1 
Backward bifurcation in HCV transmission dynamicsNazari, Fereshteh 19 August 2014 (has links)
The thesis is based on the use of mathematical theories and techniques to gain qualitative and quantitative insight into the transmission dynamics of hepatitis C virus (HCV) in an IDU (injecting drug user) population. A deterministic model, which stratifies the IDU population into eight mutuallyexclusive compartments (based on epidemiological status), is considered. Rigorous qualitative analysis of the model establishes, for the first time, the presence of the phenomenon of backward bifurcation in HCV transmission dynamics. Three routes (or causes) to such a dynamic phenomenon have been established. Furthermore, five main parameters that play a dominant role on the transmission dynamics of the disease have been identified. Numerical simulations of the model show that the reinfection of recovered individuals has marginal effect on the HCV burden (as measured in terms of the cumulative incidence and prevalence of the disease) in the IDU community.

2 
Withinhost dynamics of HIV/AIDSXie, Xinqi 03 May 2021 (has links)
This thesis first investigates withinhost HIV models for the acute stage. These models incorporate the immune responses and helper T cells produced from the activation of naive CD4 T cells. Because both naive CD4 T cells and helper T cells are susceptible classes, backward bifurcation and bistability may occur. We start with a simple model that ignores the CD8 T cell dynamics, then extend it to include this dynamics. We also extend our model to consider the latent infection of naive CD4 T cells. Backward bifurcation occurs in all these models. We numerically investigate the stability of viral equilibria, and show the bistability caused by backward bifurcation. Increasing the inflow of CTLs prevents the backward bifurcation. With a large homeostatic source of healthy naive CD4 T cells, the disease is easier to establish when the basic reproduction number is less than one. Reducing the reproduction number below one is not sufficient to control the infection of HIV. Secondly, this thesis investigates the development of AIDS caused by viral diversity, as proposed by Wodarz et al. using a model that does not include the details of immune responses. We extend their model to include density dependence, and show that the viral load increases with viral diversity. To study if this result still holds with more realistic HIV dynamics, we incorporate viral diversity into our first model. We conclude theoretically that the total viral load is positively correlated with the number of viral strains, and viral diversity can drive the development of AIDS. We also find that the total CD4 T cell count does not always decrease with viral diversity. Thus further investigation is needed to fully understand the development of AIDS. / Graduate

3 
Mathematical modeling of the population dynamics of tuberculosisAdebiyi, Ayodeji O. January 2016 (has links)
>Magister Scientiae  MSc / Tuberculosis (TB) is currently one of the major public health challenges in South Africa, and in many countries. Mycobacterium tuberculosis is among the leading causes of morbidity and mortality. It is known that tuberculosis is a curable infectious disease. In the case of incomplete treatment, however, the remains of Mycobacterium tuberculosis in the human system often results in the bacterium developing resistance to antibiotics. This leads to relapse and treatment against the resistant bacterium is extremely expensive and difficult. The aim of this work is to present and analyse mathematical models of the population dynamics of tuberculosis for the purpose of studying the effects of efficient treatment versus incomplete treatment. We analyse the spread, asymptotic behavior and possible eradication of the disease, versus persistence of tuberculosis. In particular, we
consider inflow of infectives into the population, and we study the effects of screening. A submodel will be studied to analyse the transmission dynamics of TB in an isolated population. The full model will take care of the inflow of susceptibles as well as inflow of TB infectives into the population. This dissertation enriches the existing literature with contributions in the form of optimal control and stochastic perturbation. We also show how stochastic perturbation can improve the stability of an equilibrium point. Our methods include Lyapunov functions, optimal control and stochastic differential equations. In the stability analysis of the DFE we show how backward bifurcation appears. Various phenomena are illustrated by way of simulations.

4 
Dynamics of an HIV/AIDS Model that Incorporates Preexposure ProphylaxisSimpson, Lindsay 26 August 2015 (has links)
This thesis is based on the use of mathematical theories, modelling, and simulations to study the transmission dynamics of HIV/AIDS in the presence of PrEP (preexposure prophylaxis) in the MSM (men who have sex with men) population in the United States. A new deterministic model for HIV/AIDS that incorporates PrEP is designed and used to assess the populationlevel impact of the use of PrEP on the transmission dynamics within an MSM population. Conditions for the effective control (or elimination) and persistence of HIV/AIDS in the MSM population are determined by rigorously analyzing this model. Uncertainty and sensitivity analysis is carried out to determine the effect of the uncertainties in the parameter values on the response variable (the associated reproduction number) and to identify the topfive parameters that have the most effect on the disease transmission dynamics. Numerical simulations show that HIV burden decreases with increasing PrEP coverage. / October 2015

5 
Dynamics of Multistrain Agestructured Model for Malaria TransmissionFarinaz, Forouzannia 22 August 2013 (has links)
The thesis is based on the use of mathematical modeling and analysis to gain insightinto the transmission dynamics of malaria in a community. A new deterministic
model for assessing the role of agestructure on the disease dynamics is designed.
The model undergoes backward bifurcation, a dynamic phenomenon characterized
by the coexistence of a stable diseasefree and an endemic equilibrium of the model
when the associated reproduction number is less than unity. It is shown that adding
agestructure to the basic model for malaria transmission does not alter its essential
qualitative dynamics. The study is extended to incorporate the use of antimalaria
drugs. Numerical simulations of the extended model suggest that for the case when
treatment does not cause drug resistance (and the reproduction number of each of the
two strains exceed unity), the model undergoes competitive exclusion. The impact
of various effectiveness levels of the treatment strategy is assessed.

