key: cord-281376-1xdj06m7 authors: Cirilo, E. R.; Candezano, M.; Natti, P.; Romeiro, N.; Polo, J. title: One Study of COVID-19 Spreading at The United States - Brazil - Colombia date: 2020-09-02 journal: nan DOI: 10.1101/2020.08.29.20184465 sha: doc_id: 281376 cord_uid: 1xdj06m7 The present work concerns the COVID-19's spread over The United States, Brazil and Colombia. Although countries show differences in economic development, but similarities such as continental dimension or social interaction, the spread of COVID-19 in them has some similarities. At the moment, the countries are living the disease with temporal delay. Thus, we used a database on WHO Coronavirus, Mathematical Modeling and Numerical Simulations to describe the most recent COVID-19 development patterns in these countries, which we saw. The SARS-CoV-2 coronavirus pandemic causes the COVID-19 disease, for which we have no immune response or vaccine. The origin of COVID-19 is believed to have occurred in Wuhan, China, in late 2019. From China, the disease was rapidly transmitted globally by individuals who travelled to Europe and The United States. On the America continent, the rst cases of COVID-19 appeared in The United States. On January 21, 2020, the American Center for Disease Control and Prevention (CDC) conrmed the rst case of COVID-19 in a 35-year-old man from 3% of the total number of reported cases in the American continent [4] . The United States, Brazil and Colombia are in dierent moments in the epidemiological process, Colombia is in the exponential growth phase, Brazil is probably in the peak of the epidemic, while The United States is already experiencing a second wave of SARS-CoV-2 coronavirus infection. It is noteworthy that these three countries have very dierent Human Development Indexes (HDI) and territorial dimensions and, even so, they have in common high infection rates by the SARS-CoV-2 coronavirus. The objective of this article is to carry out a mathematical study of the possible trends of the epidemic by COVID-19 in these three countries. This article was based on the data provided by the WHO [4] and the Susceptibles-Infectious-Recovered-Dead model, the SIRD model [5] [6] . The SIRD model is a classic compartmental model of the Kermack -McKendrick type [7] [8] . Compartmental models divide the population into several different compartments, for example, Susceptible population, Infectious population, Recovered population, Dead population, among others, and specify how individuals move through the compartments over time. Despite being a simple mathematical model, the SIRD is one of the most applied mathematical models to understand the current health crisis. Obviously, more realistic and complex models would better describe the dynamics of this epidemic, but data and information about COVID-19 are lacking to implement them. Reviews of epidemiological models can be found in [5] [6] . Regarding the adjustment of the parameters of the SIRD model for The United States, Brazil and Colombia, it appears that these parameters change frequently, depending on local political factors (closure of non-essential establishments, quarantine, movement restrictions...) [9] , socio-economic factors (social and hygienic behaviors, lower per-capita income...) [10] [11] , climatic factors (temperature, humidity, average wind speed, UV index...) [12] , among others. In this context, it was decided to adjust the parameters of the SIRD model over time, based on data made available by WHO, using non-linear least squares methods [13] [14] . Regarding the numerical procedures to solve the SIRD model, rst, the dis-. CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 2, 2020. . https://doi.org/10.1101/2020.08.29.20184465 doi: medRxiv preprint cretization of the ordinary dierential equation system was performed using the Finite Dierence Method. The resulting linear system of non-linear equations was solved iteratively by the Gauss-Seidel method until the convergence criterion was overcome. It was also found that all the matrices's coecients of the iterative processes satisfy the Sassenfeld convergence criterion [15] In fact the coronavirus disease is not completely understanding at the moment, certainly there are additional parameters unknown that describing better the pattern it. So, we chose to take several data sets consolidated of the 14 days to do our simulations in order to get more realistic information. In this way, we got a simulations' clustering which permitted to understanding the fundamentals of the behavior coronavirus disease over the countries. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 2, 2020. . https://doi.org/10.1101/2020.08.29.20184465 doi: medRxiv preprint There are several mathematical modelling of the COVID-19 at the moment. But, to this work we considering the SIRD's model given by equations: t, S, I, R, D are time, susceptible, infected, recovered and death variables, respectively; N is an average population value; β is an infection rate, and γ R and γ D are recover and death rates, these parameters are calculated per day. At the initial time we have and for end time, denoted by t F , we admitted The condition (2.3) was considered because at this moment the disease nished and the equations' steady state stays established. The model (2.1) has 4 ordinary dierential equations (ODE). Specically, any ODE can be written as . