Table Of ContentStudies in Systems, Decision and Control 302
Khalid Hattaf
Hemen Dutta Editors
Mathematical
Modelling
and Analysis
of Infectious
Diseases
Studies in Systems, Decision and Control
Volume 302
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Janusz Kacprzyk, Systems Research Institute, Polish Academy of Sciences,
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Khalid Hattaf Hemen Dutta
(cid:129)
Editors
Mathematical Modelling
and Analysis of Infectious
Diseases
123
Editors
KhalidHattaf Hemen Dutta
CentreRégional des Métiersdel’Education Department ofMathematics
et delaFormation (CRMEF) Gauhati University
Casablanca,Morocco Guwahati, India
ISSN 2198-4182 ISSN 2198-4190 (electronic)
Studies in Systems,DecisionandControl
ISBN978-3-030-49895-5 ISBN978-3-030-49896-2 (eBook)
https://doi.org/10.1007/978-3-030-49896-2
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Preface
This book aims to include different topics related to mathematical modelling and
analysis of infectious diseases. The emergence and re-emergence of infectious
diseases are creating new health issues and causing socio-economic problem
worldwide. This book is expected to be a valuable resource for researchers, stu-
dents, educators, scientists, professionals and practitioners associated with diverse
aspectsofdiseasesandrelatedissues.Thegeneralreadersshouldalsofindthisbook
interesting to understand the dynamics of various diseases, their control strategies
andrelatedseveralotherissues.Thisbookconsistsoftwelvechapters,andtheyare
organized as follows.
Chapter “Pathogen evolution when transmission and virulence are
stochastic” presents an analytic approach for modelling pathogen evolution by
treating the vital parameters such as transmission and virulence as random vari-
ables. Starting with a general stochastic model of evolution, it derives specific
equationsfortheevolutionoftransmissionandvirulence,andthenappliestheseto
aparticularspecialcase,theSIRmodelofpathogendynamics.Itshowsthatadding
stochasticity introduces new directional components to pathogen evolution. In
particular,twokindsofcovariationbetweentraitsemergeasimportant:covariance
across the population and covariance between random variables within an indi-
vidual.Italsoshowsthatthesedifferentkindsoftraitcovariationcanbeofopposite
sign and contribute to evolution in very different ways. It further shows that
stochasticitycaninfluencepathogenevolutionthroughdirectionalstochasticeffects,
which results from the inevitable covariance between individual fitness and mean
population fitness.
Chapter “On the relationship between the basic reproduction number and
theshapeofthespatialdomain”studiesaspatiallydiffusiveSIRepidemicmodel
with constant parameters in a bounded spatial domain and investigates the rela-
tionship between the basic reproduction number R and the shape of the spatial
0
domain.UnderthehomogeneousNeumannboundaryconditions,R isthesameas
0
thatfortheclassicalnon-diffusiveSIRepidemicmodel,andthus,itdoesnotdepend
on the shape of the spatial domain. On the other hand, under the homogeneous
Dirichlet boundary conditions, the next generation operator does not have a
v
vi Preface
constant eigenvector, and R depends on the shape of the spatial domain. By
0
numerical simulation for the two-dimensional rectangular domain X = (0, p) x
(0,1/p),p>0withconstantarea|X|=1,itshowsthatsuchR attainsitsmaximum
0
for p = 1 and decreases as the shape of the domain becomes long and narrow.
Moreover,itobservesasimilarrelationshipbetweenR andtheshapeofthespatial
0
domain in a random two-dimensional lattice model.
Chapter “Cause and control strategy for infectious diseases with nonlinear
incidenceandtreatmentrate”dealswithcauseandcontrolstrategyforinfectious
diseases with nonlinear incidence and treatment rate. Control strategies regarding
infectious diseases can be developed with the help of mathematical modelling by
includingthecauseofthespreadofsuchdiseases.Differentdiseaseshavedifferent
spreadpatterns,andamajorreasonforthespreadofdiseasescanbefoundoutwith
thehelpofincidencerates.Also,treatmenttherapiesvarywiththeseverityandtype
ofdiseases.Thefactorsliketheavailabilityofvaccinesforaparticulardiseaseand
thenumberofinfectedpeoplearecrucialtoconsiderforaneffectivetreatmentrate.
