remarkRemark hypothesisHypothesis claimClaim Competitive tri-virus model:
Endemic BehaviorS. Gracy, M. Ye, BDO Anderson, and CA Uribe

# Towards Understanding the Endemic Behavior of a Competitive Tri-Virus SIS
Networked Model††thanks: Submitted to the editors on . A preliminary version
of the paper has been accepted for publication in the Proceedings of the
$2023$ American Control Conference, [12]. Part of this material is based upon
work supported by the National Science Foundation under Grants #2211815 and
No. #2213568. The work of SG was also funded by a grant from the Rice Academy
of Fellows, Rice University, Houston, TX.

Sebin Gracy Department of Electrical and Computer Engineering, Rice
University, Houston, TX, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>Mengbin Ye Centre for Optimisation and Decision Science, Curtin University,
Australia<EMAIL_ADDRESS>Brian D.O. Anderson School of
Engineering, Australian National University, Canberra, Australia. ()
<EMAIL_ADDRESS>César A Uribe22footnotemark: 2

###### Abstract

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.

###### keywords:

Epidemic processes, competing viruses, finiteness of equilibria, coexistence
equilibrium.

34D05, 37C75, 92D30

##  1 Introduction

Spreading processes are observed in several settings. Prominent examples
include the spread of viruses, in particular, pandemics such as COVID-19 [54],
Spanish flu [18]; spread of opinions on social media [56, 4]; adoption of
products in a market [3]; and propagation of species in an ecological
environment [28]. Given these manifestations in several diverse settings, the
study of spreading processes has naturally attracted the attention of multiple
scientific communities viz. physics [32, 19, 57], biology [24, 59],
mathematics [5, 6, 14], computer science [39], to cite a few.

One of the primary objectives behind all of the aforementioned research
efforts is to understand the various equilibria of the systems involved and
the behavior of spread processes in the neighborhood of an equilibrium point.
Several models have been proposed and extensively studied in mathematical
epidemiology: notable among them being the susceptible-infected-recovered
(SIR) [30, 50]; susceptible-exposed-infected-recovered (SEIR) [26, 2];
susceptible–asymptomatic–infected–recovered–susceptible (SAIRS) [42, 34];
susceptible-infected (SI) [29]; and susceptible-infected-susceptible (SIS)
models [20, 23, 52, 22]. The aforementioned models provide rigorous guarantees
for (some of) the various outcomes of spreading processes; the conditions
involved in providing said guarantees are tractable; and finally, they can
generate insightful observations due to the low cost of simulations. Thus,
these models are extremely useful in understanding the behavior of spreading
processes and, consequently, have gained widespread acceptance in the broader
research community. Our focus in the present paper is on networked SIS models,
which consider a population of individuals that are divided into
subpopulations, called _agents_. A (possibly) directed graph captures how the
various agents are interconnected. Given that there is a virus spreading in
the whole population, it can spread both within a subpopulation (i.e., between
individuals in a subpopulation) and across subpopulations (i.e., between
individuals belonging to different subpopulations). If none of the individuals
in a subpopulation are infected, then this subpopulation is said to be in the
susceptible state; otherwise, it is said to be in the infected state.
Networked SIS models have been extensively studied in the literature; see, for
instance, [8, 22, 41, 37, 55, 11, 34].

