⚡ electrical engineering

Analysis of a Competitive Bivirus SIS Epidemic Model with Game Theoretic Social Distancing

This paper proposes a competitive bi-virus SIS model incorporating dynamic social distancing driven by public perception, demonstrating that this behavioral component breaks monotonicity and fundamentally alters equilibrium structures and stability criteria compared to classic models, while providing rigorous conditions for the global and local stability of disease-free, unilateral, and coexistence equilibria.

Original authors: Benjamin Catalano, Keith Paarporn, Sebin Gracy

Published 2026-05-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Benjamin Catalano, Keith Paarporn, Sebin Gracy

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a town where two different viruses, let's call them Virus A and Virus B, are spreading. In the old way of thinking about epidemics, scientists would just look at how fast the viruses spread and how fast people recover. They would assume people are just passive passengers in this story, getting sick or getting better without changing their behavior.

But this paper tells a different story. It suggests that people are active players. When they see a virus spreading, they make a choice: do they stay home and avoid crowds (social distancing), or do they keep living their normal lives?

Here is a simple breakdown of what the authors discovered, using some everyday analogies.

1. The "Game" of Staying Home

The authors treat social distancing like a game of strategy.

The Cost: Staying home is annoying and costs money (you can't go to work or the store). Let's call this the "Social Distancing Tax."
The Benefit: Staying home lowers your chance of getting sick.
The Decision: People constantly weigh the "Tax" against the "Risk." If the town is full of sick people, the risk is high, so staying home feels worth it. If the town is healthy, the risk is low, so people might decide the "Tax" isn't worth paying and go out again.

The paper models this using a tool called Replicator Equations. Think of this as a "copycat" mechanism. If you see your neighbor staying home and not getting sick, you might copy them. If you see your neighbor going out and getting sick, you might copy them too. The paper tracks how these choices change over time.

2. Two Viruses Fighting for the Same Room

The paper looks at a scenario where two viruses are fighting for the same group of people.

The Rule: You can only have one virus at a time. If you catch Virus A, you can't catch Virus B until you recover.
The Outcome: Usually, one virus is stronger and pushes the other out (like a bully taking a kid's lunch). But sometimes, they can coexist, like two rival gangs sharing a neighborhood, each controlling a different block.

3. The Big Surprise: It's Not a Straight Line

In classic epidemic models, things are predictable. If you add more people, the virus spreads more. It's a straight line.

The Paper's Finding: Because people are changing their behavior based on what they see, the system becomes messy and unpredictable.
The Analogy: Imagine a thermostat that doesn't just turn the heat on or off based on temperature, but also turns the heat on if the neighbors are cold, and off if the neighbors are hot. This creates a feedback loop where the system might swing wildly back and forth instead of settling down. The authors prove mathematically that this new model is not monotone, meaning you can't just guess the future by looking at the present; the system can do wild, complex things.

4. The Three Possible Endings (Equilibria)

The paper maps out the different ways this story can end up:

The "Clean Town" (Disease-Free): If the viruses are weak or the "Social Distancing Tax" is low enough, people stay home, the viruses die out, and eventually, everyone goes back to normal.
The "One-Winner" Scenario: One virus is stronger. It infects enough people that the other virus gets squeezed out.
- Twist: Depending on how scared people are, the winner might be the one that makes people stay home (because the sick people stay home) or the one that makes people ignore the rules.
The "Stalemate" (Coexistence): Under very specific conditions (where the two viruses are almost identical in strength), they can share the town forever. The paper shows these aren't just single points, but lines of balance. Imagine a seesaw where the viruses can settle anywhere along the plank, as long as they balance each other out.

5. Perception is Reality

One of the most interesting parts of the paper is how perception changes the outcome.

The paper assumes people don't know the exact math of the virus; they just see how many people are sick.
If people think Virus A is scarier than Virus B (even if it's not), they will stay home more to avoid A. This actually helps Virus B spread because people are less careful about it!
The Lesson: How the public feels about the danger of a virus can determine which virus wins, or if they both die out.

6. What Happens When People Copy Each Other?

In the final part of the paper, the authors add a "coordination" term. This means people don't just look at the sickness; they look at what everyone else is doing.

If everyone is staying home, you feel safe staying home.
If everyone is going out, you feel safe going out.
The Result: This can cause the town to swing back and forth in cycles. The virus spikes, people panic and stay home, the virus dies down, people relax and go out, and the virus spikes again. The paper shows that this "herd mentality" can create endless waves of infection.

Summary

This paper builds a mathematical model where viruses and human behavior dance together. It shows that when two viruses compete, the outcome isn't just about biology; it's about how scared people are, how much staying home costs them, and how much they copy their neighbors. The result is a complex system where the "winner" isn't always the strongest virus, but the one that best fits the mood of the crowd.

