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Networks & Heterogeneous Media

March 2019 , Volume 14 , Issue 1

Special issue on mathematical methods in systems biology

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Special issue on mathematical methods in systems biology
Monique Chyba and Benedetto Piccoli
2019, 14(1): ⅰ-ⅱ doi: 10.3934/nhm.20191i +[Abstract](1343) +[HTML](312) +[PDF](97.78KB)
Abstract:
Controlled cellular automata
Achilles Beros, Monique Chyba and Oleksandr Markovichenko
2019, 14(1): 1-22 doi: 10.3934/nhm.2019001 +[Abstract](1223) +[HTML](288) +[PDF](1859.84KB)
Abstract:

Cellular Automata have been successfully used to model evolution of complex systems based on simples rules. In this paper we introduce controlled cellular automata to depict the dynamics of systems with controls that can affect their evolution. Using theory from discrete control systems, we derive results for the control of cellular automata in specific cases. The paper is mostly oriented toward two applications: fire spreading; morphogenesis and tumor growth. In both cases, we illustrate the impact of a control on the evolution of the system. For the fire, the control is assumed to be either firelines or firebreaks to prevent spreading or dumping of water, fire retardant and chemicals (foam) on the fire to neutralize it. In the case of cellular growth, the control describes mechanisms used to regulate growth factors and morphogenic events based on the existence of extracellular matrix structures called fractones. The hypothesis is that fractone distribution may coordinate the timing and location of neural cell proliferation, thereby guiding morphogenesis, at several stages of early brain development.

Energy and implicit discretization of the Fokker-Planck and Keller-Segel type equations
Luis Almeida, Federica Bubba, Benoît Perthame and Camille Pouchol
2019, 14(1): 23-41 doi: 10.3934/nhm.2019002 +[Abstract](1026) +[HTML](225) +[PDF](835.15KB)
Abstract:

The parabolic-elliptic Keller-Segel equation with sensitivity saturation, because of its pattern formation ability, is a challenge for numerical simulations. We provide two finite-volume schemes that are shown to preserve, at the discrete level, the fundamental properties of the solutions, namely energy dissipation, steady states, positivity and conservation of total mass. These requirements happen to be critical when it comes to distinguishing between discrete steady states, Turing unstable transient states, numerical artifacts or approximate steady states as obtained by a simple upwind approach.

These schemes are obtained either by following closely the gradient flow structure or by a proper exponential rewriting inspired by the Scharfetter-Gummel discretization. An interesting fact is that upwind is also necessary for all the expected properties to be preserved at the semi-discrete level. These schemes are extended to the fully discrete level and this leads us to tune precisely the terms according to explicit or implicit discretizations. Using some appropriate monotonicity properties (reminiscent of the maximum principle), we prove well-posedness for the scheme as well as all the other requirements. Numerical implementations and simulations illustrate the respective advantages of the three methods we compare.

Optimal stopping for response-guided dosing
Jakob Kotas
2019, 14(1): 43-52 doi: 10.3934/nhm.2019003 +[Abstract](745) +[HTML](249) +[PDF](311.03KB)
Abstract:

In response-guided dosing (RGD), the goal is to make optimal dosing decisions based on the stochastic evolution of a patient's disease condition. Typically, RGD is formulated as a finite-horizon problem with decision-making occurring over a predetermined time frame. In this paper we relax the latter assumption to allow for the possibility of ending treatment early. This could occur due to remission of the disease or a finding of futility in treatment of the disease. Our framework is formulated as a stochastic dynamic program (DP) where a stop/do-not-stop decision is made in discrete sessions, and if stopping is not chosen, an optimal dose is determined for that session. Numerical simulations for rheumatoid arthritis are presented, and monotonicity of the stop/do-not-stop threshold with respect to time is proven.

A network model of immigration: Enclave formation vs. cultural integration
Yao-Li Chuang, Tom Chou and Maria R. D'Orsogna
2019, 14(1): 53-77 doi: 10.3934/nhm.2019004 +[Abstract](1029) +[HTML](291) +[PDF](1728.19KB)
Abstract:

Successfully integrating newcomers into native communities has become a key issue for policy makers, as the growing number of migrants has brought cultural diversity, new skills, but also, societal tensions to receiving countries. We develop an agent-based network model to study interacting "hosts" and "guests" and to identify the conditions under which cooperative/integrated or uncooperative/segregated societies arise. Players are assumed to seek socioeconomic prosperity through game theoretic rules that shift network links, and cultural acceptance through opinion dynamics. We find that the main predictor of integration under given initial conditions is the timescale associated with cultural adjustment relative to social link remodeling, for both guests and hosts. Fast cultural adjustment results in cooperation and the establishment of host-guest connections that are sustained over long times. Conversely, fast social link remodeling leads to the irreversible formation of isolated enclaves, as migrants and natives optimize their socioeconomic gains through in-group connections. We discuss how migrant population sizes and increasing socioeconomic rewards for host-guest interactions, through governmental incentives or by admitting migrants with highly desirable skills, may affect the overall immigrant experience.

