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. 2026 Apr 7;26:1599. doi: 10.1186/s12889-026-27111-y

SARS-CoV-2 and climate factors: a study between linear regression and quantile regression model using longitudinal count data

Nazmin Akter 1,✉, Md Mobarak Hossain Khan 1
PMCID: PMC13192041  PMID: 41947101

Abstract

Background

Regression is the most popular statistical tool used to assess the relationship between predictors and response variables. SARS-CoV-2 infections counts, however are typically right-skewed, heteroskedastic, and show episodic surges. In longitudinal epidemic data, mean-based linear regression-which focusses on the conditional mean under homoscedasticity and normality assumptions- may provide an incomplete representation of transmission dynamics, particularly when extreme outbreak periods disproportionately influence estimates. Quantile regression offers a flexible alternative by allowing covariate effects to vary across the conditional distribution of the response, thereby capturing heterogeneous effects across different transmission regimes. This feature is especially useful for understanding epidemic extremes. The aim of this study is to determine the predictors of SARS-CoV-2 infected cases, using quantile regression and comparing the models at several quantiles.

Methods

The data utilized in this study were sourced from the website of the Institute of Epidemiology Disease Control and Research (IEDCR), Dhaka Bangladesh. The daily temperature and humidity with a number of daily infected case are taken as predictors and to describe the spread of SAR-CoV-2, we consider the daily number of SAR-CoV-2 infected cases as a response variable, during the period of March 08, 2020 to May 31, 2023. Classical mean regression and Quantile regression was performed to understand the association among climate variables and the results were compared across quantiles.

Results

Under the mean and quantile regression framework we found that high temperature and high humidity have a substantial impact on spread of SARS-CoV-2 (in log scale) at a different levels of quantiles (0 < τ < 1). Temperature’s effect steadily decreases as we move from low to high infection quantiles. Temperature effect flips from positive to slightly negative (-1.8%) at the highest quantile 95th quantile and 10.2% at 25th quantile. Humidity remains positively associated across all quantiles but with decreasing strength. For τ = 0.25,0.50,0.75,0.90 and 0.95, Humidity’s effect drops significantly as move to higher quantile: 8.2%, 3.9%, 1.5%, 2.1%, 2.0%, respectively. We also compared conditional quantile models using HQC, AIC, BIC values and found quantile regression model fit at τ = 0.75 is relatively stable across all the conditional distribution.

Conclusions

The findings of this study shows that temperature and humidity have significant impact on SARS-CoV-2 transmission in Bangladesh, with heterogeneous effects across different levels of infection intensity. By modeling multiple points of the conditional distribution, quantile regression provides a more comprehensive understanding of epidemic behavior than mean-based approaches. The regression models also suggest that, even in lower infection scenarios, warmer and more humid conditions are linked to higher infection levels. However, Quantile regression is distribution-agnostic, which is both a strength and a limitation in this work. Nevertheless, as the data are longitudinal in nature and the models do not explicitly incorporate dynamic temporal structures, the results should be interpreted as associative rather than causal. Future research incorporating time-series–based quantile or autoregressive frameworks may further elucidate the role of climate in epidemic progression.

Keywords: SARS-CoV-2, Temperature, Humidity, Quantile Regression, Bangladesh

Introduction

Regression analysis is one of the most popular and helpful methods used in statistical data analysis to analyze the relationship between response and predictors by fitting a line to the observed data [1]. Three major uses for regression analysis include (1) determining the strength of predictors, (2) forecasting an effect and (3) forecasting the trend. For instance, regression models are widely used to assess data on climate change [2, 3], pollution, natural resources, and ecosystem dynamics; to model air quality [4], assess the impact of rising CO2 concentrations on global temperature, and analyze climate change scenarios [5]. In regression model, we often summarize the behavior of response variables for fixed value of predictors using the measures of central tendency (mean, median and mode) [6]. Classical regression modeling is focused on the mean, where it summarizes the association between response and predictors using conditional mean functions for response variables. Based on this idea, the core of a board family of regression models such as simple linear regression, multiple regression, and nonlinear regression model are developed. Although conditional mean models are easy to calculate and interpret, having certain attractive assumptions including normality, homoscedasticity, and independence [7]. However, these traditional conditional mean models are sensitive to skewed distributions and outliers [8]. For instance, environmental data frequently shows complex patterns; the data of health inequality in height, weight, incidence of disease, treatments, life expectancy; Demographic factors on infant’s birth weights; educational inequality in academic achievement [6, 9], such variables effect under different situations or heterogeneous variability. An alternative approach- Quantile Regression model, introduced by Koenker and Bassett (1978) [10]- is gradually becoming a popular standard econometric procedure in situations where estimation of the conditional mean function is insufficient to capture the full pattern of associations between the dependent variable and the covariates over the distribution of the former. Although, the goal of both regression models is same but quantile model predicts the conditional quantiles for sufficient data instead of conditional mean for small sample size. Applications of Quantile regression spread to many fields notably, ecology, environmental science [11, 12]; medicine and public health [13–17]; environmental health [18, 19], health service research (e.g. health care expenditure analysis) [20–23]; wage distribution and modeling [24–27]; Health economics [28, 29], Business analytics (risk management [9]) etc.

