First detection of threshold crossing events under intermittent sensing

TL;DR

This paper investigates the distribution of the first detection time in stochastic processes monitored intermittently, providing analytical results, validating with simulations, and demonstrating applications in various models.

Contribution

It introduces a framework linking first detection time to first passage time in intermittent sensing scenarios, with analytical solutions and practical inference methods.

Findings

Analytical expression for first detection time distribution.

Excellent agreement between theory and simulations.

Applications demonstrated in epidemiological and birth-death models.

Abstract

The time of the first occurrence of a threshold crossing event in a stochastic process, known as the first passage time, is of interest in many areas of sciences and engineering. Conventionally, there is an implicit assumption that the notional 'sensor' monitoring the threshold crossing event is always active. In many realistic scenarios, the sensor monitoring the stochastic process works intermittently. Then, the relevant quantity of interest is the $\textit{first detection time}$, which denotes the time when the sensor detects the threshold crossing event for the first time. In this work, a birth-death process monitored by a random intermittent sensor is studied, for which the first detection time distribution is obtained. In general, it is shown that the first detection time is related to, and is obtainable from, the first passage time distribution. Our analytical results display an excellent agreement with simulations. Further, this framework is demonstrated in several applications -- the SIS compartmental and logistic models, and birth-death processes with resetting. Finally, we solve the practically relevant problem of inferring the first passage time distribution from the first detection time.