Hopf Bifurcation and Stability of Periodic Solutions for Delay Differential Model of HIV Infection of CD4 +

This paper deals with stability and Hopf bifurcation analyses of a mathematical model of HIV infection of CD4+ T-cells. The model is based on a system of delay differential equations with logistic growth term and antiretroviral treatment with a discrete time delay, which plays a main role in changing the stability of each steady state. By fixing the time delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. We study the effect of the time delay on the stability of the endemically infected equilibrium. We derive explicit formulae to determine the stability and direction of the limit cycles by using center manifold theory and normal form method. Numerical simulations are presented to illustrate the results

Immune Response in the Study of Infectious Diseases (Co-Infection) in an Endemic Region

abstract: Diseases have been part of human life for generations and evolve within the population, sometimes dying out while other times becoming endemic or the cause of recurrent outbreaks. The long term influence of a disease stems from different dynamics within or between pathogen-host, that have been analyzed and studied by many researchers using mathematical models. Co-infection with different pathogens is common, yet little is known about how infection with one pathogen affects the host's immunological response to another. Moreover, no work has been found in the literature that considers the variability of the host immune health or that examines a disease at the population level and its corresponding interconnectedness with the host immune system. Knowing that the spread of the disease in the population starts at the individual level, this thesis explores how variability in immune system response within an endemic environment affects an individual's vulnerability, and how prone it is to co-infections. Immunology-based models of Malaria and Tuberculosis (TB) are constructed by extending and modifying existing mathematical models in the literature. The two are then combined to give a single nine-variable model of co-infection with Malaria and TB. Because these models are difficult to gain any insight analytically due to the large number of parameters, a phenomenological model of co-infection is proposed with subsystems corresponding to the individual immunology-based model of a single infection. Within this phenomenological model, the variability of the host immune health is also incorporated through three different pathogen response curves using nonlinear bounded Michaelis-Menten functions that describe the level or state of immune system (healthy, moderate and severely compromised). The immunology-based models of Malaria and TB give numerical results that agree with the biological observations. The Malaria--TB co-infection model gives reasonable results and these suggest that the order in which the two diseases are introduced have an impact on the behavior of both. The subsystems of the phenomenological models that correspond to a single infection (either of Malaria or TB) mimic much of the observed behavior of the immunology-based counterpart and can demonstrate different behavior depending on the chosen pathogen response curve. In addition, varying some of the parameters and initial conditions in the phenomenological model yields a range of topologically different mathematical behaviors, which suggests that this behavior may be able to be observed in the immunology-based models as well. The phenomenological models clearly replicate the qualitative behavior of primary and secondary infection as well as co-infection. The mathematical solutions of the models correspond to the fundamental states described by immunologists: virgin state, immune state and tolerance state. The phenomenological model of co-infection also demonstrates a range of parameter values and initial conditions in which the introduction of a second disease causes both diseases to grow without bound even though those same parameters and initial conditions did not yield unbounded growth in the corresponding subsystems. This results applies to all three states of the host immune system. In terms of the immunology-based system, this would suggest the following: there may be parameter values and initial conditions in which a person can clear Malaria or TB (separately) from their system but in which the presence of both can result in the person dying of one of the diseases. Finally, this thesis studies links between epidemiology (population level) and immunology in an effort to assess the impact of pathogen's spread within the population on the immune response of individuals. Models of Malaria and TB are proposed that incorporate the immune system of the host into a mathematical model of an epidemic at the population level.Dissertation/ThesisPh.D. Applied Mathematics for the Life and Social Sciences 201

