The spread of COVID-19 at Hot-Temperature Places With Different Curfew Situations Using Copula Models
TThe spread of COVID-19 at Hot-TemperaturePlaces With Different Curfew Situations UsingCopula Models
Fadhah Alanazi
Deanship of Educational Services,Department of General Sciences,Prince Sultan University,Riyadh, Saudi ArabiaEmail:[email protected]
BSTRACT
The infectious coronavirus disease 2019 (COVID-19) hasbecome a serious global pandemic. Different studies haveshown that increasing temperature can play a crucial role inthe spread of the virus. Most of these studies were limited towinter or moderate temperature levels and were conducted us-ing conventional models. However, traditional models are toosimplistic to investigate complex, non-linear relationships andsuffer from some restrictions. Therefore, we employed copulamodels to examine the impact of high temperatures on virustransmission. The findings from the copula models showedthat there was a weak to moderate effect of temperature onthe number of infections and the effect almost vanished undera lockdown policy. Therefore, this study provides new insightinto the relationship between COVID-19 and temperature, bothwith and without social isolation practices. Such results canlead to improvements in our understanding of this new virus.In particular, the results derived from the copula models exam-ined here, unlike existing traditional models, provide evidencethat there is no substantial influence of high temperatureson the active COVID-19 outbreak situation. In addition, theresults indicate that the transmission of COVID-19 is stronglyinfluenced by social isolation practices. To the best of theauthors’ knowledge, this is the first copula model investigationapplied to the COVID-19 pandemic.I
NTRODUCTION
In December 2019, a novel infectious disease termedcoronavirus disease 2019 (COVID-19) was discovered inWuhan city, Hubei province, China. Subsequently, andthrough human-to human transmission, this virus has causeda global pandemic. COVID-19 is characterized by clinicalfeatures similar to those caused by severe acute respiratorysyndrome coronavirus (SARS-CoV) and Middle Easternrespiratory syndrome coronavirus (MERS-CoV) infections,such as a fever and dry cough [1].Previous studies have shown that meteorological variablescan affect the transmission and survival of coronaviruses [2], [3]. Earlier research [3] found that MERS-CoV is most activeat high temperatures and low humidity.Notably, recent studies have shown that warm weatherand high humidity may be important factors for reducing thespread of COVID-19 (e.g., see [4]). Conversely, some existingstudies have found that increasing temperatures will not affectthe transmission of COVID-19 (e.g., see [5]). However, mostof these studies were limited to winter or low-temperatureweather with a small number of observations. Hence, there isstill no definitive evidence as to whether there is a negativeassociation between environmental variables and the spreadof COVID-19 in extremely hot or cold locations [4]. Besides,most previous studies were performed using traditionalmodels, which are too simplistic and may be unable todeal with complex, non-linear dependency patterns. Thus,further research to understand the activity of COVID-19under high-temperature conditions is warranted. In addition,such an association should be investigated not only in regardto weather variables, but also by taking into account thelockdown situation at these locations. Presently, copulamodels have become a favored statistical tool to describethe association between variables. These models have beenapplied in different areas, including the study of infectiousdiseases (e.g., see [6]) and environmental science (e.g., see[7]). One important benefit of using a copula