# Competitive Networked Bivirus SIS spread over Hypergraphs

Sebin Gracy, Brian D.O. Anderson, Mengbin Ye and César A. Uribe Sebin Gracy
and César A. Uribe are with the Department of Electrical and Computer
Engineering, Rice University, Houston, TX, USA<EMAIL_ADDRESS>cesar.uribe@rice.edu).
Brian D.O. Anderson is with the School of Engineering, Australian National
University, Canberra, Australia<EMAIL_ADDRESS>Mengbin Ye is
with the Centre for Optimisation and Decision Science, Curtin University,
Australia. (mengbin.ye@curtin.edu.au).

###### Abstract

The paper deals with the spread of two competing viruses over a network of
population nodes, accounting for pairwise interactions and higher-order
interactions (HOI) within and between the population nodes. We study the
competitive networked bivirus susceptible-infected-susceptible (SIS) model on
a hypergraph introduced in Cui et al. [1]. We show that the system has, in a
generic sense, a finite number of equilibria, and the Jacobian associated with
each equilibrium point is nonsingular; the key tool is the Parametric
Transversality Theorem of differential topology. Since the system is also
monotone, it turns out that the typical behavior of the system is convergence
to some equilibrium point. Thereafter, we exhibit a tri-stable domain with
three locally exponentially stable equilibria. For different parameter
regimes, we establish conditions for the existence of a coexistence
equilibrium (both viruses infect separate fractions of each population node).

##  I Introduction

The study of virus spread has been an active area of research for over two
centuries. In particular, diverse scientific communities, such as physics [2],
mathematics [3], computer science [4], automatic control [5] , etc., have
significantly aided in furthering our understanding of the complex mechanisms
behind the spread of a virus. Fundamental to this effort has been the
development of compartmental models where each individual is healthy and
susceptible (S), infected with a virus (I), or has recovered from a viral
infection (R). Two compartmental models, susceptible-infected-recovered (SIR)
and susceptible-infected-susceptible (SIS) have garnered significant attention
in several scientific disciplines, particularly in mathematical epidemiology.
In contrast to the SIR model, the SIS model allows for the possibility of
reinfection and is the focus of the present paper. More specifically, we will
deal with networked SIS models, with each node in the network being
representative of a large population, and the interconnection among the nodes
denotes the possible spreading pathways for the virus.

The existing literature on modeling virus spread typically relies on the
assumption that there is just a single virus present. However, one often
encounters scenarios where there are two viruses, say virus 1 and virus 2,
circulating in a meta-population (i.e., a network of population nodes). In
such a context, said viruses could be cooperative, i.e., infection with virus
1 (resp. virus 2) increases (resp. decreases) the likelihood of simultaneous
infection with virus 2 (resp. virus 1); see [6] for more details. Another
possibility is for the two viruses to compete; infection with virus 1 (resp.
virus 2) precludes the possibility of simultaneous infection with virus 2
(resp. virus 1) - this is the focus of the present paper. We stress that the
notion of competing viruses is not restricted to just epidemics; it manifests
itself in, among others, product adoption in a marketplace and the spread of
opinions in social networks [7].

Networked competitive multi-virus SIS models have been analyzed in substantial
depth in recent times; see [8, 9, 10, 11, 12, 13, 14, 3, 15]. A major drawback
of networked competitive bivirus SIS models studied in the aforementioned
papers is that they account only for pairwise interactions between
individuals. In reality, interactions in social groups often involve more than
two individuals - it is not unusual that an individual can _simultaneously_
interact with more than one other individual. This motivates the need for
higher-order networks such as hypergraphs 111Simplicial networks (see [16] for
more details) have also been used for studying HOI, see [17]., i.e., graphs
where an edge can connect more than two nodes, which are quite effective in
representing higher-order interactions (HOI) [17]. Inspired by the approach in
[18], an SIS model on a hypergraph has been proposed and analyzed in [19].
However, the analytic results therein relied on certain restrictions on the
network structure. Overcoming this drawback, a networked SIS model on a
hypergraph has been devised and studied in considerable detail in [20].
However, the modeling frameworks in [18, 19, 20] are restrictive in the sense
that none of these account for the possibility of more than one virus
simultaneously circulating in a given population. Addressing this shortcoming,
a competitive networked bivirus SIS model on a hypergraph has been developed
and analyzed in [1]. The set of equilibria for the model in [1] can be broadly
classified into three categories: the disease-free equilibrium (both viruses
have been eradicated), the boundary equilibria (one virus is dead, and the
other is alive); and coexistence equilibria (two viruses infect separate
fractions of every population node in the network). Nevertheless, the results
in [1] have the following limitations: a) some of the findings therein have
yet to be rigorously established, and b) the analysis, while improving our
understanding of the existence and stability of various equilibria, is not
exhaustive. The present paper aims to address the aforementioned gaps. Our
main contributions, therefore, are as follows:

  1. i)

We show that the networked bivirus SIS system with HOI has, in a generic
sense, a finite number of equilibria. Furthermore, for each equilibrium, the
associated Jacobian is a nonsingular matrix; see Theorem 1. In so doing, since
our proof of Theorem 1 does not, unlike the proof of [1, Theorem 5.5], require
the HOI infection rates to be set to zero, we establish the correctness of the
claim raised in [1, Theorem 5.5]. Building off of Theorem 1 and leveraging the
fact that the system is monotone as identified in [1, Theorem 5.5], we prove
that the typical behavior of the bivirus SIS system with HOI is convergence to
an equilibrium point; see Theorem 2.

  2. ii)

We identify a parameter regime that not only establishes the existence of
three equilibria (a single-virus endemic equilibrium corresponding to virus 1
(resp. virus 2) and the DFE) but also guarantees that all of the said
equilibria are locally exponentially stable at the same time; see Proposition
1.

  3. iii)

We identify a parameter regime, different from the one covered by Proposition
1, for the existence of a coexistence equilibrium. We do so under different
configurations of the boundary equilibria, viz. both being unstable and both
being stable; see Proposition 3 and Theorem 3, respectively.

Additionally, for the parameter regime covered by Proposition 1, we establish
existence of a coexistence equilibrium; see Proposition 4.

Notation: We denote the set of real numbers by $\mathbb{R}$ and the set of
nonnegative real numbers by $\mathbb{R}_{+}$. For any positive integer $n$, we
use $[n]$ to denote the set $\\{1,2,...,n\\}$. We use 0 and 1 to denote the
vectors whose entries all equal $0$ and $1$, respectively, and use $I$ to
denote the identity matrix. For a vector $x$, we denote the diagonal square
matrix with $x$ along the diagonal by $\operatorname{diag}(x)$. For any two
vectors $a,b\in\mathbb{R}^{n}$ we write $a\geq b$ if $a_{i}\geq b_{i}$ for all
$i\in[n]$, $a>b$ if $a\geq b$ and $a\neq b$, and $a\gg b$ if $a_{i}>b_{i}$ for
all $i\in[n]$. Likewise, for any two matrices $A,B\in\mathbb{R}^{n\times m}$,
we write $A\geq B$ if $A_{ij}\geq B_{ij}$ for all $i\in[n]$, $j\in[m]$, and
$A>B$ if $A\geq B$ and $A\neq B$. For a square matrix $M$, we use $\sigma(M)$
to denote the spectrum of $M$, $\rho(M)$ to denote the spectral radius of $M$,
and $s(M)$ to denote the spectral abscissa of $M$, i.e.,
$s(M)=\max\\{{\mathrm{Re}}(\lambda):\lambda\in\sigma(M)\\}$.

A real square matrix $A$ is called Metzler if all its off-diagonal entries are
nonnegative. A matrix $A$ is said to be an M-matrix if all of its off-diagonal
entries are nonpositive, and there exists a constant $c>0$ such that, for some
nonnegative $B$ and $c\geq\rho(B)$, $A=cI-B$. All eigenvalues of an M-matrix
have nonnegative real parts. Furthermore, if an M-matrix has an eigenvalue at
the origin, we say it is singular; if each eigenvalue has strictly positive
parts, then we say it is nonsingular. If $A(=[a_{ij}]_{n\times n})$ is a
nonnegative matrix, then $\rho(A)$ decreases monotonically with a decrease in
$a_{ij}$ for any $i,j\in[n]$. The matrix $A$ is reducible if, and only if,
there is a permutation matrix $P$ such that $P^{\top}AP$ is block upper
triangular; otherwise, $A$ is said to be irreducible. If a nonnegative $A$ is
irreducible, and $Ax=y$ for $x>\textbf{0}$, then $y>\textbf{0}$, and $y$
cannot have a zero in every position where $x$ has a zero.

##  II Problem Formulation

###  II-A Model

Consider a network of $n$ nodes. A node represents a well-mixed222Well-mixed
means that the probability of any two individuals in a node interacting with
each other is the same. population of individuals. We will assume that the
size of the population is fixed. We suppose two viruses, say virus 1 and virus
2, are spreading over such a network. Throughout this paper, we will assume
that the two aforementioned viruses are competing. Through pairwise or HOI as
described in more detail below, an otherwise healthy individual in node $i$
gets infected with virus 1 (resp. virus 2) due to contact with either other
individuals in node $i$ who are infected with virus 1 (resp. virus 2) and/or
with other individuals in node $j$ (where $j$ is a neighbor of $i$) who are
infected with virus 1 (resp. virus 2). When a single interaction is involved
(i.e., between two individuals in node $i$ or between an individual in node
$i$ and an individual in node $j$), we say that the infection is caused due to
_pairwise interactions_. An individual in node $i$ could also be infected with
virus 1 (resp. virus 2) due to _simultaneous_ interactions with infected
individuals in nodes $j$ and $\ell$, where either a) $j=i,$ and/or $\ell=i$,
or b) $j,\ell$ are neighbors of $i$. Such interactions are referred to as
_higher-order interactions_ (HOI). The notion of competition implies that no
individual can be simultaneously infected with virus 1 and virus 2.

