Mathematical Analysis of Infectious Diseases

346 Pages·2022·12.988 MB·English

by  Praveen Agarwal, Juan J. Nieto, Delfim F.M. Torres

Mathematical Analysis of Infectious
Diseases
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Mathematical Analysis of
Infectious Diseases
Edited by
Praveen Agarwal
Anand International College of Engineering
Jaipur, India
Nonlinear Dynamics Research Center (NDRC)
Ajman University
Ajman, United Arab Emirates
Peoples Friendship University of Russia (RUDN University)
Moscow, Russian Federation
Juan J. Nieto
Institute of Mathematics
CITMAga
University of Santiago de Compostela
Santiago de Compostela, Galicia, Spain
Delfim F.M. Torres
Center for Research and Development in Mathematics and Applications (CIDMA)
Department of Mathematics
University of Aveiro
Aveiro, Portugal
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Contents
Contributors...................................................................... xiii
Preface........................................................................... xvii
Chapter 1:Spatiotemporaldynamicsofthe firstwaveofthe COVID-19epidemicin
Brazil ............................................................................ 1
J.M.V.Grzybowski,R.V.daSilva,andM.Rafikov
1.1 Introduction .................................................................... 1
1.2 Materialsandmethods......................................................... 3
1.2.1 TheSEIRmodel........................................................... 3
1.2.2 Modelintegration.......................................................... 4
1.2.3 Modelcalibration.......................................................... 4
1.2.4 Parameters ................................................................. 5
1.2.5 Casestudy:Brazil ......................................................... 5
1.2.6 ObtaininganetworkstructurefromR data.............................. 8
t
1.3 Results.......................................................................... 9
1.3.1 Spatiotemporaldynamicsofcoronavirusspread......................... 9
1.3.2 Coronavirusspreadandareacoverageoftheepidemic.................. 21
1.4 Discussion...................................................................... 22
1.4.1 EarlyspreadofcoronavirusinBrazil..................................... 22
1.4.2 Spreadvelocity ............................................................ 24
1.4.3 Brazil’sfailureincontainingthepandemicatearlystages.............. 24
1.5 Finalremarks................................................................... 24
References ............................................................................. 25
Chapter 2:Transport and optimalcontrolofvaccinationdynamicsforCOVID-19. 27
MohamedAbdelazizZaitri,MohandOuamerBibi,andDelfimF.M.Torres
2.1 Introduction .................................................................... 27
2.2 Vaccinetransportmodel ....................................................... 28
2.3 InitialmathematicalmodelforCOVID-19 ................................... 29
2.4 MathematicalmodelforCOVID-19withvaccination ....................... 30
2.5 Optimalcontrol................................................................. 31
2.6 Numericalresults............................................................... 33
2.7 Conclusion...................................................................... 37
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Contents
Acknowledgments..................................................................... 37
References ............................................................................. 37
Chapter 3:COVID-19’spandemic:anewwayofthinking through linear
combinations ofproportions ...................................................... 41
AdelaideFreitas,SaraEscudeiro,JulianaReis,andCristianaJ.Silva
3.1 Introduction .................................................................... 41
3.2 Estimationoflinearcombinationsofproportions ............................ 43
3.3 Materialandmethods.......................................................... 44
3.3.1 Datasets .................................................................... 44
3.3.2 Linearfunctionsofproportions........................................... 44
3.3.3 Inference ................................................................... 47
3.3.4 Graphicalprocedure....................................................... 48
3.4 Resultsanddiscussion ......................................................... 48
3.4.1 LinearcombinationL (t)=ω p (t)− p2(t)+p3(t) ...................... 48
1 1 1 2
3.4.2 LinearcombinationL (t)=ω p (t)−p (t) ........................... 51
2 1 1 2
3.4.3 LinearcombinationL (t)=ω p (t)−p (t) ........................... 52
3 2 2 3
3.4.4 LinearcombinationL (t)=ω p (t)−p (t) ........................... 54
4 4 4 5
3.4.5 LinearcombinationL (t)=ω p (t)− p3(t)+p5(t) ...................... 55
5 4 4 2
3.5 Conclusion...................................................................... 56
Acknowledgments..................................................................... 58
References ............................................................................. 58
Chapter 4:Stochastic SICAepidemicmodelwith jumpLévyprocesses ............ 61
HoussineZine,JaouadDanane,andDelfimF.M.Torres
4.1 Introduction .................................................................... 61
4.2 Existenceanduniquenessofaglobalpositivesolution ...................... 63
4.3 Extinction....................................................................... 65
