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Dynamics of Multi-strain Age-structured 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 age-structure on the disease dynamics is designed.
The model undergoes backward bifurcation, a dynamic phenomenon characterized
by the co-existence of a stable disease-free and an endemic equilibrium of the model
when the associated reproduction number is less than unity. It is shown that adding
age-structure to the basic model for malaria transmission does not alter its essential
qualitative dynamics. The study is extended to incorporate the use of anti-malaria
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.
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Dynamics of Multi-strain Age-structured 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 age-structure on the disease dynamics is designed.
The model undergoes backward bifurcation, a dynamic phenomenon characterized
by the co-existence of a stable disease-free and an endemic equilibrium of the model
when the associated reproduction number is less than unity. It is shown that adding
age-structure to the basic model for malaria transmission does not alter its essential
qualitative dynamics. The study is extended to incorporate the use of anti-malaria
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.
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Mathematical modelling of the HIV/AIDS epidemic and the effect of public health educationVyambwera, Sibaliwe Maku January 2014 (has links)
>Magister Scientiae - MSc / HIV/AIDS is nowadays considered as the greatest public health disaster of modern time.
Its progression has challenged the global population for decades. Through mathematical
modelling, researchers have studied different interventions on the HIV pandemic, such as treatment, education, condom use, etc. Our research focuses on different compartmental models with emphasis on the effect of public health education. From the point of view of statistics, it is well known how the public health educational programs contribute towards the reduction of the spread of HIV/AIDS epidemic. Many models have been studied towards understanding the dynamics of the HIV/AIDS epidemic. The impact of ARV treatment have been observed and analysed by many researchers. Our research studies and investigates a compartmental model of HIV with treatment and education campaign. We study the existence of equilibrium points and their stability. Original contributions of this dissertation are the modifications on the model of Cai et al. [1], which enables us to use optimal control theory to identify optimal roll-out of strategies to control the HIV/AIDS. Furthermore, we introduce randomness into the model and we study the almost sure exponential stability of the disease free equilibrium. The randomness is regarded as environmental perturbations in the system. Another contribution is the global stability analysis on the model of Nyabadza et al. in [3]. The stability thresholds are compared for the HIV/AIDS in the absence of any intervention to assess the possible community benefit of public health educational campaigns. We illustrate the results by way simulation The following papers form the basis of much of the content of this dissertation, [1 ] L. Cai, Xuezhi Li, Mini Ghosh, Boazhu Guo. Stability analysis of an HIV/AIDS epidemic model with treatment, 229 (2009) 313-323. [2 ] C.P. Bhunu, S. Mushayabasa, H. Kojouharov, J.M. Tchuenche. Mathematical Analysis of an HIV/AIDS Model: Impact of Educational Programs and Abstinence in Sub-Saharan Africa. J Math Model Algor 10 (2011),31-55. [3 ] F. Nyabadza, C. Chiyaka, Z. Mukandavire, S.D. Hove-Musekwa. Analysis of an HIV/AIDS model with public-health information campaigns and individual with-drawal. Journal of Biological Systems, 18, 2 (2010) 357-375. Through this dissertation the author has contributed to two manuscripts [4] and [5], which are currently under review towards publication in journals, [4 ] G. Abiodun, S. Maku Vyambwera, N. Marcus, K. Okosun, P. Witbooi. Control and sensitivity of an HIV model with public health education (under submission). [5 ] P.Witbooi, M. Nsuami, S. Maku Vyambwera. Stability of a stochastic model of HIV population dynamics (under submission).