A case study of optimal input-output system with sampled-data control: Ding et al. force and fatigue muscular control model
Toufik Bakir, Bernard Bonnard and Jérémy Rouot
2019, 14(1): 79-100 doi: 10.3934/nhm.2019005 +[Abstract](778) +[HTML](205) +[PDF](1800.29KB)
Abstract:

The objective of this article is to make the analysis of the muscular force response to optimize electrical pulses train using Ding et al. force-fatigue model. A geometric analysis of the dynamics is provided and very preliminary results are presented in the frame of optimal control using a simplified input-output model. In parallel, to take into account the physical constraints of the problem, partial state observation and input restrictions, an optimized pulses train is computed with a model predictive control, where a non-linear observer is used to estimate the state-variables.

Stability of metabolic networks via Linear-in-Flux-Expressions
Nathaniel J. Merrill, Zheming An, Sean T. McQuade, Federica Garin, Karim Azer, Ruth E. Abrams and Benedetto Piccoli
2019, 14(1): 101-130 doi: 10.3934/nhm.2019006 +[Abstract](1176) +[HTML](236) +[PDF](596.72KB)
Abstract:

The methodology named LIFE (Linear-in-Flux-Expressions) was developed with the purpose of simulating and analyzing large metabolic systems. With LIFE, the number of model parameters is reduced by accounting for correlations among the parameters of the system. Perturbation analysis on LIFE systems results in less overall variability of the system, leading to results that more closely resemble empirical data. These systems can be associated to graphs, and characteristics of the graph give insight into the dynamics of the system.

This work addresses two main problems: 1. for fixed metabolite levels, find all fluxes for which the metabolite levels are an equilibrium, and 2. for fixed fluxes, find all metabolite levels which are equilibria for the system. We characterize the set of solutions for both problems, and show general results relating stability of systems to the structure of the associated graph. We show that there is a structure of the graph necessary for stable dynamics. Along with these general results, we show how stability analysis from the fields of network flows, compartmental systems, control theory and Markov chains apply to LIFE systems.

On the role of tumor heterogeneity for optimal cancer chemotherapy
Urszula Ledzewicz, Heinz Schättler and Shuo Wang
2019, 14(1): 131-147 doi: 10.3934/nhm.2019007 +[Abstract](1677) +[HTML](387) +[PDF](482.37KB)
Abstract:

We review results about the influence tumor heterogeneity has on optimal chemotherapy protocols (relative to timing, dosing and sequencing of the agents) that can be inferred from mathematical models. If a tumor consists of a homogeneous population of chemotherapeutically sensitive cells, then optimal protocols consist of upfront dosing of cytotoxic agents at maximum tolerated doses (MTD) followed by rest periods. This structure agrees with the MTD paradigm in medical practice where drug holidays limit the overall toxicity. As tumor heterogeneity becomes prevalent and sub-populations with resistant traits emerge, this structure no longer needs to be optimal. Depending on conditions relating to the growth rates of the sub-populations and whether drug resistance is intrinsic or acquired, various mathematical models point to administrations at lower than maximum dose rates as being superior. Such results are mirrored in the medical literature in the emergence of adaptive chemotherapy strategies. If conditions are unfavorable, however, it becomes difficult, if not impossible, to limit a resistant population from eventually becoming dominant. On the other hand, increased heterogeneity of tumor cell populations increases a tumor's immunogenicity and immunotherapies may provide a viable and novel alternative for such cases.

Steady distribution of the incremental model for bacteria proliferation
Pierre Gabriel and Hugo Martin
2019, 14(1): 149-171 doi: 10.3934/nhm.2019008 +[Abstract](1403) +[HTML](244) +[PDF](474.48KB)
Abstract:

We study the mathematical properties of a model of cell division structured by two variables – the size and the size increment – in the case of a linear growth rate and a self-similar fragmentation kernel. We first show that one can construct a solution to the related two dimensional eigenproblem associated to the eigenvalue \begin{document}$ 1 $\end{document} from a solution of a certain one dimensional fixed point problem. Then we prove the existence and uniqueness of this fixed point in the appropriate \begin{document}$ {\rm{L}} ^1 $\end{document} weighted space under general hypotheses on the division rate. Knowing such an eigenfunction proves useful as a first step in studying the long time asymptotic behaviour of the Cauchy problem.

The cardiac bidomain model and homogenization
Erik Grandelius and Kenneth H. Karlsen
2019, 14(1): 173-204 doi: 10.3934/nhm.2019009 +[Abstract](948) +[HTML](200) +[PDF](612.0KB)
Abstract:

We provide a rather simple proof of a homogenization result for the bidomain model of cardiac electrophysiology. Departing from a microscopic cellular model, we apply the theory of two-scale convergence to derive the bidomain model. To allow for some relevant nonlinear membrane models, we make essential use of the boundary unfolding operator. There are several complications preventing the application of standard homogenization results, including the degenerate temporal structure of the bidomain equations and a nonlinear dynamic boundary condition on an oscillating surface.

2018  Impact Factor: 0.871

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