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Modeling the dynamical interaction between epidemics on overlay networks

Vincent Marceau, Pierre-André Noël, Laurent Hébert-Dufresne, Antoine Allard, Louis J. Dubé

arXiv:1103.4059v2physics.soc-phcond-mat.stat-mechcs.SIq-bio.PE

TL;DR

Because epidemics may involve multiple interacting agents, this paper develops an analytic approach for two agents spreading on overlay networks. It reports that the approach captures their dynamics with great accuracy and uses it to examine how delay and network structure shape a hypothetical immunizing intervention.

  • Problem

    The paper addresses limited modeling of cases where multiple viral agents interact in one host population, beyond the extensively studied single-agent setting.

    Source: §I · p. 1

  • Method

    It maps propagation on overlay networks to progressive on-the-fly network generation and builds a compartmental analytic formalism for two agents interacting through leaky partial immunity.

    Source: §I · p. 1, §V · p. 9

  • Results

    Compared with full Monte Carlo simulations, the approach captures interacting epidemic dynamics with great accuracy, and the analysis finds positive degree correlation improves intervention efficiency while overlap facilitates undesirable-agent invasion.

    Source: §V · p. 9

  • Takeaways & Limitations

    For the hypothetical intervention, effectiveness depends on delay relative to the agent-1 epidemic timescale, with less time to react efficiently in heterogeneous populations.

    Source: A. Delayed intervention · p. 7

    In the arbitrary-overlap analysis, accuracy declines as Q increases, although it recovers when Q is close to one.

    Source: B. Overlap and degree correlation · p. 9

Abstract

from arXiv · show

Epidemics seldom occur as isolated phenomena. Typically, two or more viral agents spread within the same host population and may interact dynamically with each other. We present a general model where two viral agents interact via an immunity mechanism as they propagate simultaneously on two networks connecting the same set of nodes. Exploiting a correspondence between the propagation dynamics and a dynamical process performing progressive network generation, we develop an analytic approach that accurately captures the dynamical interaction between epidemics on overlay networks. The formalism allows for overlay networks with arbitrary joint degree distribution and overlap. To illustrate the versatility of our approach, we consider a hypothetical delayed intervention scenario in which an immunizing agent is disseminated in a host population to hinder the propagation of an undesirable agent (e.g. the spread of preventive information in the context of an emerging infectious disease).

I. INTRODUCTION … B. Interacting epidemics

The paper develops an analytical approach to simultaneous, dynamically interacting epidemics on overlay networks, using progressive network generation and allowing arbitrary joint degree distributions and overlap. It defines a two-agent SIR model in which transmission depends on the target’s state toward the other agent.

I. INTRODUCTION

  • Pathogens, disease strains, information, and computer-virus countermeasures motivate studying processes that interact within one population rather than treating epidemics as isolated.The paper notes ecological and immunological interactions, crossimmunity, disease-related fear or awareness, and contagious vaccination schemes.Source: §I · p. 1
  • Funk and Jansen analyzed agents spreading successively on overlay networks using static bond percolation, an approach that does not capture dynamic interaction.Source: §I · p. 1
  • The analytical approach models simultaneous, dynamically interacting viral agents on overlay networks through progressive network generation and permits arbitrary joint degree distributions and overlap.Source: §I · p. 1

II. INTERACTING EPIDEMICS ON OVERLAY NETWORKS

A. Overlay networks

  • Overlay networks share a node population, and their unique and common links can be represented as three non-overlapping networks whose joint degree distribution records overlap.The joint degree distribution P(k1, k2) describes degrees on the two networks, while the three-network distribution ρ(κ1, κ2, κb) captures the distribution of overlapping links.Source: A. Overlay networks · pp. 1–2, Figure 1 · p. 2

B. Interacting epidemics

  • Each agent spreads on its own network with infectious-contact rate βg and recovery rate αg; nodes move among susceptible, infectious, and recovered states, with recovery conferring immunity to that agent.Source: B. Interacting epidemics · p. 2
  • With leaky partial immunity, transmission succeeds with a probability determined by the susceptible target’s state toward the other agent, and each pair has at most one infectious contact.The paper chooses this interaction rule to retain analytical tractability while keeping the model general.Source: B. Interacting epidemics · p. 2

C. Monte Carlo simulations · III. MEAN-FIELD APPROACH · A. Networks with random overlap

The paper pairs Monte Carlo simulations on overlay networks with a mean-field approach based on generating network links on the fly. For random overlap, the stochastic process corresponds almost exactly to the simulations, and its large-network limit yields deterministic compartment equations.

