key: cord-0686524-lfiemurn authors: Dias Junior, T.; Machado, C. B. title: A Computational Model for Estimating the Evolution of COVID-19 in Rondenia-Brazil date: 2020-05-09 journal: nan DOI: 10.1101/2020.05.05.20091942 sha: 84cf2c8e235df069e9dadbc1bf231b6a3cbfe887 doc_id: 686524 cord_uid: lfiemurn In this work, the modified SEIR model was proposed to account separately for the tested and isolated cases, with severe and critical symptoms, from those not tested, with mild and moderate symptoms. Two parameters were estimated and evaluated for the cases registered in Rondenia-Brazil, between March 20 and April 22. The basic reproduction rate did not remain constant during the period, showing variations eventually due to social behavior. The results show that the increase in the proportion of testing to about 56% provided a significant decrease in the confirmed cases since the expansion of the tested cases beyond the current testing criterion (20%) would help to identify and isolate also mild and moderate cases, generally referred to as asymptomatic. In December 2019, COVID-19 (Coronavirus disease 2019), caused by SARS-CoV-2, emerged in Wuhan, China, and, despite the extensive and severe containment measures implemented by the Chinese government, it was disseminated to other regions, reaching several countries on all continents currently [1] . In Brazil, the first confirmed case for COVID-19 was a 61-year-old male patient with a history of travel from Italy registered on 02/25/2020 in São Paulo-SP. In Rondônia State (RO), Brazil, it was registered the first confirmed case on 03/20/2020, of a patient with a history of travel to Ji-Paraná / RO from São Paulo / SP [2] . On the same date, the State Government of Rondônia published the Decree No. 24,887 of 20 March 2020 that implemented comprehensive measures of social isolation [3] . In this work, we utilized the official data made available by ANVISA (National Health Surveillance Agency) for Rondônia from 03/20/2020 to 04/22/2020 in the study of the proposed mathematical model to estimate epidemiological parameters related to behavior population in response to social isolation measures enacted by the State Government. The present study aimed to propose a mathematical model that represents the quantitative epidemiological evolution of COVID-19 in Rondônia, as well as to evaluate the influence of social behavior parameters and testing criteria on the number of confirmed cases, and consequently on the need for hospitalizations of severe and critical cases and the mortality rate. For the modeling carried out in this work, we proposed a modification of the SEIR model (Susceptible-Exposed-Infected-Removed) [4] [5] [6] . In the model proposed here, the number of susceptible individuals S decreases according to the rate of effective contacts between the untested infected and the susceptible (β), assuming that the tested infected remain isolated according to the guidelines of the health authorities [7] [8] [9] . The number of individuals 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 9, 2020. . https://doi.org/10.1101/2020.05.05.20091942 doi: medRxiv preprint exposed increases in proportion to the decrease in susceptible individuals and decreases with the proportion that those exposed become infected after the average incubation period (1/ ) . In this work, unlike traditional SEIR modeling, we chose to classify those infected, recovered and deceased in two groups, those from tested and untested groups. Thus, the parameter is introduced to quantify the proportion of individuals tested, so that in the computer simulation phase it is possible to evaluate the effect of the testing strategy on the results obtained. Therefore, the number of infected individuals tested will increase with the number of exposed individuals who have undergone incubation and decrease with those who have recovered or died, according to the mortality rate and the average period of infection 1/ . In the obtained equation, it can be demonstrated, using the method of the subsequent generation matrix [4] , that the basic rate of reproduction is defined by the 0 ( ) = / , it may vary over time, since it is assumed that the rate of effective contacts ( ) depends on the strategy of social isolation or the use of collective masks. The total effective population considered in modeling viral transmission can be expressed by = + + + + + , where corresponds to susceptible individuals, to individuals exposed to the virus, and to the tested and untested infected, respectively, and and to the tested and untested recoveries, and those who do not recover die ( and ) . Thus, the mathematical model can be formulated as: 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 9, 2020. . https://doi.org/10.1101/2020.05.05.20091942 doi: medRxiv preprint For the numerical solution of the system of eight ordinary differential equations resulting from the mathematical modeling of the problem, we employed the 4th order Runge-Kutta algorithm. For this, the Python programming language was used, through the Numpy libraries [10] and Scipy [11] . The algorithm for the integration step keep the local error of integration limited by