model is thatone can model the marginal distribution independently fromthe dependency structures, which are completely captured viathe copula function. Another benefit of using a copula modelis that the margins do not need to follow the same parametricfamily. Furthermore, many copula families exist, each with itsown capability to describe the unique dependency structure.Hence, various types of associations can be discovered viacopula models.Hence, this study aimed to perform flexible statisticalmodeling with a copula model to improve our knowledgeabout the spread of the virus in hot locations with differentcurfew levels. Specifically, we investigated the impact of high a r X i v : . [ q - b i o . P E ] F e b emperatures on the number of confirmed cases in the citiesof Riyadh, Jeddah, and Mecca in Saudi Arabia, and thesecities were selected for several reasons. First, Saudi Arabiahas been strongly affected by MERS-CoV [8], [9], whichproduces a similar severe respiratory illness as COVID-19.Second, the highest numbers of confirmed cases in SaudiArabia 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 Jeddahhave been placed under a series of lockdowns for a longtime. Riyadh, however, was only placed under a curfewfor a short period. Hence, these cities represent strong tomoderate lockdown situations, which could be a factor criticalto understanding the effects of high temperatures on thespread of COVID-19. By using the data from these cities andcapitalizing on the flexibility of copula models, we aimedto provide clear evidence on the association between hightemperatures and confirmed cases of COVID-19.M ATERIALS AND METHODS
Data collection
For this study, the cities of Riyadh, Mecca, and Jeddahwere selected for the analysis. Riyadh is the capital cityof Saudi Arabia and the city most affected by COVID-19in this country. Riyadh had , confirmed cases for theobserved period from March to June . Thepopulation of Riyadh estimated for the middle of 2018 basedon demographic survey data collected in 2016 was , , [10]. Jedda and Mecca are the second most affected citiesin Saudi Arabia, with , and , confirmed cases,respectively.Daily counts of confirmed cases for the study period werecollected from official reports [11] based on information[12] for Saudi Arabia. The daily average temperature datafor the same period time were obtained from the WeatherUnderground Company [13]. Data analysis
Both the COVID-19 confirmed cases and the average tem-perature data demonstrated a non-normal distribution for allcities. Fig 1 shows the confirmed cases during the studyperiod, where the new confirmed cases of COVID-19 inRiyadh exceeded from to June . However, thehighest records for Mecca and Jeddah were generally similarand lower than those of Riyadh.
Copula
Copula is a Latin word that means joins or links. Acopula function refers to a multivariate function that joins themultivariate distribution functions to their univariate standarduniform margins [14]. Formally, a copula can be defined asfollows: copulas [15] are multivariate cumulative distributionfunctions with uniform marginal distributions on (0,1) suchthat: C : [0 , n → [0 , , n ≥ . (1) Date C a s e s ( R i y adh ) Mar 13 Mar 18 Mar 23 Mar 28 Apr 02 Apr 07 Apr 12 Apr 17 Apr 22 Apr 27 May 02 May 07 May 12 May 17 May 22 May 27 Jun 01 Jun 06 Jun 11Mar 13 Mar 18 Mar 23 Mar 28 Apr 02 Apr 07 Apr 12 Apr 17 Apr 22 Apr 27 May 02 May 07 May 12 May 17 May 22 May 27 Jun 01 Jun 06 Jun 11
Date C a s e s ( M e cc a ) Mar 13 Mar 18 Mar 23 Mar 28 Apr 02 Apr 07 Apr 12 Apr 17 Apr 22 Apr 27 May 02 May 07 May 12 May 17 May 22 May 27 Jun 01 Jun 06 Jun 11
Date C a s e s ( J eddah ) Mar 13 Mar 18 Mar 23 Mar 28 Apr 02 Apr 07 Apr 12 Apr 17 Apr 22 Apr 27 May 02 May 07 May 12 May 17 May 22 May 27 Jun 01 Jun 06 Jun 11
Fig. 1.