Mathematical modelling of cytokine-mediated immune response and autoimmunity

One of the major outstanding challenges in immunology is the development of a comprehensive, quantitative and accurate approach to understanding the causes and dynamics of immune responses. The immune system normally protects the body against infections, but at the same time it is possible that it can fail to distinguish the host’s own cells from the cells affected by the infection, which can lead to autoimmune disease. The question of what releases the auto-pathogenic potential of T lymphocytes is at the heart of understanding autoimmune disease. Among various possible causes of autoimmune disease, an important role is played by infections that can result in a breakdown of immune tolerance, primarily through the mechanism of molecular mimicry, where the introduction of pathogenic peptides that structurally resemble self-peptides, derived from infection, may induce T lymphocytes to proliferate and leave them with the ability to respond to self, as well as foreign antigens. Deterministic and stochastic models have been extensively used in the past to study the dynamics of immune responses and analyse a possible onset of autoimmunity. The main focus of this thesis is the development and analysis of mathematical models of immune response to infection, as well as the onset and progress of autoimmunity. Particular emphasis is made on developing new mathematical approaches for elucidating the roles played by various cytokines in the immune dynamics. In the first part of the thesis I develop a mathematical model for dynamics of immune response to hepatitis B. This model explicitly includes contributions from innate and adaptive immune responses, as well as from cytokines. Analysis of the model identifies parameter regimes where the model exhibits clearance of infection, maintenance of a chronic infection, or periodic oscillations. Effects of nucleoside analogues and interferon treatments are analysed, and the critical drug efficiency is determined. The second part of the thesis investigates the dynamics of immune response to a general viral infection and a possible onset of autoimmunity, which account for regulatory T cells, T cells with different activation thresholds, and cytokines. Feasibility and stability analyses of different steady states yield boundaries of stability and bifurcations in terms of system parameters. This model exhibits bi-stability and shows different regimes of normal clearance of viral infection, chronic infection, or autoimmune behaviour. Therefore, it can provide significant new insights into autoimmune dynamics. To investigate the role of stochasticity in immune dynamics, I developed a stochastic version of the model, and the major result is that adding stochasticity can lead to the emergence of sustained oscillations around deterministically stable steady states, thus providing a possible explanation for experimentally observed variations in the progression of autoimmune disease. I also have investigated stochastic dynamics in the regime of bi-stability and computed the magnitude of these fluctuations. I have also analysed the effects of different time delays, as well as the inhibiting effect of regulatory T cells on secretion of interleukin-2 on autoimmune dynamics. To this end, I have performed a systematic analysis of stability of all steady states of the corresponding model both analytically, and numerically. The identification of basins of attraction of different steady states and periodic solutions indicates that time delays can change the shape of these basins of attraction, and the new results show better qualitative agreement with the experimental observations. My thesis culminates with the last part, where I explore stochastic effects in a time-delayed model for autoimmunity. The major achievement in this part of the thesis is the development of a new methodology for deriving an Itô stochastic delay differential equation (SDDE) from delay discrete stochastic models, as well as showing the equivalency of previously proposed methods. Using this equivalence, I derived a simpler SDDE model to perform numerical simulations. I have used a linear noise approximation (LNA) to determine the magnitude of stochastic fluctuations around deterministic steady states, and to obtain insights into how the coherence of stochastic oscillations around deterministically stable steady states depends on system parameters

Dynamics Analysis of a Viral Infection Model with a General Standard Incidence Rate

The basic viral infection models, proposed by Nowak et al. and Perelson et al., respectively, have been widely used to describe viral infection such as HBV and HIV infection. However, the basic reproduction numbers of the two models are proportional to the number of total cells of the host's organ prior to the infection, which seems not to be reasonable. In this paper, we formulate an amended model with a general standard incidence rate. The basic reproduction number of the amended model is independent of total cells of the host’s organ. When the basic reproduction number R01, then the endemic equilibrium is globally asymptotically stable and the virus persists in the host

Stochastic dynamics in a time-delayed model for autoimmunity

In this paper we study interactions between stochasticity and time delays in the dynamics of immune response to viral infections, with particular interest in the onset and development of autoimmune response. Starting with a deterministic time-delayed model of immune response to infection, which includes cytokines and T cells with different activation thresholds, we derive an exact delayed chemical master equation for the probability density. We use system size expansion and linear noise approximation to explore how variance and coherence of stochastic oscillations depend on parameters, and to show that stochastic oscillations become more regular when regulatory T cells become more effective at clearing autoreactive T cells. Reformulating the model as an Itô stochastic delay differential equation, we perform numerical simulations to illustrate the dynamics of the model and associated probability distributions in different parameter regimes. The results suggest that even in cases where the deterministic model has stable steady states, in individual stochastic realisations, the model can exhibit sustained stochastic oscillations, whose variance increases as one gets closer to the deterministic stability boundary. Furthermore, in the regime of bi-stability, whereas deterministically the system would approach one of the steady states (or periodic solutions) depending on the initial conditions, due to the presence of stochasticity, it is now possible for the system to reach both of those dynamical states with certain probability. Biological significance of this result lies in highlighting the fact that since normally in a laboratory or clinical setting one would observe a single individual realisation of the course of the disease, even for all parameters characterising the immune system and the strength of infection being the same, there is a proportion of cases where a spontaneous recovery can be observed, and similarly, where a disease can develop in a situation that otherwise would result in a normal disease clearance

