Temperature Decreases Spread Parameters of the New Covid-19 Case Dynamics

(1) Background: The virulence of coronavirus diseases due to viruses like SARS-CoV or MERS-CoV decreases in humid and hot weather. The putative temperature dependence of infectivity by the new coronavirus SARS-CoV-2 or covid-19 has a high predictive medical interest. (2) Methods: External temperature and new covid-19 cases in 21 countries and in the French administrative regions were collected from public data. Associations between epidemiological parameters of the new case dynamics and temperature were examined using an ARIMA model. (3) Results: We show that, in the first stages of the epidemic, the velocity of contagion decreases with country- or region-wise temperature. (4) Conclusions: Results indicate that high temperatures diminish initial contagion rates, but seasonal temperature effects at later stages of the epidemy remain questionable. Confinement policies and other eviction rules should account for climatological heterogeneities, in order to adapt the public health decisions to possible geographic or seasonal gradients.


Introduction
Two coronavirus pandemics occurred in the last 20 years, transmitted from animals to humans: (i) in 2002, the SARS-CoV virus provoked a severe acute respiratory syndrome (SARS) and spread from China, with more than 8000 cases and 774 deaths in 30 countries (about 10% mortality) [1] and (ii) in 2012, the MERS-CoV virus caused also a respiratory syndrome and spread from Saudi Arabia, with 1589 cases and 567 deaths in 26 countries (about 30% mortality) [1]. In 2019, the SARS-CoV-2 virus (called also covid-19) epidemic started in China in December in Wuhan (Hubei province) [2]. The SARS-CoV-2 and SARS-CoV genomes are very similar [3] and the spread of SARS-CoV is temperature-dependent [4]. Hence, SARS-CoV-2 is suspected, like other coronaviruses, to have a weather-dependent virulence. The changes in weather alone would not necessarily decrease numbers of confirmed cases during the whole epidemic, but warm and humid weather could make SARS-CoV-2 less transmissible and less stable, provoking a break in the chain of transmission, and hence diminishing the contagious force of the disease [5]. Ancient [1,4,5] studies, as well as characteristics of the covid-19 disease compared to past corona infections like the SARS one [3,[6][7][8], suggest that the spread of covid-19 could diminish in warm weather, particularly at the start of the epidemic, and may have a low temperature threshold under which it could spread fastest. These seasonal changes could occur in exactly the same way as for other pathogens, like the common cold or influenza [9][10][11][12]. This phenomenon can be modelled and the deterministic as well as stochastic models [2,[13][14][15][16][17][18][19][20][21][22][23][24][25] include potentially temperature-dependent parameters, like the contagion coefficient increasing with cold, dry weather because of faster evaporation of aerosol droplets. The present paper aims to identify such parameters from the covid-19 spread dynamics.
Section 2 describes how we collected information about the covid-19 spread since the beginning of March 2020. Then, we proposed two types of modelling in which parameters can depend on the temperature and we give a statistical method for studying the anti-correlation between the spread velocity and the external temperature. In Section 3, we give the results of the correlation analysis for all French administrative regions and for 21 countries suffering the covid-19 epidemic.

