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Generalizations of the 'Linear Chain Trick': Incorporating more flexible dwell time distributions into mean field ODE models

Paul J. Hurtado, Adam S. Kirosingh

arXiv:1808.07571v1q-bio.PEmath.DS

TL;DR

Mean-field ODE models often offer limited choices for dwell-time distributions, while more flexible delay models can be harder to derive and analyze. The paper formalizes and extends the Linear Chain Trick, with analytical results for constructing ODEs from stochastic assumptions and accommodating broader dwell-time settings.

  • Problem

    ODE dwell-time options are largely tied to first-event distributions of Poisson processes, while more flexible integral or delay models can be harder to derive and analyze.

    Source: §1 · pp. 3, 5

  • Method

    The paper formalizes the LCT and develops direct ODE construction from stochastic assumptions, extending the approach to time-varying rates and conditional dwell-time relationships.

    Source: §1 · p. 5, §3 · p. 9, §3.1 · p. 9

  • Results

    The paper provides analytical results for constructing mean-field ODEs from stochastic assumptions and extending the LCT to more flexible dwell-time cases.

    Source: §1 · p. 5, §3 · p. 9

  • Takeaways & Limitations

    The framework clarifies how individual-level stochastic assumptions are reflected in mean-field ODE structure while allowing more flexible dwell-time assumptions.

    Source: §1 · p. 5

    The stated correspondence between an ODE transition rate and exponential dwell time depends on the Poisson-process stochastic model assumed for that transition.

    Source: §1 · p. 3

Abstract

from arXiv · show

Mathematical modelers have long known of a "rule of thumb" referred to as the Linear Chain Trick (LCT; aka the Gamma Chain Trick): 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. This has forced modelers to 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 equations 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 1) provide novel extensions of the LCT to various scenarios found in applications; 2) provide formulations of the LCT and it's extensions that bypass the need to derive ODEs from integral or stochastic model equations; and 3) introduce a novel Generalized Linear Chain Trick (GLCT) framework that extends the LCT to a much broader family of distributions, including the flexible phase-type distributions which can approximate distributions on $\mathbb{R}^+$ and be fit to data. These results give modelers more flexibility to incorporate appropriate dwell time assumptions into mean field ODEs, including conditional dwell time distributions, and these results help clarify connections between individual-level stochastic model assumptions and the structure of corresponding mean field ODEs.

1 Introduction

Mean-field ODEs are convenient, but their standard dwell-time assumptions can be too restrictive, while more flexible delay models are harder to derive, analyze, and simulate. The paper formalizes the Linear Chain Trick and extends it to broader dwell-time settings, including conditional and time-varying cases.

1 Introduction

  • ODE dwell-time options are limited largely to first-event times of Poisson processes, while arbitrary dwell-time distributions can require more challenging integral or delay equations.The paper notes that modelers may instead assume simplistic Poisson-based dwell times or undertake laborious derivations.Source: §1 · pp. 3–5
  • The familiar Linear Chain Trick reduces integral or integro-differential equations with Erlang dwell times, but complex models can exceed existing rules of thumb.Deriving ODEs from first principles may require first deriving the integral equations and stochastic model.Source: §1 · pp. 4–5
  • The paper develops a theoretical foundation for constructing mean-field ODEs directly from stochastic assumptions and for incorporating more flexible and conditional dwell-time distributions.It also aims to clarify how stochastic assumptions are reflected in ODE structure.Source: §1 · p. 5
  • The Generalized Linear Chain Trick extends construction to phase-type dwell-time distributions, which can be fitted to data or used to approximate other distributions.Phase-type distributions are described as hitting-time distributions for certain continuous-time Markov chains.Source: §1 · p. 5

1.1 Intuitive description of the Linear Chain Trick

  • For an Erlang dwell time, the LCT partitions state X into k exponential-dwell sub-states, each corresponding to the next event awaited in a Poisson process.The sub-state amounts follow linear or nearly linear ODEs and sum to the amount in X.Source: §1.1 · p. 6
  • The Poisson-process perspective extends the LCT to conditional sub-state transitions and to time-varying dwell-time rates.These extensions preserve the use of sub-state structure while broadening the dwell-time assumptions.Source: §1.1 · p. 6

2 Model Framework

This section frames the LCT as a way to replace mean-field integral equations with equivalent first-order ODE systems. It sets out the event-time and distribution notation used to describe constant- and time-varying dwell times.