C. Monte Carlo simulations

  • Monte Carlo overlay networks are generated with two configuration-model-based algorithms [24, 25]; the random-overlap algorithm uses a joint degree distribution and independently matches stubs on each network.The procedure generates degree sequences and rewires self-loops or repeated links.Source: C. Monte Carlo simulations · p. 2
  • For arbitrary overlap, the second algorithm generates γ1, γ2, and γb from the joint degree distribution ρ(κ), forbids a link from appearing on multiple networks, and constructs Γ1 and Γ2 from them.The construction is illustrated in Fig. 1.Source: C. Monte Carlo simulations · p. 2
  • Simulations test infectious contacts in discrete time steps, apply transmission and recovery probabilities, and introduce agent 2 at a delay τ after randomly seeding agent 1.The initial fractions are ǫ1 at t = 0 and ǫ2 at t = τ; transmission depends on the susceptible node’s state regarding the other agent.Source: C. Monte Carlo simulations · pp. 2–3
  • Unless noted otherwise, Monte Carlo simulations use N = 25 000, ∆t = 0.001, α1 = α2 = 1, ǫ1 = ǫ2 = 0.001, and 1000 runs per figure.These values define the paper’s standard simulation setup.Source: C. Monte Carlo simulations · p. 3

III. MEAN-FIELD APPROACH

  • The mean-field formalism extends on-the-fly network generation [21] to interacting epidemics on overlay networks, first for random overlap and then for arbitrary overlap.The paper also indicates how to handle similar interaction dynamics.Source: §III · p. 3

A. Networks with random overlap

  • On-the-fly generation tests unmatched stubs from infectious nodes, matches them randomly only when infectious contacts occur, and assigns the neighbor’s state afterward, without requiring prior network structure.Unreached parts of the network are never built; self-loops and repeated links are allowed, with probability decreasing as N^-1.Source: A. Networks with random overlap · p. 3, Figure 2 · p. 3
  • For random overlap, on-the-fly generation differs from the simulation algorithm only by self-loops and repeated links, whose probability decreases as N^-1; the mean-field approach takes N → ∞.The resulting deterministic equations track mean compartment fractions by viral state and the numbers of unmatched stubs on both networks.Source: A. Networks with random overlap · p. 3
  • The random-overlap ODE tracks mean node fractions by both viral states and unmatched-stub counts, with transmission, stub loss, and recovery contributing to compartment changes.For agent 1, susceptible nodes lose Γ1 stubs during infectious contacts; infectious nodes can recover or lose stubs through contacts they initiate or receive.Source: A. Networks with random overlap · pp. 3–4

B. Networks with arbitrary overlap

For networks with arbitrary overlap, the paper extends the dynamical description using a three-network decomposition and approximations, since the networks’ structures influence one another and the exact on-the-fly correspondence no longer applies. The approximation accounts for possible later agent 2 transmission after agent 1 contacts on the shared network, while increasing ODE complexity.

B. Networks with arbitrary overlap

  • Arbitrary overlap requires approximation because the networks are no longer built independently, so the exact correspondence with on-the-fly neighbor assignment cannot be established.Source: B. Networks with arbitrary overlap · p. 4
  • The arbitrary-overlap description tracks each node’s states and unmatched stubs on the two network-specific layers and their shared layer, retaining the prior agent 1 dynamics for the network-specific layer.The additional ODE contributions describe agent 1 dynamics on the shared layer.Source: B. Networks with arbitrary overlap · p. 4
  • After an agent 1 contact on the shared layer, the approximation adds an unmatched stub on network 2 only if the nodes’ states permit a later agent 2 transmission.Otherwise, the nodes lose the shared-layer stubs without gaining network 2 stubs; this preserves relevant state information despite not tracking neighbors’ states.Source: B. Networks with arbitrary overlap · p. 4
  • For arbitrary overlap, the ODE system becomes more complex, with its degrees truncated at kmax and κmax, but the approach can still significantly speed up Monte Carlo simulations.Source: B. Networks with arbitrary overlap · p. 5

C. Other types of dynamics

The section outlines adaptations of the analytical framework for multiple infectious contacts, perfect partial immunity, and SIS dynamics. These alternatives require changes to the model’s state representation or compartmentalization.

