Some novel techniques of parameter estimation for the dynamical models in biological systems

Inverse problems based on using experimental data to estimate unknown parameters of a system often arise in biological and chaotic systems. In this paper, we consider parameter estimation in systems biology involving linear and non-linear complex dynamical models, including the Michaelis–Menten enzyme kinetic system, a dynamical model of competence induction in Bacillus subtilis bacteria and a model of feedback bypass in B. subtilis bacteria. We propose some novel techniques for inverse problems. Firstly, we establish an approximation of a non-linear differential algebraic equation that corresponds to the given biological systems. Secondly, we use the Picard contraction mapping, collage methods and numerical integration techniques to convert the parameter estimation into a minimization problem of the parameters. We propose two optimization techniques: a grid approximation method and a modified hybrid Nelder–Mead simplex search and particle swarm optimization (MH-NMSS-PSO) for non-linear parameter estimation. The two techniques are used for parameter estimation in a model of competence induction in B. subtilis bacteria with noisy data. The MH-NMSS-PSO scheme is applied to a dynamical model of competence induction in B. subtilis bacteria based on experimental data and the model for feedback bypass. Numerical results demonstrate the effectiveness of our approach.

Whole-proteome phylogeny of large dsDNA viruses and parvoviruses through a composition vector method related to dynamical language model

Background The vast sequence divergence among different virus groups has presented a great challenge to alignment-based analysis of virus phylogeny. Due to the problems caused by the uncertainty in alignment, existing tools for phylogenetic analysis based on multiple alignment could not be directly applied to the whole-genome comparison and phylogenomic studies of viruses. There has been a growing interest in alignment-free methods for phylogenetic analysis using complete genome data. Among the alignment-free methods, a dynamical language (DL) method proposed by our group has successfully been applied to the phylogenetic analysis of bacteria and chloroplast genomes. Results In this paper, the DL method is used to analyze the whole-proteome phylogeny of 124 large dsDNA viruses and 30 parvoviruses, two data sets with large difference in genome size. The trees from our analyses are in good agreement to the latest classification of large dsDNA viruses and parvoviruses by the International Committee on Taxonomy of Viruses (ICTV). Conclusions The present method provides a new way for recovering the phylogeny of large dsDNA viruses and parvoviruses, and also some insights on the affiliation of a number of unclassified viruses. In comparison, some alignment-free methods such as the CV Tree method can be used for recovering the phylogeny of large dsDNA viruses, but they are not suitable for resolving the phylogeny of parvoviruses with a much smaller genome size.

Interpersonal pattern dynamics and adaptive behavior in multiagent neurobiological systems : conceptual model and data

Ecological dynamics characterizes adaptive behavior as an emergent, self-organizing property of interpersonal interactions in complex social systems. The authors conceptualize and investigate constraints on dynamics of decisions and actions in the multiagent system of team sports. They studied coadaptive interpersonal dynamics in rugby union to model potential control parameter and collective variable relations in attacker–defender dyads. A videogrammetry analysis revealed how some agents generated fluctuations by adapting displacement velocity to create phase transitions and destabilize dyadic subsystems near the try line. Agent interpersonal dynamics exhibited characteristics of chaotic attractors and informational constraints of rugby union boxed dyadic systems into a low dimensional attractor. Data suggests that decisions and actions of agents in sports teams may be characterized as emergent, self-organizing properties, governed by laws of dynamical systems at the ecological scale. Further research needs to generalize this conceptual model of adaptive behavior in performance to other multiagent populations.

Fundamental solution and discrete random walk model for time-space fractional diffusion equation

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

Numerical analysis of the time variable fractional order mobile-immobile advection-dispersion model

Many physical processes exhibit fractional order behavior that varies with time or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider the time variable fractional order mobile-immobile advection-dispersion model. Numerical methods and analyses of stability and convergence for the fractional partial differential equations are quite limited and difficult to derive. This motivates us to develop efficient numerical methods as well as stability and convergence of the implicit numerical methods for the fractional order mobile immobile advection-dispersion model. In the paper, we use the Coimbra variable time fractional derivative which is more efficient from the numerical standpoint and is preferable for modeling dynamical systems. An implicit Euler approximation for the equation is proposed and then the stability of the approximation are investigated. As for the convergence of the numerical scheme we only consider a special case, i.e. the time fractional derivative is independent of time variable t. The case where the time fractional derivative depends both the time variable t and the space variable x will be considered in the future work. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

A prototype system dynamic model for assessing the sustainability of construction projects

There is a worldwide demand for an increasingly sustainable built environment. This has resulted in the need for a more accurate evaluation of the level of sustainability of construction projects. To do this it involves the development of better measurement and benchmarking methods. One approach is to use a theoretical model to assess construction projects in terms of their sustainable development value (SDV) and sustainable development ability (SDA) for implementation in the project life cycle, where SDA measures the contribution of a project to development sustainability and as a major criterion for assessing its feasibility. This paper develops an improved SDA prototype model that incorporates the effects of dynamical factors on project sustainability. This involves the introduction of two major factors concerning technological advancement and changes in people's perceptions. A case study is used to demonstrate the procedures involved in simulation and modeling, one outcome of which is to demonstrate the greater influence of technological advancement on project sustainability than changes in perception.

