Table Of ContentMathematical 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
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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