2 Model Framework

  • The framework focuses on focal states in stochastic state-transition models whose mean-field integral equations can be replaced by equivalent first-order ODE systems.The construction is intended to proceed directly from stochastic model assumptions.Source: §2 · pp. 6–7

2.1 Distributions & notation

  • The notation covers event-time distributions under nonhomogeneous Poisson processes and survival functions, as well as categorical allocation among recipient states.The recipient-state probabilities sum to one.Source: §2.1 · pp. 7–8
  • An Erlang distribution is a gamma distribution with integer shape k, equivalently the sum of k independent exponential waiting times with rate r.This event-time interpretation underlies the LCT.Source: §2.1 · p. 7
  • To match a target gamma mean and variance, the rate and shape can be parameterized from those moments, but an Erlang distribution requires rounding the shape to an integer.That rounding adjusts the assumed variance up or down.Source: §2.1 · p. 7

3 Results

The results give a route from stochastic state-transition assumptions to mean-field ODEs without first deriving equivalent integral equations. They establish the Poisson-rate correspondence, a time-shifted memorylessness property, and LCT formulations for Erlang dwell times and recipient states.

3 Results

  • The results provide a way to construct mean-field ODEs directly from stochastic assumptions without first deriving equivalent integral equations.The focus is on one or more states within a potentially larger transition model.Source: §3 · p. 9

3.1.1 Per capita transition rates in ODEs reflect underlying Poisson process rates

  • For a state with time-varying Poisson exit rate r(t) and influx I(t), the ODE’s per-capita loss rate corresponds to the underlying process intensity.This connects the stochastic exit assumption to the mean-field ODE representation.Source: §3.1.1 · pp. 9–10

3.1.2 Weak memoryless property of Poisson process 1st event time distributions

  • The weak memorylessness property says that, conditional on no event by time s, the remaining first-event time follows the same Poisson-process form shifted to rate r(t+s).For constant rate, this recovers exponential memorylessness.Source: §3.1.2 · pp. 10–11

3.2 Simple case of the LCT

3.3 Standard LCT

  • The standard LCT retains the Erlang delay from X to Y while allowing arbitrary survival-based dwell times in Y and additional inflows to both states.The theorem assumes nonnegative, integrable input rates and nonnegative initial conditions.Source: §3.3 · p. 13

X IX(t)

The Standard LCT constructs mean field ODEs for Erlang dwell times, including examples that vary the dwell-time assumptions across states. The results also allow ODE construction directly from model assumptions, without first deriving and differentiating integral equations.

X IX(t)

  • Figure 2 applies the LCT to Erlang dwell times in X and shows alternatives for Y: a per-capita loss rate µ(t) or an Erlang delay with shape κ.The figure also identifies the inflows IX(t) and IY(t).Source: Figure 2 · p. 14
  • In the particle example, individuals move from W to X after an exponential delay, remain in X for an Erlang(r, k) delay, then enter Y and leave after an exponential delay.The resulting system is stated as a mean field model.Source: X IX(t) · p. 15
  • For the SIR example, an Erlang infected-state dwell time with mean 1/γ yields staged infected compartments, and shape k = 1 recovers the original exponential-dwell-time equations.The example assumes an integer-valued shape parameter.Source: X IX(t) · p. 16
  • Theorem 2 constructs mean field ODEs directly from stochastic assumptions, avoiding the intermediate derivation and differentiation of integral equations.The paper describes this route as more straightforward for relaxing an implicit exponential dwell-time assumption.Source: X IX(t) · p. 16

3.4 Extended LCT for Poisson process kth event time distributed dwell times

The Extended LCT generalizes Erlang-based results to dwell times given by kth event times of a Poisson process with a time-varying rate. It yields an equivalent mean field ODE system for the resulting state transitions.

3.4 Extended LCT for Poisson process kth event time distributed dwell times

  • The Extended LCT covers dwell times given by the kth event time of a Poisson process with rate r(t), generalizing the Standard LCT to time-varying rates.The paper identifies this as a time-dependent or state-dependent rate extension.Source: §3.4 · pp. 17–18
  • Theorem 3 expresses the corresponding mean field integral equations as an equivalent ODE chain, with stage equations for xj using rate r(t).The y equation may also be further reduced to ODEs, for example using Corollary 1.Source: §3.4 · p. 18
  • The kth-event extension generalizes Erlang-based results and opens the analysis to more complex assumptions about the underlying stochastic state-transition model.The paper contrasts the Standard LCT and the extension through their corresponding equations.Source: §3.4 · pp. 17, 19

3.5 Transitions to multiple states

This section develops mean field ODE formulations for transitions to multiple recipient states, addressing how dwell-time assumptions and particle assignment shape the resulting models. It extends the Linear Chain Trick to proportional outputs, heterogeneous dwell-time mixtures, and competing Poisson processes, including an equivalence under first-event dwell times.

