

$${VaR}_{p}={x}_{p}=inf\left\{\left(x|\mathrm{F\alpha }(\mathrm{x})\ge \mathrm{p}\right)\right\}$$
(20)

where, inf is the smallest real number. Hence, the tail behavior of the cumulative distribution function of Fα(x) or its quantile is condition necessary for approaching VaR calculation (Tsay 2005).

3.5 Linguistic summarization using type 1 fuzzy set

We define a set of M objects \({\mathbb{Y}}=\left\{{y}_{1},{y}_{2},\dots ,{y}_{M}\right\}\) and a set of N criteria \({\mathbb{V}}=\left\{{v}_{1},{v}_{2},\dots ,{v}_{N}\right\}\).Let \({\mathbb{X}}_{n}(n=\mathrm{1,2},\dots ,N)\) be the domain of \({v}_{n}\).Then \({v}_{n}{(y}_{m})\equiv {v}_{n}^{m}\in {\mathbb{X}}_{n}\) is the value of the nth object (m = 1,2,…,M) (Wu et al. 2010). Hence the data set \({\mathbb{D}}\), which collects information about elements from \({\mathbb{Y}}\), is in the form of

$${\mathbb{D}}=\left\{<{v}_{1}^{1},{v}_{2}^{1},{v}_{N}^{1}>,\dots ,<{v}_{1}^{M},{v}_{2}^{M},\dots ,{v}_{N}^{M}>\right\}\equiv \left\{{d}_{1},{d}_{2},\dots ,{d}_{M}\right\}$$
(21)

where \({{d}_{m}=<v}_{1}^{m},{v}_{2}^{m},\dots ,{v}_{N}^{m}>\) is a complete record about object \({y}_{m}\).

Only single-antecedent single-consequent rules are considered in this subsection. Because we are interested in generating IF–THEN rules from a dataset, our canonical form for LS using type 1 fuzzy sets is:

IF \({\mathbb{X}}_{1}\) is/has S1, THEN \({\mathbb{X}}_{2}\) is Has S2.

where S1 and S2 are words modeled by type 1 fuzzy sets (Wu et al. 2010).

4 Application

The study aims to evaluate and interpret countries' levels of exposure to the COVID-19 outbreak based on cumulative total cases, cases recorded in the last 24 h, and cumulative total deaths, deaths recorded in the last 24 h. In the application part of the study, data collection is to be used, as the weighting method and the application steps of the TOPSIS method, which is one of the MCDM techniques, to the blurred environment. On the other hand, the TOPSIS method is a very useful method for evaluating the criteria values (data). The figures related to the COVID-19 pandemic are very dependent on the size of the country. We certainly trust that the data standardization process considers the population size effects. The data normalization step (vector normalization in Eqs. (5–8)) in the fuzzy TOPSIS method is suitable for the standardization of the collected values. WHO systematically publishes reports throughout the COVID-19 outbreak. These reports numerically show the cumulative daily total cases of countries, cases recorded in the last 24 h, cumulative total deaths, and deaths recorded in the last 24 h. In the study, 11 countries were selected depending on the density of cases between March 2, 2020, and August 16, 2020, and these countries are Germany, the United States of America, Argentina, Brazil, India, England, Spain, Italy, Colombia, Russia, and Turkey, respectively.

To calculate the criteria weights for the TOPSIS the collected data are arranged within the date range selected based on criteria. The edited data were plotted graphically (Figs. 3, 4, 5, 6) and received linguistic expressions based on expert opinion. The decision-makers can use fuzzy triangular numbers when attempting to predict the weight of the criteria. The center value of the fuzzy triangular number is the closest number to the crisp equivalent. Therefore, the lower and upper values of the fuzzy number are one unit apart. Triantaphyllou and Lin (1996) offer a fuzzy scale that appropriates Saaty’s scale (Table 1). Fuzzy number equivalents of the linguistic expressions in the graphics above are shown in Table 1.

Fig. 3
figure 3

Cumulative total cases

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Fig. 4
figure 4

Total reported new cases

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Fig. 5
figure 5

Cumulative total number of deaths

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Fig. 6
figure 6

Total reported new deaths

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Table 1 Linguistic expressions and fuzzy numbers (Triantaphyllou and Lin 1996)
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The evaluation criteria, along with the abbreviations used, are shown in Table 2. They include the cumulative total cases, cases recorded in the last 24 h, the total number of cumulative deaths, and the number of deaths recorded in the last 24 h. These four criteria are all published by the World Health Organization (WHO) on their website and are all cost-type. We use the distances between the ideal and negative-ideal solutions to be calculated based on these cost criteria. The minimum value of the criterion represents the positive ideal solution value, while the maximum criteria values represent the negative-ideal solution for the TOPSIS model.