4.4 Persistenceinthemean ........................................................ 67
4.5 Numericalresults............................................................... 69
4.6 Conclusion...................................................................... 70
Acknowledgments..................................................................... 71
References ............................................................................. 71
Chapter 5:Examining the correlationbetweenthe weather conditionsand COVID-
19 pandemicinGalicia ........................................................... 73
LucaPiccotti,GuidoIgnacioNovoa-Flores,andJuanJ.Nieto
5.1 Introduction .................................................................... 73
5.2 Fuzzysets....................................................................... 75
5.3 Results.......................................................................... 75
5.4 Conclusions .................................................................... 79
References ............................................................................. 79
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Contents
Chapter 6:Afractional-order malariamodelwithtemporaryimmunity ........... 81
RamSinghandAttiqulRehman
6.1 Introduction .................................................................... 81
6.2 Preliminariesonfractionalcalculus........................................... 82
6.3 Modeldescription.............................................................. 83
6.3.1 Classicalintegermodel.................................................... 84
6.3.2 Fractionalordermathematicalmodel..................................... 84
6.4 BasicpropertiesoftheABCmalariamodel .................................. 85
6.4.1 Existenceanduniqueness ................................................. 85
6.4.2 Invariantregionandattractivity........................................... 87
6.4.3 Positivityandboundedness ............................................... 87
6.5 Theanalysis .................................................................... 88
6.5.1 Stabilityanalysis........................................................... 89
6.6 Numericalsolutionoffractionalmalariamodel.............................. 92
6.7 Discussion...................................................................... 94
6.8 Conclusion...................................................................... 96
References ............................................................................. 100
Chapter 7:Parameter identification inepidemiologicalmodels .................... 103
AnaCarpioandEmilePierret
7.1 Introduction .................................................................... 103
7.2 SEIJRmodelsforclosedsystems ............................................. 104
7.3 UncertaintyquantificationbyBayesiantechniques .......................... 105
7.3.1 BayesianformulationforSEIJRcoefficients............................. 106
7.3.2 Priorselection.............................................................. 107
7.3.3 MarkovChainMonteCarlosampling.................................... 109
7.4 Effectofnonpharmaceuticalactions .......................................... 111
7.4.1 PiecewiseSEIJRsystem:lockdown ...................................... 113
7.4.2 PiecewiseSEIJRsystem:release ......................................... 114
7.4.3 Equilibrium ................................................................ 119
7.5 SEIJRmodelincludingmigration............................................. 119
7.6 Optimizationapproachtocontrol ............................................. 120
7.7 Conclusions .................................................................... 123
Acknowledgments..................................................................... 123
References ............................................................................. 123
Chapter 8:Lyapunov functions and stability analysis offractional-ordersystems .. 125
AdnaneBoukhouima,HoussineZine,ElMehdiLotfi,MarouaneMahrouf,DelfimF.M.Torres,and
NouraYousfi
8.1 Introduction .................................................................... 125
8.2 Preliminaries ................................................................... 126
8.3 Usefulfractionalderivativeestimates......................................... 128
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Contents
8.4 Anapplication.................................................................. 131
8.5 Conclusion...................................................................... 134
Acknowledgments..................................................................... 135
References ............................................................................. 135
Chapter 9:Somekey conceptsofmathematical epidemiology...................... 137
JaafarElKarkriandMohammedBenmir
9.1 Introduction .................................................................... 137
9.2 Ashorthistoricalintroduction................................................. 138
9.3 Equilibria,thebasicreproductionnumberandfinalsizerelation............ 142
9.3.1 Equilibriaandstability .................................................... 142
9.3.2 Thebasicreproductionnumber........................................... 143
9.3.3 Thefinalsizerelations .................................................... 145
9.4 Sojourntime,delay,andincidenceforms..................................... 146