A recent pandemic caused by novel coronavirus was identified in Wuhan, Hubei Province, China [30], in December 2019 which caused a highly transferable viral disease brought about by severe acute respiratory condition coronavirus 2 (official name: SARS-CoV2). It caused over 700 million cases with 7.01 million deaths worldwide to date [31] and became as a Public Health Emergency of International concern by World Health Organization (WHO) [32]. Bangladesh also faced this crisis moment due to pandemic SARS-CoV-2 and almost 2.04 million people were infected by SARS-CoV-2 in Bangladesh [31]. The transmission of viruses can be influenced by many meteorological factors, such as temperature and humidity, wind speed, precipitation, radiation; however, the effect of meteorological parameters on SARS-CoV-2 remains controversial. Irfan et al. (2022), Sarwar et al. (2021) have shown that the meteorological factors, for instance, humidity and temperature, may drive the pace of the SARS-CoV-2 infections using quantile regression analysis [33, 34]. Another studies such as Karim and Nazmin (2022) illustrated that climate factors (temperature and humidity) have significant impact on mortality due to SARS-CoV-2 transmission using a parametric (GAM) and semiparametric (GAMLSS) regression model [19, 34]. Guo et. al. (2021) and H Ai et al. (2022) showed that higher temperatures were linked to a reduction in the effective reproductive number of COVID-19 in both China and the U.S using distributed lag non-linear model [35, 36]. Many existing QR applications to COVID-19 focus on asymmetric impacts of weather or financial variables on outcomes [33, 37], There’s less work on purely epidemiological quantile regression models infection rates with climate covariates. Most QR based study treated data as cross-sectional (e.g., climate vs. cases) or basic time components [38] and most research focuses on a few quantile levels (e.g., 25th, 50th, 75th). While previous studies have primarily relied on mean regression approaches to identify determinants of SARS-CoV-2 transmission, such methods implicitly assume homogeneous effects across the distribution of transmission intensity. This study addresses this limitation by employing quantile regression to uncover heterogeneous determinants across low, median, and higher transmission regimes, and by systematically comparing these results across quantiles.

In this study, we discussed quantile regression from an applied perspective. In particular, we illustrate the concept, advantages and disadvantages of both linear and quantile regression using SARS-CoV-2 datasets as an example and uncover whether the climatic determinants of SARS-CoV-2 transmission differ across the conditional distribution (e.g. lower vs. upper tails dynamics) to understand how do predictors affect mild vs. severe infection scenarios. Although Poisson and Negative Binomial models are standard approaches for count data, they focus on modeling the conditional mean of infections [19]. Our objective was to examine whether the effects of temperature and humidity differ across low and high transmission regimes. Given the strong right-skewness, heteroscedasticity, and episodic surges in SARS-CoV-2 cases, quantile regression allows covariate effects to vary across the conditional distribution, providing insights into heterogeneous transmission dynamics that cannot be captured by mean-based count models. This study included 2,023,058 SARS-CoV-2-infected cases in Bangladesh during the observation period March 08, 2020, to May 31, 2023. In the following section, we will go over the data sources and procedures used for developing statistical models in detail. Sect. "Data and Methods" explained how the Linear Regression Model and Quantile Regression will be used to study the impact of predictor variables on the response variable. Sect. "Results and Discussions" provides a summary of the data analysis and findings.

Data and methods

General linear regression model

Linear regression is a linear approach for modeling the relationship between the dependent variable and one or more than one explanatory variable [1]. These regression analyses which has been studied rigorously, and to be used extensively in practical applications [39]. Suppose (Y1, X1), (Y2, X2),…,(Yn, Xn) be a random sample of size n drawn from population and the linear regression model can be written as follows:

graphic file with name d33e408.gif 1

Where, Inline graphic be a set of response variables, XT = (1, X1,X2, … ……,Xd) be a set of observations of predictors and θT = (θ0,θ1,…,θd)T be the unknown parameters and Inline graphic is a random disturbance term in the regression model. A traditional regression is summarized the relationship between the average value of the response variable and a set of covariates by the estimation of the conditional mean function E(Yi/Xi) (See, for example, [7]. There are certain assumptions of linear model are 1) Linearity 2) Multivariate normality 3) No or little multicollinearity 4) No auto-correlation 5) Homoscedasticity and It also assume that the error term satisfies the zero conditional mean assumption e.g. Inline graphic 0), ensuring that the regressors are exogenous and that the OLS estimator provides unbiased estimates of the regression coefficients. Therefore, the conditional mean of these function is written as:

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Ordinary Least square (OLS) method minimizes the sum of square of error (called residuals) between the observed and predictors values of response variable. To obtain the estimate of the parameter Inline graphic we take its derivative with respect to Inline graphic and set it to zero. All mathematical description see [1, 40]. The linearity assumption can best be tested with scatter plots. Furthermore, the linear regression model requires that all variables be multivariate normal. A goodness of fit test can be used to assess normality.

Quantile regression model

Koenker and Bassett (1978) proposed a general linear model by introducing “regression quantiles” of linear combination of sample quantiles and trimmed mean along with robustness properties [10, 41]. Basically, Quantiles are cut points in a data distribution that divide the data set into equal size groups with equal probabilities. Koenker (2005) and Hao and Naiman (2007) added more excellent introduction to the theory and applications of quantile regression model estimation techniques and properties in literature [6, 8]. Quantile regression (QR) model uses least-absolute-distance estimation method instead of least square estimation and has greater computing power capabilities than OLS method [6].

For a given probability threshold τ, the quantile corresponding to τ defined as Inline graphic Algebraically, the quantile Inline graphic of level Inline graphic of a random variable Y with cumulative distribution function (cdf) Inline graphic as

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Thus the quantile function QY (τ) of the random variable Y is the inverse of its cdf. In general, Quantile regression is an extension of linear regression, which is used when the criteria of linear region are not met (normality, homoscedasticity), it is used. In Eq. (2), Inline graphic represents the marginal change in the mean of the response variable to a marginal change in the predictor variable X. The conditional quantile function is thus defined as

graphic file with name d33e584.gif

The linear quantile regression model takes the form:

graphic file with name d33e589.gif 3

where, Y= Inline graphic be a real valued random variable with FY (y) = P(Y ≤ y) and Inline graphic, and θ(τ) = (θ0(τ),θ1(τ),…,θd(τ))T is unknown quantile coefficients which depends on τ-th quantile which measures the marginal change in the τth quantile of Y due to a marginal change in X. If we assume that Inline graphic then Eq. (3) becomes.

graphic file with name d33e682.gif 4

Thus there are many regression models in (4) with each corresponding to a given quantile of interest. For example, Inline graphic = 0.5 gives us the median regression with model: Inline graphic X Inline graphic and so on.

To obtain the estimate of Inline graphic at Inline graphic quantile, minimizes the risk associated with the asymmetric absolute loss function. For mathematical description see [8].

