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Competing epidemics on complex networks
Brian Karrer, M. E. J. Newman
TL;DR
The paper asks how two fully cross-immune diseases spreading concurrently over the same network divide the population. Using analytic arguments and numerical simulation in a two-disease Reed–Frost model, it derives the phase diagram and expected final outbreak sizes. The system exhibits dominance transitions, strong finite-size sensitivity near the growth-rate boundary, and a substantial coexistence regime.
Problem
The paper asks whether and how far two fully cross-immune disease strains spread concurrently as transmissibility and network structure vary.
Method
The authors combine analytic results, heuristic arguments, and numerical simulation for a two-disease Reed–Frost model on a static contact network.
Results
The phase diagram has four regions, including coexistence and dominance regimes, with a discontinuous growth-rate boundary and strong early-fluctuation effects nearby.
Takeaways & Limitations
Both diseases can infect extensive fractions of the network in the coexistence regime, while elsewhere one disease dominates and the other reaches only a vanishing fraction.
Takeaways & Limitations
Analytic results assume configuration-model networks, no early extinction, and large-network behavior becomes unreliable near the growth-rate boundary.
Abstract
from arXiv · showhide
Human diseases spread over networks of contacts between individuals and a substantial body of recent research has focused on the dynamics of the spreading process. Here we examine a model of two competing diseases spreading over the same network at the same time, where infection with either disease gives an individual subsequent immunity to both. Using a combination of analytic and numerical methods, we derive the phase diagram of the system and estimates of the expected final numbers of individuals infected with each disease. The system shows an unusual dynamical transition between dominance of one disease and dominance of the other as a function of their relative rates of growth. Close to this transition the final outcomes show strong dependence on stochastic fluctuations in the early stages of growth, dependence that decreases with increasing network size, but does so sufficiently slowly as still to be easily visible in systems with millions or billions of individuals. In most regions of the phase diagram we find that one disease eventually dominates while the other reaches only a vanishing fraction of the network, but the system also displays a significant coexistence regime in which both diseases reach epidemic proportions and infect an extensive fraction of the network.
I. INTRODUCTION
The paper frames disease spreading as a network-dependent process and studies two concurrently spreading, fully cross-immune SIR-type diseases. It uses a two-disease Reed–Frost model to examine how transmissibility, network structure, and relative spreading speed shape outcomes.
- I. INTRODUCTION: Network structure affects disease-spreading dynamics, motivating approaches that account for contact-network effects.Traditional propagation theories largely ignored networks, while recent research has examined their role.
- I. INTRODUCTION: The SIR model represents infection, transmission, recovery, and permanent immunity, so each individual can be infected at most once.It is presented as a simplified but reasonably accurate framework for many real-world diseases and as a foundational network-epidemiology model.
- I. INTRODUCTION: Two SIR-type diseases can compete on the same contact network when infection with either strain confers immunity to both.The central question is how far each strain spreads as transmissibility and network structure vary.
- I. INTRODUCTION: Earlier work considered diseases spreading sequentially, where the first disease leaves a residual network that may or may not support the second epidemic.This analysis identified a coexistence threshold: an upper limit on the first disease’s transmissibility compatible with spread of the second.
- I. INTRODUCTION: This paper studies the more general case of concurrent spread using the Reed–Frost framework, because relative disease dynamics matter when outbreaks overlap in time.The model includes transmissibilities and relative time-step scales, yielding three parameters rather than two in the one-disease version.
II. PRINCIPAL RESULTS
The model yields a four-region phase diagram governed by epidemic thresholds, relative growth rates, and coexistence conditions. Finite network size and early stochastic fluctuations substantially affect outcomes near the growth-rate boundary.
- Phase diagram: β = 1 forms a surface separating regimes in which blue or red grows faster and assumes the first-disease role in the equivalent time-separated picture.The faster disease reaches the network before the slower disease becomes significant in the large-n limit.
- Coexistence: No coexistence occurs on the side where blue is faster, because coexistence would require both Tr > Tb and Tb > Tr.This contradiction follows from α ≤ 1.
- Thresholds: Tr < φc prevents a red epidemic, leaving red with only O(1) infected individuals.More generally, nontrivial two-disease outcomes require Tr, Tb > φc.
- Phase diagram: The phase diagram has four regions: one coexistence region and three regions where one disease dominates while the other infects a vanishing fraction.The dominance transition is discontinuous, unlike the continuous percolation and coexistence transitions.
- Coexistence: For Poisson random graphs, coexistence requires α ≤ 1/2, meaning red must spread at least twice as fast as blue.Above a maximum α, the coexistence region disappears.
- Finite-size effects: Finite-size effects round the discontinuous boundary and can make slower diseases competitive when growth rates are similar or when they receive a head start.With β = 0.9 and A = 10, the network would need n ≫ 10^10 to suppress these effects; stochastic fluctuations can also produce dominance or coexistence contrary to infinite-n predictions.
III. EPIDEMIC GROWTH RATES
The paper derives each disease’s early exponential growth from network reproductive numbers, then uses the ratio of growth rates to identify which disease dominates on large networks.
- The basic reproductive ratio R0 is the average number of additional infections caused by a newly infected individual.
- Both diseases grow exponentially above threshold, at rates ln Rb for blue and α^-1 ln Rr for red.These rates are positive when Rb, Rr > 1.
- β is the ratio of the diseases’ exponential growth rates and can be calculated from Tr, Tb, α, and the network degree distribution.
- On large networks, the faster-growing disease fills an O(n) fraction before the slower disease exceeds a vanishing fraction, except where finite-size effects matter.The β = 1 growth-rate boundary separates blue-dominant and red-dominant regions.