Plots of the confirmed cases during the study period (from March to June . (top): Riyadh, (middle): Mecca, (bottom)Jeddah. 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 distri-bution function with univariate margins F , F , ....., F n , thenthere exists an n -variate copula function, C , such that ∀ x = ( x , .., x n ) (cid:48) ∈ R n : F ( x , x , ...., x n ) = C ( F ( x ) , F ( x ) , ...., F n ( x n )) . (2)f the margins are continuous, then the copula C ( u , u , ...., u n ) = F ( F − ( u ) , F − ( u ) , ..., F − n ( u n )) (3)is unique, where F − is the inverse function of the marginsand u ∈ [0 , n . Conversely, if F , ..., F n are the marginaldistribution functions and C is a copula function, then thefunction F (defined by equation (2)) is a joint distributionfunction with margins F , ..., F n .In accordance with Sklar’s theorem (2), a copula modelsthe marginal distributions separately from the dependencypattern, with no restriction on the type of margins.In this study, we consider an arbitrary number of copula typesincluding the Joe, Gumbel, and Clayton copulas, as wellas their rotation types. In addition, we consider the Frank,Gaussian, t-students, and other two-parametric copulas, suchas the Joe-Frank (BB8) copula. The following text providesdetails on some commonly used copula families. • Frank copula is a one-parametric symmetricArchimedean copula with generator function ϕ ( t ) = − ln[ e − θt − e − θ − ] , with θ ∈ ( −∞ , ∞ ) \ { } . TheFrank copula can control both the negative and positivedependency pattern, where the strongest dependencyoccurs at the center of the distribution. However, in theFrank copula, the extremes are independent.The distribution function of the Frank copula canbe given by: C θ ( u , u ) = − θ ln [1+ ( e − θu − e − θu − e − θ − , (4)and its density function is: c ( u , u ) = θ ( e − θ − e − θ ( u + u ) e − θ − e − θu − e − θu − . (5) • Clayton copula is a one-parametric ( θ > ) non-symmetric Archimedean copula. It is a lower positive taildependence copula with generator ϕ ( t ) = θ ( t − θ − . Itsdistribution is given by: C ( u , u ) = [ u − θ + u − θ − − θ , (6)and its density function is: c ( u , u ) = (1 + θ )( u u ) − − θ ( u − θ + u − θ − − θ − . (7) • Joe copula , in contrast to the Clayton copula, this isa one-parametric upper tail Archimedean copula withgenerator ϕ ( t ) = ln[1 − (1 − t ) θ ] . Its distribution functionis: C ( u , u ) = 1 − [(1 − u ) θ +(1 − u ) θ − (1 − u ) θ (1 − u ) θ ] θ , (8) and its density function is: c ( u , u ) = [(1 − u ) θ + (1 − u ) θ − (1 − u ) θ (1 − u ) θ ] θ − × (1 − u ) θ − (1 − u ) θ − [ θ − − u ) θ +(1 − u ) θ − (1 − u ) θ (1 − u ) θ ] . (9) • Rotated copula refers to a rotation version of asymmetriccopulas. This rotation includes , , and rotationdegrees, with arguments (1 − u , u ) , ( u , − u ) , and (1 − u , − u ) , respectively. The rotation degree pro-duces a corresponding survival copula family. However,rotations by and degrees provide correspondingcopulas to deal with negative dependencies. For moredetails on rotated copulas, see for example, [17], [18],[19], and [20]. Pseudo maximum-likelihood method:
In this study, we ap-plied the so-called pseudo maximum-likelihood method ( PML )to estimate the parameters for the selected copula func-tion.
PML is introduced by [21] as a two-step estimationmethod. With this method, the margins are estimated non-parametrically via their empirical cumulative distribution func-tion at first, and then, the copula parameter ( θ c ) is estimatedat the second step. By using PML , the copula parameter isestimated by maximizing the copula density, i.e., L MPL ( θ c ) = n (cid:88) i =1 log[ c ( u i , u i ; θ c )] , (10)where u = ˆ F ( x ; α ) and u = ˆ F ( x ; α ) are the empiricalprobability integral transform of variable X and X , respec-tively. A simulation study of [22] showed that the performanceof PML is better than that of the full maximum likelihoodestimation method and
Inference Function of Margins of [23]if the margins are unknown, which is the case in almost allreal life applications.
A. Goodness-of-fit test
As there is a wide range of copula functions, it is necessaryto test the copula shape with the best fit. Therefore, we willuse the Akaike Information Criterion (
AIC ) of [24] and theBayesian Information Criterion (
BIC ) of [25] to select theright copula.
AIC and
BIC can be given by:
AIC = − ln L (ˆ θ ) + 2 P , (11) BIC = − ln L (ˆ θ ) + P ( ln (N)) , (12)where ˆ θ is the estimated value of the parameters, and P isthe number of the model parameters.The summary of the full inference steps of copula modelsused in this study is as follows: • Transform the continuous variable of the observed datato copula data. • Calculate the cumulative density function for the discretevariable of the observed data.