Epidemic Data Extraction
We used web sites giving world weather data (like https://www.weather-atlas.com or http://data. un.org/Data.aspx?d=CLINO&f=ElementCode%3A11). Analyses at country level used mean annual temperatures for that country (from https://en.wikipedia.org/wiki/List_of_countries_by_average_ yearly_temperature), and mean daily temperatures for the first half of March in the administrative capital of each French administrative region.
Covid-19 spread data exist in databases such as www.who.int/emergencies/diseases/novelcoronavirus-2019, www.cdc.gov/coronavirus/2019-ncov, www.cia.gov/library/publicapublications/ the-world-factbook/fields, www.worldometers.info/coronavirus, www.fr.statista.com/statistiques/ 1101324/morts-coronavirus-monde, www.santepubliquefrance.fr/recherche/#search=COVID-19%20: %20point%20epidemiologique&sort=date). These repositories are updated daily and offer free access to information concerning new cases, deaths and recovered cases, which is sufficient to build and assess mathematical models. Data for countries are from www.worldometers.info/coronavirus and for French administrative regions from www.statista.com/statistics/1101388/coronavirus-france-confirmed-cases. Data collection was constrained by the date of website accession-third week of March-and the availability of adequate data for specific countries at that date. This implies the inclusion of the first record with at least 100 cases after February 15, which excludes some countries for which the spread of the virus was more advanced at that period. Figure 1 gives an example of data extracted from these databases. Epidemiological kinetics vary among countries, and could be due to differences in weather variables like temperature.
Biology 2020, 9,94 2 of 10 include potentially temperature-dependent parameters, like the contagion coefficient increasing with cold, dry weather because of faster evaporation of aerosol droplets. The present paper aims to identify such parameters from the covid-19 spread dynamics. Section 2 describes how we collected information about the covid-19 spread since the beginning of March 2020. Then, we proposed two types of modelling in which parameters can depend on the temperature and we give a statistical method for studying the anti-correlation between the spread velocity and the external temperature. In Section 3, we give the results of the correlation analysis for all French administrative regions and for 21 countries suffering the covid-19 epidemic.

Epidemic Data Extraction
We used web sites giving world weather data (like https://www.weather-atlas.com or http://data.un.org/Data.aspx?d=CLINO&f=ElementCode%3A11). Analyses at country level used mean annual temperatures for that country (from https://en.wikipedia.org/wiki/List_of_countries_ by_average_yearly_temperature), and mean daily temperatures for the first half of March in the administrative capital of each French administrative region.
Covid-19 spread data exist in databases such as www.who.int/emergencies/diseases/novelcoronavirus-2019, www.cdc.gov/coronavirus/2019-ncov, www.cia.gov/library/publica publications/ the-world-factbook/fields, www.worldometers.info/coronavirus, www.fr.statista.com/statistiques/ 1101324/morts-coronavirus-monde, www.santepubliquefrance.fr/recherche/#search=COVID-19%20: %20point%20epidemiologique&sort=date). These repositories are updated daily and offer free access to information concerning new cases, deaths and recovered cases, which is sufficient to build and assess mathematical models. Data for countries are from www.worldometers.info/coronavirus and for French administrative regions from www.statista.com/statistics/1101388/coronavirus-france-confirmed-cases. Data collection was constrained by the date of website accession-third week of March-and the availability of adequate data for specific countries at that date. This implies the inclusion of the first record with at least 100 cases after February 15, which excludes some countries for which the spread of the virus was more advanced at that period. Figure 1 gives an example of data extracted from these databases. Epidemiological kinetics vary among countries, and could be due to differences in weather variables like temperature.

Modelling the Epidemic Spread with Temperature-Dependent Parameters
The classical epidemic modelling uses the continuous differential approach, which describes the infinitesimal change in the size of population of susceptible (S), infective (I) and recovering (R) individuals between times t and t + dt. This model explains the dynamics of epidemic spread, following the classical Bernoulli-d'Alembert-Ross equations [14]: where r is a renewal coefficient (depending on natality and immigration), v's are diffusion coefficients (depending on population displacements), b is the contagion coefficient, which can depend on temperature, and the constants k 1 and k 2 are, respectively, the natural death rate and the specific epidemic death rate. An example of this continuous differential approach, applied to the covid-19 spread, has been developed by P. Magal [2], and more sophisticated approaches would take into account more general demographic (age-classes), sociologic (socio-economic categories), geographic (latitude and altitude) and climatic (temperature and humidity) variations explaining the heterogeneity of the data between the regions of the same countries and between the countries of the same geo-climato-demographic cluster. Some parameter dependencies are described in [14][15][16], like the dependency of v on altitude, b on temperature and humidity and k's on age. The applications described in [14,16] have been performed, respectively, on classical Black Death data from St Anthony's Order and Mali malaria data from O.K. Doumbo.