We assume that the pairwise infection (resp. HOI) rate with respect to virus
$k$ is the same for all nodes, denoted by $\beta_{1}^{k}$ (resp.
$\beta_{2}^{k}$) for all $i\in[n]$ and $k\in[2]$ 333Indeed, it is far more
natural to have possibly different infection rates for each node; it is
standard in the literature on classic SIS bivirus networked systems [10, 8, 9,
11, 12, 21, 13]. As evident below, we do not impose constraints on the values
of the nonnegative matrices capturing the interactions, and hence the analysis
does not differ materially. We choose this particular notation to remain
consistent with earlier literature on epidemic models with HOI [20]. . An
individual infected with virus $k$ recovers from said infection at a healing
rate $\delta_{i}^{k}$ and immediately becomes susceptible to virus 1 or by
virus 2. All individuals within a node have the same healing rate with respect
to virus $k$; individuals in different nodes possibly have different healing
rates. We say that node $i$ is healthy if all individuals in node $i$ are
healthy; otherwise, we say it is infected. Within the same node, it is
possible for there to simultaneously exist a fraction of individuals that are
infected with virus $1$ and for a different fraction that is infected with
virus $2$.

As mentioned previously, diseases could spread due to pairwise interactions
and HOI. In case of the former, if an individual in node $j$ can infect an
individual in node $i$ with virus $k$, then, with $a_{ij}^{k}(\geq 0)$
denoting the strength of interactions between an individual in node $j$ and an
individual in node $i$ with respect to spread of virus $k$, we have that
$a_{ij}^{k}>0$; otherwise $a_{ij}^{k}=0$. For the case of HOI, if an
individual in node $i$ gets infected with virus $k$ due to simultaneous
interactions with individuals in nodes $j$ and $\ell$, then, with
$b_{ij\ell}^{k}$ denoting the strength of interaction that nodes $j$ and
$\ell$ together have on node $i$ with respect to the spread of virus $k$, we
have that $b_{ij\ell}^{k}>0$; else, $b_{ij\ell}^{k}=0$. Let $x_{i}^{k}(t)$
denote the fraction of individuals infected with virus $k$ in agent $i$ at
time instant $t$. The evolution of this fraction can, therefore, be
represented by the following scalar differential equation [1, Section 5],
where, for $i=1,2,\ldots,n$, we have

$\displaystyle\dot{x}_{i}^{1}=$
$\displaystyle-\delta_{i}^{1}x_{i}^{1}+\beta_{1}^{1}(1-x_{i}^{1}-x_{i}^{2})\textstyle\sum_{j=1}^{n}a_{ij}^{1}x_{j}^{1}+$
$\displaystyle~{}~{}\beta_{2}^{1}(1-x_{i}^{1}-x_{i}^{2})\textstyle\sum_{j,\ell=1}^{n}b_{ij\ell}^{1}x_{j}^{1}x_{\ell}^{1}$
$\displaystyle\dot{x}_{i}^{2}=$
$\displaystyle-\delta_{i}^{2}x_{i}^{2}+\beta_{1}^{1}(1-x_{i}^{1}-x_{i}^{2})\textstyle\sum_{j=1}^{n}a_{ij}^{2}x_{j}^{2}+$
$\displaystyle~{}~{}\beta_{2}^{2}(1-x_{i}^{1}-x_{i}^{2})\textstyle\sum_{j,\ell=1}^{n}b_{ij\ell}^{2}x_{j}^{2}x_{\ell}^{2}$
(1)

Define $D^{1}=\operatorname{diag}(\delta_{i}^{1})$, where $i\in[n]$, and
define $D^{2}$ analogously. Define $X^{1}=\operatorname{diag}(x_{i}^{1})$,
where $i\in[n]$, and define $X^{2}$ analogously. Let
$A^{1}=[a_{ij}^{1}]_{n\times n}$, and $A^{2}=[a_{ij}^{2}]_{n\times n}$. Let
$B_{i}^{k}=[b_{ij\ell}^{k}]_{n\times n}$, for each $i\in[n]$ and $k\in[2]$.
Let
$x^{k}=\small\begin{bmatrix}x_{1}^{k}&&x_{2}^{k}&&\ldots&&x_{n}^{k}\end{bmatrix}^{\top}$
for $k=1,2$.

Therefore, in vector form, equation (II-A) can be written as:

$\displaystyle\dot{x}^{1}=$
$\displaystyle-D^{1}x^{1}+\beta_{1}^{1}(I-X^{1}-X^{2})A^{1}x^{1}+$
$\displaystyle~{}~{}~{}~{}\beta_{2}^{1}(I-X^{1}-X^{2})((x^{1})^{\top}B_{1}^{1}x^{1},(x^{1})^{\top}B_{2}^{1}x^{1},\ldots,(x^{1})^{\top}B_{n}^{1}x^{1})^{\top}$
$\displaystyle\dot{x}^{2}=$
$\displaystyle-D^{2}x^{2}+\beta_{1}^{2}(I-X^{1}-X^{2})A^{2}x^{2}+$
$\displaystyle~{}~{}~{}~{}\beta_{2}^{2}(I-X^{1}-X^{2})((x^{2})^{\top}B_{1}^{2}x^{2},(x^{2})^{\top}B_{2}^{2}x^{2},\ldots,(x^{2})^{\top}B_{n}^{2}x^{2})^{\top}$
(2)

Throughout this document, we will drop the superscript $k$ while considering
the single-virus case.

We note that system (II-A) is a special case of [1, system 5.5] in the
following sense: System (II-A) only accounts for a) the case where, for
$k=1,2$, $\beta_{i}^{k}$ and $\beta_{2}^{k}$ is identical for every node $i$,
$i=1,2,\ldots,n$, and b) the case where virus 1 (resp. virus 2) spread only
due to contact with the infected individuals. In contrast, the model in [1]
(see [1, system 5.5]) allows for the possibility of $\beta_{i}^{k}$ and
$\beta_{2}^{k}$ being not necessarily the same for every node. Furthermore, it
also allows for the possibility of the viruses to spread through additional
mediums such as a water distribution network, a public transit network, etc.

###### Remark 1

Note that setting $\beta_{2}^{k}=0$ for $k=1,2$ results in system (II-A)
coinciding with the classic networked bivirus SIS model studied in, among
others, [10, 11, 9, 8, 13, 12]. Setting $x^{1}(0)=\textbf{0}$ (resp.
$x^{2}(0)=\textbf{0}$) results in system (II-A) coinciding with the model used
for studying the spread of a single virus over hypergraphs in [20].

The model in system (II-A) has three kinds of equilibria, viz. healthy state
or disease-free equilibrium (DFE), $(\textbf{0},\textbf{0})$; single-virus
endemic equilibria corresponding to virus $k$, of the form
$(\bar{x}^{k},\textbf{0})$, where $\textbf{0}\ll\bar{x}^{k}\ll\textbf{1}$ for
$k=1,2$; and coexisting equilibria, $(\bar{x}^{1},\bar{x}^{2})$, where, as we
will show in Lemma 1, $\textbf{0}\ll\bar{x}^{1},\bar{x}^{2}\ll\textbf{1}$,
and, furthermore, $\bar{x}^{1}+\bar{x}^{2}\ll\textbf{1}$. It is unknown
whether the single-virus endemic equilibria corresponding to virus $k$ are
unique, in contrast to the classic bivirus SIS network model without HOI.

The Jacobian of system (II-A) evaluated at an arbitrary point,
$(x^{1},x^{2})$, in the state space is as given in (3).

$J(x^{1},x^{2})=\begin{bmatrix}J_{11}&&J_{22}\\\ J_{21}&&J_{22}\end{bmatrix},$
(3)

where

$\displaystyle J_{11}=$
$\displaystyle-D^{1}+\beta_{1}^{1}(I-X^{1}-X^{2})A^{1}-\operatorname{diag}(\beta_{1}^{1}A^{1}x^{1})+$
$\displaystyle~{}~{}~{}~{}\beta_{2}^{1}(I-X^{1}-X^{2})O_{1}(x^{1})-\beta_{2}^{1}O_{2}(x^{1})$
(4) $\displaystyle J_{12}=$
$\displaystyle-\operatorname{diag}(\beta_{1}^{1}A^{1}x^{1})-\beta_{2}^{1}\operatorname{diag}((x^{1})^{\top}B_{i}^{1}x^{1})_{i=1,2,\ldots,n}$
(5) $\displaystyle J_{21}=$
$\displaystyle-\operatorname{diag}(\beta_{1}^{2}A^{2}x^{2})-\beta_{2}^{2}\operatorname{diag}((x^{2})^{\top}B_{i}^{2}x^{2})_{i=1,2,\ldots,n}$
(6) $\displaystyle J_{22}=$
$\displaystyle-D^{2}+\beta_{1}^{2}(I-X^{1}-X^{2})A^{2}-\operatorname{diag}(\beta_{1}^{2}A^{2}x^{2})+$
$\displaystyle~{}~{}~{}~{}\beta_{2}^{2}(I-X^{1}-X^{2})O_{3}(x^{2})-\beta_{2}^{2}O_{4}(x^{2})$
(7)

The terms $O_{1}(x^{1})$, $O_{2}(x^{1})$, $O_{3}(x^{2})$ and $O_{4}(x^{2})$
are as given in (8), (9), (10) and (11), respectively.

$\displaystyle O_{1}(x^{1})=$
$\displaystyle\begin{bmatrix}(B_{1}^{1}+(B_{1}^{1})^{\top})x^{1}&&(B_{2}^{1}+(B_{2}^{1})^{\top})x^{1}&&\ldots&&(B_{n}^{1}+(B_{n}^{1})^{\top})x^{1}\end{bmatrix}$
(8) $\displaystyle O_{2}(x^{1})=$
$\displaystyle\operatorname{diag}((x^{1})^{\top}B_{i}^{1}x^{1})_{i=1,2,\ldots,n}$
(9) $\displaystyle O_{3}(x^{2})=$
$\displaystyle\begin{bmatrix}(B_{1}^{2}+(B_{1}^{2})^{\top})x^{2}&&(B_{2}^{2}+(B_{2}^{2})^{\top})x^{2}&&\ldots&&(B_{n}^{2}+(B_{n}^{2})^{\top})x^{2}\end{bmatrix}$
(10) $\displaystyle O_{4}(x^{2})=$
$\displaystyle\operatorname{diag}((x^{2})^{\top}B_{i}^{2}x^{2})_{i=1,2,\ldots,n}$
(11)

We will need the following assumptions to ensure the model is well-defined.

###### Assumption 1

The matrix $D^{k}$, for $k=1,2$, is a positive diagonal matrix. The matrix
$A^{k}$, for $k=1,2$, is nonnegative. The matrix $B_{i}^{k}$ is nonnegative
for all $i\in[n]$ and $k\in[2]$.