So, nonlinear treatment rates can vary from disease to disease. Thus, it concludes
that the nonlinear incidence and treatment rate can play a vital role in suggesting
effective therapies to health agencies to control the spread of disease.
Chapter “Global stability of a delay virus dynamics model with mitotic
transmissionandcurerate”studiestheglobalpropertiesofabasicmodelforviral
infectionwithmitotictransmission,“cure”ofinfectedcells,saturationinfectionrate
andadiscreteintracellulardelay.Inconnectionwiththeproposedmodel,itderives
somethresholdparametersandestablishesasetofconditionswhicharesufficientto
determinetheglobaldynamicsofthemodels.ItusessuitableLyapunovfunctionals
and Lyapunov–LaSalle-type theorem for delay systems to prove the global
asymptoticstabilityofallequilibriaofthemodel.Italsoestablishestheoccurrence
of a Hopf bifurcation and determines conditions for the permanence of model and
the length of delay to preserve stability. Finally, it incorporates numerical simu-
lations to illustrate the analytical results.
Chapter“Dynamicsofafractional-orderhepatitisBepidemicmodelandits
solutions by nonstandard numerical schemes” aims to propose and analyse a
fractional-order hepatitis B epidemic model. It studies dynamical properties of the
proposed fractional-order model as well as its numerical solutions. It first estab-
lishes positivity and boundedness of the proposed model, and then asymptotic
stability of the model is investigated by the Lyapunov stability theorem for frac-
tional dynamical systems and numerical simulations. Finally, it constructs
positivity-preserving nonstandard finite difference (NSFD) schemes for the
fractional-order model.Numerical simulationshavebeenperformedtoconfirmthe
validity of the theoretical results and show advantages and superiority of NSFD
schemes over the standard one.
Chapter “On SICA models for HIV transmission” aims to revisit the
Susceptible-Infectious-Chronic-AIDS(SICA)mathematicalmodelfortransmission
dynamicsofthehumanimmunodeficiencyvirus(HIV)withvaryingpopulationsize
inahomogeneouslymixingpopulation.ItconsidersSICAmodelsgivenbysystems
of ordinary differential equations and some generalizations given by systems with
Preface vii
fractional and stochastic differential operators. Local and global stability results
have been proved for deterministic, fractional and stochastic-type SICA models.
Two case studies, in Cape Verde and Morocco, have been investigated.
Chapter“AnalyticalandnumericalsolutionsofaTB-HIV/AIDSco-infection
model via fractional derivatives without singular kernel” aims to analyse a
TB-HIV/AIDS co-infection model. The model has been extended to the Caputo–
Fabrizio fractional derivative obtained using the exponential function. Then, it
investigates for the uniqueness solutions with the help of a fixed-point theorem.
Thereafter, the uniqueness solution of the model has been obtained by assuming
certain parameters and its stability analyses have also been carried out. Finally,
numerical solutions of the mathematical model have been obtained and also per-
formed numerical simulations.
Chapter“Developingamultiparametricriskindexfordenguetransmission”
aims todiscussthe frameworkof developing a multiparametric index todetermine
the transmission risk of dengue in urban zones in Sri Lanka and predict it by
consideringthevariation ofappropriatefactors.Ituses literaturereview toidentify
risk factors for dengue transmission and risk levels. Fuzzy analytic hierarchy
process (AHP) has been used to weight the risk factors. The constructed Haddon
matrices have been used to identify the risk strata of dengue transmission. The
obtained results have been compared with the other records to check the validity
of the model. Sensitivity analysis has been carried out to identify impacts and
variation about the contribution of the risk factors towards dengue transmission.
Chapter “The effect of delay and diffusion on the dynamics of wild Aedes
aegypti mosquitoes” presents a study on the effect of time delay and diffusion on
the dynamics of Aedes aegypti mosquitoes’ invasion with quiescent female phase.
