# On the Transmission Dynamics of SARS-CoV-2 in a Temperate Climate

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Model

_{A}(t) + I

_{S}(t) + R(t) + D(t). S(t) denotes the number of susceptible individuals at time t, E(t) is the number of exposed (infected but not infectious) individuals, I

_{A}(t) is the number of asymptomatic infective individuals, I

_{S}(t) is the number of infected individuals with symptoms, R(t) is the number of individuals who recovered and were removed, and D(t) the number of fatalities during the outbreak. We assumed that the total population remains constant during the outbreak. The assumption of a constant population is reasonable when the infection spreads fast through the population, as this is the case for the COVID-19 pandemic. Therefore, we did not include demographic effects such as births and natural deaths in the model. The modelling approach also implies that the populations are homogeneously mixed. We also ignored high-risk groups. We explicitly included the characteristic incubation period between infection and the appearance of clinical symptoms. This period consists of the latency and the preclinical state (Figure 1). The pathogen is transmitted to a susceptible person through person-to–person contact (via generation of respiratory droplets containing infectious pathogens) with either an asymptomatic infected person or an infected person who developed clinical symptoms (illness). Susceptible individuals (S), once infected, enter the state of exposed individuals (E) harbouring a latent infection but not being infectious. All exposed individuals enter the state of being asymptomatic and having subclinical infections. Asymptomatic individuals (I

_{A}) are infectious and there are two options. An individual may remain asymptomatic for the duration of the infection and recover without ever developing clinical symptoms entering the recovery state (R). Alternatively, an individual may remain asymptomatic (preclinical) for a period of time before ultimately entering the clinical state and becoming an infected individual with clinical symptoms (I

_{S}). For those becoming ill with clinical symptoms a fraction recovers and some may die. We assumed that recovered individuals obtain immunity, and they are removed from the infection process. Figure 2 shows the transitions between the different states of the infection pathway. The equations of the model are written as follows:

_{0}; E(0) = 0; I

_{A}(0) = I

_{A}

_{0}; I

_{S}(0) = I

_{S}

_{0}; R(0) = 0; D(0) = 0. One may start the epidemic with a few infected persons with clinical symptoms and/or asymptomatic individuals. The equation for the fatalities does not contribute to the dynamics of the system. It simply counts the number of fatalities.

_{A}) than that to an infected person who develops symptoms (β

_{S}). The rationale behind this assumption is that as observed not every infection leads to illness. In addition, there is probably a difference, though difficult to quantify, in transmissibility between transmission from asymptomatic infected to susceptible persons and transmission from symptomatic infected to susceptible persons. An infected person with clinical illness sheds more virus that one with subclinical infection [52]. Moreover, a clinically ill person has symptoms (e.g., coughing, rhinorrhea) that contribute to the generation of infectious pathogen laden droplets of all sizes. On the other side, infected persons with clinical symptoms may show reduced virus transmissibility if they are confined to bed due to severity of illness. Therefore, the magnitude of difference between the two transmission rates depends on factors such as age, severity of illness, social behavior, living conditions in households (common rooms, share bedrooms) and others.

_{A}) to the symptomatic infected state (I

_{S}) with a rate δ denoting the period from onset of infectiousness to onset of clinical symptoms and may remain asymptomatic (preclinical infection) for a period of 1/δ before developing symptoms. Alternatively, an asymptomatic individual may never develop symptoms and recover after a period 1/γ

_{A}. Therefore, the average time of an asymptomatic individual in state I

_{A}is 1/(δ + γ

_{A}). The incubation period is denoted as the period from infection to onset of symptoms (1/α) + (1/δ). Asymptomatic infected persons are expected to recover faster than those who have developed clinical symptoms. Symptomatic infected individuals are likely to have greater infectivity due to higher virus loads higher viral shedding due the severity of their clinical conditions. Thus, following the onset of symptoms, the recovery period for a person with symptoms is 1/γ

_{S}. Finally, the fraction of persons who developed severe illness and died is denoted with f

_{s}.

#### 2.2. The Basic Reproduction Number

_{0}) represents the average number of secondary infections where an infected individual can cause in an entirely susceptible population. R

_{0}can be calculated using the disease-free equilibrium of the above system of ordinary differential equations. It can be derived using the next generation matrix using the methods described in [53]. In this case the R

_{0}is the sum of the basic reproduction number for the infections caused by an asymptomatic individual and that of an individual with symptoms.

_{0}is a threshold. If ${R}_{0}>1$ then the number of infective individuals first increases before decreasing to zero representing a full-blown epidemic curve. If ${R}_{0}1$ then the number of infected individuals decreases monotonically to zero.