###### Assumption 2

The matrix $A^{k}$, for $k=1,2$, is irreducible.

We define the set $\mathcal{D}$ as follows:

$\mathcal{D}:=\\{(x^{1},x^{2})\mid
x^{k}\geq\textbf{0},k=1,2,\textstyle\sum_{k=1}^{2}x^{k}\leq\mathbf{1}\\}.$
(12)

It is known that the set $\mathcal{D}$ is positively invariant, and that the
DFE is always an equilibrium for system (II-A); see [1, Lemma 5.1]. The fact
that $\mathcal{D}$ is positively invariant guarantees that the state values
$x^{k}_{i},k\in[2],i\in[n]$, always stay in the $[0,1]$ interval. Since the
states represent fractions of an infected population node, if the states were
to take values outside the $[0,1]$ interval, then those would not correspond
to physical reality.

###  II-B Problem Statements

With respect to system (II-A), we aim to answer the following questions in
this paper conclusively:

  1. i)

What is the typical behavior the trajectories exhibit as time goes to
infinity?

  2. ii)

Can we identify a parameter regime such that multiple equilibria are
simultaneously stable?

  3. iii)

Can we identify sufficient conditions for the existence of a coexistence
equilibrium? Furthermore, can we establish the stability properties of such an
equilibrium based on knowledge of the stability properties of the boundary
equilibria?

###  II-C Preliminary Lemmas and analysis of healthy state

In this subsection, we will establish certain preliminary results on the
nature of equilibria of system (II-A), and recall some of the results on
irreducible matrices - all of these will aid in the development of the main
results of the paper.

###### Lemma 1

Consider system (II-A) under Assumptions 1 and 2. If
$\bar{x}=(\bar{x}^{1},\bar{x}^{2})\in\mathcal{D}$ is an equilibrium of (II-A),
then, for each $k\in[2]$, either $\bar{x}^{k}=\textbf{0}$, or
$\textbf{0}\ll\bar{x}^{k}\ll\textbf{1}$. Moreover,
$\textstyle\sum_{k=1}^{2}\bar{x}^{k}\ll\textbf{1}$.

The proof is inspired from [11, Lemma 3.1].
Proof: It is clear that $(\textbf{0},\textbf{0})$ is an equilibrium of (II-A).
Therefore, in the rest of the proof, we will show that any non-zero
equilibrium $\bar{x}=(\bar{x}^{1},\bar{x}^{2})$ of (II-A) must satisfy, for
each $k\in[2]$, $\textbf{0}\ll\bar{x}^{k}\ll\textbf{1}$ and
$\textstyle\sum_{k=1}^{2}\bar{x}^{k}\ll\textbf{1}$. We start off by showing
that $\bar{x}^{1}+\bar{x}^{2}\ll\textbf{1}$. For any $i\in[n]$, observe that
the following is satisfied:

$\displaystyle\dot{\bar{x}}_{i}^{1}+\dot{\bar{x}}_{i}^{2}=$
$\displaystyle-\delta_{i}^{1}\bar{x}^{1}_{i}-\delta_{i}^{2}\bar{x}^{2}_{i}+\beta_{1}^{1}(1-\bar{x}_{i}^{1}-\bar{x}_{i}^{2})\textstyle\sum_{j=1}^{n}a_{ij}^{1}\bar{x}_{j}^{1}$
$\displaystyle~{}+\beta_{2}^{1}(1-\bar{x}_{i}^{1}-\bar{x}^{2}_{i})\textstyle\sum_{j,\ell=1}^{n}b_{ij\ell}^{1}\bar{x}_{j}^{1}\bar{x}_{\ell}^{1}$
$\displaystyle~{}+\beta_{1}^{2}(1-\bar{x}_{i}^{1}-\bar{x}_{i}^{2})\textstyle\sum_{j=1}^{n}a_{ij}^{2}\bar{x}_{j}^{2}$
$\displaystyle~{}+\beta_{2}^{2}(1-\bar{x}_{i}^{1}-\bar{x}^{2}_{i})\textstyle\sum_{j,\ell=1}^{n}b_{ij\ell}^{2}\bar{x}_{j}^{2}\bar{x}_{\ell}^{2}$
(13)

Suppose that, for some $i\in[n]$, $\bar{x}_{i}^{1}+\bar{x}^{2}_{i}=1$.
Therefore, since, by Assumption 1, $\delta_{i}^{k}>0$ for $k=1,2$, and since
$\bar{x}_{i}^{k}\in\mathcal{D}$, from (II-C), it is clear that
$\dot{\bar{x}}_{i}^{1}+\dot{\bar{x}}_{i}^{2}<0$. However, since by assumption
$\bar{x}=(\bar{x}^{1},\bar{x}^{2})$ is an equilibrium, it must be that
$\dot{\bar{x}}_{i}^{1}+\dot{\bar{x}}_{i}^{2}=0$, which is a contradiction.
Therefore, for all $i\in[n]$, $\bar{x}_{i}^{1}+\bar{x}^{2}_{i}<1$, which
implies that $\textstyle\sum_{k=1}^{2}x^{k}\ll\textbf{1}$; thus guaranteeing
that $\bar{x}^{k}\ll\textbf{1}$ for $k=1,2$.
We are left to show that $\bar{x}^{k}\gg\textbf{0}$ for $k=1,2$. To this end,
suppose that $\bar{x}^{1}>\textbf{0}$ is an equilibrium point for which there
exists at least one (but possibly more) $i\in[n]$ such that
$\bar{x}_{i}^{1}=0$. Note that the equilibrium version of the first line of
equation (II-A) yields the following:

$\displaystyle\dot{\bar{x}}^{1}=$
$\displaystyle-D^{1}\bar{x}^{1}+\beta_{1}^{1}(I-\bar{X}^{1}-\bar{X}^{2})A^{1}\bar{x}^{1}+$
$\displaystyle~{}~{}~{}~{}\beta_{2}^{1}(I-\bar{X}^{1}-\bar{X}^{2})((\bar{x}^{1})^{\top}B_{1}^{1}\bar{x}^{1},(\bar{x}^{1})^{\top}B_{2}^{1}\bar{x}^{1},\ldots,(\bar{x}^{1})^{\top}B_{n}^{1}\bar{x}^{1})^{\top}$
(14)

By noting that $\bar{x}^{1}$ is an equilibrium point, and by a suitable
rearrangement of terms, we obtain:

$\displaystyle\bar{x}^{1}=$ $\displaystyle S\bar{x}^{1},$ (15)

where

$\displaystyle S=$
$\displaystyle(D^{1})^{-1}\beta_{1}^{1}(I-\bar{X}^{1}-\bar{X}^{2})A^{1}+$
$\displaystyle(D^{1})^{-1}\beta_{2}^{1}(I-\bar{X}^{1}-\bar{X}^{2})((\bar{x}^{1})^{\top}B_{1}^{1},\ldots,(\bar{x}^{1})^{\top}B_{n}^{1}.$
(16)

By Assumptions 1 and 2, it is clear that the matrix $S$ is nonnegative and
irreducible. Since, by assumption, $\bar{x}^{1}>\textbf{0}$, from (15) and
coupled with a property of a nonnegative matrix that is irreducible we have
the following: i) $S\bar{x}^{1}>\textbf{0}$, and ii) there is at least one
$i\in[n]$ such that $\bar{x}^{1}_{i}=0$ but $(S\bar{x}^{1})_{i}>0$. Note that
ii) contradicts (15). Therefore, if $\bar{x}^{1}>\textbf{0}$ is an equilibrium
point, then it must be that $\bar{x}^{1}\gg\textbf{0}$. By an analogous
argument, it can be shown that $\bar{x}^{2}\gg\textbf{0}$, thus completing the
proof. $\Box$

###### Lemma 2

[10, Proposition 1]] Suppose that $\Lambda$ is a negative diagonal matrix and
$N$ is an irreducible nonnegative matrix. Let $M$ be the irreducible Metzler
matrix $M=\Lambda+N$. Then, $s(M)<0$ if and only if
$\rho(-\Lambda^{-1}N)<1,s(M)=0$ if and only if $\rho(-\Lambda^{-1}N)=1$, and
$s(M)>0$ if and only if, $\rho(-\Lambda^{-1}N)>1$.

###### Lemma 3

[22, Proposition 2] Let $A\in\mathbb{R}^{n\times n}$ be Metzler. Then, $A$ is
Hurwitz if, and only if, there exists an $x\in\mathbb{R}^{n}$ such that
$x\gg\textbf{0}$ and $Ax\ll 0$.

###### Lemma 4

[23, Chapter 8.3] [24, Theorem 2.7] Suppose that $N$ is an irreducible
nonnegative matrix. Then,

  1. (i)

$r=\rho(N)$ is a simple eigenvalue of $N$.

  2. (ii)

There is an eigenvector $\zeta\gg\textbf{0}$ corresponding to the eigenvalue
$r$.

  3. (iii)

$x>\textbf{0}$ is an eigenvector only if $Nx=rx$ and $x\gg\textbf{0}$.

  4. (iv)

If $A$ is a nonnegative matrix such that $A<N$, then $\rho(A)<\rho(N)$.
$\blacksquare$

It can be seen that $(\textbf{0},\textbf{0})$ is an equilibrium of (II-A), and
is referred to as the disease-free equilibrium (DFE). We recall a sufficient
condition for convergence to the DFE.

###### Lemma 5

[1, Theorem 5.2, statement 1] Consider system (II-A) under Assumptions 1 and
2. If, for $k=1,2$, $\rho(\beta_{1}^{k}(D^{k})^{-1}A^{k})<1$, then the DFE is
locally stable.

Note that the guarantees provided by Lemma 5 are only local. It turns out that
the DFE, under appropriate conditions, is endowed with stronger stability
guarantees. We define for $k=1,2$ the following matrices:

$R^{k}:=\tiny\begin{bmatrix}\textbf{1}^{\top}B_{1}^{k}\\\
\textbf{1}^{\top}B_{2}^{k}\\\ \vdots\\\
\textbf{1}^{\top}B_{n}^{k}\end{bmatrix}.$

With the matrices $R^{k}$, $k=1,2$, in hand, we can recall the following
result.