Statistical Time Series Modelling
The statistical time series modelling has been introduced by N. Wiener for prediction and forecasting [26]. Its parametric approach assumes that the underlying stationary stochastic process of the covid-19 new daily cases N(t) can be described using a small number of parameters using the autoregressive ARIMA model N(t) = Σ i=1,s a(i) N(i) + W(t), where W is a random residue whose variance is to minimize. The autocorrelation analysis is done by calculating the correlation A(k) between the N(t)'s and N(t − k)'s (t belonging to a moving time window) by using the formula: where E denotes the expectation and σ the standard deviation. The autocorrelation function A allows examining the serial dependence of the N(t)'s.

Statistical Analyses for French Administrative Regions and Selected Countries
Daily rates of new cases in French administrative regions and selected countries were correlated with mean temperatures in administrative capitals, using Pearson's correlation coefficient r.

Temperature Decreases Initial Negative Autocorrelation Slope of Epidemic Spread in Five Countries
By using the classical ARIMA approach on the new cases' time series in five countries with different weather (mild for France, Mediterranean for Italy, continental for Germany, oceanic for Chile and continental for China), we show that, for all countries, the regression minimizing the standard deviation (STD) of the residue W is of order 6 ( Table 1), which is in agreement with the duration of the presence of virus in urine and blood after the mean incubation time of 6 days, which corresponds to the maximum of contagion for sputum, stool and swab (Figure 2a).
From the data of weather-atlas, we can calculate a mean temperature, obtained as the mean of the highest temperatures minus the mean of the lowest temperatures observed each day during February until 14 March 2020. The autocorrelation curves of the ARIMA regression have temperature-dependent shapes and the negative initial slope of their autocorrelation function decreases with the mean temperature (Figure 2b,c), reinforcing the plausibility of our hypothesis that the spread of covid-19 could diminish in warm weather, particularly at the start of the epidemic. February until 14 March 2020. The autocorrelation curves of the ARIMA regression have temperature-dependent shapes and the negative initial slope of their autocorrelation function decreases with the mean temperature (Figure 2b,c), reinforcing the plausibility of our hypothesis that the spread of covid-19 could diminish in warm weather, particularly at the start of the epidemic.   Table 2 presents the mean temperature in the first half of March in the 16 administrative regions of metropolitan France, numbers of confirmed covid-19 cases on 4 th March, and ulterior daily rates as  Table 2 presents the mean temperature in the first half of March in the 16 administrative regions of metropolitan France, numbers of confirmed covid-19 cases on 4 March, and ulterior daily rates as compared to data from the previous publication date. The last row shows Pearson correlation coefficients between regional rates of infection and mean temperatures across the period. Correlations show a slower increase in infection rates in warmer regions. The strength of this tendency decreases towards the end of this short period of rapid increase. This suggests that temperature most affects the early phases of epidemic dynamics. Figure 3 presents correlation analyses for two dates, 6 and 15 March 2020. Note that Figure 3 includes R 2 , which is the square of the Pearson correlation coefficient R. These adjustments are along logarithmic and exponential models for 6 and 15 March, respectively. We give the best (highest R 2 ) models for these dates among linear (y = a.T + b), logarithmic (y = a.ln(T) + b), exponential (ln(y) = a.T + b) and power (ln(y) = a.ln(T) + b) regression models. Last row is the Pearson correlation coefficient R of above data in that column with mean temperature. For column 3, the latter correlation is not with a rate, as the date of first infection in that region varies among regions. All R < −0.47615 and R < −0.63385 have p < 0.05 and p < 0.01, respectively, according to one-tailed tests. Results indicate that negative effects of temperature on infection rates are strongest at the beginning of regional epidemics.  Table 3 presents the cumulative numbers of confirmed cases for countries with more than 100 cases in at least two days as of 14 March (last measure, besides for South Korea (10 March)). The constant and slope are for the exponential regression model to these data ln(N) = a × D + b, where D is the number of days since N crossed 100 confirmed cases in that country. Figure 4 shows the slope in the last column of Table 3 as a function of the mean annual temperature in that country. Slopes are lower and higher than expected by temperature for Canada and Spain, respectively. The causes for this might range from differences in containment policies and/or populational compliances, to other climatic factors (for example, humidity).  Table 3 presents the cumulative numbers of confirmed cases for countries with more than 100 cases in at least two days as of 14 March (last measure, besides for South Korea (10 March)). The constant and slope are for the exponential regression model to these data ln(N) = a × D + b, where D is the number of days since N crossed 100 confirmed cases in that country. Figure 4 shows the slope in the last column of Table 3 as a function of the mean annual temperature in that country. Slopes are lower and higher than expected by temperature for Canada and Spain, respectively. The causes for this might range from differences in containment policies and/or populational compliances, to other climatic factors (for example, humidity). Daily change in case numbers

