An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

We formulate and analyze an unconditionally stable nonstandard finite difference method for a mathematical model of HIV transmission dynamics. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves the positivity of the solution, which is one of the essential requirements when modeling epidemic diseases. Robust numerical results confirming theoretical investigations are provided. Comparisons are also made with the other conventional approaches that are routinely used for such problems.IS

Dynamical analysis and consistent numerics for a delay model of viral infection in phytoplankton population

In this article, the effects of time delay and carrying capacity in the dynamics of viral phytoplankton and consistence numerics are studied. Basic properties and stabilities of equilibria are rigorously analyzed and conditions for stability switches are found. A dynamically consistent nonstandard finite difference scheme for the continuous delay model is designed to verify the results and reveal some interesting features of the model. Some ecological implications and interpretations are provided.South African DST/NRF SARChI chair on Mathematical Models and Methods in Bioengineering and Biosciences (M3B2), Department of Mathematics and Applied Mathematics, University of Pretoria and MacArthur Foundation, Bayero University, Kano, Nigeria.https://link.springer.com/journal/133702019-03-01hj2018Mathematics and Applied Mathematic

Mathematical modelling of virus RSV: qualitative properties, numerical solutions and validation for the case of the region of Valencia

El objetivo de esta memoria se centra en primer lugar en la modelización del comportamiento de enfermedades estacionales mediante sistemas de ecuaciones diferenciales y en el estudio de las propiedades dinámicas tales como positividad, periocidad, estabilidad de las soluciones analíticas y la construcción de esquemas numéricos para las aproximaciones de las soluciones numéricas de sistemas de ecuaciones diferenciales de primer orden no lineales, los cuales modelan el comportamiento de enfermedades infecciosas estacionales tales como la transmisión del virus Respiratory Syncytial Virus (RSV). Se generalizan dos modelos matemáticos de enfermedades estacionales y se demuestran que tiene soluciones periódicas usando un Teorema de Coincidencia de Jean Mawhin. Para corroborar los resultados analíticos, se desarrollan esquemas numéricos usando las técnicas de diferencias finitas no estándar desarrolladas por Ronald Michens y el método de la transformada diferencial, los cuales permiten reproducir el comportamiento dinámico de las soluciones analíticas, tales como positividad y periocidad. Finalmente, las simulaciones numéricas se realizan usando los esquemas implementados y parámetros deducidos de datos clínicos De La Región de Valencia de personas infectadas con el virus RSV. Se confrontan con las que arrojan los métodos de Euler, Runge Kutta y la rutina de ODE45 de Matlab, verificándose mejores aproximaciones para tamaños de paso mayor a los que usan normalmente estos esquemas tradicionales.Arenas Tawil, AJ. (2009). Mathematical modelling of virus RSV: qualitative properties, numerical solutions and validation for the case of the region of Valencia [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/8316Palanci

Construction and analysis of efficient numerical methods to solve mathematical models of TB and HIV co-infection

Philosophiae Doctor - PhDThe global impact of the converging dual epidemics of tuberculosis (TB) and human immunodeficiency virus (HIV) is one of the major public health challenges of our time, because in many countries, human immunodeficiency virus (HIV) and mycobacterium tuberculosis (TB) are among the leading causes of morbidity and mortality. It is found that infection with HIV increases the risk of reactivating latent TB infection, and HIV-infected individuals who acquire new TB infections have high rates of disease progression. Research has shown that these two diseases are enormous public health burden, and unfortunately, not much has been done in terms of modeling the dynamics of HIV-TB co-infection at a population level. In this thesis, we study these models and design and analyze robust numerical methods to solve them. To proceed in this direction, first we study the sub-models and then the full model. The first sub-model describes the transmission dynamics of HIV that accounts for behavior change. The impact of HIV educational campaigns is also studied. Further, we explore the effects of behavior change and different responses of individuals to educational campaigns in a situation where individuals may not react immediately to these campaigns. This is done by considering a distributed time delay in the HIV sub-model. This leads to Hopf bifurcations around the endemic equilibria of the model. These bifurcations correspond to the existence of periodic solutions that oscillate around the equilibria at given thresholds. Further, we show how the delay can result in more HIV infections causing more increase in the HIV prevalence. Part of this study is then extended to study a co-infection model of HIV-TB. A thorough bifurcation analysis is carried out for this model. Robust numerical methods are then designed and analyzed for these models. Comparative numerical results are also provided for each model.South Afric

Modeling, analysis and numerical method for HIV-TB co-infection with TB treatment in Ethiopia

