key: cord-0816523-xvfab24g authors: Fokas, A. S.; Dikaios, N.; Kastis, G. A. title: COVID-19: Predictive Mathematical Models for the Number of Deaths in South Korea, Italy, Spain, France, UK, Germany, and USA date: 2020-05-12 journal: nan DOI: 10.1101/2020.05.08.20095489 sha: c131e435329bf6d365369c98dc3ec9448b34e957 doc_id: 816523 cord_uid: xvfab24g We have recently introduced two novel mathematical models for characterizing the dynamics of the cumulative number of individuals in a given country reported to be infected with COVID-19. Here we show that these models can also be used for determining the time-evolution of the associated number of deaths. In particular, using data up to around the time that the rate of deaths reaches a maximum, these models provide estimates for the time that a plateau will be reached signifying that the epidemic is approaching its end, as well as for the cumulative number of deaths at that time. The plateau is defined to occur when the rate of deaths is 5% of the maximum rate. Results are presented for South Korea, Italy, Spain, France, UK, Germany, and USA. The number of COVID-19 deaths in other counties can be analyzed similarly. unprecedented mobilization of the scientific community has already led to remarkable progress 38 towards combating this threat, such as understanding significant features of the virus at the 39 molecular level, see for example (1) and (2). In addition, international efforts have intensified 40 towards the development of specific pharmacological interventions; they include, clinical trials 41 using old or relatively new medications and the employment of specific monoclonal antibodies, as to viruses, and that complement deposits are abundant in the lung biopsies from SARS-CoV-2 48 patients indicating that this system is presumably overacting (6), it has been suggested that anti-49 complement therapies may be beneficial to SARS-CoV-2 patients. Further support for this 50 suggestion is provided by earlier studies showing that the activation of various components of the 51 complement system exacerbates the acute respiratory distress syndrome disease associated with 52 SARS-CoV (6). The Federal Drug Administration of USA has granted a conditional approval to 53 the anti-viral medication Remdesivir (7). Unfortunately, the combination of the anti-viral 54 medications lopinavir and ritonavir that are effective against the human immunodeficiency virus 55 has not shown any benefits (4). Similarly, the combination of the anti-malarial medication 56 hydroxychloroquine and the antibiotic azithromycin, is not only ineffective but can be harmful (5). 57 The scientific community is also playing an important role in advising policy makers of possible 58 non-pharmacological approaches to limit the catastrophic impact of the pandemic. For example, 59 following the analysis in (8) of two possible strategies, called mitigation and suppression, for 60 combating the epidemic, UK switched from mitigation to suppression. Within this context, in order 61 to design a long-term strategy, it is necessary to be able to predict important features of the COVID-62 19 epidemic, such as the final accumulative number of deaths. Clearly, this requires the 63 development of predictive mathematical models. 64 In a recent paper (9) we presented a model for the dynamics of the accumulative number of 65 individuals in a given country that are reported at time t to be infected by COVID-19. This model 66 is based on a particular ordinary differential equation of the Riccati type, which is specified by a In the particular case that the associate time-dependent function is a constant, the explicit solution 76 of the above Riccati equation becomes the classical logistic formula. It was shown in (9) that 77 although this formula provides an adequate fit of the given data, is not sufficiently accurate for 78 predictive purposes. In order to provide more accurate predictions, we introduced two novel 79 models, called rational and birational. Here we will show that the Ricatti equation introduced in (9) can also be used for determining the 82 time evolution of the number, N(t), of deaths in a given country caused by the COVID-19 epidemic. (which was not certified by peer review) is the author/funder, who has granted medRxiv a license to display the preprint in perpetuity. It turns out that the birational formula, in general, yields better predictions that the rational, which 100 in turn provides better predictions than the logistic. Also, in general, the birational curve is above 101 the curve obtained from the data, whereas the rational curve is below. Thus, the birational and (which was not certified by peer review) 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 May 12, 2020. For the epidemic of the UK (Fig. 3D) For the epidemic of Germany (Fig. 3E ), the logistic model predicts a plateau on May 17, 2020 (59 186 days after the day that 25 deaths were reported) with 6,830 deaths; the rational model predicts a 187 plateau on June 16, 2020 (day 89) with 8,702 deaths. For the epidemic of the USA (Fig. 3F) , the 188 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 May 12, 2020. It is evident from the above that whereas each of the three equations (3) The fact that a is now a function of t has important implications. In particular, it made it possible 220 to construct the rational and birational models. All two models, as well as the logistic model provide 221 good fits for the available data. However, as discussed in detail above, the rational and birational 222 models provide more accurate predictions. Furthermore, the birational model may provide an 223 upper bound of the actual N(t), whereas the rational model yields a better lower bound that the 224 logistic model. 