Epidemic analysis of COVID-19 in Brazil by a generalized SEIR model
EE PIDEMIC ANALYSIS OF
COVID-19 IN B RAZIL BY AGENERALIZED
SEIR
MODEL
A P
REPRINT
Suzete Maria Silva Afonso
Universidade Estadual Paulista (UNESP)Instituto de Geociências e Ciências Exatas, 13506-900, Rio Claro-SP, Brasil. [email protected]
Juarez dos Santos Azevedo
Universidade Federal da Bahia (UFBA)Instituto de Ciências, Tecnologia e Inovação, Centro, 42802-721, Camaçari-BA, Brazil [email protected]
Mariana Pinheiro Gomes da Silva
Universidade Federal do Recôncavo da Bahia (UFRB)Centro de Ciências Exatas e Tecnológicas, Centro, 44380-000, Cruz das Almas-BA, Brazil. [email protected] th May, 2020 A BSTRACT
We shall apply a generalized SEIR model to study the outbreak of COVID-19 in Brazil. In particular,we would like to explain the projections of the increase in the level of infection over a long period oftime, overlapping large local outbreaks in the most populous states in the country. A time-dependentdynamic SEIR model inspired in a model previously used during the outbreak in China was usedto analyses the time trajectories of infected, recovered, and deaths. The model has parameters thatvary with time and are fitted considering a nonlinear least-squares method. The simulations startingfrom April 8, 2020, concluded that the time for a peak in Brazil will be in July 21, 2020 with totalcumulative infected cases around 982K people; in addition, an estimated total death case will reachto 192K in the end. Besides that, Brazil will reach a peak in terms of daily new infected cases anddeath cases around the middle of July with 50K cases of infected and almost 6.0K daily deaths. K eywords Epidemic Model · Covid-19 · SEIR model · Analysis in Brazil
In recent times, COVID-19 has become one of humanity’s biggest epidemic problems, gaining pandemic status and,having a significant impact on mortality rates in both rich and poor countries. The current balance (May 18, 2020)of World Health Organization (WHO) records that a total of around 5 million people were infected by the COVID-19 causing 320K deaths [3]. In Brazil, since the identification of the first case of COVID-19, in February, 2020,approximately 310 thousand cases of the disease have already been identified and almost 20 thousand deaths (Ministryof Health of Brazil, in the last epidemiological bulletin at (May 18, 2020)) [2]. Currently, this epidemic is one of themain public health problems in the world. The alarming numbers of this epidemic have encouraged several authors towork on epidemic models that can predict the progress of this disease.Some mathematical models are proposed for studying the spread of viruses among humans, such as SIR (Susceptible,Infected, Recovered) model [5, 9, 14, 18] and SEIR (Susceptible, Exposed, Infected, Recovered) model [4, 7, 15, 17]. a r X i v : . [ q - b i o . P E ] M a y PREPRINT - 26 TH M AY , 2020The SEIR model derives from the SIR model and require a large amount of data over time to execute. Here, we followthe adapted SEIR model developed by Tan and Chen in [13], which is the most reliable response to emergency publichealth actions and takes into account the population exposed to the virus.More specifically, we use the generalized version of the classical SEIR known as SEIRDP (Susceptible, Exposed,Infected, Recovered, Death, Insusceptible (P)) which is a particular case of the model proposed by [10]. In this model itis possible to include key epidemic parameters for COVID-19, such as the latent time, infected time, protection rateand time evolution of the recovery. This allows us to estimate the inflection point, ending time and total infected casesin Brazil and its states according to the Ministry of Health of Brazil data. The data are estimated considering a timewindow for collecting cure and mortality rates, which provides more accurate information on the evolution of localcases.The search for a reliable estimate of the number of infected and deaths is crucial so that public health systems canact to respond quickly to the needs presented. For this reason, many studies aim to estimate the contagion curve, thespreading speed and the prediction of deaths. Recently, Imperial College [12] released a study on the current situationof COVID-19 in Brazil and estimate that human-to-human transmission of Sars-CoV-2, virus of the coronavirus familycausing the disease COVID-19, in Brazil averages from 1 to 2.8 people. They use a semi-deterministic Bayesianhierarchical model that analyzes the impacts of interventions aimed at restricting the transmission of the virus but it isnot clear how effective these measures were. The model adopted follows a function of patterns in urban mobility toestimate deaths, infections, and transmissions. Some models can also predict the evolution of contaminants in real-time,such as the IHME [1] with some important variables such as the transmission rate and the recovery rate, neither staticnor variable over time, which makes the result very broad and somewhat vague.Local health policies and actions can also be taken