A matlab code to compute reproduction numbers with applications to the Covid-19 outbreak
Paulo R. Zingano, Janaina P. Zingano, Alessandra M. Silva, Carolina P. Zingano
aa r X i v : . [ q - b i o . P E ] J un A matlab code to compute reproduction numberswith applications to the Covid-19 outbreak
Paulo R. Zingano, Jana´ına P. Zingano,
Institute of Mathematics and StatisticsUniversidade Federal do Rio Grande do SulPorto Alegre, RS 91509-900, Brazil
Alessandra M. Silva
Companhia de Planejamento do Distrito FederalGoverno de Bras´ıliaBras´ılia, DF 70620-080, Brazil andCarolina P. Zingano,
School of MedicineUniversidade Federal do Rio Grande do SulPorto Alegre, RS 90035-003, Brazil
Abstract
We discuss the generation of various reproduction ratios or numbers thatare very useful to monitor an ongoing epidemic like Covid-19 and examine theeffects of intervention measures. A detailed SEIR algorithm is described fortheir computation, with applications given to the current Covid-19 outbreaksin a number of countries (Argentina, Brazil, France, Italy, Mexico, Spain, UKand USA). The corresponding matlab script , complete and ready to use, isprovided for free downloading. Key words:
Covid-19 outbreak, SARS-Cov-2 coronavirus, reproduction numbers,SEIR deterministic models, parameter uncertainties, robust methods
Matlab code:
A complete matlab source code to compute reproduction num-bers of Covid-19 or other epidemics is freely available by clicking here: find Rt.m. . Introduction
The monitoring of the evolving state of a serious epidemic can be done duringand after its outbreak by estimating the daily values of basic ratios generally knownas reproductive or reproduction numbers [5, 6, 7, 13]. While not properly geared toallow serious predictions of future values of the epidemic, they are nevertheless ableto display the past and present history with amazing clarity. However, as their calcu-lation depends on the values of various mathematical parameters (like the length oftransmission and incubation periods), this ability may be impaired by inaccuraciesin their estimation. This is particularly true for the widely used basic reproductionnumber , which measures the average number of secondary cases generated by a typ-ical infectious individual in a full susceptible population (Figure 1). t = 20 T t = 10 Fig. 1:
Time evolution of standard basic reproduction numbers of Co-vid-19 in Brazil since the date of100 cases reported ( t = 0), showingthe effect of two distinct hypothet-ical transmission periods ( T t = 20and T t = 10, resp.). In this example, t = 0 corresponds to 03/13/2020.(Data source: covid.saude.gov.br) On the other hand, once some mathematical model has been chosen to simulatethe disease dynamics and its parameters determined, several alternative reproduc-tive numbers become automatically available at no additional computational cost,many showing very little dependence on key parameters like transmission or incuba-tion times. We will illustrate this fact in the context of deterministic
SEIR models,but our approach can be adapted to other mathematical models (deterministic orstochastic) as well.The idea is most easily explained by considering the simplest
SEIR model of all,defined by the equations (1.1) below. This model divides the entire population inquestion into four classes: the susceptible individuals (class S ), those exposed (class E ,formed by infected people who are still inactive (i.e., not yet transmitting the dis-ease), the active infected or infectious individuals (class I ) and the removed ones.The latter class is formed by those who have recovered from the disease (class R ) orwho have died from it (class D ). The dynamics between the various classes is givenin the universal language of calculus by the differential equations1 dSdt = − β S ( t ) N I ( t ) ,dEdt = β S ( t ) N I ( t ) − δ E ( t ) ,dIdt = δ E ( t ) − ( r + γ ) I ( t ) ,dRdt = γ I ( t ) ,dDdt = r I ( t ) , (1.1)see e.g. [2, 4, 7, 12] for a detailed discussion of the various terms and their meanings.The parameters β ( average transmission rate ) and r ( average lethality rate of the population I due to the disease) vary with t (time, here measured in days ),but δ and γ are typically positive constants given by γ = 1 T t , δ = 1 T i , (1.2)where T t denotes the average transmission period and T i stands for the meanincubation time , which will be taken as 14 and 5.2, respectively [9, 10, 14]). In thesystem (1.1), N denotes the full size of the susceptible population initially exposed,so that we have S ( t ) + E ( t ) + I ( t ) + R ( t ) + D ( t ) = N , where t denotes theinitial time. Observing that, by the equations (1.1), the sum S ( t ) + E ( t ) + I ( t ) + R ( t )+ D ( t ) is invariant, it follows the