C. Other types of dynamics

  • With multiple infectious contacts, use a local description that tracks neighbors by number and infection state, rather than unmatched stubs.The state variables count 1-susceptible and 1-infectious neighbors on Γ1 and 2-susceptible and 2-infectious neighbors on Γ2; tracking recovered neighbors is optional.Source: C. Other types of dynamics · pp. 5–6
  • Perfect partial immunity [18] can be modeled by adding a fourth compartment, Vg, for g-susceptible nodes perfectly immune to agent g.A g-susceptible node in state Yˆg regarding the other agent ˆg has probability γ of being perfectly immune to agent g.Source: C. Other types of dynamics · p. 6
  • For SIS dynamics, the framework requires a local description sorting nodes by the number and state of their neighbors, because on-the-fly neighbor assignment is not equivalent.The analytical approach could accommodate SIS dynamics after slight changes to the compartmentalization scheme.Source: C. Other types of dynamics · p. 6

IV. VALIDATION THROUGH CASE STUDIES

The case studies test the mean-field approach on two host populations with different network heterogeneity, finding close agreement in the single-agent case and similar incidence at β1=0.66.

IV. VALIDATION THROUGH CASE STUDIES

  • Population A has a Poisson degree distribution, whereas population B has a power-law degree distribution, with both networks’ mean degree set to approximately 3.5.The power-law network is defined over degrees 1 through 20.Source: §IV · p. 6
  • The incidence comparison sets ε2=0, reducing the dynamics to a single-agent SIR epidemic.Source: §IV · p. 6
  • At β1=0.66, the single-agent epidemics in populations A and B reach approximately the same incidence, and the mean-field approach captures their behavior with great accuracy.The two populations’ networks have the same mean degree, ⟨k1⟩≃3.5, despite different topologies.Source: Figure 3 · p. 6, §IV · p. 6

A. Delayed intervention

Delayed agent 2 intervention is more time-sensitive in heterogeneous population B under full immunity, while partial immunity leaves substantial agent 1 spread there even at zero delay. The analytical predictions agree closely with simulations.

a. Full immunity

  • Under full immunity, agent 1 incidence rises faster with intervention delay in population B than in A, leaving less time to intervene effectively.Figure 4 shows this contrast in total incidence; the shorter epidemic timescale associated with greater network heterogeneity helps explain it.Source: Figure 4 · p. 7, A. Delayed intervention · p. 7
  • With full immunity, delays τ=0 and 1 inhibit agent 1 initially in both populations; at τ=5 restraint persists in A but B nearly completes its epidemic, and at τ=10 intervention has minimal effect in both.The agent 1 epidemic's timescale is crucial to delayed-intervention efficiency and decreases with network heterogeneity.Source: A. Delayed intervention · p. 7, Figure 5 · p. 7

b. Partial immunity

  • With partial immunity (σ=0.5), incidence in A behaves much like full immunity but rises slightly faster, whereas in B even τ=0 leaves agent 1 reaching half as many nodes as without intervention.Nodes reached by agent 2 are only partially protected against agent 1.Source: A. Delayed intervention · p. 7, Figure 4 · p. 7
  • Under partial immunity, a degree-k1 node remains agent 1-susceptible with probability (1 − σ)^k1 after infectious contacts with all its neighbors, weakening protection for high-degree nodes that are involved early.This helps explain why agent 1 can invade a significant fraction of population B despite a short-delay intervention.Source: A. Delayed intervention · p. 7

a. Full immunity

  • The mean-field predictions agree excellently with Monte Carlo simulations; the approach is expected to be exact for infinite configuration-model overlay networks with random overlap.Small discrepancies are attributed to finite-size effects, including early stochastic extinction and restrictions on self-loops and repeated links.Source: A. Delayed intervention · pp. 7–8

