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A two-state recurrent stochastic model with time-dependent transition rates.

S S Kenley¹, C L Chiang, R J Brand

¹Syntex Research, Palo Alto, California 94303.

Mathematical Biosciences

|October 1, 1992

Summary

This study generalizes a two-state recurrent stochastic model to include time-dependent transition rates. New formulas are derived for state probabilities and expectations under these dynamic conditions.

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Area of Science:

Stochastic modeling
Mathematical biology
Statistical physics

Background:

Recurrent stochastic models are crucial for understanding dynamic systems.
Existing models often assume time-independent transition rates, limiting their applicability.
Real-world processes frequently exhibit rates that vary with external calendar time.

Purpose of the Study:

To generalize the two-state recurrent stochastic model by incorporating time-dependent transition rates.
To provide a mathematical framework for analyzing systems where rates change over time.
To derive fundamental formulas for time-varying stochastic processes.

Main Methods:

Extension of the two-state recurrent stochastic model.
Mathematical derivation of formulas for time-dependent transition rates.

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Analysis of arbitrary functions of external time.

Main Results:

Formulas for state transition probabilities with time-dependent rates.
Derivation of the proportion of individuals in a specific state at time t.
Formulas for the distribution function and expectation of individuals in a state over time.

Conclusions:

The generalized model accurately captures systems with time-varying dynamics.
The derived formulas provide essential tools for analyzing complex stochastic processes.
This work enhances the applicability of recurrent stochastic models to real-world phenomena.