conservation law S ( t ) + E ( t ) + I ( t ) + R ( t ) + D ( t ) = N, ∀ t > t , (1.3)since, for simplicity, the model neglects any changes in the population due to birth,migration or death by other causes during the period of the epidemic (of the order ofa few months). To well define the model (1.1), besides informing the functions β ( t )and r ( t ) we need to provide the initial values S ( t ), E ( t ), I ( t ), R ( t ), D ( t ), whichis not a trivial task, since not all of these variables are reported, and those reportedmay be in error — which may well be large in case of significant underreporting.It thus seems clear that predicting reasonably right values for the variables S ( t ), E ( t ), I ( t ), R ( t ) and D ( t ) at future times is not a simple problem, especially in thelong time range. The situation becomes even more complicated for more complex(i.e., stratified) models, which add other variables and parameters to be determined.Calibrating many parameters can quickly become a nightmare. For all its simplicity,models with few variables and parameters like (1.1) can yield surprisingly goodresults and thus should not be overlooked, as will be seen in the sequel.2 . Implementing the SEIR model Having introduced the
SEIR equations (1.1), we now describe an implementationof this model that is suitable for the computation of reproduction numbers.( i ) assigning a value to the population parameter N In the case of C ovid-19, which can be considered a new virus ( SARS - C o V -2), it hasbeen common to assume the entire population susceptible and assign its whole valueto N . This is highly debatable, since this parameter refers to that particular fractionof the susceptible population that is effectively subject to infection. For determinis-tic models, this introduces the possibility that an outbreak might not happen afterthe introduction or reintroduction of a few infected individuals, as it has been longrecognized in the stochastic literature [1, 9]. In any case, it turns out that N is notso much important for the short range dynamics as it proves to be in the long run(see Figures 2 a and 2 b ), so that for our present purposes this is not a serious issue.We have therefore taken for N the full population of the region under consideration.
50 100 150 20001234567 x 10 t (days) Reported New Cases predicted by SEIR: 2 scenarios N = 50e6 N = 20e6 Fig. 2 a : Prediction by model (1.1) of the daily number of new cases of Covid-19 expectedto be reported in Brazil between the initial time t = t = 60 (April 25th) and t =200 (September 12th), considering susceptible populations of N = 20 million (redcurve) and N = 50 million (black curve). Note the appreciable difference betweenthe predicted peak values (34 and 70 thousand, resp.) and their respective dates,June 6th and July 4th. Actual data points are shown in blue. (Computed fromdata available at the official site https://covid.saude.gov.br.) t (days) Reported New Cases predicted by SEIR: 2 scenarios N = 50e6 N = 20e6 Fig. 2 b : Thirty day prediction by model (1.1) of the daily number of new cases of Covid-19to be reported in Brazil between the initial time t = t = 60 (04/25) and t = 90(05/25), considering susceptible exposed populations of N = 20 million (red curve)and N = 50 million (black curve). Note the very close similarity of the two 30Dpredictions in spite of the appreciable difference in the values of N . Points shownin blue are the official values reported (cf.https://covid.saude.gov.br.) ( ii ) generation of initial data S ( t ), E ( t ), I ( t ), R ( t ), D ( t )Initial values S , E , I , R , D for the five variables are generated from a startingdate t s on, which is taken so as to meet some minimum value chosen of total reportedcases (typically, 100). Denoting by C r ( t ) the total amount of reported cases up tosome time t , and letting EIR ( t ) be the sum of the populations E ( t ), I ( t ) and R ( t ),we set EIR ( t s ) = f c · ( C r ( t s ) − D ( t s )) , (2.1)where f c ≥ correction factor to account for likely underreportings onthe official numbers given. (In (2.1), we have neglected possible underreportings onthe number of deaths, which could of course be similarly accounted for if desired.)Again, this factor will not play an important role in this paper and could be safelyignored, but it should be carefully considered in the case of long time predictions.Having estimated EIR ( t s ), we then set E ( t s ) = E ( t s ) := a · (1 − b ) · EIR ( t s ), (2.2 a ) I ( t s ) = I ( t s ) := (1 − a ) · (1 − b ) · EIR ( t s ), (2.2 b )4 ( t s ) = R ( t s ) := b · EIR ( t s ), (2.2 c ) S ( t s ) = S ( t s ) := N − (cid:0) E ( t s ) + I ( t s ) + R ( t s ) + D ( t s ) (cid:1) , (2.2 d )where a = T i / ( T i + T t ) and b = 0 .