Available models account for the first instance of the spread of only one
virus. However, one encounters scenarios where multiple viruses are
simultaneously circulating in a population. In such a context, the viruses
involved could either be cooperative (see [10] and references therein for
details), or they could be competitive - the present paper focuses on the
competitive case. The notion of competing viruses can be observed in several
settings; some examples include spread of multiple strains of a virus [33, 24,
59], spread of competing opinions on social networks[38, 58], and that of
products competing in a market [49]. More concretely, supposing that two
viruses (say virus 1 and virus 2) are competing, an individual can be either
infected with virus 1 or with virus 2 or with neither. In contrast, no
individual can be infected with _both_ viruses simultaneously. The dynamics of
competing viruses are far richer than the case of single-virus propagation
[32]. Specifically, in multi-competitive virus settings, some possibilities
include one virus dominating the rest and all (or a subset of all) viruses
existing in some balance – a phenomenon referred to as _coexistence_. Various
mathematical models have been proposed in the literature to deal with multi-
competitive viruses. Prime among them is the susceptible-infected-recovered
(SIR) model (proposed in [32] and improved upon in [64]) and the susceptible-
infected-susceptible (SIS) model (first proposed in [5]). This paper focuses
on the continuous-time multi-competitive networked SIS model.

The study of multi-competitive networked SIS models has been pursued in a
plethora of works starting with [5, 6], and, more recently, in [43, 17, 44,
62, 27, 61, 35, 1, 7, 12]. Note that the vast majority of the literature in
this area, namely [43, 44, 7, 17, 27, 62, 1], focuses on the special case when
two viruses compete with each other to infect a given population. The more
general case where $m>2$ viruses are competing has been relatively less well
studied; see [35] (resp. [36]) for a preliminary analysis of (resp. discrete-
time) multi-competitive networked SIS model. While [35, 36] provide the
general modeling framework for multi-competitive networked SIS models, in that
the models therein admit the possibility of more than two viruses competing,
the analysis of the possible endemic behavior in said papers is rather
limited; in particular most of the results in [36] pertain to the existence of
boundary equilibria. Thus, to our knowledge, an overarching analysis
accounting for $m$ competing viruses, where $m$ is some arbitrary but finite
positive integer, is not available in the existing literature. Therefore, as a
first step, in this direction, the present paper deals with the case where
$m=3$ (tri-virus system). The equilibria of the tri-virus system can be
broadly classified into three categories: the disease-free equilibrium (DFE)
(all three viruses have been eradicated); the boundary equilibria (two viruses
are dead, and one is endemic); and $k$-coexistence equilibria (where $k\geq 2$
viruses infect separate fractions of every population node in the network).

We make the following key contributions:

  1. i)

We show that it (i.e., the tri-virus system) is not monotone; see Theorem 1.
Consequently, one cannot use the existing tools in the literature on
competitive bivirus systems, which are deeply rooted in monotone dynamical
systems (see [15, 46]) to study the limiting behavior of our model.

  2. ii)

We show, using the Parametric Transversality Theorem of differential topology
[25, page 145], that it has a finite number of equilibria, in a generic 111A
precise definition of the term “generic” is provided in Section 3. sense; see
Proposition 4. Additionally, Proposition 4 also establishes that the Jacobian
matrices evaluated at each of the equilibrium points are nonsingular.

  3. iii)

We identify a sufficient condition, and multiple necessary conditions, for
local exponential convergence to a boundary equilibrium; see Theorem 2.

  4. iv)

We identify a sufficient condition for the nonexistence of any 3-coexistence
equilibria, and nonexistence of certain forms of 2-coexistence equilibria; see
Theorem 3 and Theorem 4, respectively.

  5. v)

Leveraging that a bivirus system is monotone, we identify a novel and simpler
sufficient condition for a 2-coexistence equilibrium; see Proposition 7.

  6. vi)

Using the notion of saturated fixed points, we identify a sufficient condition
for the existence of a 3-coexistence equilibrium; see Corollary 1.

  7. vii)

We identify a special scenario (with respect to the class of system
parameters) that gives rise to the existence of, and, subject to the
fulfillment of certain conditions, local convergence to, a line of coexistence
equilibria; see Theorem 6.

  8. viii)

Assuming that the three viruses are identical, we show that, regardless of the
initial non-zero infection levels, the dynamics of the tri-virus system
converge to a plane of coexisting equilibria; see Theorem 7.

