Mathematical Models for Describing and Predicting the COVID-19 Pandemic Crisis
MM ATHEMATICAL M ODELS FOR D ESCRIBING AND P REDICTINGTHE
COVID-19 P
ANDEMIC C RISIS
Cintra, P. H. P. ∗ Instituto de FísicaUniversidade de BrasiliaBrasilia, DF, Brasil, 70910-900 [email protected]
Citeli, M. F.
Instituto de FísicaUniversidade de BrasiliaBrasilia, DF, Brasil, 70910-900 [email protected]
Fontinele, F. N.
Instituto de FísicaUniversidade de BrasiliaBrasilia, DF, Brasil, 70910-900 [email protected]
June 5, 2020 A BSTRACT
The present article studies the extension of two deterministic models for describing the novelcoronavirus pandemic crisis, the SIR model and the SEIR model. The models were studied andcompared to real data in order to support the validity of each description and extract importantinformation regarding the pandemic, such as the basic reproductive number R , which might provideuseful information concerning the rate of increase of the pandemic predicted by each model. Wenext proceed to making predictions and comparing more complex models derived from the SEIRmodel with the SIRD model, in order to find the most suitable one for describing and predicting thepandemic crisis. Aiming to answer the question if the simple SIRD model is able to make reliablepredictions and deliver suitable information compared to more complex models. K eywords COVID-19 · coronavirus · SEIR model · SIR model · epidemic model Back on 2015, a group of researches described the potential for a SARS coronavirus circulating inside bats to mutate tohumans [1]. Early on 2020 the world suffers from an new pandemic crisis caused by the novel coronavirus, SARS-CoV-2, belonging to the
Betacoronavirus genus and with probable origin on bats [2]. The first cases of the novel virus dateback to December 2019 at the Food Market of Wuhan, China [3], where bats are sold among other exotic animals, sincethen the virus has been spreading throughout China, later Asia, Europe, Africa and America, causing a global scaleeconomic crisis and being notified by the World Health Organization as an pademic on March 11th.COVID-19 is a respiratory disease caused by the Sars-Cov-2 virus, currently in human-to-human sustained transmission[4]. We know that the contamination mainly occurs by a close interaction with infected individuals, as the viral chargeis carried by respiratory droplets that can remain in suspension in air or deposited on surfaces of common contact. As anovel strain of the
Coronaviridae family, it is not expected that any individual has antibodies against it, which causesthe entire population to be susceptible to infection. As an individual is exposed to the virus, the incubation periodbegins, with no symptoms and a small chance of contaminating others. When the virus is onset, the infected individualshow symptoms in a varied range of intensities and may develop severe acute respiratory syndrome. COVID-19 has ageneral mortality rate bellow 5% [5], with an average of 2.3%. The behavior of the disease is age dependent, with thehigher risk group being older populations, that present a mortality rate of 8% for individuals between 70-79 years and14.8% for people older than 80 years [6]. However, even with a low mortality rate, the number of hospitalizations isquite high, with 5% of the cases being critical and 14% being severe [7], presenting an challenge to health care systemsof some countries. ∗ Use footnote for providing further information about author (webpage, alternative address)— not for acknowledging fundingagencies. a r X i v : . [ phy s i c s . s o c - ph ] J un athematical models predicted the potential for an international outbreak early on [8] and described how Wuhanbecame the center of an epidemic crisis on China. The outbreak quickly spread throughout mainland China and othercountries. Although the pandemic crisis began on Wuhan, today the United States of America is the epicenter of thepandemic crisis on the world.Many models are widely used from scenario prediction [9, 10, 11, 12] to data analysis [13]. Among all those models,the most used ones are the SIR and SEIR models [14], including their modifications to include hospitalizations,asymptomatic cases and other compartments [15]. We develop here a comparative study between modified SIR andSEIR models; in order to provide support for the accuracy of both models, we compare the differences on the fittingprocess between the simple SIRD and simple SEIRD models and the accuracy of prediction generated by the simpleSIRD and the SEIRD model with age division, using data from Germany and the Republic of Korea. The choice wasbased on the accuracy of the data for representing the true scale and dynamics of the pandemic, other countries whopresented a much lower testing rate such as Brazil [16] are not reliable sources for testing models describing the disease.There are also countries such as Taiwan and Iceland, which kept track of the disease; however the number of cases inthose countries were much lower, escaping the deterministic nature of the models described here. Mathematical models for disease epidemic are either deterministic or stochastic [17], where the first may be consideredsome sort of thermodynamic limit of the second. An analogy made with thermodynamics, where given a big enoughnumber of particles in a gas, for example, you find deterministic equations to describe the behavior of the gas givenby the laws of thermodynamics without the need to know the exact behavior of each particle. Otherwise, when yournumber of molecules is low or you try to compute too many interactions between particles of your system, the randombehavior and fluctuations start taking place and you get to a stochastic model.We describe here a simple extensions of models constantly used on literature [18, 19, 10] and with it show some possiblebehaviors of a disease outbreak.
