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Bi-Virus SIS Epidemic Propagation under Mutation and Game-theoretic Protection Adoption

Urmee Maitra, Ashish R. Hota, Vaibhav Srivastava

TL;DR

This work analyzes a bi-virus SIS model with two strains $\mathtt{H}$ and $\mathtt{L}$, incorporating mutation between strains and a game-theoretic protection adoption by susceptibles. It derives precise conditions for the disease-free equilibrium to be globally stable and proves the existence and global stability of a unique endemic equilibrium under a mutation-rate assumption, using a Lyapunov function and Dulac's criterion. The authors then characterize the stationary Nash equilibrium for protection adoption, showing how protection cost $C_P$ and infection impact shape adoption, and demonstrate monotone decreases in infection levels with higher protection. They further specialize to uni-directional mutation, revealing the impossibility of coexistence in one direction and the persistence of either coexistence or single-strain dominance in the other, depending on mutation rates. Numerical simulations validate the theory and illustrate how mutation rates and protection costs shape infection trajectories and equilibrium states, highlighting practical implications for protection incentives in multi-strain epidemics. All results are expressed with rigorous stability analyses and exploit the Metzler structure and monotonicity properties of the system.

Abstract

We study a bi-virus susceptible-infected-susceptible (SIS) epidemic model in which individuals are either susceptible or infected with one of two virus strains, and consider mutation-driven transitions between strains. The general case of bi-directional mutation is first analyzed, where we characterize the disease-free equilibrium and establish its global asymptotic stability, as well as the existence, uniqueness, and stability of an endemic equilibrium. We then present a game-theoretic framework where susceptible individuals strategically choose whether to adopt protection or remain unprotected, to maximize their instantaneous payoffs. We derive Nash strategies under bi-directional mutation, and subsequently consider the special case of unidirectional mutation. In the latter case, we show that coexistence of both strains is impossible when mutation occurs from the strain with lower reproduction number and transmission rate to the other strain. Furthermore, we fully characterize the stationary Nash equilibrium (SNE) in the setting permitting coexistence, and examine how mutation rates influence protection adoption and infection prevalence at the SNE. Numerical simulations corroborate the analytical results, demonstrating that infection levels decrease monotonically with higher protection adoption, and highlight the impact of mutation rates and protection cost on infection state trajectories.

Bi-Virus SIS Epidemic Propagation under Mutation and Game-theoretic Protection Adoption

TL;DR

This work analyzes a bi-virus SIS model with two strains and , incorporating mutation between strains and a game-theoretic protection adoption by susceptibles. It derives precise conditions for the disease-free equilibrium to be globally stable and proves the existence and global stability of a unique endemic equilibrium under a mutation-rate assumption, using a Lyapunov function and Dulac's criterion. The authors then characterize the stationary Nash equilibrium for protection adoption, showing how protection cost and infection impact shape adoption, and demonstrate monotone decreases in infection levels with higher protection. They further specialize to uni-directional mutation, revealing the impossibility of coexistence in one direction and the persistence of either coexistence or single-strain dominance in the other, depending on mutation rates. Numerical simulations validate the theory and illustrate how mutation rates and protection costs shape infection trajectories and equilibrium states, highlighting practical implications for protection incentives in multi-strain epidemics. All results are expressed with rigorous stability analyses and exploit the Metzler structure and monotonicity properties of the system.

Abstract

We study a bi-virus susceptible-infected-susceptible (SIS) epidemic model in which individuals are either susceptible or infected with one of two virus strains, and consider mutation-driven transitions between strains. The general case of bi-directional mutation is first analyzed, where we characterize the disease-free equilibrium and establish its global asymptotic stability, as well as the existence, uniqueness, and stability of an endemic equilibrium. We then present a game-theoretic framework where susceptible individuals strategically choose whether to adopt protection or remain unprotected, to maximize their instantaneous payoffs. We derive Nash strategies under bi-directional mutation, and subsequently consider the special case of unidirectional mutation. In the latter case, we show that coexistence of both strains is impossible when mutation occurs from the strain with lower reproduction number and transmission rate to the other strain. Furthermore, we fully characterize the stationary Nash equilibrium (SNE) in the setting permitting coexistence, and examine how mutation rates influence protection adoption and infection prevalence at the SNE. Numerical simulations corroborate the analytical results, demonstrating that infection levels decrease monotonically with higher protection adoption, and highlight the impact of mutation rates and protection cost on infection state trajectories.

Paper Structure

This paper contains 15 sections, 10 theorems, 54 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and Background
  3. SIS Mutation Model Coupled with Protection Adoption Behavior
  4. Equilibria and their Stability under Bi-directional Mutation
  5. Stability of Disease-free Equilibrium
  6. Characterization of Endemic Equilibrium
  7. Game-Theoretic Protection Adoption under Bi-directional Mutation
  8. Specialization to Uni-directional Mutation
  9. Mutation from Strain $\mathtt{H}$ to $\mathtt{L}$
  10. Mutation from Strain $\mathtt{L}$ to $\mathtt{H}$
  11. Numerical Simulations
  12. Conclusion
  13. Proof of Theorem \ref{['thm:existence']}
  14. Proof of Lemma \ref{['prop:states_mon_dec']}
  15. Proof of Corollary \ref{['cor:GAS_qlh=0']}

Key Result

Proposition 1.1

Suppose $\Lambda \in \mathbb{R}^{n \times n}$ is a diagonal matrix with all entries being strictly negative, and $N \in \mathbb{R}^{n \times n}$ is an irreducible nonnegative matrix. Let $M = \Lambda + N$. Then, following relations hold:

Figures (6)

  • Figure 1: Evolution of infection states of an individual in the SIS model with two mutant strains.
  • Figure 2: Variation of steady-state infection levels $(\mathtt{I}_{\mathtt{H}}^{\star}, \mathtt{I}_{\mathtt{L}}^{\star})$, and Nash equilibrium $(z_{\mathtt{S}}^{\mathtt{NE}})$ with protection cost $(\mathtt{C}_{\mathtt{P}})$, for mutation rates (left) $q_\mathtt{HL} = 0.05, q_\mathtt{LH} = 0.1$; (middle) $q_\mathtt{HL} = 0.1, q_\mathtt{LH} = 0.1$; and (right) $q_\mathtt{HL} = 0.18, q_\mathtt{LH} = 0.1$.
  • Figure 3: Variation of steady-state infection levels $(\mathtt{I}_{\mathtt{H}}^{\star}, \mathtt{I}_{\mathtt{L}}^{\star})$ and Nash equilibrium $(z_{\mathtt{S}}^{\mathtt{NE}})$ with protection cost $(\mathtt{C}_{\mathtt{P}})$, for mutation rate (left) $q_\mathtt{HL} = 0.01$; (middle) $q_\mathtt{HL} = 0.13$; and (right) $q_\mathtt{HL} = 0.19$.
  • Figure 4: Figure depicting a closed curve, and the associated vector fields of dynamics \ref{['eq:bidir_ihdot']} and \ref{['eq:bidir_ildot']}.
  • Figure 5: Three-quarter circular arc (magnified) with center at origin and both eigenvalues being positive, i.e., $\theta_{1} > \theta_{2}$.
  • ...and 1 more figures

Theorems & Definitions (28)

  • Proposition 1.1: Proposition 1, liu2019analysis
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Remark 3.4
  • Remark 3.6
  • Theorem 3.7
  • ...and 18 more