Table Of ContentTarek I. Zohdi
Modeling and
Simulation
of Infectious
Diseases
Microscale Transmission,
Decontamination and Macroscale
Propagation
Modeling and Simulation of Infectious Diseases
Tarek I. Zohdi
Modeling and Simulation
of Infectious Diseases
Microscale Transmission, Decontamination
and Macroscale Propagation
TarekI.Zohdi
UniversityofCalifornia
Berkeley,CA,USA
ISBN 978-3-031-18052-1 ISBN 978-3-031-18053-8 (eBook)
https://doi.org/10.1007/978-3-031-18053-8
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature
SwitzerlandAG2022
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse
ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
InRemembrance:Dedicatedtomyfather,
MagdE.Zohdi(1933–2020) andthemillions
ofvictimsoftheCOVID-19pandemic.
Preface
TheCOVID-19pandemicthatstartedin2019–2020hasbeenresponsibleformillions
of deaths and has led to a huge interest in modeling and simulation of infectious
diseases.Thismonographisanoutgrowthofthisera.Inordertomakethismonograph
asconciseandaccessibleaspossibletoawideaudience,itiscomprisedofashort
introductorychapteronbasicmathematics,optimizationandmachinelearningand
thenfoursubsequentpartsthatfocusonaspectsofmodernmodelingandsimulation
toolsfortheanalysisofinfectiousdiseases.Specifically,themonographisorganized
asfollows:
• Preliminaries-BasicMathematics,OptimizationandMachinelearning:The
systemsencounteredinthisfieldofstudyinvariablyleadtomodelswithdifferen-
tialequationsthathavemanyparameters.Thisrequirestoolsfornumericalsimu-
lationandparameteridentification,whichareformulatedasinverse/optimization
problemsforcost/errorfunctionminimization.Thecostfunctionsassociatedwith
parameteridentificationforcomplexphysicalsystemsareusuallynonconvexand
nonsmooth,makingapplicationofgradient-basedmethodsextremelydifficultor
impossible.Thismotivatesnonderivative,machinelearning-based,algorithms.
• Part 1-Macroscale Disease Propagation: The increase in readily available
computationalpowerraisesthepossibilitythatdirectagent-basedmodelingcan
playakeyroleintheanalysisofepidemiologicalpopulationdynamics.Specif-
ically, Part 1’s objective is to first introduce the reader to classical qualitative
approachesfordiseasespreadandthentopresentmoremodernmethods,based
onagent-basedmethodstoinvestigatetheemergentstructureofSIR-type(Suscep-
tible–Infected–Removed/Recovered)populations,onaglobalplanetaryscale.As
anillustrativemodelproblem,aplanet-widesystemisdeveloped,basedoninter-
actionbetweendiscreteentities(agents),whereeachagentonthesurfaceofthe
planetisinitiallyuninfected.Infectionsarethenseededontheplanet.Contracting
aninfectiondependsonthecharacteristicsofeachagent,i.e.,theirsusceptibility
and contact with the seeded, infected, agents. Agent mobility on the planet is
dictated by social policies, for example “shelter in place,” and “complete lock-
down”. The global population is then allowed to evolve according to infected
vii
viii Preface
statesofagents,overmanygenerations,leadingtoanSIRpopulation.Thischapter
followsworkfoundinZohdi[173].
• Part2-MicroscaleDiseaseTransmissionandVentilationSystemDesign:One
of the central issues in the propagation of infectious diseases is the potential
infectionzoneproducedbyarespiratoryemission,suchasavirus-ladencough.
In Part 2, mathematical models are developed to simulate the progressive time
evolutionofthedistributionofparticlesproducedbyrespiratoryemissions.The
models ascertain the range, distribution and settling time of the particles under
theinfluenceofgravityanddragfromthesurroundingair,whichareneededfor
constructing social distancing policies and workplace protocols. This leads to
anotherkeyissue—thedesignofventilationsystemstomitigateairbornetrans-
missionofinfectiousdiseases,producedbyrespiratoryemissions.Accordingly,
themodelisthenextendedtocouple:
– the progressive time evolution of the spatiotemporal distribution of particles
producedbyrespiratoryemissions,
– theuseofmasksimpederespiratoryparticleemissionpropagationand
– the use of genetic-based machine learning to optimally design ventilation
systemstocaptureparticlesnotstoppedbymasks.
ThischapterfollowsworkfoundinZohdi[172,176].
• Part3-UltravioletViralDecontamination:Part3focusesonviraldecontami-
nationbyultraviolet(UV)irradiationtechnologies.Specifically,Part3develops
anefficientandrapidcomputationalmethodtosimulateaUVpulse,inorderto
ascertainthedecontaminationefficacyofUVirradiationforasurface.Itisbased
ondecompositionofapulseintogroupsofrays,whicharethentrackedasthey
progress toward the target contact surface. This allows one to quickly quantify
the decontamination efficacy across the topology of the structure. This chapter
followsworkfoundinZohdi[174].
• Part4-VaccineDesignandImmuneResponse:InPart4,acomputationalframe-
workisdevelopedthatresearchersinthefieldcaneasilyimplementandsubse-
quentlyuseasanefficienttooltostudytheimmuneresponsetoavaccineinjection.
Therearethreemaincomponents:
– Digital-twin construction: The work develops an approach that efficiently
simulates the time-transient proliferation of cells/antibodies (proteins) and
regulator/antigens (deactivated toxin) to an injected vaccine within tissue
possessingcomplexheterogeneousmicrostructure.