The minimization problem for obtaining the estimate of quantile parameter is done by using Linear Programming Problem (LPP) which are well implemented in many software (R, SAS, Python). LPP are solved by numerous methods namely: Simplex Method, Frisch-Newton Interior Point Method, Interior Point Method with Prepossessing, Spare Regression Quantile fitting Method, etc. All information can be found in Koenker (2005) [8]. Koenker–Machado specification test [42] is also used in this study to assess whether regression coefficients vary across quantiles.

While the linear regression specifies the changes in conditional mean of response variable, the quantile model specifies changes in the conditional median. This approach is also refereed as conditional median model which is a natural extension of linear regression model. Although, the goal of both regression model is same but quantile model predicts the conditional median for sufficient data instead of conditional mean for small sample size. Unlike OLS, which focuses on a single summary measure, quantile regression reveals how predictors influence low, median, and high response counts differently, thus offering insight into whether relationships vary across epidemic intensity [43]. This conceptual distinction means that while OLS captures average epidemic behavior, quantile regression uncovers distribution-dependent effects that a mean model might overlook, making it particularly valuable in skewed longitudinal epidemic contexts [44].

Comparison standards

The Information Criterion measures the deviation of a model specified by the probability distribution from the true distribution [45]. There are several information criteria for model selection. Out of them HQC, AIC, BIC are commonly used in regression analysis. Useful reviews of these model selection criteria are found in [46, 47].

If Inline graphic is the log likelihood that measures how well the model describes the data, K is the number of parameter in the model and n is the total number of observations then.

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Hannan–Quinn information criterion (HQC)

The Hannan-Quinn information criterion (HQC) measures the goodness of fit of a statistical model. It is often used as a criterion for model selection among a finite set of models [48]. It is essential to remember that the HQC can be used to compare estimated models only when the numerical values of the dependent variable are identical for all estimates being compared. It is computed by following formula:

graphic file with name d33e776.gif

Akaike Information Criteria (AIC)

Akaike Information Criterion (AIC) measure used to evaluate models by balancing fit and complexity, which help to select the best model without overfitting. It is calculated according to the following formula:

graphic file with name d33e783.gif

Schwartz Bayesian Information Criteria (BIC)

The Bayesian Information Criterion (BIC) is comparable to the AIC but has a stricter approach to model complexity. It is based on Bayesian statistics, which impose a higher penalty for models with more parameters, especially when the dataset is large. It is calculated according to the following formula:

graphic file with name d33e790.gif

Data descriptions: SARS-CoV-2 case study

A study conducted on SARS-CoV-2 infected cases due to severity infected cases of disease in Bangladesh. Data of SARS-CoV-2 cases are retrieved from daily report of the national data bases of the Institute of Epidemiology Disease Control and Research (IEDCR), Dhaka, Bangladesh, during the period of March 08, 2020 to May 31, 2023 (website link: https://old.dghs.gov.bd/index.php/bd/home/5371). The number of daily infected people with daily temperature (°C) and daily humidity (%) of 1134 days are provided for featuring these report having no missing data. The daily temperature measured in (°C) and humidity (%) of Bangladesh are drawn from the website https://www.timeanddate.com/weather/bangladesh/dhaka. The research variables can be explained as follows: Y: The dependent variable represents the log(number of the infected patients) due to SARS-CoV-2 and the data was log-transformed to as quantile regression requires a real-valued response variable (see Koenker (2005)). The independent variables are: X1: Temperature, X2: Humidity. We used log transformation on predictor to make it continuous and to achieve normality as well since quantile regression model requires continuous response variable [8].

For the classical linear regression model for the given predictors:

graphic file with name d33e828.gif

We shall formulate the quantile regression models at following quantiles for given predictors:

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We shall use the classical linear regression model and the quantile linear regression models above to fit the data on the variables using the R programming software. The results and discussions are portrayed in following section.

Results and discussions

Exploratory data analysis

To explore the raw data and find an indication for selecting the more sophisticated statistical model, we have provided descriptive statistics and some graphical presentation of the variables in this section. Due to SARS-CoV-2 spread in Bangladesh the total number of infected people from 8 March 2020 to 31 May 2023 is 2,023,058. The descriptive statistics of number of infected people due to SARS-CoV-2 with climatology parameters such as temperature and humidity for 1134 days are summarized in Table 1. The average number of daily infected people was 1786. Also, the daily average Temperature and humidity during this period were 30.38 °C and 63.01% respectively. In addition, it also found that, Temperature exhibited a weak linear correlation with COVID-19 cases (r = 0.07), indicating a negligible linear association despite statistical significance, and Humidity showed a weak to modest correlation (r = 0.335), suggesting limited linear association. Moreover, the correlation between temperature and humidity is found very week negative (Pearson’s r = −0.0834) indicating no meaningful linear association and suggesting that multicollinearity is unlikely to bias the regression estimates.

Table 1.

Descriptive statistics of daily number. of SARS-CoV-2 infected case, temperature, and humidity for March 08, 2020, to May 31, 2023

Variables Infected case Temperature (°C) Humidity (%)
Mean (SD) 1786 (2873.509) 30.38 (3.99) 63.01 (16.32)
Coefficient of Variation (%) 160.89 13.14 25.90
Skewness 2.764 −0.771 −0.149
Kurtosis 8.089 0.749 −0.564
25th percentile 100 28.00 50.00
Median 641 31.00 65.00
75th percentile 2131 33.00 74.00
90th percentile 4379.60 35.00 84.00
95th percentile 8430.20 35.00 89.00
Lowest 0 10.00 20.00
Highest 16,230 40.00 100.00
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During this pandemic period the highest temperature was 40°and the lowest temperature was 10°whereas the lowest humidity was 20% and the highest humidity was 100%. The histogram and density plot of number of daily SARS-CoV-2 infected people have found a skewed pattern which are illustrated in Fig. 1. The scatter diagram among the variables are also plotted in Fig. 2 which indicates towards the non-linear relationship with a number of outliers.

Fig. 1.

Fig. 1

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a Histogram of number of infected new people; b Histogram of log (number of infected people) due to the SARS-CoV-2 the period March 08, 2020– May 31, 2023

Fig. 2.