6 
Dynamics of Multistrain Agestructured Model for Malaria TransmissionForouzannia, Farinaz 22 August 2013 (has links)
The thesis is based on the use of mathematical modeling and analysis to gain insightinto the transmission dynamics of malaria in a community. A new deterministic
model for assessing the role of agestructure on the disease dynamics is designed.
The model undergoes backward bifurcation, a dynamic phenomenon characterized
by the coexistence of a stable diseasefree and an endemic equilibrium of the model
when the associated reproduction number is less than unity. It is shown that adding
agestructure to the basic model for malaria transmission does not alter its essential
qualitative dynamics. The study is extended to incorporate the use of antimalaria
drugs. Numerical simulations of the extended model suggest that for the case when
treatment does not cause drug resistance (and the reproduction number of each of the
two strains exceed unity), the model undergoes competitive exclusion. The impact
of various effectiveness levels of the treatment strategy is assessed.

7 
Analysis of TumorImmune Dynamics in an Evolving Dendritic Cell Therapy ModelJanuary 2020 (has links)
abstract: Cancer is a worldwide burden in every aspect: physically, emotionally, and financially. A need for innovation in cancer research has led to a vast interdisciplinary effort to search for the next breakthrough. Mathematical modeling allows for a unique look into the underlying cellular dynamics and allows for testing treatment strategies without the need for clinical trials. This dissertation explores several iterations of a dendritic cell (DC) therapy model and correspondingly investigates what each iteration teaches about response to treatment.
In Chapter 2, motivated by the work of de Pillis et al. (2013), a mathematical model employing six ordinary differential (ODEs) and delay differential equations (DDEs) is formulated to understand the effectiveness of DC vaccines, accounting for cell trafficking with a blood and tumor compartment. A preliminary analysis is performed, with numerical simulations used to show the existence of oscillatory behavior. The model is then reduced to a system of four ODEs. Both models are validated using experimental data from melanomainduced mice. Conditions under which the model admits rich dynamics observed in a clinical setting, such as periodic solutions and bistability, are established. Mathematical analysis proves the existence of a backward bifurcation and establishes thresholds for R0 that ensure tumor elimination or existence. A sensitivity analysis determines which parameters most significantly impact the reproduction number R0. Identifiability analysis reveals parameters of interest for estimation. Results are framed in terms of treatment implications, including effective combination and monotherapy strategies.
In Chapter 3, a study of whether the observed complexity can be represented with a simplified model is conducted. The DC model of Chapter 2 is reduced to a nondimensional system of two DDEs. Mathematical and numerical analysis explore the impact of immune response time on the stability and eradication of the tumor, including an analytical proof of conditions necessary for the existence of a Hopf bifurcation. In a limiting case, conditions for global stability of the tumorfree equilibrium are outlined.
Lastly, Chapter 4 discusses future directions to explore. There still remain open questions to investigate and much work to be done, particularly involving uncertainty analysis. An outline of these steps is provided for future undertakings. / Dissertation/Thesis / Doctoral Dissertation Applied Mathematics 2020

8 
Mathematical modelling and analysis of HIV/AIDS and trichomonas vaginalis coinfectionMumba, Chibale K. January 2017 (has links)
Deterministic models for the transmission dynamics of HIV/AIDS and trichomonas vaginalis
(TV) in a human population are formulated and analysed. The models which
assumed standard incidence formulations are shown to have globally asymptotically stable
(GAS) diseasefree equilibria whenever their associated reproduction number is less
than unity. Furthermore, both models possess a unique endemic equilibrium that is GAS
whenever the associated reproduction number is greater than unity. An extended model
for the coinfection of TV and HIV in a human population is also designed and rigorously
analysed. The model is shown to exhibit the phenomenon of backward bifurcation,
where a stable diseasefree equilibrium (DFE) coexists with a stable endemic equilibrium
whenever the associated reproduction number is less than unity. This phenomenon can be
removed by assuming that the coinfection of individuals with HIV and TV is negligible.
Furthermore, in the absence of coinfection, the DFE of the model is shown to be GAS
whenever the associated reproduction number is less than unity. This study identifies
a sufficient condition for the emergence of backward bifurcation in the model, namely
TVHIV coinfection. The endemic equilibrium point is shown to be GAS (for a special
case) when the associated reproduction number is greater than unity. Numerical simulations
of the model, using initial and demographic data, show that increased incidence of
TV in a population increases HIV incidence in the population. It is further shown that
control strategies, such as treatment, condomuse and counselling of individuals with TV
symptoms, can lead to the effective control or elimination of HIV in the population if
their effectiveness level is high enough. / Dissertation (MSc)University of Pretoria, 2017. / DSTNRF SARChI Chair in
Mathematical Models and Methods in Biosciences and Bioengineering (M3B2) / Mathematics and Applied Mathematics / MSc / Unrestricted

