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Simulating non-Markovian stochastic processes.

Marian Boguñá1, Luis F Lafuerza2, Raúl Toral3

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This summary is machine-generated.

We developed a generalized Gillespie algorithm for simulating non-Markovian discrete stochastic processes. This new method accurately models complex systems, revealing significant impacts of non-exponential event times on epidemic spreading and biochemical reactions.

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Area of Science:

  • Computational Biology
  • Complex Systems Modeling
  • Stochastic Processes

Background:

  • Stochastic processes are fundamental to modeling complex systems in biology and epidemiology.
  • Existing Gillespie algorithms are limited to Markovian processes, where event rates are independent of time since the last event.
  • Non-Markovian processes, with time-dependent event rates, are crucial for accurately representing many real-world phenomena like epidemic dynamics and biochemical reactions with delays.

Purpose of the Study:

  • To introduce a general and statistically correct framework for simulating non-Markovian discrete stochastic processes.
  • To develop an efficient and practical algorithm analogous to the Gillespie algorithm for non-Markovian systems.
  • To investigate the impact of non-exponential interevent time distributions on epidemic spreading and biochemical systems.

Main Methods:

  • Developed a generalized Gillespie stochastic simulation methodology for non-Markovian processes.
  • Provided an exact analytical solution for these processes.
  • Applied the algorithm to the susceptible-infected-susceptible (SIS) model of epidemic spreading and a biochemical reaction system with time delays.

Main Results:

  • Demonstrated that subtle differences in modeling non-Markovian processes can drastically alter the global behavior of complex systems.
  • Unveiled significant effects of non-exponential interevent time distributions on epidemic spreading dynamics.
  • Showcased the algorithm's effectiveness in simulating biochemical reactions with time delays, outperforming existing methods in generality and ease of implementation.

Conclusions:

  • The generalized Gillespie algorithm offers a powerful and versatile tool for simulating non-Markovian stochastic systems.
  • Accurate modeling of non-Markovian dynamics is essential for understanding and predicting the behavior of complex systems.
  • This methodology has broad applicability in fields ranging from epidemiology to systems biology.