Fig. 2

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Scatter diagram (a). Number of infected people due to SARS-CoV-2 vs. Daily temperature; b Number of infected people due to SARS-CoV-2 vs. Daily humidity during the period March 08, 2020– May 31, 2023

The skewness statistic, which measures the degree of asymmetry in infected new cases, is more than one and positive. It means that the response variable, COVID-19 infected new cases, has a skew distribution. Positive skewness means that the tail on the right side is larger than the one on the left. The kurtosis statistic, which measures the relative peakness or fatness of a distribution in comparison to the normal distribution, for infected new cases is larger than 3 (See the detailed results in Table 1). It shows that the data set has larger tails than a normal distribution (see Fig. 1a). In this scenario, if we applied a classical regression model of the outcome variable log (number of infected new cases) caused by COVID-19 to determine the effect of covariates, the results may be deceptive because the normality condition is not met. Quantile regression, which does not require a normal distribution, could be an ideal option for dealing with this situation.

Correlation test

To assess the relationship between these variables pearson’s correlation coefficient (r) can be calculated. A correlation coefficient different from 0 in the sample does not mean that the correlation is significantly different from 0 in the population. This needs to be tested with a hypothesis test or correlation test. The hypothesis is: H0: ρ = 0 meaning that there is no linear relationship between two variables. The scatter plots reveal curved patterns, indicating a nonlinear connection between these factors. Following Table 2 shows that temperature, humidity and infected people are significantly correlated with a correlation coefficient of 0.072 and 0.335 respectively. Therefore, the sample contains sufficient evidence to reject the null hypothesis and conclude that the correlation coefficient does not equal 0, so the relationship exists in the datasets.

Table 2.

Correlation test for Number of infected people due to SARS-CoV-2, temperature and humidity from March 08, 2020 to May 31, 2023

Covariates r t df p-value 95% CI
Infected people, Temperature 0.072 2.403 1131 < 0.05 [0.013, 0.128]
Infected people, Humidity 0.335 11.961 1131 < 0.05 [0.282, 0.385]
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The scatter plot of the number of infected people due to SARS-CoV-2 against temperature and humidity for the period from March 08, 2020 to May 31, 2023 are drawn in Fig. 2. We clearly see a nonlinear relationship between the response variables and the explanatory variables. These graphs demonstrated that the response variable and explanatory variables have a clear nonlinear relationship.

Although the data were longitudinal, the primary objective of this study was to compare conditional mean and conditional quantile regression frameworks in assessing heterogeneous climate effects. The models were therefore estimated without explicitly specifying autoregressive or time-series correlation structures. The estimated effects should be interpreted as conditional distributional associations.

Linear regression model

The linear regression model for the response variable Y = log (number of infected people) is estimated and conditional relationship of the response variables is:

graphic file with name d33e1140.gif

Looking at Table 3, we find that (p-value < 0.05) for the calculated (F) value is less than the 5% significance level, which indicates the importance of the estimated general linear regression model. It also turns out that the independent variables (temperature, humidity) exert a significant influence on the dependent variable because the marginal regression coefficients have probability values smaller than 0.05, moreover, temperature and humidity only explain 12.06% of variation in dependent variable of infected cases.

Table 3.

Summary statistics of OLS model for log (no. of infected people), temperature and humidity for March 08, 2020 to May 31, 2023

Dependent variable: Y = log (number of infected people + 1.1)
Estimate Inline graphic se (Inline graphic) t − value p − value F − value p − value Adjusted R2
Intercept 1.432 0.545 2.631 < 0.05 78.65 < 0.05 0.1206
Temperature 0.054 0.015 3.572 < 0.05
Humidity 0.046 0.003 12.279 < 0.05
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Therefore, linearity assumptions are tested, the Fig. 3 represents the standard diagnostic plots used to evaluate the assumptions of the linear regression model. The residuals versus fitted values plot (Fig. 3a) indicates that residuals are generally centered around zero, suggesting an overall adequate model fit. However, the presence of a slight downward trend in the smoothing line implies a minor deviation from strict linearity. The deviations of normal Q-Q plot of Fig. 3b suggest mild non-normality in the residual distribution, particularly in the tails. However, the plot of scale location (Fig. 3c) is indicating that the variance of residuals decreases as fitted values increase. This suggests weak heteroscedasticity and the assumption of equal variance is marginally violated. Finally, the residuals versus leverage plot (Fig. 3d), including Cook’s distance contours, indicates that most observations have low leverage. While a small number of points exhibit moderately higher leverage, none exceed conventional thresholds for influential observations, suggesting that no single data point unduly influences the regression estimates. Overall, Variance appears to increase (heteroscedasticity) slightly with fitted values in residual plot and middle line reveals that the model mostly fits the central trend well. However, the residuals are not perfectly normal as well.

Fig. 3.

Fig. 3

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Diagnostic plots for the linear regression model: a Residuals versus Fitted values, b Normal Q–Q plot of standardized residuals, c Scale–location plot, and d Residuals versus Leverage with Cook’s distance for log (number of infected people) due to the SARS-CoV-2 during the period March 08, 2020– May 31, 2023

Test of heteroscedasticy

The Breusch-Pagan (BP) test is a statistical test used to detect heteroscedasticity in a regression model to check if the variance of errors in a regression model changes based on the predictors. The test value of BP test is found 126.30 with p-value < 0.05, which indicates strong statistical evidence of heteroscedasticity in model residuals. Since the quantile regression model does not assume homoscedasticity (mentioned in Sect. "Data and Methods"), Heteroscedasticity is another reasons to use quantile regression later.

Quantile regression model

The quantile regression requires a real-valued response variable (see Koenker and Bassett (1978) [41]). As a result, the daily number of infected cases is not directly usable in the quantile regression analysis described in Sect. "Data and Methods". Many researchers use the logarithmic transformation of the "number of infected new cases" in mean regression analysis (see for example [49, 50]). We also use the logarithmic transformation of "number of infected cases" caused by SARS-CoV-2 to change a real-valued response variable for quantile regression analysis. Because the quantile regression estimators have the equivariance properties, we add 1.1 to the response variable before the logarithmic transformation to avoid a log-scale transformation of zero. This change of the response variables with 1.1 extra value has no effect on the final results (see Koenker 2005).

