Mathematical Modeling of Biological Systems, Volume II: Epidemiology, Evolution and Ecology,Immunology, Neural Systems and the Brain, and Innovative Mathematical Methods PDF
Preview Mathematical Modeling of Biological Systems, Volume II: Epidemiology, Evolution and Ecology,Immunology, Neural Systems and the Brain, and Innovative Mathematical Methods
Mathematical Modeling of Biological Systems, Volume II Epidemiology, Evolution and Ecology, Immunology, Neural Systems and the Brain, and Innovative Mathematical Methods Andreas Deutsch Rafael Bravo de la Parra Rob J. de Boer Odo Diekmann Peter Jagers Eva Kisdi Mirjam Kretzschmar Petr Lansky Hans Metz Editors Birkha¨user Boston • Basel • Berlin RafaelBravodelaParra RobJ.deBoer AndreasDeutsch DepartamentodeMatema´ticas DepartmentofTheoretical CenterforInformationServices UniversidaddeAlcala´ Biology andHighPerformanceComputing 28871Alcala´deHenares UtrechtUniversity TechnischeUniversita¨tDresden Spain Padualaan8 01062Dresden 3584CDUtrecht Germany TheNetherlands OdoDiekmann PeterJagers EvaKisdi MathematischInstituut MathematicalSciences DepartmentofMathematics UtrechtUniversity ChalmersUniversityofTechnology andStatistics Budapestlaan6 Go¨teborgUniversity UniversityofHelsinki 3584CHUtrecht 41296Go¨teborg GustafHa¨llstro¨minkatu2b TheNetherlands Sweden 00014Helsinki Finland HansMetz MirjamKretzschmar PetrLansky SectionTheoretical Fakulta¨tfu¨rGesundheitswis- InstituteofPhysiology EvolutionaryBiology senschaften AcademyofSciencesof InstituteofEvolutionaryand Universita¨tBielefeld theCzechRepublic EcologicalSciences Universita¨tsstr.25 Videnska1083 RijksuniversiteitLeiden 33615Bielefeld 14220Prague4 Kaiserstraat63 Germany CzechRepublic 2311GPLeiden TheNetherlands MathematicsSubjectClassification:00A71,37N25,46N60,47N60,62P10,76Zxx,78A70,92-XX, 92Bxx,92B05,92Cxx,92C05,92C15,93A30 LibraryofCongressControlNumber:2007925791 ISBN-13:978-0-8176-4555-7 e-ISBN-13:978-0-8176-4556-4 Printedonacid-freepaper. (cid:1)c2008Birkha¨userBoston Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewrit- tenpermissionofthepublisher(Birkha¨userBoston,c/oSpringerScience+BusinessMediaLLC,233 SpringStreet,NewYork,NY10013,USA),exceptforbriefexcerptsinconnectionwithreviewsor scholarlyanalysis.Useinconnectionwithanyformofinformationstorageandretrieval,electronic adaptation,computersoftware,orbysimilarordissimilarmethodologynowknownorhereafterde- velopedisforbidden. Theuseinthispublicationoftradenames,trademarks,servicemarksandsimilarterms,evenifthey arenotidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyare subjecttoproprietaryrights. 9 8 7 6 5 4 3 2 1 www.birkhauser.com (LM/EB) Preface This edited volume contains a selection of chapters that are an outgrowth of the Eu- ropean Conference on Mathematical and Theoretical Biology (ECMTB05, Dresden, Germany, July 2005). The peer-reviewed contributions show that mathematical and computational approaches are absolutely essential to solving central problems in the lifesciences,rangingfromtheorganizationallevelofindividualcellstothedynamics ofwholepopulations. The contributions indicate that theoretical and mathematical biology is a diverse and interdisciplinary field, ranging from experimental research linked to mathemati- calmodellingtothedevelopmentofmoreabstractmathematicalframeworksinwhich observations about the real world can be interpreted and with which new hypotheses fortestingcanbegenerated.Today,muchattentionisalsopaidtothedevelopmentof efficientalgorithmsforcomplexcomputationsandvisualization,notablyinmolecular biologyandgenetics.Thefieldoftheoreticalandmathematicalbiologyandmedicine hasprofoundconnectionstomanycurrentproblemsofgreatrelevancetosociety.The medical, industrial, and social interests in its development are in fact undisputable. Insights and predictions from mathematical modelling are used increasingly in deci- sionsupportinmedicine(e.g.,immunologyandspreadofinfectiousdiseases,cancer research,cardiovascularresearch,neurologicalresearch,optimizationofmedicaltreat- ments,imaging),environmentalandnaturemanagement,climateproblems,agriculture andmanagementofnaturalresources.Fastdevelopmentsinareassuchasbiotechnol- ogy (e.g., genome projects, genetic modification, tissue engineering) continue to add newfocalpointsofactivitytothefield.Thecontributionsofthisvolumecapturesome ofthesedevelopments. The volume contains five parts: epidemiology, evolution and ecology, immunol- ogy,neuralsystemsandthebrain,and,finally,innovativemathematicalmethodsand education. PartIdealswithepidemiologyandcontainsthreechapters. Smith discusses implications of new malaria vaccines. Recent breakthroughs in malaria vaccines have given new hope that a safe, effective malaria vaccine may be found. In particular, the following epidemiological questions are addressed: 1. What vi Preface level of vaccination coverage is required to offset the limitations of an imperfect disease-modifying vaccine? 