Consider arbitrary types of bivariate copula functions forthe assumed model. • Select the best fit bivariate copula type among all fittedcopula functions using
AIC and
BIC .R ESULTS AND DISCUSSION
Descriptive results
Table I shows the summary statistics for the daily data ontemperature and COVID-19 confirmed cases in the cities ofRiyadh, Mecca, and Jeddah. With average values of . °C(Riyadh), °C (Mecca), and . °C (Jeddah), the temper-ature in the three cities was very high. Importantly, this studytakes into account the daily average temperature, and not themaximum temperature. TABLE I
Summary statistics for the daily average temperature and COVID-19confirmed cases in the cities of Riyadh, Mecca, and Jeddah. Zerosvalues indicate no new confirmed cases for the corresponding dateduring the observation period.
Variable N Mean Standard deviation Min MaxTemperature (Riyadh) (°C)
95 29 .
788 4 .
947 18 37 . Temperature (Mecca) (°C) 95 24.983 3.472 18 32Temperature (Jeddah) (°C) 95 29.014 3.258 22.9 36Confirm cases (Riyadh) 95 392.042 427.393 2 1,735Confirm cases (Mecca) 95 213.137 160.938 0 623Confirm cases (Jeddah) 95 222.653 174.632 0 586
Discussion of the Copula model
This study used a copula model to investigate therelationship between high temperatures and confirmed casesof COVID-19 for three cities in Saudi Arabia. The existingnumber of copulas is large, and to select the most appropriatefitted model for each city, bicop() function of R ( [26])package [27] was used. As bicop() allows one to considerdifferent selection criteria at each run, it was applied to eachdata 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) presentthe surface and contour plots for the selected copula families.
TABLE II
Selected copula families (for each city) based on BIC by bicop()
Type Parameter ( τ ) AIC BICRiyadhFrank θ = 8 . ( τ = 0 . ) − . − . MeccaSurvival Joe (rotation degree ) θ = 1 . ( τ = 0 . ) − . − . JeddahFrank θ = 7 . ( τ = 0 . ) − . − . Surface plot of Frank copula ( q = , t = ) (Jeddah city) u u Surface plot of Frank copula ( q = , t = ) (Riyadh city) u u Surface plot of Survival Joe copula ( q = , t = ) (Mecca city) u u Fig. 2.
Surface plots of the Frank copulas for Riyadh (top) and Jeddah(middle), and the Clayton copula for Mecca (Bottom) as the best-fit copulafamilies for each city.
In accordance with Table (II), Frank copulas with amoderate positive dependency ( θ = 8 . , τ = 0 . ) and(( θ = 7 . , τ = 0 . ) were selected for Riyadh and Jeddah,respectively. The results indicate that there is a positiverelationship between temperature and the spread of COVID-19 in moderate and high temperatures. However, these two ontour plot of Frank copula ( q = , t = ) (Jeddah city) z z −3 −2 −1 0 1 2 3 − − − Contour plot of Frank copula ( q = , t = ) (Riyadh city) z z −3 −2 −1 0 1 2 3 − − − Contour plot of Survial Joe copula ( q = , t = ) (Mecca city) z z −3 −2 −1 0 1 2 3 − − − Fig. 3.
Contour plots of the Frank copulas for Riyadh (top) and Jeddah(middle), and the Clayton copula for Mecca (Bottom) as the best-fit copulafamilies for each city.
