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Edge-Based Compartmental Modeling for Infectious Disease Spread Part I: An Overview

Joel C. Miller, Anja C. Slim, Erik M. Volz

arXiv:1106.6320v1q-bio.PEphysics.bio-ph

TL;DR

The paper addresses the mass action SIR model’s assumptions of homogeneous contact rates and fleeting contacts. It develops edge-based compartmental ODE models using neighbor-transmission probabilities, finding a simple framework that incorporates social heterogeneity and arbitrary contact duration across network settings.

  • Problem

    Mass action SIR assumes homogeneous contact rates and negligible contact duration, while existing tractable approaches often discard detailed contact information by reducing it to averages.

  • Method

    The paper derives edge-based differential-equation models that calculate population states through the probability that a randomly chosen test node’s neighbors have transmitted infection.

  • Results

    The approach develops conceptually and mathematically simple models incorporating social heterogeneity and arbitrary contact duration, while forming a broad family that includes known models as special cases.

  • Takeaways & Limitations

    The framework enables analytical study of contact-duration effects, overlapping contacts, and other behaviors previously inaccessible to analytic treatment.

  • Takeaways & Limitations

    The approach assumes neighbors’ infections are independent and therefore cannot be directly applied to SIS diseases or populations with substantial clustering.

Abstract

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The primary tool for predicting infectious disease spread and intervention effectiveness is the mass action Susceptible-Infected-Recovered model of Kermack and McKendrick. Its usefulness derives largely from its conceptual and mathematical simplicity; however, it incorrectly assumes all individuals have the same contact rate and contacts are fleeting. This paper is the first of three investigating edge-based compartmental modeling, a technique eliminating these assumptions. In this paper, we derive simple ordinary differential equation models capturing social heterogeneity (heterogeneous contact rates) while explicitly considering the impact of contact duration. We introduce a graphical interpretation allowing for easy derivation and communication of the model. This paper focuses on the technique and how to apply it in different contexts. The companion papers investigate choosing the appropriate level of complexity for a model and how to apply edge-based compartmental modeling to populations with various sub-structures.

1 Introduction

The paper addresses limitations of the mass action SIR model by developing edge-based models that represent heterogeneous contact rates and arbitrary contact duration while retaining mathematical simplicity.

  • Motivation: Mass action SIR is widely used because it is conceptually and mathematically simple, but it assumes homogeneous contact rates and negligible contact duration.These assumptions can make predictions differ from reality, particularly because early infections tend to have more contacts.
  • Motivation: Existing approaches often reduce measured contact patterns to average rates because no coherent mathematical framework captures social heterogeneity and contact duration together.This discards potentially relevant detail collected in real-world contact studies.
  • Contribution: The paper develops conceptually and mathematically simple models that incorporate social heterogeneity and arbitrary contact duration.The approach provides a unifying framework for existing models and expands the range of models that can be analyzed.
  • Method: Instead of directly tracking population proportions, the method asks whether a randomly chosen test node is susceptible, infected, or recovered.Because the node is randomly chosen, these probabilities equal the corresponding population proportions; knowing S(t) determines I(t) and R(t) through the stated equations.
  • Method: The method focuses on the probability that a random neighbor is infected, accounting for the neighbor’s contact number, changing state, and transmission history.A random neighbor may have more contacts than a random node, so its infection probability is not given by the infected fraction alone.
  • Scope: The paper develops exact differential equations for large populations on static and dynamic actual-degree and expected-degree networks.Actual-degree networks use Configuration Model structures, while expected-degree networks use Mixed Poisson, or Chung-Lu, structures.

2 Configuration Model Epidemics

For Configuration Model networks, the paper derives an edge-based ODE system by tracking whether neighbors have transmitted infection, then compares its predictions with simulations across degree distributions.

