Competing epidemics on complex networks

Abstract: 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.

• The paper analyzes the dynamics of two competing SIR epidemics with cross-immunity on complex networks, using bond percolation to map transmission.
• Key findings include a phase diagram with a novel growth-rate boundary predicting dominance shifts and conditions under which both diseases can coexist.
• The study highlights the importance of network structure, transmissibility, and stochastic effects for understanding simultaneous outbreaks and informing public health strategies.

Analysis of Competing Epidemics on Complex Networks

The study elucidated the dynamics of simultaneous competing epidemics within a shared network structure, specifically exploring the spread of two diseases where recovery from either yields immunity against both. This critical investigation hinges on a nuanced understanding of the network's architecture and its influence on disease dissemination.

The susceptible-infected-recovered (SIR) model forms the theoretical backbone of this research, mapping disease transmission to bond percolation processes on complex networks. The study primarily focuses on the configuration model, assuming a random graph framework with a predefined degree distribution.

Key Findings

1. Phase Diagram and Growth-Rate Boundary:
   • The research identifies a phase diagram demonstrating distinct regions based on the relative growth rates and transmissibilities of the two competing diseases.
   • A novel dynamic transition, termed the growth-rate boundary, distinguishes the dominance of one disease over the other. This boundary marks the conditions where the exponential growth rates are equal, resulting in a discontinuity in the number of eventual infections for both diseases in the limit of large network sizes.
2. Coexistence Regime:
   • Notably, the study delineates conditions under which both diseases can coexist and affect a substantial fraction of the network population. Coexistence necessitates that the slower-growing disease exhibits a higher transmissibility, preventing mutual exclusivity from immutably determining the epidemic's extent.
3. Epidemic Thresholds and Percolation Analysis:
   • The paper further utilizes a bond percolation approach to calculate thresholds determining whether an epidemic will occur based on the transmissibility and structural properties of the residual network after the initial (faster) epidemic.
   • The positioning of these thresholds plays a crucial role in defining the conditions under which one disease can subsequently spread following the initial epidemic's vestige.
4. Finite-Size Effects and Stochastic Fluctuations:
   • While the theoretical predictions hold under the assumption of infinite network size, the study acknowledges significant finite-size effects in real-world networks. This phenomenon is amplified near the growth-rate boundary, where stochastic fluctuations during the initial stages of an outbreak considerably influence the epidemics' outcomes.

Implications and Future Directions

The model's insights offer profound implications for understanding concurrent epidemic dynamics, particularly in scenarios involving pathogens with cross-immunity effects. The research contributes substantially to network epidemiology by challenging the preconceived notion of sequential spreading being a requisite for substantial epidemic outcomes.

From a practical standpoint, this understanding can inform public health strategies prioritizing interventions based on the interplay between epidemic transmissibility and network connectivity. The results underscore the importance of considering both network structure and stochastic dynamics when preparing for and responding to simultaneous infectious disease outbreaks.

Future investigative avenues could include:

• Extending the current SIR model framework to consider more complex network topologies and interaction dynamics that better reflect real-world contact networks.
• Exploring scenarios incorporating partial instead of complete immunity following infection and recovery.
• Analyzing the impact of varied transmission and recovery time distributions to better capture realistic disease dynamics.

Overall, this study lays a foundational understanding of competing epidemic dynamics on complex networks, opening up new vistas for theoretical explorations and practical applications in public health and epidemiological modeling.