A simple mathematical model for disease epidemic can been built dividing the population in 3 groups: susceptibleindividuals (S), infected individuals (I) and recovered individuals (R). The model composed by these groups is called the
SIR model . In this article, however, we consider also individuals who have died by the disease, denote by D. Followingthe same arguments of the
SIR model, the
SIRD model can be described by the set of four differential equations: dSdt = − βN I ( t ) S ( t ) (1) dIdt = βN I ( t ) S ( t ) − ( γ + µ ) I ( t ) (2) dRdt = γI ( t ) (3) dDdt = µI ( t ) (4)Last equation is easily understood by thinking that the variation of the number of deaths may be proportional to theinfected individuals, where the proportionality constant is denoted by µ . The constants γ and β are, respectively, therecovery rate and the number of infected, where µ and γ are given in terms of the infection fatality rate (IFR) or thecase fatality rate (CFR); that is, the number of people who contracted the disease and died according to the total numberof infections (IFR) or the registered number of infections (CFR), and the average time taken from symptoms onset torecovery, τ r , or death τ d , formally µ = P CF R /τ d and γ = (1 − P CF R ) /τ r . The equations are simply a mathematicalway to describe how individuals passes from one group to the other according to the following chain of events: Asusceptible individual becomes infected by the virus, and from this point, it either dies or recovers (Figure 1).2 usceptible S(t) Infected I(t)
Recovered R(t) Dead D(t) βI ( t ) S ( t ) γI ( t ) µI ( t ) Figure 1: Representation of a SIRD model, a susceptible person gets infected and either dies or recovers from thedisease.Summing the four equations we get S ( t ) + I ( t ) + R ( t ) + D ( t ) = const , (5)where the constant may represent the total number of individuals, N . Before proceeding, we propose the initialcondition that, when t goes to zero, I ( t ) = I , R ( t ) = D ( t ) = 0 , and, therefore, S ( t ) = S = N − I ≈ N . Such anassumption is based on the fact that the entire population is susceptible to the SARS-CoV-2 virus.Since R ( t ) and D ( t ) are both data updated day by day in Germany and Korea, it would be helpful to write I ( t ) asfunction of them so as to predict its behavior, obtaining, for example, the maximum number of infected individuals. Forreasons that may be clear soon, we first write I ( t ) in terms of S ( t ) . An intuitive step is to divide equation (2) by (1).Thus, dI/dtdS/dt = − k S , (6)where k = ( γ + µ ) /β . Eliminating the temporal dependence, we get a separable differential equation, that is, dI = − dS + k dSS , (7)which the solution is easily verified to be I ( t ) = − S ( t ) + k ln S ( t ) + const . (8)Applying the initial condition, we obtain I = − ( N − I ) + k ln( N − I ) + const (9) → const = N − k ln( N − I ) . Hence, equation (8) may be written as I ( t ) = N − S ( t ) + k ln S ( t ) N − I . (10)We can visualize here, that depending on the combination of γ and µ , I reaches 0 before the entire population S becomes infected (Figure 2). 3 I n f e c t e d I max ⇒ S = γ + µβ ⇐ γ = µ = 0 Figure 2: Plot of equation (10) with different conbinations of γ and µ .Next, we may write S ( t ) in terms of R ( t ) and D ( t ) . For this purpose, we begin by dividing equation (1) by (3) and (1)by (4), dSdR = − βγ S ( t ) and (11) dSdD = − βµ S ( t ) . (12)Adding these two equations and writing S ( R, D ) as S ( R, D ) = f ( R ) g ( D ) , we get f dfdR + 1 g dgdD = − β (cid:18) γ + 1 µ (cid:19) (13)Since (13) is a separable equation, the well-known solution is given by f ( R ) = Ae aR and (14) g ( D ) = Be bD . (15)Therefore, S ( t ) can be written as S ( t ) = Ce aR ( t ) + bD ( t ) , (16)where we absorbed both constants A and B into C . By the initial condition, we find that C = N − I . To find a and b ,we must derive (16) in time under the condition that it may return to equation (1). In this way, we