In this thesis, a mathematical model for HIV and TB co-infection with TB treatment among populations of Ethiopia is developed and analyzed. The TB model includes an age of infection. We compute the basic reproduction numbers RTB and RH for TB and HIV respectively, and the overall repro- duction number R for the system. We find that if R 1; then the disease-free and the endemic equilibria are locally asymptotically stable, respectively. Otherwise these equilibria are unstable. The TB-only endemic equilibrium is locally asymptotically stable if RTB > 1, and RH < 1. How- ever, the symmetric condition, RTB 1, does not necessarily guarantee the stability of the HIV-only equilibrium, but it is possible that TB can coexist with HIV when RH > 1: As a result, we assess the impact of TB treatment on the prevalence of TB and HIV co-infection. To derive and formulate the nonlinear differential equations models for HIV and TB co-infection that accounts for treatment, we formulate and analyze the HIV only sub models, the TB-only sub models and the full models of HIV and TB combined. The TB-only sub model includes both ODEs and PDEs in order to describe the variable infectiousness and e ect of TB treatment during the infectious period. To analyse and solve the three models, we construct robust methods, namely the numerical nonstandard definite difference methods (NSFDMs). Moreover, we improve the order of convergence of these methods in their applications to solve the model of HIV and TB co-infection with TB treatment at the population level in Ethiopia. The methods developed in this thesis work and show convergence, especially for individuals with small tolerance either to the disease free or the endemic equilibria for first order mixed ODE and PDE as we observed in our models.Mathematical SciencesPh. D. (Applied Mathematics

Analysis and implementation of robust numerical methods to solve mathematical models of HIV and Malaria co-infection

Philosophiae Doctor - PhDThere is a growing interest in the dynamics of the co-infection of these two diseases. In this thesis, firstly we focus on studying the effect of a distributed delay representing the incubation period for the malaria parasite in the mosquito vector to possibly reduce the initial transmission and prevalence of malaria. This model can be regarded as a generalization of SEI models (with a class for the latently infected mosquitoes) and SI models with a discrete delay for the incubation period in mosquitoes. We study the possibility of occurrence of backward bifurcation. We then extend these ideas to study a full model of HIV and malaria co-infection. To get further inside into the dynamics of the model, we use the geometric singular perturbation theory to couple the fast and slow models from the full model. Finally, since the governing models are very complex, they cannot be solved analytically and hence we develop and analyze a special class of numerical methods to solve them.South Afric

Investigating the Role of Hydration and DNA Dynamic Alterations in DNA Recognition by A Heterocyclic Diamidine

Target recognition by DNA-binding ligands, such as drugs, occurs in an aqueous environment, in which water (near unit mole fraction, ~55 M) dominates every solute. A quantitative account of how water molecules are disposed in DNA/ligand binding is indispensable for understanding the driving forces that confer high-affinity and selectivity. We are investigating the DNA sequence selectivity of a model DNA minor groove-binding heterocyclic diamidine, DB1976, which shows therapeutic activity in acute myeloid leukemia, systemic fibroses, and obesity-related liver disorders in vivo. The DNA minor groove is richly populated with water molecules. Studies based on explicit-solvent MD simulation have shown distinct DNA dynamics upon drug-DNA complexes. We have cooperated the role of hydration and conformational dynamics in contributing to drug selectivity. Moving forward, our goal is to evaluate the structure-hydration relationships of designed diamidines to site-specific and nonspecific DNA as part of their biophysical characterization as potential therapeutic agents

Controlling microbial community dynamics through engineered metabolic dependencies

Metabolic cross-feeding is an important process that can broadly shape microbial communities. Comparative genomic analysis of >6000 sequenced bacteria from diverse environments provides evidence to suggesting that amino acid biosynthesis has been broadly optimized to reduce individual metabolic burden in favor of enhanced cross-feeding to support synergistic growth across the biosphere. Still, little is known about specific cross-feeding principles that drive the formation and maintenance of individuals within a mixed population. Here, we devised a series of synthetic syntrophic communities to probe the complex interactions underlying metabolic exchange of amino acids. We experimentally analyzed multi-member, multi-dimensional communities of Escherichia coli of increasing sophistication to assess the outcomes of synergistic cross-feeding. We find that biosynthetically costly amino acids including methionine, lysine, isoleucine, arginine and aromatics, tend to promote stronger cooperative interactions than amino acids that are cheaper to produce. Furthermore, cells that share common intermediates along branching pathways yielded more synergistic growth, but exhibited many instances of both positive and negative epistasis when these interactions scaled to higher-dimensions. This system enabled the identification of synergistic pairings and optimal expression levels of amino acid exporters of arginine, threonine and aromatics towards drastic improvements of ecosystem productivity. Tradeoffs identified in these mutualistic systems between secretion, relative abundance and absolute community productivity have implication in the evolution of cooperative behaviors. Long-term evolution of these synthetic communities highlight transporter over-expression, amino acid pool redistribution, and perturbations to nitrogen regulation as strategies to circumvent imposed metabolic dependencies. To address this potentially problematic genomic plasticity, a genetically reassigned organism is leveraged to investigate synthetic metabolic dependencies showing improved biocontainment and potential for microbial consortia control. These results improve our basic understanding of microbial syntrophy while also highlighting the utility and limitations of current approaches to modeling and controlling the dynamic complexities of microbial ecosystems. This work sets a foundation for future endeavors in microbial ecology and evolution, and presents a platform to develop better and more robust engineered synthetic communities for industrial biotechnology