The conditional median function qτ(x) at different level of quantiles (0 < τ < 1) are as follows. At different quantiles (0.25, 0.50, 0.75, 0.90 and 0.95) we have found temperature gives mostly significant and few insignificant results regarding to the response variable Y = log (number of infected people + 1.1). However, humidity gives the significant relationship with the response variables during SARS-CoV-2 pandemic in Bangladesh. Table 4 shows the results of the estimated value of the meteorological variables Temperature and Humidity at different levels of quantile which indicate the significance, except the temperature at the 90% and 95% quantiles. Additionally, the Koenker–Machado specification test [42] was performed which indicated that significant heterogeneity in the effects of temperature and humidity across the conditional distribution of COVID-19 cases. Moreover, the fitted quantile regression implies that the τ-th conditional quantile of the residuals is zero, indicating that residuals are centered around zero at the estimated quantile level (see Fig. 4).

Table 4.

Summary statistics of Quantile Regression model for log( no. of infected people), temperature and humidity from March 08, 2020 to May 31, 2023

quantile Estimate Inline graphic se (Inline graphic) t-value p-value F-value p-value Pseudo-Inline graphic
Inline graphic Intercept −3.667 0.795 −4.613 < 0.05 134.50 < 0.05 0.155
Temperature 0.097 0.023 4.429 < 0.05
Humidity 0.079 0.005 16.306 < 0.05
Inline graphic Intercept 1.949 0.742 2.628 < 0.05 29.640 < 0.05 0.0377
Temperature 0.066 0.018 3.545 < 0.05
Humidity 0.038 0.005 7.115 < 0.05
Inline graphic Intercept 4.824 0.474 10.159 < 0.05 20.997 < 0.05 0.0326
Temperature 0.062 0.013 4.730 < 0.05
Humidity 0.014 0.003 4.923 < 0.05
Inline graphic Intercept 6.600 0.965 6.838 < 0.05 10.704 < 0.05 0.0380
Temperature 0.017 0.025 0.675 0.499
Humidity 0.022 0.004 4.546 < 0.05
Inline graphic Intercept 8.220 0.643 12.772 < 0.05 11.975 < 0.05 0.0578
Temperature −0.018 0.018 –1.013 0.311
Humidity 0.020 0.005 4.605 < 0.05
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Fig. 4.

Fig. 4

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Histogram of Residuals from the quantile regression models

25th Percentile regression model

The estimated conditional quantile function when τ = 0.25,

graphic file with name d33e1704.gif

The above quantile equation at τ = 0.25 showed that every 1 °C increase in temperature, the number of infected people multiplies by about 1.102 (= e0.097) or the number of infected people increases by approximately 10.2% per 1-unit increase in temperature. For each 1-unit increase in humidity, the number of infected people multiplies by about 1.082 (= e0.079) means 8.2% increase per 1-unit rise in humidity. Also, it turns out that the independent climate variables (explained by 15.50%) have a significant effect on the dependent variable, also the marginal slope parameters (θ0,θ1,θ2) has probability values smaller than 0.05.

50th Percentile regression model

The estimated conditional quantile function when τ = 0.50,

graphic file with name d33e1756.gif

The above quantile equation at τ = 0.50 showed that every 1 °C increase in temperature, the number of infected people multiplies by about 1.068 (= e0.066). In other word, at median level of infection a 1-unit increase in temperature corresponds to about a 6.8% increase in the number of infected people. For each 1-unit increase in humidity, the number of infected people multiplies by about 1.039 (= e0.038) means about a 3.9% increase in the median number of infected people. Furthermore, it turns out that the independent climate variables by (3.80%) have a significant effect on the dependent variable because the marginal slope parameters (θ0,θ1,θ2) has probability values smaller than 0.05.

75th Percentile regression model

The estimated conditional quantile function when τ = 0.75,

graphic file with name d33e1808.gif

The above quantile equation at τ = 0.75 showed that 1 °C rise in temperature is associated with about a 6.4% increase in the number of infected people, since the number of infected people multiplies by about 1.064 (= e0.062). For each 1-unit increase in humidity corresponds to about a 1.5% increase in the number of infected people at the 75th percentile since the number of infected people multiplies by about 1.015 (= e0.015). Furthermore, it turns out that the independent climate variables by (3.26%) have a significant effect on the dependent variable because the marginal slope parameters (θ0,θ1,θ2)) have probability values smaller than 0.05.

90th Percentile regression model

The estimated conditional quantile function when τ = 0.90,

graphic file with name d33e1860.gif

The above quantile model revels that at the very high infection levels (90th percentile), a 1 °C increase in temperature corresponds to about a 1.7% increase in the number of infected people., since the number of infected people multiplies by about 1.017 (= e0.017). For a 1-unit increase in humidity corresponds to about a 2.1% increase in the number of infected people at the 90th percentile, since the number of infected people multiplies by about 1.021 (= e0.021). Furthermore, it turns out that the independent climate variables by 3.80% have a significant effect on the dependent variable because the marginal slope parameters (θ0,θ2) has probability values smaller than 0.05 except quantile coefficient for temperature (θ1).

95th Percentile regression model

The estimated conditional quantile function when τ = 0.95,

graphic file with name d33e1907.gif

The above quantile equation at τ = 0.95 showed that a 1 °C rise in temperature corresponds to about a 1.8% decrease in the number of infected people at the very high (95th percentile) infection level, since the number of infected people multiplies by about 0.982 (= e−0.018). For a 1-unit increase in humidity corresponds to about a 2.0% increase in infections at the 95th percentile, since the number of infected people multiplies by about 1.020 (= e0.020). Furthermore, it turns out that the independent climate variables by almost 6% have a significant effect on the dependent variable because the marginal slope parameters (θ0,θ2) has probability values smaller than 0.05 except quantile coefficient for temperature (θ1).