see that aγ + bµ = − β. (17)By the other hand, substituting equations (14) and (15) in (13), we get a + b = − β (cid:18) γ + 1 µ (cid:19) . (18)4olving this system, a = − βγ (1 − γ/µ ) (19) b = − βµ (1 − µ/γ ) (20)Hence, I ( t ) can be finally written as I ( t ) = N − ( N − I ) e aR ( t )+ bD ( t ) + γµ γ + µγ − µ (cid:18) R ( t ) γ − M ( t ) µ (cid:19) (21)With this equation, see that as t → ∞ , I does not approaches N necessarily, depending on the recovery and death rates, I does not reach N .The last important quantity extracted from this model is the basic reproduction number R , given by [14]: R = βγ + µ . (22)This quantity, is of vital importance of the study of a disease outbreak. Another deterministic mathematical model possible is the SEIRD model, in which we consider the population N of agiven region as divided in 5 groups. At time t , there are those who are susceptible to get infected S ( t ) , the ones whohave already been exposed the virus but does not present symptoms yet E ( t ) , people who are already infected andpresent the symptoms I ( t ) , the ones that have already recovered from the disease R ( t ) and those who are dead due tothe infection D ( t ) . This model is a good approximation to a short epidemic, so the population of a region is roughlyconstant throughout the epidemic period. Also, since this is a deterministic model, we assume N to be a big numbercompared to the number of people associated with the infection of a single person. The final consideration is that wealso assume that people that are recovered from the disease acquire immensity and does not become susceptible tobecome infected again.The rate of infection λ is proportional to the number of people infected, λ ( t ) = βI ( t ) , where the constant β representsthe effectiveness of the infection, the rate of cure γ = P :) τ − r , where P :) is the probability of recovery and τ r is the average time taken for an infected person to recover. Similarly the rate of death is µ = P CF R τ − d , where P CF R = 1 − P :) is the probability of death, given by the CFR and τ d is the average time taken for an infected person todie. Figure 3 carries an visual representation of the SEIRD model. Susceptible S(t)
Infected I(t)
Recovered R(t) Dead D(t)
Exposed E(t) βI ( t ) S ( t ) + kE ( t ) S ( t ) cE ( t ) γI ( t ) µI ( t ) Figure 3: Representation of a SEIRD model, a susceptible person gets exposed to the virus, being infected afterwardsand either dies or recovers from the disease. 5he differential equations representing the evolution of the populations are given by dSdt = − (1 − P exp ) βN I ( t ) S ( t ) − P exp βN E ( t ) S ( t ) (23) dEdt = (1 − P exp ) βN I ( t ) S ( t ) + P exp βN E ( t ) S ( t ) − cE ( t ) (24) dIdt = cE ( t ) − γI ( t ) − µI ( t ) (25) dRdt = γI ( t ) (26) dDdt = µI ( t ) (27)We first turn our attention to the construction of an appropriate formula for calculating R with this model. For that wefollow the method derived on [20]. The study develops a mathematical generalization for writing R depending on thetype of epidemiological model. R is defined as R = ρ ( F V − ) (28)where ρ ( X ) means the spectral radius of the matrix X, that is, the largest absolute eigenvalue. Both F and V arethe matrices of the derivatives of the functions defining the behavior of the disease population, with respect to eachpopulation compartment.To get to these matrices, we first note that the set of equations regarding the dynamics of the SEIRD model can beexpressed as follow: Consider (cid:126)x the vector of populations, that is (cid:126)x = ( x , x , x , x , x ) where x = E , x = I , x = S , x = R and x = D . Analogously, d(cid:126)x/dt is the vector of the first derivatives. Then, we can write thedynamics of the populations as d(cid:126)xdt = F − V (29)where F is the vector that relates the appearance of new infections on the disease populations due to contamination,and V is the input and output of members in all populations due to all