9.4.1 Sojourntime ............................................................... 146
9.4.2 Delaysinepidemiologicalmodels........................................ 147
9.4.3 Differentformsofincidence.............................................. 149
9.5 Numericalsimulations......................................................... 150
9.5.1 Numericalmethods........................................................ 150
9.5.2 Agent-basedsimulationmodeling........................................ 153
9.5.3 Neuralnetworks ........................................................... 154
9.6 Herpesmodeling ............................................................... 157
9.7 Conclusion...................................................................... 159
Acknowledgment...................................................................... 160
References ............................................................................. 160
Chapter 10: Analytical solutions and parameterestimationofthe SIRepidemic
model............................................................................. 163
DimiterProdanov
10.1 Introduction .................................................................... 163
10.2 TheSIRmodel ................................................................. 164
10.3 Second-ordersystemsequivalenttoSIR...................................... 166
10.3.1 Asecondorderdifferentialequationforthei-variable .................. 166
10.3.2 Asecondorderdifferentialequationforthes-variable.................. 167
10.4 Indeterminateanalyticalsolution.............................................. 169
10.4.1 Thes-variable.............................................................. 169
10.4.2 Thei-variable.............................................................. 170
10.4.3 Ther-variable.............................................................. 170
10.5 Inverseparametricsolution.................................................... 171
10.5.1 Peakvalueparametrization................................................ 173
10.5.2 Initialvalueparametrization .............................................. 174
10.6 Analysisoftheincidencevariable............................................. 175
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Contents
10.7 AsymptoticanalysisoftheSIRmodel........................................ 178
10.8 Numericalapproximation...................................................... 180
10.9 CaststudyI:applicationtoinfluenzaA....................................... 181
10.10 CaststudyII:applicationtoCOVID-19 ...................................... 182
10.11 Discussionandconclusions.................................................... 182
Appendix10.A TheLambertWfunctionandrelatedintegrals .................... 184
Appendix10.B Differentialfields.................................................... 186
References ............................................................................. 188
Chapter 11: Globalstability ofadiffusive SEIRepidemicmodelwithdistributed
delay ............................................................................. 191
AbdesslemLamraniAlaoui,MoulayRchidSidiAmmi,MouhcineTilioua,andDelfimF.M.Torres
11.1 Introduction .................................................................... 191
11.2 Mathematicalmodel ........................................................... 192
11.3 Analysisofthemodel.......................................................... 194
11.3.1 Well-posedness ............................................................ 194
11.3.2 Equilibriaandthebasicreproductionnumber ........................... 197
11.3.3 Globalstabilityofthediseasefreeequilibrium.......................... 198
11.3.4 Globalstabilityoftheendemicequilibrium.............................. 201
11.4 Numericalsimulations......................................................... 204
11.5 Concludingremarks............................................................ 207
Acknowledgment...................................................................... 208
References ............................................................................. 208
Chapter 12: Applicationoffractional orderdifferential equations inmodelingviral
diseasetransmission .............................................................. 211
ShahramRezapourandHakimehMohammadi
12.1 Introduction .................................................................... 211
12.2 Preliminaries ................................................................... 212
12.3 MathematicalmodeloftheAH1N1/09influenzatransmission.............. 214
12.4 Equilibriumpoints ............................................................. 215
12.4.1 Stabilityofequilibriumpoint ............................................. 216
12.5 Existenceofsolution........................................................... 217
12.5.1 ExistenceofsolutionbythePicard-Lindelofapproach.................. 218
12.6 Optimalcontrolapproach...................................................... 220
12.7 Numericalresults............................................................... 222
12.7.1 Numericalmethod......................................................... 222
12.7.2 Numericalsimulation...................................................... 223
12.7.3 Reproductionnumbersensitivity ......................................... 225
12.8 Conclusion...................................................................... 227
References ............................................................................. 228
ix