The Fig. 5 shows that higher temperature is associated with slightly lower SARS-CoV2 infected cases, particularly in the higher quantiles of the distribution means weak and statistically insignificant evidence. Additionally, Humidity may have a mildly positive effect at lower quantiles, but its impact diminishes at higher quantiles. However, the effect is not significant, which may not strong enough to conclude a real pattern since Fig. 3 shows residuals are non-normal or heteroscedastic, therefore quantile regression models are more robust and reliable.

Fig. 5.

Fig. 5

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The Trend of Coefficient of predictors, on Response Variable at different quantiles

Quantiles comparison

The fitted linear regression model provides the HQC, AIC and BIC 4863.68, 4856.057 and 4876.215 respectively. Since HQC, AIC, BIC comparisons between OLS and quantile regression are not formally justified, as quantile regression is not likelihood-based in the classical sense, Moreover it derived from the asymmetric Laplace pseudo-likelihood (see Koenker, 2005). These criteria are employed descriptively to assess relative fit within the class of quantile regression models and are not used to compare quantile regression with likelihood-based models such as OLS. For better blooming, both regression lines are plotted together in Fig. 6, and to investigate distributional heterogeneity quantile models comparison standards are shown in Table 5.

Fig. 6.

Fig. 6

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Linear regression model vs Quantile regression model (a). Number of Infected case vs Temperature; b Number of Infected case vs Humidity, during March 08, 2021 to May 31, 2023

Table 5.

Comparison between the OLS model and the quantile regression model

Dependent variable: Y1=log (number of infected people+1.1)
Model OLS q0.25 q0.50 q0.75 q0.90 q0.95
HQC 4863.689 5220.433 5063.959 4849.845 4984.494 5068.238
AIC 4856.085 5313.493 5057.020 4842.905 4977.554 5061.298
BIC 4876.215 5233.419 5076.946 4862.831 4997.480 5081.224
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In Fig. 6a shows that temperature has a stronger effect on typical infected cases, whereas OLS understates the relationship by averaging across the distributions along constant variance and homogeneity effects. However, the quantile models overcome the homogeneous effect by revealing heterogeneity effects at different levels of condition distributions. At low quartile to median, the infected cases and temperature show a positive effect and with high transmission of SARS-CoV-2, the temperature effects become flattens. This weaker effect may suggest nonlinear or threshold effects of temperature. Similarly, in 6(b) the traditional OLS model shows constant effect across all levels, and it couldn’t capture the real difference in how humidity affects daily number of infected people (high/low) due to SARS-CoV-2 scenarios. However, the upper quintiles model (95th, 90th) indicate stronger humidity effects in high spread of viruses and the median (50th) quantile fits better to describe the relationship among variables.

From Table 5 among all three model selection criteria, we found that 75th conditional quantile model of log (number of infected people) for relative goodness of fit which provides the lowest HQC, AIC and BIC value among others quantile models. It means temperature and humidity might have their predictive and strongest effects on upper infection levels. Though the median quantile model is also close but it doesn’t better than 75th percentile model.

Conclusions

Quantile regression provides an alternative method for accounting for the asymmetries that the classical linear regression model ignores [8]. In this study, we used the quantile regression model for highly skewed data using the temperature and humidity effect on spread of SARS-CoV-2 Virus among the people in Bangladesh. To investigate the effect, a sample of size 1134 days are provided from the government website of IEDCR, Bangladesh and per day’s temperature, daily humidity and daily infected number of people are taken as variables. Therefore, applied mean and quantile regression model for investigating the significant effects among the variables. The both regression model findings revealed the significant effect among the variables with non-linear relationships that indicates the positive impact on Transmission on the SARS-Cov-2 virus. The quantile regression model provides more accurate results at different level of quantile (0 < τ < 1) than traditional linear regression model, however some temperature effects are insignificant at 75% and 90% location of percentiles with daily temperature in Bangladesh. Humidity’s effect decreases sharply from 25th to 75th percentile but slightly rises again at the 90th percentile. At extremely high infection levels, higher temperatures might slightly reduce infection counts, while humidity still tends to increase them. This study indicates that climatic factors like temperature and humidity have stronger influence on lower to median infection levels, but at very high infection counts, other factors might dominate. Our study indicates high temperature and humidity are more favorable for spread of disease due to SARS-CoV-2 (crowding indoors, air retention etc.) and this finding could imply that temperature might help limit the highest infection counts, while humidity consistently supports infection spread. However, for better explaining of actual behavior of spread of SARS-CoV-2, the potential impact of temperature, humidity on the number of daily infected people may not be strong enough to reduce the spread of disease, only controlled laboratory experiments can decisively prove it. However, other factors such as people’s lifestyle and their social activities during these such pandemic have to be considered for analyzing the statistical model intensively. The research limitations suggest the improved generalizability, integration of more variables, comparison analyses, and intervention studies. Future research should address these limitations (including temporal autocorrelation) to enhance understanding of the associated factors influencing the disease transmission.

Limitation of the study and future work

Quantile regression model is primarily applicable for continuous (at-least quasi continuous) predictors and over dispersion (variance > mean) in dataset. In this study our response variable is discrete (non-negative integers) often skewed and over dispersed. However, to suit quantile regression, we applied log transformation on predictor which smooths the distribution and reduces the skewness as well. As results, quantities become more stable and meaningful. Since quantile regression does not model count generation process directly, (like Poisson and Negative Binomial regression can do) it’s distribution-agnostic, which is both a strength and a limitation for this study. Additionally, this study analyzes daily SARS-CoV-2 case counts over time; therefore, the data are inherently longitudinal and may exhibit serial dependence due to epidemic persistence and temporal clustering. The regression models applied in this study did not explicitly incorporate autoregressive or temporal correlation structures. Ignoring longitudinal dependence may influence standard error estimation and the statistical significance of covariate effects. Consequently, results should be interpreted as associative rather than dynamic causal relationships. Future research may benefit from employing time-series quantile regression, autoregressive distributed lag models, mixed-effects quantile approaches, or other longitudinal frameworks that explicitly account for serial correlation in epidemic data. Our findings also do not report causal association between climatic factors and SARS-CoV-2. Controlled Laboratory based experiments will be required to address these. For better understanding and interpretation of discrete data, quantile regression with Bayesian methods can be combined in future research work. Nevertheless, Distribution Regression can be also employed for handling discreteness with excess zeros having strong theoretical foundation.

