Academic Articles & Supplements

Discrete Gompertz Scheme and Bootstrap Analysis for Early Monitoring of the COVID-19 Pandemic in Cuba

María Teresa Perez Maldonado, Julian Bravo Castillero, Ricardo Mansilla, Rogelio Oscar Caballero Perez

Author

María Teresa Perez Maldonado, Julian Bravo Castillero, Ricardo Mansilla, Rogelio Oscar Caballero Perez

Title

Discrete Gompertz Scheme and Bootstrap Analysis for Early Monitoring of the COVID-19 Pandemic in Cuba

Description

The discrete version of the Gompertz model is used for early monitoring and short-term forecasting of the spread of an epidemic in a region. Real data of COVID-19 infection in Cuba is used. The proposed methodology was implemented for the first thirty-five days and was used to predict accurately the data reported for the following twenty days.

Discrete Gompertz Scheme and Bootstrap Analysis for Early Monitoring of the COVID-19 Pandemic in Cuba

María Teresa Pérez-Maldonado1 https://orcid.org/0000-0003-3658-3861
Julián Bravo-Castillero2 https://orcid.org/0000-0002-7499-3821
Ricardo Mansilla3 https://orcid.org/0000-0002-1248-0959
Rogelio Oscar Caballero-Pérez4 https://orcid.org/0000-0002-5474-174X

1 Universidad de La Habana, Facultad de Física. Departamento de Física Teórica
2 Universidad Nacional Autónoma de México, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas, Unidad Académica del IIMAS en el Estado de Yucatán
3 Universidad Nacional Autónoma de México, Centro Peninsular en Humanidades y Ciencias Sociales
4 Universidad Nacional Autónoma de México, ENES Mérida, Universidad Politécnica de Yucatán

Abstract

The COVID-19 pandemic has motivated a resurgence in the use of phenomenological growth models for predicting the early dynamics of infectious diseases. These models assume that time is a continuous variable, whereas in the present contribution the discrete version of the Gompertz model is used for early monitoring and short-term forecasting of the spread of an epidemic in a region. The time-continuous models are represented mathematically by first-order differential equations, while their discrete versions are represented by first-order difference equations that involve parameters that should be estimated prior to forecasting. The methodology for estimating such parameters is based on a bootstrap technique. Real data of COVID-19 infection in Cuba is used. The proposed methodology was implemented for the first thirty-five days and was used to predict accurately the data reported for the following twenty days.

(*Initialdata*)l={3,3,4,4,4,5,7,11,16,23,33,38,46,55,65,78,117,137,168,184,210,231,267,286,318,348,394,455,513,562,618,667,724,764,812,860,921,984,1033,1085,1135,1187,1233,1283,1335,1367,1387,1435,1465,1499,1535,1609,1647,1666,1683,1701,1727,1739,1752,1764,1781};(*Step1.Window-averaging*)lp=Table[N[Sum[l[[k]]/7,{k,i-3,i+3}]],{i,4,58}];(*Step2.Calibrationperiodandlistofpairs*)Ene=35;lpr=Take[lp,Ene];prs=Table[{lpr[[n]],lpr[[n+1]]},{n,1,Ene-1}];(*Step3.FittingtoGompertzfunction*)func=NonlinearModelFit[prs,x(A-gammaLog[x]),{gamma,A},x];A1=A/.func["BestFitParameters"];gamma1=gamma/.func["BestFitParameters"]Ka=Exp[(A1-1)/gamma1](*Step4.Predictedtimeseries*)sol1=RecurrenceTable[{aa[n+1]aa[n](1+gamma1Log[Ka/aa[n]]),aa[1]lpr[[1]]},aa,{n,55}];Show[ListPlot[sol1,PlotStyleRed,AxesLabel{"n (day number)","

"}],ListPlot[lp],PlotRange->All](*Bootstrap*)M=100000;(*Step5.Errorsandgenerationofreplicas*)muu=Table[sol1[[n+1]]-sol1[[n]],{n,1,54}];realiz=Table[Drop[sol1,1]+Table[RandomVariate[PoissonDistribution[muu[[n]]]],{n,1,54}],{m,1,M}];A2=List[];gamma2=List[];(*Step6.FittingtoGompertzfunction.Meanandstandarddeviations*)Do[prsn=Table[{realiz[[m]][[n]],realiz[[m]][[n+1]]},{n,1,Ene-1}];funcn=NonlinearModelFit[prsn,x(Be-deltaLog[x]),{delta,Be},x];AppendTo[A2,Be/.funcn["BestFitParameters"]];AppendTo[gamma2,delta/.funcn["BestFitParameters"]],{m,1,M}]A2=Exp[(A2-1)/gamma2];(*Step7.Meansandstandarddeviations*)Mean[A2]StandardDeviation[A2]Mean[gamma2]StandardDeviation[gamma2](*Graphicsandhistograms*)FactorConfidence=1.96;Klow=Mean[A2]-FactorConfidence*StandardDeviation[A2];Khigh=Mean[A2]+FactorConfidence*StandardDeviation[A2];gammalow=Mean[gamma2]-FactorConfidence*StandardDeviation[gamma2];gammahigh=Mean[gamma2]+FactorConfidence*StandardDeviation[gamma2];sollowgamma=RecurrenceTable[{aa[n+1]aa[n](1+gammalowLog[Mean[A2]/aa[n]]),aa[1]lpr[[1]]},aa,{n,55}];solhighgamma=RecurrenceTable[{aa[n+1]aa[n](1+gammahighLog[Mean[A2]/aa[n]]),aa[1]lpr[[1]]},aa,{n,55}];ListPlot[{sollowgamma,solhighgamma,sol1,lp},PlotLegendsPlaced[{None,None,"Non-Linear Model Fit","Window-averaged data"},{Left,Top}],Filling{1{2}},FillingStyleGray,PlotStyle{None,None,Black,Black},Joined{True,True,True,False},Frame{True,True,False,False},FrameLabel{"n","

"}]Histogram[gamma2,35,Frame{True,True,False,False},FrameLabel{"γ",None}]sollowA=RecurrenceTable[{aa[n+1]aa[n](1+Mean[gamma2]Log[Klow/aa[n]]),aa[1]lpr[[1]]},aa,{n,55}];solhighA=RecurrenceTable[{aa[n+1]aa[n](1+Mean[gamma2]Log[Khigh/aa[n]]),aa[1]lpr[[1]]},aa,{n,55}];ListPlot[{sollowA,solhighA,sol1,lp},PlotLegendsPlaced[{None,None,"Non-Linear Model Fit","Window-averaged data"},{Left,Top}],Filling{1{2}},FillingStyleGray,PlotStyle{None,None,Black,Black},Joined{True,True,True,False},Frame{True,True,False,False},FrameLabel{"n","

"}]Histogram[A2,35,Frame{True,True,False,False},FrameLabel{"K",None}]

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Cite this as: María Teresa Perez Maldonado, Julian Bravo Castillero, Ricardo Mansilla, Rogelio Oscar Caballero Perez, "Discrete Gompertz Scheme and Bootstrap Analysis for Early Monitoring of the COVID-19 Pandemic in Cuba" from the Notebook Archive (2022), https://notebookarchive.org/2022-05-1fu9m2t