Modeling and Analysis of a Coupled SIS Bi-Virus Model

TL;DR

This work develops a coupled bivirus SIS framework to capture simultaneous infection by two viruses on directed networks, deriving a mean-field ODE model from a $4^n$-state Markov process and introducing coupling parameters $\epsilon^{(m)}$ to regulate cross-infection effects. It establishes rigorous stability results for the disease-free state via spectral conditions, and derives existence and stability results for single-virus endemic equilibria, including a corollary on the case where both viruses pervade. The analysis shows that certain coexisting equilibria cannot exist (notably when $\epsilon^{(m)}>0$) and provides necessary conditions for other coexisting configurations, revealing the model is not monotone and thus not amenable to standard competitive bivirus techniques. Simulations on line, star, complete, and large-scale networks validate the mean-field model against the full Markov chain and illustrate diverse outcomes (extinction, single-virus persistence, or potential coexistence) under varying coupling strengths, underscoring the complex dynamics of coinfection and the limitations of monotone-system methods in this setting.

Abstract

The paper deals with the setting where two viruses (say virus 1 and virus 2) coexist in a population, and they are not necessarily mutually exclusive, in the sense that infection due to one virus does not preclude the possibility of simultaneous infection due to the other. We develop a coupled bi-virus susceptible-infected-susceptible (SIS) model from a 4n-state Markov chain model, where n is the number of agents (i.e., individuals or subpopulation) in the population. We identify a sufficient condition for both viruses to eventually die out, and a sufficient condition for the existence, uniqueness and asymptotic stability of the endemic equilibrium of each virus. We establish a sufficient condition and multiple necessary conditions for local exponential convergence to the boundary equilibrium (i.e., one virus persists, the other one dies out) of each virus. Under mild assumptions on the healing rate, we show that there cannot exist a coexisting equilibrium where for each node there is a nonzero fraction infected only by virus 1; a nonzero fraction infected only by virus 2; but no fraction that is infected by both viruses 1 and 2. Likewise, assuming that healing rates are strictly positive, a coexisting equilibrium where for each node there is a nonzero fraction infected by both viruses 1 and 2, but no fraction is infected only by virus 1 (resp. virus 2) does not exist. Further, we provide a necessary condition for the existence of certain other kinds of coexisting equilibria. We show that, unlike the competitive bivirus model, the coupled bivirus model is not monotone. Finally, we illustrate our theoretical findings using an extensive set of in-depth simulations.

Figures (18)

• Figure 1: The possible states and transitions for node $i$. State $S$ stands for being healthy but susceptible. States $I^{(1)}$, $I^{(2)}$, and $I^{(1,2)}$ stand for being infected by virus 1, virus 2, and both virus 1 and virus 2, respectively.
• Figure 2: Graph structures: a) line, b) star, c) complete.
• Figure 3: A plot of $\|[v^{(1)}(T); v^{(2)}(T); v^{(3)}(T)] -[x^{(1)}(T); x^{(2)}(T); z(T)]\|$ for the line graph, $T=10,000$. Results from using the different initial conditions \ref{['eq:initial1']}, \ref{['eq:initial2']}, and \ref{['eq:initial3']} are depicted by the blue lines, red dashed lines, and black dash-dot lines, respectively.
• Figure 4: A plot of $\|[v^{(1)}(T); v^{(2)}(T); v^{(3)}(T)] -[x^{(1)}(T); x^{(2)}(T); z(T)]\|$ for the star graph, $T=10,000$. Results from using the different initial conditions \ref{['eq:initial1']}, \ref{['eq:initial2']}, and \ref{['eq:initial3']} are depicted by the blue lines, red dashed lines, and black dash-dot lines, respectively.
• Figure 5: A plot of the error $\|[v^1(T); v^2(T); v^3(T)] -[x^{(1)}(T); x^{(2)}(T); z(T)]\|$ for the complete graph, $T=10,000$. Results from using the different initial conditions \ref{['eq:initial1']}, \ref{['eq:initial2']}, and \ref{['eq:initial3']} are depicted by the blue lines, red dashed lines, and black dash-dot lines, respectively.

Theorems & Definitions (24)

• Remark 1
• Lemma 1
• Proposition 1
• Theorem 1
• Theorem 2
• Corollary 1
• Corollary 2
• Theorem 3
• Proposition 2
• Remark 2