other causes, such as recovery from the disease,development of symptoms after an incubation period, etc. In our case, since all newly infected members go to the E population F = (1 − P exp ) βIS + P exp βES (30)while V = βcE − cE + γI + µI (1 − P exp ) βIS + P exp βES − γI − µI . (31)Now, we know that the situation of a disease free equilibrium (DFE), meaning no disease is happening, is achived bythe vector (cid:126)x = (0 , , S , , , where S = N . According now to [20] we can calculate F and V as6 = (cid:18) ∂ F i ∂x j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x = x ≤ i (32) V = (cid:18) ∂ V i ∂x j (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) x = x j ≤ m (33)(34)being x j the vector components of (cid:126)x related to the populations with the disease, in our case E and I , and m is thenumber of populations related to infectious beings. Here m = 2 . Performing the derivatives, we conclude F = (cid:18) P exp β (1 − P exp ) β (cid:19) (35) V = (cid:18) c − c γ + µ (cid:19) (36)The next step is to find the inverse matrix of V , fortunately V is a 2x2 matrix and the formula for it’s inverse isstraightforward V − = 1 c ( γ + µ ) (cid:18) γ + µ c c (cid:19) , (37)and we proceed to the last step of combining F V − in order to retrieve ρ ( F V − ) and find R . F V − = 1 c ( γ + µ ) × (38) × (cid:18) P exp β ( γ + µ ) + (1 − P exp ) βc (1 − P exp ) βc (cid:19) , therefore, by computing the eingenvalues of F V − we find R = P exp β [ γ + (1 − P exp ) β ] + (1 − P exp ) βcc ( γ + µ ) (39)Having R in our hands, we continue to the study of some behaviors of this model.The set of equations describing the model is subjected to the initial conditions. When t → , I ( t ) − → I , S ( t ) → S = N − I , R ( t ) → , D ( t ) → and E ( t ) → E , where I is the initial number of infected, E is the initial number ofexposed in the population and no deaths or recoveries are assumed at t = 0 . Without vaccines or efficient medicine against the disease, non-pharmaceutical interventions are the only effectiveway to prevent further increase of the pandemic [13]. These interventions take different approaches such as socialdistancing, social isolation and lockdown of the population. Despite the differences, they all carry the same objective,decreasing the infection rate β . It is convenient to implement the effect of these interventions on the model, whenmaking predictions. Here, we model this effect by a logistic function, where β starts at a initial value β i and at somecritical time t c a intervention is imposed and beta decreases to β f = P dec β i , where P dec is the fraction of β i decreasedby the intervention. In France, studies estimate that the intervention decreased β i by 77% [21], therefore P dec = 0 . in France. β ( t ) = (1 − P dec ) β i τ e t − t c + P dec β i (40)7here τ is a constant related to the time taken for the intervention to have the effect desired. Such model reconstruct thegeneral behavior of interventions against the spread of the disease (Figure 4) N u m b e r o f p e o p l e Without intervention70% reduction of β i
50% reduction of β i Deaths without interventionMortes 70% reduction of β i Mortes 50% reduction of β i Figure 4: Visual representation of the effect of non-pharmaceutical interventions on the infection curve, depending onthe efficiency of the intervention, given by P dec . Since the case fatality rate (CFR) of COVID-19 is different among age groups [6, 7, 22], we propose here a modificationon both models, including the age distribution of the population and the social aspects of close contact between membersof the population. The modification is describe as follow: Each compartment is divided into M age groups, whereeach i -th group has a P CF R i associated to it, that is, the probability of death associated to the i -th age group. The β parameter is now described as the average number of daily contacts between a member belonging to the i -th age groupto the j -th age group, multiplied by