30, consistently with the literature (see e.g. [14]).The arbitrariness in this choice of weights gets eventually corrected as we computemore values S ( t ) , E ( t ) , I ( t ) , R ( t ) , D ( t ) at later initial times t = t s + 1 ,..., t F ,where t F stands for the final (i.e., most recent) date of reported data available. Foreach t , the solution of the equations (1.1) with the previously obtained initial dataat t − J ( t ) = [ t − , t ], t = min { t − d , t F } ,with constant parameters β = β ( t − r = r ( t −
1) determined so that the com-puted values for C r ( t ), D ( t ) best fit the reported data for these variables on [ t , t ] inthe sense of least squares [12]. (Here, d ∈ [ 2 ,
10 ] is chosen according to the dataregularity.) Once this solution (
S, E, I, R, D )( t ) is obtained, we set S ( t ) := S ( t ), E ( t ) := E ( t ), I ( t ) := I ( t ), R ( t ) := R ( t ), D ( t ) := D ( t ) and move on to thenext time level t + 1, repeating the procedure until t F is reached.( iii ) computing the solution on some final interval [ t , T ] ( prediction phase )Having completed the previous steps, we can address the possibility of prediction .Although this is not important for our present goals, it is included for completeness.Choosing an initial time t ∈ ( t s , t F ], we then take the initial values S ( t ) = S ( t ) , E ( t ) = E ( t ) , I ( t ) = I ( t ) , R ( t ) = R ( t ) , D ( t ) = D ( t ) . In order to predict the values of the variables S ( t ) , E ( t ) , I ( t ) , R ( t ) , D ( t ) for t > t ,it is important to have good estimates for the evolution of the key parameters β ( t )and r ( t ) beyond t . This is the most computationally intensive part of the algorithmand is better executed in large computers. Such estimates can be given in the form β ( t ) = β + a β e − λ β ( t − t ) (2.3 a ) r ( t ) = r + a r e − λ r ( t − t ) (2.3 b )where β , a β , λ β , r , a r , λ r ∈ R are determined so as to minimize the maximum sizeof weighted relative errors in the computed values for C r ( t ) , D ( t ) as comparedto the official data reported for these variables on some previous interval [ t − τ , t ](weighted Chebycheff problem ) for some chosen τ > ≤ τ ≤ β ( t ) , r ( t ) computed in the step ( ii ) above. The resultis illustrated in Figure 3 for the case of β ( t ), with similar considerations for r ( t ).5 β ( t ) Estimation of β values beyond the initial time (t = 70) β (t) Fig. 3:
Estimation of future values of the transmission parameter β ( t ) beyond the initial time t = 70 (05/05/2020) for the outbreak of Covid-19 in Brazil, assuming the basic form (2.3 a ),after solving the Chebycheff problem (red curve). The data points in the interval [ 40 ,
70 ],shown here in blue, are values of the function β ( t ) computed in step ( ii ), which are usedto obtain the first approximation to β ( t ). Values of β ( t ) previous to t = 40 (04/05/2020),shown in black, are disregarded. The golden points beyond t = 70 are future values of β ( t ),not available on 05/05/2020, displayed to allow comparison with the predicted values β ( t ). Once β ( t ), r ( t ) have been obtained, the equations (1.1) are finally solved (Figure 4).
