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A framework for epidemic spreading in multiplex networks of metapopulations

David Soriano-Paños, Laura Lotero, Jesús Gómez-Gardeñes, Alex Arenas

arXiv:1802.03969v1physics.soc-phq-bio.PE

TL;DR

Analytical metapopulation models often rely on simplified mobility patterns or mean-field assumptions, while incorporating realistic recurrent movement in a general theory remains an open challenge. This paper develops a patch-level framework using multiplex layers for heterogeneous agents and analyzes SIS and SIR epidemics. Its equations agree with Monte Carlo simulations, and its exact threshold depends on mobility patterns and other multiplex properties.

  • Problem

    Existing theoretical approaches often simplify mobility or rely on mean-field assumptions, leaving a general framework that incorporates realistic mobility patterns as an open challenge.

    Source: §I · p. 1

  • Method

    The framework treats patch variables independently and uses multiplex layers to represent agent classes with different recurrent mobility patterns.

    Source: §I · p. 2, §VI · p. 11

  • Results

    Markovian equations agree with Monte Carlo simulations, and the epidemic threshold is related to the maximum eigenvalue of a supra-matrix combining mobility patterns, local census, and mobility degree.

    Source: §VI · p. 11

  • Takeaways & Limitations

    In Medellín, increasing mobility raises the epidemic threshold, an effect attributed to homogenization of the demographic distribution.

    Source: B. Real Multiplex Metapopulations · p. 9

Abstract

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We propose a theoretical framework for the study of epidemics in structured metapopulations, with heterogeneous agents, subjected to recurrent mobility patterns. We propose to represent the heterogeneity in the composition of the metapopulations as layers in a multiplex network, where nodes would correspond to geographical areas and layers account for the mobility patterns of agents of the same class. We analyze both the classical Susceptible-Infected-Susceptible and the Susceptible-Infected-Removed epidemic models within this framework, and compare macroscopic and microscopic indicators of the spreading process with extensive Monte Carlo simulations. Our results are in excellent agreement with the simulations. We also derive an exact expression of the epidemic threshold on this general framework revealing a non-trivial dependence on the mobility parameter. Finally, we use this new formalism to address the spread of diseases in real cities, specifically in the city of Medellin, Colombia, whose population is divided into six socio-economic classes, each one identified with a layer in this multiplex formalism.

I. INTRODUCTION

Fast-moving outbreaks require models that connect local contagion with mobility across places, yet realistic mobility patterns remain difficult to incorporate into general analytical frameworks. The paper proposes a multiplex metapopulation framework for heterogeneous agents and analyzes SIS and SIR spreading with an exact epidemic-threshold expression.

I. INTRODUCTION

  • Emergent outbreaks can move rapidly from one region to a global epidemic, motivating efforts to understand their spread and design local and global containment strategies.The passage cites SARS, H1N1, Ebola, and Zika as examples.Source: §I · p. 1
  • Metapopulation models combine well-mixed contagion within patches with agent mobility between patches represented as a network.Patches can differ in population size, and mobility links indicate possible movement between locations.Source: §I · p. 1
  • Existing approaches trade realism against analytical feasibility, and a general theoretical framework incorporating realistic mobility patterns remains an open challenge.Prior work addressed recurrent, spatially constrained movements but relied on mean-field assumptions or simple mobility networks.Source: §I · pp. 1–2
  • A multiplex represents different agent classes as layers sharing geographical nodes, with each layer encoding that class's mobility network.This formalism captures the coexistence of agents with different mobility preferences within each subpopulation.Source: §I · p. 2
  • The proposed framework treats each patch's variables independently, accommodates mobility multiplexes and population distributions, and analyzes SIS and SIR dynamics with simulations and an exact threshold expression.The threshold depends nontrivially on the different mobility patterns represented in the multiplex.Source: §I · p. 2

II. METAPOPULATION MODEL

The model represents a metapopulation as patches connected by a weighted, directed network, with agents associated with home patches. At each time step, agents move, interact within their current patches, and return home.

