A Novel Exponential Regression Model for Analyzing Dengue Fever Case Rates in the Federal District of Brazi

1. Introduction

Dengue fever represents a major challenge in epidemiology around the world, especially in Brazil, where recurring epidemics happen in some endemic territories. The uncontrollable increase in the metropolitan population and the lack of information and control are factors that are determinants of the occurrence of the disease and the burden on the healthcare system. More than 100 countries have dengue fever as an endemic problem, which affects millions of people each year. One of the most important factors contributing to the global spread of the virus nowadays is climate change, which is growing fast.

In this regard, many applications for generalized extreme value (GEV) distributions can be found in epidemiological studies. [1] presented a review study on the relationship between dengue fever and meteorological parameters, as well as a meta-analysis to investigate the impact of ambient temperature and precipitation on dengue fever. Dengue case counts during outbreaks in Thailand were modeled using extreme value theory (EVT) [2]. A zero-inflated GEV regression model is used in an application using Vietnam dengue data [3]. The estimate of the risk of infection for the individuals is based on the covariates of age and weight.

Likewise, [4] presented a GEV approach to study the frequency and intensity of extreme novel epidemics similar to COVID-19. As well, extreme value statistics are used to predict in real-time severe influenza epidemics [5]. Moreover, [6] investigated the extreme correlation between infectious illness outbreaks and crude oil futures. [7] estimated the disease burden of dengue in endemic regions to study the influence pattern of socioeconomic factors. They recommend allocating more resources to areas with high population expansion and urbanization.

Further, [8] explored the characteristics and studied the temporal-spatial cases of the overseas imported dengue fever in outbreak provinces of China. [9] studied myocardial manifestations related to dengue fever cases in a systematic review. In [10], dengue fever cases in the Brazilian state of Alagoas were modeled monthly using GEV distribution. The findings underline the significance of ongoing monitoring and assistance in this area.

Therefore, the study is based on the dengue fever cases of the epidemiological weeks in 2022 in the Federal District of Brazil. A regression analysis is applied to study extreme events (an epidemiological event that affects health centers and the economy) and the maximum likelihood method is used to estimate the parameters. The accuracy of the estimators is confirmed by means of Monte Carlo simulations. Some local influence measures are used, as are residual analyses, to validate the goodness-of-fitness of the proposed model.

Considering this, the studies present a novel bimodal regression model based on the generalized odd log-logistic exponential (GOLLE) distribution [11] and [12], namely the log generalized odd log-logistic exponential (LGOLLE) regression model. The GOLL-G class’s broad flexibility, which allows modeling its tails and asymetry, combined with the exponential distribution, which is used in many EVT applications and has a closed mathematical form, makes the novel regression a relevant model to apply in many areas, as well as a potential application in extreme events.

The rest of the work is organized as follows. Section 2 presents the GOLLE distribution [11] and [12]. Moreover, reviews the linear representation and some of its mathematical features. Maximum likelihood estimation is discussed, as well as a Monte Carlo simulation study to show the estimators’ consistency. The new LGOLLE distribution is introduced, maximum likelihood is used and some simulations assess the consistency of the maximum likelihood estimates (MLEs) in Section 3. In Section 4, a novel regression model based on the GOLLE distribution, using location parametrization, is proposed and simulations examine the behavior of the estimates. The model’s fit is assessed using global influence metrics and residual analysis. Section 5 demonstrates the usefulness of the new regression model using an epidemiological data set. The results and a discussion are presented in Section 6. Additionally, diagnostic and residual analyses are provided. Section 7 contains some final observations.

2. Materials and Methods

2.1. The GOLLE Distribution

Recently, the development of new distributions using well-known distributions aims to capture accurately the underlying distribution of the data and obtain more precise estimates or key quantities of interest.

In this context, the generalized odd log-logistic-G (GOLL-G) family, pioneered by [13], is a versatile class of continuous distributions for modeling various types of data. In their study, [13] elucidated the advantages of the introduced family of distributions and its applicability across various fields, highlighting its superiority over well-known generators. This particular distribution offers advantages compared to other competing models, as elaborated in the upcoming sections.

This family is based on the transformer-transformer (T-X) generator defined by [14]. Consider a baseline cdf $G\left(x\right)=G(x;\xi )$, where $\xi$ denotes an unknown parameter vector. The GOLL-G cdf is defined by integrating the log-logistic density function, namely

$$F\left(y\right)={\int }_0^{{\displaystyle \frac{G{\left(x\right)}^{\theta }}{1-G{\left(x\right)}^{\theta }}}}{\displaystyle \frac{\alpha w^{\alpha -1}}{{(1-w)}^2}}dw={\displaystyle \frac{G{\left(x\right)}^{\alpha \theta }}{G{\left(x\right)}^{\alpha \theta }+{[1-G{\left(x\right)}^{\theta }]}^{\alpha }}},$$

(1)

where $\alpha$ > 0 and $\theta$ > 0 are two extra shape parameters.