the infection probability P infc β i I = N (cid:88) j =1 C ij I j P infc , (41) β i E = N (cid:88) j =1 C ij E j P infc , (42)where C ij is called the social contact matrix and we included I j and E j inside β now to place everything on the samesum. The age distribution among the population is retrieved from the UN prospects [23] and the social contact matrix forthose countries was measured on previous studies [24, 25]. The specific contact matrix for the Republic of Korea wasnot found, however, [26] finds evidences of cultural clusters in the world, where countries belonging to the same clustershare cultural similarities; thus, we use this fact to justify the use of Hong Kong’s social contact matrix to describe theRepublic of Korea. That way, we include cultural and population aspects for each of those countries, increasing theodds of a successful prediction. This type of model was used recently to describe the coronavirus outbreak on largecities in Brazil [27]. To test the SIRD and SEIRD model we first compare them to the pandemic crisis on the Republic of Korea, running anumerical solution for the differential equations (23) - (27) we adjust the general behavior of the populations to Koreandata acquired from [28] since 15/02/2020. The data from the Republic of Korea consists of the infection curve anddeath curve. To prevent problems with initial guess on the fitting process, both models used the same values for theinitial guess, except E , which is found only on the SEIRD model. S = N was also left as a free parameter of theadjustment instead of set to the total population of the country, which is justified by a limitation in both models, wherethe population is assumed homogeneously spread, which does not correspond to reality. Thus, N does not represent the8otal population, instead it represents an effective population smaller than the total population, due to non-homogeneousdistribution throughout the territory, the interpretation of N as the disease evolves is discussed on the discussion session.The parameter was chosen to be k = 0 . β , we considered a study which estimated that presyntomatic cases caused44% of infections [29], while for c we used an average of several clinical studies shown on table 1.incubation time 95% confidence Reference6.4 days 5.6-7.7 [30]5.2 days 4.1-7 [3]5 days – [31]4 days – [32]5.1 days 4.5 - 5.8 [33]Table 1: Incubation time of the disease according to other studies.Since the Korean government did not impose a lockdown or social isolation, we set P dec = 0 in both models. Figures 5and 6 show the result of the fitting process and table 2 includes the acquired values for all parameters for each model. N u m b e r o f p e o p l e DataDataInfected adjustment SEIRDIead adjustment SEIRD
Figure 5: Fit for the infected and deaths by SARS-CoV-2on the Republic of Korea using the SEIRD model. N u m b e r o f p e o p l e DataDataInfected adjustment SIRDIead adjustment SIRD
Figure 6: Fit for the infected and deaths by SARS-CoV-2on the Republic of Korea using the SIRD model.Parameter SIRD SEIRD χ τ r ± ± τ d ± ± I ± ± E – 62 ± β ± ± N ±
57 11218 ± τ d = 18 or days.Comparing the accuracy of the fitting with the data, both models resulted the same value of χ . The parameter E presents a large margin of error, which is expected given the lack of real data concerning the exposed population.Proceeding to the calculation of R (cid:48) for both models, using equations (39) and (22) we found9 (cid:48) SEIRD = 1 . ± . (43) R (cid:48) SIRD = 2 . ± . (44)The value for R according to other studies ranges from 2 to 3 [37, 38, 39, 40], therefore, both models yield acceptablevalues for R (cid:48) being the one predicted by the SEIRD model lower. We now proceed to test the prediction accuracy of both models. We used the first third of the data for fitting bothmodels and extracting parameters, after having the parameters, we compare the prediction for the next days with theseparameters with the rest of the dataset.
