Mathematical Modelling in Health, Social and Applied Science

325 Pages·2020·10.185 MB·English

by  Hemen Dutta

Forum for Interdisciplinary Mathematics
Hemen Dutta   Editor
Mathematical
Modelling in
Health, Social
and Applied
Sciences
Forum for Interdisciplinary Mathematics
Editor-in-Chief
P. V. Subrahmanyam, Department of Mathematics, Indian Institute of Technology
Madras, Chennai, Tamil Nadu, India
Editorial Board
Yogendra Prasad Chaubey, Department of Mathematics and Statistics, Concordia
University, Montreal, QC, Canada
Jorge Cuellar, Principal Researcher, Siemens AG, München, Bayern, Germany
Janusz Matkowski, Faculty of Mathematics, Computer Science and Econometrics,
University of Zielona Góra, Zielona Góra, Poland
Thiruvenkatachari Parthasarathy, Chennai Mathematical Institute, Kelambakkam,
Tamil Nadu, India
Mathieu Dutour Sikirić, Institute Rudjer Boúsković, Zagreb, Croatia
BhuDevSharma,ForumforInterdisciplinaryMathematics,Meerut,UttarPradesh,
India
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Hemen Dutta
Editor
Mathematical Modelling
in Health, Social and Applied
Sciences
123
Editor
Hemen Dutta
Department ofMathematics
Gauhati University
Guwahati, Assam, India
ISSN 2364-6748 ISSN 2364-6756 (electronic)
Forumfor Interdisciplinary Mathematics
ISBN978-981-15-2285-7 ISBN978-981-15-2286-4 (eBook)
https://doi.org/10.1007/978-981-15-2286-4
MathematicsSubjectClassification(2010): 97Mxx,68Uxx,91G70,90-XX,41-XX
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Preface
This book is targeted to graduate students, teachers and researchers interested in
various mathematical models associated with health, social and applied sciences.
The readers should find several useful tools and techniques necessary to develop
mathematicalmodelsandalsovariouswaystosolvethem.Theinterpretationofthe
findings shouldhelp readers tounderstand bettertheseveral issuesassociated with
health, social and applied sciences. The book consists of 10 chapters as follows:
Thechapteron“ViralImmunology:ModelingandAnalysis”aimstomodeland
analyse the interactions between viruses and immune system by proposing two
mathematical models that describe the role of adaptive immune response in
infectious diseases caused by viruses such as the human immunodeficiency virus,
hepatitisBvirusandhepatitisCvirus.Thefirstmodelisbasedondelaydifferential
equationsandthesecondonpartialdifferentialequations.Themodelsintegratethe
two main modes of virus propagation, namely virus-to-cell infection and direct
cell-to-cell transmission. The dynamical behaviours of the models were further
examinedby fivethreshold parameters,and the biological aspects of theanalytical
results were also presented.
Thechapteron“ModelingtheStochasticDynamicsofInfluenzaEpidemicswith
VaccinationControl,andtheMaximumLikelihoodEstimationofModelParameters”
discusses a family of stochastic models for the dynamics of influenza in a closed
humanpopulation.Itconsiderstreatmentforthediseaseintheformofvaccination
and incorporates the periods of effectiveness of the vaccine and infectiousness for
individualsinthepopulation.Themethodofmaximumlikelihoodandexpectation–
maximization algorithm has been applied in finding estimates for the parameters.
Estimators for some special epidemiological control parameters, such as the basic
reproduction number, are also computed. Further, a numerical simulation example
hasbeenincorporatedtofindthemaximumlikelihoodestimatorsoftheparameters
ofthemodel.
Thechapteron“ATwo-DimensionalDynamicalSystemforLocalTransmission
ofDenguewithTimeInvariantMosquitoDensity”discussesamathematicalmodel
describing the transmission of dengue. The critical parameters of the model
equations are determined by the climate variation and techniques to model the
v
vi Preface
parameters are developed considering their uncertainty behaviour. The stability
analysisofthemodeliscarriedout,andthemodelequationsarenumericallysolved
by using time-invariant parameters. Further, the simulated dengue infections are
validated with the actual cases in Colombo, Sri Lanka.
