key: cord-0450094-bj7zcozl authors: Alanazi, Fadhah Amer title: The spread of COVID-19 at Hot-Temperature Places With Different Curfew Situations Using Copula Models date: 2021-02-20 journal: nan DOI: nan sha: 0fb5e3c8702a7c7ff7dd1847193ffda8dafffbf9 doc_id: 450094 cord_uid: bj7zcozl The infectious coronavirus disease 2019 (COVID-19) has become a serious global pandemic. Different studies have shown that increasing temperature can play a crucial role in the spread of the virus. Most of these studies were limited to winter or moderate temperature levels and were conducted using conventional models. However, traditional models are too simplistic to investigate complex, non-linear relationships and suffer from some restrictions. Therefore, we employed copula models to examine the impact of high temperatures on virus transmission. The findings from the copula models showed that there was a weak to moderate effect of temperature on the number of infections and the effect almost vanished under a lockdown policy. Therefore, this study provides new insight into the relationship between COVID-19 and temperature, both with and without social isolation practices. Such results can lead to improvements in our understanding of this new virus. In particular, the results derived from the copula models examined here, unlike existing traditional models, provide evidence that there is no substantial influence of high temperatures on the active COVID-19 outbreak situation. In addition, the results indicate that the transmission of COVID-19 is strongly influenced by social isolation practices. To the best of the author knowledge, this is the first copula model investigation applied to the COVID-19 pandemic. In December 2019, a novel infectious disease termed coronavirus disease 2019 (COVID-19) was discovered in Wuhan city, Hubei province, China. Subsequently, and through human-to human transmission, this virus has caused a global pandemic. COVID-19 is characterized by clinical features similar to those caused by severe acute respiratory syndrome coronavirus (SARS-CoV) and Middle Eastern respiratory syndrome coronavirus (MERS-CoV) infections, such as a fever and dry cough [1] . Previous studies have shown that meteorological variables can affect the transmission and survival of coronaviruses [2] , [3] . Earlier research [3] found that MERS-CoV is most active at high temperatures and low humidity. Notably, recent studies have shown that warm weather and high humidity may be important factors for reducing the spread of COVID-19 (e.g., see [4] ). Conversely, some existing studies have found that increasing temperatures will not affect the transmission of COVID-19 (e.g., see [5] ). However, most of these studies were limited to winter or low-temperature weather with a small number of observations. Hence, there is still no definitive evidence as to whether there is a negative association between environmental variables and the spread of COVID-19 in extremely hot or cold locations [4] . Besides, most previous studies were performed using traditional models, which are too simplistic and may be unable to deal with complex, non-linear dependency patterns. Thus, further research to understand the activity of COVID-19 under high-temperature conditions is warranted. In addition, such an association should be investigated not only in regard to weather variables, but also by taking into account the lockdown situation at these locations. Presently, copula models have become a favored statistical tool to describe the association between variables. These models have been applied in different areas, including the study of infectious diseases (e.g., see [6] ) and environmental science (e.g., see [7] ). One important benefit of using a copula model is that one can model the marginal distribution independently from the dependency structures, which are completely captured via the copula function. Another benefit of using a copula model is that the margins do not need to follow the same parametric family. Furthermore, many copula families exist, each with its own capability to describe the unique dependency structure. Hence, various types of associations can be discovered via copula models. Hence, this study aimed to perform flexible statistical modeling with a copula model to improve our knowledge about the spread of the virus in hot locations with different curfew levels. Specifically, we investigated the impact of high temperatures on the number of confirmed cases in the cities of Riyadh, Jeddah, and Mecca in Saudi Arabia, and these cities were selected for several reasons. First, Saudi Arabia has been strongly affected by MERS-CoV [8] , [9] , which produces a similar severe respiratory illness as COVID-19. Second, the highest numbers of confirmed cases in Saudi Arabia have been recorded in Riyadh, Jeddah, and Mecca, which are three of the hottest areas in Saudi Arabia. Third, because of the transmission of COVID-19, Mecca and Jeddah have been placed under a series of lockdowns for a long time. Riyadh, however, was only placed under a curfew for a short period. Hence, these cities represent strong to moderate lockdown situations, which