B. Overlap and degree correlation

Degree correlation and overlap affect agent 1’s invasion, while the partial-overlap approximation loses accuracy as overlap increases but becomes accurate again near full overlap.

a. Overlap versus degree correlation

  • With random overlap, agent 1 invades more easily without degree correlation in both populations, and the correlation effect is stronger in B.The comparison is shown for random overlap with no degree correlation versus random overlap with full degree correlation.Source: Figure 6 · p. 8, B. Overlap and degree correlation · p. 8
  • For a given joint degree distribution, full overlap facilitates agent 1’s epidemic compared with random overlap and full degree correlation.When Γ1 = Γ2, nodes reached by agent 1 that have not already been reached by agent 2 are more likely to connect to nodes agent 2 has not reached.Source: Figure 6 · p. 8, B. Overlap and degree correlation · p. 8
  • Differences among configurations are largest when the agents’ epidemic timescales are comparable; at very low β1 invasion is difficult, while at β1 ≫ β2 agent 1 is almost unaffected.Source: B. Overlap and degree correlation · p. 8
  • The results generalize the findings of on how positive degree correlation and increasing overlap affect the epidemic threshold to simultaneously spreading, dynamically interacting processes.The paper notes that threshold variations are hard to analyze when both processes spread simultaneously.Source: B. Overlap and degree correlation · p. 8

b. Partial overlap

  • As overlap Q increases, the partial-overlap approach becomes less accurate, but it becomes accurate again near Q = 1 as the propagation correspondence is nearly exact.At Q = 1, the network links are on γb, so the i and j indices indicate only newly granted stubs rather than mixing real and supposed unmatched stubs.Source: Figure 7 · p. 9, B. Overlap and degree correlation · p. 9
  • For full immunity, the modified partial-overlap approach describes the epidemic scenario better, but it is not valid for interactions involving partial immunity.The modified approach is represented by the dashed curve in Fig. 7.Source: B. Overlap and degree correlation · p. 9

V. CONCLUSION

The paper introduces a network-based compartmental model for two interacting viral agents on overlay networks, capturing their complete time evolution. It also applies the approach to delayed immunization interventions and identifies how degree correlation and overlap affect intervention efficiency.

V. CONCLUSION

  • The formalism captures complete epidemic time evolution and closely matches Monte Carlo simulations across overlay networks with varying heterogeneity, degree correlation, and overlap.It models leaky partial immunity and builds on an on-the-fly network-generation correspondence [21], unlike earlier bond-percolation approaches [18].Source: §V · p. 9
  • Positive degree correlation increases delayed-intervention efficiency, while overlap facilitates invasion of the undesirable process.The paper considers immunizing interventions across homogeneous and heterogeneous networks and discusses the effects of delay and immunity level.Source: §V · p. 9
  • Future work includes extending the model to community-structured networks and applying it to case studies such as other sexually transmitted infections’ influence on HIV spread.The conclusion also highlights the broader modeling scheme from which the formalism stems [21].Source: §V · p. 9

SUPPLEMENTARY DOCUMENT

The supplementary document provides the full equations for the analytical approach, introducing notation before presenting equations for random and arbitrary overlap.

NOTATION

The section introduces notation used in the document and notes that a summation convention applies throughout.

  • Table I is identified as the document’s notation reference.Source: Table I · p. 11
  • A summation convention is used throughout the document.The supplied passage does not specify the convention.Source: NOTATION · p. 11

OVERLAY NETWORKS WITH RANDOM OVERLAP

For overlay networks with random overlap, the time evolution of the [X1Y2]i j compartments is governed by a system of ODEs that conserves nodes. Agent 1 starts at t = 0, and agent 2 is introduced at time τ through state substitutions.

OVERLAY NETWORKS WITH ARBITRARY OVERLAP

For overlay networks with arbitrary overlap, the model specifies ODE evolution for the [X1Y2]ijk compartments and verifies node conservation. Agent 1 is initialized at t=0, and agent 2 is introduced at t=τ≥0 through prescribed compartment substitutions.

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