###### Lemma 6

[1, Theorem 5.2, statement 2] Consider system (II-A) under Assumptions 1 and
2. If, for $k=1,2$,
$\rho(\beta_{1}^{k}(D^{k})^{-1}A^{k}+\beta_{2}^{k}(D^{k})^{-1}R^{k})<1$, then
the DFE is globally exponentially stable.

##  III Monotone dynamical systems and competitive bivirus networked SIS
models with HOI

Monotone dynamical systems (MDS) are a class of systems that has found
resonance in mathematical epidemiology; one of the major reasons for this is
the fact that MDS, assuming that they generically have a finite number of
equilibria, converge to a (stable) equilibrium point for almost all initial
conditions. Here, the term “almost all” means: for all but a set of parameter
values that has measure zero. An algebraic or semi-algebraic set defines this
set of exceptional values. It is known that under Assumptions 1 and 2, system
(II-A) is monotone; see [1, Theorem 5.5]. That is, suppose that
$(x_{A}^{1}(0),x_{A}^{2}(0))$ and $(x_{B}^{1}(0),x_{B}^{2}(0))$ are two
initial conditions in $\textrm{int}(\mathcal{D})$ satisfying i)
$x_{A}^{1}(0)>x_{B}^{1}(0)$ and ii) $x_{A}^{2}(0)<x_{B}^{2}(0)$. Since system
(II-A) is monotone, it follows that, for all $t\in\mathbb{R}_{\geq 0}$, i)
$x_{A}^{1}(t)\gg x_{B}^{1}(t)$ and ii) $x_{A}^{2}(t)\ll x_{B}^{2}(t)$. Note
that [1, Theorem 5.5] also claims that system (II-A), in a generic sense, has
a finite number of equilibria. However, the proof for said assertion is the
proof for the case where $\beta_{2}^{k}=0$, for $k=1,2$, in [11, Theorem 3.6],
which is only a nongeneric case. Since the proof for finiteness of equilibria
in [1, Theorem 5.5] is not complete, it leaves open the issue of generic
convergence to an equilibrium point. To remedy this, we provide a different
proof for generic finiteness of equilibria that does not rely on
$\beta_{2}^{k}=0$ for $k=1,2$.

Given that nonlinear systems can have complex equilibria patterns, including a
continuum of equilibria for the classic bivirus network model, we establish
that for generic parameter matrices, system (II-A) has a finite number of
equilibria. We use arguments very much like those in [12]. Essentially because
the healthy equilibrium and the single-virus boundary equilibria can be
conveniently studied using single-virus techniques, it is easily established
that there are no continua of equilibria confined to any boundary, i.e. any
continuum of equilibria necessarily includes a continuum of coexistence
equilibria. Therefore, we focus on showing that such equilibria cannot exist
for generic parameter values. The tool is the Parametric Transversality
Theorem, see [25, see p. 145] and [26, see p.68]. The main result is as
follows:

###### Theorem 1

Consider the model of (II-A), under Assumptions 1 and 2. With any fixed
matrices $A^{k}$ and nonnegative $B^{k}_{i}$, and the exclusion of a set of
values for the entries of $D^{1},D^{2}$ of measure zero, the number of
coexistence equilibrium points is finite, and the associated vector field zero
is nondegenerate, i.e. the associated Jacobian is nonsingular. Similarly, with
any fixed $D^{1},D^{2}$ and $B^{k}_{i}$, and the exclusion of a set of values
for the entries of $A^{1},A^{2}$ of measure zero, the same properties of
equilibrium points hold.

Proof: See Appendix. $\Box$

Theorem 1, coupled with the fact that system (II-A) is monotone, allows us to
leverage Hirsch’s generic convergence theorem [27] to draw conclusions on the
limiting behavior of system (II-A) outside of the specific conditions
identified in Lemma 5. We have the following result.

###### Theorem 2

Consider system (II-A) under Assumptions 1 and 2. For all initial conditions
$(x^{1}(0),x^{2}(0))\in\mathcal{D}$ except possibly for a set of measure zero,
the system (II-A) will converge to an equilibrium. If the system does not
converge to an equilibrium, it is on a nonattractive limit cycle.

In words, Theorem 2 establishes that the typical behavior of system (II-A) is
convergence to _some_ equilibrium; this could be healthy, or (one of the
possibly many) single-virus boundary equilibria, or a coexistence equilibrium.
It further says that limit cycles, if any, are nonattractive. No more
complicated behavior is allowed; chaos can be ruled out, see [28]. Thus,
Theorem 2 answers question i) raised in Section II.
Theorem 2 strengthens the result in [11, Theorem 3.6] by extending the generic
convergence behavior to bi-virus SIS models that also account for HOI.
Furthermore, it establishes the correctness of a similar claim raised in [1,
Theorem 5.5].

##  IV Existence and local stability of boundary equilibria

In this section, we identify a parameter regime that permits three equilibria
of the bivirus system (II-A) to be simultaneously locally exponentially
stable. Subsequently, for a parameter regime different from the one mentioned
above, we identify a condition for the existence and instability of a boundary
equilibrium. Finally, when there is only one virus, we identify a condition
for the existence and local exponential stability of an endemic equilibrium.

###### Proposition 1

Consider system (II-A) under Assumptions 1 and 2, and $B_{i}^{k}\geq 0$ for
all $i\in[n]$ and $k\in[2]$. Define, for $k=1,2$,
$\mathbf{1}_{B^{k}}\in\\{0,1\\}^{n}$ by $(\mathbf{1}_{B^{k}})_{i}=1$ if
$B_{i}^{k}\neq\mathbf{0}$; otherwise $(\mathbf{1}_{B^{k}})_{i}=0$. Suppose
that the following conditions are fulfilled for $k=1,2$:

  1. a)

$\rho(\beta_{1}^{k}(D^{k})^{-1}A^{k})<1$, and

  2. b)

$\min\limits_{i.s.t.B_{i}^{k}\neq\mathbf{0}}\big{(}\frac{\beta_{1}^{k}}{\delta_{i}^{k}}(A^{k}\mathbf{1}_{B^{k}})_{i}+\frac{\beta_{2}^{k}}{2\delta_{i}^{k}}\mathbf{1}_{B^{k}}^{\top}B_{i}\mathbf{1}_{B^{k}}\big{)}>2$.

Then, the following statements are true:

  1. i)

The DFE is locally exponentially stable.

  2. ii)

there exist equilibria $\bar{x}^{k}\gg\textbf{0}$ such that
$\bar{x}_{i}^{k}\geq\frac{1}{2}$ for $k=1,2$, for any $i$ such that
$B_{i}^{k}\neq\mathbf{0}$.

  3. iii)

Any such equilibrium point $(\bar{x}^{1},\textbf{0})$ is locally exponentially
stable; and

  4. iv)

Any such equilibrium point $(\textbf{0},\bar{x}^{2})$ is locally exponentially
stable.

The proof is inspired from [20, Theorem 5.1, statements iv) and v)].
Proof of statement i): Note that the Jacobian evaluated at the DFE is as
follows:

$\displaystyle
J(\textbf{0},\textbf{0})=\scriptsize\begin{bmatrix}-D^{1}+\beta_{1}^{1}A^{1}&\textbf{0}\\\
\textbf{0}&-D^{2}+\beta_{1}^{2}A^{2}\end{bmatrix}.\normalsize$

By assumption, $\beta_{1}^{k}\rho(D^{k})^{-1}A^{k})<1$, for $k=1,2$.
Therefore, from Lemma 2, it must be that $s(-D^{k}+\beta_{1}^{k}A^{k})<0$ for
$k=1,2$, which, since $J(\textbf{0},\textbf{0})$ is a block diagonal matrix,
and since the matrices $-D^{1}+\beta_{1}^{1}A^{1}$ and
$-D^{2}+\beta_{1}^{2}A^{2}$ are the only blocks along the main diagonal,
implies that $s(J(\textbf{0},\textbf{0}))<0$. Local exponential stability of
the DFE, then, follows from [29, Theorem 4.15 and Corollary 4.3].

Proof of statement ii): See [20, Theorem 5.1, statement iv)].

Proof of statement iii): Consider the equilibrium point
$(\bar{x}^{1},\textbf{0})$, and observe that the Jacobian evaluated at this
equilibrium is as follows:

$J(\bar{x}^{1},\textbf{0})=\begin{bmatrix}\bar{J}_{11}&&\bar{J}_{12}\\\
\textbf{0}&&\bar{J}_{22}\end{bmatrix},$ (17)

where

$\displaystyle\bar{J}_{11}$
$\displaystyle=-D^{1}+\beta_{1}^{1}(I-\bar{X}^{1})A^{1}-\operatorname{diag}(\beta_{1}^{1}A^{1}\bar{x}^{1})+$
$\displaystyle~{}~{}~{}~{}~{}~{}\beta_{2}^{1}(I-\bar{X}^{1})O_{1}(\bar{x}^{1})-\beta_{2}^{1}O_{2}(\bar{x}^{1})$
$\displaystyle\bar{J}_{12}$
$\displaystyle=-\operatorname{diag}(\beta_{1}^{1}A^{1}\bar{x}^{1})-\beta_{2}^{1}\operatorname{diag}((\bar{x}^{1})^{\top}B_{i}^{1}\bar{x}^{1})_{i=1,\ldots,n}$
$\displaystyle\bar{J}_{22}$
$\displaystyle=-D^{2}+\beta_{1}^{2}(I-\bar{X}^{1})A^{2}.$

The terms $O_{1}(\bar{x}^{1})$ and $O_{2}(\bar{x}^{1})$ are as defined in (8)
and (9). We will establish the exponential stability of the 11 and 22 blocks
(i.e., $\bar{J}_{11}$ and $\bar{J}_{22}$) separately. Observe that

$\displaystyle\bar{J}_{11}=$
$\displaystyle-D^{1}+\beta_{1}^{1}(I-\bar{X}^{1})A^{1}-\operatorname{diag}(\beta_{1}^{1}A^{1}\bar{x}^{1})\hskip
5.69054pt+$
$\displaystyle~{}~{}~{}\beta_{2}^{1}(I-\bar{X}^{1})O_{1}(\bar{x}^{1})-\beta_{2}^{1}\begin{bmatrix}\tiny(\bar{x}^{1})^{\top}B_{1}^{1}\bar{x}^{1}&&\\\
&\ddots&\\\ &&(\bar{x}^{1})^{\top}B_{n}^{1}\bar{x}^{1}\end{bmatrix}.$