II. METAPOPULATION MODEL

  • The model begins with one agent type distributed across N patches, with each agent assigned to a home patch and each patch i associated with n_i agents.The total population is the sum of the patch populations.Source: §II · p. 2
  • Patch connections form a weighted, directed graph in which W_ij represents the interaction weight from node i to node j.The passage identifies the network as the graph connecting patches.Source: §II · p. 2
  • In the movement stage, each agent moves with probability p or stays home with probability 1−p, choosing a connected destination in proportion to W_ij.The movement probabilities are encoded by the network adjacency matrix.Source: §II · p. 2
  • After movement, agents update their epidemic state through interactions with agents in the same patch, then return to their home patches before the next time step.These stages are movement, interaction, and return (MIR).Source: §II · p. 2
  • Observed trips between locations can supply the transition rates, making the model applicable to recurrent commuting patterns in cities or countries.The model uses the number of observed trips between two locations to construct its transition rates matrix.Source: §II · p. 2

III. POPULATION-BASED MARKOVIAN DYNAMICS IN COMPLEX NETWORKS

The paper formulates patch-level Markovian equations for metapopulation spreading while keeping each patch's static and dynamical variables independent. It considers SIS and SIR dynamics and uses the framework to compare with simulations and analyze general mobility networks.

III. POPULATION-BASED MARKOVIAN DYNAMICS IN COMPLEX NETWORKS

  • The paper considers SIS and SIR models, whose reaction laws use infection probability λ and recovery probability µ, with recovery leading to susceptibility in SIS or immunity in SIR.In the SIS model, recovered agents become susceptible again; in SIR, they become immunized.Source: §III · p. 2
  • Heterogeneous mean-field approximations group patches with the same degree as having identical properties, an assumption reported as useful for predicting thresholds at small epidemic sizes.The assumption equates patches with the same number of connections even when their neighbors differ.Source: §III · pp. 2–3
  • The Markovian framework describes one time step as movement between patches, well-mixed interactions, and return to home patches.The schematic illustrates three patches and SIS status changes during the interaction stage.Source: Figure 1 · p. 3
  • The framework formulates independent patch-level variables to compare directly with microscopic Monte Carlo simulations and derive results for particular mobility networks.This extends the Markovian approach to complex metapopulations.Source: §III · p. 3

A. SIS model

The SIS equations track infection among agents associated with each patch, accounting for recovery and infection acquired through exposure at home or while moving. The section also specifies how local infection probability is calculated and compares equation-based predictions with simulations.

A. SIS model

  • For patch i, ρ_i(t) is the infected fraction among agents associated with that patch, who may be located elsewhere during a time step.The dynamics update the patch-associated fraction rather than simply the infected agents currently present at the home patch.Source: A. SIS model · p. 3
  • The update for ρ_i(t) combines infected agents who do not recover with healthy agents who become infected by the next time step.The infection contribution is weighted by Π_i(t), the probability that a healthy agent associated with i becomes infected.Source: A. SIS model · p. 3
  • Π_i(t) combines infection probability at home with infection probability after moving to a neighboring patch.The two terms distinguish staying at home from moving to a neighbor.Source: A. SIS model · p. 3
  • P_i(t) is the probability that a healthy agent currently in patch i becomes infected through contact with infected agents present there.The equation set uses this probability to represent infection following within-patch exposure.Source: A. SIS model · p. 3
  • The figure plots steady-state infected counts against λ for ER and SF networks, comparing Markovian-equation curves with Monte Carlo points across movement probabilities.Each node has 5,000 individuals, and λ is rescaled by the p=0 critical value of 4·10^-4.Source: Figure 2 · p. 4
  • The transition matrix can be constructed from observed flows between patches, while node populations can be set from local census data.Source: A. SIS model · p. 4

B. SIR dynamics

The SIR formulation adds a recovered fraction for each patch to the Markovian dynamics, while retaining the SIS infection-probability expression. Its predictions for recovered agents are compared with Monte Carlo simulations across ER and scale-free networks.

B. SIR dynamics

  • The SIR model adds N patch-specific recovered-agent variables, producing 2N Markovian equations.These variables are the recovered fractions r_i(t).Source: B. SIR dynamics · p. 4
  • Figure 3 compares Markovian predictions of recovered agents with Monte Carlo results across ER and scale-free networks and mobility probabilities.The plotted infection probability is rescaled by the well-mixed critical value at p = 0, λc = 4 × 10^-4.Source: Figure 3 · p. 4
  • The probability that a healthy agent in patch i becomes infected has the same form in SIR as in SIS.The SIR dynamics are described by the infection-probability equation together with the recovered-state equations.Source: B. SIR dynamics · p. 4

C. Validation of the Markovian equations

The Markovian equations reproduce macroscopic SIS and SIR outcomes and closely track microscopic SIR patterns in Monte Carlo simulations. Their accuracy is high at stationarity, although errors peak during the spread of contagions, and population heterogeneity can alter how mobility affects epidemic onset.