#### 2.3. The Monte Carlo Uncertainty Analysis and the Error Optimization

_{S}) + NMSE(D), where NMSE is the mean squared error (MSE) normalized with the mean of the corresponding observations (i.e., infected or dead). All parameters were optimized separately before, during and after the introduction of intervention, in this case social distancing in form of a population-based lockdown. Table 1 summarizes the range of prior distribution of the parameter values. We applied the model using the time series of cases as they were unfolded in Greece and as they were reported in the official web site of the national public health authority [55]. We run the simulations of the model with an initial population of susceptibles (S

_{0}= 1 million people) and a small number of infected persons with clinical symptoms (I

_{S}

_{0}= 1). Two cycles of Monte Carlo simulations were adopted. At the screening phase, we assumed the parameters followed a uniform distribution. The model simulations were compared with observations to identify the uncertainty range that minimized the error. At the implementation phase, the parameters were sampled from a normal distribution centred at their mode identified during the screening phase.

#### 2.4. The Forecast Horizon

#### 2.5. Clustering of COVID-19 Data

_{v}) and calculated the percentage of events within each group. We standardized the number of events within each cell with the number of weather records falling inside it (Probability of Infection). Further, we converted the magnitudes into number of human cases per million and estimated the median, to quantify the severity within each group.

## 3. Results

#### 3.1. Environmental Clustering of COVID-19 Data

^{−3}to 25 g m

^{−3}for absolute humidity. For temperatures up to 25 °C, the absolute humidity reached its upper ceiling (saturation value) given by the Clausius-Clapeyron equation. In addition, half of the occurrence points were located close to the saturation curve for temperatures between 0 and 25 °C, suggesting that at this temperature range high relative humidity favors the persistence of SARS-CoV-19 and thus the emergence of COVID-19 cases. Therefore, there existed a tendency for more occurrences as the climate became cooler and more humid. Although temperature and humidity may influence virus inactivation in the environment, a causal mechanism has not been shown yet [58,59].

^{−3}(37.4%), while for temperature 71.3% of the reported cases were observed between 10 °C and 30 °C (Figure 3). With respect to their joint probability, the zone $\{5\le {\rho}_{v}\left({\mathrm{g}\text{}\mathrm{m}}^{-3}\right)\le 10,$ 10 ≤ T(°C) ≤ 20} had the highest frequency of COVID-19 cases which accounts for 22.8% of them. This classification had similarities with the findings of [43] who reported temperature in the range $5\u201311$ °C and absolute humidity in the range $4\u20137{\mathrm{g}\text{}\mathrm{m}}^{-3}$ at cities with substantial transmission. This was not too different from our result, considering the uncertainty due to the different datasets (point measurements in our study versus gridded reanalysis). This result was also aligned with other studies, which indicated that a favorable area around 10 °C/5${\mathrm{g}\text{}\mathrm{m}}^{-3}$ existed.

#### 3.2. Model Calibration for Greece: March-April-May 2020

_{0}) was around 2.7 and the subsequent effective reproduction number fell below the critical value of one after the intervention. The reproduction number remained below the critical value of one when the lockdown was lifted. Those numbers were in agreement with the reproduction number officially reported from the COVID-19 committee of Greece [60]. Overall, the experiment MC1 replicated the observed curves (total cases, deaths) with small uncertainty and also demonstrated good accuracy and phasing with the observed reproduction number curves.

_{A}) had an optimal value of 0.24/day. The corresponding value of the transmission rate between an infected symptomatic and a susceptible (β

_{S}) was 0.61/day and thus 2.5 times higher than that between asymptomatic infected individuals and susceptibles in the optimal case. After the intervention measures, the transmission rate of an infected symptomatic dropped to 0.16/day, i.e., close to the level of β

_{A}prior to the intervention. The transmission rate between asymptomatics and susceptibles dropped to 0.07/day. The relation β

_{S}/β

_{A}remained similar during the intervention. When the lockdown was lifted, β

_{S}and β

_{A}dropped further to 0.15/day and 0.05/day. See also Table 3.

_{A}) and individuals with clinical symptoms (I

_{S}) reached a maximal value that was interrupted by the introduction of the interventions leading to a permanent decrease of their population with a short time delay between I

_{A}and I

_{S}. The intervention period changed the curvature of the recovered (R) and the fatalities (D) populations.

#### 3.3. The Variation of Transmissibility: From March to December 2020

_{S}). Therefore, we kept all factors but β

_{S}fixed and performed another cycle of Monte Carlo simulations where we sought the intra-annual variability of the optimal β

_{S}. We split the period 15/02/2020–15/12/2020 into temporal windows of equal size and calibrated the model within each period following a slightly different approach than before for reasons explained hereafter. The COVID-19 tests performed weekly varied from 7000 in March-April to 65,000 in July–August to 130,000 in September–October 2020. After August, the total number of weekly tests included a portion of rapid tests having lower sensitivity and specificity than polymerase chain reaction (PCR) tests. Moreover, the sampling gradually included an increasing number of random tests. To get over with those uncertainties as we do not know the number of asymptomatic individuals detected daily nor the implemented sampling strategy, the calibration for the whole period was based on the number of deaths, which were not affected by the abovementioned uncertainties.