Table 2 Evaluation criteria
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The display of evaluation criteria graphs with linguistic expressions was shown verbally in Table 3 by creating a decision matrix, which was then transformed into numerical values using fuzzy number equivalents. The linguistic expressions were obtained with the help of a physician who is an intensive care specialist at a hospital in Turkey, as shown in Table 3. The time series was observed and evaluated with linguistic expressions according to the time series in the graphs given in Figs. 3, 4, 5, 6, respectively; (high important, high important, low important), (very high important, high important, low important), (very low important, medium important, high important), and (medium important, very high important, high important). To obtain the normalized fuzzy matrix, the geometric means were first calculated. However, we also provide a discussion on why such a primitive indicator could be competitive with the multidimensional reflection of the problem. The issue is very complicated, and we used a successful approach called Value at Risk (VaR) to validate the advantage of such a simplistic approach for the decision-making process. On the other hand, it creates the illusion that the complex assessment of the phenomena could be simplified without discussing the consequences of such action. The calculated VaR values for the criteria are shown in Table 4.

Table 3 Criteria weighting
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Fig. 7
figure 7

Graphical illustrations for VaR

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Table 4 Calculated VaR for selected time frame
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In Table 4, the highest VaR result is 0.10816 for K1 in March. However, K1 is the “minimum value is better” type criteria and, the 0.10816 indicates the better VaR for March. So, the minimum VaR appears in March. Conversely, the maximum VaR appears in August for K1. Also, VaR is increasing from March to August for K1. IF–THEN fuzzy rules are determined according to the principles summarized in Sect. 3.5 and listed below:

If VaR is extremly increasing/decrease from the time frame for the criteria, then importance is “very highly important.”

If VaR is highly increasing/decrease from the time frame for the criteria, then importance is “highly important.”

If VaR is a steady-state from the time frame for the criteria, then importance is “moderately important.”

If VaR is slightly increasing/decrease from the time frame for the criteria, then importance is “low important.”

If VaR is an ordinary increase/decrease from the time frame for the criteria, then importance is “very low important.”

We can easily assign linguistic values for the importance level of the criteria according to the graphics in Fig. 7. Once graphical illustrations are given for the VaR (see Fig. 7), the antecedents and consequents of the rules can be determined. A decision-maker needs to specify the linguistic terms used for each antecedent and consequent and also their corresponding fuzzy set models, as shown in Table 3. The proposed weighting procedure is based on the data, which can be related to the trend of the pandemic. In the future, we may see different proportions of importance, possibly even a negative proportion. Therefore, IF–THEN rules are a necessary technique for modeling the criteria weights using linguistic evaluation in the proposed model.

The data from the selected dates (10th, 20th, and 30th of the month) were used to construct the decision matrix (World Health Organization 2020a-m). The data on the decision matrix was transformed into a fuzzy decision matrix. The \(\widetilde{D}\) matrix created for March 02, 2020, was prepared as in Table 5. As an example of the K1 criteria, daily data values are shown in Appendix 1. Some of the data have zero value on some days in Appendix 1, as there are no COVID-19 case records for some countries on those days. However, all countries have COVID-19 cases based on the spread of the pandemic if we consider the data from the starting day to the final day in Appendix 1. We used data from March 02, 2020, to August 10, 2021, so the pandemic spread has volatility. In this perspective, the number of new cases is crucial when the pandemic spread is increasing, but in the case of decreasing, the cumulative cases are more important for the management of the pandemic by health authorities. Therefore, it is necessary to develop a unique model, such as TOPSIS, to monitor the pandemic trend in a country. All criteria are useful in the TOPSIS model, but sometimes one criterion is more important than the others. In this case, we use the weighting process in the TOPSIS model. This specification provides flexibility to monitor the impacts of the pandemic from different perspectives.

Table 5 Decision matrix
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The distance of the alternatives to the ideal solution was calculated using Eqs. 15 and 16. Finally, the ranking values were calculated using Eq. 17 and shown in Table 6 and Fig. 8, respectively. The elements of the weighted normalized decision matrix are deffuzified using Eq. (12). Furthermore, positive ideal (A*) and negative ideal (A -) solutions are obtained by using the Eqs. (13) and (14), and the results are shown in Table 6. The steps applied for the data dated March 02, 2020, were repeated in order, taking the 10th, 20th, and 30th of each month (10th of the month for August only) for the dates included in the data set. Table 7 was created by arranging all the calculated TOPSIS scores in a common table. The graphic of the created table is shown in Fig. 9. According to Fig. 9, the USA, Brazil, and India are the most affected countries by the COVID-19 pandemic because their scores dramatically changed from March 2020 to August 2020. However, Turkey, Italy, England, Colombia, and Russia are the less affected countries by the COVID-19 pandemic from March 2020 to August 2020 time frame. The proposed fuzzy TOPSIS model has successfully ranked the countries according to the selected criteria. The proposed model can use by the authorities to monitor the COVID-19 pandemic, especially during the monthly periods.