  • Configuration Model construction: A Configuration Model network assigns each node a random degree, creates corresponding stubs, and pairs stubs randomly into edges.A randomly selected neighbor is degree-biased because the chance of connection is proportional to the neighbor’s degree.
  • Edge-based variables: The neighbor state is partitioned into φS, φI, and φR for susceptible, infected, and recovered neighbors that have not transmitted, plus 1 −θ for transmission having occurred.The flow diagram encodes infection and recovery among neighbor states and transmission from φI to 1 −θ.
  • Edge-based variables: The model defines θ as the probability that a randomly chosen neighbor has not transmitted infection to a test node.Susceptibility is then calculated from the probability that none of the test node’s neighbors has transmitted.
  • ODE system: For a degree distribution with generating function ψ, the Configuration Model equations use S = ψ(θ), I = 1 −S −R, and ˙R = γI.The derivation obtains φI from θ, φS, and φR, with transmission rate β and recovery rate γ governing the flows.
  • Model properties: The resulting system captures substantially more population structure than mass action with only marginally more complexity.It avoids the O(M) or O(M^2) ODE systems required by some alternative approaches, where M is the maximum degree.
  • Example: Across four Configuration Model networks with 5 × 10^5 nodes and average degree ⟨K⟩ = 5, model predictions match simulations while the average-degree mass action model fails.Different degree distributions significantly alter spread: greater heterogeneity produces an earlier peak but generally a smaller epidemic.

3 Actual Degree Models

The actual-degree models extend configuration-model epidemics from static contacts to rapidly changing, edge-swapping, and dormant-contact networks while retaining compact ODE formulations. Flow diagrams track infection, recovery, and edge dynamics, and simulations support the model predictions.

  • Mean Field Social Heterogeneity: The MFSH model assumes each stub continually chooses a new neighbor, making neighbor status changes arise from changing contacts rather than status transitions.
  • Mean Field Social Heterogeneity: For durationless contacts, the model tracks θ and stub-status probabilities πS, πI, and πR, with S = ψ(θ), I = 1 − S − R, and R(∞) = 1 − ψ(θ(∞)).
  • Mean Field Social Heterogeneity: The MFSH model was tested on a population with degrees 1, 5, and 25, with equal numbers of stubs in each degree class; theory was compared with a 5 × 10^5-node simulation.
  • Dynamic Fixed-Degree: The DFD model interpolates between static configuration-model and MFSH limits by breaking edges at rate η and immediately swapping the freed stubs.
  • Dynamic Fixed-Degree: Because φS cannot be calculated explicitly in the DFD model, the equations determine the φS-to-φI infection flux using ODEs and the conditional infection rate.
  • R0 and final size: The actual-degree framework retains mathematical simplicity, but no simple final-size expression is available and numerical ODE solution is required.
  • Dynamic Fixed-Degree: The DFD equations accurately predicted spread using β = 5/4, γ = 1, and η = 1/2, compared against the average of 102 simulated epidemics.
  • Dormant Contacts: The Dormant Contact model adds a dormant stub phase after edge breaking, representing serial monogamy and reducing to the paper’s other models under appropriate limits.

4 Expected Degree Models

The expected-degree models apply edge-based compartmental reasoning to Mixed Poisson and dynamic mean-field networks, where contacts are formed probabilistically from expected degrees. Their equations preserve a common structure, match simulations, and connect equivalent formulations.

  • Expected-degree networks: Expected-degree networks assign each node a possibly noninteger expected degree κ, with edge probabilities proportional to the expected degrees of the two nodes.
  • Mixed Poisson: The Mixed Poisson model uses a test node and Θ, the probability an edge has not transmitted infection, to derive S = Ψ(Θ), I = 1 − S − R, and Ṙ = γI.
  • Mixed Poisson: The Mixed Poisson and configuration-model equations are nearly identical, with Ψ and Θ replacing ψ and θ, because Mixed Poisson networks are a special case of configuration-model networks.
  • Mixed Poisson example: Mixed Poisson model predictions matched a single simulated epidemic in a network with 5 × 10^5 nodes, with time aligned at 1% cumulative incidence.
  • Mean Field Social Heterogeneity: For expected-degree MFSH networks, edge existence is independently resampled over time, so current-edge status probabilities equal the probabilities for newly formed edges.
  • Mean Field Social Heterogeneity: The expected-degree MFSH equations differ from the Mixed Poisson equations only in the first term of the Θ̇ equation, while the formulations are equivalent under appropriate limits.
  • Mean Field Social Heterogeneity: The expected-degree MFSH final infected proportion is R(∞) = 1 − Ψ(Θ(∞)), where Θ(∞) is determined by the model equations.
  • Results: The final size is larger than in the DFD model because additional transmission routes from individuals with more contacts outweigh the reduction from individuals with fewer contacts.