_{S}which exhibits three discrete peaks: two related with the timing of the spring and autumn epidemic waves and another one in between.

_{S}which we analyzed next in predictive mode.

#### 3.4. The Variation of Predictability: From March to December 2020

_{S}that optimized the error, as performed before. Then, using the optimal β

_{S}, we continued the simulation for 30 days beyond the last day of the window (T

_{E}) and we searched for the day after T

_{E}when the NRMSE exceeded 0.1 (i.e., the forecast error becomes 10%).

_{S}, closely replicated the coarse pattern seen earlier using non-moving temporal windows (Figure 7), implying the internal dynamics were invariant under different model configurations. Three peaks were evident in β

_{S}occurring with decreasing magnitude; apart from the spring (strongest: 0.71/day) and autumn (weakest: 0.38/day) signals, there existed an intermediate peak at the beginning of July (0.58/day). The transmission rate initially decreased gradually following the intervention measures and reached a plateau. The abrupt increase in the β

_{S}at the beginning of July was probably related to the relaxation of social distancing measures such as the maximum number of participants in social events and the opening of the touristic season, which occurred few weeks earlier. The change was also evident at the curve of the new cases. The forecast horizon depended on the β

_{S}magnitude and its day-to-day variability. Large and variable β

_{S}resulted in smaller forecast horizons. The same held true for the upslope portion of the epidemic curve, where the fastest changes were linked to the least predictability. In the case of the Greek data, the implemented error tolerance resulted in a minimum forecast horizon of roughly one week, observed twice during the periods of the steepest change in the epidemic curve (March, November). At the opposite end, the forecast horizon exceeded four weeks during the period April–July when the variability was smoother. Therefore, the current epidemiological state is a significant indicator of the length of time into the future over which the forecasts are reliable [63].

