Dynamic Analysis of Genetic Regulatory Networks with Delays

Guan, Zhi‐Hong; Ling, Guang

doi:10.1007/978-3-662-55663-4_14

Cited by 2 publications

(2 citation statements)

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“…2000; Takashima et al. 2011; Guan and Ling 2018), physiological processes such as glucose-insulin regulation (Makroglou et al. 2006, and references therein), cell proliferation and differentiation (Özbay et al.…”

Section: Introductionmentioning

confidence: 99%

“…Readers unfamiliar with IEs and IDEs are referred to Burton (2005) or similar texts. IEs and IDEs have proven to be quite useful models in biology, e.g., they have been used to model chemical kinetics (Roussel 1996), gene expression (Smolen et al 2000;Takashima et al 2011;Guan and Ling 2018), physiological processes such as glucose-insulin regulation (Makroglou et al 2006, and references therein), cell proliferation and differentiation ( Özbay et al 2008;Clapp and Levy 2015;Yates et al 2017), cancer biology and treatment (Piotrowska and Bodnar 2018;Krzyzanski et al 2018;Câmara De Souza et al 2018), pathogen and immune response dynamics (Fenton et al 2006), infectious disease transmission (Anderson and Watson 1980;Lloyd 2001a,b;Feng and Thieme 2000;Wearing et al 2005;Lloyd 2009;Feng et al 2007;Ciaravino et al 2018), and population dynamics (MacDonald 1978a;Blythe et al 1984;Metz and Diekmann 1986;Boese 1989;Nisbet et al 1989;Cushing 1994;Wolkowicz et al 1997;Gyllenberg 2007;Wang and Han 2016;Lin et al 2018;Robertson et al 2018). See also Campbell and Jessop (2009) and the applications reviewed therein.…”

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confidence: 99%

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Generalizations of the ‘Linear Chain Trick’: incorporating more flexible dwell time distributions into mean field ODE models

2019

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In this paper we generalize the Linear Chain Trick (LCT; aka the Gamma Chain Trick) to help provide modelers more flexibility to incorporate appropriate dwell time assumptions into mean field ODEs, and help clarify connections between individual-level stochastic model assumptions and the structure of corresponding mean field ODEs. The LCT is a technique used to construct mean field ODE models from continuous-time stochastic state transition models where the time an individual spends in a given state (i.e., the dwell time) is Erlang distributed (i.e., gamma distributed with integer shape parameter). Despite the LCT’s widespread use, we lack general theory to facilitate the easy application of this technique, especially for complex models. Modelers must therefore choose between constructing ODE models using heuristics with oversimplified dwell time assumptions, using time consuming derivations from first principles, or to instead use non-ODE models (like integro-differential or delay differential equations) which can be cumbersome to derive and analyze. Here, we provide analytical results that enable modelers to more efficiently construct ODE models using the LCT or related extensions. Specifically, we provide (1) novel LCT extensions for various scenarios found in applications, including conditional dwell time distributions; (2) formulations of these LCT extensions that bypass the need to derive ODEs from integral equations; and (3) a novel Generalized Linear Chain Trick (GLCT) framework that extends the LCT to a much broader set of possible dwell time distribution assumptions, including the flexible phase-type distributions which can approximate distributions on and can be fit to data.

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