The pdf corresponding to (1) can be expressed as

$$f\left(y\right)={\displaystyle \frac{\alpha \theta g\left(x\right)G{\left(x\right)}^{\alpha \theta -1}{[1-G{\left(x\right)}^{\theta }]}^{\alpha -1}}{{\{G{\left(x\right)}^{\alpha \theta }+{[1-G{\left(x\right)}^{\theta }]}^{\alpha }\}}^2}},$$

(2)

where $g\left(x\right)=g(x;\xi )$ is the baseline pdf. Its hazard rate function (hrf) is easily found as $\tau \left(y\right)=f\left(y\right)/[1-F(y\left)\right]$.

These equations define some characteristics of the GOLL-G family, allowing it to effectively model a wide range of data types (skewed, bimodal, asymmetric, etc.) The parameters $\alpha$ and $\theta$ play an important role in shaping the distribution. In addition, Equations (1) and (2) do not involve complex mathematical functions, unlike the gamma and beta classes.

Table 1 reports some sub-models of Equation (1).

The cumulative distribution function (cdf) and the probability density function (pdf) of the GOLLE distribution are defined in [11] and [12], respectively, as

$$F(y;\alpha ,\theta ,\lambda )={\displaystyle \frac{{\left(1-{\mathrm{e}}^{-\lambda y}\right)}^{\alpha \theta }}{{\left(1-{\mathrm{e}}^{-\lambda y}\right)}^{\alpha \theta }+{\left[1-{\left(1-{\mathrm{e}}^{-\lambda y}\right)}^{\theta }\right]}^{\alpha }}}$$

(3)

and

$$f(y;\alpha ,\theta ,\lambda )={\displaystyle \frac{\alpha \theta \lambda {\mathrm{e}}^{-\lambda y}{\left(1-{\mathrm{e}}^{-\lambda y}\right)}^{\alpha \theta -1}{\left[1-{\left(1-{\mathrm{e}}^{-\lambda y}\right)}^{\theta }\right]}^{\alpha -1}}{{\left\{{\left(1-{\mathrm{e}}^{-\lambda y}\right)}^{\alpha \theta }+{\left[1-{\left(1-{\mathrm{e}}^{-\lambda y}\right)}^{\theta }\right]}^{\alpha }\right\}}^2}}.$$

(4)

The hazard rate function (hrf) can be obtained as follows

$$\tau (y;\alpha ,\theta ,\lambda )={\displaystyle \frac{\alpha \theta \left(\lambda {\mathrm{e}}^{-\lambda x}\right){\left(1-{\mathrm{e}}^{-\lambda x}\right)}^{\alpha \theta -1}}{\left[1-{\left(1-{\mathrm{e}}^{-\lambda x}\right)}^{\theta }\right]\left\{{\left(1-{\mathrm{e}}^{-\lambda x}\right)}^{\alpha \theta }+{[1-{\left(1-{\mathrm{e}}^{-\lambda x}\right)}^{\theta }]}^{\alpha }\right\}}}.$$

(5)

Here, in Table 2, the sub-models obtained from Equation (4) are provided. Its capacity to handle data fitting across a wide range of distributions demonstrates its versatility and application.

Figure 1 and Figure 2 show plots of Y’s hrfs and histograms for selected parameters. One of the most notable properties of the GOLLE distribution is its ability to generate a wide range of hazard shapes, in contrast to the exponential hrf’s constant behavior over time.

Figure 1 illustrates many forms, including the inverse J-shape, increasing-decreasing, decreasing-increasing and bathtub. Figure 2 demonstrates that the model is effective for modeling non-normal data sets with diverse histogram patterns (e.g., asymmetric, heavy tail, multimodal).

2.2. Main Properties

This Section presents the GOLLE distribution’s linear representation of the density function, including the qf, moments, and mgf, as defined in [12].

2.3. Representation

 Definition 1.

The GOLLE density (4) can be represented linearly using exponential densities as

$$g\left(y\right)=\sum _{k,m=0}^{\infty }{h}_{k,m}g(y;{\lambda }^{*}),$$

(6)

where $g(·)$ is the exponential density with a shared parameter ${\lambda }^{*}={\lambda }^{*}(\lambda ,m)=\lambda (m+1)$ and $h_{k,m}$ is defined below

$$h_{k,m}={\displaystyle \frac{{(-1)}^m(k+1)\left(\genfrac{}{}{0pt}{}{k}{m}\right)}{(m+1)}}{b}_k.$$

2.4. Quantile function

 Definition 2.