Define summands

$\displaystyle Q_{1}:=$
$\displaystyle-D^{1}+\beta_{1}^{1}(I-\bar{X}^{1})A^{1}+\beta_{2}^{1}(I-\bar{X}^{1})\begin{bmatrix}\tiny(\bar{x}^{1})^{\top}B_{1}^{1}\\\
\vdots\\\ (\bar{x}^{1})^{\top}B_{n}^{1}\\\ \end{bmatrix},\text{ and }$
$\displaystyle Q_{2}:=$
$\displaystyle\beta_{2}^{1}(I-\bar{X}^{1})\begin{bmatrix}\tiny(\bar{x}^{1})^{\top}(B_{1}^{1})^{\top}\\\
\vdots\\\ (\bar{x}^{1})^{\top}(B_{n}^{1})^{\top}\\\ \end{bmatrix}-$
$\displaystyle~{}~{}~{}\operatorname{diag}(\beta_{1}^{1}A^{1}\bar{x}^{1})-\beta_{2}^{1}\begin{bmatrix}\tiny(\bar{x}^{1})^{\top}B_{1}^{1}\bar{x}^{1}&&\\\
&\ddots&\\\ &&(\bar{x}^{1})^{\top}B_{n}^{1}\bar{x}^{1}\end{bmatrix}.$

It is immediate that $\bar{J_{11}}=Q_{1}+Q_{2}$, which implies that
$\bar{J}_{11}\bar{x}^{1}=Q_{1}\bar{x}^{1}+Q_{2}\bar{x}^{1}$. Since
$\bar{x}^{1}$ is a single-virus endemic equilibrium corresponding to virus 1,
by taking recourse to the equilibrium version of the first line of equation
(II-A), it is clear that $Q_{1}\bar{x}^{1}=\textbf{0}$. Hence,
$\bar{J}_{11}\bar{x}^{1}=Q_{2}\bar{x}^{1}$.

Note that

$\displaystyle Q_{2}\bar{x}^{1}$
$\displaystyle=\beta_{2}^{1}(I-\bar{X}^{1})\begin{bmatrix}\tiny(\bar{x}^{1})^{\top}(B_{1}^{1})^{\top}\bar{x}^{1}\\\
\vdots\\\ (\bar{x}^{1})^{\top}(B_{n}^{1})^{\top}\bar{x}^{1}\\\ \end{bmatrix}-$
$\displaystyle~{}~{}~{}\operatorname{diag}(\beta_{1}^{1}A^{1}\bar{x}^{1})\bar{x}^{1}-\beta_{2}^{1}\begin{bmatrix}\tiny(\bar{x}^{1})^{\top}B_{1}^{1}\bar{x}^{1}&&\\\
&\ddots&\\\
&&(\bar{x}^{1})^{\top}B_{n}^{1}\bar{x}^{1}\end{bmatrix}\bar{x}^{1}.$ (18)

Denote by $(Q_{2}\bar{x}^{1})_{i}$ the $i^{th}$ entry of the vector
$Q_{2}\bar{x}^{1}$. Therefore, in view of (18), we have the following:

$(Q_{2}\bar{x}^{1})_{i}=-\beta_{1}^{1}\Big{(}\textstyle\sum_{j=1}^{n}a_{ij}^{1}\bar{x}^{1}_{j}\Big{)}\bar{x}_{i}^{1}+\beta_{2}^{1}(1-2\bar{x}_{i}^{1})((\bar{x}^{1})^{\top}B_{i}^{1}\bar{x}^{1})$
(19)

We consider its sign under two circumstances. Suppose first that
$B_{i}^{1}=\mathbf{0}$. Then, in view of (19), since by Assumption 2 the
matrix $A^{1}$ is irreducible, $\beta_{1}^{1}>0$, and from statement ii) we
know that $\bar{x}^{1}\gg\textbf{0}$, it must be that
$(Q_{2}\bar{x}^{1})_{i}<0$. Suppose secondly that $B_{i}^{1}\neq\mathbf{0}$.
Since from statement ii) we know that $\bar{x}^{1}_{i}\geq\frac{1}{2}$, it
follows that $1-2\bar{x}^{1}_{i}<0$; thus implying that
$(Q_{2}\bar{x}^{1})_{i}<0$. Note that the choice of index $i$ was arbitrary,
and therefore again, we have $(Q_{2}\bar{x}^{1})_{i}<0$ for all $i\in[n]$.
Hence, since $\bar{J}_{11}\bar{x}^{1}=Q_{2}\bar{x}^{1}$, it follows that
$(\bar{J}_{11}\bar{x}^{1})_{i}<0$ for all $i\in[n]$. Note that Assumptions 1
and 2 guarantee that the matrices $Q_{1}$ and $Q_{2}$ are irreducible Metzler
matrices; hence, the matrix $\bar{J}_{11}$ is an irreducible Metzler matrix.
Therefore, from Lemma 3, it must be that the matrix $\bar{J}_{11}$ is Hurwitz.

Turning our attention to the matrix $\bar{J}_{22}$, consider the matrices
$\beta_{1}^{2}(D^{2})^{-1}A^{2}$ and
$\beta_{1}^{2}(D^{2})^{-1}(I-\bar{X}^{1})A^{2}$. From Assumption 1, it is
clear that $\beta_{1}^{2}(D^{2})^{-1}A^{2}$ is a nonnegative matrix. Since,
from statement ii), $\bar{x}^{1}$ satisfies
$\textbf{0}\ll\bar{x}^{1}\ll\textbf{1}$, it is also clear that
$\beta_{1}^{2}(D^{2})^{-1}(I-\bar{X}^{1})A^{2}$ is a nonnegative matrix.
Furthermore, we also immediately obtain the following:

$\beta_{1}^{2}(D^{2})^{-1}(I-\bar{X}^{1})A^{2}<\beta_{1}^{2}(D^{2})^{-1}A^{2}.$

Therefore, since the spectral radius of a nonnegative matrix decreases
monotonically with a decrease in any entry of said matrix, it follows that
$\rho(\beta_{1}^{2}(D^{2})^{-1}(I-\bar{X}^{1})A^{2})\leq\rho(\beta_{1}^{2}(D^{2})^{-1}A^{2})$.
By assumption, $\rho(\beta_{1}^{2}(D^{2})^{-1}A^{2})<1$, which implies that
$\rho(\beta_{1}^{2}(D^{2})^{-1}(I-\bar{X}^{1})A^{2})<1$, and consequently,
from Lemma 2, we have that $s(-D^{2}+(I-\bar{X}^{1})A^{2})<0$. Therefore,
since $J(\bar{x}^{1},\textbf{0})$ is block upper triangular, and since we have
already established that $\bar{J}_{11}$ is Hurwitz, it follows that
$s(J(\bar{x}^{1},\textbf{0}))<0$. Local exponential stability of
$(\bar{x}^{1},\textbf{0})$, then, follows from [29, Theorem 4.15 and Corollary
4.3].

Proof of statement iv): The proof is analogous to that of statement iii).
$\Box$

Proposition 1 answers question ii) raised in Section II. Proposition 1
guarantees the existence and simultaneous local exponential stability of three
equilibria, whereas [1, Theorem 5.3], assuming that an endemic equilibrium
exists, guarantees its local stability. Furthermore, the possibility of the
DFE being locally stable simultaneously is alluded to; see [1, Remark 10]. On
the other hand, for a parameter regime different from the one covered in
Proposition 1, assuming that the terms corresponding to HOI are sufficiently
small, [1, Theorem 5.3] secures global stability of the endemic equilibrium.
The following remarks are in order.

###### Remark 2

Proposition 1 sheds light on an interesting phenomenon that bivirus spread
over hypergraph exhibits (but bivirus spread over a normal graph does not
exhibit): identification of a parameter regime that permits three equilibria,
namely the DFE and the two boundary equilibria, to be simultaneously stable.
This is an extension to the single virus case studied in [20], which permitted
the simultaneous stability of the DFE and (since there is only one in this
case) an endemic equilibrium.

###### Remark 3

It is known that, assuming $\beta_{2}^{k}=0$ for $k=1,2$, the condition
$\rho(\beta_{1}^{k}(D^{k})^{-1}A^{k})\leq 1$ guarantees that the DFE is the
only equilibrium of system (II-A); see [21, Lemma 2]. However, as Proposition
1 shows, that is not necessarily true when considering bivirus SIS spread over
hypergraphs.

Proposition 1 guarantees existence of boundary equilibria for the case when
$\rho(\beta_{1}^{k}(D^{k})^{-1}A^{k})<1$. It is natural to ask if one is
assured of existence even if the spectral radii of relevant quantities are
larger than one. The following proposition addresses this issue.

###### Proposition 2

Consider system (II-A) under Assumptions 1 and 2. Suppose that, for all
$k\in[2]$, $\rho(\beta_{1}^{k}(D^{k})^{-1}A^{k})>1$. Then system (II-A) has at
least three equilibria, namely the DFE, a single virus endemic equilibrium
corresponding to virus $1$ $(\bar{x}^{1},\textbf{0})$, and a single virus
endemic equilibrium corresponding to virus $2$ $(\textbf{0},\bar{x}^{2})$.
Furthermore, if $s(-D^{i}+\beta_{1}^{i}(I-\bar{X}^{k})A^{i})>0$ for
$i,k\in[2]$ such that $i\neq k$, then the equilibrium points
$(\bar{x}^{1},\textbf{0})$ and $(\textbf{0},\bar{x}^{2})$ are unstable.