Mathematical modelling of In-vivo HIV optimal therapy and management

Thesis submitted in total fulfillment of the requirements for the Degree of Doctor of Philosophy in Biomathematics of Strathmore UniversityHuman Immunodeficiency Virus (HIV) remains the main cause of premature death globally. In 2013, Kenya was the fourth largest endemic HIV country in the world having over 1.6 million people living with the virus. The fact that there is an increase in the rate of new-HIV infections in Africa and especially in Kenya underscores the need for adequate strategies to cope with this deadly disease and to achieve vision 2020. Where the government envision that, by 2020, 90% of all people living with HIV will know their HIV status, 90% of all people with diagnosed HIV infection will receive sustained antiretroviral therapy and 90% of all people receiving antiretroviral therapy will have viral suppression. Currently, there is no known cure for HIV, hence the most optimal way is the management of HIV infected people to prevent virus progression and HIV transmission. Although there has been progress in management of HIV by the use of antiretroviral drugs (ARTs), long-term use of these ARTs leads to overwhelming challenges. These challenges are: toxicity of the medication, nonadherence problems as a result of inaccessibility of comprehensive care centres, drug resistance-mutations and significant financial burdens. This study aimed at formulating and analysing mathematical in-vivo models for the interaction between HIV virions, CD4+ T-cells, CD8+ T-cells and the optimal control for effective therapy, whose numerical simulations would assist in giving more insight about the challenges aforementioned.Various mathematical methods including ordinary differential equations, Runge-Kutta forth order scheme and optimal control theory have been applied in the development and the analysis of the model. Analysis of the formulated model indicates existence of multiple equilibria whose stability and bifurcation analysis have been presented. From the simulated results, we have noted that early initiation of HIV treatment reduce viral replication in HIV infected people. In particular, highly active antiretroviral therapy (HAART) which include the combination therapy of Fusion inhibitor (FI), Reverse Transcriptase inhibitor (RTI) and Protease inhibitor (PI) in different proportions have been found to be more effective in treating HIV than a single drug therapy. The model simulations show how to best choose the proportions of FI, RTI and PI in order to maintain an acceptable level of CD4+ T-cells and, at the same time, reduce the side effects associated with their long term use. In addition, the most optimal way of administering ART drugs that lead to maximum benefit has been predicted from optimal control simulation. The findings give a significant explanation of why late initiation of ARTs might not be helpful to an HIV infected person and suggest that the controls ought to be optimal at the acute phase of infection where the viral replication is extremely high. If the controls are well implemented, many potential infections would be averted by lowering the viral load and increasing the number of the T-helper cells. This, in turn, will also lead to reduction in HIV transmission. Therefore, there is need for increased awareness campaigns to encourage people to know their HIV status and adhere to the prescribed treatment.The research outcomes in this study emphasizes the importance of “Anza Sasa”campaign that was launched on 15th July 2016 by the Government of Kenya through the Ministry of Health in collaboration with the National AIDS and STI Control program

Methods in Computational Biology

Modern biology is rapidly becoming a study of large sets of data. Understanding these data sets is a major challenge for most life sciences, including the medical, environmental, and bioprocess fields. Computational biology approaches are essential for leveraging this ongoing revolution in omics data. A primary goal of this Special Issue, entitled “Methods in Computational Biology”, is the communication of computational biology methods, which can extract biological design principles from complex data sets, described in enough detail to permit the reproduction of the results. This issue integrates interdisciplinary researchers such as biologists, computer scientists, engineers, and mathematicians to advance biological systems analysis. The Special Issue contains the following sections:•Reviews of Computational Methods•Computational Analysis of Biological Dynamics: From Molecular to Cellular to Tissue/Consortia Levels•The Interface of Biotic and Abiotic Processes•Processing of Large Data Sets for Enhanced Analysis•Parameter Optimization and Measuremen

