Equation-Free Multiscale Computational Analysis of Individual-Based Epidemic Dynamics on Networks

The surveillance, analysis and ultimately the efficient long-term prediction and control of epidemic dynamics appear to be one of the major challenges nowadays. Detailed atomistic mathematical models play an important role towards this aim. In this work it is shown how one can exploit the Equation Free approach and optimization methods such as Simulated Annealing to bridge detailed individual-based epidemic simulation with coarse-grained, systems-level, analysis. The methodology provides a systematic approach for analyzing the parametric behavior of complex/ multi-scale epidemic simulators much more efficiently than simply simulating forward in time. It is shown how steady state and (if required) time-dependent computations, stability computations, as well as continuation and numerical bifurcation analysis can be performed in a straightforward manner. The approach is illustrated through a simple individual-based epidemic model deploying on a random regular connected graph. Using the individual-based microscopic simulator as a black box coarse-grained timestepper and with the aid of Simulated Annealing I compute the coarse-grained equilibrium bifurcation diagram and analyze the stability of the stationary states sidestepping the necessity of obtaining explicit closures at the macroscopic level under a pairwise representation perspective

Multiscale Computations on Neural Networks: From the Individual Neuron Interactions to the Macroscopic-Level Analysis

We show how the Equation-Free approach for multi-scale computations can be exploited to systematically study the dynamics of neural interactions on a random regular connected graph under a pairwise representation perspective. Using an individual-based microscopic simulator as a black box coarse-grained timestepper and with the aid of simulated annealing we compute the coarse-grained equilibrium bifurcation diagram and analyze the stability of the stationary states sidestepping the necessity of obtaining explicit closures at the macroscopic level. We also exploit the scheme to perform a rare-events analysis by estimating an effective Fokker-Planck describing the evolving probability density function of the corresponding coarse-grained observables

Approximation of slow and fast dynamics in multiscale dynamical systems by the linearized Relaxation Redistribution Method

In this paper, we introduce a fictitious dynamics for describing the only fast relaxation of a stiff ordinary differential equation (ODE) system towards a stable low-dimensional invariant manifold in the phase-space (slow invariant manifold - SIM). As a result, the demanding problem of constructing SIM of any dimensions is recast into the remarkably simpler task of solving a properly devised ODE system by stiff numerical schemes available in the literature. In the same spirit, a set of equations is elaborated for local construction of the fast subspace, and possible initialization procedures for the above equations are discussed. The implementation to a detailed mechanism for combustion of hydrogen and air has been carried out, while a model with the exact Chapman-Enskog solution of the invariance equation is utilized as a benchmark.Comment: accepted in J. Comp. Phy

Some Aspects of Time-Reversal in Chemical Kinetics

Chemical kinetics govern the dynamics of chemical systems leading towards chemical equilibrium. There are several general properties of the dynamics of chemical reactions such as the existence of disparate time scales and the fact that most time scales are dissipative. This causes a transient relaxation to lower dimensional attracting manifolds in composition space. In this work, we discuss this behavior and investigate how a time reversal effects this behavior. For this, both macroscopic chemical systems as well as microscopic chemical systems (elementary reactions) are considered

An iterative method for the approximation of fibers in slow-fast systems

In this paper we extend a method for iteratively improving slow manifolds so that it also can be used to approximate the fiber directions. The extended method is applied to general finite dimensional real analytic systems where we obtain exponential estimates of the tangent spaces to the fibers. The method is demonstrated on the Michaelis-Menten-Henri model and the Lindemann mechanism. The latter example also serves to demonstrate the method on a slow-fast system in non-standard slow-fast form. Finally, we extend the method further so that it also approximates the curvature of the fibers.Comment: To appear in SIAD