5 Discussion

The paper presents edge-based compartmental modeling as a simple framework that jointly represents social heterogeneity and contact duration across several network settings. Its test-node perspective and flow-diagram formulation support analytical epidemic modeling while relying on explicit independence and deterministic-growth assumptions.

  • 5 Discussion: Edge-based compartmental modeling simultaneously incorporates heterogeneous contact propensities and arbitrary contact duration in analytically tractable epidemic models.The approach produces equations comparable in simplicity to the mass action model.
  • 5 Discussion: A flow-diagram interpretation guides straightforward derivations that require tracking only a small number of compartments and no closure approximations.The approach is presented as a way to derive and communicate models across multiple network populations.
  • 5 Discussion: The method assumes infections arriving from different neighbors of a test node are independent, which limits its application to SIS diseases with nonzero contact duration.In SIS settings, reinfection through neighbors can make the independence assumption substantially alter predictions.
  • 5 Discussion: The deterministic test-node equivalence assumes epidemic size grows deterministically, so it is false when stochastic effects can prevent or delay an outbreak.The approach also uses the assumption that the test node does not cause infections when calculating its state probabilities.
  • 5 Discussion: The framework treats population proportions as probabilities that a randomly selected test node is susceptible, infected, or recovered.This equivalent problem shifts analysis from tracking population fractions directly to tracking the test node’s state.

D R0, early growth, and final size

The paper analyzes R0, early growth, and final size using branching-process arguments and linearized edge-based ODEs. It identifies threshold behavior and notes that several calculations assume an occurring epidemic with an initially vanishing infected proportion.

  • D R0, early growth, and final size: The analysis combines branching-process calculations of R0 with linearized ODEs to derive early growth rates, initial conditions, and final-size relations.For most models, the final size is obtained from an equilibrium after disease spread.
  • D R0, early growth, and final size: R0 is defined using the number of cases caused early in an outbreak by an average infected individual, whose properties may differ from a random population member.On networks, early infected nodes typically have higher degree than randomly selected nodes.
  • D R0, early growth, and final size: R0 < 1 produces negative early growth and only an equilibrium corresponding to no large-scale transmission, whereas R0 > 1 yields an additional equilibrium for final-size calculation.These threshold results are reported for the edge-based models’ early-growth and equilibrium analyses.
  • D R0, early growth, and final size: Most early-growth and final-size calculations assume an epidemic occurs and the initial infected proportion approaches zero.These results are therefore inappropriate for R0 < 1, and initial-condition effects remain significant in decaying epidemics.

D.1.1 CM

For static Configuration Model networks, the edge-based formulation derives R0, early growth, initial conditions, and final epidemic size from the degree distribution.

  • R0: R0 equals the expected infections caused by a newly infected node, accounting for its k−1 susceptible neighbors and transmission probability β/(β + γ).The degree distribution is size-biased for newly infected nodes, and the infector is excluded from the susceptible-neighbor count.
  • R0: For Poisson degree distributions, the higher-degree selection and loss of one susceptible neighbor balance, yielding R0 = β⟨K⟩/(β + γ).This simplification does not hold for more general degree distributions.
  • Early Growth and Initial Conditions: The disease-free equilibrium loses stability exactly when R0 crosses 1, so branching-process and ODE analyses predict the same epidemic threshold.The threshold is identified by linearizing the deterministic equations around θ = 1.
  • Early Growth and Initial Conditions: Initial values need not precisely separate I and R: any convenient assignment with I + R = 1 − ψ(θ) produces rapidly decaying initial-condition error.The variable θ determines the subsequent solution, while errors in I and R decay exponentially quickly.
  • Final Size: When R0 > 1, the smaller fixed-point solution θ(∞) gives the epidemic final size R(∞) = 1 − ψ(θ(∞)).The larger solution θ = 1 represents the pre-disease equilibrium; the smaller solution represents an epidemic that spreads and dies out.