The GOLLE quantile function (qf) of Y is expressed by

$$Q\left(u\right)=-{\displaystyle \frac{1}{\lambda }}log[1-{\epsilon }_{\alpha .\theta }\left(u\right)],$$

(7)

where

$${\epsilon }_{\alpha ,\theta }\left(u\right)={\left[{\displaystyle \frac{{\left(\frac{u}{1-u}\right)}^{1/\alpha }}{1+{\left(\frac{u}{1-u}\right)}^{1/\alpha }}}\right]}^{1/\theta }.$$

Figure 3 displays Galton’s skewness and Moors’ kurtosis for different $\alpha$ and $\theta$, with $\lambda =1.58$. These plots show how both parameters influence the distribution shape. As $\alpha$ and $\theta$ increase, the skewness and kurtosis measures reduce to a minimum value.

2.5. Moments

 Definition 3.

The nth moment of the GOLLE distribution is defined by

$${\mu }_n^{\prime }=E\left(Y^n\right)=\sum _{k,m=0}^{\infty }{\displaystyle \frac{1}{{\lambda }^{*}}}{h}_{k,m}=\sum _{k,m=0}^{\infty }{\displaystyle \frac{{(-1)}^m(k+1)\left(\genfrac{}{}{0pt}{}{k}{m}\right)}{(m+1){\lambda }^{*}}}{b}_k.$$

2.6. Generating Function

 Definition 4.

The moment generating function (mgf) of the GOLLE density can be expressed as

$$M_Y\left(t\right)=\sum _{k,m=0}^{\infty }{\displaystyle \frac{{\lambda }^{*}}{{\lambda }^{*}-t}}{h}_{k,m}=\sum _{k,m=0}^{\infty }{\displaystyle \frac{{(-1)}^m(k+1){\lambda }^{*}}{{\lambda }^{*}-t}}{b}_k,t<{\lambda }^{*}.$$

2.7. Estimation

The MLEs of the GOLLE parameters vector $\tau ={(\alpha ,\theta ,\lambda )}^{\top }$ are calculated from a complete sample $y_1,\dots ,y_n$ by maximizing

$$\begin{array}{cc}\hfill {\mathcal{L}}_n\left(\tau \right)& =nlog\left(\alpha \theta \lambda \right)-\lambda \sum _{i=1}^n{y}_i+(\alpha \theta -1)\sum _{i=1}^{n}log(1-{\mathrm{e}}^{-\lambda y_i})+(\alpha -1)\sum _{i=1}^{n}log\left[1-{(1-{\mathrm{e}}^{-\lambda y_i})}^{\theta }\right]\hfill \\ \hfill & -2\sum _{i=1}^{n}log\left\{{(1-{\mathrm{e}}^{-\lambda y_i})}^{\alpha \theta }+{\left[1-{(1-{\mathrm{e}}^{-\lambda y_i})}^{\theta }\right]}^{\alpha }\right\}.\hfill \end{array}$$

(8)

Let’s consider

$$A_i\left(\lambda \right)=A_i=1-{\mathrm{e}}^{\lambda y_i}.$$

Therefore, the elements of the score vector can be formulated as follows

$$\begin{array}{ccc}\hfill U_{\alpha }& =& {\displaystyle \frac{n}{\alpha }}+\theta \sum _{i=1}^{n}log\left(A_i\right)+\sum _{i=1}^{n}log\left(1-{A_i}^{\theta }\right)\hfill \\ & -& 2\sum _{i=1}^n{\displaystyle \frac{\theta log\left(A_i\right){A_i}^{\alpha \theta }+{(1-{A_i}^{\theta })}^{\alpha }log(1-{A_i}^{\theta })}{{A_i}^{\alpha \theta }+{(1-{A_i}^{\theta })}^{\alpha }}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill U_{\theta }& =& {\displaystyle \frac{n}{\theta }}+\alpha \sum _{i=1}^{n}log\left(A_i\right)-(\alpha -1)\sum _{i=1}^n{\displaystyle \frac{{A_i}^{\theta }log\left(A_i\right)}{1-{A_i}^{\theta }}}\hfill \\ & +& \sum _{i=1}^n{\displaystyle \frac{\alpha {A_i}^{\alpha \theta }log\left(A_i\right)+{(1-{A_i}^{\alpha })}^{\theta }log(1-{A_i}^{\alpha })}{{A_i}^{\alpha \theta }+{(1-{A_i}^{\alpha })}^{\theta }}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill U_{\lambda }& =& {\displaystyle \frac{n}{\lambda }}-\sum _{i=1}^n{y}_i+(\alpha \theta -1)\sum _{i=1}^n{\displaystyle \frac{(1-A_i)}{A_i}}-\theta (\alpha -1)\sum _{i=1}^n{\displaystyle \frac{(1-A_i)A_i^{\theta -1}}{1-A_i^{\theta }}}\hfill \\ & -& 2\alpha \theta \sum _{i=1}^n{\displaystyle \frac{(1-A_i)[A_i^{\alpha \theta -1}-A_i^{\theta -1}{(1-A_i^{\theta })}^{\alpha -1}]}{A_i^{\alpha \theta }+{(1-A_i^{\theta })}^{\alpha }}}.\hfill \end{array}$$

