Reaction networks are systems in which the populations of a finite number of species evolve through predefined interactions. Such networks are found as modeling tools in many disciplines, spanning biochemistry, epidemiology, pharmacology, ecology and social networks . It is now well-established that, for small population sizes, stochastic models for  reaction networks are necessary to capture randomness in the interactions. The tools for analyzing them , however, still lag far behind their deterministic counterparts. In this paper, we bridge this gap by developing a constructive framework for examining the long-term behavior and stability properties of the reaction dynamics in a stochastic setting. In particular, we address the problems of determining ergodicity of the reaction dynamics, which is analogous to having a globally attracting fixed point for deterministic dynamics , and determining moment bounds for the underlying stochastic process . Theoretical and computational solutions for these problems are obtained by utilizing a blend of ideas and techniques from probability theory, linear algebra , polynomial analysis and optimization theory. We demonstrate that stability properties of a wide class of  networks can be assessed from theoretical results that can be recast as efficient and scalable linear programs, well-known for their tractability. It is notably shown that the computational complexity is often linear in the number of species , but worst-case quadratic . We illustrate the validity, the efficiency and the universality of our results on several reaction networks arising in fields such as biochemistry, epidemiology and ecology  .
Reaction networks are systems in which the populations of a finite number of species evolve through predefined interactions. Such networks are found as modeling tools in many biological disciplines such as biochemistry, ecology, epidemiology, immunology, systems biology and synthetic biology . It is now well-established that, for small population sizes, stochastic models for biochemical reaction networks are necessary to capture randomness in the interactions. The tools for analyzing such models , however, still lag far behind their deterministic counterparts. In this paper, we bridge this gap by developing a constructive framework for examining the long-term behavior and stability properties of the reaction dynamics in a stochastic setting. In particular, we address the problems of determining ergodicity of the reaction dynamics, which is analogous to having a globally attracting fixed point for deterministic dynamics . We also examine when the statistical moments of the underlying process remain bounded with time and when they converge to their steady state values. The framework we develop relies on a blend of ideas from probability theory, linear algebra and optimization theory. We demonstrate that stability properties of a wide class of biological networks can be assessed from theoretical results that can be recast as efficient and scalable linear programs, well-known for their tractability. It is notably shown that the computational complexity is often linear in the number of species . We illustrate the validity, the efficiency and the universality of our results on several reaction networks arising in biochemistry, systems biology, epidemiology and ecology . The biological implications of the results as well as an example of a non-ergodic biological network are also discussed .
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disciplines, spanning biochemistry, epidemiology, pharmacology, ecology and social networks
biological disciplines such as biochemistry, ecology, epidemiology, immunology, systems biology and synthetic biology
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biochemical
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them
such models
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, and determining moment bounds for the underlying stochastic process . Theoretical and computational solutions for these problems are obtained by utilizing
. We also examine when the statistical moments of the underlying process remain bounded with time and when they converge to their steady state values. The framework we develop relies on
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and techniques
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, polynomial analysis
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biological
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, but worst-case quadratic
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fields such as biochemistry,
biochemistry, systems biology,
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. The biological implications of the results as well as an example of a non-ergodic biological network are also discussed