On a Class of Decentralized Feedback Controllers for the Networked Bivirus SIS Model

I. INTRODUCTION

• In the uncontrolled single-population model, generic endemic outcomes are winner-take-all: the virus with the stronger infection-to-healing ratio drives out the other.The paper’s outline distinguishes its single-population bifurcation analysis from its networked equilibrium-stability analysis.Source: §I · pp. 1–2
• The proposed controls raise healing monotonically with infection, targeting lower infected fractions at endemic equilibria.The paper examines how this control changes equilibria and convergence in the two-virus setting.Source: §I · p. 1
• Extending decentralized feedback from the single-virus case is nontrivial because two viruses have coupled spreading and competitive dynamics.The prior controller class could reduce, but not eliminate, the endemic equilibrium infection level in the single-virus setting.Source: §I · p. 1
• With control, a coexistence equilibrium can be almost globally attractive, so the uncontrolled winner-take-all equilibrium picture may fail.The paper notes that both viruses may persist instead of one becoming extinct.Source: §I · p. 2
• For linear feedback functions, the shift from winner-take-all to possible coexistence is associated with a transcritical bifurcation as the scaling parameter changes.The paper also establishes that the considered decentralized controllers cannot stabilize an unstable boundary equilibrium in the networked case.Source: §I · p. 2

II. MODEL

• For each virus, healing parameters are diagonal in D_k and spreading between populations is encoded by B_k.The populations form a two-layer contact network, with a separate edge set for each virus.Source: §II · p. 2
• The model assumes positive healing parameters and nonnegative spreading matrices, with each virus’s network layer strongly connected.Strong connectivity is expressed as irreducibility of each B_k.Source: §II · p. 2
• The decentralized local state feedback uses bounded, smooth, nondecreasing functions that raise healing with infection and vanish at zero infection.The controlled dynamics retain a state-nonlinear term.Source: §II · p. 2
• The admissible state region D is positively invariant under control, and interior initial states remain strictly inside its boundaries at every finite positive time.The region restricts population fractions to [0, 1] while the two virus fractions sum to at most one.Source: §II · pp. 2–3
• At equilibrium, each virus is either absent or present in every population, and the two virus fractions together remain strictly below one.Boundary equilibria are the disease-free state or a single-virus endemic state; single-virus endemic equilibria exist uniquely for each virus.Source: §II · p. 3
• The controlled system is strongly monotone, and if it has finitely many equilibria, almost all initial conditions converge to a stable equilibrium.Any limit cycles must be nonattractive, and other behaviors such as chaos are ruled out.Source: §II · p. 3

III. THE SINGLE POPULATION CASE

• The single-population model is used as a tractable baseline for motivating extensions to the networked case.The section contrasts the equilibrium possibilities of controlled and uncontrolled dynamics.Source: §III · p. 3
• For generic parameters, the uncontrolled single-population model has no coexistence equilibrium, so when both viruses persist, one must win.The winner is the virus whose single-virus endemic equilibrium is stable.Source: §III · p. 4
• Under control, a coexistence equilibrium is possible, yielding a state in which neither virus wins the winner-take-all contest.The supplied figure caption identifies simulations for the n = 1 case and distinguishes the viruses by line color.Source: §III · p. 4, Figure 1 · p. 4

A. An illustrative example

• With δ1 = δ2 = 1, b1 = 3, and b2 = 2, the uncontrolled system converges to the virus-1 boundary equilibrium from all interior initial conditions.This is the behavior shown in Fig. 1a.Source: A. An illustrative example · p. 4
• With h1(x1) = αx1 and h2(x2) = 0, increasing α shifts convergence to a coexistence equilibrium in the simulated cases.The control is applied only to virus 1 in this example.Source: A. An illustrative example · p. 4
• The example suggests a threshold near α = 1, below which the virus-1 boundary equilibrium remains and above which coexistence occurs.The uncontrolled system cannot admit coexistence for these generic parameters.Source: A. An illustrative example · p. 4

• As α increases, E0 and E2 remain fixed while E1 and E3 move on intersecting lines; E3 is an additional solution for α ≠ 0 that enters D above a critical value.Source: B. Formal Analysis · p. 4
• The stability character of E0 and E2 is independent of α: they remain an unstable node and a saddle node, respectively.Source: B. Formal Analysis · p. 4
• As α rises, E1 loses stability and a stable coexistence equilibrium appears in a transcritical bifurcation; when both boundary equilibria are unstable, monotone systems theory guarantees coexistence.Source: B. Formal Analysis · p. 5
• Across stable coexistence equilibria, the total infected fraction stays constant as α changes, which instead shifts the mix of infections between the viruses.Source: B. Formal Analysis · p. 5

IV. EQUILIBRIA STABILITY ANALYSIS FOR THE GENERAL CASE - DFE AND BOUNDARY EQUILIBRIA

• The disease-free equilibrium is unique exactly when s(−Dk + Bk) ≤ 0 for both viruses; under that condition, each virus infection converges to zero from every initial state in D.For each virus separately, the same spectral condition ensures its infection converges to zero.Source: §IV · p. 5
• When s(−D1 + B1) > 0, the controlled system has a nonzero virus-1 boundary equilibrium in addition to the disease-free equilibrium.This follows by restricting the dynamics to initial states with virus 2 absent.Source: §IV · p. 6
• Under the stated assumptions, controlled single-virus boundary equilibria are smaller than their uncontrolled counterparts, and control cannot stabilize an unstable boundary equilibrium.The instability result applies to either virus.Source: §IV · pp. 6–7
• In the single-population case, control on the dominant virus can make its stable boundary equilibrium unstable while a stable coexistence equilibrium appears.The paper associates this transition with a transcritical bifurcation when the feedback functions are linear.Source: §IV · p. 7
• In one two-population simulation, increasing control on virus 1 made its boundary equilibrium unstable and produced an attractive coexistence equilibrium.Simulations from several initial conditions suggested that the coexistence equilibrium was globally attractive; they did not converge to the virus-2-only equilibrium.Source: §IV · p. 7
• In another two-population simulation, control with α = 0.1 made the virus-1 boundary equilibrium unstable, and trajectories converged to the virus-2 boundary equilibrium except when already at another equilibrium.The uncontrolled system had two locally stable boundary equilibria and one unstable coexistence equilibrium.Source: §IV · p. 7

V. CONCLUSION

• In the single-population case, control can produce isolated coexistence equilibria, which the uncontrolled bivirus system does not admit.For general networked systems, the controller cannot stabilize an unstable boundary equilibrium of the uncontrolled system.Source: §V · p. 7
• Control can replace a stable winner-take-all equilibrium with stable coexistence through a transcritical bifurcation.The conclusion describes this as a form of instability introduced by control.Source: §V · p. 7
• Future work aims to derive equilibrium and Jacobian expressions under simultaneous control of both viruses and possibly extend the controller to time-varying SIS epidemics.Source: §V · p. 7