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SUMMARY
The spread of diseases has always been a threat to public health. It has not only
caused serious problems for the survival of human beings and other species but also
for the economic and social development of the human society. To prevent and to
control these diseases, it is important to understand the mechanism of the spread and
the transmission dynamics of the diseases and establish better strategies for useful
predictions and guidance. Mathematical Epidemiology is an important approach to
investigate the transmission dynamics of infectious diseases. It includes formulation
of mathematical models to describe the mechanisms of disease transmission and
dynamics of agents that cause the diseases. The mathematical models on diseases are
based on population dynamics, behavior of disease transmissions, features of the
infectious agents, and the connections with other social and physiologic factors.
Mathematical models can give us good understanding of how infectious diseases
spread and help us to identify more important and sensitive parameters, to make
reliable predictions and provide useful prevention and control strategies and guidance
through quantitative and qualitative analysis, sensitivity analysis, and numeric
simulations.
Keeping all the above points in view, this thesis is divided into seven chapters
that deal with a mathematical model of infectious diseases like H 1N 1 flu, a
mathematical model of malaria and some mathematical models on cancer growth and
its treatment. These mathematical models are analysed using theory of nonlinear
ordinary differential equations and computer simulations. Mathematical analyses of
the model equations include boundedness of solutions, nature of equilibrium points
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and their local and global study. To substantiate the analytical findings, the models
are studied numerically using fourth order Runge-Kutta method and results are
displayed graphically. A brief introduction of the chapters of the thesis is as follows:
CHAPTER 1: INTRODUCTION
This chapter introduces the basic concepts and ideas of mathematical modeling
and mathematical epidemiology with some historical background in this area. In
addition, this chapter also provides brief descriptions of terms, like region of
attraction, equilibrium point, stability, Routh-Hurwitz stability criterion and Hopf
bifurcation that are used in subsequent chapters.
CHAPTER 2: MODELING H1N1 FLU WITH CONTACT TRACING AND
ISOLATION
This chapter deals with a nonlinear mathematical model that is proposed and
analyzed to study the dynamics of 2009 H1N1 flu in a homogeneous population with
constant immigration of susceptibles. The effect of contact tracing and isolation
strategies in reducing the spread of H1N1 flu is considered. The model monitors the
dynamics of five sub-populations namely susceptible with high infection risk,
susceptible with reduced risk of infection, infective, quarantined and recovered
individuals. The model analyses include the determination of equilibrium points and
their stability in terms of the threshold parameter R0 . The analytical and numerical
simulation results demonstrate that the maximum implementation of contact tracing
and isolation strategies help in reducing infective class size and hence act as effective
intervention strategies to control the disease from spreading.
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CHAPTER 3: A MATHEMATICAL MODEL FOR THE DYNAMICS OF
MALARIA WITH IMMIGERATION OF INDIVIDUALS IN DIFFERENT
COMPARTMENTS
This chapter considers a host-vector mathematical model for the spread of
malaria that incorporates recruitment of human population through a constant
immigration, with a fraction of infective and exposed immigrants. It is found that in
the presence of infective and exposed immigrants; a unique endemic equilibrium
point exists for which the disease persists in the population but disease free equilibria
cannot exist. However, when immigration of infective and exposed population is not
allowed, disease free equilibrium also exists which is stable if basic reproduction ratio
is less than one. Local stability conditions for the equilibria of the system with and
without immigration of infectives and exposed population are determined by RouthHurwitz criterion. It is found that due to immigration of infective and exposed
population in the community infective human population level rises. Our study shows
that infective human population can be reduced by preventing mosquitoes to bite
infective humans and by reducing the probability of transmission of infection from
human to mosquitoes.
CHAPTER 4: A GENERALISED PREY - PREDATOR MODEL OF CANCER
GROWTH WITH THE EFFECT OF IMMUNOTHERAPY
This chapter deals with the analysis of the system of nonlinear ordinary
differential equations describing the interaction between cancer and immune cells
during immunotherapy. It is observed that cancer cell population decreases
considerably due to proliferation of lymphocytes mediated by immunotherapy. It is
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further found from our analysis that cancer population can be controlled easily if
cancer is immunogenic that is, cancer cells possess distinctive surface markers called
tumor-specific antigens.
