A Discrete-time Networked Competitive Bivirus SIS Model

The paper deals with the analysis of a discrete-time networked competitive bivirus susceptible-infected-susceptible (SIS) model. More specifically, we suppose that virus 1 and virus 2 are circulating in the population and are in competition with each other. We show that the model is strongly monotone, and that, under certain assumptions, it does not admit any periodic orbit. We identify a sufficient condition for exponential convergence to the disease-free equilibrium (DFE). Assuming only virus 1 (resp. virus 2) is alive, we establish a condition for global asymptotic convergence to the single-virus endemic equilibrium of virus 1 (resp. virus 2) -- our proof does not rely on the construction of a Lyapunov function. Assuming both virus 1 and virus 2 are alive, we establish a condition which ensures local exponential convergence to the single-virus equilibrium of virus 1 (resp. virus 2). Finally, we provide a sufficient (resp. necessary) condition for the existence of a coexistence equilibrium

General SIS diffusion process with indirect spreading pathways on a hypergraph

While conventional graphs only characterize pairwise interactions, higher-order networks (hypergraph, simplicial complex) capture multi-body interactions, which is a potentially more suitable modeling framework for a complex real system. However, the introduction of higher-order interactions brings new challenges for the rigorous analysis of such systems on a higher-order network. In this paper, we study a series of SIS-type diffusion processes with both indirect and direct pathways on a directed hypergraph. In a concrete case, the model we propose is based on a specific choice (polynomial) of interaction function (how several agents influence each other when they are in a hyperedge). Then, by the same choice of interaction function, we further extend the system and propose a bi-virus competing model on a directed hypergraph by coupling two single-virus models together. Finally, the most general model in this paper considers an abstract interaction function under single-virus and bi-virus settings. For the single-virus model, we provide the results regarding healthy state and endemic equilibrium. For the bi-virus setting, we further give an analysis of the existence and stability of the healthy state, dominant endemic equilibria, and coexisting equilibria. All theoretical results are finally supported by some numerical examples

Towards Understanding the Endemic Behavior of a Competitive Tri-Virus SIS Networked Model

This paper studies the endemic behavior of a multi-competitive networked susceptible-infected-susceptible (SIS) model. Specifically, the paper deals with three competing virus systems (i.e., tri-virus systems). First, we show that a tri-virus system, unlike a bi-virus system, is not a monotone dynamical system. Using the Parametric Transversality Theorem, we show that, generically, a tri-virus system has a finite number of equilibria and that the Jacobian matrices associated with each equilibrium are nonsingular. The endemic equilibria of this system can be classified as follows: a) single-virus endemic equilibria (also referred to as the boundary equilibria), where precisely one of the three viruses is alive; b) 2-coexistence equilibria, where exactly two of the three viruses are alive; and c) 3-coexistence equilibria, where all three viruses survive in the network. We provide a necessary and sufficient condition that guarantees local exponential convergence to a boundary equilibrium. Further, we secure conditions for the nonexistence of 3-coexistence equilibria (resp. for various forms of 2-coexistence equilibria). We also identify sufficient conditions for the existence of a 2-coexistence (resp. 3-coexistence) equilibrium. We identify conditions on the model parameters that give rise to a continuum of coexistence equilibria. More specifically, we establish i) a scenario that admits the existence and local exponential attractivity of a line of coexistence equilibria; and ii) scenarios that admit the existence of, and, in the case of one such scenario, global convergence to, a plane of 3-coexistence equilibria.Comment: arXiv admin note: substantial text overlap with arXiv:2209.1182