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Estudo qualitativo de um modelo de propagação de dengue / Qualitative study of a dengue disease transmission modelSantos, Bruna Cassol dos 25 July 2016 (has links)
Em epidemiologia matemática, muitos modelos de propagação de doenças infecciosas em populações têm sido analisados matematicamente e aplicados para doenças específicas. Neste trabalho um modelo de propagação de dengue é analisado considerando-se diferentes hipóteses sobre o tamanho da população humana. Mais precisamente, estamos interessados em verificar o impacto das variações populacionais a longo prazo no cálculo do parâmetro Ro e no equilíbrio endêmico. Vamos discutir algumas ideias que nortearam o processo de definição do parâmetro Ro a partir da construção do Operador de Próxima Geração. Através de um estudo qualitativo do modelo matemático, obtivemos que o equilíbrio livre de doença é globalmente assintoticamente estável se Ro é menor ou igual a 1 e instável se Ro>1. Para Ro>1, a estabilidade global do equilíbrio endêmico é provada usando um critério geral para estabilidade orbital de órbitas periódicas associadas a sistemas autônomos não lineares de altas ordens e resultados da teoria de sistemas competitivos para equações diferenciais ordinárias. Também foi desenvolvida uma análise de sensibilidade do Ro e do equilíbrio endêmico com relação aos parâmetros do modelo de propagação. Diversos cenários foram simulados a partir dos índices de sensibilidade obtidos nesta análise. Os resultados demonstram que, de forma geral, o parâmetro Ro e o equilíbrio endêmico apresentam considerável sensibilidade a taxa de picadas do vetor e a taxa de mortalidade do vetor. / In mathematical epidemiology many models of spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. In this work a dengue propagation model is analyzed considering different assumptions about the size of the human population. More precisely, we are interested to verify the impact of population long-term variations in the calculation of the parameter Ro and endemic equilibrium. We will discuss some ideas that guided the parameter setting process Ro from the construction of the Next Generation Operator. Through a qualitative study of the mathematical model, we found that the disease-free equilibrium is globally asymptotically stable if Ro is less or equal than 1 and unstable if Ro> 1. For Ro> 1 the global stability of the endemic equilibrium is proved using a general criterion for orbital stability of periodic orbits associated with nonlinear autonomous systems of higher orders and results of the theory of competitive systems for ordinary differential equations. Also a sensitivity analysis of the Ro and the endemic equilibrium with respect to the parameters of the propagation model was developed. Several scenarios were simulated from the sensitivity index obtained in this analysis. The results demonstrate that in general the parameter Ro and the endemic equilibrium are the most sensitive to the vector biting rate and the vector mortality rate.
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Mathematics of HSV-2 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 (HSV-2), a sexually-transmitted 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 virus-free equilibrium of the model is globally-asymptotically stable whenever a
certain biological threshold, known as the reproduction number, is less than unity. Furthermore, the model has at least one virus-present 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 HSV-2 vaccine that enhances cell-mediated immune response will be effective in
curtailling HSV-2 burden in vivo.
A new single-group model for the spread of HSV-2 in
a homogenously-mixed sexually-active population is also designed. The disease-free equilibrium of the model is globally-asymptotically stable when its associated reproduction number is less
than unity. The model has a unique endemic equilibrium, which is shown to be
globally-stable 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 vaccine-induced 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
sub-populations of the model). Furthermore, it is shown that the use of such an
imperfect vaccine could lead to a positive or detrimental population-level impact (depending on the sign of a certain threshold quantity).
The model is extended to incorporate the effect of variability in HSV-2 susceptibility due to gender differences. The resulting two-group (sex-structured) model is shown to have essentially the
same qualitative dynamics as the single-group model. Furthermore, it is shown that adding periodicity to the corresponding autonomous two-group model does not alter the dynamics of the autonomous two-group model (with respect to the elimination of the disease). The model is used to evaluate the impact of various anti-HSV control strategies.
Finally, the two-group model is further extended to address the effect of risk structure (i.e., risk of acquiring or transmitting HSV-2). Unlike the two-group model described above, it
is shown that the risk-structured 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 HSV-2 in a population.
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Mathematics of HSV-2 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 (HSV-2), a sexually-transmitted 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 virus-free equilibrium of the model is globally-asymptotically stable whenever a
certain biological threshold, known as the reproduction number, is less than unity. Furthermore, the model has at least one virus-present 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 HSV-2 vaccine that enhances cell-mediated immune response will be effective in
curtailling HSV-2 burden in vivo.
A new single-group model for the spread of HSV-2 in
a homogenously-mixed sexually-active population is also designed. The disease-free equilibrium of the model is globally-asymptotically stable when its associated reproduction number is less
than unity. The model has a unique endemic equilibrium, which is shown to be
globally-stable 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 vaccine-induced 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
sub-populations of the model). Furthermore, it is shown that the use of such an
imperfect vaccine could lead to a positive or detrimental population-level impact (depending on the sign of a certain threshold quantity).
The model is extended to incorporate the effect of variability in HSV-2 susceptibility due to gender differences. The resulting two-group (sex-structured) model is shown to have essentially the
same qualitative dynamics as the single-group model. Furthermore, it is shown that adding periodicity to the corresponding autonomous two-group model does not alter the dynamics of the autonomous two-group model (with respect to the elimination of the disease). The model is used to evaluate the impact of various anti-HSV control strategies.