Mathematical Modelling of Tuberculosis Infection

Tuberculosis is one of the leading causes of death by infectious disease in the world today. However, the majority of individuals infected with Mycobacterium tuberculosis are able to contain bacterial growth and establish a latent infection. The aim of this thesis is to develop mathematical models to study the progression of disease in individuals infected with Mycobacterium tuberculosis. This work focuses on understanding bacterial and host defence mechanisms that govern the outcome of infection, and on identifying factors that affect the outcome of anti-tuberculosis chemotherapy. A detailed model of human tuberculosis infection in the lung and peripheral draining lymph node is developed that builds on models published in the literature. Analysis of this model suggests a differential role for innate and adaptive immune responses in determining the outcome of infection, and a possible role for an intracellular bacterial population in establishing a persistent infection. For certain parameter values this system has multiple steady states so the outcome of infection may also depend on initial conditions. This model is then modified to incorporate the effect of treatment with the antimycobacterial agent rifampicin. The model is used to investigate different treatment regimens and simulation results suggest that the length of tuberculosis therapy can be reduced by further optimizing the standard rifampicin dosing regimen. Simple predator-prey type models of infection are constructed to gain further insight into the mechanisms that control the establishment and maintenance of latency. These models support observations made from the full disease model regarding the roles of innate and adaptive immunity in fighting infection and the influence of an intracellular bacterial population that is protected from the innate immune system. The addition of a population of non-replicating or slow growing bacteria contributes to the establishment of latent infection and generally makes latency a more robust and stable state

An investigation into the effects of commencing haemodialysis in the critically ill

<b>Introduction:</b> We have aimed to describe haemodynamic changes when haemodialysis is instituted in the critically ill. 3 hypotheses are tested: 1)The initial session is associated with cardiovascular instability, 2)The initial session is associated with more cardiovascular instability compared to subsequent sessions, and 3)Looking at unstable sessions alone, there will be a greater proportion of potentially harmful changes in the initial sessions compared to subsequent ones. <b>Methods:</b> Data was collected for 209 patients, identifying 1605 dialysis sessions. Analysis was performed on hourly records, classifying sessions as stable/unstable by a cutoff of >+/-20% change in baseline physiology (HR/MAP). Data from 3 hours prior, and 4 hours after dialysis was included, and average and minimum values derived. 3 time comparisons were made (pre-HD:during, during HD:post, pre-HD:post). Initial sessions were analysed separately from subsequent sessions to derive 2 groups. If a session was identified as being unstable, then the nature of instability was examined by recording whether changes crossed defined physiological ranges. The changes seen in unstable sessions could be described as to their effects: being harmful/potentially harmful, or beneficial/potentially beneficial. <b>Results:</b> Discarding incomplete data, 181 initial and 1382 subsequent sessions were analysed. A session was deemed to be stable if there was no significant change (>+/-20%) in the time-averaged or minimum MAP/HR across time comparisons. By this definition 85/181 initial sessions were unstable (47%, 95% CI SEM 39.8-54.2). Therefore Hypothesis 1 is accepted. This compares to 44% of subsequent sessions (95% CI 41.1-46.3). Comparing these proportions and their respective CI gives a 95% CI for the standard error of the difference of -4% to 10%. Therefore Hypothesis 2 is rejected. In initial sessions there were 92/1020 harmful changes. This gives a proportion of 9.0% (95% CI SEM 7.4-10.9). In the subsequent sessions there were 712/7248 harmful changes. This gives a proportion of 9.8% (95% CI SEM 9.1-10.5). Comparing the two unpaired proportions gives a difference of -0.08% with a 95% CI of the SE of the difference of -2.5 to +1.2. Hypothesis 3 is rejected. Fisher’s exact test gives a result of p=0.68, reinforcing the lack of significant variance. <b>Conclusions:</b> Our results reject the claims that using haemodialysis is an inherently unstable choice of therapy. Although proportionally more of the initial sessions are classed as unstable, the majority of MAP and HR changes are beneficial in nature