to mitigate the virus. Some authors have also studied the impactof lockdown on the spread of the virus [11]. Others, who used these models, consider the study of public policiesto mitigate the disease in specific locations [16]. Our simulations are based on general data from Brazil and wheredifferent policies have been applied in each state according to hospital capacity and speed of contamination. In this way,we focus on responses to general data from Brazil and in some Brazilian states with a higher rate of cases in order toassist in medical and political actions.In this work, we analyzed the case data of COVID-19 from the states of São Paulo, Rio de Janeiro, Ceará, EspíritoSanto, Bahia, Amazonas, Pará, Pernambuco, and Maranhão. To validate our results through the SEIRDP model, wecarefully collect the epidemic data from the Ministry of Health of Brazil [2] from April 08th till May 21th, 2020 andstudied the approximation of the observed and computed data considering the infected, recovered and death cases. Then,we were able to estimate the parameters that characterize the natural history of this disease and allows us to investigatethe projection of the epidemic in certain periods. The SEIRDP (Susceptible, Exposed, Infected, Recovered, Death, Insusceptible (P)) model used to analyze the coron-avirus epidemic in Brazil has the following variables: • S ( t ) : Susceptible population; • E ( t ) : Population who are exposed to the virus, but not yet infectious in latent period; • I ( t ) : Population who get laboratory positive confirmation and with infectious capacity; • R ( t ) : Recovery cases; • D ( t ) : Death number; • P ( t ) : Insusceptible cases; • N = S + P + E + I + R + D is the total population; • α : Protection rate (include people exposed to the infectious patients and people exposed to the asymptomaticpatients); • β : Infection rate (include exposed people catch COVID-19, people get COVID-19 with diagnosed confirmationand people; without symptoms but can transmit it to others); • γ − : Average latent time; • λ ( t ) : Coefficient used in the time-dependent cure rate;2 PREPRINT - 26 TH M AY , 2020 • κ ( t ) : Coefficient used in the time-dependent mortality rate.The variable described above are related through the following ODE system: dS ( t ) dt = − β S ( t ) I ( t ) N − αS ( t ) ,dE ( t ) dt = β S ( t ) I ( t ) N − γE ( t ) ,dI ( t ) dt = γE ( t ) − λ ( t ) I ( t ) − κ ( t ) I ( t ) ,dR ( t ) dt = λ ( t ) I ( t ) ,dD ( t ) dt = κ ( t ) I ( t ) ,dP ( t ) dt = αS ( t ) . (1)The model is time-dependent, because the cure rate λ ( t ) and mortality rate κ ( t ) are analytical functions of the timedefined by λ ( t ) = λ / [1 + exp( − λ ( t − λ ))] , (2) κ ( t ) = κ exp( − κ t ) , (3)respectively, which are fitted according to the number of cases for t ∈ [ t , t f ] , where t and t f are the initial andfinal periods, respectively. Equations (2) and (3) indicate that cure rate or recovery rate λ ( t ) goes up exponentiallytoward a threshold value and the mortality rate κ ( t ) should be zero after an infinite time. Moreover, the parameters { α, β, γ − , λ ( t ) , κ ( t ) } were fitted in the least square sense.The parameters { α , β , γ − , λ , λ , λ , κ ( t ) , κ ( t ) } are computed simultaneously by a nonlinear least-squares solver [8].We can replace them in the model (1) and calculate the time-histories of the different states { S ( t ) , P ( t ) , E ( t ) , I ( t ) ,R ( t ) , D ( t ) } . In this case, we use the standard fourth–order Runge–Kutta process [6]. For this, the system of equationsis written as a single ODE d Y dt = A · Y + F (4)such that Y = [ S ( t ) E ( t ) I ( t ) R ( t ) D ( t ) P ( t )] T , A = − α − γ γ − λ ( t ) − κ ( t ) 0 0 00 0 λ ( t ) 0 0 00 0 κ ( t ) 0 0 0 α , F = S ( t ) · I ( t ) · (cid:2) − βN βN (cid:3) T . Anchored by the information from the public data of the Ministry of Health of Brazil [2], we reliably estimate keyepidemic parameters and make projections on the inflection point considering fitted and actual data. Experiments showthe spatial distribution of COVID-19 cases in Brazil and in some states of the federation, namely, São Paulo, Ceará, andRio de Janeiro. These states stand out for the highest incidence of COVID-19 in Brazil. We also projected cases ofCOVID-19 in other states in the country: Espírito Santo, Bahia, Amazonas, Pará, Pernambuco and Maranhão that had alarge number of confirmed cases in relation to other states in the federation, about
I > . each one, according in thelast epidemiological bulletin of Ministry of Health of Brazil in May 18, 2020. Fig. 1(a) shows the first modeling and the projection for Brazil from late early April to mid-May 2020 for the currentlytotal, infected, recovery and death cases, while Fig. 1(b) presents its respective relative error solutions. According to3