The chapter on “A Mathematical Study of a Model for HPV with Two High-
Risk Strains” aims to design a new two-sex deterministic model for two strains
(HPVtype-16/18andtype-31/45)ofHPVinfectionandanalyseforgaininginsights
intoitstransmissiondynamics.Themodelisclaimedtoexhibitthephenomenonof
backward bifurcation, where a stable disease-free equilibrium coexists with one or
more stable endemic equilibria when the associated reproduction number is less
thanunity.Itisfurtherclaimedthatthebackwardbifurcationphenomenoniscaused
due to the imperfect vaccine as well as the re-infection of individuals who recover
naturally from previous infection with the same strain of the disease. Numerical
simulations of the model are also carried out, which reveal that increasing the
fraction of vaccinated females against strain 1 (HPV type-16/18) infection can
significantly bring down the burden of strain 2 (HPV type-31/45) infection.
Thechapteron“TheImpactofFractionalDifferentiationinTermsofFittingfor
aProstateCancerModel UnderIntermittentAndrogenSuppressionTherapy”aims
to introduce fractional calculus as a prospective mathematical tool for cancer
dynamicsandforprostatecancermodelling,inparticular.Itfirstattemptstohandle
theproblemontheroleofandrogensforprostatecancerdevelopment.Basedonthe
hypothesis of authors, a new mathematical model consisting of conventional
logisticgrowthphenomenahasbeenconstructed.Anotherprospectivemodelbased
on an ecological phenomenon, cell quota, has also been developed. It compared
both the models demonstrating the mean squared error values for androgen- and
prostate-specificantigenforthefirst1.5cyclesofintermittentandrogensuppression
therapy administered to 62 selected patients from Vancouver Prostate Center
(Vancouver, BC, Canada). It also generates the fractional version of the model to
reduce the mean squared error values and verified that fractional differentiation
provides nearly better data fitting for mathematical modelling.
The chapter on “Toward the Realization of the ‘Europe 2020’ Agenda for
Economic Growth in the European Union: An Empirical Analysis Based on Goal
Programming” proposes a weighted goal programming model that can be used to
determine the optimal allocation of labour in each economic sector in order to
minimize the deviation from the goals offour different criteria which model eco-
nomic, environmental, energetic and social objectives. The model was applied to
each country of the European Union and measured their performance with respect
to the Europe 2020 agenda. The model claimed to provide insights and policy
recommendations such as a better integration of the incoming workforce in a
context of increasing immigration flows, development of renewable sources of
energy and green sustained transformation of national economic environments.
Thechapteron“OnthePoincaré-Andronov-MelnikovMethodforModellingof
Grazing Periodic Solutions in Discontinuous Systems” aims to derive Melnikov-
like condition for the persistence of a periodic and grazing solution under auton-
omousperturbationfordiscontinuoussystems.ThegrazingPoincarémapisderived
Preface vii
to fulfil the purpose. Then, its fixed point is investigated to determine desired
solutions leading to Melnikov-like conditions. It was further emphasized to illus-
tratethetheorybymeansofanexamplemodellingthetypeofsolutionsdiscussed.
Thechapteron“ModellingandAnalysisofPredationSystemwithNonlocaland
Nonsingular Operator” aims to model a novel system of predation involving two
individuals or species which interact in a nonlinear fashion with the Atangana–
Baleanu fractional derivative of order 0 < c < 1 in the sense of Caputo. This
derivativehasbeenusedtomodelsomeimportantreal-lifephenomenasuchasheat
flow,fractals,diffusionandgroundwaterflows,amongmany others.Thelocaland
globalstabilityanalysisofsuchmodelisgiventoaccuratelyprovideagoodchoice
of parameters when numerically simulating the full process. Relevant numerical
results for different instances offractional powerare also discussed inthis chapter.