For Germany, the fitting data corresponds to the cases and deaths until 17th March. However, until the peak is reached,both models find presents very large margin of error for N , to avoid this problem, we varied N manually, from 0 to 10%of the local population; which was taken from a united nation prospect for the year of 2020 [23], at steps of 0.05%. Ateach step, we fit the initial data and reject the fitting if the χ value is lower than 0.995, this χ method for validating thegoodness of a data adjustment had already been used for epidemiological models [41]. We than plotted the maximumand minimum acceptable fits to generate the margin of prediction, comparing it with the complete dataset. We alsodecided to use τ r and τ d according to clinical studies when performing the prediction, instead of leave them as freeparameters for the fitting, I was also determined a priori according to the first registered number on 15/02/2020. Theresulting free parameters for fitting the training set are P infc and E .With the SEIRD model, we found the maximum and minimum values of N to be N min = 0 . of the Germanpopulation and N max = 0 . of the German population. The P infc parameter varied from 15.5% to 16.2%Using the SIRD, leaving β and I as free parameters. The limit values of N were N min = 0 . and N max = 0 . ,while β varied from . to . and I from 11 to 30. Figures 7 and 8 show the result of both simulations, with themaximum N max and minimum N min curves. The shaded region is the region between N max and N min . N u m b e r o f p e o p l e Simulated SEIRDGerman data
Figure 7: Prediction for Germany in comparison with realdata using the SEIRD model with age division N u m b e r o f p e o p l e simulated SIRGerman data Figure 8: Prediction for Germany in comparison with realdata using the simple SIRD model.
The training set consisted of 20 days, corresponding to the infections from 15/02 to 06/03. The SEIRD model found N min = 0 . and N max = 0 . of the total Korean population, while P infc varied between 80 to 85%.The simple SIRD model found N min = 0 . and N max = 0 . , β went from 0.345 to 0.436, and I was between23 to 65. Figures 9 and 10 present the result for prediction of both models.10
50 100 150 200 250Days since 15/0202000400060008000 N u m b e r o f p e o p l e Simulation SEIRDKorean data
Figure 9: Prediction for the Republic of Korea in com-parison with real data using the SEIRD model with agedivision N u m b e r o f p e o p l e simulated SIRKorean data Figure 10: Prediction for the Republic of Korea in compar-ison with real data using the simple SIRD model.