Table 6 Defuzzification and positive and negative ideal solutions
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Fig. 8
figure 8

TOPSIS scores of countries

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Table 7 Fuzzy TOPSIS results for selected dates
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Fig. 9
figure 9

Change of TOPSIS scores of countries by dates

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The method applied in the study was re-applied for the data in the WHO report dated April 25, 2021, which is an up-to-date data set, and in Table 8. The obtained results were combined with the results of March 2, 2020's results (Fig. 10). The USA, Brazil, and India were the most affected countries by the epidemic. Germany, Italy, Russia, and Turkey were affected by the pandemic. Also, the least affected countries were Argentina, England, Spain, and Colombia, respectively. When the data set used in the study is updated, we can determine the level of exposure of the countries to the COVID-19 pandemic. Figure 8 illustrates March 2020 data. This date is the beginning of the pandemic. So, it is the expected result that most countries are rated at 1. As we can see in Fig. 10, the rating scores are diversified based on the spread of the COVID-19 pandemic based on the spread of pandemic all over the world.

Table 8 Data set for April 25, 2021
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Fig. 10
figure 10

Ranking comparison for March 02 and April 25

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By using the normalized fuzzy numbers and fuzzy weights, a weighted normalized matrix was created as shown in Table 9.

Table 9 Weighted normalized matrix
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5 Discussion

5.1 Practical perspective

In this paper, we propose a methodology based on a published data set for predicting the impacts of the COVID-19 pandemic on different countries. The management of pandemics is a complex, unpredictable, and challenging task that can be worrying. Therefore, usable decision-support tools are crucial for simplifying the management of pandemics. In chaotic processes like the COVID-19 pandemic, producing, collecting, and publishing data is crucial for developing suitable decision support tools. WHO is one of the organizations responsible for publishing data on the spread of the COVID-19 pandemic. Firstly we collected all daily published data from the WHO’s website. Then we analyzed their suitabilities to construct a decision support tool to measure the COVID-19 impacts on the countries. As an example of the K1, some daily data values have zero numbers for some days (Appendix 1).

Some days, there are no COVID-19 case records for the related criterion in some countries. However, entire countries have COVID-19 cases based on the spread of the pandemic if we consider the data from the starting day to the final day. Therefore, the spread of the pandemic is volatile. In light of this, it is important to develop a decision support tool that can effectively process this data and monitor the spread of the COVID-19 pandemic and its effects on different countries. If we only use daily and volatile data, it becomes difficult to predict the extent of the pandemic's impact on a country, as the cases can vary greatly from one day to another. So, we developed an easy-to-use, modifiable, flexible, reusable, and robust model for monitoring the COVID-19 spread. For example, the number of new cases is crucial when the pandemic's spread is increasing. So, it is necessary to develop a robust model, like TOPSIS, to effectively monitor the pandemic trend in a country from one date to another. In this manner, we proposed a fuzzy TOPSIS model to effectively manage the COVID-19 pandemic’s effect on the countries that are useful for the responsible institutions and their authorities.

5.2 Comparative analysis

We compared the fuzzy TOPSIS results with the Multi-Objective Optimization by Ratio Analysis (MOORA) results in this section. The MOORA method was developed by Brauers (2008). It is a multi-objective optimization method. The application steps of the MOORA method are simple and easier to understand. After the Eq. (12) in the fuzzy TOPSIS method, we can develop the MOORA index by using the Eq. (22):

$${Y}_{i}=\sum_{j=1}^{g}{v}_{ij}-\sum_{j=g+1}^{n}{v}_{ij}$$
(22)

where g is the number of criteria to be maximized, (n-g) is the number of criteria to be minimized, and (Yi) is the ranking score of ith country. A larger (Yi) value for the countries is better performance. As we used a cost-type criteria in our model, we should use a modified equation for Eq. (22) is as follows:

$${\mathrm{Y}}_{\mathrm{i}}=-\sum_{\mathrm{j}=1}^{\mathrm{n}}{\mathrm{v}}_{\mathrm{ij}}$$
(23)

For comparative analysis of the ranking results, the statistical significance of the differences between the rankings obtained by the fuzzy TOPSIS model and the MOORA method using Spearman’s rank correlation test (Table 10). This test is applied when “the related values of compared ranking data are substituted which the values occupy in the respective samples” (Parkan and Wu 1999). In the application to test the null hypothesis (H0: There is no similarity between the corresponding ranks), a test statistic, Z, is calculated using Eqs. (24) and (25) and compared with a pre-obtained significance level value (a). For the a = 0.05, if the test statistic is computed by Eq. (25) exceeds 1.645, the null hypothesis is rejected, and it is to be concluded that ‘H1: The two ranking results are similar’ is accepted.