Using a Newton-Raphson type method and setting the score equations $U_{\alpha }=U_{\theta }=U_{\lambda }=0$, the MLEs are calculated. The optim procedure can also be used to numerically maximize Equation (8).

2.7.1. Simulation Study

In two scenarios, Monte Carlo simulations generated by $1,000$ samples of varied sizes of the GOLLE distribution are utilized to assess the accuracy of MLEs. For each sample size, $n=\{50,100,200,400,800,1,000\}$, the average estimates (AEs), absolute biases (ABs) and root mean square errors (RMSEs) are computed, for each $(\alpha ,\theta ,\lambda )$.

As predicted by the consistency requirement, the results in Table 10 indicate that AEs approximate to the real values and ABs and RMSE approach zero as n increases. It is notable that for scenario 1, all the estimates obtained when $n=50$ were overestimated, while for scenario 2, the parameters $\theta$ and $\lambda$ were overestimated. This shows the sensitivity of the model’s parameters to some values, but in general, as the sample size increases, convergence towards the true values is achieved.

3. The proposal LGOLLE distribution

Let be a rv Y following the GOLLE density function (4), here is defined $W=log\left(Y\right)$. Setting $\lambda =-exp\left(\mu \right)$, the density function of W is expressed as (for $y\in \mathbb{R}$)

$$f(y;\alpha ,\theta ,\mu )={\displaystyle \frac{\alpha \theta {\mathrm{e}}^{[(y-\mu )-{\mathrm{e}}^{y-\mu }]}{\left[1-{\mathrm{e}}^{-{\mathrm{e}}^{y-\mu }}\right]}^{\alpha \theta -1}{\left[1-{\left(1-{\mathrm{e}}^{-{\mathrm{e}}^{y-\mu }}\right)}^{\theta }\right]}^{\alpha -1}}{{\left\{{\left(1-{\mathrm{e}}^{-{\mathrm{e}}^{y-\mu }}\right)}^{\alpha \theta }+{\left[1-{\left(1-{\mathrm{e}}^{-{\mathrm{e}}^{y-\mu }}\right)}^{\theta }\right]}^{\alpha }\right\}}^2}},$$

(9)

where $\mu \in \mathbb{R}$.

The Equation (9) is the log generalized odd log-logistic exponential (LGOLLE) distribution, namely $W\sim$ LGOLLE($y;\alpha ,\theta ,\mu$) where $\mu$ is the location parameter. Thus, if $Y\sim$ GOLLE($y;\alpha ,\theta$) then $W=log\left(Y\right)\sim$ LGOLLE($y;\alpha ,\theta ,\mu$). The density function in Figure 4 is displayed for select values of the parameters $\alpha ,\theta$ and $\mu$. These plots show the versatility of the new distribution with its skewed and bimodal shapes.

4. The New LGOLLE Regression Model

Let be $W\sim$ LGOLLE($\alpha ,\theta ,\mu$) the density function, thus the rv $Z=Y-\mu$ is given by

$$f(z;\alpha ,\theta ,\mu )={\displaystyle \frac{\alpha \theta {\mathrm{e}}^{[z-{\mathrm{e}}^z]}{\left[1-{\mathrm{e}}^{-{\mathrm{e}}^z}\right]}^{\alpha \theta -1}{\left[1-{\left(1-{\mathrm{e}}^{-{\mathrm{e}}^z}\right)}^{\theta }\right]}^{\alpha -1}}{{\left\{{\left(1-{\mathrm{e}}^{-{\mathrm{e}}^z}\right)}^{\alpha \theta }+{\left[1-{\left(1-{\mathrm{e}}^{-{\mathrm{e}}^z}\right)}^{\theta }\right]}^{\alpha }\right\}}^2}},$$

(10)

and the rv $Z\sim$ LGOLLE($\alpha ,\theta$,0).