The model proposed in this chapter is given by three delay differential equations
and its corresponding reaction–diffusion equations, which describe the interaction
betweenthreesub-populations,viz.eggs,pupaeandfemale.Itfocusesonstudying
the effect of quiescent female phase represented by time delay. The existence of
periodic oscillations around the persistent positive equilibrium when time delay
crossessome critical value, nooccurrence ofTuring bifurcationand thesensitivity
analysisofparametershavebeenestablished.Thestabilityofthebifurcatingbranch
ofperiodicoscillations hasbeen shownby using normalformandcentremanifold
theory. Finally, numerical simulations have been carried out to support the theo-
retical results.
Chapter“ModelingthedynamicsofhepatitisBvirusinfectioninpresenceof
capsids and immunity” proposes three generalized systems of differential equa-
tionstomodelthedynamicsofhepatitisBvirus(HBV)infectioninthepresenceof
HBVDNA-containingcapsidsandimmunitymediated bycytotoxic Tlymphocyte
(CTL)cells.Theglobalpropertiesofthreeproposedmodelshavebeeninvestigated.
Moreover,manypreviousstudiesexistingintheliteraturehavealsobeenclaimedto
improve and generalize.
Chapter“AclassofEbolavirusdiseasemodelswithpost-deathtransmission
and environmental contamination” proposes two mathematical Ebola virus dis-
ease (EVD) models that incorporate the three modes of transmission. The modes
viii Preface
have been modelled by three general incidence functions that cover many types of
incidence rates existing in the literature. The first model is formulated by ordinary
differential equations, and the second one is governed by partial differential equa-
tions in order to describe the evolution of EVD in time and space. The qualitative
analysisofboth modelshas beeninvestigatedindetail.Further,an applicationhas
been given and numerical simulations have also been performed to support the
analytical results.
Chapter “A survey on sufficient optimality conditions for delayed optimal
control problems” presents a survey on recent sufficient optimality conditions for
optimal control problems with time delays in both state and control variables. The
results have been obtained by transforming delayed optimal control problems into
equivalent non-delayed problems. It isclaimed thatsuchanapproachallowsusing
of standard theorems that ensure sufficient optimality conditions for non-delayed
optimalcontrolproblems.Exampleshavebeenincorporatedtoillustratetheresults.
We sincerely acknowledge the cooperation and patience of contributors during
the entire process of editing this book. Reviewers deserve the most sincere thanks
for their valuable contribution in a timely manner. We are thankful to numerous
colleaguesand friendsfor their continuous encouragementsto developsuch books
for the benefit of several kinds of readers. We also thankfully acknowledge the
cooperation and support of editorial staff at Springer.
Morocco Khalid Hattaf
India Hemen Dutta
April 2020
Contents
Pathogen Evolution When Transmission and Virulence
are Stochastic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Pooya Aavani and Sean H. Rice
On the Relationship Between the Basic Reproduction Number
and the Shape of the Spatial Domain. . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Toshikazu Kuniya
Cause and Control Strategy for Infectious Diseases with Nonlinear
Incidence and Treatment Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Nilam
Global Stability of a Delay Virus Dynamics Model with Mitotic
Transmission and Cure Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Eric Avila-Vales, Abraham Canul-Pech, Gerardo E. García-Almeida,
and Ángel G. C. Pérez
Dynamics of a Fractional-Order Hepatitis B Epidemic Model
and Its Solutions by Nonstandard Numerical Schemes. . . . . . . . . . . . . . 127
Manh Tuan Hoang and Oluwaseun Francis Egbelowo
On SICA Models for HIV Transmission . . . . . . . . . . . . . . . . . . . . . . . . 155
Cristiana J. Silva and Delfim F. M. Torres
Analytical and Numerical Solutions of a TB-HIV/AIDS Co-infection
Model via Fractional Derivatives Without Singular Kernel . . . . . . . . . . 181
Mustafa Ali Dokuyucu and Hemen Dutta
Developing a Multiparametric Risk Index for Dengue Transmission . . . 213
I. T. S. Piyatilake and S. S. N. Perera
The Effect of Delay and Diffusion on the Dynamics of Wild Aedes
Aegypti Mosquitoes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
R. Yafia and M. A. Aziz Alaoui
ix