In addition, we make several auxiliary contributions. More specifically, we
show that for low-dimensional systems, i.e., $n=1$ or $n=2$, certain kinds of
coexistence equilibria can, generically, never exist; see Propositions 5 and
6. Using the notion of saturated fixed points, we prove that either a stable
boundary equilibrium exists and/or a saturated 2-coexistence equilibrium
and/or a 3-coexistence equilibrium; see Theorem 5. We identify a special
setting, but one that subsumes the setting in Theorem 7, that permits the
existence of a plane of coexistence equilibria; see Proposition 8.

A subset of the results of this paper has been accepted for publication in the
Proceedings of the $2023$ American Control Conference; see [12]. The present
paper is a significant extension from [12]; Theorem 3, Theorem 4, Corollary 1,
Theorem 5, Propositions 4, 5, 6, 7 and 8 were not provided in [12].

### Paper Outline

The paper unfolds in the following manner: We use the rest of this section to
collect notations used in the sequel. The formal presentation of the tri-virus
model and the questions of interest in the present paper are provided in
Section 2. We investigate the monotonicity of the tri-virus system in Section
3, whereas Section 4 deals with the question of the finiteness of the number
of equilibria (in a generic sense). Note that the proof for the result in
Section 4 is relegated to the Appendix since it uses different tools compared
to the rest of the results in the paper. Section 5 presents a necessary and
sufficient condition for local exponential stability of the boundary
equilibrium. Sections 6 and 7 deal with the nonexistence and existence of
various kinds of coexistence equilibria. Certain special setting(s), all of
which have measure zero, that yield a line (resp. plane) of coexistence
equilibria, are provided in Section 8 (resp. Section 9). Simulations
illustrating our theoretical findings are provided in Section 10. A summary of
the paper, along with certain future research directions, is provided in
Section 11.

### Notations

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 $\\{1,2,...,n\\}$. The $i^{\rm{th}}$ entry of a vector $x$
is denoted by $x_{i}$. The element in the $i^{\rm{th}}$ row and $j^{\rm{th}}$
column of a matrix $M$ is denoted by $M_{ij}$. 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, while the sizes of the vectors and matrices are to
be understood from the context. For a vector $x$ we denote the square matrix
with $x$ along the diagonal by $\operatorname{diag}(x)$. For any two real
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 real 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 $A\neq B$, and $A\gg B$ if $A_{ij}>B_{ij}$ for all
$i\in[n]$, $j\in[m]$. 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 largest real part among the eigenvalues of $M$, i.e.,
$s(M)=\max\\{\rm{Re}(\lambda):\lambda\in\sigma(M)\\}$.

A square matrix $A$ is said to be Hurwitz if $s(A)<0$. A real square matrix
$A$ is called Metzler if all its off-diagonal entries are nonnegative. A real
square matrix $A$ is said to be a Z-matrix if all of its off-diagonal entries
are nonpositive. A Z-matrix is an M-matrix if all its eigenvalues have
nonnegative real parts. An M-matrix is singular or nonsingular if it has an
eigenvalue at the origin or if all its eigenvalues have strictly positive
parts. The matrix $A$ is positive semidefinite if $x^{\top}Ax\geq 0$ for all
vectors $x$; we denote this by $A\succeq 0$.

##  2 Problem Formulation

This section presents a model representing the possible spread of multiple
competing viruses over a population network. Thereafter, we state the
requisite assumptions and definitions needed for the theoretical developments
of this paper. Lastly, we will formally state the problems of interest.

###  2.1 Model

We consider a network of $n\geq 2$ nodes222As an aside, multivirus SIS models
where $n=1$ have been studied in, among others, [34, Section 2].. Each node
represents a well-mixed population of individuals (i.e., any two individuals
in the population can interact with the same positive probability) with a
large and constant size. One of the central assumptions underpinning this
model is that of homogeneity within the population node and (possible)
heterogeneity outside the population node. That is, all individuals within a
population node have the same infection (resp. healing) rates, but individuals
in different population nodes need not necessarily have the same healing
(resp. infection) rate [23]. We suppose that $m$ viruses compete with each
other to infect the nodes. The fact that the viruses are competing implies
that at least two (but possibly more) viruses are simultaneously circulating
in the population.