In order to introduce a new regression structure in the class of models (10), the parameter ${\mu }_i$ is assumed to vary across observations through a regression structure expressed as

$$y_i={\mu }_i+z_i,i=1,\dots ,n,$$

(11)

where the random error $z_i$ has density function (10) and ${\mu }_i$ is parameterized by

$${\mu }_i={\mu }_i\left({\mathbf{x}}_{\mathbf{i}}^{\top }\beta \right),$$

where $\beta ={(\beta ,\dots ,{\beta }_p)}^{\top }$ is the parameter vector of dimension p associated with the explanatory variables ${\mathbf{x}}_{\mathbf{i}}^{\top }=(x_{i1},\dots ,x_{ip})$ for the location parameter.

4.1. Estimation

Except for ${\mu }_i$, the components of the score vector for $U_{\alpha }$ and $U_{\theta }$ are the same as those obtained from the Equations presented in Subsection (Section 2.7). The score component of the location parameter ${\mu }_i$ is defined to add the regression part as follows

$$\begin{array}{ccc}\hfill U_{{\mu }_i}& =& \sum _{i=1}^{\infty }{\displaystyle \frac{{\partial }_{\beta }g(z_i;{\mu }_i)}{g(z_i;{\mu }_i)}}+(\alpha \theta -1)\sum _{i=1}^{\infty }{\displaystyle \frac{{\partial }_{\beta }G(z_i;{\mu }_i)}{G(z_i;{\mu }_i)}}+\theta (1-\alpha )\sum _{i=1}^{\infty }{\displaystyle \frac{{\partial }_{\beta }G(z_i;{\mu }_i)G{(z_i;{\mu }_i)}^{\theta -1}}{1-G{(z_i;{\mu }_i)}^{\theta }}}\hfill \\ & -& 2\sum _{i=1}^{\infty }{\partial }_{\beta }G(z_i;{\mu }_i){\displaystyle \frac{G{(z_i;{\mu }_i)}^{\alpha \theta -1}-G{(z_i;{\mu }_i)}^{\theta -1}{[1-G{(z_i;{\mu }_i)}^{\theta }]}^{\alpha -1}}{G{(z_i;{\mu }_i)}^{\alpha \theta }+{[1-G{(z_i;{\mu }_i)}^{\theta }]}^{\alpha }}},\hfill \end{array}$$

where $G(·)$ and $g(·)$ are the cdf and pdf of the exponential distribution, respectively, ${\partial }_{\beta }g(z_i;{\mu }_i)={\partial }_{{\mu }_i}g(z_i;{\mu }_i){\partial }_{\beta }{\mu }_i(z_i;\beta )$ and ${\partial }_{\beta }G(z_i;{\mu }_i)={\partial }_{{\mu }_i}G(z_i;{\mu }_i){\partial }_{\beta }{\mu }_i(z_i;\beta )$ denotes the derivatives of the parameter ${\mu }_i$ using the chain rule.

The MLE $\widehat{\tau }$ of $\tau$ of the regression model is calculated by setting the score equations $U_{\alpha }=U_{\theta }=U_{{\mu }_i}=0$ or using the optim routine in [18].

4.2. Regression Simulation Study

To show the accuracy of the MLEs for $\alpha =0.25$, $\theta =9.75$, ${\beta }_0=0.89$ and ${\beta }_1=1.15$, 1,000 samples of size $n=\{25,\dots ,1,000\}$ were generated. The study is based on the measurements: biases, MSEs and ALs. The measures are (for $ϵ=\alpha ,\theta ,\lambda$)

$$Bia{s}_{ϵ}\left(n\right)={\displaystyle \frac{1}{N}}\sum _{i=1}^N({\widehat{ϵ}}_i-ϵ),MS{E}_{ϵ}\left(n\right)={\displaystyle \frac{1}{N}}\sum _{i=1}^N{({\widehat{ϵ}}_i-ϵ)}^2\mathrm{and}A{L}_{ϵ}\left(n\right)={\displaystyle \frac{3.919928}{N}}\sum _{i=1}^N{s}_{{\widehat{ϵ}}_i}.$$

Figure 5, Figure 6 and Figure 7 show the values of these measures in relation to n. Biases, MSEs and ALs tend to zero as the sample size increases. In the biases, the estimate of ${\beta }_0$ is overestimated, while the estimate of ${\beta }_1$ shows oscillatory behavior. These indicated potential concerns with optimization for certain values and sample sizes. Despite this, the biases converge to zero as expected. These findings corroborate the consistency of the MLEs.