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Ethics Committee Approval

## References

- Furukawa, N.W.; Brooks, J.T.; Sobel, J. Evidence supporting transmission of severe acute respiratory syndrome coronavirus 2 while presymptomatic or asymptomatic. Emerg. Infect. Dis.
**2020**, 26. [Google Scholar] [CrossRef] [PubMed] - Bai, Y.; Yao, L.; Wei, T.; Tian, F.; Jin, D.Y.; Chen, L.; Wang, M. Presumed asymptomatic carrier transmission of COVID-19. JAMA
**2020**, 323, 1406–1407. [Google Scholar] [CrossRef][Green Version] - Pan, X.; Chen, D.; Xia, Y.; Wu, X.; Li, T.; Ou, X.; Zhou, L.; Liu, J. Asymptomatic cases in a family cluster with SARS-CoV-2 infection. Lancet Infect. Dis.
**2020**, 20, 410–411. [Google Scholar] [CrossRef] - Lytras, T.; Dellis, G.; Flountzi, A.; Hatzianastasiou, S.; Nikolopoulou, G.; Tsekou, K.; Diamantis, Z.; Stathopoulou, G.; Togka, M.; Gerolymatos, G.; et al. High prevalence of SARS-CoV-2 infection in repatriation flights to Greece from three European countries. J. Travel Med.
**2020**, 27, taaa054. [Google Scholar] [CrossRef][Green Version] - Wei, W.E.; Li, Z.; Chiew, C.J.; Yong, S.E.; Toh, M.P.; Lee, V.J. Pre-symptomatic transmission of SARS-CoV-2-Singapore, January 23–March 16, 2020. Morb. Mortal.
**2020**, 69, 411–415. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hoehl, S.; Rabenau, H.; Berger, A.; Kortenbusch, M.; Cinatl, J.; Bojkova, D.; Behrens, P.; Böddinghaus, B.; Götsch, U.; Naujoks, F.; et al. Evidence of SARS-CoV-2 infection in returning travelers from Wuhan, China. N. Engl. J. Med.
**2020**, 382, 1278–1280. [Google Scholar] [CrossRef] - Arons, M.M.; Hatfield, K.M.; Reddy, S.C.; Kimball, A.; James, A.; Jacobs, J.R.; Taylor, J.; Spicer, K.; Bardossy, A.C.; Oakley, L.P.; et al. Presymptomatic SARS-CoV-2 infections and transmission in a skilled nursing facility. N. Engl. J. Med.
**2020**, 382, 2081–2090. [Google Scholar] [CrossRef] [PubMed] - Zou, L.; Ruan, F.; Huang, M.; Liang, L.; Huang, H.; Hong, Z.; Yu, J.; Kang, M.; Song, Y.; Xia, J.; et al. SARS-CoV-2 viral load in upper respiratory specimens of infected patients. N. Engl. J. Med.
**2020**, 382, 1177–1179. [Google Scholar] [CrossRef] - Du, Z.; Xu, X.; Wu, Y.; Wang, L.; Cowling, B.J.; Meyers, L.A. Serial interval of COVID-19 among publicly reported confirmed cases. Emerg. Infect. Dis.
**2020**, 26, 1341–1343. [Google Scholar] [CrossRef] [PubMed] - Li, R.; Pei, S.; Chen, B.; Song, Y.; Zhang, T.; Yang, W.; Shaman, J. Substantial undocumented infection facilitates the rapid dissemination of novel coronavirus (SARS-CoV-2). Science
**2020**, 368, 489–493. [Google Scholar] [CrossRef] [PubMed][Green Version] - He, X.; Lau, E.H.Y.; Wu, P.; Deng, X.; Wang, J.; Hao, X.; Lau, Y.C.; Wong, J.Y.; Guan, Y.; Tan, X.; et al. Temporal dynamics in viral shedding and transmissibility of COVID-19. Nat. Med.
**2020**, 26, 672–675. [Google Scholar] [CrossRef] [PubMed][Green Version] - Nishiura, H.; Kobayashi, T.; Miyama, T.; Suzuki, A.; Jung, S.-M.; Hayashi, K.; Kinoshita, R.; Yang, Y.; Yuan, B.; Akhmetzhanov, A.R.; et al. Estimation of the asymptomatic ratio of novel coronavirus infections (COVID-19). Int. J. Infect. Dis.
**2020**, 94, 154–155. [Google Scholar] [CrossRef] - Mizumoto, K.; Kagaya, K.; Zarebski, A.; Chowell, G. Estimating the asymptomatic proportion of coronavirus disease 2019 (COVID-19) cases on board the Diamond Princess cruise ship, Yokohama, Japan. Eurosurveillance
**2020**, 25, 2000180. [Google Scholar] [CrossRef][Green Version] - Pan, A.; Liu, L.; Wang, C.; Guo, H.; Hao, X.; Wang, Q.; Huang, J.; He, N.; Yu, H.; Lin, X.; et al. Association of public health interventions with the epidemiology of the COVID-19 outbreak in Wuhan, China. JAMA
**2020**, 323, 1915. [Google Scholar] [CrossRef][Green Version] - Wu, J.T.; Leung, K.; Bushman, M.; Kishore, N.; Niehus, R.; De Salazar, P.M.; Cowling, B.J.; Lipsitch, M.; Leung, G.M. Estimating clinical severity of COVID-19 from the transmission dynamics in Wuhan, China. Nat. Med.
**2020**, 26, 506–510. [Google Scholar] [CrossRef][Green Version] - Fraser, C.; Riley, S.; Anderson, R.M.; Ferguson, N.M. Factors that make an infectious disease outbreak controllable. Proc. Natl. Acad. Sci. USA
**2004**, 101, 6146–6151. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lipsitch, M.; Donnelly, C.A.; Fraser, C.; Blake, I.M.; Cori, A.; Dorigatti, I.; Ferguson, N.M.; Garske, T.; Mills, H.L.; Riley, S.; et al. Potential biases in estimating absolute and relative case-fatality risks during outbreaks. PLoS Negl. Trop. Dis.
**2015**, 9, e0003846. [Google Scholar] [CrossRef][Green Version] - Delamater, P.L.; Street, E.J.; Leslie, T.F.; Yang, Y.T.; Jacobsen, K.H. Complexity of the basic reproduction number (R0). Emerg. Infect. Dis.
**2019**, 25, 1–4. [Google Scholar] [CrossRef] [PubMed][Green Version] - Heesterbeek, J.A.P.; Dietz, K. The concept of Roin epidemic theory. Stat. Neerl.