10 20 30 40 50 60 70 80 90 10000.511.522.533.544.555.5 x 10 t (days) C r ( t )
30D prediction of Total Cases reported in Brazil from May 5th on predictedofficial data
Fig. 4:
Computation of C r ( t ) = (cid:0) E ( t ) + I ( t ) + R ( t ) (cid:1) /f c + D ( t ) for t > t = 70 (05/05/2020), withinitial data C r ( t ) = (cid:0) E ( t ) + I ( t ) + R ( t ) (cid:1) /f c + D ( t ), after obtaining β ( t ), r ( t ) – seeFig. 3 for β ( t ). The numerical solution of equations (1.1) is easily obtained by any method. . Reproduction numbers A natural by-product of the results generated by the algorithm is the estimate of reproduction numbers of the epidemic, which measure the intensity of transmissionat various times and, in doing so, are useful indicators to monitor the situation andthe effects of intervention procedures that may have been taken. Using the genericsymbol R t to denote such quantities, they signal a rise in the number of infectionsin the case R t >
1, their decrease when R t <
1, and temporary steadiness if R t = 1.For instance, rewriting the equation for the critical population I ( t ) in the form dIdt = α ( t ) I ( t ) , α ( t ) := δ · E ( t ) /I ( t ) − r ( t ) − γ, (3.1 a )we see that I ( t ) will increase if α ( t ) >
0, decrease when α ( t ) < α ( t ) = 0 — or, in terms of the ratio R t := δ · E ( t ) /I ( t ) r ( t ) + γ , (3.1 b )whether we have R t > R t < R t = 1, respectively. Another natural possibilitywould be to consider basic ratios like R t := I ( t + d ) I ( t − d ) , R t := E ( t + d ) + I ( t + d ) E ( t − d ) + I ( t − d ) (3.2)for some chosen d >
0. For example, the choice d = T t / corresponds to the standard basic reproduction number , or the mean number of secondary infections caused by atypical infected individual during his transmission period [9, 12]. The correspondingexpressions would be, using the calculations performed in step ( ii ) of the algorithm, R (1) t := δ · E ( t ) /I ( t ) r ( t ) + γ , (3.3)where r ( t ) denotes the lethality rates computed there, or else R (2) t := I ( t + 3) I ( t − , R (3) t := E ( t + 3) + I ( t + 3) E ( t −
3) + I ( t − , (3.4)and so forth. These indicators point to similar scenarios (Figura 5), with R (1) t seem-ingly more influenced by seasonal (weekly) variations in the data. We have found R (2) t particularly useful, with numerical results that are consistent with previous analyses The notation R t is natural in stochastic models, and is adopted here as we have already used R ( t ), R ( t ) with other meanings (size of the recovered population and their initial values, resp.). C ovid-19, the choice d = 3 is good tozoom in the scenario and facilitate the reading (Figure 6), while not compromisingrobustness (Figure 7). R t Time evolution of Covid−19 in Brazil as seen by R t(1) , R t(2) , R t(3) R t(1) R t(2) R t(3) Fig. 5:
Comparison of the time evolution of Covid-19 in Brazil (since 100 cases reported) as seenby the indicators defined in (3.3), (3.4), pointing to similar scenarios. In the three cases itis clear that Brazil has not yet reached a state of control of the epidemic ( R t < R t Time evolution of Covid−19 in Brazil as seen by indicators with different time span d = 3 d = 2 d = 1
Fig. 6:
Comparison of the time evolution of Covid-19 in Brazil (since 100 cases reported) as seenby R t = I ( t + d ) /I ( t − d ) for different values of d , showing similar scenarios. In the threecases it is clear that Brazil has not yet reached a state of control of the epidemic ( R t <
10 20 30 40 50 60 70 80 9001234567 t (days) R t Values of R t(2) for Covid−19 in Brazil assuming different transmission periods T t = 20 T t = 10 Fig. 7:
Robustness of R (2) t with respect to large uncertainties on the value of transmission time.Date zero refers to 100 cases reported, that is: 03/13/2020. (As in Fig. 5 and Fig. 6 above,calculations were based upon official data reported at https://covid.saude.gov.br.)
4. Applications
In this section we will illustrate the use of reproduction values by examining theevolution of Covid-19 in various countries around the world under the view of suchnumbers — choosing for definiteness the numeric ratio R (2) t defined in (3.4) aboveas our basic indicator, unless explicitly stated otherwise. Thus, we set R t = I ( t + 3) I ( t −
3) (4.1)where I ( s ) is the size of the active infected population at time s as computed inthe step ( ii ) of the SEIR algorithm (see Section 2).Taking right decisions about intervention or relaxation measures is a very difficultand complex process that involves a careful consideration of various mathematicalindicators and a lot of other factors including many health, economic and socialissues. In the following examples we consider only the single factor given by repro-duction numbers. For all the simplicity and obvious limitations of this approach, itoffers nevertheless precious insight and information about the disease dynamics andevolution.
Acknowledgements.