The case where $m=1$ (meaning there is no competition) has been well-studied
in the literature; see, for instance, [22, 23, 9, 51], whereas the case where
$m=2$ (i.e., two competing viruses) has been explored relatively well in
recent times, see, for example, [43, 44, 62, 27, 35]. This paper, for reasons
mentioned in Section 1, focuses on the case where $m=3$. Hence, within each
population node, individuals can be partitioned into four mutually exclusive
health compartments: susceptible, infected with virus 1, infected with virus
2, and infected with virus 3. Note that no individual can be _simultaneously_
infected by more than one virus [24, 59]. A population node is termed
_healthy_ if all individuals in said node belong to the susceptible
compartment; otherwise, we say it is infected. An individual belonging to
population node $i$ (where $i\in[n]$), in the susceptible compartment, as a
consequence of coming into contact with its infected neighbors, transitions to
the “infected with virus $k$” (for $k\in[m]$) compartment at a rate
$\beta_{i}^{k}>0$. An individual in population $i$ that is infected with virus
$k$ recovers from it based on said individual’s healing rate with respect to
virus $k$, i.e., $\delta_{i}^{k}>0$.

We model the spread of $m$-competing viruses using an $m$-layer graph $G$; the
vertices of the graph represent the population nodes. The contact graph for
the spread of virus $k$, for each $k\in[m]$, is denoted by the
$k^{\textrm{th}}$ layer. More specifically, for the graph $G$, there exists a
directed edge from node $j$ to node $i$ in layer $k$ if, assuming an
individual in population $j$ is infected with virus $k$, then said individual
can infect at least one (but possibly more) healthy individual in node $i$.
The edge set corresponding to the $k^{\textrm{th}}$ layer of $G$ is denoted by
$E^{k}$, whereas $A^{k}$ denotes the weighted adjacency matrix corresponding
to layer $k$ (where $a_{ij}^{k}\geq 0$). The elements in $A^{k}$ are in one-
to-one correspondence with the existence (or lack thereof) of edges in layer
$k$. That is, $(i,j)\in E^{k}$ if, and only if, $a_{ji}^{k}\neq 0$. We use
$x_{i}^{k}(t)$ to represent the fraction of individuals infected with virus
$k$ in population $i$ at time instant $t$. Given that the viruses are
circulating in the population nodes, the infection level $x_{i}^{k}(t)$
possibly changes over time. Hence, the evolution of said fraction can, then,
be represented by the following scalar differential equation [35, Equation 4]:

(1)
$\dot{x}_{i}^{k}(t)=-\delta_{i}^{k}x_{i}^{k}(t)+\big{(}1-\textstyle\sum_{l=1}^{m}x_{i}^{l}(t)\big{)}\textstyle\sum_{j=1}^{n}\beta_{ij}^{k}x_{j}^{k}(t),$

where $\beta_{ij}^{k}=\beta_{i}^{k}a_{ij}^{k}$.

Define $x^{k}(t)=[x_{1}^{k}(t),\ldots,x_{n}^{k}(t)]^{\top}$,
$D^{k}=\operatorname{diag}{(\delta_{i}^{k})}$, and
$B^{k}=[\beta_{ij}^{k}]_{n\times n}$. Therefore, the $n$ coupled equations in
(1) for $i\in[n]$ can be written as

(2)
$\dot{x}^{k}(t)=\Big{(}-D^{k}+\big{(}I-\textstyle\sum_{l=1}^{m}\operatorname{diag}(x^{l}(t))\big{)}B^{k}\Big{)}x^{k}(t),$