IV. PERCOLATION ANALYSIS
The paper maps sequential disease spread onto percolation on the residual network left by the first epidemic. This analysis yields epidemic and coexistence thresholds and conditions for extensive spread by both diseases.
- Percolation setup: When one disease spreads first, the second is analyzed by bond percolation on the residual network left after the first epidemic.The residual network contains individuals not infected by the first disease and therefore not immune to the second.
- Residual network: The residual network remains a configuration-model network, but removing infected vertices changes the degree distribution of their uninfected neighbors.
- Outbreak size: The second giant cluster occupies C(1 − S) of the original network when it fills fraction C of the residual network.
- Thresholds: The second disease’s threshold is computed from degree averages or generating functions for the residual network.The residual-network threshold φb depends on the first disease’s transmissibility through u.
- Coexistence: Coexistence requires the second disease to have higher transmissibility than the first, so equal transmissibilities cannot produce coexistence.
- Coexistence: If blue grows faster, coexistence is impossible; the coexistence regime instead requires red to grow faster while blue has higher transmission probability.The authors note that these contrary criteria may be uncommon for real diseases.
- Poisson networks: For Poisson networks, coexistence requires red to transmit at least twice as fast as blue, with a stronger requirement for larger Tb or c.
V. NUMERICAL RESULTS
Numerical simulations generally converge to the analytic scaling and epidemic-size predictions, but finite-size effects and early stochastic fluctuations strongly affect outcomes near the growth-rate boundary. Simulations also display both the discontinuous dominance switch and the continuous coexistence transition.
- Numerical results generally converge to the expected large-n scaling as network size increases, confirming the analytic calculations in most parameter regimes.The simulations average red and blue infection counts over 1000 networks and compare them with the expected scaling slopes.
- Near the growth-rate boundary, averaged infection counts show intermediate scaling between the expected O(n^β) and O(n), because early fluctuations can let the slower disease dominate.Reported slopes include 0.70, 0.66, and 0.76, reflecting the influence of finite-size effects and stochastic early growth.
- Increasing network size does not quickly restore asymptotic behavior near the growth-rate boundary, while adding initial carriers reduces fluctuations but increases finite-size effects.The paper concludes that there is no easy way to reach the asymptotic scaling regime near this boundary.
- For fixed transmissibilities outside coexistence, epidemic sizes are independent of α except at a discontinuous growth-rate boundary where the dominant disease switches.Away from the boundary, analytic and numerical results agree well; near it, chance fluctuations produce poorer agreement.
- When parameter ranges include coexistence, simulations show both the discontinuous growth-rate transition and the continuous coexistence transition.Finite-size rounding is most evident near the coexistence transition for smaller networks, whereas deviations are more dramatic around the growth-rate boundary.
VI. EPIDEMIC SIZES CLOSE TO THE GROWTH-RATE BOUNDARY
Near the growth-rate boundary, finite-size effects and early stochastic fluctuations prevent unique prediction of epidemic sizes, although the two final sizes remain mathematically related. Simulations show dominance, coexistence, and fluctuation-driven reversals across parameter regimes.
- Finite-size effects: Finite-size effects and early stochastic fluctuations cause significant deviations from large-n predictions near the growth-rate boundary.In this regime, the large-n theory cannot predict either epidemic’s final size uniquely.
- Analytic relation: Eq. (29) constrains the two epidemic sizes even when chance fluctuations leave their individual values undetermined.Knowing either probability determines the other through the equation’s relation.
- Dominance: As network size increases, dominance becomes progressively better defined when parameters lie outside the coexistence region.Finite networks can still allow the slower disease to win occasionally near the boundary.
- Coexistence: Below the coexistence threshold, one disease dominates outside the growth-rate boundary, whereas within the coexistence region both infect significant network fractions even for large n.The figure’s third row shows coexistence, while its fourth row shows blue dominance.
- Early growth: The faster-growing disease can dominate early, with the slower disease’s final size then determined by the fast disease’s outbreak size.Simulations support this relationship when the early slow-to-fast ratio is small.
- Finite-size scaling: Larger systems shift the typical slow-to-fast ratio leftward because they take longer to reach the 1/n point.The ratio dwindles over time as the network grows toward its epidemic scale.
VII. CONCLUSION
The paper analyzes concurrently spreading, fully cross-immune diseases using analytic arguments, heuristics, and simulations to derive their phase structure and epidemic sizes. It identifies a discontinuous growth-rate boundary, while emphasizing model and network-scope limitations and unresolved finite-size behavior.
- Main findings: The authors derive a four-phase diagram and calculate expected numbers infected by two concurrently spreading diseases with complete cross-immunity.The study combines analytic results, heuristic arguments, and numerical simulation on a static contact network.
- Main findings: The growth-rate boundary is a discontinuous transition between dominance by one disease and dominance by the other in the large-system limit.Finite systems blur this transition through strong finite-size effects.
- Scope and extensions: The analysis uses the Reed–Frost model, while extensions to other SIR-style dynamics may change the boundary’s shape without changing its qualitative transition.This expected robustness is presented as a possible generalization, not as a result directly established here.
- Limitations: Analytic results are derived for the configuration model, and other network structures might exhibit different behavior.This is an explicit scope boundary of the paper’s calculations.
- Limitations: Near the growth-rate boundary, finite-size deviations remain strong and the epidemic-growth exponents are not presently calculable from the analysis.The paper also assumes neither disease dies out during early growth, leaving some outcome probabilities unresolved.