Finally, the two-group model is further extended to address the effect of risk structure (i.e., risk of acquiring or transmitting HSV-2). Unlike the two-group model described above, it
is shown that the risk-structured 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 HSV-2 in a population.
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Estudo qualitativo de um modelo de propagação de dengue / Qualitative study of a dengue disease transmission modelBruna Cassol dos Santos 25 July 2016 (has links)
Em epidemiologia matemática, muitos modelos de propagação de doenças infecciosas em populações têm sido analisados matematicamente e aplicados para doenças específicas. Neste trabalho um modelo de propagação de dengue é analisado considerando-se diferentes hipóteses sobre o tamanho da população humana. Mais precisamente, estamos interessados em verificar o impacto das variações populacionais a longo prazo no cálculo do parâmetro Ro e no equilíbrio endêmico. Vamos discutir algumas ideias que nortearam o processo de definição do parâmetro Ro a partir da construção do Operador de Próxima Geração. Através de um estudo qualitativo do modelo matemático, obtivemos que o equilíbrio livre de doença é globalmente assintoticamente estável se Ro é menor ou igual a 1 e instável se Ro>1. Para Ro>1, a estabilidade global do equilíbrio endêmico é provada usando um critério geral para estabilidade orbital de órbitas periódicas associadas a sistemas autônomos não lineares de altas ordens e resultados da teoria de sistemas competitivos para equações diferenciais ordinárias. Também foi desenvolvida uma análise de sensibilidade do Ro e do equilíbrio endêmico com relação aos parâmetros do modelo de propagação. Diversos cenários foram simulados a partir dos índices de sensibilidade obtidos nesta análise. Os resultados demonstram que, de forma geral, o parâmetro Ro e o equilíbrio endêmico apresentam considerável sensibilidade a taxa de picadas do vetor e a taxa de mortalidade do vetor. / In mathematical epidemiology many models of spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases. In this work a dengue propagation model is analyzed considering different assumptions about the size of the human population. More precisely, we are interested to verify the impact of population long-term variations in the calculation of the parameter Ro and endemic equilibrium. We will discuss some ideas that guided the parameter setting process Ro from the construction of the Next Generation Operator. Through a qualitative study of the mathematical model, we found that the disease-free equilibrium is globally asymptotically stable if Ro is less or equal than 1 and unstable if Ro> 1. For Ro> 1 the global stability of the endemic equilibrium is proved using a general criterion for orbital stability of periodic orbits associated with nonlinear autonomous systems of higher orders and results of the theory of competitive systems for ordinary differential equations. Also a sensitivity analysis of the Ro and the endemic equilibrium with respect to the parameters of the propagation model was developed. Several scenarios were simulated from the sensitivity index obtained in this analysis. The results demonstrate that in general the parameter Ro and the endemic equilibrium are the most sensitive to the vector biting rate and the vector mortality rate.
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Mathematical Analysis of an SEIRS Model with Multiple Latent and Infectious Stages in Periodic and Non-periodic EnvironmentsMelesse, Dessalegn Yizengaw 30 August 2010 (has links)
The thesis focuses on the qualitative analysis of a general class of SEIRS models in periodic and non-periodic environments. The classical SEIRS model, with standard incidence function, is, first of all, extended to incorporate multiple infectious stages. Using Lyapunov function theory and LaSalle's Invariance Principle, the disease-free equilibrium (DFE) of the resulting SEI<sup>n</sup>RS model is shown to be globally-asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, this model has a unique endemic equilibrium point (EEP), which is shown (using a non-linear Lyapunov function of Goh-Volterra type) to be globally-asymptotically stable for a special case. The SEI<sup>n</sup>RS model is further extended to incorporate arbitrary number of latent stages. A notable feature of the resulting SE<sup>m</sup>I<sup>n</sup>RS model is that it uses gamma distribution assumptions for the average waiting times in the latent (m) and infectious (n) stages. Like in the case of the SEI<sup>n</sup>RS model, the SE<sup>m</sup>I<sup>n</sup>RS model also has a globally-asymptotically stable DFE when its associated reproduction threshold is less than unity, and it has a unique EEP (which is globally-stable for a special case) when the threshold exceeds unity. The SE<sup>m</sup>I<sup>n</sup>RS model is further extended to incorporate the effect of periodicity on the disease transmission dynamics. The resulting non-autonomous SE<sup>m</sup>I<sup>n</sup>RS model is shown to have a globally-stable disease-free solution when the associated reproduction ratio is less than unity. Furthermore, the non-autonomous model has at least one positive (non-trivial) periodic solution when the reproduction ratio exceeds unity. It is shown (using persistence theory) that, for the non-autonomous model, the disease will always persist in the population whenever the reproduction ratio is greater than unity. One of the main mathematical contributions of this thesis is that it shows that adding multiple latent and infectious stages, gamma distribution assumptions (for the average waiting times in these stages) and periodicity to the classical SEIRS model (with standard incidence) does not alter the main qualitative dynamics (pertaining to the persistence or elimination of the disease from the population) of the SEIRS model.