Technical Summary: Analysis of a Competitive Bi-virus SIS Epidemic Model with Game Theoretic Social Distancing

Problem Statement
Mathematical epidemiology has traditionally focused on single-virus spread or static behavioral assumptions. However, real-world scenarios often involve the simultaneous circulation of competing viruses (e.g., influenza and the common cold) and dynamic public responses to outbreaks, such as social distancing. Existing competitive bi-virus models typically fail to account for the co-evolution of individual decision-making and disease transmission. Specifically, they do not model how public perception of viral threats and the associated costs of mitigation strategies dynamically alter the effective infectivity of competing strains. This paper addresses the gap by proposing a model that integrates competitive bi-virus dynamics with evolutionary game theory to describe how individuals dynamically choose to social distance or not based on the prevalence of two competing viruses.

Methodology
The authors construct a novel dynamical system consisting of five coupled non-linear ordinary differential equations (ODEs). The state variables represent the fractions of the population infected by Virus 1 ( $y_1$ ), Virus 2 ( $y_2$ ), and the fractions of susceptible ( $z_S$ ) and infected ( $z_1, z_2$ ) individuals practicing social distancing.

Epidemic Dynamics: The model uses a Susceptible-Infected-Susceptible (SIS) framework for two competitive viruses. Infection rates are modulated by a social distancing factor $q \in (0,1)$ , which reduces contact rates.
Behavioral Dynamics: Individual decisions to social distance are modeled using replicator equations, a standard tool in evolutionary game theory.
- Susceptible Individuals: Their payoff for social distancing depends on the economic/social cost ( $c_D$ ) versus the perceived risk of infection, which is a linear function of the current prevalence of both viruses ( $r_1 y_1 + r_2 y_2$ ).
- Infected Individuals: Their payoff depends on the cost of social distancing ( $c_D$ ) versus a perceived cost of not distancing ( $c_i$ ), representing the risk of infecting others.
Analysis Tools: The authors employ stability analysis techniques including Jacobian linearization, spectral radius analysis, and Center Manifold Theory. They specifically investigate the monotonicity of the system and the stability of various equilibrium classes: Disease-Free Equilibria (DFE), Unilateral Equilibria (single virus endemic), and Coexistence Equilibria (both viruses endemic).

Key Contributions
The paper makes several theoretical contributions to the understanding of multi-virus systems with dynamic behavior:

Non-Monotonicity: Unlike classic competitive bi-virus SIS models, the authors prove (Theorem 3.1) that their coupled system is not a monotone dynamical system (MDS). This implies that standard convergence guarantees for MDS do not apply, and more complex behaviors (such as limit cycles or chaos) cannot be ruled out without further analysis.
Equilibrium Classification: The paper exhaustively classifies the equilibria of the system:
- DFEs: Identifies eight possible DFEs based on the binary choices of social distancing for susceptible and infected populations.
- Unilateral Equilibria: Characterizes states where one virus persists and the other dies out, with varying levels of social distancing (none, all, or partial).
- Coexistence Equilibria: Demonstrates that coexistence equilibria do not appear as isolated points but as continua (lines or curves) of equilibria, occurring only when the base reproduction numbers of both viruses are identical ( $R_0^1 = R_0^2$ ).
Stability Conditions:
- DFE Stability: Provides necessary and sufficient conditions for the almost global exponential stability of DFEs, linking stability to the effective reproduction number being less than unity.
- Unilateral Stability: Derives conditions for the stability of single-virus endemic states, showing that stability depends on the dominant virus's effective reproduction number and the population's payoff for social distancing.
- Coexistence Stability: Uses Center Manifold Theory to establish local stability conditions for the continua of coexistence equilibria.

Results
The theoretical findings are supported by numerical simulations using Runge-Kutta methods. Key results include:

Behavioral Impact on Eradication: The model illustrates that public perception and the cost of social distancing can shift the system from an endemic state to a disease-free state, even if the base reproduction numbers ( $R_0$ ) are greater than 1.
Dominance and Exclusion: In unilateral scenarios, the virus with the higher effective reproduction number (mediated by social distancing behavior) tends to dominate and exclude the other, provided specific cost-benefit thresholds are met.
Continuum of Coexistence: When viruses have identical base reproduction numbers, the system admits a line of coexistence equilibria. The stability of this line depends on whether the effective reproduction number is sufficiently high or low relative to the cost of social distancing.
Complex Dynamics: In Section 9.2, the authors explore a modified model with a "coordination term" (where social distancing is socially reinforced). This modification introduces the possibility of limit cycles, demonstrating that the interplay between opinion dynamics and epidemic spread can generate oscillatory behavior not present in the base model.

Significance
The paper claims significance in rigorously illustrating how social behavior mediates the effective infectivity of competing viruses. By moving beyond static assumptions, the model reveals that:

Public perception of threat levels is a critical determinant in whether viruses persist or die out.
The distinction between the perception of different viruses is necessary to understand multi-virus scenarios.
The introduction of game-theoretic dynamics fundamentally alters the stability landscape compared to classic models, moving the system away from monotonicity and opening the door for more complex dynamical behaviors.

The work provides a framework for analyzing how mitigation policies (like social distancing) interact with viral competition, suggesting that the success of such policies depends heavily on the dynamic feedback between infection prevalence and individual cost-benefit calculations.