Thresholds for extinction and proliferation in a stochastic tumour-immune model with pulsed comprehensive therapy

Periodical applications of immunotherapy and chemotherapy play significant roles in cancer treatment and studies have shown that the evolution of tumour cells is subject to random events. In order to capture the effects of such noise we developed a stochastic tumour-immune dynamical model with pulsed treatment to describe combinations of immunotherapy with chemotherapy. By using theorems of the impulsive stochastic dynamical equation, the tumour free solution and the global positive solution of the proposed system were investigated. We then show that the expectations of the solutions are bounded. Furthermore, threshold conditions for extinction, non-persistence in the mean, weak persistence and stochastic persistence of tumour cells are provided. The results reveal that comprehensive therapy or noise can dominate the evolution of tumours. Finally, biological implications are addressed and a conclusion is presented

Stochastic Gene Expression in a Lentiviral Positive Feedback Loop: HIV-1 Tat Fluctuations Drive Phenotypic Diversity

Stochastic gene expression has been implicated in a variety of cellular processes, including cell differentiation and disease. In this issue of Cell, Weinberger et al. (2005) take an integrated computational-experimental approach to study the Tat transactivation feedback loop in HIV-1 and show that fluctuations in a key regulator, Tat, can result in a phenotypic bifurcation. This phenomenon is observed in an isogenic population where individual cells display two distinct expression states corresponding to latent and productive infection by HIV-1. These findings demonstrate the importance of stochastic gene expression in molecular "decision-making."Comment: Supplemental data available as q-bio.MN/060800

A continuous strain-space model of viral evolution within a host

Viruses rapidly evolve, and HIV in particular is known to be one of the fastest evolving human viruses. It is now commonly accepted that viral evolution is the cause of the intriguing dynamics exhibited during HIV infections and the ultimate success of the virus in its struggle with the immune system. To study viral evolution, we use a simple mathematical model of the within-host dynamics of HIV which incorporates random mutations. In this model, we assume a continuous distribution of viral strains in a one-dimensional phenotype space where random mutations are modelled by di ffusion. Numerical simulations show that random mutations combined with competition result in evolution towards higher Darwinian fitness: a stable traveling wave of evolution, moving towards higher levels of fi tness, is formed in the phenoty space

Nonlinear Systems

Open Mathematics is a challenging notion for theoretical modeling, technical analysis, and numerical simulation in physics and mathematics, as well as in many other fields, as highly correlated nonlinear phenomena, evolving over a large range of time scales and length scales, control the underlying systems and processes in their spatiotemporal evolution. Indeed, available data, be they physical, biological, or financial, and technologically complex systems and stochastic systems, such as mechanical or electronic devices, can be managed from the same conceptual approach, both analytically and through computer simulation, using effective nonlinear dynamics methods. The aim of this Special Issue is to highlight papers that show the dynamics, control, optimization and applications of nonlinear systems. This has recently become an increasingly popular subject, with impressive growth concerning applications in engineering, economics, biology, and medicine, and can be considered a veritable contribution to the literature. Original papers relating to the objective presented above are especially welcome subjects. Potential topics include, but are not limited to: Stability analysis of discrete and continuous dynamical systems; Nonlinear dynamics in biological complex systems; Stability and stabilization of stochastic systems; Mathematical models in statistics and probability; Synchronization of oscillators and chaotic systems; Optimization methods of complex systems; Reliability modeling and system optimization; Computation and control over networked systems

A Study of Infectious Disease Models with Switching

Infectious disease models with switching are constructed and investigated in detail. Modelling infectious diseases as switched systems, which are systems that combine continuous dynamics with discrete logic, allows for the use of methods from switched systems theory. These methods are used to analyze the stability and long-term behaviour of the proposed switched epidemiological models. Switching is first incorporated into epidemiological models by assuming the contact rate to be time-dependent and better approximated by a piecewise constant. Epidemiological models with switched incidence rates are also investigated. Threshold criteria are established that are sufficient for the eradication of the disease, and, hence, the stability of the disease-free solution. In the case of an endemic disease, some criteria are developed that establish the persistence of the disease. Lyapunov function techniques, as well as techniques for stability of impulsive or non-impulsive switched systems with both stable and unstable modes are used. These methods are first applied to switched epidemiological models which are intrinsically one-dimensional. Multi-dimensional disease models with switching are then investigated in detail. An important part of studying epidemiology is to construct control strategies in order to eradicate a disease, which would otherwise be persistent. Hence, the application of controls schemes to switched epidemiological models are investigated. Finally, epidemiological models with switched general nonlinear incidence rates are considered. Simulations are given throughout to illustrate our results, as well as to make some conjectures. Some conclusions are made and future directions are given

Threshold Dynamics of a Stochastic S

A stochastic SIR model with vertical transmission and vaccination is proposed and investigated in this paper. The threshold dynamics are explored when the noise is small. The conditions for the extinction or persistence of infectious diseases are deduced. Our results show that large noise can lead to the extinction of infectious diseases which is conducive to epidemic diseases control

Mathematical models for chemotaxis and their applications in self-organisation phenomena

Chemotaxis is a fundamental guidance mechanism of cells and organisms, responsible for attracting microbes to food, embryonic cells into developing tissues, immune cells to infection sites, animals towards potential mates, and mathematicians into biology. The Patlak-Keller-Segel (PKS) system forms part of the bedrock of mathematical biology, a go-to-choice for modellers and analysts alike. For the former it is simple yet recapitulates numerous phenomena; the latter are attracted to these rich dynamics. Here I review the adoption of PKS systems when explaining self-organisation processes. I consider their foundation, returning to the initial efforts of Patlak and Keller and Segel, and briefly describe their patterning properties. Applications of PKS systems are considered in their diverse areas, including microbiology, development, immunology, cancer, ecology and crime. In each case a historical perspective is provided on the evidence for chemotactic behaviour, followed by a review of modelling efforts; a compendium of the models is included as an Appendix. Finally, a half-serious/half-tongue-in-cheek model is developed to explain how cliques form in academia. Assumptions in which scholars alter their research line according to available problems leads to clustering of academics and the formation of "hot" research topics.Comment: 35 pages, 8 figures, Submitted to Journal of Theoretical Biolog