Proof: Observe that the DFE is always an equilibrium of system (II-A). Suppose
that for some $k\in[2]$, $\rho(\beta_{1}^{k}(D^{k})^{-1}A^{k})>1$, then from
[20, Theorem 5.1, statement vii)] we know that there exists an endemic
equilibrium, $\bar{x}^{k}$, where $\textbf{0}\ll\bar{x}^{k}\ll\mathbf{1}$.
Since, by assumption, $\rho(\beta_{1}^{k}(D^{k})^{-1}A^{k})>1$ for all
$k\in[2]$, it is also immediate that there exist equilibria,
$(\bar{x}^{1},\textbf{0})$ and $(\textbf{0},\bar{x}^{2})$, where
$\textbf{0}\ll\bar{x}^{k}\ll\mathbf{1}$, for $k=1,2$.
It can be verified that $J(\bar{x}^{1},\textbf{0})$ is block upper triangular,
with the matrix $-D^{1}+\beta_{1}^{1}(I-\bar{X}^{2})A^{1}$ being one of the
blocks along the diagonal. By assumption,
$s(-D^{1}+\beta_{1}^{1}(I-\bar{X}^{2})A^{1})>0$, which implies that
$s(J(\bar{x}^{1},\textbf{0}))>0$. Consequently, instability of
$(\bar{x}^{1},\textbf{0})$ follows from [29, Theorem 4.7, statement ii)]. The
instability of $(\textbf{0},\bar{x}^{2})$ can be shown analogously, thus
completing the proof. $\Box$

###### Remark 4

Proposition 2 (resp. Proposition 1) guarantees the existence (resp. existence
and local exponential stability) of the equilibrium points,
$(\bar{x}^{1},\textbf{0})$ and $(\textbf{0},\bar{x}^{2})$. It turns out that
it is possible to compute these points iteratively; see [20, Theorem 5.3].

##  V (Non)existence of Coexistence equilibria

This section identifies sufficient conditions for the existence (resp.
nonexistence) of coexistence equilibria. Specifically, for investigating
existence, we consider two parameter regimes, viz. for $k=1,2$, i)
$s(-D^{k}+\beta_{i}^{k}A^{k})>0$, and ii) $s(-D^{k}+\beta_{i}^{k}A^{k})<0$.
Further, for the parameter regime i), we consider two stability configurations
of the boundary equilibria, viz. a) both being unstable and b) both being
stable; for parameter regime ii), we consider the case where both boundary
equilibria are stable.

###### Proposition 3

Consider system (II-A) under Assumptions 1 and 2. Let
$(\bar{x}^{1},\textbf{0})$ and $(\textbf{0},\bar{x}^{2})$ denote a single-
virus endemic equilibrium corresponding to virus 1 and virus 2, respectively.
Suppose that the following conditions are satisfied:

  1. i)

$s(-D^{1}+\beta_{1}^{1}A^{1})>0$;

  2. ii)

$s(-D^{2}+\beta_{1}^{2}A^{2})>0$;

  3. iii)

$s(-D^{1}+\beta_{1}^{1}(I-\bar{X}^{2})A^{1})>0$; and

  4. iv)

$s(-D^{2}+\beta_{1}^{2}(I-\bar{X}^{1})A^{2})>0$.

Then there exists at least one equilibrium of the form
$(\hat{x}^{1},\hat{x}^{2})$ such that
$\textbf{0}\ll\hat{x}^{1},\hat{x}^{2}\ll\textbf{1}$ and
$\hat{x}^{1}+\hat{x}^{2}\ll\textbf{1}$.

Before proving the claim in Proposition 3, we need the following background
material. In line with the terminology of [30], given an equilibrium point of
system (II-A), we classify the same as saturated or unsaturated. We say that
an equilibrium is saturated (resp. strictly saturated) if the diagonal block
corresponding to the zero entries of said equilibrium possibly has a single
eigenvalue at the origin (resp. has every eigenvalue to be strictly less than
zero) and unsaturated otherwise [30]. A boundary equilibrium of (II-A) is
saturated if and only if said boundary equilibrium is locally exponentially
stable; this follows immediately by noting the structure of the Jacobian
matrix, evaluated at a boundary equilibrium, see (17). The definition also
implies that every fixed point in the interior of $\mathcal{D}$, irrespective
of its stability properties, is saturated [30]; therefore, from Lemma 1, we
have that every coexistence equilibrium of system (II-A) is saturated.

Proof: Assumptions i) and ii) of Proposition 3 guarantee existence of boundary
equilibria, $(\bar{x}^{1},\textbf{0})$ and $(\textbf{0},\bar{x}^{2})$; see
Proposition 2. Observe that [1, Lemma 5.1] guarantees that, for each
$k\in[2]$, $x^{k}(0)\geq\textbf{0}$ implies that $x^{k}(t)\geq\textbf{0}$ for
all $t\in\mathbb{R}_{\geq 0}$, and that the set $\mathcal{D}$ (which is
compact) is forward invariant. Therefore, from [30, Theorem 2], it follows
that system (II-A) has at least one saturated fixed point. There are two cases
to consider.
Case 1: Suppose the aforementioned saturated fixed point is in the interior of
$\mathcal{D}$. Note that any fixed point in the interior of $\mathcal{D}$ is
of the form $(\hat{x}^{1},\hat{x}^{2})$, where
$\textbf{0}\ll(\hat{x}^{1},\hat{x}^{2})\ll\textbf{1}$, thus implying that
$(\hat{x}^{1},\hat{x}^{2})$ is a coexistence equilibrium. From Lemma 1, it
must necessarily satisfy
$\textbf{0}\ll(\hat{x}^{1},\hat{x}^{2})\ll\textbf{1}$, and
$\hat{x}^{1}+\hat{x}^{2}\ll\textbf{1}$.
Case 2: Suppose, but we will demonstrate a contradiction, that there are no
fixed points in the interior of $\mathcal{D}$. This implies that there must be
a saturated fixed point on the boundary of $\mathcal{D}$ [30]. Therefore, at
least one of the single-virus boundary equilibria is saturated.
However, from Proposition 2, it is clear that assumptions iii) and iv)
guarantee that the boundary equilibria are unstable; thus implying that they
are unsaturated, and the contradiction is obtained. $\Box$

Proposition 3 is implied by [1, Theorem 5.4], which, assuming
$\beta_{2}^{k}=0$ for $k=1,2$, is the same as [9, Theorem 5]. The proof
technique in [1, Theorem 5.4] is quite involved since it primarily relies on
fixed point mapping, Perron Frobenius theory, etc. Our proof is significantly
shorter. Note that [30, Theorem 2] is a key ingredient of our proof strategy.
In light of Theorem 1, one could perhaps leverage [30, Theorem 2] to obtain a
lower bound on the number of coexistence equilibria for the stability
configuration of boundary equilibria given in Proposition 3, as has been done
for classic bivirus networked SIS models; see [12, Corollary 3.9, statement
2]. Subsequently, one could possibly exploit the properties of MDS to conclude
that there must exist a locally exponentially stable coexistence equilibrium.

Observe that in Propostion 3 the demonstration of the existence of a
coexistence equilibrium point $(\hat{x}^{1},\hat{x}^{2})$ relies on the
assumption that both boundary equilibria are unstable. We now present a
different condition that also guarantees the existence of a coexistence
equilibrium point $(\hat{x}^{1},\hat{x}^{2})$ even when both boundary
equilibria are stable.

###### Theorem 3

Consider system (II-A) under Assumptions 1 and 2. Let
$(\bar{x}^{1},\textbf{0})$ and $(\textbf{0},\bar{x}^{2})$ denote a single-
virus endemic equilibrium corresponding to virus 1 and virus 2, respectively.
Suppose that the following conditions are satisfied:

  1. i)

$s(-D^{1}+\beta_{1}^{1}A^{1})>0$;

  2. ii)

$s(-D^{2}+\beta_{1}^{2}A^{2})>0$;

Suppose that both $(\bar{x}^{1},\textbf{0})$ and $(\textbf{0},\bar{x}^{2})$
are locally exponentially stable. Then there exists at least one equilibrium
of the form $(\hat{x}^{1},\hat{x}^{2})$ such that
$\textbf{0}\ll\hat{x}^{1},\hat{x}^{2}\ll\textbf{1}$ and
$\hat{x}^{1}+\hat{x}^{2}\ll\textbf{1}$, such that $(\hat{x}^{1},\hat{x}^{2})$
is either neutrally stable or unstable.444Assuming that equilibria of system
(II-A) are hyperbolic, a stronger conclusion can be drawn: for generic
parameter matrices, the coexistence equilibrium is unstable.

Proof: By assumption, $s(-D^{k}+\beta_{1}^{k}A^{k})>0$ for $k=1,2$. Therefore,
from Proposition 2, it follows that there exists a single-virus endemic
equilibrium corresponding to virus 1, $\bar{x}^{1}\gg\textbf{0}$, and a
single-virus endemic equilibrium corresponding to virus 2,
$\bar{x}^{2}\gg\textbf{0}$. By assumption, both $(\bar{x}^{1},\textbf{0})$ and
$(\textbf{0},\bar{x}^{2})$ are locally exponentially stable.
The condition $s(-D^{1}+\beta_{1}^{1}A^{1})>0$ implies that the origin is
unstable; this can be observed from the proof of statement i) in Proposition
1. We are left to show that the stable manifold of the origin does not lie in
the interior of $\mathcal{D}$. We will rely on the proof technique of [12,
Lemma 3.8]. It suffices to show that for the (linear) system

$\begin{bmatrix}\dot{x}^{1}\\\
\dot{x}^{2}\end{bmatrix}=\begin{bmatrix}-D^{1}+\beta_{1}^{1}A^{1}&\textbf{0}\\\
\textbf{0}&-D^{2}+\beta_{1}^{2}A^{2}\end{bmatrix}\begin{bmatrix}x^{1}\\\
x^{2}\end{bmatrix}$ (20)

no trajectory starting in the interior of $\mathcal{D}$ converges to the
origin. First, consider $x^{1}(t)$. Let $w^{\top}$ be the left eigenvector
associated with $s(-D^{1}+\beta_{1}^{1}A^{1})$ so all its entries sum to one.
Define $z:=w^{\top}x^{1}$, and observe that $\dot{z}=w^{\top}\dot{x}^{1}$,
which, from (20), further implies that
$\dot{z}=s(-D^{1}+\beta_{1}^{1}A^{1})z$. Since, by assumption,
$s(-D^{1}+\beta_{1}^{1}A^{1})>0$, it is clear that the projection onto $w$
(which is a positive vector) of the points of (20) in the interior of
$\mathcal{D}$ is away from $x^{1}=\textbf{0}$. An analogous argument can be
made for $x^{2}(t)$, since, by assumption, $s(-D^{2}+\beta_{1}^{2}A^{2})>0$.
Therefore, the stable manifold of the origin does not lie in the interior of
$\mathcal{D}$. Consequently, since we know that system (II-A) is monotone (see
[1, Theorem 5.5]) and the monotone condition
$\bar{x}^{1}\gg{\bf{0}},{\bf{0}}\ll\bar{x}^{2}$ relates the two exponentially
stable equilibrium points, from [31, Proposition 2.9] it follows that there
exists an equilibrium point of the form $(\hat{x}^{1},\hat{x}^{2})$ such that
$\textbf{0}\ll\hat{x}^{1},\hat{x}^{2}\ll\textbf{1}$ and
$\hat{x}^{1}+\hat{x}^{2}\ll\textbf{1}$. Furthermore, the point
$(\hat{x}^{1},\hat{x}^{2})$ satisfies $s(J(\hat{x}^{1},\hat{x}^{2}))\geq 0$,
thus concluding the proof. $\Box$

Proposition 3 and Theorem 3 partially answer question iii) raised in Section
II-B. Observe that neither of these results covers the case where one boundary
equilibrium is locally exponentially stable, and the other is unstable.
We next consider a different parameter regime, namely
$s(-D^{k}+\beta_{1}^{k}A^{k})<0$, and identify a sufficient condition for the
existence of an unstable coexistence equilibrium. We have the following
result.