Acknowledgements

Not applicable.

Code availability

The R codes are available. It will be provided if anyone requires this.

Authors’ contributions

NA: Conceptualization; data curation; formal analysis; investigation; methodology; software; visualization; writing—original draft, review and editing. MMHK: review, editing.

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

Data availability

Data of SARS-CoV-2 cases are retrieved from daily report of the national data bases of the Institute of Epidemiology Disease Control and Research (IEDCR), Dhaka, Bangladesh, during the period of March 08, 2020 to May 31, 2023 (website link: https://old.dghs.gov.bd/index.php/bd/home/5371). The number of daily infected people with daily temperature (°C) and daily humidity (%) of 1134 days are provided for featuring these report having no missing data. The daily temperature measured in (°C) and humidity (%) of Bangladesh are drawn from the website https://www.timeanddate.com/weather/bangladesh/dhaka.

Declarations

Ethics approval and consent to participate

We have conducted ourselves with integrity, fidelity, and honesty. We have not intentionally engaged in or participated in malicious harm to another person or animal.

Consent for publication

Not Applicable.

Competing interests

The authors declare no competing interests.

Footnotes

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

  • 1.Gujarati DN. Essentials of econometrics. Sage Publications; 2021.
  • 2.Baig MA, Zaman Q, Baig SA, Qasim M, Khalil U, Khan SA, et al. Regression analysis of hydro-meteorological variables for climate change prediction: A case study of Chitral Basin, Hindukush region. Sci Total Environ. 2021;793:148595. [DOI] [PubMed] [Google Scholar]
  • 3.Gyawali R, Griffis VW, Watkins DW, Fennessey NM. Regional regression models for hydro‐climate change impact assessment. Hydrol Process. 2015;29(8):1972–85. [Google Scholar]
  • 4.Zhang L, Tian X, Zhao Y, Liu L, Li Z, Tao L, et al. Application of nonlinear land use regression models for ambient air pollutants and air quality index. Atmos Pollut Res. 2021;12(10):101186. [Google Scholar]
  • 5.Han H, Zeeshan Z, Talpur BA, Sadiq T, Bhatti UA, Awwad EM, et al. Studying long term relationship between carbon emissions, soil, and climate change: insights from a global earth modeling framework. Int J Appl Earth Obs Geoinf. 2024;130:103902.
  • 6.Hao L, Naiman DQ. Quantile regression. Sage; 2007, no. 149.
  • 7.Gujarati DN, Porter DC. Basic econometrics. McGraw-hill, 2009.
  • 8.Koenker R. Quantile regression, vol. 38. Cambridge university press; 2005. [Google Scholar]
  • 9.Koenker R. Quantreg: quantile regression. 2009 http://CRAN.R-project.org/package=quantreg.
  • 10.Rodriguez RN, Yao Y. Five things you should know about quantile regression. In Proceedings of the SAS Global Forum 2017 Conference, Orlando, FL, USA, 2017, pp. 2–5.
  • 11.Azimi M, Feng F, Zhou C. Air pollution inequality and health inequality in China: an empirical study. Environ Sci Pollut Res. 2019;26(12):11962–74. [DOI] [PubMed] [Google Scholar]
  • 12.Yang Z, Ren J, Ma S, Chen X, Cui S, Xiang L. The emission-inequality nexus: empirical evidence from a wavelet-based quantile-on-quantile regression approach. Front Environ Sci. 2022;10:871846. [Google Scholar]
  • 13.Zeleke AJ, Moscato S, Miglio R, Chiari L. Length of stay analysis of COVID-19 hospitalizations using a count regression model and quantile regression: a study in Bologna, Italy. Int J Environ Res Public Health. 2022;19(4):2224. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 14.Ghosh S, Sharma SK, Bhattacharya D. Deciphering disparities in childhood stunting in an underdeveloped state of India: an investigation applying the unconditional quantile regression method. BMC Public Health. 2020;20(1):1549. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 15.Schwarz L, Bruckner T, Ilango SD, Sheridan P, Basu R, Benmarhnia T. A quantile regression approach to examine fine particles, term low birth weight, and racial/ethnic disparities. Environ Epidemiol. 2019;3(4):e060. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 16.Bigio RS, Verly Junior E, d. Castro MA, C´esar CLG, Fisberg RM, Marchioni DML. Determinants of fruit and vegetable intake in adolescents using quantile regression. Rev Saude Publica. 2011;45:448–56. [DOI] [PubMed] [Google Scholar]
  • 17.Lipsitz SR, Fitzmaurice GM, Molenberghs G, Zhao LP. Quantile regression methods for longitudinal data with drop-outs: application to cd4 cell counts of patients infected with the human immunodeficiency virus. J Roy Stat Soc: Series C (Appl Stat). 1997;46(4):463–76. [Google Scholar]
  • 18.Magzamen S, Amato MS, Imm P, Havlena JA, Coons MJ, Anderson HA, et al. Quantile regression in environmental health: early life lead exposure and end-of-grade exams. Environ Res. 2015;137:108–19. [DOI] [PubMed] [Google Scholar]
  • 19.Karim R, Akter N. Effects of climate variables on the covid-19 mortality in Bangladesh. Theor Appl Climatol. 2022;150(3):1463–75. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 20.Peng X, Zhang JH. Socioeconomic inequality in public satisfaction with the healthcare system in China: a quantile regression analysis. Arch Public Health. 2022;80(1):165. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 21.Wang K-M, Nguyen Thi T-B. Quantile panel-type analysis for income inequality and healthcare expenditure. Econ Res-Ekon Istraz. 2022;35(1):873–93. [Google Scholar]
  • 22.Zhao Y, Atun R, Anindya K, McPake B, Marthias T, Pan T, van Heusden A, Zhang P, Duolikun N, Lee JT. Medical costs and out-of-pocket expenditures associated with multimorbidity in china: quantile regression analysis. BMJ Global Health. 2021;6(2). [DOI] [PMC free article] [PubMed]