3.5.1 Transition to multiple states independent of the X dwell time distribution

  • Theorem 4 extends the Linear Chain Trick to particles assigned among multiple recipient states upon leaving X, independently of their dwell time in X.The recipient probabilities may vary over time; the resulting integral equations can be reduced to ODEs.Source: §3.5.1 · pp. 20–21

3.5.2 Transition from sub-states of X with differing dwell time distributions and differing output distributions across states Yj

  • Theorem 5 represents heterogeneous dwell times and substate-dependent recipient probabilities by partitioning X into sub-states with separate Poisson-process event-time chains.Particles are assigned to a sub-state upon entering X, and each sub-state has its own dwell-time distribution and output probabilities.Source: §3.5.2 · p. 22

3.5.3 Extended LCT for dwell times given by finite mixtures of Poisson process event time distributions

  • Corollary 2 gives ODEs for a finite mixture of independent Poisson-process event-time dwell distributions in X when particles enter a single recipient state.The corresponding model uses a separate linear chain for each mixture component.Source: §3.5.3 · pp. 24–25

3.5.4 Transition to multiple states following ‘‘competing” Poisson processes

  • For competing Poisson-process dwell times, Lemma 3 generalizes the sub-state recursion, and Theorem 6 extends the LCT to transitions whose recipient probabilities depend on which process finishes first.The dwell time is the minimum of independent event times; X is partitioned by progress along each process before one reaches its target event.Source: §3.5.4 · pp. 25–27
  • For competing Poisson processes, Theorem 6 partitions X into substates tracking awaited events; their shared first-event waiting-time structure generalizes to more processes and underpins the GLCT.In the two-process example, each substate’s remaining wait is governed by a first-event rate r(t) = r1(t) + r2(t).Source: §3.5.4 · pp. 29–30, Figure 7 · p. 29

3.5.5 Mean field equivalence of proportional outputs & competing Poisson processes

  • Theorem 7 establishes that competing Poisson first-event dwell times yield the same mean field model as proportional outputs, with total rate r(t) = sum_i r_i(t).The equivalence holds when the probability of entering each output state accounts for which competing event occurred first.Source: §3.5.5 · pp. 30–31
  • This equivalence can simplify mean field ODEs while retaining Poisson first-event time distributions as building blocks for LCT generalizations.The section identifies this flexibility as important to the GLCT.Source: §3.5.5 · pp. 30–31

3.6 Modeling intermediate state transitions: Reset the clock, or not?

Intermediate-state transitions can either reset the overall dwell-time clock or preserve it, and the corresponding LCT extensions produce different mean field ODE structures. The distinction matters in applications such as quarantine models, where entering an intermediate state can affect disease transmission while the infectious-period distribution may or may not change.

3.6 Modeling intermediate state transitions: Reset the clock, or not?

  • The two clock assumptions matter in applications such as Ebola hospitalization models, where epidemic output depends strongly on whether hospitalization changes the infected-state dwell-time distribution.The extensions cover multiple intermediate and recipient states reached through competing Poisson processes.Source: §3.6 · p. 32

3.6.1 Intermediate states that reset dwell time distributions

  • When intermediate-state dwell time is independent of time already spent in X0, entering XI resets the clock and alters the overall dwell time in X.Theorem 8 extends the LCT to competing Poisson-process exits from X0 and then models each intermediate state with its own dwell-time distribution.Source: §3.6.1 · p. 32, Figure 9 · p. 33

3.6.2 Intermediate states that preserve dwell time distributions

  • To preserve X’s overall dwell-time distribution, condition the XI dwell time on how many of the k0 events have occurred in X0; the remaining time follows a (k0−k)th event-time distribution at rate r0(t).The weak memorylessness property supports this construction, formalized in Theorem 9.Source: §3.6.2 · p. 35

3.7 Generalized Linear Chain Trick (GLCT)

The Generalized Linear Chain Trick (GLCT) provides a framework for constructing mean field ODEs from stochastic state-transition models with a broad range of dwell-time assumptions. With constant rates and transition probabilities, it includes phase-type dwell times, and it can also represent examples such as the maximum of independent Erlang times.

3.7 Generalized Linear Chain Trick (GLCT)

  • The GLCT treats earlier LCT extensions as special cases and constructs mean field ODEs for models satisfying its stated sub-state, inflow, transition, and recipient-state assumptions.Its transition probabilities and rates may vary with time.Source: §3.7 · pp. 37–38
  • With constant rates and transition probabilities, the GLCT yields dwell times given by CTMC hitting-time distributions, including continuous phase-type distributions.The listed examples include hypoexponential, hyper-exponential, hyper-Erlang, and generalized Coxian distributions.Source: §3.7 · pp. 38–39
  • For a dwell time equal to the maximum of two independent Erlang(ri, 2) variables, the GLCT partitions X into sub-states indexed by progress under both Poisson processes.The resulting sub-state rates vary: some are r1+r2, while others are r1 or r2.Source: Figure 11 · p. 39, §3.7 · p. 40

4 Discussion

The paper extends the Linear Chain Trick and introduces the GLCT to construct mean field ODEs for broader dwell-time distributions and sub-state configurations. These methods support more flexible distributional assumptions, while raising implementation and interpretation considerations.