PREPRINT - 26 TH M AY , 2020these results we can say that the model has a good correspondence with the data provided. This means that the stateparameters found by the least-squares solution should be suitable for projecting future dates. A test was done with thetraditional SEIR and the results were overestimated (see Fig. 1(c) ).In Fig. 2(a), through the SEIRDP model, we made the projections in the period from April to August. According to thismodel, the infected cases in Brazil will reach to the peak at middle of July 2020 with about 982K cases. In this sense,the period also represents the peak time for medical resources get ready. According to the Ministry of Health, Brazilhas a total of 441,811 hospital beds distributed in the country’s numerous regions. From the number of infected, we cansee that the public health system in Brazil may collapse in the middle of June. The number of victims is also expectedto grow in the same period. It should be noted that these results are only predictions based on data and populationmitigation policies. These results may change as the population’s isolation patterns change, as studies in this directionindicate, [11, 16]. (a) (b)(c) Figure 1: (a) Modeling of the differential SEIRDP model from April 8 to May 21. (b) The relative mean errors ofprojection for Brazil between actual data and SEIRDP model involving cases of infected, recovered and death. (c)Modeling of the differential SEIR model from April 8 to May 21.Fig. 2(b) and Fig. 2(c) presents daily newly confirmed infected cases and daily death cases for Brazil over the sameperiod respectively. The red lines indicate the actual data and the blue lines are predicted data from May to early ofAugust 2020. As we can see that the daily infected cases seem reaching the peak around July 21 with about 55K perday and then start to decrease. From Fig. 2(c), we can see that the daily death is converging to approximately 6000daily deaths without a projection of decrease in this period.4
PREPRINT - 26 TH M AY , 2020 (a)(b) (c) Figure 2: (a) Predictions of the differential SEIRDP model for Brazil from early April to final August, 2020. (b) Thedaily of infected cases and (c) deceased measured and projected for Brazil early April to mid-August, 2020.
São Paulo is the second metropolis in Latin America and the main financial, commercial and corporate center in SouthAmerica and where the first case of contamination in Brazil was detected. Currently, São Paulo concentrates the largestnumber of infected with the new Coronavirus. For São Paulo state, the modeling and relative errors from the sameperiod of Brazil are shown in Figs. 3. We notice that the modeled curves present a good approximation with thecollected data. In addition, a large part of the relative errors in infected cases, recovery, and death cases is below .In Fig. 4(a) we present an estimate of accumulative infected cases, recovery, and death cases. Projections show that thecurrently infected cases will reach a peak around July 20 close to Brazil, with about 114K cases. Fig. 4(b) and Fig. 4(c)show the daily confirmed infected cases and the death cases for São Paulo over the mentioned period respectively.Estimates show that the peak will reach in July 20 (about 5K) for daily infected and converging to 436 daily deaths tillAugust 19, 2020. The daily and cumulative peak has shown to be very close to the results obtained in the projection ofBrazil.
Another important study presented is on the state of Ceará, which recently showed a great increase in the number ofcases. For Ceará state, the modeling and relative errors from the same period of Brazil are shown in Figs. 5. Lookingat the curves modeled for Ceará state, we see that the error curves of infected, recovered and deceased cases also areoscillating around less than 20% in the period from early April to mid-May. Therefore, it is possible to infer that theSEIRDP model provides a good approximation of the results with respect to the collected data.5
PREPRINT - 26 TH M AY , 2020 (a) (b) Figure 3: (a) Modeling of the differential SEIRDP model from April, 08 to May 21, 2020. (b) The relative mean errorsof projection for Brazil from April 08 to May 21, 2020 between actual data and SEIRDP model involving cases ofinfected, recovered and death. (a)(b) (c)
Figure 4: (a) Predictions of the differential SEIRDP model for São Paulo state from early April to mid-August, 2020.(b) The daily of infected cases and (c) deceased measured and projected for São Paulo state early April to mid-August,2020. 6
PREPRINT - 26 TH M AY , 2020In Fig. 6(a) we provide an estimate of accumulative infected cases, recovery and death cases. Projections show thatthe currently infected cases will reach a peak around July 20 with about 80K cases. Fig. 6(b) and Fig. 6(c) present thedaily confirmed infected cases and the death cases for Ceará over mentioned period, respectively. By the estimates, wecan conjecture that the peak will reach in July 07 (about 1665) for daily infected, converging to 1000 daily deaths tillAugust 19, 2020. (a) (b) Figure 5: (a) Predictions of the differential SEIRDP model from April 08 to May 21, 2020. (b) The relative mean errorsof projection for Ceará state from April 08 to May 21, 2020 between actual data and SEIRDP model involving cases ofinfected, recovered and death.
