Mathematical Models for Therapeutic Approaches to Control Psoriasis

100 Pages·2019·2.913 MB·English

by  Priti Kumar Roy, Abhirup Datta

SPRINGER BRIEFS IN APPLIED SCIENCES AND
TECHNOLOGY  MATHEMATICAL METHODS
Priti Kumar Roy
Abhirup Datta
Mathematical
Models for
Therapeutic
Approaches to
Control Psoriasis
123
SpringerBriefs in Applied Sciences
and Technology
Mathematical Methods
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Priti Kumar Roy Abhirup Datta
(cid:129)
Mathematical Models
for Therapeutic Approaches
to Control Psoriasis
123
Priti KumarRoy Abhirup Datta
Department ofMathematics Department ofMathematics
JadavpurUniversity JadavpurUniversity
Kolkata, West Bengal, India Kolkata, West Bengal, India
ISSN 2191-530X ISSN 2191-5318 (electronic)
SpringerBriefs inApplied SciencesandTechnology
ISSN 2365-0826 ISSN 2365-0834 (electronic)
SpringerBriefs inMathematical Methods
ISBN978-981-13-9019-7 ISBN978-981-13-9020-3 (eBook)
https://doi.org/10.1007/978-981-13-9020-3
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This book is globally dedicated to the
pandemic victims suffering from psoriasis.
Preface
Psoriasis is a type 1 cytokine-mediated chronic, autoimmune inflammatory skin
disease, in which both genetic and environmental influences have considerable
roles.Ithasadeepdisagreeableimpactonthepatient’sphysical,social,andmental
happiness.Themostcommonformoflesiongenerallyoccursontheuppersurface
oftheskinwitharedandsilverwhitishscalypatch.Theseverityofdiseasecanbe
seenin2%–3%ofthetotalworldpopulation.Thepathogenesisofpsoriasisisbased
on the immunological imbalance of the human immune system.
To control this skin disease, progress in the medicinal behavior or a perfect
strategy is required. Scientists and researchers have persistently devoted themselves
towards gaining exceptional knowledge of the immunological outbreak of this dis-
ease. In the present era, the use of various mathematically supported tools and
techniques is one of the furthermost momentous weapons along with the immuno-
logical study and knowledge of immunologists and experimentalists to keep the
diseaseundercontrol.Itwillfacilitateustogovernthediseasedynamics.Therefore,
inthisbook,wehavefocusedonthediseasedynamicsandvariouscontrolanalyses
of psoriasis, through which ultimately our civic body would find a new insight
globally. Thus, a new horizon, which can make a bridge between mathematics and
biology,wouldenlightenourlearnedreaders.Italsoaffordsnecessarymethodologies
and techniques for students, junior scientists, and researchers, who are interested in
mathematical modeling under disease-related study. It is our moral responsibility to
thinkofcontrollingthediseasethroughpropermathematicalunderstandingsothatits
relevance is more favorable to our humanity.
Thebookiscomprisedofseveralmathematicalmodelsemphasizingthedisease
dynamicsofpsoriasisanditspossiblecontrol.Thisbookhelpsconsidertheprocess
of cell-biological behavioral pattern in case of disease progression and different
controlmechanisms.Readerscanfinddifferentapproachestocontrolthegrowthof
keratinocyte cell population, which is the causal effect of psoriasis, in this book.
Theeffectofinclusionofarealistictime delayintheformationofpsoriasis isalso
highlighted in the proper scope of the book. During the formation of an autoim-
munedisease,immunecellsmaybeenhancedeffectively.Theprocessisidentified
as T-Cell proliferation. Keeping this thought in mind, the concept of T-Cell
vii
viii Preface
proliferation in the psoriatic system is introduced under mathematical formulation.
The system is studied through implicit and explicit ways and measures the effect
of the drug on the system through impulsive drug therapy. Readers will gather
knowledgeontheeffectofcytokinereleaseintroducedinthedifferentmathematical
models of psoriasis. Readers will be informed of two different mathematical
pathways:oneisthroughanordinarydifferentialequation(ODE)modelsystemand
the other is through a fractional-order differential equation (FODE) model system.
Inthisbook,theinteractionbetweendendriticcellsandCD8+T-Cellsisconsidered
toknowtheroleofCD8+T-Cellsinpsoriasis.Thisbookalsoconsistsofanoptimal
control approach on the fractional-order system for controlling the excess pro-
ductionofkeratinocytecellpopulationinamoresignificantway.Thisbookcovers
mathematical modeling emphasizing psoriasis to the human immune system
including its responses to various available drug therapies. As mathematicians, we
thinkthatfortheeradicationofthedisease,theoutlineofclinicalandexperimental
observations under proper mathematical understanding and its application will be
morebeneficialtothesociety.Socialworkerswhoareworkinginthisfieldwillget
prior knowledge about the application of drugs to the patients with psoriasis. We
expect this book to be very much helpful and useful for the researchers.
Undergraduate and postgraduate students and biomathematicians, who are
studying and working in the field of mathematical modeling of an autoimmune
disease like psoriasis, will benefit from this book. Different social welfare organi-
zations,NGOs,andgovernmentemployees,whoareengagedinthedomainofthis
rigorous disease, will gather primary and basic knowledge and information for the
eradication of psoriasis. Graduate students in the fields of applied mathematics,
health informatics, and applied statistics will find this textbook useful. Scientists
andresearchersworkingontheexperimentalmodelinganditsindebtanalysisbased
on mathematical modeling will also benefit from the same.
West Bengal, India Prof. Priti Kumar Roy
May 2019 Dr. Abhirup Datta
Acknowledgements
Wewouldliketoconveymygratefulnesstothemanypeople,whohavevisualized
usthroughthisbook,providedsupport,talkedthingsover,read,offeredcomments,
allowed us to quote their remarks, and assisted in the editing, proofreading, and
design of the manuscript.
At first, we would like to articulate my gratitude to Shamim Ahmad, Senior
EditorofMathematicsandStatistics,SpringerIndia,forofferingthechancetowrite
a book on this subject and the entire production team of Springer for proofreading
and their valuable corrections made during the production process.
We grateful to Prof. Xue-Zhi Li, Department of Mathematics, Xinyang Normal
University, People’s Republic of China, and Prof. Cao Xianbing, College of
Science,BeijingTechnologyandBusinessUniversity,People’sRepublicofChina,
fromwhomwehaveenrichedourknowledgeregardingthebiologyofpsoriasisand
their application in mathematical biology. Indeed, we thankful to all my students:
Dr. Amar Nath Chatterjee, Dr. Jayanta Mondal, Dr. Nikhilesh Sil, Dr. Fahad Al
Basir,Dr.MithunKumarGhosh,Dr.DibyenduBiswas,Mr.ShubhankarSaha,Mr.
JahangirChowdhury,Mr.SudipChakraborty,Mr.AmitKumarRoy,Mr.Arunabha
Sengupta,Mr.SumanDolai,andMr.SalilGhosh.Also,wegratefultoallofthose
with whom we have had the pleasure of working in during this project and other
related projects.
We were greatly encouraged by the Council of Scientific and Industrial
Research, Government of India. This work would not have been possible without
their financial assistance (Reference Number: 38(1320)/12/EMR-II, dated April 3,
2012).
ix
x Acknowledgements
Finally, we would like to express my apologies to those whom we could not
mention individually. Without your advice, support, and encouragement, this
journeywouldneverhavecometoasuccessfulend.Itisapleasantresponsibilityto
express my sincerest thanks to all who have contributed in many ways to make us
reach my goal and this achievement.
May 2019 Prof. Priti Kumar Roy
Dr. Abhirup Datta