4 Discussion

  • The GLCT and related results construct mean field ODEs for broad dwell-time distributions and sub-state configurations, including conditional dwell times.They include phase-type distributions and time-varying analogues, and can be applied directly from stochastic assumptions without first deriving integral equations.Source: §4 · p. 41
  • The GLCT also supplies a framework for scenarios beyond the stated results, illustrated by a dwell time equal to the maximum of multiple Erlang distributions.Source: §4 · p. 41
  • Phase-type distributions can approximate dwell-time distributions or be fitted to data, supporting their incorporation into mean field ODEs.The family includes mixtures, minima and maxima of Erlang distributions, hypoexponential distributions, and generalized Coxian distributions.Source: §4 · p. 42
  • More sub-states can create computational and analytical challenges, and need not correspond to real sub-state structure in the modeled system.Some results assume particles initially enter only the first sub-state, though the GLCT can accommodate modified initial conditions.Source: §4 · p. 42
  • The framework can support comparisons of detailed and simplified dwell-time assumptions to assess whether simpler models are warranted.The authors note that some applications may be better served by analyzing stochastic dynamics directly rather than using mean field models.Source: §4 · p. 42

Appendix A Deterministic Models as Mean Field Equations

The appendix explains mean field models as deterministic descriptions of expected state counts arising from stochastic transitions. It outlines how expected counts and cohort survival yield continuous-time equations.

Appendix A Deterministic Models as Mean Field Equations

  • A mean field model averages stochastic state transitions to describe deterministic dynamics of the system’s mean state.Source: Appendix A · p. 43
  • The derivation targets the expected number of particles in each state, obtained from integer-valued stochastic counts.The continuous-time model is formed by considering arbitrary time steps and taking their limit to zero.Source: Appendix A · p. 43
  • An integral equation for a state sums incoming cohorts weighted by the proportion that survives there until the evaluation time.The cohort contribution uses its instantaneous input rate and the dwell-time survival function.Source: Appendix A · p. 43

A.1 Derivation of mean field equations in Example 3.2

Example 3.2 derives mean field equations from a discrete-time stochastic model by taking the time-step limit. The derivation uses cohort counts and dwell-time survival to obtain integral equations for states with non-exponential dwell times.

A.1 Derivation of mean field equations in Example 3.2

  • The derivation starts from a discrete-time approximation and takes the time-step limit to obtain the mean field integral equations and ODEs.The linear ODEs for W and Z follow from expected transitions over a short interval and division by ∆t before taking ∆t→0.Source: §A.1 · p. 44
  • The example assumes transitions W→X after an exponential duration with rate a, X→Y after an Erlang(r, k) duration, and Y→Z after an exponential duration with rate µ.Source: §A.1 · p. 44
  • X(t) is computed by summing entering cohorts across M equal-width intervals, weighted by the probability that each cohort remains in X until t.The bin width is ∆t=t/M, and the survival probability is given by the Erlang dwell-time distribution.Source: §A.1 · pp. 44–45, Figure 12 · p. 44
  • The integral-equation derivation treats each interval’s incoming particles as a cohort and accumulates those remaining in the state at time t.Source: §A.1 · p. 45
  • For Y, each cohort contribution accounts for entering X, transitioning from X to Y in a later interval, and persisting in Y until t.The derivation represents these contributions as compound distributions over cohorts and intervals.Source: §A.1 · p. 45

Appendix B Erlang mixture approximation of Gamma(α, β)

The appendix approximates a non-integer-shape gamma distribution with a mixture of two Erlang distributions matched to its mean and variance. Numerical comparisons suggest the approximation is good for shape values above roughly 3 to 5.

Appendix B Erlang mixture approximation of Gamma(α, β)

  • Approximating empirical or named distributions with Erlang mixtures or other phase-type distributions is an established approach cited by the authors.Source: Appendix B · p. 46
  • The two-Erlang mixture matches the target gamma distribution’s mean and variance by using shapes rounded down and up from β.The Erlang components share the target mean; their rates and mixture proportion are selected to match the variance.Source: Appendix B · pp. 46–47
  • For β values larger than roughly 3 to 5, the mixture is suggested to approximate gamma(α, β) very well, depending to a lesser extent on α.Source: Appendix B · p. 47
  • The mixing probabilities could instead be selected using probability-distance or information-theoretic criteria.Examples include the L∞ distance between CDFs and KL or Jensen-Shannon divergence.Source: Appendix B · p. 47
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