2. Could the introduction of a low-efficacy malaria vac- cineleadtoanincreaseinthenumberofsecondaryinfections?3.Whatcharacteristics ofsuchavaccinewillhavethegreatesteffectontheoutcome?Amathematicalmodel isdevelopedforadisease-modifyingmalariavaccinethatisgivenoncepriortoinfec- tion,andtheminimumcoveragelevelfordiseaseeradicationisestablished.Itisshown thatthereisathresholddependingontherelativerateofinfection,theefficacyofthe vaccine,andthedurationofinfection.Vaccineswhichreducetherateanddurationof infection will always result in a decrease in secondary infections. More surprisingly, thereisaduration“shoulder,”suchthatvaccinesthatincreasethedurationofinfection slightlywillstillleadtoadecreaseinsecondaryinfections,eveniftherateofinfection isunchanged.Beyondthis,thenumberofsecondaryinfectionswillincreaseunlessthe rateofinfectionissufficientlylowered.Thisiscriticalforlow-efficacyvaccines. Burie,etal.introduceamodelfortheinvasionofafungaldiseaseoveravineyard. In particular, the spatiotemporal spreading of a fungal disease over a vineyard is in- vestigatedusingaSEIR-typemodelcoupledwithasetofpartialdifferentialequations describing the dispersal of the spores. The model takes into account both short and long range dispersal of spores and growth of the foliar surface. Results of numerical simulationsarepresented,andamathematicalresultfortheasymptoticbehaviorofthe solutionsisgiven. Stollenwerk and Mikolajczyk present an algorithm for parameter estimation in nosocomial infections. Parameter estimation in nosocomial infections poses specific problems for estimation techniques. The mathematical description of the spread of nosocomial infections incorporates transmission as the dynamic part; the outcome is discrete,andtheamountofavailableinformationisusuallysmall.Theauthorstrans- fer an estimation technique developed previously for plant epidemics to nosocomial infectionsanddemonstrateitsapplicationtoadatasetrelatedtomethicillin-resistant Staphyloccocusaureus(MRSA). PartIIfocusesonevolutionandecologyandconsistsofeightchapters. Broom, et al. discuss evolutionarily stable investment in anti-predatory defenses and aposematic signalling. Many species possess defenses (such as toxins) against predator attack which cannot be observed by the predator prior to attack, but which might be beneficial for the predator to avoid. Often, such animals are brightly col- oredorhavesomeotherwayofsignallingthattheyaredefended(aposematism).Inan earlierpapertheauthorsexaminedtheevolutionandmaintenanceofdefenseandcon- spicuousness,thebrightnessofthedefensesignal,insuchpreyspeciesusingagame theoretic model. Here, they develop the model further, and, in particular, expand on themoretheoreticalresultswithexamplesdemonstratingthetypeofsolutionswhich can occur. The authors categorize eight possible configurations of solution states for simple solutions. Finally, it is shown that there is another class of solutions possible where there is strong between-individual variation in appearance between conspicu- ous,poorlydefendedprey,andoneexampleofthiscomplexsolutionisdemonstrated. Laird, et al. introduce an overview of the Tangled Nature model of evolutionary ecology. The model focuses on the effect of evolution and multiple interactions on Preface vii ecologicalandevolutionaryobservables.Furthermore,themodelisindividualbased, andecologicalstructures,suchasspecies,areemergentquantities.Thedynamicscon- sistsofasimplisticmutation-pronemultiplicationinwhichtheprobabilityofproduc- inganoffspringisdeterminedbytheoccupancyingenotypespace.Themacroscopic long-time dynamics is intermittent and exhibits a slow decrease in the macroscopic extinction rate. Ecological quantities such as the species-abundance distribution