###### Proposition 4

Consider system (II-A) under Assumptions 1 and 2. Define, for $k=1,2$,
$\mathbf{1}_{B^{k}}\in\\{0,1\\}^{n}$ by $(\mathbf{1}_{B^{k}})_{i}=1$ if
$B_{i}^{k}\neq\mathbf{0}$; otherwise $(\mathbf{1}_{B^{k}})_{i}=0$. Suppose
that the following conditions are fulfilled for $k=1,2$ :

  1. a)

$\rho(\beta_{1}^{k}(D^{k})^{-1}A^{k})<1$, and

  2. b)

$\min\limits_{i.s.t.B_{i}^{k}\neq\mathbf{0}}\big{(}\frac{\beta_{1}^{k}}{\delta_{i}^{k}}(A^{k}\mathbf{1}_{B^{k}})_{i}+\frac{\beta_{2}^{k}}{2\delta_{i}^{k}}\mathbf{1}_{B^{k}}^{\top}B_{i}\mathbf{1}_{B^{k}}\big{)}>2$.

Then there exists at least one equilibrium of the form
$(\hat{x}^{1},\hat{x}^{2})$ such that
$\textbf{0}\ll\hat{x}^{1},\hat{x}^{2}\ll\textbf{1}$ and
$\hat{x}^{1}+\hat{x}^{2}\ll\textbf{1}$ that is either neutrally stable or
unstable.

Proof: Suppose that the conditions in Proposition 4 are fulfilled. Therefore,
it follows that there exist boundary equilibria $(\bar{x}^{1},\textbf{0})$ and
$(\textbf{0},\bar{x}^{2})$, and that both are locally exponentially stable;
see statements iii) and iv) in Proposition 1. Therefore, since we know that
system (II-A) is monotone (see [1, Theorem 5.5]), from [31, Proposition 2.9]
it follows that there exists (at least) one equilibrium point of the form
$(\hat{x}^{1},\hat{x}^{2})$ such that
$\textbf{0}\ll\hat{x}^{1},\hat{x}^{2}\ll\textbf{1}$ and
$\hat{x}^{1}+\hat{x}^{2}\ll\textbf{1}$. Furthermore,
$s(J(\hat{x}^{1},\hat{x}^{2}))\geq 0$, thus delivering the claim. $\Box$

Note that Proposition 4 guarantees the existence of at least one coexistence
equilibrium. Given that system (II-A) is monotone, and since, from Theorem 1,
it is known that for each of the equilibrium points the associated Jacobian is
nonsingular, the conditions in Proposition 4 guarantee the existence of an odd
number of coexistence equilibria, each of which must be unstable. The proof
for the same follows from a Brouwer degree argument; see [32]. In fact, for
the special case where $\beta_{2}^{k}=0$ for $k=1,2$, for the same stability
configuration as in Theorem 3 and Proposition 4, a lower bound on the number
of coexistence equilibria has been recently provided; see [12, Corollary 3.9,
statement 3].

##  VI Numerical Examples

We present a series of simulations highlighting interesting phenomena that can
emerge when HOIs are incorporated. We use the following bivirus system with
HOIs. The network has $n=5$ nodes, and we set $D^{1}=D^{2}=I$. The pairwise
interactions are captured by two-cycle graphs with self-loops, with infection
matrices:

$A^{1}=\begin{bmatrix}1&0&0&0&1\\\ 1&1&0&0&0\\\ 0&1&1&0&0\\\ 0&0&1&1&0\\\
0&0&0&1&1\end{bmatrix}\,,\qquad A^{2}=(A^{1})^{\top}.$ (21)

The HOI are captured by the following set of hyperedges with unit weight:

$\displaystyle\text{virus }1:(1,2,3),(2,3,1),(3,2,1),(1,4,5),(4,5,1),(5,4,1)$
$\displaystyle\text{virus }2:(1,2,4),(2,4,1),(4,2,1),(1,3,5),(3,5,1),(5,3,1).$

In other words, this corresponds to the following $b^{k}_{ij\ell}$ entries
being equal to $1$, with all other entries of $B^{k}_{i}$ equal to $0$:
$b^{1}_{123}$, $b^{1}_{231}$, $b^{1}_{321}$, $b^{1}_{145}$, $b^{1}_{451}$,
$b^{1}_{541}$, and $b^{2}_{124}$, $b^{2}_{241}$, $b^{2}_{421}$, $b^{2}_{135}$,
$b^{2}_{351}$, $b^{2}_{531}$. In our simulations, we randomly sample
$x^{k}_{i}(0)$ from a uniform distribution $(0,1)$, and then normalize the
vectors $x^{1}(0)$ and $x^{2}(0)$ to ensure that
$(x^{1}(0),x^{2}(0))\in\text{int}(\mathcal{D})$. The $\beta^{k}_{i}$ are
varied to yield different stability properties for the system in (II-A).

Example 1: We set $\beta^{1}_{1}=\beta^{1}_{2}=0.2$ and
$\beta^{2}_{1}=\beta^{2}_{2}=5$. This ensures the inequalities of both
conditions for Proposition 1 are satisfied. As can be observed from Fig. 1(a),
for initial conditions close to the DFE, the trajectories converge to the
locally exponentially stable DFE, $(x^{1}={\bf 0},x^{2}={\bf 0})$. In Figs.
1(b) and 1(c), the initial conditions are further in the interior of
$\mathcal{D}$, and depending on the particular initial condition, we observe
convergence to a boundary equilibrium where one of the two viruses is extinct,
$(\bar{x}^{1},{\bf 0})$ or $({\bf 0},\bar{x}^{2})$ for some positive
$\bar{x}^{1}>0.5\times{\bf 1}$ and $\bar{x}^{2}>0.5\times{\bf 1}$. That is,
both boundary equilibria are simultaneously locally exponentially stable.
Interestingly, without HOIs, it is impossible for a bivirus system to have the
DFE, $(\bar{x}^{1},{\bf 0})$, and $({\bf 0},\bar{x}^{2})$ all locally
exponentially stable [33, Section E].

(a)

(b)

(c)

Figure 1: Trajectories of the system (II-A), for different initial conditions
in Example 1.

Example 2: We set $\beta^{1}_{1}=\beta^{1}_{2}=2$ and $\beta^{2}_{1}=3$ and
$\beta^{2}_{2}=2.4$. As illustrated in Figs. 2(a) and 2(b), there are two
locally exponentially stable two boundary equilibria $(\bar{x}^{1},{\bf 0})$
or $({\bf 0},\bar{x}^{2})$, and we converge to either depending on the initial
conditions. However, the DFE is unstable, and no trajectories in $\mathcal{D}$
converge there except if one starts at the DFE. This simulation highlights are
interesting observation: for a standard bivirus system with no HOIs, examples
of systems with two locally stable boundary equilibria have not been
identified until recently and are not straightforward to construct [12, 34].

(a)

(b)

Figure 2: Trajectories of the system (II-A), for different simulation
parameters and initial conditions. 2(a) and 2(b) correspond to different
initial conditions for Example 2.

We conclude by remarking that, for each of the simulations presented, the
bivirus system exhibits dynamical phenomena when HOIs are present that is not
observed when HOIs are not present. In other words, HOIs unlock new
possibilities in the competition of two viruses spreading over networks and
suggest a significant amount of understanding remains to be unveiled.

##  VII Conclusion

This paper analyzed a networked competitive bivirus SIS model that also
accounts for the possibility of HOI among the nodes. By taking recourse to the
Parametric Transversality Theorem of differential topology, we showed that the
bivirus system with HOI has, for generic parameter values, a finite number of
equilibria. Furthermore, the Jacobian matrices associated with each of the
equilibria are nonsingular. This finding, coupled with the knowledge that the
system is monotone, enabled us to establish that the typical behavior that our
system exhibits is convergence to some equilibrium. Subsequently, we
identified a parameter regime that ensures the existence of multiple boundary
equilibria and simultaneous stability of the same along with that of the DFE.
For the special case where only one virus is circulating in the meta-
population, we guarantee the existence and local stability of an endemic
equilibrium; our result does not impose any restrictions on the model
parameters besides those covered by Assumptions 1 and 2. Thereafter, for
different parameter regimes, we identified conditions that guarantee the
existence of a coexistence equilibrium.