TABLE III
Selected copula families (for each city) based on AIC by bicop() . Type Parameter ( τ ) AIC BICRiyadhBB8 (Joe-Frank) θ = 8 ( α = 0 . ) − . − . Meccatll . df − . − . JeddahFrank θ = 7 . ( τ = 0 . ) − . − . variables became independent at extreme values. Therefore,the results provide clear evidence that SARS-CoV-2 canstill remain an active virus in hot places. In the case ofMecca, the survival Joe copula with a low dependency level( θ = 1 . , τ = 0 . ) was selected as the most appropriatecopula function. With these data, there was only a very weakdependency pattern between high temperatures and confirmedcases at the low values. This relationship reflects the period before the series of lockdowns in the city of Mecca. Duringthe curfew, there was no relationship detected between thespread of COVID-19 and the temperature.The question that now remains is, do high temperatures affectthe transmission of COVID-19? To answer this question,we need to mention some of the main similarities anddifferences among these cities regarding the (1) temperature,(2) confirmed cases of COVID-19, (3) copula model results,and (4) lockdown situations. First, the temperatures in Riyadhand Jeddah were almost the same and slightly higher thanthose in Mecca. However, the number of confirmed cases inRiyadh was higher than that in Jeddah and Mecca. In addition,Mecca and Jeddah had almost the same number of confirmedcases. Hence, the same temperature levels were associatedwith different numbers of new confirmed cases within thethree cities. Thus, COVID-19 can spread differently at thesame temperature level. Second, the fitted copula modelsshow that the dependency between the temperature andnumber of COVID-19 cases was very similar for Riyadh andJeddah, while it was very low for Mecca. Given the first andsecond points mentioned above, there was another importantfactor driving the active situation of this new virus, namely,the lockdown situation. During the study period, the citiesof Jeddah and Mecca were placed under a lockdown on March and March , respectively. Then, Meccawas subjected to a hour curfew on April . Later,on April , various areas in Jeddah were placed undera hour curfew. After about one day, Riyadh too wasplaced under a hour curfew for approximately days.Then, on April , Jeddah and Riyadh were placedunder partial curfews, while Mecca still remained under a lockdown. Hence, Mecca experienced a long lockdownperiod, while duration of curfew in Riyadh was the shortest.These factors may explain the similarity in copula results forRiyadh and Jeddah and the low dependency pattern in thecase of Mecca. In consideration of these last findings, we canconclude that high temperatures had only a weak to moderateeffect on the transmission of COVID-19 if there was a partialcurfew policy in place. However, this effect vanished underthe condition of strong social isolation. Hence, even in hotplaces, COVID-19 can still spread readily when no socialdistancing is implemented.C ONCLUSION
This study examined the effect of high temperatures on thespread of COVID-19 in hot climates under different curfewsituations using copula models. We applied the models tothe cities of Riyadh, Jeddah, and Mecca in Saudi Arabia.For Riyadh and Jeddah, which had almost the same averagetemperature level, the association between temperature andconfirmed cases of COVID-19 reflected a moderate positiveFrank copula. However, the number of COVID-19 cases inRiyadh was higher than the number in Jeddah. Hence, thetransmission of this virus in these two cities may have beenaffected by the curfew level and not by the high temperature.n the case of Mecca, which had a temperature level (slightly)less than that of Riyadh and Jeddah, there was a very weakdependency 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. Inaddition, Mecca was under a strong hour lockdown formore than half of the observed data set. Therefore, there isclear evidence that high temperatures are not able to stop thespread of this virus if there is no social isolation. Clearly,lockdowns represent the most effective strategy to prevent thespread of this virus. To the best of our knowledge, this studydescribes the first copula model fitted to COVID-19 data. Theresults of this study, derived using copula models, are unlikethose derived using existing traditional methods and indicatethat the association between COVID-19 and temperature isweak and no substantial decreases in the number of COVID-19 cases can be expected in response to high temperatures.Rhour lockdown formore than half of the observed data set. Therefore, there isclear evidence that high temperatures are not able to stop thespread of this virus if there is no social isolation. Clearly,lockdowns represent the most effective strategy to prevent thespread of this virus. To the best of our knowledge, this studydescribes the first copula model fitted to COVID-19 data. Theresults of this study, derived using copula models, are unlikethose derived using existing traditional methods and indicatethat the association between COVID-19 and temperature isweak and no substantial decreases in the number of COVID-19 cases can be expected in response to high temperatures.R