In the following examples, the computation of the R t curveswas based on data available for each country at the site worldometers/coronavirus.9
20 40 60 80 10000.511.522.533.54 t (days)Time evolution of 7D reproduction number of Covid−19 in Argentina R t Example 1:
Time evolution of Co-vid-19 in Argentina since 03/18/2020 ( t = 0), the date of 97 totalcases reported. Strong containmentmeasures had begun 3 days earlier( t = −
3) and managed to keep thenumber of cases and deaths downlow, with R t decreasing continuallyuntil 05/04/2020 ( t = 47), when itreached a minimum value of 1.08.Following that, the situation dete-riorated with R t increasing to 1.54on 05/24/2020 ( t = 67), despite thereinforcement of most interventionprocedures. Partial relaxation of some of these measures was introduced on 06/01/2020( t = 75) and, in this new period, R t has remained relatively stable at 1.30 (yellow band). Bringing the epidemic to a state of nationwide control ( R t <
1) still seems far away.This example illustrates the basic fact that having low numbers of infections anddeaths does not necessarily mean having the epidemic under control.
Example 2:
Time evolution of Co-vid-19 in Brazil since 03/13/2020,the date of 98 total cases reported( t = 0). With a poor coordinationbetween the central and regionalauthorities and different levels ofintervention in the various states ofthe country, the decreasing of R t after reaching 1.5 by mid-April pro-ceeded very slowly (green band) dueto the spread of the epidemic andthe emergence of new infection foci.Relaxation measures began to be im-plemented on different dates accord-ing to the individual regions, but can be traced back to 06/01/2020 ( t = 80) on the av-erage. Despite the encouraging behavior of R t shown in the last fortnight (yellow band),the indicator is likely to resume increasing due to further disease development in less af-fected areas of the country, particularly the southern and central western states. Anothernegative factor is that flexibilization of control measures has been introduced before the arious regions had attained a state of epidemic control ( R t < not ideal. t Example 3:
Time evolution of Co-vid-19 in France since 02/29/2020( t = 0), the date of 100 total casesreported. Containment measuresbegan relatively late on 03/16/2020( t = 16), with a strict eight-weeklockdown that reduced the value of R t down to 0.81 (green band). Re-strictions were afterwards relaxed(yellow band), with R t stable fora couple of weeks, when it beganincreasing. A peak value of 0.99 wasreached on 05/30/2020, followed bya reduction to its present value 0.92.The situation requires constant monitoring, with the possibility of having to reimposesome restrictions to keep the epidemic under control ( R t < t Example 4:
Time evolution of Co-vid-19 in Italy since 02/22/2020,the date of 79 total cases reported( t = 0). Containment measures be-gan fifteen days later, with astrict eight-week national lockdownimposed on 03/10/2020 ( t = 17).The strong intervention succeededin continually reducing R t down toa safe value of 0.80 on 05/18/2020( t = 86), when some of the conten-tion rules began being relaxed (yel-low band). The descent continued fornineteen days, reaching a bottomvalue of 0.77 on 06/06/2020 ( t = 105). After this, a steady and very slow increase set inleading to the present value of 0.81 ( t = 116).
20 40 60 80 10000.511.522.533.54 t (days)Time evolution of 7D reproduction number of Covid−19 in Mexico R t Example 5:
Time evolution of Co-vid-19 in Mexico since 03/18/2020,the date of 93 total cases reported( t = 0). After containment mea-sures began on 03/22/2020 ( t = 4),the value of R t continually de-creased to 1.20 (green band), whenrestrictions began to be relaxedon 06/01/2020 (yellow band).Relaxing measures have seeminglynot changed the behavior of R t afterwards, but reaching a state ofcontrol ( R t <
1) still looks far away.Similarly to Argentina and Brazil,the flexibilization started before the country had properly entered the safe zone R t < t Example 6:
Time evolution of Co-vid-19 in Spain since 03/01/2020,the date of 84 total cases reported( t = 0). After containment measuresbegan on 03/13/2020 ( t = 12), thevalue of R t continually decreased to0.89 on 05/11/2020 ( t = 71), whenrestrictions began to be relaxed(yellow band). A minimum valueof 0.74 was finally reached on06/07/2020 ( t = 98), after which aslow, steady increase set in towardsthe present value of 0.79 ( t = 108),in a similar way to Italy.