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 2, 2020. . https://doi.org/10.1101/2020.08.29.20184465 doi: medRxiv preprint so that therefore, all deductive numeric procedure at this work will be over equation (2.4) . The temporal discret grid considered is showed at the Figure (1) . We denoted the time lapse like ∆t, and the end time t F = (s − 1) ∆t. The value k is a time counter, thereby k = s means temporal nodes' total value. and the Neumann condition (2.3) we have too With respect to temporal derivative of the equations (2.4), they are approximated by forward second order nite dierence in the node k dΦ l dt similarly, the equations (2.6) are approximated by backward second order nite dierence at the last node s dΦ l dt These approximations were used to obtain improved numerical results. . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 2, 2020. . https://doi.org/10.1101/2020.08.29.20184465 doi: medRxiv preprint (2.10) The equality (2.10) is an our temporal dierence equation, that describe the COVID-19 behaviour. We want to explain that the equation (2.10) is the temporal evolution for Φ l . Numerically, from Figure (2), setting s we have to vary k = 1, . . . , s − 2 to obtain the dierences' equations at the points k = 2, . . . , s − 1. . (2.11) . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 2, 2020. . https://doi.org/10.1101/2020.08.29.20184465 doi: medRxiv preprint Note that the system (2.11) is not a linear system. The terms values were explicit. But they are unknown! Thus, we use a iterative strategy to solve the system (2.11) how presented in [23] . Without loss of generality, if s = 4 the system (2.11) becomes l whose the iterative process associated can be like where IT = 0, 1, . . . , IT M AX , and IT M AX means iteration maximum number. Particularly, we have a linear system to each specic IT . Furthermore, we choose the Gauss-Seidel method to solve (2.12) because the convergence is guaranteed. it is easy to see that the Sassenfeld criterium is veried. The stopped criteria value of the Gauss-Seidel method used in this work was 10 −7 . When IT = 0 the system (2.12) becomes so that: • Φ 1 l is known (initial condition); • Ψ 1 l = Ψ 1 l Φ 1 l is known, of course; • 0 Ψ 2 l = Ψ 1 l is admitted how our prediction hypothesis. Thus, we have the linear system's solution Gauss-Seidel method. Now, if IT = 1 we nd so that 1 Ψ 2 l = Ψ l 1 Φ l is our prediction hypothesis again. It solving (2.14), we get After this, if the 2 Φ 2 is less than one (which means an end to the spread of covid-19) we accept the 2 Φ l ≡ Φ l as numerical solution of the SIRD's (2.4) at the time t = 2∆t. Otherwise, we set s = 5 and do the all process again to a new time t = 3∆t, and so on. The numerical process nishes if we get the m Φ 2 < 1 to some 0 < m < IT M AX . Finally, we have some gains with strategy provided here: . CC-BY-NC-ND 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 2, 2020. . https://doi.org/10.1101/2020.08.29.20184465 doi: medRxiv preprint • the governing equations system (2.4) is solved by a convergent methodology to any time; • our implicit methodology does not need a severe restriction on the ∆t value. The implicit method together with Gauss-Seidel and Lax's law guarantee it. Besides, the t F time is discovered in the computational run's time. To performance all simulation we developed a fortran 90 code that solve the SIRD's governing equations from auxiliary conditions. The parameters' optimization in the SIRD (2.1) is done as in [24, 25] , that is, solving a nonlinear least square problem. We dene the vector function u(t) = (S(t), I(t), R(t), D(t)), the vectors of parameters q = (β, γ R , γ D ) and the known data y at times t 1 , . . . t n . Given a function F (u, q), we estimate the parameters q by solving the following nonlinear least square problem The trust-region based method adapted in the lsqnonlin Matlab function is used for the constrained optimization problem (2.15). In this section we introduce our analyse of the COVID-19 and simulations as well. The reader will see here that the country's richness, its continental dimension or is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 2, 2020. . https://doi.org/10.1101/2020.08.29.20184465 doi: medRxiv preprint is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 2, 2020. . https://doi.org/10.1101/2020.08.29.20184465 doi: medRxiv preprint peak is being attenuating at the country. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 2, 2020. . https://doi.org/10.1101/2020.08.29.20184465 doi: medRxiv preprint so far. Colombia, whose simulations started in the date 2020-05-14, presents the same patter of Brazil's susceptible, see the Figure is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 2, 2020. . https://doi.org/10.1101/2020.08.29.20184465 doi: medRxiv preprint is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted September 2, 2020. . https://doi.org/10.1101/2020.08.29.20184465 doi: medRxiv preprint Finally, we emphasize that the COVID-19's dynamic of the Brazil and Colombia are the same, but as delay. Of course the next days are crucials, the countries cannot relax your ght actions against COVID-19. The our simulations predict that there are some control over COVID-19 disease in theses countries, yet. Clearly, we do not know the specic proceedings which provide it. But if they relax, the countries could have a second wave infection similar to the one the United States is currently experiencing. Resumo. O presente trabalho trata do espalhamento da COVID-19 sobre os Estados Unidos, Brasil e Colômbia. Embora os países exibam diferenças no desenvolvimento econômico, mas similaridades tais como dimensão continental ou interação social, a propagação do COVID-19 neles tem algumas semelhanças. Agora, os países estão vivendo a doença com atraso temporal. Assim, usamos um banco de dados de Coronavirus da OMS, Modelagem Matemática e Simulações Numéricas para descrever o padrão mais recente de desenvolvimento do COVID-19 nesses países, que nós observamos. Palavras-chave. Modelagem Matemática, Simulações Numéricas, COVID-19. 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