Dynamics and synchronisation of two coupled parametric pendula

One of the most important discoveries in the study of nonlinear dynamical systems in the last decade is that chaotic systems can be controlled and synchronised. Chaos synchronisation can be viewed as a particular problem of chaos control in the sense that by introducing a coupling term between two independent chaotic systems, we can provide a controlling mechanism in one or both systems (unidirectional or multi-directional coupling) that will eventually cause their trajectories to converge onto each other and then remain synchronised. But in most dynamical systems, chaotic attractors coexist with periodic attractors for a given set of parameters. This guarantees the coexistence of competing synchronous behaviours (chaotic and periodic synchronisation). Therefore in order to fully understand the synchronisation regimes that can occur to a given coupled dynamical system, we need to consider both the chaotic synchronisation component of the dynamics as well as periodic synchronisation and the transition between them. In this thesis we study both periodic and chaotic synchronisation of coupled dynamical systems. We introduce the subject of synchronisation of coupled dynamical systems in chapter 1. In chapters 2, 3 and 4 we study the oscillating, rotating and chaotic solutions of the single parametrically excited pendulum. The study of both periodic and chaotic synchronisation of two coupled parametrically excited pendula (sometimes called pendulums) is considered in chapters 5 and 6 respectively. Then we summarise our main findings in chapter 7 together with some proposals for future research directions

The research and development process for multiscale models of infectious disease systems.

Multiscale modelling of infectious disease systems falls within the domain of complexity science-the study of complex systems. However, what should be made clear is that current progress in multiscale modelling of infectious disease dynamics is still as yet insufficient to present it as a mature sub-discipline of complexity science. In this article we present a methodology for development of multiscale models of infectious disease systems. This methodology is a set of partially ordered research and development activities that result in multiscale models of infectious disease systems built from different scientific approaches. Therefore, the conclusive result of this article is a methodology to design multiscale models of infectious diseases. Although this research and development process for multiscale models cannot be claimed to be unique and final, it constitutes a good starting point, which may be found useful as a basis for further refinement in the discourse for multiscale modelling of infectious disease dynamics

A complete categorization of multiscale models of infectious disease systems

Modelling of infectious disease systems has entered a new era in which disease modellers are increasingly turning to multiscale modelling to extend traditional modelling frameworks into new application areas and to achieve higher levels of detail and accuracy in characterizing infectious disease systems. In this paper we present a categorization framework for categorizing multiscale models of infectious disease systems. The categorization framework consists of five integration frameworks and five criteria. We use the categorization framework to give a complete categorization of host-level immuno-epidemiological models (HL-IEMs). This categorization framework is also shown to be applicable in categorizing other types of multiscale models of infectious diseases beyond HL-IEMs through modifying the initial categorization framework presented in this study. Categorization of multiscale models of infectious disease systems in this way is useful in bringing some order to the discussion on the structure of these multiscale models

A primer on multiscale modelling of infectious disease systems

The development of multiscale models of infectious disease systems is a scientific endeavour whose progress depends on advances on three main frontiers: (a) the conceptual framework frontier, (b) the mathematical technology or technical frontier, and (c) the scientific applications frontier. The objective of this primer is to introduce foundational concepts in multiscale modelling of infectious disease systems focused on these three main frontiers. On the conceptual framework frontier we propose a three-level hierarchical framework as a foundational idea which enables the discussion of the structure of multiscale models of infectious disease systems in a general way. On the scientific applications frontier we suggest ways in which the different structures of multiscale models can serve as infrastructure to provide new knowledge on the control, elimination and even eradication of infectious disease systems, while on the mathematical technology or technical frontier we present some challenges that modelers face in developing appropriate multiscale models of infectious disease systems. We anticipate that the foundational concepts presented in this primer will be central in articulating an integrated and more refined disease control theory based on multiscale modelling - the all-encompassing quantitative representation of an infectious disease system. Keywords: Multiscale models of infectious diseases, Immuno-epidemiological models, Linking individual/lower/micro and population/upper/macro scales, Comparative effectiveness researc

Optimal Control of Combined Therapy in a Single Strain HIV-1 Model

Highly active antiretroviral therapy (HAART) is administered to symptomatic human immunodeficiency virus (HIV) infected individuals to improve their health. Various administration schemes are used to improve patients?lives and at the same time suppressing development of drug resistance, reduce evolution of new viral strains, minimize serious side effects, improve patient adherence and also reduce the costs of drugs. We deduce an optimal drug administration scheme useful in improving patients? health especially in poor resourced settings. In this paper we use the Pontryagin?s Maximum Principle to derive optimal drug dosages based on a mathematical dynamical model. We use methods of optimal control to determine optimal controls analytically, and then use the Runge-Kutta scheme of order four to numerically simulate different therapy effects. We simulate the different effects of a drug regimen composed of a protease inhibitor and a nucleoside reverse transcriptase inhibitor. Our results indicate that for highly toxic drugs, small dosage sizes and allowing drug holidays make a profound impact in both improving the quality of life and reducing economic costs of therapy. The results show that for drugs with less toxicity, continuous therapy is beneficial