20 40 60 80 10000.511.522.533.544.5 t (days)Time evolution of 7D reproduction number of Covid−19 in UK R t Example 7:
Time evolution of Co-vid-19 in the UK since 03/04/2020,the date of 87 total cases reported( t = 0). After containment measuresbegan relatively late on 03/20/2020( t = 16), including strict nationallockdown and other rules three dayslater, the value of R t continuallydecreased to 0.98 on 05/13/2020( t = 70), when restrictions began tobe relaxed, and then further downto 0.86 nineteen days later, when thelockdown was removed (yellow band).Despite successfully bringing the epi-demic under control, the number of reported cases and deaths was very high due to theinitial delay in taking intervention action. t Example 8:
Time evolution of Co-vid-19 in the US since 03/02/2020,the date of 100 total cases reported( t = 0). After containment measuresbegan on 03/15/2020 ( t = 13), R t successfully decreased continually to0.97 on 05/15/2020 ( t = 74), whenrestrictions began to be relaxed,and then slightly down to 0.96 on05/27/2020 ( t = 86), followed bya slow and steady ascent to thepresent value of 1.04 (yellow band).With a poor coordination betweencentral and local authorities in thebeginning, the country suffered a high mortality rate (0 .
037 %) and number of infections(2.4 million cases reported). Despite their efforts, the United States have not yet succeededin bringing the epidemic under nationwide control. eferences [1] L. S. Allen , An introduction to stochastic epidemic models , in: F. Brauer etal (Eds),
Mathematical Epidemiology , Lecture Notes in Mathematics, vol. 1945,Springer, New York, 2008, pp. 81-130.[2]
F. Brauer, P. van den Driessche and J. Wu (eds),
Mathematical Epidemiology ,Lecture Notes in Mathematics, vol. 1945, Springer, New York, 2008.[3]
O. Diekmann, J. P. Heesterbeek and J. J. Metz , On the definition and thecomputations of the basic reproduction ratio R in models for infectious diseases inheterogeneous populations , J. Math. Biol. 28 (1990), 365-382.[4] P. van den Driessche and J. Watmough , Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission ,Math. Biosci. 180 (2002), 29-48.[5]
P. van den Driessche and J. Watmough , Further notes on the basic reproduc-tion number , in: F. Brauer et al (Eds),
Mathematical Epidemiology , Lecture Notesin Mathematics, vol. 1945, Springer, New York, 2008, pp. 159-178.[6]
J. M. Heffernan, R. J. Smith and L. M. Wahl , Perspectives on the basic re-productive ratio , J. R. Soc. Interface, 2 (2005), 281-293.[7]
H. W. Hethcote , The mathematics of infectious diseases , SIAM Rev. 42 (2000),599-653.[8]
S. Kim, Y. B. Seo and E. Jung , Prediction of Covid-19 transmission dynamicsusing a mathematical model considering behavior changes in Korea , Epidemiologyand Health, 42 (2020), DOI: 10.4178/epih.e2020026.[9]
A. J. Kucharski, T. W. Russell, C. Diamond, Y. Liu, J. Edmunds, S. Funkand R. M. Eggo , Early dynamics of transmission and control of COVID-19 : amathematical modelling study , Lancet Infectious Diseases 2020, 20:553-558, DOI:10.1016/S1473-3099(20)30144-4.[10] S. A. Lauer, K. H. Grantz, Q. Bi, F. L. Jones, Q. Zheng, H. A. Meredith et al,
The incubation period of coronavirus disease 2019 ( Covid-19 ) from publiclyreported confirmed cases : estimation and application , Ann. Intern. Med. 172 (2020),577-582.[11] Q. Li, X. Guan, P. Wu, L. Zhou et al.,
Early transmission dynamics in Wuhan,China, of novel coronavirus-infected pneumonia , New Engl. J. Med. 2020, 382:1199-1207, DOI: 10.1056/NEJMoa2001316.[12]
M. Martcheva , An Introduction to Mathematical Epidemiology , Springer, NewYork, 2015. A. Mellan, H. H. Hoeltgebaum, S. Mishra, C. Whittaker et al.,
EstimatingCOVID-19 cases and reproduction number in Brazil , Report
S. Vaid, C. Cakan and M. Bhandari , Using machine learning to estimate un-observed COVID-19 infections in North America , J. Bone Joint Surg. Am. 2020,00:1-5 (DOI: 10.2106/JBJS.20.00715).15