4.3. Model Checking

Several approaches for analyzing outliers have been documented in the literature, including [19,20] and [21]. Diagnostic methods such as observation exclusion are used to detect influential observations in the proposed regression model.

In this context, for the proposed systematic component, the exclusion of observations follows

$${\mu }_l=\mu \left({\mathbf{x}}_{\mathbf{l}}^{\top }{\beta }_{\mathbf{j}}\right),l=1,\dots ,n,l\ne i.$$

(12)

For investigating the influential observations, the generalized Cook’s distance is given by

$$GC{D}_i=\left({\widehat{\tau }}_{\left(i\right)}-\widehat{\tau }{)}^{\top }\left[\ddot{\mathbf{L}}\left(\widehat{\tau }\right)\right]({\widehat{\tau }}_{\left(i\right)}-\widehat{\tau }\right),$$

(13)

and the likelihood distance, as

$$L{D}_i=2\left\{\mathcal{L}\left(\widehat{\tau }\right)-\mathcal{L}\left({\widehat{\tau }}_{\left(i\right)}\right)\right\},$$

(14)

where the subscript i denotes the observation deleted from the dataset and $\ddot{\mathbf{L}}\left(\tau \right)$ is the observed information matrix.

Moreover, the objective of residual analysis is to find trends or characteristics in the residuals that may affect the model’s validity. Therefore, the deviance residuals have been commonly used to assess the goodness-of-fit of regression models [22]. It follows that the deviance residuals are given by

$$r_{D_i}=\mathrm{sgn}\left({\widehat{r}}_{M_i}\right){\left\{-2[{\widehat{r}}_{M_i}+log(1-{\widehat{r}}_{M_i})]\right\}}^{1/2},$$

(15)

where

$${\widehat{r}}_{M_i}=1+log\left\{1-{\displaystyle \frac{{(1-{\mathrm{e}}^{-{\mathrm{e}}^{y-\widehat{\mu }}})}^{\widehat{\alpha }\widehat{\theta }}}{{(1-{\mathrm{e}}^{-{\mathrm{e}}^{y-\widehat{\mu }}})}^{\widehat{\alpha }\widehat{\theta }}+{[1-{(1-{\mathrm{e}}^{-{\mathrm{e}}^{y-\widehat{\mu }}})}^{\widehat{\theta }}]}^{\widehat{\alpha }}}}\right\},$$

(16)

are the martingale residuals and sign(·) is the signal function with a value $+1$ if the argument is positive and $-1$ if the argument is negative.

[23] proposed the construction of envelopes to support the analysis of the residuals with normal probability plots. For these envelopes, confidence bands are simulated, and if the model fits well, the majority of the points will be randomly distributed inside.

5. Application: Dengue Fever Cases Data

To demonstrate the potential of the GOLLE distribution, Table 4 illustrates some alternative distributions of well-known generators, in addition to the nested model.

The distributions are presented (for $x>0$), respectively, as

$$F_{\mathrm{KwFr}}\left(x\right)={\{1-{\left[F_{\mathrm{Fr}}\left(x\right)\right]}^a\}}^b,$$

$$F_{\mathrm{KwE}}\left(x\right)={\{1-{\left[G\left(x\right)\right]}^a\}}^b,$$

$$F_{\mathrm{GFr}}\left(x\right)={\displaystyle \frac{\gamma \{a,-log[1-F_{\mathrm{Fr}}\left(x\right)]/b\}}{\Gamma \left(a\right)}},$$

$$F_{\mathrm{GE}}\left(x\right)={\displaystyle \frac{\gamma \{a,-log[1-G\left(x\right)]/b\}}{\Gamma \left(a\right)}},$$

$$F_{\mathrm{BE}}\left(x\right)=I_{G\left(x\right)}(a,b)={\displaystyle \frac{1}{B(a,b)}}{\int }_0^{G\left(x\right)}{w}^{a-1}{(1-w)}^{b-1}dw$$

and

$$F_{\mathrm{Fr}}\left(x\right)=exp[-{(x-a)}^{-b}],$$

where all of the parameters are positive and $G\left(x\right)$ and $F_{\mathrm{Fr}}\left(x\right)$ represent the exponential and Fréchet distributions, respectively. The goodness.fit function of AdequacyModel package (see, [29]) computes the MLEs (standard errors (SEs) in parentheses) for all fitted models using the BFGS approach.