Temperature Decreases Country-Wise Initial Rates of Epidemic Spread
Mean regional temperature, first half of March   Table 3 as a function of mean annual temperature in that country. The Pearson correlation coefficient is R = −0.568, one-tailed p = 0.0036.

Discussion
A study (https://www.accuweather.com/en/health-wellness/coronavirus-expert-says-the-viruswill-burn-itself-out-in-about-6-months/679415) by J. Nicholls from the Sun Yat-sen University in Guangzhou, the capital of south China's Guangdong province, has determined how the spread of the new coronavirus might be affected by changes in season and temperature.
We show in the present study that the negative initial slope of the autocorrelation curve related to the new daily cases N(t) of covid-19 spread and the duration of the positive autocorrelation period  Table 3 as a function of mean annual temperature in that country. The Pearson correlation coefficient is R = −0.568, one-tailed p = 0.0036.

Discussion
A study (https://www.accuweather.com/en/health-wellness/coronavirus-expert-says-the-viruswill-burn-itself-out-in-about-6-months/679415) by J. Nicholls from the Sun Yat-sen University in Guangzhou, the capital of south China's Guangdong province, has determined how the spread of the new coronavirus might be affected by changes in season and temperature.
We show in the present study that the negative initial slope of the autocorrelation curve related to the new daily cases N(t) of covid-19 spread and the duration of the positive autocorrelation period decreases when the weather temperature increases, which corresponds to a shorter duration of the period of contagiousness. The entropy of the distribution of daily R0's during this period would decrease if the period becomes shorter, for the same overall R0. For example, if the daily R0's tend to be the same (without peak), the entropy is maximum, equal to the logarithm of the number of contagious days. The decrease in the duration of the contagiousness period is considered favorable and corresponds to the "mitigation" of the contagion. The calculation of the entropy needs a precise estimation of the ARIMA coefficients a(i)'s in the development, N(t) = Σ i=1,s a(i) N(i) + W(t), which is difficult during the transient start of the spread, due to the weak number of cases.
The two independent datasets, analysed at the level of French metropolitan administrative regions and of countries, show that temperature affects the increase in the number of cases at relatively local, as well as global, levels. The mechanisms by which temperature decreases the rates of detected cases are unknown. Higher temperatures might prevent the spread of droplets that transmit viruses, perhaps through faster evaporation. Other factors, like the decrease in the virus survival time in atmosphere probably also affect these rates, in particular at the level of the unexplained variation among countries shown in Figure 4. These results have to be considered with caution. They indicate that temperature affects early rates of spreading. It is unclear whether ulterior increases in seasonal temperatures will decrease rates [28][29][30][31]. Indeed, once a dynamic is set, temperature might not affect this dynamic anymore, or affect it only marginally.
Some papers on influenza show that if the warm weather period is followed by a cold season, the rebound of the epidemic could be severe, due to the loss of immune defense [8,15], and the present results have to serve as model for building a system of systematic surveillance along the future months of the covid-19 spread.

Conclusions
For more than fifty years, the relationship between weather and diseases has been studied [32]. Such studies are important for predicting viral disease spread, in particular if this leads to pandemics like in case of covid-19, in order to help decisions in public health policies at the world level.