The transmission mechanism theory of disease dynamics: Its aims, assumptions and limitations

Most of the progress in the development of single scale mathematical and computational models for the study of infectious disease dynamics which now span over a century is build on a body of knowledge that has been developed to address particular single scale descriptions of infectious disease dynamics based on understanding disease transmission process. Although this single scale understanding of infectious disease dynamics is now founded on a body of knowledge with a long history, dating back to over a century now, that knowledge has not yet been formalized into a scientific theory. In this article, we formalize this accumulated body of knowledge into a scientific theory called the transmission mechanism theory of disease dynamics which states that at every scale of organization of an infectious disease system, disease dynamics is determined by transmission as the main dynamic disease process. Therefore, the transmission mechanism theory of disease dynamics can be seen as formalizing knowledge that has been inherent in the study of infectious disease dynamics using single scale mathematical and computational models for over a century now. The objective of this article is to summarize this existing knowledge about single scale modelling of infectious dynamics by means of a scientific theory called the transmission mechanism theory of disease dynamics and highlight its aims, assumptions and limitations

A Multiscale Model for the World’s First Parasitic Disease Targeted for Eradication: Guinea Worm Disease

Guinea worm disease (GWD) is both a neglected tropical disease and an environmentally driven infectious disease. Environmentally driven infectious diseases remain one of the biggest health threats for human welfare in developing countries and the threat is increased by the looming danger of climate change. In this paper we present a multiscale model of GWD that integrates the within-host scale and the between-host scale. The model is used to concurrently examine the interactions between the three organisms that are implicated in natural cases of GWD transmission, the copepod vector, the human host, and the protozoan worm parasite (Dracunculus medinensis), and identify their epidemiological roles. The results of the study (through sensitivity analysis of R0) show that the most efficient elimination strategy for GWD at between-host scale is to give highest priority to copepod vector control by killing the copepods in drinking water (the intermediate host) by applying chemical treatments (e.g., temephos, an organophosphate). This strategy should be complemented by health education to ensure that greater numbers of individuals and communities adopt behavioural practices such as voluntary reporting of GWD cases, prevention of GWD patients from entering drinking water bodies, regular use of water from safe water sources, and, in the absence of such water sources, filtering or boiling water before drinking. Taking into account the fact that there is no drug or vaccine for GWD (interventions which operate at within-host scale), the results of our study show that the development of a drug that kills female worms at within-host scale would have the highest impact at this scale domain with possible population level benefits that include prevention of morbidity and prevention of transmission

A two strain tuberculosis transmission model with therapy and quarantine

A two strain tuberculosis model with treatment which allows TB patients with the drug sensitive of strain Mycobacterium tuberculosis to be cured is presented. The model is further extended to incorporate quarantine for active TB cases with multi‐drug resistant TB strains. The model assumes that latently infected individuals develop active disease as a result of endogenous activation and exogenous reinfection. Qualitative analysis of the model including positivity, boundedness and persistence of solutions are presented. The thresholds and equilibria quantities for the models are determined and stability of the solution is analyzed. From the study we conclude that quarantine of the multi‐drug resistant tuberculosis cases reduces the multi‐drug resistant tuberculosis induced reproduction number to values below unit, thus this intervention strategy can control the development of multi‐drug resistant tuberculosis epidemic. Also effective chemoprophylaxis and treatment of infectives result in a reduction of multi‐drug resistant tuberculosis cases since most multi‐drug resistant tuberculosis cases are a result of inappropriate treatment. First published online: 14 Oct 201

Application of the replication–transmission relativity theory in the development of multiscale models of infectious disease dynamics

Despite the existence of a powerful theoretical foundation for the development of multiscale models of infectious disease dynamics in the form of the replication–transmission relativity theory, the majority of current modelling studies focus more on single-scale modelling. The explicit aim of this study is to change the current predominantly single-scale modelling landscape in the design of planning frameworks for the control, elimination and even eradication of infectious disease systems through the exploitation of multiscale modelling methods based on the application of the replication–transmission relativity theory. We first present a structured roadmap for the development of multiscale models of infectious disease systems. The roadmap is tested on hookworm infection. The testing of the feasibility of the roadmap established a fundamental result which can be generalized to confirm that the complexity of an infectious disease system is encapsulated with a level of organization spanning a microscale and a macroscale