**1996**, 50, 89–110. [Google Scholar] [CrossRef] - Koo, J.R.; Cook, A.R.; Park, M.; Sun, Y.; Sun, H.; Lim, J.T.; Tam, C.; Dickens, B.L. Interventions to mitigate early spread of SARS-CoV-2 in Singapore: A modelling study. Lancet Infect. Dis.
**2020**, 20, 678–688. [Google Scholar] [CrossRef][Green Version] - Longini, I.M.; Halloran, M.E.; Nizam, A.; Yang, Y. Containing pandemic influenza with antiviral agents. Am. J. Epidemiol.
**2004**, 159, 623–633. [Google Scholar] [CrossRef][Green Version] - Arino, J.; Brauer, F.; Driessche, P.V.D.; Watmough, J.; Wu, J. A model for influenza with vaccination and antiviral treatment. J. Biol.
**2008**, 253, 118–130. [Google Scholar] [CrossRef] [PubMed] - McCaw, J.M.; Wood, J.G.; McCaw, C.T.; McVernon, J. Impact of emerging antiviral drug resistance on influenza containment and spread: Influence of subclinical infection and strategic use of a stockpile containing one or two drugs. PLoS ONE
**2008**, 3, e2362. [Google Scholar] [CrossRef] [PubMed] - Stilianakis, N.I.; Perelson, A.S.; Hayden, F.G. Emergence of drug resistance during an influenza epidemic: Insights from a mathematical model. J. Infect. Dis.
**1998**, 177, 863–873. [Google Scholar] [CrossRef] [PubMed] - Regoes, R.R.; Bonhoeffer, S. Emergence of Drug-Resistant Influenza Virus: Population Dynamical Considerations. Science
**2006**, 312, 389–391. [Google Scholar] [CrossRef][Green Version] - Débarre, F.; Bonhoeffer, S.; Regoes, R.R. The effect of population structure on the emergence of drug resistance during influenza pandemics. J. R. Soc. Interface
**2007**, 4, 893–906. [Google Scholar] [CrossRef][Green Version] - Alexander, M.E.; Bowman, C.S.; Feng, Z.; Gardam, M.; Moghadas, S.M.; Röst, G.; Wu, J.; Yan, P. Emergence of drug resistance: Implications for antiviral control of pandemic influenza. Proc. R. Soc. B Boil. Sci.
**2007**, 274, 1675–1684. [Google Scholar] [CrossRef][Green Version] - Robinson, M.; Stilianakis, N.I. A model for the emergence of drug resistance in the presence of asymptomatic infections. Math. Biosci.
**2013**, 243, 163–177. [Google Scholar] [CrossRef] - Weber, T.P.; Stilianakis, N.I. Inactivation of influenza a viruses in the environment and modes of transmission: A critical review. J. Infect.
**2008**, 57, 361–373. [Google Scholar] [CrossRef] - Biryukov, J.; Boydston, J.A.; Dunning, R.A.; Yeager, J.J.; Wood, S.; Reese, A.L.; Ferris, A.; Miller, D.; Weaver, W.; Zeitouni, N.E.; et al. Increasing temperature and relative humidity accelerates inactivation of SARS-CoV-2 on surfaces. MSphere
**2020**, 5. [Google Scholar] [CrossRef] - Schuit, M.; Ratnesar-Shumate, S.; Yolitz, J.; Williams, G.; Weaver, W.; Green, B.; Miller, D.; Krause, M.; Beck, K.; Wood, S.; et al. Airborne SARS-CoV-2 is rapidly inactivated by simulated sunlight. J. Infect. Dis.
**2020**, 222, 564–571. [Google Scholar] [CrossRef] - Fisman, D. Seasonality of viral infections: Mechanisms and unknowns. Clin. Microbiol. Infect.
**2012**, 18, 946–954. [Google Scholar] [CrossRef][Green Version] - Moriyama, M.; Hugentobler, W.J.; Iwasaki, A. Seasonality of respiratory viral infections. Annu. Rev. Virol.
**2020**, 7, 83–101. [Google Scholar] [CrossRef] - Dowell, S.F. Seasonal variation in host susceptibility and cycles of certain infectious diseases. Emerg. Infect. Dis.
**2001**, 7, 369–374. [Google Scholar] [CrossRef] [PubMed] - Shaman, J.; Kohn, M. Absolute humidity modulates influenza survival, transmission, and seasonality. Proc. Natl. Acad. Sci. USA
**2009**, 106, 3243–3248. [Google Scholar] [CrossRef] [PubMed][Green Version] - Cannell, J.J.; Vieth, R.; Umhau, J.C.; Holick, M.F.; Grant, W.B.; Madronich, S.; Garland, C.F.; Giovannucci, E. Epidemic influenza and vitamin D. Epidemiol. Infect.
**2006**, 134, 1129–1140. [Google Scholar] [CrossRef] [PubMed] - Azziz-Baumgartner, E.; Dao, C.N.; Nasreen, S.; Bhuiyan, M.U.; Mah-E-Muneer, S.; Al Mamun, A.; Sharker, M.A.Y.; Zaman, R.U.; Cheng, P.; Klimov, A.I.; et al. Seasonality, timing, and climate drivers of influenza activity worldwide. J. Infect. Dis.
**2012**, 206, 838–846. [Google Scholar] [CrossRef] [PubMed][Green Version] - Shaman, J.; Pitzer, V.E.; Viboud, C.; Grenfell, B.T.; Lipsitch, M. Absolute humidity and the seasonal onset of influenza in the continental United States. PLoS Biol.
**2010**, 8, e1000316. [Google Scholar] [CrossRef] - Peci, A.; Winter, A.-L.; Li, L.; Gnaneshan, S.; Liu, J.; Mubareka, S.; Gubbay, J.B. Effects of absolute humidity, relative humidity, temperature, and wind speed on influenza activity in Toronto, Ontario, Canada. Appl. Environ. Microbiol.
**2019**, 85. [Google Scholar] [CrossRef][Green Version] - Tamerius, J.; Nelson, M.I.; Zhou, S.Z.; Viboud, C.; Miller, M.A.; Alonso, W.J. Global influenza seasonality: Reconciling patterns across temperate and tropical regions. Environ. Health Perspect.
**2011**, 119, 439–445. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bukhari, Q.; Jameel, Y. Will coronavirus pandemic diminish by summer? Electron. J.
**2020**. [Google Scholar] [CrossRef] - Neher, R.A.; Dyrdak, R.; Druelle, V.; Hodcroft, E.B.; Albert, J. Potential impact of seasonal forcing on a SARS-CoV-2 pandemic. Swiss Med. Wkly.
**2020**, 150, w20224. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kissler, S.M.; Tedijanto, C.; Goldstein, E.; Grad, Y.H.; Lipsitch, M. Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period. Science
**2020**, 368, 860–868. [Google Scholar] [CrossRef] - Sajadi, M.M.; Habibzadeh, P.; Vintzileos, A.; Shokouhi, S.; Miralles-Wilhelm, F.; Amoroso, A. Temperature, humidity, and latitude analysis to estimate potential spread and seasonality of coronavirus disease 2019 (COVID-19). JAMA Netw. Open
**2020**, 3, e2011834. [Google Scholar] [CrossRef] - Araujo, M.B.; Naimi, B. Spread of SARS-CoV-2 coronavirus likely to be constrained by climate. MedRxiv
**2020**. [Google Scholar] [CrossRef][Green Version] - Wu, J.T.; Leung, K.; Leung, G.M. Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: A modelling study. Lancet
**2020**, 395, 689–697. [Google Scholar] [CrossRef][Green Version] - Kucharski, A.J.; Russell, T.W.; Diamond, C.; Liu, Y.; Edmunds, J.; Funk, S.; Eggo, R.M.; Sun, F.; Jit, M.; Munday, J.D.; et al. Early dynamics of transmission and control of COVID-19: A mathematical modelling study. Lancet Infect. Dis.
**2020**, 20, 553–558. [Google Scholar] [CrossRef][Green Version] - Kissler, S.; Tedijanto, C.; Lipsitch, M.; Grad, Y. Social distancing strategies for curbing the COVID-19 pandemic. MedRxiv
**2020**. [Google Scholar] [CrossRef][Green Version] - Arino, J.; Portet, S. A simple model for COVID-19. Infect. Dis. Model.
**2020**, 5, 309–315. [Google Scholar] [CrossRef] - Gatto, M.; Bertuzzo, E.; Mari, L.; Miccoli, S.; Carraro, L.; Casagrandi, R.; Rinaldo, A. Spread and dynamics of the COVID-19 epidemic in Italy: Effects of emergency containment measures. Proc. Natl. Acad. Sci. USA
**2020**, 117, 10484–10491. [Google Scholar] [CrossRef][Green Version] - Liu, Z.; Magal, P.; Seydi, O.; Webb, G. A COVID-19 epidemic model with latency period. Infect. Dis. Model.
**2020**, 5, 323–337. [Google Scholar] [CrossRef] [PubMed] - Liu, Y.; Yan, L.-M.; Wan, L.; Xiang, T.-X.; Le, A.; Liu, J.-M.; Peiris, M.; Poon, L.L.M.; Zhang, W. Viral dynamics in mild and severe cases of COVID-19. Lancet Infect. Dis.
**2020**, 20, 656–657. [Google Scholar] [CrossRef][Green Version] - Driessche, P.V.D.; Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci.
**2002**, 180, 29–48. [Google Scholar] [CrossRef] - Bar-On, Y.M.; Flamholz, A.; Phillips, R.; Milo, R. SARS-CoV-2 (COVID-19) by the numbers. eLife
**2020**, 9. [Google Scholar] [CrossRef] [PubMed] - COVID19 Greek Government. Available online: https://covid19.gov.gr/covid19-live-analytics/ (accessed on 15 December 2020).
- Dbouk, T.; Drikakis, D. Weather impact on airborne coronavirus survival. Phys. Fluids
**2020**, 32, 093312. [Google Scholar] [CrossRef] - University of Wisconsin. Available online: http://weather.uwyo.edu (accessed on 8 May 2020).
- Marr, L.C.; Tang, J.W.; Van Mullekom, J.; Lakdawala, S.S. Mechanistic insights into the effect of humidity on airborne influenza virus survival, transmission and incidence. J. R. Soc. Interface
**2019**, 16, 20180298. [Google Scholar] [CrossRef] - Carlson, C.J.; Gomez, A.C.R.; Bansal, S.; Ryan, S.J. Misconceptions about weather and seasonality must not misguide COVID-19 response. Nat. Commun.
**2020**, 11, 1–4. [Google Scholar] [CrossRef] - Greek Ministry of Health. Available online: https://www.moh.gov.gr/articles/ministry/grafeio-typoy/press-releases/ (accessed on 15 December 2020).
- McAloon, C.; Collins, A.; Hunt, K.; Barber, A.; Byrne, A.W.; Butler, F.; Casey, M.; Griffin, J.; Lane, E.; McEvoy, D.; et al. Incubation period of COVID-19: A rapid systematic review and meta-analysis of observa-tional research. BMJ Open
**2020**, 10, e039652. [Google Scholar] [CrossRef] - Johansson, M.A.; Quandelacy, T.M.; Kada, S.; Prasad, P.V.; Steele, M.; Brooks, J.T.; Slayton, R.B.; Biggerstaff, M.; Butler, J.C. SARS-CoV-2 transmission from people without COVID-19 Symptoms. JAMA Netw. Open
**2021**, 4, e2035057. [Google Scholar] [CrossRef] - Castro, M.; Ares, S.; Cuesta, J.A.; Manrubia, S. The turning point and end of an expanding epidemic cannot be precisely forecast. Proc. Natl. Acad. Sci. USA
**2020**, 117, 26190–26196. [Google Scholar] [CrossRef] [PubMed] - Palmer, T.N.; Gelaro, R.; Barkmeijer, J.; Buizza, R. Singular vectors, metrics and adaptive observations. J. Atmos. Sci.
**1998**, 55, 633–653. [Google Scholar] [CrossRef] - Kioutsioukis, I.; Galmarini, S. De praeceptis ferentis: Good practice in multi-model ensembles. Atmos. Chem. Phys.
**2014**, 14, 11791–11815. [Google Scholar] [CrossRef][Green Version]