and thespecies-arearelationshipcomparequalitativelywellwithobservations,asdoesthe relationbetweeninteractionanddiversity.Theeffectofcorrelationsbetweenparents andmutantshasbeenstudied,aswellastheeffectofaconservedresource. Garaydiscussestherelativeadvantageandfundamentaltheoremsofnaturalselec- tion. According to the tenet of Darwinian selection, a phenotype will spread only if itsfitnessisgreaterthanthemeanfitnessoftheentirepopulation.Itisthereforenat- ural to introduce the notion of relative advantage of a replicator, which is defined as theexpectedfitnessofthisreplicatorminustheaveragefitnessoftheentirereplicator population.Forgeneralreplicatordynamics,itisshownthattherelativeadvantageof an offspring population over its parent population is proportional to the variance in fitness.Therelationshipbetweentheproposedandearlierversionsofthefundamental theoremofnaturalselectionisalsodiscussed. Konconsiderscompetitiveexclusionbetweenyear-classesinasemelparousbien- nial population. In particular, competitive exclusion between two reproductively iso- latedyear-classesintheLesliematrixmodelforasemelparousbiennialpopulationis investigated.Theresultsshowthatcompetitiveexclusionoccursifcompetitionismore severebetweenthanwithinyear-classes.Acriterionissuggestedwhichisapplicable evenifthemodelexhibitscomplexbehavior. Nedorezov,etal.studytheimpactofwinterconditionsonthedynamicsofaniso- latedpopulation.Inparticular,thischapterisdevotedtotheanalysisofsingle-species population dynamics models with overlapping and nonoverlapping generations. It is assumedthattherearenoactivitiesofindividualsduringthewintertime(as,forexam- ple,isthecaseforforestinsectpopulationsintheborealzone),andchangesinpopula- tionsizeatthesemomentsaredescribedwithabrokentrajectory(“jumpdown”).Fur- thermore,itisassumedthatthefecundityofindividualsisconstantandthatthequota ofindividualssurvivingwinterdependsonthewithin-yearpopulationdynamics.The dynamics of the models, which are determined by the influence of winter conditions on the survival of individuals and by the influence of intra-population self-regulative mechanisms,areanalyzed.Forsomeparticularcasestheconditionsforpopulationex- tinctionandforstabilizationatanonzerolevelaredetermined;itisshownnumerically thatchaoticregimesexistinsomemodels.Inaddition,theconditionsforthereduction ofthemodelstosomewell-knowndiscretemodelsareobtained. Fuller, et al. consider the topic of planning for biodiversity conservation using stochastic programming. Rapid species extinctions and the loss of other biodiversity features worldwide have prompted the development of a systematic planning frame- workfortheconservationofbiodiversity.Limitedresources(∼40millionUSDannu- ally)areavailableforconservation,particularlyinthedevelopingcountriesthatcon- tain many of the world’s hotspots of species diversity. Thus, conservation planning problemsareoftenrepresentedasmathematicalprogramsinwhichtheobjectiveisto viii Preface select sites to serve as conservation areas so that the cost of the plan is as small as possible and adequate habitat is protected for each species. Here, the authors gener- alizethisapproachtoallowforuncertaintyintheplanningprocess.Inparticular,itis assumedthatthespeciestobeprotecteddisperseaftertheconservationareasareestab- lished and that planners cannot anticipate with certainty the species’ future locations whenselectingtheconservationareas.Thisuncertaintyismodeledbyincludingran- domvariablesinthemathematicalprogram.Theapproachisillustratedbydesigning anetworkofconservationareasforbirdsinsouthernQuebec. Eberl and Schraft present a diffusion-reaction model of a mixed culture biofilm arising in food safety studies. Bacterial biofilms are communities of microorganisms thatdeveloponinterfacesinaqueousenvironments.Theauthorsformulateadensity- dependentdiffusion-reactionmodelforthegrowthofadual-speciesbiofilm.Bothbac- teriaresponddifferentlytotheirenvironmentanddevelopdifferenttypesofbiofilms: one is a classical aerobic biofilm former that produces the characteristic cluster-and- channelbiofilmmorphology;theotheronealsodevelopsunderanaerobicconditions andtendstoformflat,creepingbiofilms.Apreviouslydevelopednonstandardfinite- differenceschemeisadaptedforcomputersimulation.Inanumericalexperimentitis shownhowvariationsofasingleparameter(growthrate)cantriggerdifferentspatial