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Mathematical Analysis of an SEIRS Model with Multiple Latent and Infectious Stages in Periodic and Non-periodic EnvironmentsMelesse, Dessalegn Yizengaw 30 August 2010 (has links)
The thesis focuses on the qualitative analysis of a general class of SEIRS models in periodic and non-periodic environments. The classical SEIRS model, with standard incidence function, is, first of all, extended to incorporate multiple infectious stages. Using Lyapunov function theory and LaSalle's Invariance Principle, the disease-free equilibrium (DFE) of the resulting SEI<sup>n</sup>RS model is shown to be globally-asymptotically stable whenever the associated reproduction number is less than unity. Furthermore, this model has a unique endemic equilibrium point (EEP), which is shown (using a non-linear Lyapunov function of Goh-Volterra type) to be globally-asymptotically stable for a special case. The SEI<sup>n</sup>RS model is further extended to incorporate arbitrary number of latent stages. A notable feature of the resulting SE<sup>m</sup>I<sup>n</sup>RS model is that it uses gamma distribution assumptions for the average waiting times in the latent (m) and infectious (n) stages. Like in the case of the SEI<sup>n</sup>RS model, the SE<sup>m</sup>I<sup>n</sup>RS model also has a globally-asymptotically stable DFE when its associated reproduction threshold is less than unity, and it has a unique EEP (which is globally-stable for a special case) when the threshold exceeds unity. The SE<sup>m</sup>I<sup>n</sup>RS model is further extended to incorporate the effect of periodicity on the disease transmission dynamics. The resulting non-autonomous SE<sup>m</sup>I<sup>n</sup>RS model is shown to have a globally-stable disease-free solution when the associated reproduction ratio is less than unity. Furthermore, the non-autonomous model has at least one positive (non-trivial) periodic solution when the reproduction ratio exceeds unity. It is shown (using persistence theory) that, for the non-autonomous model, the disease will always persist in the population whenever the reproduction ratio is greater than unity. One of the main mathematical contributions of this thesis is that it shows that adding multiple latent and infectious stages, gamma distribution assumptions (for the average waiting times in these stages) and periodicity to the classical SEIRS model (with standard incidence) does not alter the main qualitative dynamics (pertaining to the persistence or elimination of the disease from the population) of the SEIRS model.
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Mathematical modeling and analysis of HIV/AIDS control measuresGbenga, Abiodun J. January 2012 (has links)
>Magister Scientiae - MSc / In this thesis, we investigate the HIV/AIDS epidemic in a population which experiences a significant flow of immigrants. We derive and analyse a math-
ematical model that describes the dynamics of HIV infection among the im-
migrant youths and intervention that can minimize or prevent the spread of
the disease in the population. In particular, we are interested in the effects of
public-health education and of parental care.We consider existing models of public-health education in HIV/AIDS epidemi-ology, and provide some new insights on these. In this regard we focus atten-tion on the papers [b] and [c], expanding those researches by adding sensitivity analysis and optimal control problems with their solutions.Our main emphasis will be on the effect of parental care on HIV/AIDS epidemi-ology. In this regard we introduce a new model. Firstly, we analyse the model without parental care and investigate its stability and sensitivity behaviour.We conduct both qualitative and quantitative analyses. It is observed that
in the absence of infected youths, disease-free equilibrium is achievable and is
asymptotically stable. Further, we use optimal control methods to determine
the necessary conditions for the optimality of intervention, and for disease
eradication or control. Using Pontryagin’s Maximum Principle to check the
effects of screening control and parental care on the spread of HIV/AIDS, we
observe that parental care is more effective than screening control. However,
the most efficient control strategy is in fact a combination of parental care and screening control. The results form the central theme of this thesis, and are included in the manuscript [a] which is now being reviewed for publication.
Finally, numerical simulations are performed to illustrate the analytical results.
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