9 
Mathematical modelling and analysis of HIV transmission dynamicsHussaini, Nafiu January 2010 (has links)
This thesis firstly presents a nonlinear extended deterministic SusceptibleInfected (SI) model for assessing the impact of public health education campaign on curtailing the spread of the HIV pandemic in a population. Rigorous qualitative analysis of the model reveals that, in contrast to the model without education, the full model with education exhibits the phenomenon of backward bifurcation (BB), where a stable diseasefree equilibrium coexists with a stable endemic equilibrium when a certain threshold quantity, known as the effective reproduction number (Reff ), is less than unity. Furthermore, an explicit threshold value is derived above which such an education campaign could lead to detrimental outcome (increase disease burden), and below which it would have positive populationlevel impact (reduce disease burden in the community). It is shown that the BB phenomenon is caused by imperfect efficacy of the public health education program. The model is used to assess the potential impact of some targeted public health education campaigns using data from numerous countries. The second problem considered is a SusceptibleInfectedRemoved (SIR) model with two types of nonlinear treatment rates: (i) piecewise linear treatment rate with saturation effect, (ii) piecewise constant treatment rate with a jump (Heaviside function). For Case (i), we construct travelling front solutions whose profiles are heteroclinic orbits which connect either the diseasefree state to an infected state or two endemic states with each other. For Case (ii), it is shown that the profile has the following properties: the number of susceptible individuals is monotone increasing and the number of infectives approaches zero, while their product converges to a constant. Numerical simulations are shown which confirm these analytical results. Abnormal behavior like travelling waves with nonmonotone profile or oscillations are observed.

10 
Mathematics of HSV2 DynamicsPodder, Chandra Nath 26 August 2010 (has links)
The thesis is based on using dynamical systems theories and techniques to study the qualitative dynamics of herpes simplex virus type 2 (HSV2), a sexuallytransmitted disease of major public health significance.
A deterministic model for the interaction of the virus with the immune system in the body of an infected individual (in vivo) is designed first
of all. It is shown, using Lyapunov function and LaSalle's Invariance Principle, that
the virusfree equilibrium of the model is globallyasymptotically stable whenever a
certain biological threshold, known as the reproduction number, is less than unity. Furthermore, the model has at least one viruspresent equilibrium when the threshold quantity exceeds unity. Using persistence theory, it is shown that the virus will always be present in vivo whenever the reproduction threshold exceeds unity. The analyses (theoretical and numerical) of this model show that a future HSV2 vaccine that enhances cellmediated immune response will be effective in
curtailling HSV2 burden in vivo.
A new singlegroup model for the spread of HSV2 in
a homogenouslymixed sexuallyactive population is also designed. The diseasefree equilibrium of the model is globallyasymptotically stable when its associated reproduction number is less
than unity. The model has a unique endemic equilibrium, which is shown to be
globallystable for a special case, when the reproduction number exceeds unity.
The model is extended to incorporate an imperfect vaccine with some therapeutic benefits.
Using centre manifold theory, it is shown that the resulting vaccination model undergoes a vaccineinduced backward bifurcation (the epidemiological
importance of the phenomenon of backward bifurcation is that the
classical requirement of having the reproduction threshold less than unity is, although necessary, no longer sufficient for disease elimination. In such a case, disease elimination depends upon the initial sizes of the
subpopulations of the model). Furthermore, it is shown that the use of such an
imperfect vaccine could lead to a positive or detrimental populationlevel impact (depending on the sign of a certain threshold quantity).
The model is extended to incorporate the effect of variability in HSV2 susceptibility due to gender differences. The resulting twogroup (sexstructured) model is shown to have essentially the
same qualitative dynamics as the singlegroup model. Furthermore, it is shown that adding periodicity to the corresponding autonomous twogroup model does not alter the dynamics of the autonomous twogroup model (with respect to the elimination of the disease). The model is used to evaluate the impact of various antiHSV control strategies.
Finally, the twogroup model is further extended to address the effect of risk structure (i.e., risk of acquiring or transmitting HSV2). Unlike the twogroup model described above, it
is shown that the riskstructured model undergoes backward
bifurcation under certain conditions (the backward bifurcation property can be removed if the susceptible population is not stratified according to the risk of acquiring infection). Thus, one of the main findings of this thesis is that risk structure can induce the phenomenon of backward bifurcation in the transmission dynamics of HSV2 in a population.

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