Computation of saddle type slow manifolds using iterative methods

This paper presents an alternative approach for the computation of trajectory segments on slow manifolds of saddle type. This approach is based on iterative methods rather than collocation-type methods. Compared to collocation methods, that require mesh refinements to ensure uniform convergence with respect to $\epsilon$, appropriate estimates are directly attainable using the method of this paper. The method is applied to several examples including: A model for a pair of neurons coupled by reciprocal inhibition with two slow and two fast variables and to the computation of homoclinic connections in the FitzHugh-Nagumo system.Comment: To appear in SIAM Journal of Applied Dynamical System

Development of Reduced-Order Models for Engine Applications

Detailed chemical kinetics is critical for accurate prediction of complex flame behaviors, such as ignition and extinction in engine applications, but difficult to be applied in multi-dimensional flame simulations due to their large sizes. Reduced-order models are needed in such cases to enable high fidelity combustion simulations. This dissertation is focused on developing new model reduction strategies and reduced-order models for engine combustion applications. First, a linearized error propagation (LEP) method for skeletal mechanism reduction is proposed. LEP is based on Jacobian analysis of perfectly stirred reactors (PSR) and can more accurately predict the propagation of small reduction errors compared with the previous methods of directed relation graph (DRG) and DRG with error propagation (DRGEP). Skeletal models generated by using LEP are further validated for auto-ignition and 1-D laminar premixed flames to demonstrate the feasibility of reaction state sampling using only PSR for mechanism reduction. Second, a direct method is developed to accurately and efficiently compute the ignition and extinction turning points of PSR by solving a local optimization problem formulated based on analytic Jacobian. It is shown that the direct method features significantly better accuracy and efficiency compared with the continuation methods that march along the S-curves. Third, reduced and skeletal mechanisms for gasoline surrogates with and without ethanol are developed based on a 1389-species detailed mechanism developed by the Lawrence Livermore National Laboratory (LLNL). The skeletal reduction was performed with DRG, sensitivity analysis, isomer lumping, and the time-scale based reduction is based on linearized quasi-steady-state approximations. The skeletal and reduced mechanisms are extensively validated against the detailed mechanism and available experimental data for ignition delay time and flame speed. The skeletal mechanism is employed in cooperative fuel research engine simulations and the results agree well with experimental data. Lastly, skeletal mechanisms are generated for three gasoline/bio-blend-stock surrogates respectively based on a 2878-species detailed LLNL mechanism for engine simulations. An upgraded solver combining analytical Jacobian and sparse matrix techniques is employed to accelerate the reduction process, such that the reduction time becomes linearly proportional to the mechanism size and a speedup factor of approximately 100 is achieved

Lumping methods for model reduction

ix, 93 leaves ; 29 cmModelling a chemical or biochemical system involves the use of differential equations which often include both fast and slow time scales. After the decay of transients, the behaviour of these differential equations usually rests on a low-dimensional surface in the phase space which is called the slow invariant manifold (SIM) of the flow. A model has been effectively reduced if such a manifold can be obtained. In this study, we develop a method that introduces the lumping process (a technique whereby chemical species are grouped into pseudo-reagents (lumps) to simplify modelling) into the invariant equation method, and this new method effectively reduces complex models as well as preserving the structure and underlying mechanism of the original system. We also apply these methods to simple models of metabolic pathways. This method of model reduction would be of great importance for industrial application

An efficient iterative algorithm for the approximation of the fast and slow dynamics of stiff systems

The relation between the iterative algorithms based on the computational singular perturbation (CSP) and the invariance equation (IE) methods is examined. The success of the two methods is based on the appearance of fast and slow time scales in the dynamics of stiff systems. Both methods can identify the low-dimensional surface in the phase space (slow invariant manifold, SIM), where the state vector is attracted under the action of fast dynamics. It is shown that this equivalence of the two methods can be expressed by simple algebraic relations. CSP can also construct the simplified non-stiff system that models the slow dynamics of the state vector on the SIM. An extended version of IE is presented which can also perform this task. This new IE version is shown to be exactly similar to a modified version of CSP, which results in a very efficient algorithm, especially in cases where the SIM dimension is small, so that significant model simplifications are possible. (c) 2005 Elsevier Inc. All rights reserved