Existence of travelling wave solutions for a model of tumour invasion

The existence of travelling wave solutions to a haptotaxis dominated model is analysed. A version of this model has been derived in Perumpanani et al. (1999) to describe tumour invasion, where diffusion is neglected as it is assumed to play only a small role in the cell migration. By instead allowing diffusion to be small, we reformulate the model as a singular perturbation problem, which can then be analysed using geometric singular perturbation theory. We prove the existence of three types of physically realistic travelling wave solutions in the case of small diffusion. These solutions reduce to the no diffusion solutions in the singular limit as diffusion as is taken to zero. A fourth travelling wave solution is also shown to exist, but that is physically unrealistic as it has a component with negative cell population. The numerical stability, in particular the wavespeed of the travelling wave solutions is also discussed.

A geometric construction of travelling wave solutions to the Keller–Segel model

We study a version of the Keller–Segel model for bacterial chemotaxis, for which exact travelling wave solutions are explicitly known in the zero attractant diffusion limit. Using geometric singular perturbation theory, we construct travelling wave solutions in the small diffusion case that converge to these exact solutions in the singular limit.

A model of workplace safety incorporating worker interactions and simple interventions

Although there was substantial research into the occupational health and safety sector over the past forty years, this generally focused on statistical analyses of data related to costs and/or fatalities and injuries. There is a lack of mathematical modelling of the interactions between workers and the resulting safety dynamics of the workplace. There is also little work investigating the potential impact of different safety intervention programs prior to their implementation. In this article, we present a fundamental, differential equation-based model of workplace safety that treats worker safety habits similarly to an infectious disease in an epidemic model. Analytical results for the model, derived via phase plane and stability analysis, are discussed. The model is coupled with a model of a generic safety strategy aimed at minimising unsafe work habits, to produce an optimal control problem. The optimal control model is solved using the forward-backward sweep numerical scheme implemented in Matlab.

Novel solutions for a model of wound healing angiogenesis

We prove the existence of novel, shock-fronted travelling wave solutions to a model of wound healing angiogenesis studied in Pettet et al (2000 IMA J. Math. App. Med. 17 395–413) assuming two conjectures hold. In the previous work, the authors showed that for certain parameter values, a heteroclinic orbit in the phase plane representing a smooth travelling wave solution exists. However, upon varying one of the parameters, the heteroclinic orbit was destroyed, or rather cut-off, by a wall of singularities in the phase plane. As a result, they concluded that under this parameter regime no travelling wave solutions existed. Using techniques from geometric singular perturbation theory and canard theory, we show that a travelling wave solution actually still exists for this parameter regime. We construct a heteroclinic orbit passing through the wall of singularities via a folded saddle canard point onto a repelling slow manifold. The orbit leaves this manifold via the fast dynamics and lands on the attracting slow manifold, finally connecting to its end state. This new travelling wave is no longer smooth but exhibits a sharp front or shock. Finally, we identify regions in parameter space where we expect that similar solutions exist. Moreover, we discuss the possibility of more exotic solutions.

A semi-alternating direction method for a 2-D fractional FitzHugh–Nagumo monodomain model on an approximate irregular domain

A FitzHugh-Nagumo monodomain model has been used to describe the propagation of the electrical potential in heterogeneous cardiac tissue. In this paper, we consider a two-dimensional fractional FitzHugh-Nagumo monodomain model on an irregular domain. The model consists of a coupled Riesz space fractional nonlinear reaction-diffusion model and an ordinary differential equation, describing the ionic fluxes as a function of the membrane potential. Secondly, we use a decoupling technique and focus on solving the Riesz space fractional nonlinear reaction-diffusion model. A novel spatially second-order accurate semi-implicit alternating direction method (SIADM) for this model on an approximate irregular domain is proposed. Thirdly, stability and convergence of the SIADM are proved. Finally, some numerical examples are given to support our theoretical analysis and these numerical techniques are employed to simulate a two-dimensional fractional Fitzhugh-Nagumo model on both an approximate circular and an approximate irregular domain.

An approximate method for the solution of an influence function foil rolling model

Fleck and Johnson (Int. J. Mech. Sci. 29 (1987) 507) and Fleck et al. (Proc. Inst. Mech. Eng. 206 (1992) 119) have developed foil rolling models which allow for large deformations in the roll profile, including the possibility that the rolls flatten completely. However, these models require computationally expensive iterative solution techniques. A new approach to the approximate solution of the Fleck et al. (1992) Influence Function Model has been developed using both analytic and approximation techniques. The numerical difficulties arising from solving an integral equation in the flattened region have been reduced by applying an Inverse Hilbert Transform to get an analytic expression for the pressure. The method described in this paper is applicable to cases where there is or there is not a flat region.