could be a factor critical to understanding the effects of high temperatures on the spread of COVID-19. By using the data from these cities and capitalizing on the flexibility of copula models, we aimed to provide clear evidence on the association between high temperatures and confirmed cases of COVID-19. For this study, the cities of Riyadh, Mecca, and Jeddah were selected for the analysis. Riyadh is the capital city of Saudi Arabia and the city most affected by COVID-19 in this country. Riyadh had 37, 244 confirmed cases for the observed period from 13 March 2020 to 15 June 2020. Both the COVID-19 confirmed cases and the average temperature data demonstrated a non-normal distribution for all cities. Fig 1 shows the confirmed cases during the study period, where the new confirmed cases of COVID-19 in Riyadh exceeded 1500 from 08 to 15 June 2020. However, the highest records for Mecca and Jeddah were generally similar and lower than those of Riyadh. Copula is a Latin word that means joins or links. A copula function refers to a multivariate function that joins the multivariate distribution functions to their univariate standard uniform margins [14] . Formally, a copula can be defined as follows: copulas [15] are multivariate cumulative distribution functions with uniform marginal distributions on (0,1) such that: Sklar's theorem [16] is the key rule of the copula function, and it can be introduced as follows: Theorem 0.1 (Sklar's theorem): If F is an n-variate distribution function with univariate margins F 1 , F 2 , ....., F n , then there exists an n-variate copula function, C, such that ∀ x = (x 1 , .., x n ) ∈ R n : If the margins are continuous, then the copula is the inverse function of the margins and u ∈ [0, 1] n . Conversely, if F 1 , ..., F n are the marginal distribution functions and C is a copula function, then the function F (defined by equation (2)) is a joint distribution function with margins F 1 , ..., F n . In accordance with Sklar's theorem (2), a copula models the marginal distributions separately from the dependency pattern, with no restriction on the type of margins. In this study, we consider an arbitrary number of copula types including the Joe, Gumbel, and Clayton copulas, as well as their rotation types. In addition, we consider the Frank, Gaussian, t-students, and other two-parametric copulas, such as the Joe-Frank (BB8) copula. The following text provides details on some commonly used copula families. • Frank copula is a one-parametric symmetric Archimedean copula with generator function The Frank copula can control both the negative and positive dependency pattern, where the strongest dependency occurs at the center of the distribution. However, in the Frank copula, the extremes are independent. The distribution function of the Frank copula can be given by: and its density function is: . (5) • Clayton copula is a one-parametric (θ > 0) nonsymmetric Archimedean copula. It is a lower positive tail dependence copula with generator ϕ(t) = 1 θ (t −θ − 1). Its distribution is given by: and its density function is: (7) • Joe copula, in contrast to the Clayton copula, this is a one-parametric upper tail Archimedean copula with generator ϕ(t) = ln[1 − (1 − t) θ ]. Its distribution function is: and its density function is: • Rotated copula refers to a rotation version of asymmetric copulas. This rotation includes 90, 180, and 270 rotation degrees, with arguments (1 − u 1 , u 2 ), (u 1 , 1 − u 2 ), and (1−u 1 , 1−u 2 ), respectively. The 180 rotation degree produces a corresponding survival copula family. However, rotations by 90 and 270 degrees provide corresponding copulas to deal with negative dependencies. For more details on rotated copulas, see for example, [17] , [18] , [19] , and [20] . Pseudo maximum-likelihood method: In this study, we applied the so-called pseudo maximum-likelihood method (PML) to estimate the parameters for the selected copula function. PML is introduced by [21] as a two-step estimation method. With this method, the margins are estimated nonparametrically via their empirical cumulative distribution function at first, and then, the copula parameter (θ c ) is estimated at the second step. By using PML, the copula parameter is estimated by maximizing the copula density, i.e., where u 1 =F 1 (x 1 ; α 1 ) and u 2 =F 2 (x 2 ; α 2 ) are the empirical probability integral transform of variable X 1 and X 2 , respectively. A simulation study of [22] showed that the performance of PML is better than that of the full maximum likelihood estimation method and Inference Function of Margins of [23] if the margins are unknown, which is the case in almost all real life applications. As there is a wide range of copula functions, it is necessary to test the copula shape with the best fit. Therefore, we will use the Akaike Information Criterion (AIC) of [24] and the Bayesian Information Criterion (BIC) of [25] to select the right copula. AIC and BIC can be given by: whereθ is the estimated value of the parameters, and P is the number of the model parameters. The summary of the full inference steps of copula models used in this study is as follows: • Transform the