5.1. Descriptive of the Data

The data set was extracted from the Health Problem and Notification Information System (SINAN) (https://datasus.saude.gov.br/acesso-a-informacao/doencas-e-agravos-de-notificacao-de-2007-em-diante-sinan/, accessed on 02nd July 2024). This system maintains a repository of patient notifications encompassing diseases, injuries and public health incidents listed as nationally mandatory for reporting. Notably, this includes more than 40 diseases (dengue fever, chikungunya fever, pandemic influenza, etc). The data is composed of notifications related to dengue fever cases documented within the Federal District, Brazil, spanning all 49 epidemiological weeks (observations) throughout the year 2022: 689, 1205, 938, 1121, 1523, 1469, 1508, 1999, 2468, 2827, 3196, 3651, 3550, 4142, 4118, 4178, 3853, 2700, 6726, 2183, 2581, 1616, 1126, 898, 548, 752, 622, 415, 309, 291, 396, 476, 411, 360, 500, 402, 418, 313, 385, 475, 406, 277, 323, 433, 505, 574, 465, 1.682.

The focus of the study is on the variables below:

• $y_i$: total dengue fever cases of a epidemiological week (DG) (response variable);
• $m_{ij}$: month (levels: 0 - January to 11 - December). Thus, for $i=1,\dots ,49$ and $j=0,\dots ,11$, dummy variables.

The proposed model provides both advantages and disadvantages over counting models. The exponential distribution used as a baseline has several important characteristics, including applicability in some epidemiological contexts, memorylessness, a good fit with real data with heavy tails, flexibility and a simple density form. On the other hand, a variety of concerns might arise, including a lack of flexibility for trend modeling, violations of the independence assumption, which can result in inaccurate models, and restrictions with inflated zero data, among others. Nonetheless, the model captures the significance of the exploratory variables as well as an extreme event involving dengue fever patients in the temporal scenario.

Table 5 presents descriptive statistics. The number of dengue fever cases varied from very low (277) to high (6726). The standard deviation of 1445,35 indicates increased variability in dengue fever cases over time. The distribution is skewed to the right (1.509), indicating that there are more severe values near the top of the scale and the kurtosis indicates heavier tails (4.998).

Figure 8 shows the histogram and time series of the data. Figure 8(a) demonstrated heavy tail behavior, consistent with extreme event data. Figure 8(b) shows unusual event behavior between May and June. This is a record compared to the same period since 1998 (https://www.correiobraziliense.com.br/cidades-df/2022/06/5017446-casos-atingem-maior-numero-desde-1998.html, accessed on 02nd July 2024), demonstrating the atypical behavior of the observations, which deviate significantly from the historical average, indicating an unusual outbreak, or, in epidemiology, an extreme event of dengue fever that can have an impact on both the health system and the economy. In addition, the plot shows an increase in tendency between February and June, when the disease is most likely to develop in the Federal District and a decrease in tendency between July and December, which is the drought period.

Brazil is a continental country with different patterns, however the Midwest has the highest incidence of dengue fever, according to the arbovirus monitoring panel of the Ministry of Health (https://www.gov.br/saude/pt-br/assuntos/saude-de-a-a-z/a/aedes-aegypti/monitoramento-das-arboviroses, accessed on 02nd July 2024), which is the region of the study data.

6. Results

6.1. Findings from GOLLE Distribution

The analysis of time series data requires a detailed inspection, which includes identifying relationships between the observations. Failure to account for these associations may result in an inadequate model that neglects temporal relationships, potentially leading to incorrect forecasts and interpretations. To identify serial correlation, it is essential to analyze the autocorrelation function (ACF) and partial autocorrelation function (PACF) plots. Figure 9 shows ACF and PACF for an autoregressive integrated moving average model with one-lag in the differenced series ARIMA(1,1,0).

Despite the fact that the data set demonstrates a correlation, i.e., dependence, the model can be employed if this assumption is relaxed given the small number of observations and the employing of diagnostic and residual analyses.

Table 6 presents the results of the fitted distributions to the current data and shows that the GOLLE distribution is the best fit. Figure 10(a) and Figure 10(b) show histograms and plots of estimated density functions, as well as empirical cdf estimated ones. The Fréchet distribution density is commonly used to model extreme occurrences and these findings suggest that the KwFr distribution (the second-best model) is competitive with the presented model.

LR tests were used to compare the GOLLE distribution and its nested models. Table 7 shows that adding more parameters has a significant influence on accurately modeling the existing data.

6.2. Findings from LGOLLE Regression Model

To assess the LGOLLE model, Figure 11 shows the ACF and PACF plots for a model with one-lag in the differenced series ARIMA(0,1,0), which indicates a cumulative sum of an i.i.d. process, which itself is known as ARIMA(0,0,0) (random walk). Therefore, the proposed regression model can be employed in the current data set.