Table 10 Comparative analysis result
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$${r}_{s}=1-\lceil\frac{6\times \sum_{i=1}^{n}{\left({d}_{i}\right)}^{2}}{n\times \left({n}^{2}-1\right)}\rceil$$
(24)
$$Z={r}_{s}\sqrt{n-1}$$
(25)

where di is the ranking differences for each alternative, n is the number of alternatives to be compared, and rs presents Spearman’s rank-correlation coefficient.

The Z-score for the Spearman's rank correlation test for the fuzzy TOPSIS and MOORA method's rankings on the March 2, 2020 case is 0.661, which is lower than the critical value of 1.645. This implies that the differences between the rankings generated by the two methods are statistically significant. On the other hand, the Z-score for April 25, 2021, case is 2.904, which is higher than the critical value of 1.645. This means that the differences between the rankings generated by the two methods are statistically insignificant. Based on the test results, we can conclude that the proposed model is robust when the data are sufficiently accessible fuzzy TOPSIS (i.e., April 25, 2021 case). Additionally, the results indicate that the rankings generated by the MOORA method are similar to the ones generated by the fuzzy TOPSIS method. The TOPSIS model is used to calculate the distances between the negative and positive ideal solutions for the alternatives. However, the MOORA method does not use distance-based or compromise solution steps. The TOPSIS method is the preferred method for this study as it uses a distance-based MCDM procedure. For future studies, if a benefit-type criterion is available for evaluation, the TOPSIS method has the advantage of modeling the problem.

5.3 Sensitivity analysis

In this section, we present a sensitivity analysis to analyze how robust are the presented results obtained from the fuzzy TOPSIS model. We analyzed some scenarios, such as if one of the criteria is deleted, how does the ranking? And deletion of a criterion or deletion of a country reverse the country rankings? We used six different cases as follows:

- Deleting the K3 criterion on the March 2 case (starting day for the analysis).

- Deleting the K1 criterion on the March 2 case (starting day for the analysis).

- Deleting the K3 criterion on the April 25 case (ending the day for the analysis).

- Deleting the K1 criterion on the April 25 case(ending day for the analysis).

- Deleting the USA alternative on the April 25 case (ending the day for the analysis/for the largest population country).

- Deleting the Brazil alternative on the April 25 case (ending day for the analysis/for the second largest population country).

There are no significant ranking differences among the cases for March 2, 2020, as shown in Table 11. Also, Table 11 presents the test results between the different cases for April 25, 2021. The last rows in Table 11 show Spearman's rank correlation coefficients and Z-scores for the ranking differences. All the Z-scores, 3.119, 3.119, 2.745, and 2.964, are higher than the critical value of 1.645, which implies that the differences in the rankings are statistically insignificant. Based on the test results, it can be concluded that the rankings obtained by the proposed model are statistically similar to the rankings obtained from sensitivity analysis cases. Therefore, it can be concluded that the proposed model is robust as the results of sensitivity analysis indicate that the rankings obtained by the model are consistent across different cases. The rankings change little across both cases of March 2 and April 25, which further supports the conclusion that the proposed model is robust.

Table 11 Sensitivity analysis results
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6 Conclusions

We present the decision-making model, using the fuzzy TOPSIS for monitoring the country's COVID-19 pandemic trend monthly basis. The proposed model is easily adapted for the statistical data by using the effective processing capability of the TOPSIS model. On the other hand, in linguistic evaluations, the graphical representation of the criteria trends is analyzed by the decision-makers for a selected time frame. This cognitive evaluation is converted to the fuzzy number integrated with Value at Risk (VaR) combined IF Then rules, and all modeling steps of the TOPSIS method are handled in the Excel Program. This model can be converted into a computer program to evaluate the COVID-19 pandemic daily analysis.

The COVID-19 pandemic has become endemic in most countries. However, the proposed methodology could also be extended to other possible diseases in the near future. Although the COVID-19 pandemic is not currently the most pressing problem in the world, its impact is a hot topic.

Also, the COVID-19 pandemic illustrates that the new potential pandemics would be impacted the population. So, our paper is the exemplified case study that illustrates how the new pandemic disease can be modeled with an MCDM methodology to determine its possible impacts on the countries.

All pandemics have unique characteristics. So, our model cannot be generalized to all pandemics. But, it can be modified with the new pandemic's specifications. The proposed approach can be adapted to incorporate the unique characteristics of future pandemics.