– Efficient computation: A technique based on voxel (3D “volume pixels”)
representation of tissue microstructure and corresponding digital solution
methods is developed for the calculations, allowing extremely fast methods
to be used to construct derivatives and to iteratively solve the system with
minimalmemoryrequirements.
Preface ix
– Machine learning: The rapid and efficient computation allows for many
vaccinestobetestedquicklyandusesagenetic-basedmachinelearningalgo-
rithm to optimize the system. This is particularly useful for rapid design of
next-generationvaccinesandboostersfordiseasestrainmutations.
ThischapterfollowsworkfoundinZohdi[179].
A key concept that arises throughout the monograph is that of a “digital-twin”
of physical reality, i.e., a digital replica of a complex system that can be inexpen-
sivelyandsafelymanipulatedandoptimizedinavirtualsettingandthendeployed
inthephysicalworldafterward,thusreducingthepotentialcostsofexperimentsand
acceleratingthedevelopmentofnewtechnologies.
Berkeley,USA TarekI.Zohdi
September2022
Acknowledgements This work has been partially supported by the University of California
Berkeley,CollegeofEngineering,theWillC.FamilyHallChairinEngineering,theUSDAAI
InstituteforNextGenerationFoodSystems(AIFS),USDAawardnumber2020-67021-32855and
LamResearch.ManythanksgotoAidanGould,MayaHoriiandZacharyYunfortheireffortsin
proofreadingdraftsofthiswork.
Contents
1 Preliminaries: Basic Mathematics, Optimization
andMachine-Learning ......................................... 1
1.1 ElementaryNotationandMathematicalOperations .............. 1
1.1.1 Vectors,ProductsandNorms .......................... 1
1.1.2 BasicLinearAlgebra ................................. 2
1.1.3 IntegralTransformations .............................. 3
1.2 TemporalDiscretizationMethods ............................. 5
1.2.1 IsolatingaSingleParticle ............................. 5
1.3 BasicMachine-LearningandOptimization ..................... 7
1.3.1 Gradient-BasedMethods .............................. 7
1.3.2 Difficulties .......................................... 8
1.4 Genetic-BasedMachine-LearningAlgorithm ................... 9
1.4.1 AlgorithmicSpecifics ................................. 9
2 Part1:MacroscaleDiseasePropagation .......................... 13
2.1 Introduction ............................................... 13
2.1.1 ClassicalBasicModels ............................... 14
2.1.2 SIRSub-populationModels ........................... 15
2.1.3 GeneralizationoftheSIRFamilyofModels ............. 16
2.1.4 Agent-BasedModelsandObjectives .................... 18
2.2 DirectAgent-BasedInteractionModels ........................ 19
2.2.1 Agent-to-AgentInteractionandRulesofEngagement ..... 19
2.2.2 Algorithm ........................................... 21
2.2.3 ComputationalAcceleration ........................... 21
2.3 AModelProblem .......................................... 22
2.4 Extensions ................................................ 26
2.4.1 AnExampleofaSwarmFormulation ................... 27
2.5 AnAlgorithmforMovementinaRegion ...................... 29
2.5.1 PreliminaryNumericalExample ....................... 30
xi
xii Contents
3 Part 2: Microscale Disease Transmission and Ventilation
SystemDesign ................................................. 33
3.1 Introduction ............................................... 33
3.2 AnalyticalCharacterization:SimplifiedStokesianModel ......... 35
3.2.1 AnalysisofParticleVelocities ......................... 35
3.2.2 AnalysisofParticlePositions .......................... 37
3.2.3 Settling(Airborne)Time .............................. 37
3.3 ComputationalApproachesforMoreComplexModels .......... 38
3.3.1 MoreDetailedCharacterizationoftheDrag .............. 38
3.3.2 SimulationParameters ................................ 39
3.4 VentilationSystemDesign ................................... 43
3.4.1 Assumptions ........................................ 46
3.4.2 IncorporationofMasks ............................... 47
3.4.3 IncorporationofVents ................................ 47
3.4.4 OverallModel ....................................... 47
3.5 Genetic-BasedMachine-LearningVentilationOptimization ....... 48
3.5.1 ModelProblem ...................................... 48
3.6 SummaryandExtensions .................................... 50
4 Part3:UltravioletViralDecontamination ........................ 55
4.1 Introduction ............................................... 55
4.1.1 Objectives .......................................... 56
4.2 ElectromagneticEnergyPropagation .......................... 57
4.2.1 Beam-RayDecomposition ............................. 57
4.2.2 ReflectionandAbsorptionofRays ..................... 58
4.3 ElectromagneticWavePropagationandRays ................... 59
4.3.1 PlaneHarmonicWaveFronts .......................... 60
4.3.2 Natural (Random) Electromagnetic Energy
Propagation ......................................... 61
4.3.3 ReflectionandAbsorptionofEnergy-FresnelRelations .... 61
4.3.4 Reflectivity ......................................... 63
4.4 ModelProblemandResponseTrends ......................... 64
4.4.1 TrackingofBeam-DecomposedRays ................... 65
4.4.2 TestSurface ......................................... 65
4.5 Numerical/QuantitativeExamples ............................. 66
4.6 SummaryandDiscussion .................................... 67
5 Part4:VaccineDesignandImmune-SystemResponse ............. 71
5.1 Introduction ............................................... 72
5.1.1 BriefHistory ........................................ 72
5.1.2 TypesofVaccines .................................... 72
5.1.3 VaccineEfficacy ..................................... 73
5.1.4 ObjectivesofThisWork .............................. 74
5.2 AFlexibleImmune-ResponseDigital-Twin .................... 75
5.3 RapidVoxelBasedComputation .............................. 77
5.4 NumericalSimulationoftheCoupledSystem .................. 79