C. Validation of the Markovian equations

  • The validation uses 103-node ER and SF networks with 5,000 agents per node and averages Monte Carlo results over 102 realizations.The simulations initialize infections with probability 10^-3 per agent; the microscopic test instead seeds one node in a 200-node SF network.Source: C. Validation of the Markovian equations · pp. 4–5, Figure 4 · p. 5
  • SIS and SIR macroscopic indicators agree almost exactly with Markovian equations across disease-free and epidemic regimes.The agreement is reported for ER and scale-free networks; for SIR it holds before and after the epidemic threshold.Source: C. Validation of the Markovian equations · p. 5
  • With homogeneous patch populations, mobility decreases epidemic onset, while demographic heterogeneity may instead raise the threshold as mobility increases.The authors note that the homogeneous-population examples differ from real metapopulations such as cities.Source: C. Validation of the Markovian equations · p. 5
  • After roughly 50 time steps, the seeded SIR outbreak reaches a frozen state with a nonuniform recovered-agent density across patches.The node-level Markovian solution fairly reproduces the agent-based simulation’s spatiotemporal pattern; the test uses an SF network with patch sizes proportional to out-strength.Source: C. Validation of the Markovian equations · p. 5
  • The microscopic error peaks at 8% just after contagions spread across the metapopulation, then settles around 1%.At stationarity, this indicates high accuracy in reproducing the observed pattern.Source: C. Validation of the Markovian equations · pp. 5–6

IV. MULTIPLEX METAPOPULATIONS

The multiplex metapopulation represents each agent type’s mobility pattern in a separate network layer, with corresponding patch nodes mixing agents during contagion. The model tracks infection and recovery by layer and patch, with a simplifying assumption of type-independent infection probabilities.

IV. MULTIPLEX METAPOPULATIONS

  • A patch’s population is the sum of agents across its L coexisting mobility types.The probability that a type-α agent visits patch j from patch i generalizes the earlier mobility expression; link weights represent observed trips by that type.Source: §IV · p. 6
  • Each mobility type forms a multiplex layer with its own network matrix, while each geographical patch appears in every layer.Corresponding nodes across layers are virtually connected because their agents mix during contagion.Source: §IV · p. 6
  • The multiplex model multiplies the Markovian variables and equations by L, tracking infected and recovered fractions for each type and patch.The type-specific SIR and SIS formulations use these layer-level state variables.Source: §IV · p. 6
  • The simplified analysis assumes infection probability λ is identical across agent-type pairs.This sets λαβ = λ for all α and β.Source: §IV · p. 6

A. Validation or the Markovian equations

The Markovian equations are tested against simulations on synthetic multiplexes and in a heterogeneous ER–SF setting. They reproduce epidemic dynamics while revealing mobility-dependent onset patterns and distinct spatial outcomes across layers.

A. Validation or the Markovian equations

  • The validation covers three two-layer multiplex types—ER–ER, SF–SF, and ER–SF—with 10^3 patches and 500 agents per layer at each patch.The link weights are assigned uniformly from [1, 50].Source: A. Validation or the Markovian equations · pp. 6–7
  • The Markovian equations reproduce simulated SIR and SIS epidemic diagrams, with mobility advancing onset and ER–SF showing an intermediate anticipation effect.The ER–ER, ER–SF, and SF–SF multiplexes are compared; ER–SF lies between the other two in onset anticipation.Source: A. Validation or the Markovian equations · p. 7, Figure 6 · p. 7
  • In an ER–SF multiplex seeded in the ER layer, the equations capture the delayed SF-layer onset, which follows from localized seeding and weak cross-type contagion.The cross-type infection probability is 0.025µ/500, below the well-mixed threshold µ/500; this does not prevent invasion of the SF layer.Source: A. Validation or the Markovian equations · pp. 7–8
  • The model reproduces patch-level recovered fractions, which are nearly uniform across ER patches but more heterogeneous across SF patches at stationarity.The comparison uses Monte Carlo simulations and solutions of the Markovian equations.Source: A. Validation or the Markovian equations · p. 8, Figure 7 · p. 8

B. Real Multiplex Metapopulations

The framework is applied to Medellín as a six-layer multiplex representing socioeconomic classes across 413 neighborhoods. It agrees with simulations and shows that mobility raises the full system's epidemic threshold, while class-level patterns reveal heterogeneous population distribution and interclass spread.