**Figure 2.**Transition diagram of model populations. Susceptible (S) persons become infected after exposure to respiratory droplets exhaled from asymptomatic infected persons (I

_{A}) or infected persons with clinical symptoms (I

_{S}) with rates β

_{A}and β

_{S}. Exposed and infected persons (E) go through a period of latency during which they are not infectious. Exposed persons become asymptomatic infected and infectious (I

_{A}) with a rate α. Asymptomatic infected persons develop clinical symptoms and become infected with symptoms (illness) with a rate δ or they recover (R) with a rate γ

_{A}. Persons with clinical symptoms can recover (R) with a rate (1 − f

_{S}) γ

_{S}or die (D) with a rate f

_{S}γ

_{S}.

**Figure 3.**Temperature and humidity conditions for all reported COVID-19 human cases worldwide (01/01/2020–07/05/2020). In the same figure, univariate histograms of the scattered-data are plotted. The frequency of the joint distribution is given in Table 2 [left]. Percentiles (1, 5, 10, 25, 50, 75, 90, 95, 99) of humidity as a function of temperature for all reported COVID-19 human cases worldwide (01/01/2020–07/05/2020). The thickest line corresponds to the median. In the same figure, the maximum possible value of humidity at each temperature, estimated from the Clausius-Clapeyron equation, is plotted with the dotted line [right].

**Figure 4.**[Top row] Dynamic evolution of the ensemble of simulations (boxplot (light blue) with interquartile range (deep blue part), which set the uncertainty limits of the forecast, together with the optimal simulation (solid line) and the observations (red circles). Left panel shows the symptomatic infected persons (I

_{S}), middle panel the deaths, right panel the R

_{0}. [Bottom row] Optimum transmission rates identified from the MC experiments (left panel). Dynamic evolution of the corresponding fraction of the population who recovered (middle panel). Fraction of the asymptomatic infected persons (I

_{A}) among all infected cases (right panel). See text for explanation.

**Figure 5.**Model population dynamics (per million) with the optimal configuration for Susceptibles (S), Exposed individuals (E), Asymptomatic infected individuals (I

_{A}), Infected individuals with symptomatics (I

_{S}), Recovered (R), Dead (D).

**Figure 6.**Dynamic evolution of the simulation with optimal transmission rate (β

_{S}) at each time block. [top row] modelled number of symptomatic infected persons (I

_{S}) (blue, left panel) and deaths (blue, right panel) together with the observed number of positive cases (red circles) and the observed number of deaths (red circles). [bottom row] Modelled R

_{0}(blue, left panel) and β

_{S}(blue, right panel) together with the polynomial fit (right). See text for explanation.

**Figure 7.**The forecast horizon and its variability (top panel), together with the evolution of the new cases (middle; from observations) and the β

_{S}(bottom panel). The lockdown periods are shaded. See text for explanation.