225 It is noted that our approach has the capacity for increasing continuously the accuracy of the 226 predictions: as soon as the epidemic in a given country passes the time T, the rational model can be 227 used; furthermore, when the sigmoidal part of the curve is approached, the rational model can be 228 supplemented with the birational model. Also, as more data become available, the parameters of 229 the rational and of the birational models can be re-evaluated; this will yield better predictions. (which was not certified by peer review) 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 May 12, 2020. (21). In any case, whatever strategy is followed, it is natural to expect that the number of reported 248 infected individuals as well as the number of deaths will begin to grow. At this stage the predictions 249 of the models proposed in (9) and here will not be valid. However, these works will still be valuable: 250 they can be used to compute the additional number of reported infected individuals and deaths 251 caused by easing the lockdown measures. Let us hope that a prudent exit strategy is adopted so that 252 these numbers will not be staggering. The stability of the fitting procedure was established by using the following simple criterion: 272 different fitting attempts based on the use of a fixed number of data points, must yield curves which 273 have the same form beyond the above fixed points. In this way, it was established that the rational 274 formula could be employed provided that data were available until around the time T, whereas the 275 birational formula could be used only for data available well beyond T. The fitting accuracy of each model was evaluated by fitting the associated formula on all the 278 available data in a specified set. The relevant parameters specifying the logistic, rational, and 279 birational models are given on table 1. 280 We assume that the function N(t) satisfies the ordinary differential equation 281 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 May 12, 2020. . https://doi.org/10.1101/2020.05.08.20095489 doi: medRxiv preprint Page 7 of 13 = ( ) − . (2) 282 The general solution of this equation is given by (9) 283 284 304 If b, c, d, k, are close to b1, c1, d1, k1, then Nf is close to c1, and hence the value of α(t) after t=X is 305 close to the value of α(t) before t= X. The constant T can be computed by solving the equation obtained by equating to zero the second 308 derivative of N. This implies that for the logistic and the rational models T is given respectively by 309 310 = ln( ) , (1 + ) = − 1 + 1 . 311 All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 May 12, 2020. All rights reserved. No reuse allowed without permission. (which was not certified by peer review) 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 May 12, 2020. (which was not certified by peer review) 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 May 12, 2020. . https://doi.org/10.1101/2020.05.08.20095489 doi: medRxiv preprint A pneumonia outbreak associated with a new 319 coronavirus of probable bat origin Structural 321 basis of receptor recognition by SARS-CoV-2 Infectious Diseases Society of America Guidelines on the 325 Treatment and Management of Patients with COVID-19 Infection A trial of lopinavir-ritonavir in adults hospitalized with severe Outcomes of hydroxychloroquine usage in United States veterans hospitalized with Complement as a target in COVID-19? Impact of nonpharmaceutical interventions (NPIs) 346 to reduce COVID-19 mortality and healthcare demand Response Team Predictive mathematical models for the number of 349 individuals infected with COVID-19. medRxiv Modeling and forecasting the early evolution of the Covid-19 351 pandemic in Brazil Modeling Covid-19 dynamics for real-time estimates and projections: an 355 application to Albanian data The first 100 days: modeling the evolution of the 357 COVID-19 pandemic A mathematical model for the novel coronavirus epidemic in Wuhan No reuse allowed without permission. (which was not certified by peer review) 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 Projected Development of COVID-19 in Louisiana A mathematical model for the spatiotemporal epidemic 364 spreading of A spatial model of CoVID-19 366 transmission in England and Wales: early spread and peak timing Tracing DAY-ZERO and Forecasting the Fade out of the COVID-19 Outbreak in Italy: A Compartmental Modelling and Numerical Optimization Approach Data-based analysis, modelling and 373 forecasting of the COVID-19 outbreak The psychological impact of quarantine and how to reduce it: rapid review of the 376 evidence Two alternative scenarios for easing 378 COVID-19 lockdown measures: one reasonable and one catastrophic (preprint) Bound Constrained Optimization Convergence Properties of the Simplex Method in Low Dimensions The Levenberg-Marquardt Algorithm: Implementation and Theory An Interior, Trust Region Approach for Nonlinear Minimization 391 Subject to Bounds Competing interests: The authors declare no competing interests. 407 Data and materials availability: All data needed to evaluate the conclusions in the paper 408 are present in the paper and/or the Supplementary Materials. 409 All rights reserved. No reuse allowed without permission.(which was not certified by peer review) 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 May 12, 2020. (which was not certified by peer review) 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 May 12, 2020. 453 All rights reserved. No reuse allowed without permission.(which was not certified by peer review) 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 May 12, 2020. . https://doi.org/10.1101/2020.05.08.20095489 doi: medRxiv preprint