We also present a more detailed study of the second largest state in population of Brazil and the second state to recordcases of contaminated by the new Coronavirus which is Rio de Janeiro. The modeling and projection results from thesame period of Brazil are shown in Fig. 7(a) for infected, recovered and deaths. The results from April to middle Maybring consistent response. The respective relative errors are shown in Fig. 7(b)).According to Fig. 8(a), the state of Rio de Janeiro projects that infected cases will reach to the peak around August02 with about 191K cases, which is close to period of the peak estimation of Brazil. This plot also suggests that thenumber of deaths will continue with the same growth trend in the cumulative total if the population is not subject tonew rules of protection.Fig. 8(b) and Fig. 8(c) show the daily confirmed infected cases, the death cases and the total accumulative infectedcases for Rio de Janeiro state over mentioned period, respectively. Rio de Janeiro state will reach the peak around July21 for both the daily infected increase (about 9K per day). But for daily death cases, it will reach the peak of 800 perday around August 19.
In this study, a time-dependent SEIRDP model was used to analyze the evolution of currently active, recovered anddeaths cases in states in Brazil whose infected toll exceeds 8.000 cases actually. The states Espírito Santo, Bahia,Amazonas, Pará, Pernambuco and Maranhão marked together with São Paulo, Ceará, and Rio de Janeiro have morethan 90 percent of the total fatalities in the country. As in previous experiments, theses projections take into accountmitigation and self-protective measures implemented by the government and the population, in the periods givenpreviously.Fig.9 shows the prediction of infected, recovered, and death from April to early August. According to this figure, thestate of Pernambuco is close to reaching the peak of patients infected by COVID-19 in mid-June, giving the authoritiestime to plan control and mitigation measures during this period. On the other hand, the number of deaths maintains aprojected growth in all states considered in this experiment.7
PREPRINT - 26 TH M AY , 2020 (a)(b) (c) Figure 6: (a) Predictions of the differential SEIRDP model for Ceará state from early April to early August, 2020. (b)The daily of infected cases and (c) deceased measured and projected for Ceará state early April to early August, 2020.
We consider an extension of the classical SEIR model known as SEIRDP with the aim of modeling COVID-19 epidemicsand introduce an estimation algorithm for Brazil. These parameters are found by applying the least-squares estimate inorder to fit the SEIR differential equations to the daily and cumulative cases observed by symptoms onset. The resultsrevealed that the model (1) provides a good approximation with respect to the collected data, since a large part of therelative errors in infected cases, recovery, and death is below .It should be noted that we need a reasonable set of data so that the estimates are more reliable. Fig. 10 illustrates howbad can be the prediction if the data set is too small. The case of Brazil is used in the following display. Note that thediscrepancies are significant when few points are used to measure the prediction of results, but these discrepancies tendto reduce as the number of points increases.According to the estimates obtained through the SEIRDP model, the states of São Paulo, Ceará, Amazonas, Maranhão,Espírito Santo, Pará and Bahia will reach to the peak of patients infected in mid-July, while Rio de Janeiro in earlyaugust. In addition, the daily number of deaths projects growth until August 19, 2020.It is worth mentioning that the first cases were diagnosed in the country, suspected cases, and those who had contactswith confirmed cases were tested. Due to the large scale of contamination and low testing capacity, the Ministry ofHealth started to recommend since mid-March that in places with community transmission only serious cases should betested. Consequently, many cases of coronavirus may have been underreported, underestimating the results of the model.Another intrinsic problem raised by this study was the delay between the occurrences of deaths and their recordingin official data released by the Ministry of Health. Unfortunately, in some states, much information arrived late, this8
PREPRINT - 26 TH M AY , 2020 (a) (b) Figure 7: (a) Predictions of the differential SEIRDP model from Match 20 to May 17, 2020. (b) The relative meanerrors of projection for Rio de Janeiro State from April 08 to May 17, 2020 between actual data and SEIRDP modelinvolving cases of infected, recovered and death. (a)(b) (c)
Figure 8: (a) Predictions of the differential SEIRDP model for Rio de Janeiro state from early April to early August,2020. (b) The daily of infected cases and (c) deceased measured and projected for Rio de Janeiro state early April tomid-August, 2020. 9
PREPRINT - 26 TH M AY , 2020 (a) (b)(c) (d)(e) (f) Figure 9: Predictions of the differential SEIRDP model for Amazonas (AM), Pará (PA), Espirito Santos (ES),Pernambuco (PE), Maranhão (MA) and Bahia (BA) from early April to early August, 2020.can compromise the projection results. Despite these problems, we hope that the estimates obtained will minimize theinfluence of Coronavirus in Brazil, especially in large urban centers whose probability of accumulating serious cases inthe short term is high. 10
PREPRINT - 26 TH M AY , 2020 (a) (b) Figure 10: (a) Prediction of the differential SEIRDP model for Brazil from April-08 to April-10. (b) Several predictionsof the differential SEIRDP model for Brazil from April-8 to April-26.
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