D.1.2 Actual Degree MFSH

For actual-degree mean-field social heterogeneity networks, the model derives R0, early growth, initial conditions, and final size while allowing each infected node to retain k susceptible neighbors throughout infection.

  • R0: A newly infected node of degree k causes infections at rate βk throughout its infection, giving an expected βk/γ infections before recovery.Averaging βk/γ over the newly infected degree distribution gives R0 for the model.
  • Early Growth and Initial Conditions: The early-growth equilibrium loses stability exactly when R0 = 1, and the remaining calculations parallel the Configuration Model case.The eigenvalue governing growth differs from the Configuration Model value.
  • Early Growth and Initial Conditions: Any reasonable initial condition with θ close to 1, S = ψ(θ), and I + R = 1 − S is acceptable.The authors retain the practical initial-condition strategy used for the Configuration Model.
  • Final Size: When R0 > 1, solving the nontrivial θ fixed point gives the total infected fraction R(∞) = 1 − ψ(θ(∞)).The θ = 1 solution represents no transmission, whereas the solution with 0 < θ < 1 is the epidemic outcome.
  • Connection to Mass Action: When ψ(x) = x^k, the model reduces to the mass-action case with contact rate kβ and R0 = kβ/γ.The corresponding final-size equation is the well-known mass-action relation.

D.1.3 DFD

The dynamic fixed-degree formulation accounts for contact turnover by allowing an infected node’s source edge to break and reform, producing a higher-dimensional dynamical system.

  • R0: A source edge can generate additional infections after breaking and reforming, with transmission and reformation probabilities determining its expected contribution.The expected infections from one such stub are [β/(β + γ)][1 + r/(1 − r)].
  • Early Growth and Initial Conditions: Early growth is analyzed by linearizing the higher-dimensional system around the disease-free equilibrium and using its dominant eigenvalue for initial conditions.The equilibrium includes θ, φS, φI, πS, and πI; the initial perturbation is scaled by a small ε.
  • Interpretation: The dynamic formulation avoids the earlier models’ unphysical return of θ to 1 when R0 < 1 after transmission has already occurred.Transmission that has happened does not decay; a zero eigenvector represents the persistent post-transmission component.
  • Final Size: The final size has no closed-form expression and must be obtained numerically by solving the ODE system from initial conditions representing a small number of cases.Multiple equilibria require an additional constraint to identify the equilibrium reached after disease introduction.

D.1.4 DC

The dynamic contact formulation models active and dormant stubs that switch states, and computes transmission, growth, and final size through a higher-dimensional ODE system.

  • R0: The model distinguishes active, dormant, and source-connected stubs, whose expected infection contributions are represented by ra, rd, and rs.Each quantity is defined by the stub’s state when the node becomes infected.
  • R0: The active-stub contribution is ra = β(η1 + γ)(η2 + γ) / [γ(γ + η1 + η2)(β + γ + η2)].This expression follows by combining the active-to-dormant and dormant-to-active transition equations.
  • R0: An infected node of degree k_m is expected to cause rs + (k−1)ξra + (k−1)(1−ξ)rd infections, averaged over the size-biased degree distribution.The active-stub probability is ξ = η1/(η1 + η2), with dormant probability 1 − ξ.
  • Early Growth and Initial Conditions: The authors do not calculate the early growth rate, instead using approximate initial conditions or a linear-system eigenvalue to initialize the ODEs.The approximate initialization is expected to converge relatively quickly to the appropriate eigenvalue.
  • Final Size: The final size requires numerical solution because the additional constraint selecting the appropriate equilibrium is unavailable analytically.This mirrors the dynamic fixed-degree case’s need for numerical final-size prediction.

D.2.1 MP

The MP formulation computes epidemic thresholds, early growth, and final size using expected-degree structure while retaining simple governing equations. Its predicted growth threshold coincides exactly with R0 = 1, and its final-size expression parallels the configuration-model result.