a linear combination of the relative (10 −3 ) and absolute tolerances (10 −6 ). All initial values for the variables are null, except for the number of susceptible individuals that is equal to the population in 2019 estimated by the IBGE (Brazilian Institute of Geography and Statistics) for the region under study minus the number of untested cases that was one. Thus, we considered a healthy population fully susceptible to have had initial contact with only one infected and untested case. We obtained the system solutions for some sets of parameters corresponding to each case analyzed, considering, for example, the initial population of susceptible individuals, the mortality rate, the proportion of individuals tested, and other parameters, according to the hypothetical case established. To estimate the evolution of COVID-19 cases in Rondônia, the official data published by ANVISA (National Health Surveillance Agency) on 04/22/2020 [12] , containing the numbers of new and accumulated cases and new and accumulated deaths, were used. The initial number of susceptible individuals was considered equal to the population of Rondônia estimated by the IBGE for 2019 [13] . We estimate the proportion of tested and untested cases at = 0.2 and the mortality rate at = 0.02, according to official data for Rondônia. To (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 9, 2020. . https://doi.org/10.1101/2020.05.05.20091942 doi: medRxiv preprint the Erlang distribution with equal to two, obtaining periods greater than or equal to one variable around the suggested means [6, 14, 15] . The model estimated the rate of effective contacts from the basic reproduction rate and the period of infection ( = 0 ). We performed the simulations for three values of the basic reproduction rate 0 admitted constant over time and another with variable 0 with , as a result of changes in social isolation behavior or the collective use of masks, allowing intervals for 0 ( ) or phases. For variable 0 ( ), we evaluated the influence of the proportion of testing on the number of accumulated known cases ( cases that present clinical symptoms: fever and at least one respiratory symptom and one of 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 9, 2020. . https://doi.org/10.1101/2020.05.05.20091942 doi: medRxiv preprint the epidemiological criteria: a history of travel to an area with local transmission or history of close contact with a suspected case to COVID-19, in the 14 days before the onset of symptoms [17] . In this context, the WHO (World Health Organization) estimated that, in China, 80% of confirmed patients developed mild to moderate symptoms, 13.8% had severe symptoms, and 6.1% critical symptoms [18] . Therefore, considering the testing criteria used and the incidence of severe and critical cases mentioned, we estimated the proportion of those tested at 20% ( = 0.2). The estimated lethality rate was 2%. To assess the influence of the basic reproduction rate on the adequacy of the proposed theoretical model to confirmed cases registered in Rondônia, we simulated the mathematical model for 40 days for three situations with different arbitrated values for the basic reproduction rate ( 0 ) constant during the simulated period, the first day was that with the appearance of the first confirmed case. In Figure 1 , the higher the value of 0 , the greater the number of cases accumulated during the simulated period. Official data for confirmed cases are shown [12] , and the proposed model captured its exponential trend, demonstrating in general terms the adequacy 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 9, 2020. . https://doi.org/10.1101/2020.05.05.20091942 doi: medRxiv preprint of the model to the registered reality. However, the simulations performed for 0 constant in the considered time interval do not fit precisely with the data, suggesting that the basic reproduction rate did not remain constant in the recorded period. Figure 2 shows the same simulations and real data for the logarithmic scale of cases, displaying the slopes of the curves from the simulations and the variations in the slope of the recorded data. Note that until the 15th day the slope of the trend of confirmed cases tends to approach the curve 0 = 2.2, between the 15th and the 25th, it approaches the curve 0 = 2.3, after the 25th day it approaches the curve 0 = 2.4 until the 31st day, and then following the trend of approaching the curve 0 = 2.6. From the analysis of Figure 2 , we could roughly estimate the values for the basic reproduction rate as varying in the period considered according to eq. (2), which establishes four intervals or phases. Figure 3 shows, in a logarithmic scale, that the simulation result approximates more accurately to the recorded data, validating the estimate of the variation of the basic reproduction rate in the considered interval. It is important to note that the rate of 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 9, 2020. Although it is a projection, considering the current trend that can change according to social behavior, the projected cases serve as estimated information for logistical decisions. Between these, provisioning beds for hospitalization of severe and critical cases (6.1%), the number of kits for tests, as well as morgue infrastructure, taking into account that the mortality rate, currently estimated at 2%, may increase as the adequate treatment capacity is saturated and the public health collapses. 