When concerning the adjustment process for acquisition of parameters with both models, there were no difference onthe accuracy of the fit, and both models yielded very close values for the parameters. However, τ d is super estimated inboth models, being slightly lower on the SEIRD model. The value of τ r is acceptable inside the variation of clinicalmeasures.The SEIRD model yields a slower growth rate than the SIRD model, that might happen due to the incubation periodon the SEIRD model, which slows down the propagation of the virus towards other individuals. The main differencebetween the growth rate predicted by both models is better visualized by figure 11, the action of the incubation periodslows down the rate of infection, as seen by the adjustments, but also decreases the peak of infections. However, thecumulative numbers of infection, deaths and recoveries are the same.Figure 11: Comparison between SIRD and SEIRD models. The parameters chosen were the same for both models,except for c = 1 / . on the SEIRD. β = 0 . , γ = 0 . , µ = 0 . , k = 0 . β and N = 6000000 N could be understood as the population susceptible to the first pandemic wave, due to the non-homogeneous distributionof the population, not everyone is susceptible to the disease right at the start. With such an interpretation, N tends to11ncrease with time and approach the total population, here. Comparing predictions generated by the SIRD model withthe SEIRD model with age division, the SEIRD model becomes a little more precise, although both simulations fail topredict the slower decrease of Korean data, that might be explained by an increase on N as time passes, resulting in newcases registered and therefore, slowing down the rate of decrease. Such hypothesis is well acceptable since the Republicof Korea did not adopt any lockdown or social isolation measure, making the disease able to propagate towards otherregions, increasing N with time. Even with better prediction, the SEIRD model is far more complicated than the SIRDmodel and the use of the later should probably not compromise any data analysis. The same must hold true for simpleSIR and SEIR models, when deaths are not a population to be accounted for, instead are just represented with a rate ofremoval for individuals.The social isolation model developed here shows good results on the predictions, indicating that the description of β should be close to reality. Here we find a huge advantage of the SEIRD model with age division in comparison with theSIRD model; by including age division, it is possible to simulate the effect of specific non-pharmaceutical interventions,such as school closure, which in principle would decrease β i I and β i E for the age groups between 0 to 19 years only.Another possibility is to include isolation of only elderly individuals. Several non-pharmaceutical measures have beenalready described in literature [42], other studies show how the total number of infected might be changed due to theefficiency of non-pharmaceutical measures [43].Other models might present more complete analysis of the disease, including hospitalizations and even asymptomaticcases, which are difficult to track and seem to vary a lot from place to place, the Diamond Princess cruise shipfound 17.9% of asymptomatic infections [44], while an airplane flight found 11.2% of cases being asymptomatic. AnItalian village presented 50 to 70% of cases being asymptomatic [45]. There are yet the problem of assuring that theasymptomatic cases registered on studies are really asymptomatic and not presymtomatic, that is, are people still on theincubation period.Of course, any mathematical model is only as good as the data allows, using mathematical models to describe thedisease on countries with low testing rates might yield unrealistic predictions. For example, [46] estimates 86% ofinfections being undocumented on China, at the early stages of the outbreak.Another consideration we did not take, was the possibility of reinfection, where individuals leave the recoveredgroup and re-enter the susceptible compartment. However, since other coronaviruses belonging to the same genus betacoronavirus such as the SARS-CoV and the MERS-CoV does not present a high enough mutation rate to causereinfection in short term [47], the only cause of reinfection would be the loss of antibodies to fight the virus; nevertheless,on both diseases, the infected person acquires antibodies enough to prevent reinfection for a period of 2 - 3 years [48].With those considerations, we did not assume reinfection was probable on short-term. Future studies may be conductedto study the possibility of reinfection of individuals on the long-term. Mathematical models of a disease outbreak such as the COVID-19 are able to predict the behavior of the infection. Bothmodels have proved to be efficient tools for acquiring data and forecast the future situation. Despite the limitations, themodels made it possible to achieve a value of R in good agreement with other studies, providing evidence in favor ofthe validity of the model.However, the present models do not take into consideration the spatial distribution of the population, reflecting on someuncertainties that made the window of prediction larger.The age division does not change the prediction drastically, suggesting that in the case of a simple prediction or analysis,SIRD models are useful. The age division SEIRD model provides an advantage when requiring specific simulations onspecific groups of the population. References [1] Vineet D Menachery, Boyd L Yount Jr, Kari Debbink, Sudhakar Agnihothram, Lisa E Gralinski, Jessica A Plante,Rachel L Graham, Trevor Scobey, Xing-Yi Ge, Eric F Donaldson, et al. A sars-like cluster of circulating batcoronaviruses shows potential for human emergence.
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