patternsandorganizationofthebiofilmcommunity. Iwata, et al. discuss the periodical population dynamics of lottery models with undevelopedseeds.Themechanismthatpromotescoexistenceofspecieshasnotbeen completelyclarifiedyet.Theauthorsproposethattheamountofnutrientcanbeoneof thefactorsthatpromotescoexistenceofspecies.Plantspecieshavetoreproduceseeds toproducedescendants.Evenifplantspeciesdoreproduceseeds,itisnotensuredthat every seed will bud. The amount of seeds that can bud successfully depends on the amount of nutrient: if the nutrient is scarce, then not every seed can bud, but if the nutrient is rich, then every seed can bud. It is also assumed that the amount of seeds reproducedbyoneplantindividualdependsontheamountofnutrient.Itisshownthat inthissituationthepopulationdynamicsofplantsexhibitsacomplexbehavior,which promotescoexistenceofspecies. PartIIIdealswiththeimmunesystemandhasfourchapters. Zanlungo, et al. present an automata-based microscopic model for the clonal ex- pansion process. The model is based on a repertoire of antigens and T lymphocytes interacting via the APC cells which present the antigen peptides. Each cell is repre- sentedbyanautomatonmovingrandomlyonatwo-dimensionallattice.Thissimplified modelisusedinordertointroducelocalandspatialconsiderationsinthemathematical modelsofclonalexpansionbasedondifferentialequations,andatthesametimetoat- temptananalyticalinterpretationoftheresultsofcomputersimulations.Furthermore, ameanfieldtheoryisderived,whoseresultsareingoodagreementwiththesolutions ofthemicroscopicmodel,atleastforsituationsthatarenottoofarfromequilibrium. Thismodelmaybeusedasthebasisofamorerealisticonethatcouldfollowtheclonal expansionprocessonasimplifiedversionofthelymphaticnetwork. VogelandBehnfocusonTh1–Th2regulationandallergyandpresentabifurcation analysis of the nonautonomous system. A previously proposed mathematical model Preface ix based on a simplified scheme of Th1–Th2 regulation mediated by the cytokine net- workwhichdescribesthepopulationdynamicsofallergen-specificnaiveTcells,Th1 and Th2 cells, autocrine and cross-suppressive cytokines, and allergen is further in- vestigated. The model provides a theoretical explanation of the switch from a Th2- dominatedresponsetoaTh1-dominatedresponsetoallergeninallergicindividualsas aresultofahyposensitizationtherapy.Theauthorsfocushereonthebifurcationanal- ysis of the nonautonomous dynamical system driven by periodic allergen injections. Thestabilityofthefixedpointsofastroboscopicmapisinvestigated.Thesetofunsta- blefixedpointsformsthedynamicalseparatrixbetweentheregionsofTh2-dominated response and Th1-dominated response which is crossed during a successful therapy. Themaintenancephaseofthetherapyholdsthesystemnearthestablefixedpointof thestroboscobicmap. SchmidtchenandBehndiscussthearchitectureofrandomlyevolvingidiotypicnet- works.Blymphocytesexpressontheirsurfacereceptors(antibodies)ofagivenspeci- ficity (idiotype). Crosslinking these receptors by complementary structures, antigens or antibodies, stimulates the lymphocyte. Thus, a large functional network of inter- actinglymphocytes,theidiotypicnetwork,emerges.Idiotypicnetworks,conceivedby NielsJerne30yearsago,experiencearenewedinterest,e.g.,inthecontextofautoim- munediseases.Inapreviouslyproposedminimalisticmodel,idiotypesarerepresented bybitstrings.Thepopulationdynamicsoftheidiotypeclonesisreducedtoazero-one scheme. An idiotype survives only if it meets enough but not too many complemen- tary structures. The authors investigate the random evolution of the network towards a highly organized functional architecture which is driven by the influx of new idio- types,randomlygeneratedinbonemarrow.Theverticescanbeclassifiedintodifferent groups,whichareclearlydistinguished,e.g.,bythemeanlifetimeoftheoccupiedver- tices. They include densely connected core groups and peripheral groups of isolated vertices,resemblingthecentralandperipheralpartsofthebiologicalnetwork.Theau- thorshavedeterminedtheconstructionprinciplesoftheobservedpatternsandpropose a description of their architecture, which is easily transferable to other patterns and applicabletodifferentsystemsizes. Sannikova presents an analysis of infectious mortality by means of the individu- alizedriskmodel.Thegoaloftheworkistodescribethemechanismunderlyingthe