A histogram and time series of the log data are displayed in Figure 12. Figure 12(a) showed a behavior that was bimodal. Extreme occurrences between May and June and a trend behavior during the time scenario are shown in Figure 12(b).

The LGOLLE distribution is the best fit, as demonstrated by Table 8, which displays the results of the fitted nested distributions to the log data. In addition to histograms and the estimated density functions in Figure 13(a), Figure 13(b) displays the empirical cdf estimated ones.

LR tests are employed to compare the LGOLLE distribution’s nested models. Table 9 shows that adding extra parameters has a considerable influence on modeling performance with the given data.

The new LGOLLE regression model for the dengue fever cases data in Federal District, Brazil can be expressed as follows

$$y_i={\mu }_i+z_i,$$

where $z_1,z_2,\dots ,z_{49}$ are independent rv variables with density function (10) and the systematic structure is defined by

$${\mu }_i={\beta }_0+\sum _{j=1}^{11}{\beta }_j{m}_j.$$

The results (MLEs, SEs, and p-values) of the fitted LGOLLE regression model are presented in Table 10. Therefore, using a significance level of $5\%$, the months of the temporal scenario are significant and can be utilized to model the location.

Figure 14 presents two potentially significant observations based on the LD and GCD measures. It’s worth noting that the 20th and 49th observations, respectively, correspond to the beginning of June and the last epidemiological week. The first matches to the record (https://www.metropoles.com/distrito-federal/boletim-revela-aumento-dos-casos-de-dengue-em-todas-as-regioes-do-df, accessed on 02nd July 2024) of dengue fever cases in the Federal District. The second relates to the end-of-year vacation/recess period, which results in a backlog of alerts due to a lack of health care professionals available to notify cases and input data into the system.

Nonetheless, Figure 15 indicates that the index deviation residuals behave randomly over the range and are within the simulated envelope, implying that the observations have a minimal effect on the regression model.

6.3. Discussion

The findings indicate that the LGOLLE regression model is suitable for explaining the weekly dengue fever cases in the Federal District. Table 10 presents parameter estimates for the LGOLLE regression model, which becomes (for $i=1,\dots ,49$)

$$\begin{array}{cc}\hfill {\widehat{\mu }}_i& =15.7495+0.5511m_{i1}+1.0827m_{i2}+1.6788m_{i3}+1.7125m_{i4}\hfill \\ \hfill & +1.4393m_{i5}-0.5881m_{i7}-0.4623m_{i8}-0.5778m_{i9}-0.5749m_{i10}.\hfill \end{array}$$

(17)

The following discussion examines the systematic structure, using January as the month of reference.

Interpretations for $\widehat{\mu }$

• Except for the covariates $m_6$ and $m_{11}$, referring to the months July and December, all other covariates are significant at a $5\%$ level of significance. This indicates that there is a difference in dengue fever cases registered in the Federal District between the other months and January. The months of July and December are probably not significant due to their similar behavior to the reference month;
• The months of February to June have positive estimates, which is significant, showing an increase in comparison to January. This may be seen in Figure 12(b), which shows an extreme event (between May and June) in that window data scenario;
• August to November have negative values, indicating a decline in dengue fever cases compared to January. During this period, the Federal District experiences a drought that corroborates the study’s findings (https://portal.inmet.gov.br/uploads/notastecnicas/Estado-do-clima-no-Brasil-em-2022-OFICIAL.pdf, accessed on 02nd July 2024).

7. Conclusions

The paper defined the generalized odd log-logistic exponential distribution (see [11] and [12]) and introduced a new bimodal regression model based on the GOLLE distribution with a location-systematic structure to investigate weekly dengue fever cases in the Federal District in 2022. Some mathematical properties are reviewed, the parameters are estimated by the maximum likelihood method and the consistency criterion is evaluated by means of Monte Carlo simulations. The consistency of the MLEs of the regression model is evaluated by means of simulations using some measures. Some global influence measures and residual analysis are addressed to investigate the fit of the new model.

Some important findings are addressed. Except for the July and December months, the remaining months are significant at a $5\%$ significance level. February to June exhibit positive estimates, suggesting a positive impact on dengue fever cases during this period. This is corroborated by the time scenario of the climatological effects that increase the cases in these months. August to November experience a drought, supporting the negative estimates during this period, which have a negative effect on weekly dengue fever cases.

The epidemiology data set showed that the novel regression model is more flexible than other nested and competitively well-known models. As a result, the proposed model enhances comprehension of dengue fever cases in the Federal District, as well as an extreme event that occurred during the study period.