continuous variable of the observed data to copula data. • Calculate the cumulative density function for the discrete variable of the observed data. • Consider arbitrary types of bivariate copula functions for the assumed model. • Select the best fit bivariate copula type among all fitted copula functions using AIC and BIC. Table I shows the summary statistics for the daily data on temperature and COVID-19 confirmed cases in the cities of Riyadh, Mecca, and Jeddah. With average values of 29.788°C (Riyadh), 24°C (Mecca), and 29.014°C (Jeddah), the temperature in the three cities was very high. Importantly, this study takes into account the daily average temperature, and not the maximum temperature. This study used a copula model to investigate the relationship between high temperatures and confirmed cases of COVID-19 for three cities in Saudi Arabia. The existing number of copulas is large, and to select the most appropriate fitted model for each city, bicop() function of R ( [26] ) package [27] was used. As bicop() allows one to consider different selection criteria at each run, it was applied to each data set twice, one with BIC and the other one with AIC. The results of these models are provided in Tables (II, III) . Boldfont indicates the selected copula type. Figs (2, 3) present the surface and contour plots for the selected copula families. In accordance with Table (II) , Frank copulas with a moderate positive dependency (θ = 8.51, τ = 0.62) and ((θ = 7.47, τ = 0.57) were selected for Riyadh and Jeddah, respectively. The results indicate that there is a positive relationship between temperature and the spread of COVID-19 in moderate and high temperatures. However, these two This study examined the effect of high temperatures on the spread of COVID-19 in hot climates under different curfew situations using copula models. We applied the models to the cities of Riyadh, Jeddah, and Mecca in Saudi Arabia. For Riyadh and Jeddah, which had almost the same average temperature level, the association between temperature and confirmed cases of COVID-19 reflected a moderate positive Frank copula. However, the number of COVID-19 cases in Riyadh was higher than the number in Jeddah. Hence, the transmission of this virus in these two cities may have been affected by the curfew level and not by the high temperature. In the case of Mecca, which had a temperature level (slightly) less than that of Riyadh and Jeddah, there was a very weak dependency between temperature and the number of COVID-19 cases. However, the number of confirmed cases of COVID-19 in Mecca was very close to the number in Jeddah. In addition, Mecca was under a strong 24 hour lockdown for more than half of the observed data set. Therefore, there is clear evidence that high temperatures are not able to stop the spread of this virus if there is no social isolation. Clearly, lockdowns represent the most effective strategy to prevent the spread of this virus. To the best of our knowledge, this study describes the first copula model fitted to COVID-19 data. The results of this study, derived using copula models, are unlike those derived using existing traditional methods and indicate that the association between COVID-19 and temperature is weak and no substantial decreases in the number of COVID-19 cases can be expected in response to high temperatures. Clinical features of patients infected with 2019 novel coronavirus in Wuhan The effects of temperature and relative humidity on the viability of the SARS coronavirus The pattern of Middle East respiratory syndrome coronavirus in Saudi Arabia: a descriptive epidemiological analysis of data from the Saudi Ministry of Health High Temperature and High Humidity Reduce the Transmission of COVID-19 Association between ambient temperature and COVID-19 infection in 122 cities from China Infectious disease prediction with kernel conditional density estimation Development in copula applications in forestry and environmental sciences High prevalence of MERS-CoV infection in camel workers in Saudi Arabia Stability of Middle East respiratory syndrome coronavirus (MERS-CoV) under different environmental conditions An Introduction to Copulas Probabilistic metric spaces Fonctions de répartitioná n dimensions et leurs marges. Publications de lâC™Institut Statistique de lâC™Université de Paris Copula-based top-down approaches in financial risk aggregation Modeling dependence between error components of the stochastic frontier model using copula: application to intercrop coffee production in Northern Thailand Stepwise estimation of D-Vines with arbitrary specified copula pairs and EDA Tools. Diploma thesis Selecting and estimating regular vine copulae and application to financial returns A Semiparametric estimation procedure of dependence parameters in multivariate families of distributions Comparison of semiparametric and parametric methods for estimating copulas Multivariate models and dependence concepts Information theory and an extension of the maximum likelihood principle Estimating the dimension of a model R: A language and environment for statistical computing High performance algorithms for vine copula Modeling