key: cord-0939591-22as0chj authors: Vazquez, A. title: Superspreaders and lockdown timing explain the power law dynamics of COVID-19 date: 2020-07-24 journal: nan DOI: 10.1101/2020.07.23.20160531 sha: 2d02ec4b81d3dfbbbbc38bc6e65986a0d1139bbf doc_id: 939591 cord_uid: 22as0chj Infectious disease outbreaks are expected to grow exponentially in time, when left unchecked Measures such as lockdown and social distancing can drastically alter the growth dynamics of the outbreak. Indeed the 2019-2020 COVID-19 outbreak is characterized by a power law growth. Strikingly however, the power law exponent is different across countries. Here I illustrate the relationship between these two extreme scenarios, exponential and power law growth, based on the impact of superspreaders and lockdown strategies to contain the outbreak. The theory predicts an inverse relationship between the power law exponent and the speed of the lockdown that is validated by the observed COVID-19 data across different countries. In earlier work I investigated the influence of superspreaders on infectious disease outbreaks 11 . These analyses showed that superspreaders can lead to a new type of infectious disease dynamics that is better described by a power law rather than the canonical exponential growth. To understand how that happens let us have a look at the two trees of disease transmission in Fig. 1A Here ' represents the number of patients zero. ' =1 for the country where the outbreak originated, but it can be larger than 1 in countries where multiple infected cases arrive and start a new outbreak. ' is the average number of secondary cases generated by a patient zero, is the average number of secondary cases generated by infected individuals other than patient zero and ' *+, is the number of new cases at generation d of the outbreak. D is the final generation, when the outbreak ends due to natural extinction or interventions strategies. The remaining part of equation (1) translates generations into infection times. It is basically the time interval distribution of a chain of d disease transmission events, where T is the average time from being infected to disease transmission. When D is very large, equation (1) represents the Taylor expansion of the exponential . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 24, 2020. If > 1 then the outbreak grows exponentially over time, else if > 1 the outbreak decays exponentially. This is the canonical expectation of infectious disease dynamics. In this scenario the key quantity is the reproductive number R and interventions strategies are focused on bringing it below 1. However, there are a number of assumptions behind equation (2) that make it inadequate to model all infectious diseases outbreaks. First, we have just assumed that D is large, i.e. that the free spreading of the disease goes over several generations. That would typically be true for infectious diseases with mild symptoms such as the common cold, but it is not the case for COVID-19. The mortality and hospitalization rate of COVID-19 infections have led governments to impose strict lockdown measures. As a consequence, the tree of disease transmission is truncated after a few generations, as shown in Fig. 1B . Second, the distinction between patient zero and the other infected individuals needs a deeper analysis. The disease spreading introduces a bias towards individuals with larger daily person-to-person proximity contacts. For example, if the disease is transmitted via the daily proximity patterns between individuals and each individual is in contact with others at a rate Λ = , then 12 where () denotes the expectation based on the distribution of Λ = in the population. If Λ = exhibits small variations around its expected value then ≈ ' . However, the existence of superspreaders tell us that there are a few individuals with very large Λ = . In reality (Λ E ) ≫ (Λ) E 6 , implying that ≫ ' . . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 24, 2020. . https://doi.org/10.1101/2020.07.23.20160531 doi: medRxiv preprint When these two elements are taken into consideration, a small number of generations D and the existence of superspreaders, then equation (1) In short, the number of infected cases from one generation to the next can increase so dramatically that the number of new daily infectious will be dominated by the time the individuals at the last layer get infected. The power law behaviour predicted by equation (4) is so different from the readily explainable exponential behaviour (2) that it has been neglected for 14 years. However, recent reports indicate that the COVID-19 outbreak is in fact better fitted by a power law growth rather than an exponential growth 1,2 . Equation (4) makes a testable prediction, that the exponent of the power law growth, ( )~H, depends on D, In the COVID-19 context, the number D of generations, or rounds of unhindered spread, in a population is determined the time between contact with patients zero and a lockdown measure imposed. Therefore D can be estimated as the number generations of disease transmission from patient zero to the implementation of lockdown, . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. The copyright holder for this preprint this version posted July 24, 2020. . https://doi.org/10.1101/2020.07.23.20160531 doi: medRxiv preprint from being infected to disease transmission. Combining equations (5) and (6) then yields To test the equation above I collected data for the power law exponents 2 and the date of first confirmed case (World Health Organization, WHO) associated with the COVID-19 outbreak in several countries. Excluding China, there is a clear inverse correlation between the date the first confirmed case was reported and the estimated power law exponent (Fig. 2) . To test equation (7), T was estimated as the COVID-19 incubation time, which is approximately 5 days 13 . I have also assumed that most countries (China excluded) implemented lockdown measures around March 20 th (79 days from Jan 1 st ) 14 . Based on these parameter estimates we obtain the theoretical line predicted by equation (7), which is in very good agreement with the data in Fig. 2 . In conclusion, the power law dynamics of the COVID-19 outbreak and the relationship between the power law exponent and the time interval between first case and lockdown, are a validation of the new power law of infectious disease spreading 11 . Again this study also underscores the crucial importance of early lockdown timing in the control of an infectious disease in a population with superspreaders. The mathematical formulation leading to equation (1) was reported in Ref. 11 . The time of first confirmed case were retrieved from the WHO website at https://covid19.who.int/. . CC-BY-NC 4.0 International license It is made available under a is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. (which was not certified by peer review) The copyright holder for this preprint this version posted July 24, 2020. . https://doi.org/10.1101/2020.07.23.20160531 doi: medRxiv preprint Fractal kinetics of COVID-19 pandemic The COVID-19 pandemic: growth patterns, power law scaling, and saturation Emergence of scaling in random networks The web of human sexual contacts Epidemic spreading in scale-free networks Modelling disease outbreaks in realistic urban social networks Super-spreaders in infectious diseases Superspreading SARS events Discrete Methods in Epidemiology DIMACS Series in Discrete Mathematics and Theoretical Computer Science The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application The time of lockdown is reported in Ref 14 and available at https://www.politico.eu/article/europescoronavirus-lockdown-measures-compared/. This work was supported by Cancer Research UK C596/A21140. AV conceived the project and wrote the manuscript. The authors declare no competing interests.