  • R0: R0 is obtained by averaging infections caused by newly infected nodes over the expected-degree distribution ρn(κ).For a node with expected degree κ, the expected number of infections is κβ/(γ + β).
  • Early Growth and Initial Conditions: The model’s governing equations relate recovery, susceptibility, and infection through ˙R = γI, S = Ψ(Θ), and I = 1 − S − R.These equations support the early-growth and initial-condition calculations.
  • Early Growth and Initial Conditions: The early-growth threshold occurs exactly where R0 = 1.Thus the sign change in the growth rate and the basic-reproduction threshold coincide.
  • Final Size: The final susceptible and recovered proportions are obtained from the limiting value Θ(∞), with S(∞) = Ψ(Θ(∞)) and R(∞) = 1 − S(∞).This calculation is described as analogous to the configuration-model final-size calculation.
  • MFSH Comparison: The MFSH threshold likewise has λ = 0 exactly when R0 = 1.The same threshold relation is stated for the expected-degree formulation.
  • MFSH Comparison: The MFSH final size follows from Θ(∞), giving R(∞) = 1 − Ψ(Θ(∞)).The limiting equation for Θ(∞) is solved as part of the final-size calculation.

D.2.3 DVD

The DVD formulation accounts for both existing and newly created contacts during infectiousness. Its predicted growth changes sign exactly at R0 = 1, while final size is computed from the limiting edge-survival variable Θ(∞).

  • R0: A newly infected node can infect both original neighbors and contacts created before recovery.The probability of edge-based transmission before breaking or recovery is β/(β + η + γ).
  • R0: The expected number of infections caused by a node with expected degree κ is [β/(β + η + γ)]κ[1 + η/γ].The factor 1 + η/γ accounts for contacts existing at infection or created before recovery.
  • Early growth: The early-growth system tracks both node-level and edge-level susceptible, infected, and recovered quantities.The governing equations include Π variables alongside S, I, and R.
  • Early growth: λ > 0 exactly when R0 > 1, while λ < 0 exactly when R0 < 1.The equivalence follows by relating the dominant eigenvalue conditions to the determinant and R0.
  • Initial conditions: Initial conditions may use any sufficiently small reasonable perturbation because the particulars of that choice are unimportant.Alternatively, the early solution approaches Ce^λtv along the dominant eigenvector.
  • Final Size: At epidemic end, infected nodes vanish and the total recovered fraction is determined by Θ(∞) through Ψ′(Θ(∞))/Ψ′(1).Θ(∞) can be found using iterative methods.

E Equivalence of MFSH models with pre-existing models

The paper shows that edge-based equations are equivalent to pre-existing MFSH formulations, while deriving them through variables with clearer meanings. This provides a direct connection between the simplified systems and other edge-based models.

  • E Equivalence of MFSH models with pre-existing models: The usual MFSH equations are introduced as the comparison system for the equivalence analysis.The comparison includes the actual-degree formulation ˙R = γI, S = ψ(θ), and I = 1 − S − R.
  • E Equivalence of MFSH models with pre-existing models: The reduction proceeds by solving the Sk equations as Sk = α^k and showing that α satisfies the same equation as θ.The derivation tracks ζ and integrates its relation to α.
  • E Equivalence of MFSH models with pre-existing models: The expected-degree and fixed-degree MFSH systems are equivalent because both yield the same susceptible-state formula Sk = α^k.The paper states that α and θ solve the same equation.
  • E Equivalence of MFSH models with pre-existing models: The derivation approach is new even though earlier authors had already simplified the usual system to a handful of equations.The resulting equations are equivalent but use slightly different variables.
  • E Equivalence of MFSH models with pre-existing models: The paper’s variables connect more easily to meaningful quantities and can be related directly to other edge-based compartmental models.This is presented as the advantage of the new derivation over the prior change-of-variables approach.
  • E Equivalence of MFSH models with pre-existing models: A continuous-contact-rate formulation is also described as equivalent to the expected-degree MFSH equations.The equivalence is established by a similar approach.
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