  • 23.Le Cook B, Manning WG. Thinking beyond the mean: a practical guide for using quantile regression methods for health services research. Shanghai Arch of Psychiatry. 2013;25(1):55. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 24.Olaoye OO, Al-Faryan MAS, Tabash MI. Fiscal policy and income inequality in SA: a quantile regression. Int J Soc Econ. 2025;52(5):698–714.
  • 25.Shih Y-CT, Konrad TR. Factors associated with the income distribution of fulltime physicians: a quantile regression approach. Health Serv Res. 2007;42(5):1895–925. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 26.Lee B-J, Lee MJ. Quantile regression analysis of wage determinants in the korean labor market. The journal of the Korean economy. 2006;7(1):1–31. [Google Scholar]
  • 27.Buchinsky M. The dynamics of changes in the female wage distribution in the usa: a quantile regression approach. J Appl Econom. 1998;13(1):1–30. [Google Scholar]
  • 28.Lamichhane DK, Lee S-Y, Ahn K, Kim KW, Shin YH, Suh DI, et al. Quantile regression analysis of the socioeconomic inequalities in air pollution and birth weight. Environ Int. 2020;142:105875. [DOI] [PubMed] [Google Scholar]
  • 29.Jones AM, Lo´pez-Nicola´s A. The importance of individual heterogeneity in the´ decomposition of measures of socioeconomic inequality in health: an approach based on quantile regression. 2002.
  • 30.Guan W-j, Ni Z-y, Hu Y, Liang W-h, Ou C-q, He J-x, et al. Clinical characteristics of coronavirus disease 2019 in china. N Engl J Med. 2020;382(18):1708–20. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 31.Worldometers. COVID-19 Coronavirus Pandemic. 2025. Online; Accessed 03-August-2025. Available from: https://www.worldometers.info/coronavirus/.
  • 32.Sohrabi C, Alsafi Z, O’neill N, Khan M, Kerwan A, Al-Jabir A, et al. World health organization declares global emergency: a review of the 2019 novel coronavirus (covid-19). Int J Surg. 2020;76:71–6. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 33.Irfan M, Razzaq A, Suksatan W, Sharif A, Elavarasan RM, Yang C, et al. Asymmetric impact of temperature on covid-19 spread in india: Evidence from quantile-on-quantile regression approach. J Therm Biol. 2022;104:103101. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 34.Sarwar S, Shahzad K, Fareed Z, Shahzad U. A study on the effects of meteorological and climatic factors on the covid-19 spread in Canada during 2020. J Environ Health Sci Eng. 2021;19(2):1513–21. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 35.Guo C, Bo Y, Lin C, Li HB, Zeng Y, Zhang Y, et al. Meteorological factors and COVID-19 incidence in 190 countries: an observational study. Sci Total Environt. 2021;757:143783. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 36.Ai H, Nie R, Wang X. Evaluation of the effects of meteorological factors on COVID-19 prevalence by the distributed lag nonlinear model. J Transl Med 2022;20(1):170. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 37.Roux J, Faisant M, François D, Retel O, Le Tertre A. Age at death during the Covid-19 lockdown in French metropolitan regions: a non-parametric quantile regression approach. BMC Public Health. 2024;24(1):1251. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 38.Jiang F, Zhao Z, Shao X. Modelling the COVID-19 infection trajectory: a piecewise linear quantile trend model. J R Stat Soc Ser B: Stat Methodol. 2022;84(5):1589–607. [Google Scholar]
  • 39.Yan X, Su X. Linear regression analysis: theory and computing. World Scientific, 2009.
  • 40.Bingham NH, Fry JM. Regression: Linear models in statistics. Springer Science & Business Media. 2010.
  • 41.Koenker R, Bassett Jr G. Regression quantiles. Econometrica: journal of the Econometric Society. 1978:33–50.
  • 42.Koenker R, Machado JA. Goodness of fit and related inference processes for quantile regression. J Am Stat Assoc. 1999;94(448):1296–310. [Google Scholar]
  • 43.Francis R, Nwakuya M. A comparative analysis of ordinary least squares and quantile regression estimation technique. Am J Math Comput Model. 2022;7(4):49–54. [Google Scholar]
  • 44.Gnardellis C. Quantile regression in epidemiology: capturing heterogeneity beyond the mean. Methods and Protocols. 2025;9(1):2. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 45.Akaike H. Akaike’s information criterion. In International encyclopedia of statistical science. Springer, 2025, pp. 41–42.
  • 46.Akaike H. Information theory and an extension of the maximum likelihood principle. In Selected papers of hirotugu akaike. Springer, 1998, pp. 199–213.
  • 47.Amemiya T. Selection of regressors. Int Economic Rev. 1980:331–354.
  • 48.Lu H. ¨tkepohl and M. Kr¨atzig, Applied time series econometrics. Cambridge university press, 2004.
  • 49.Mecenas P, Bastos RTdRM, Vallinoto ACR, Normando D. Effects of temperature and humidity on the spread of COVID-19: a systematic review. PLoS ONE. 2020;15(9):e0238339. [DOI] [PMC free article] [PubMed] [Google Scholar]
  • 50.Karim MR, Akter MB, Haque S, Akter N. Do temperature and humidity affect the transmission of sars-cov-2? -a flexible regression analysis. Ann of Data Sci. 2022;9(1):153–73. [DOI] [PMC free article] [PubMed] [Google Scholar]

Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Data Availability Statement

Data of SARS-CoV-2 cases are retrieved from daily report of the national data bases of the Institute of Epidemiology Disease Control and Research (IEDCR), Dhaka, Bangladesh, during the period of March 08, 2020 to May 31, 2023 (website link: https://old.dghs.gov.bd/index.php/bd/home/5371). The number of daily infected people with daily temperature (°C) and daily humidity (%) of 1134 days are provided for featuring these report having no missing data. The daily temperature measured in (°C) and humidity (%) of Bangladesh are drawn from the website https://www.timeanddate.com/weather/bangladesh/dhaka.

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