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 9, 2020. As shown in Figure 1 , for the model used, the increase in the basic reproduction rate causes an increase in the accumulated number of tested cases and, consequently, deaths, maintaining the same criteria and proportion of testing and estimated mortality rate. To assess the effect of the influence of the basic reproduction rate on the calibrated model, three cases and values of 0 ( ) for ≥ 31 were established: maintenance of current isolation attitudes 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 9, 2020. . https://doi.org/10.1101/2020.05.05.20091942 doi: medRxiv preprint ( 0 ( ) = 2.6), relaxation ( 0 ( ) = 2.8) and intensification ( 0 ( ) = 2.0). Table 1 shows that small changes in the basic reproduction rate, after the 31st day, result in significant changes in the number of days required to reach the peak of active cases, demonstrating the need for social isolation and use of homemade masks to reduce and delay the peak of new cases as shown in Figure 5 , in the absence of new treatments or vaccines against the virus. 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 9, 2020. . https://doi.org/10.1101/2020.05.05.20091942 doi: medRxiv preprint Another critical parameter is the proportion of testing ( ) because, in the model, it represents the proportion of infected people tested and submitted to quarantine, preventing their contact with the rest of the population. Here, we estimated it at 20% due to the criteria and prevalence of severe cases and critics of COVID-19 reported in the literature. We assessed this testing parameter influence and, therefore, the influence of the testing criteria in the accumulated cases obtained by the model; the parameter varied between 0.1 and 0.65. 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 9, 2020. . https://doi.org/10.1101/2020.05.05.20091942 doi: medRxiv preprint 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 9, 2020. The results show that the basic reproduction rate did not remain constant during the analyzed period, showing variations in social behavior and variations in the flow of testing or in the flow of official daily records, which we cannot estimate with the official information available. The temporal projections showed that the decrease in the basic reproduction rate for the next days, after the recorded data period, has a significant effect on the peak of active cases, as well as in the period in which it will occur in the future. Regarding the proportion of testing, the results demonstrate the effectiveness of its increase beyond 40%, which causes a decrease in infected cases, since they can be immediately isolated and reducing the transmission speed. Thus, changing the testing criteria to include cases considered mild or moderate, easily confused, and described as asymptomatic, has beneficial effects. And, despite testing costs, increased testing can bring economic benefits if used in conjunction with selective isolation monitoring and implementation tools. During the period evaluated, according to the testing criteria, we conservatively estimated that the proportion of infected people tested was 20%, and according to the results obtained for the proposed model, its beneficial effects become very significant from about 56% of the infected population, thus including, in addition to all severe and critical cases, more mild and moderate cases in the testing criteria. In conclusion, despite the limitations and simplifications inherent to any process of mathematical modeling of natural phenomena, the proposed model can be used as a basis to assist the planning of intensification or relaxation actions of social isolation to match the demand for treatment with infrastructure and adequate service capacity. 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 9, 2020. . https://doi.org/10.1101/2020.05.05.20091942 doi: medRxiv preprint COVID-19 situation reports n Vigilância em Saúde -Confirmado o primeiro caso de coronavírus em Rondônia -Governo do Estado de Rondônia n Reproduction numbers of infectious disease models A deterministic epidemic model for the emergence of COVID-19 in China A Computational Model for Estimating the Progression of COVID-19 Cases in the US West and East Coasts Modeling the Spreading Risk of 2019-nCoV n A decision-support framework to optimize border control for global outbreak mitigation A contribution to the mathematical theory of epidemics The NumPy array: A structure for efficient numerical computation SciPy 1.0: fundamental algorithms for scientific computing in Python The Incubation Period of Coronavirus Disease 2019 (COVID-19) From Publicly Reported Confirmed Cases: Estimation and Application Plano Estadual de Contingência ao Coronavírus Report of the WHO-China Joint Mission on Coronavirus Disease The authors declare that they have no competing interests.