**Table 1.**Prior Distributions based on a uniform distribution U (min, max) during the screening phase and normal distribution N (μ, σ) during the implementation phase. Transmission rate (β

_{S}) values refer to those before, during and after the intervention (lockdown).

Screening Phase | Implementation Phase | ||||
---|---|---|---|---|---|

Parameter | Description | Min | Max | μ | σ |

α | Latency rate (day^{−1}) | 0.3 | 5 | 0.5 | 0.03 |

β_{S} | Transmission rate between I_{s} and S (day^{−1}) | 0.3 | 1.5 | 0.6 | 0.03 |

0.2 | 0.03 | ||||

0.2 | 0.03 | ||||

μ = β_{A}/β_{S} | Transmission rate ratio (day^{−1}) | 0.2 | 1 | 0.35 | 0.037 |

γ_{A} | Recovery rate from subclinical infection (day^{−1}) | 0.07 | 0.5 | 0.25 | 0.03 |

γ_{S} | Recovery rate from clinical symptoms (day^{−1}) | 0.05 | 0.2 | 0.15 | 0.02 |

δ | Transition rate at which I_{A} becomes I_{S} (day^{−1}) | 0.07 | 0.5 | 0.25 | 0.05 |

f_{S} | Deaths (%) | 0.01 | 0.09 | 0.05 | 0.005 |

**Table 2.**Probability of infection (estimated from event occurrence) and human cases per million (estimated from detected cases) with respect to temperature and absolute humidity, based on reported COVID-19 human cases worldwide (01/01/2020–07/05/2020). Shaded cells contain more than 5% of the data.

Probability of Infection (%) | Cases per Million (Median) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

ρ_{v}\T_{AVG} | −10–0 | 0–10 | 10–20 | 20–30 | 30–40 | −10–0 | 0–10 | 10–20 | 20–30 | 30–40 |

0–5 | 17 | 38 | 72 | 10 | 24 | 3.6 | 11.7 | 10.8 | 0.6 | 0.8 |

5–10 | 32 | 50 | 51 | 70 | 3.1 | 5.0 | 1.7 | 2.3 | ||

10–15 | 52 | 35 | 77 | 2.3 | 1.2 | 0.9 | ||||

15–20 | 29 | 25 | 58 | 1.4 | 0.6 | |||||

20–25 | 28 | 37 | 1.1 | 1.8 |

**Table 3.**Posterior distributions of the optimum model parameters (1/day). The numbers in parenthesis denote the interquartile range (IQR) of the 0.5th percentile. Values before, during and after the intervention (MC1: 15/02/2020–31/05/2020). The values in the last column are used in the MC2 runs (15/02/2020–15/12/2020).

Parameter | MC1 | MC2 | ||
---|---|---|---|---|

Before Intervention | During Intervention | After Intervention | ||

α | 0.52 (0.48 0.52) | 0.50 (0.48 0.52) | 0.47 (0.47 0.50) | 0.52 |

β_{S} | 0.61 (0.59 0.62) | 0.16 (0.16 0.19) | 0.15 (0.14 0.20) | |

β_{A} | 0.24 (0.22 0.24) | 0.07 (0.07 0.08) | 0.05 (0.05 0.07) | β_{S}/3 |

γ_{A} | 0.23 (0.22 0.26) | 0.22 (0.22 0.26) | 0.26 (0.24 0.27) | 0.24 |

γ_{S} | 0.15 (0.13 0.15) | 0.15 (0.14 0.16) | 0.15 (0.15 0.16) | 0.15 |

δ | 0.28 (0.24 0.28) | 0.25 (0.23 0.27) | 0.27 (0.23 0.27) | 0.25 |

f_{S} (%)
| 0.041 (0.041 0.046) | 0.057 (0.055 0.058) | 0.059 (0.056 0.060) | 0.03 |

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**MDPI and ACS Style**

Kioutsioukis, I.; Stilianakis, N.I.
On the Transmission Dynamics of SARS-CoV-2 in a Temperate Climate. *Int. J. Environ. Res. Public Health* **2021**, *18*, 1660.
https://doi.org/10.3390/ijerph18041660

**AMA Style**

Kioutsioukis I, Stilianakis NI.
On the Transmission Dynamics of SARS-CoV-2 in a Temperate Climate. *International Journal of Environmental Research and Public Health*. 2021; 18(4):1660.
https://doi.org/10.3390/ijerph18041660

**Chicago/Turabian Style**

Kioutsioukis, Ioannis, and Nikolaos I. Stilianakis.
2021. "On the Transmission Dynamics of SARS-CoV-2 in a Temperate Climate" *International Journal of Environmental Research and Public Health* 18, no. 4: 1660.
https://doi.org/10.3390/ijerph18041660