## References

  * [1] S. Cui, F. Liu, H. Jardón-Kojakhmetov, and M. Cao, “General SIS diffusion process with indirect spreading pathways on a hypergraph,” _arXiv preprint arXiv:2306.00619_ , 2023.
  * [2] P. Van Mieghem, J. Omic, and R. Kooij, “Virus spread in networks,” _IEEE/ACM Transactions On Networking_ , vol. 17, no. 1, pp. 1–14, 2008.
  * [3] C. Castillo-Chavez, H. W. Hethcote, V. Andreasen, S. A. Levin, and W. M. Liu, “Epidemiological models with age structure, proportionate mixing, and cross-immunity,” _Journal of Mathematical Biology_ , vol. 27, no. 3, pp. 233–258, 1989.
  * [4] B. A. Prakash, A. Beutel, R. Rosenfeld, and C. Faloutsos, “Winner takes all: competing viruses or ideas on fair-play networks,” in _Proceedings of the 21st International Conference on World Wide Web_ , 2012, pp. 1037–1046.
  * [5] C. Nowzari, V. M. Preciado, and G. J. Pappas, “Analysis and control of epidemics: A survey of spreading processes on complex networks,” _IEEE Control Systems Magazine_ , vol. 36, no. 1, pp. 26–46, 2016.
  * [6] S. Gracy, P. E. Paré, J. Liu, H. Sandberg, C. L. Beck, K. H. Johansson, and T. Başar, “Modeling and analysis of a coupled SIS bi-virus model,” _arXiv:2207.11414_ , 2023.
  * [7] A. Nedić, A. Olshevsky, and C. A. Uribe, “Graph-theoretic analysis of belief system dynamics under logic constraints,” _Scientific Reports_ , vol. 9, no. 1, pp. 1–16, 2019.
  * [8] F. D. Sahneh and C. Scoglio, “Competitive epidemic spreading over arbitrary multilayer networks,” _Physical Review E_ , vol. 89, no. 6, p. 062817, 2014\.
  * [9] A. Janson, S. Gracy, P. E. Paré, H. Sandberg, and K. H. Johansson, “Networked multi-virus spread with a shared resource: Analysis and mitigation strategies,” _arXiv:2011.07569_ , 2020.
  * [10] J. Liu, P. E. Paré, A. Nedić, C. Y. Tang, C. L. Beck, and T. Başar, “Analysis and control of a continuous-time bi-virus model,” _IEEE Transactions on Automatic Control_ , vol. 64, no. 12, pp. 4891–4906, 2019.
  * [11] M. Ye, B. D. Anderson, and J. Liu, “Convergence and equilibria analysis of a networked bivirus epidemic model,” _SIAM Journal on Control and Optimization_ , vol. 60, no. 2, pp. S323–S346, 2022.
  * [12] B. D. Anderson and M. Ye, “Equilibria analysis of a networked bivirus epidemic model using Poincaré-Hopf and manifold theory,” _SIAM Journal on Applied Dynamical Systems_ , to appear.
  * [13] A. Santos, J. M. Moura, and J. M. Xavier, “Bi-virus SIS epidemics over networks: Qualitative analysis,” _IEEE Transactions on Network Science and Engineering_ , vol. 2, no. 1, pp. 17–29, 2015.
  * [14] S. Gracy, M. Ye, B. Anderson, and C. A. Uribe, “Towards understanding the endemic behavior of a competitive tri-virus SIS networked model,” 2023. [Online]. Available: arXiv:2303.16457
  * [15] C. Castillo-Chavez, W. Huang, and J. Li, “Competitive exclusion and coexistence of multiple strains in an SIS STD model,” _SIAM Journal on Applied Mathematics_ , vol. 59, no. 5, pp. 1790–1811, 1999.
  * [16] A. Hatcher, _Algebraic Topology_ , 2005.
  * [17] C. Bick, E. Gross, H. A. Harrington, and M. T. Schaub, “What are higher-order networks?” _SIAM Review_ , vol. 65, no. 3, pp. 686–731, 2023.
  * [18] I. Iacopini, G. Petri, A. Barrat, and V. Latora, “Simplicial models of social contagion,” _Nature Communications_ , vol. 10, no. 1, p. 2485, 2019.
  * [19] G. F. de Arruda, G. Petri, and Y. Moreno, “Social contagion models on hypergraphs,” _Physical Review Research_ , vol. 2, no. 2, p. 023032, 2020\.
  * [20] P. Cisneros-Velarde and F. Bullo, “Multigroup SIS epidemics with simplicial and higher order interactions,” _IEEE Transactions on Control of Network Systems_ , vol. 9, no. 2, pp. 695–705, 2021.
  * [21] P. E. Paré, J. Liu, C. L. Beck, A. Nedić, and T. Başar, “Multi-competitive viruses over time-varying networks with mutations and human awareness,” _Automatica_ , vol. 123, p. 109330, 2021.
  * [22] A. Rantzer, “Distributed control of positive systems,” in _Proceedings of the 50th IEEE Conference on Decision and Control_ , 2011, pp. 6608–6611.
  * [23] C. Meyer, _Matrix Analysis and Applied Linear Algebra_ , ser. Other Titles in Applied Mathematics. SIAM, 2000.
  * [24] R. Varga, _Matrix Iterative Analysis_ , ser. Springer Series in Computational Mathematics. Springer Berlin Heidelberg, 1999.
  * [25] J. M. Lee, _Introduction to Smooth Manifolds_. Springer, 2013.
  * [26] V. Guillemin and A. Pollack, _Differential Topology_. American Mathematical Soc., 2010, vol. 370.
  * [27] M. W. Hirsch, “Systems of differential equations that are competitive or cooperative ii: Convergence almost everywhere,” _SIAM Journal on Mathematical Analysis_ , vol. 16, no. 3, pp. 423–439, 1985.
  * [28] E. D. Sontag, “Monotone and near-monotone biochemical networks,” _Systems and Synthetic Biology_ , vol. 1, no. 2, pp. 59–87, 2007.
  * [29] H. Khalil, _Nonlinear Systems_. Prentice Hall, 2002.
  * [30] J. Hofbauer, “An index theorem for dissipative semiflows,” _The Rocky Mountain Journal of Mathematics_ , vol. 20, no. 4, pp. 1017–1031, 1990.
  * [31] H. L. Smith, “Systems of ordinary differential equations which generate an order preserving flow. a survey of results,” _SIAM Review_ , vol. 30, no. 1, pp. 87–113, 1988.
  * [32] ——, “Competing subcommunities of mutualists and a generalized kamke theorem,” _SIAM Journal on Applied Mathematics_ , vol. 46, no. 5, pp. 856–874, 1986.
  * [33] M. Ye and B. D. Anderson, “Competitive epidemic spreading over networks,” _IEEE Control Systems Letters_ , vol. 7, pp. 545–552, 2022.
  * [34] M. Ye, B. D. O. Anderson, A. Janson, S. Gracy, and K. H. Johansson, “Competitive epidemic networks with multiple survival-of-the-fittest outcomes,” 2021. [Online]. Available: arXiv:2111.06538

## Appendix

### Proof of Theorem 1

We treat first the case where the $A^{k}$ and $B^{k}_{i}$ are fixed. Consider
(II-A) in the shorthand form $\dot{x}=f(x)$, and to emphasize the possibility
of variations in the entries of $D^{k}$, show this explicitly by writing the
equations as

$\dot{x}=f_{\delta}(x,\delta)$ (22)

where $\delta$ is a vector containing all the diagonal entries of the $D^{k}$.
It is easily established that

$\frac{\partial
f_{\delta}(x^{1},x^{2},D^{1},D^{2})}{\partial\delta}=\frac{\partial
f_{\delta}(x^{1},x^{2},D^{1},D^{2})}{\partial(D^{1},D^{2})}=\begin{bmatrix}X^{1}&0\\\
0&X^{2}\end{bmatrix}$

At a coexistence equilibrium, this matrix has full row rank. Consequently, the
matrix $\frac{\partial f_{\delta}(x,\delta)}{\partial(x,\delta)}$ (which is
obtained simply by adding further columns) also has full row rank at a
coexisitence equilibrium.

Similarly, in preparation for considering the second claim of the theorem, we
can consider $f$ depending explicitly on the possibly variable $A^{k}$. There
are a finite number of patterns of zero entries for the $A^{k}$ for which the
$A^{k}$ are irreducible. For each matrix separately, arbitrarily choose any
one such pattern, and let $\alpha^{k}$ denote the vector of nonzero entries of
the $A^{k}$; also set
$\alpha=[(\alpha^{1})^{\top}\;(\alpha^{2})^{\top}]^{\top}$. Denote the
associated $f$ as $f_{\alpha}(x,\alpha)$. Then one can show that

$\frac{\partial
f_{\alpha}(x,\alpha)}{\partial\alpha}=\begin{bmatrix}I-X^{1}-X^{2}&0\\\
0&I-X^{1}-X^{2}\end{bmatrix}\begin{bmatrix}\frac{\partial(A^{1}x^{1})}{\partial\alpha^{1}}&0\\\
0&\frac{\partial(A^{2}x^{2})}{\partial\alpha^{2}}\end{bmatrix}$ (23)

A detailed calculation as set out in [12] shows that the second matrix in the
product has full row rank. Examination of the system equation (II-A)
immediately reveals that no equilibrium can occur where any diagonal entry of
$I-X^{1}-X^{2}=0$, which means that the first matrix of the product is
nonsingular. Hence the matrix on the left has full row rank at any
equilibrium. The matrix $\frac{\partial
f_{\alpha}(x,\alpha)}{\partial(x,\alpha)}$, obtained by the addition of
further columns to $\frac{\partial f_{\alpha}(x,\alpha)}{\partial\alpha}$,
then has full row rank.

We now complete the proof for the first part of the theorem. (Proof of the
second part is virtually identical). Without loss of generality, suppose that
for some $\bar{\delta}$, all diagonal entries $\delta_{i}$ of the $D^{k}$
satisfy $0<\bar{\delta}<\delta_{i}<\bar{\delta}^{-1}$. Denote the set of such
$\delta$ as $\mathcal{D}$. Call the set of $x$ in the interior of the region
of interest $\mathcal{X}$. Then $\mathcal{X}\times\mathcal{D}$ is a manifold,
being a product of open sets, and $f_{\delta}(x,\delta)$ is a mapping of the
manifold $\mathcal{X}\times\mathcal{D}$. Let the image be denoted by
$\mathcal{Y}$, and let $\mathcal{Z}$ be the submanifold of $\mathcal{Y}$
consisting of the single point ${\bf{0_{2n}}}$. (The set
$f_{\delta}^{-1}(\mathcal{Z})$ is the set of zeros of $f$.) Since the Jacobian
of $f_{\delta}$ with respect to $(x,\delta)$ has full row rank on this set,
this means that the map $f_{\delta}$ is transversal to $\mathcal{Z}$. By the
Parametric Transversality Theorem, this means that for all choices of
$\delta=\delta^{*}$ excluding a set of measure zero, the Jacobian
$\frac{\partial f_{\delta^{*}}(x,\delta^{*})}{\partial x}$ will be nonsingular
at a preimage of $\mathcal{Z}$, i.e at a zero. This is equivalent to saying
that the zeros are nondegenerate. The boundedness of the region $\mathcal{X}$
then guarantees that there can only be a finite number of such zeros.