B. Real Multiplex Metapopulations

  • The Medellín multiplex has six socioeconomic layers across 413 neighborhoods, with each layer encoding a class's population distribution and mobility patterns.The network is constructed from data cited as.Source: B. Real Multiplex Metapopulations · p. 9
  • In Medellín, simulations and the model agree on total epidemic incidence, and increasing mobility raises the epidemic threshold through demographic homogenization.The reported mobility effect is detrimental to epidemic onset in both SIS and SIR diagrams.Source: Figure 8 · p. 8, B. Real Multiplex Metapopulations · p. 9
  • Although mobility shifts the full multiplex critical point upward, that threshold shift is not reported for individual layers; increasing mobility makes disease impact more homogeneous across layers.The layer-level SIS diagrams are compared for p = 0, 0.2, and 1.Source: B. Real Multiplex Metapopulations · p. 9, Figure 9 · p. 9
  • At zero mobility, classes 1–3 occupy the most populated nodes, while class 6 resides in sparsely populated neighborhoods and is practically isolated from other classes.The class-level epidemic onset in the static case is used to infer this demographic distribution.Source: B. Real Multiplex Metapopulations · p. 9, Figure 9 · p. 9
  • Localized SIR outbreaks show cascade-like contagion among nearby income classes and correlated outbreaks in middle classes 2, 3, and 4.The cascade patterns differ for seeds in classes 1 and 5, and the Markovian solution captures their interactions.Source: B. Real Multiplex Metapopulations · pp. 9–10

V. DEDUCTION OF THE EPIDEMIC THRESHOLD

The epidemic threshold is determined by the largest eigenvalue of a supra-matrix encoding interactions across multiplex metapopulations. Its predicted onset agrees with Monte Carlo simulations, and the threshold decreases as mobility increases.

V. DEDUCTION OF THE EPIDEMIC THRESHOLD

  • Each entry of M quantifies cross-layer, cross-patch contacts through within-node, between-patch, and third-place infection opportunities.The three contributions are weighted by (1 − p)², p(1 − p), and p², respectively.Source: §V · p. 10
  • The threshold λc = μ/Λmax(M) agrees with Monte Carlo epidemic onset, and λc(p) decreases monotonically as mobility increases.Figure 11 plots the threshold curves against SIS epidemic diagrams for three synthetic multiplex architectures.Source: §V · pp. 10–11, Figure 11 · p. 11
  • Setting L = 1 recovers the epidemic-threshold formulation for a single metapopulation, with M reduced to an N × N matrix.In that case, each matrix entry represents the probability that an agent at one patch contacts an agent from another patch.Source: §V · p. 11

VI. CONCLUSIONS

The conclusions present a Markovian framework for epidemics in multiplex metapopulations with recurrent mobility and heterogeneous populations. The authors report simulation agreement, an eigenvalue-based threshold, and a Medellín case study with different epidemic detriment in the full multiplex and individual layers.

VI. CONCLUSIONS

  • The framework accommodates weighted, directed origin-destination matrices and local patch census without heterogeneous mean-field assumptions about node attributes and epidemic variables.The formalism is applied to recurrent mobility patterns across populations.Source: §VI · p. 11
  • The Markovian equations agree with Monte Carlo simulations at macroscopic and microscopic levels for monoplex and multiplex metapopulations.For monoplex networks, they also reproduce spatiotemporal patterns, including local patch epidemic onset.Source: §VI · p. 11
  • The global epidemic threshold is tied to the maximum eigenvalue of a supra-matrix combining mobility patterns, local census, and mobility degree.The authors report complete agreement between the analytical threshold prediction and numerical simulations.Source: §VI · p. 11
  • In Medellín, the full multiplex structure shows epidemic detriment, whereas the individual socio-economic layers do not.The case study uses mobility patterns for different socio-economic classes as layers.Source: §VI · p. 12
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