Thechapteron“NewAspectsofFractionalEpidemiologicalModelforComputer
Viruses with Mittag–Leffler Law” aims to examine a fractional epidemiological
modelwithstrongmemoryeffects.ItusesafractionalderivativewithMittag-Leffler
typekerneltomoderatetheepidemiologicalmodelinordertointerpretthespreading
and controlling of computer viruses. The solution to the mathematical model has
been obtained by using q-HATM, a numerical algorithm used for solving epi-
demiological model of arbitrary order for computer viruses associated with the
Mittag-Leffler type kernel. The existence and uniqueness of the solution are
examined by employing the fixed point theory. Further, numerical simulations are
carried out for demonstrating theresults.
The chapter on “Numerical Simulation of Nonlinear Ecological Models with
NonlocalandNonsingularFractionalDerivative”aimstofocusonbothnon-spatial
andspatiallyextendedpredator–preysystemswhosedynamicsaredescribedbythe
Holling type-IV functional responses. It replaces the classical time derivative in
such models by the Atangana–Baleanu fractional derivative with nonlocal and
nonsingular properties. It claimed to formulate a two-step scheme based on the
fractional Adams–Bashforth method for approximating this derivative. A brief
linear stability analysis has been presented for the non-diffusive system and
reported the Hopf and Turing bifurcation analysis for the spatial case. Further,
numerical experiments are carried out to obtain range pattern results for different
parameter values of a in (0, 1] as well as numerical simulation to justify the
difference between integer and non-integer order results.
I am grateful to the contributors for their timely contribution and cooperation
during the entire process of reviewing and editing the chapters. The reviewers
deservesinceregratitudeforvoluntarilyofferingtheirserviceforthesuccessofthe
book. The editor and staff at Springer also deserve special thanks for their coop-
eration. I would like to acknowledge the encouragement of several friends and
well-wishers for bringing out such a book.
Guwahati, India Hemen Dutta
November 2019
Contents
Viral Immunology: Modeling and Analysis . . . . . . . . . . . . . . . . . . . . . . 1
Khalid Hattaf
Modeling the Stochastic Dynamics of Influenza Epidemics with
Vaccination Control, and the Maximum Likelihood Estimation
of Model Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Divine Wanduku, C. Newman, O. Jegede and B. Oluyede
A Two-Dimensional Dynamical System for Local Transmission
of Dengue with Time Invariant Mosquito Density . . . . . . . . . . . . . . . . . 73
W. P. T. M. Wickramaarachchi and S. S. N. Perera
A Mathematical Study of a Model for HPV with Two High-Risk
Strains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
A. Omame, D. Okuonghae and S. C. Inyama
The Impact of Fractional Differentiation in Terms of Fitting
for a Prostate Cancer Model Under Intermittent Androgen
Suppression Therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
Ozlem Ozturk Mizrak, Cihan Mizrak, Ardak Kashkynbayev
and Yang Kuang
Toward the Realization of the “Europe 2020” Agenda for Economic
GrowthintheEuropeanUnion:AnEmpiricalAnalysisBasedonGoal
Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
Cinzia Colapinto, Davide La Torre, Danilo Liuzzi and Aymeric Vié
On the Poincaré-Andronov-Melnikov Method for Modelling
of Grazing Periodic Solutions in Discontinuous Systems . . . . . . . . . . . . 241
Flaviano Battelli and Michal Fečkan
Modelling and Analysis of Predation System with Nonlocal
and Nonsingular Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
Kolade M. Owolabi and Hemen Dutta
ix
x Contents
New Aspects of Fractional Epidemiological Model for Computer
Viruses with Mittag–Leffler Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Devendra Kumar and Jagdev Singh
Numerical Simulation of Nonlinear Ecological Models
with Nonlocal and Nonsingular Fractional Derivative . . . . . . . . . . . . . . 303
Kolade M. Owolabi