age-specificincreaseindeathriskrelatedtoimmunosenescenceandtodeterminethe cause-specifichazardrateasafunctionofimmunesystemcharacteristics.Therefore, a mathematical model that allows for the estimation of the age-specific risk of death causedbyinfectiousdiseaseshasbeendeveloped.Themodelconsistsofthreeparts: (1)amodelofimmunosenescence,(2)amodelofinfectiousdisease,and(3)amodel giving the relationship between disease severity and the risk of death. The proposed modelmakesitpossibletoanalyzeage-specificmortalityfrominfectiousdiseasesand topredictfuturechangesinmortalityduetopublichealthactivity.Atthesametimeit canbeusedforindividualizedriskassessment. PartIVdealswithneuralsystemsandthebrainandincludesninechapters. Schierwagen,etal.focusonneuromorphologicalphenotypingintransgenicmice. 3D morphological data have been used to quantitatively characterize the morpholog- x Preface ical phenotype of pyramidal neurons in transgenic mice. The authors calculated the multiscale fractal dimension (MFD) of reconstructed neuronal cells. Changes in the complexityofneuronalmorphologyduetopermanentactivationofp21Rasinthepri- marysomatosensorycortexoftransgenicmicecorrelatewithchangesintheMFDof dendritesofpyramidalneurons.Transgenicneuronsseemslightlylesscomplex(i.e., havelowerpeakfractaldimension)ifcomparedwiththewildtype.Ontheotherhand, it is shown that the enhanced p21Ras activity in transgenic mice may lead to greater varietyinthecellmorphologicalphenotype. Gibson,etal.introduceaquantitativemodelofATP-mediatedcalciumwaveprop- agationinastrocytenetworks.Inthepast,attentionhasmainlybeenfocusedonneu- rons and the role they play, both individually and as parts of networks, in the func- tioning of the brain and nervous system. However, glial cells outnumber neurons in the brain, and it is now becoming apparent that, far from just performing support- iveandhousekeepingtasks,theyarealsoactivelyengagedininformationprocessing and possibly even learning. Communication in glial cells is manifested by waves of calcium ions (Ca2+) that are released from internal stores, and these waves are ob- served experimentally using fluorescent markers attached to the ions. The waves can beinitiatedbystimulationofasinglecell,andinitiallyitwasassumedthatthetrans- mission mechanism involved the passage of an intercellular signalling agent through gapjunctionsconnectingthecells.However,asurprisingfeatureisthatinmanycases thecalciumwavescancrosscell-freezones,thusindicatingthepresenceofanextra- cellular messenger. The authors have constructed a mathematical model of calcium wave propagation in networks of model astrocytes, these being a subclass of glial cells. The extracellular signalling agent is ATP (adenosine triphosphate), and it acts onmetabotropicpurinergicreceptorsontheastrocytes,initiatingaG-proteincascade leadingtotheproductionofinositoltrisphosphate(IP )andthesubsequentreleaseof 3 Ca2+fromintracellularstoresviaIP -sensitivechannels.Stimulationofonecell(bya 3 pulseofATPorbyraisingtheIP level)leadstotheregenerativereleaseofATPboth 3 fromthiscellandfromneighboringcells,andhenceaCa2+ wave.Resultsaregiven forthepropagationofCa2+ wavesintwo-dimensionalarraysofmodelastrocytesand alsoinlaneswithcell-freezonesinbetween.Thesetheoreticalconsiderationssupport theconceptofextracellularpurinergictransmissioninastrocytenetworks. AtayandHuttanalyzethedynamicsofneuralfieldswithdistributedtransmission speeds. In particular, the continuous field model of neural populations is considered withtheadditionofadistributionoftransmissionspeeds.Thespeeddistributionarises as a result of the natural variability of the properties of axons, such as their degree ofmyelination.Theauthorsanalyzethestabilityandbifurcationsofequilibriumsolu- tionsfortheresultingfielddynamics.Usingaperturbationapproach,itisshownthat the speed distribution affects the frequency of bifurcating periodic solutions and the phase speed of traveling waves. The theoretical findings are illustrated by numerical calculations. Hampelfocusesontheestimationofdifferentialentropyforpositiverandomvari- ablesanditsapplicationincomputationalneuroscience.Thischaptertakesessentials stepstowardthegoalofadifferentialentropyconceptandprovidesasetofmethods relatedtodifferentialentropyestimation.Atthebeginning,theauthordefinesthebasic
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