CHAPTER 5: A MATHEMATICAL MODEL OF CANCER GROWTH WITH THE
EFFECT OF DELAY IN CELLULAR INTERACTION
In this chapter, we propose and analyze a nonlinear mathematical model to
study the interaction between tumor and immune cells. The model is extended by the
inclusion of an intracellular delay effect. Stability of the steady states of model
systems without and with delay is determined. Conditions for Hopf bifurcation of the
system are determined. Moreover, critical value of delay is also found which acts as a
bifurcation parameter. We have found that the key role is being played by the rate of
conversion of resting cells to hunting cells and antigenicity of tumor cells in our
model. It is also observed that the tumor cell population is large for low rate of
conversion of resting cells to hunting cells or for small concentration of cytokines.
The effect of tumor antigenicity is studied and it is observed from our analysis that
lower is the antigenicity nastier is the tumor. For certain parameter values limit cycles
are obtained.
CHAPTER 6: MATHEMATICAL MODELING AND ANALYSIS OF CANCER
VIROTHERAPY
In chapter sixth, we study a nonlinear mathematical model to study the effect
oncolytic virotherapy. We analyse complex interaction among growing cancer cells
and replicating virus population. It is observed from our study that replication of
oncolytic virus within cancer cells and their transmission to other cancer cells is
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responsible for the infection among more cancer cells and hence increase in the
number of infected cancer cells. In this process number of uninfected cancer cells
decrease automatically. Further, it is found that as cytotoxicity rate of oncolytic virus
increase infected cancer cells decrease on the other hand uninfected cancer cells
increase. Equilibrium level of infected cancer cells decrease because oncolytic virus
kill cancer cells by infecting them. However, it is not able to annihilate the
proliferation of cancer cells. Therefore, number of uninfected cancer cells keeps on
proliferating in the human body. In addition, the model is extended by including the
effect of antiviral immune response on the model. It is found that active proliferation
of immune cells is helpful in eliminating cancer cells from the body.
CHAPTER 7: CORRELATION BETWEEN HEPATITIS AND CANCER: A
MATHEMATICAL MODEL
In this chapter, a non – linear mathematical model is proposed and analyzed to
demonstrate the relation between hepatitis and cancer with constant immigration of
cancer patients in the community. Both the horizontal and vertical mode of
transmission of hepatitis in the population is considered. Sensitivity analysis of the
endemic equilibrium to changes in the value of the different parameters associated
with the system is done. Modeling the effect of hepatitis virus infections among
cancer patients and their impact on the increase in spread of hepatitis among the
population is the novel feature of our model. It is concluded from the analysis that if
the rate of transmission of hepatitis infection increases, the endemic level of infective
population increases which can further be enhanced if risk of hepatitis infection
among cancer patients also increases. Further, it is found from our analysis that
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hepatitis infection leads to an increase in number of cancer patients in the population
because of the progression of hepatitis B and C infection to liver cancer.
RESEARCH PAPERS PUBLISHED/ACCEPTED/COMMUNICATED
1. Modeling H1N1 Flu with Contact Tracing and Quarantine, International Journal
of Biomathematics, World Scientific Publishing Company, Vol. 5, No. 5,
(September 2012), 1250038 (19 pages), ISSN (printed): 1793-5245.
2. A mathematical model for the dynamics of Malaria with immigration of different
Compartments published in International Journal of Logic Based Intelligent
Systems, © International Science Press, ISSN: 0975-4776, Vol. 4, No. 2, (JulyDecember 2010), pp.91-105.
3. A Generalised Prey - Predator Model of Cancer Growth with the Effect of
Immunotherapy, accepted for publication in International Journal of Engineering,
Science and Technology, ISSN No
4. Mathematical Modeling of Interaction between Tumor and Immune Cells
(communicated).
5. Mathematical Modeling and Analysis of Tumor therapy with Oncolytic Virus,
published in Applied Mathematics, Scientific research, ISSN Print: 2152-7385,
ISSN Online: 2152-7393, Vol.2, No.1, (January 2011),pp.131-140.
6. Mathematical Modeling and Analysis of Tumor Therapy with Oncolytic Virus and
Immune Cells (communicated).
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7.
Correlation between Hepatitis and Cancer: A Mathematical Model published in
International Journal of Mathematics and Scientific Computing, ISSN No. 22315330, Vol. 1, No. 2, (2011), pp.79-86.
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