Defining $x(t):=[x^{1}(t),\dots,x^{m}(t)]^{T}$, and
$R^{k}(x(t)):=\big{(}-D^{k}+(I-\textstyle\sum_{l=1}^{m}\operatorname{diag}(x^{l}(t)))B^{k}\big{)}$,
the dynamics of the system of all $m$ viruses are given by

(3)
$\displaystyle\dot{x}(t)=\begin{bmatrix}R^{1}\big{(}x(t)\big{)}&0&\dots&0\\\
0&R^{2}\big{(}x(t)\big{)}&\dots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\dots&R^{m}\big{(}x(t)\big{)}\end{bmatrix}x(t).$

Define, for $k\in[3]$, $X^{k}=\operatorname{diag}(x^{k})$. Based on (2), the
dynamics of the tri-virus system can be written as follows:

(4) $\displaystyle\dot{x}^{1}(t)$
$\displaystyle=\Big{(}\big{(}I-(X^{1}+X^{2}+X^{3})\big{)}B^{1}-D^{1}\Big{)}x^{1}(t),$
(5) $\displaystyle\dot{x}^{2}(t)$
$\displaystyle=\Big{(}\big{(}I-(X^{1}+X^{2}+X^{3})\big{)}B^{2}-D^{2}\Big{)}x^{2}(t)$
(6) $\displaystyle\dot{x}^{3}(t)$
$\displaystyle=\Big{(}\big{(}I-(X^{1}+X^{2}+X^{3})\big{)}B^{3}-D^{3}\Big{)}x^{3}(t).$

###  2.2 Assumptions

We need the following assumptions to ensure that the aforementioned model is
well-defined. Further, note that these assumptions are standard in the
literature on (multi-competitive) networked SIS models; see, for instance,
[23, 9, 22, 27].

######  Assumption 1.

Suppose that $\delta_{i}^{k}>0,\beta_{ij}^{k}\geq 0$ for all $i,j\in[n]$ and
$k\in[3]$.

######  Assumption 2.

The matrix $B^{k}$, for $k\in[3]$ is irreducible.

Under Assumption 1, for all $k\in[3]$, $B^{k}$ is a nonnegative matrix, and
$D^{k}$ is a positive diagonal matrix. Moreover, recall that a square
nonnegative matrix $M$ has the irreducibility property if, and only if,
supposing $M$ is the (un)weighted adjacency matrix of a graph, the said graph
is strongly connected. Then, noting that non-zero elements in $B^{k}$
represent directed edges in the set $E^{k}$, we see that $B^{k}$ is
irreducible whenever the $k^{\rm{th}}$ layer of the multi-layer network $G$ is
strongly connected.

As a consequence of Assumption 1, our analysis of the model in (4)-(6) is
limited to the sets $\mathcal{D}:=\\{x(t):x^{k}(t)\in[0,1]^{n},\forall
k\in[3],\sum_{k=1}^{3}x^{k}\leq\textbf{1}\\}$ and
$\mathcal{D}^{k}:=\\{x^{k}(t)\in[0,1]^{n}\\}$. Given that $x_{i}^{k}(t)$ is
interpreted as a fraction of a population, the aforementioned sets represent
the sensible domain of the system. That is, if $x^{k}(t)$ takes values outside
of $\mathcal{D}^{k}$, then those values will not correspond to physical
reality. The following lemma shows that $x(t)$ never leaves the set
$\mathcal{D}$.

######  Lemma 1.

[35, Lemma 1] Let Assumption 1 hold. Then $\mathcal{D}$ is positively
invariant with respect to (3).

######  Lemma 2.

[17, Lemma 7] Let Assumption 1 hold. Then
$\mathcal{D}\setminus\\{\textbf{0}\\}$ is positively invariant with respect to
system (4)-(6).

###  2.3 Various Equilibria of the Model

Clearly, $(\textbf{0},\textbf{0},\textbf{0})$ is an equilibrium of (4)-(6),
and is referred to as the DFE. A sufficient condition for global exponential
stability (GES) of the DFE is as follows:

######  Proposition 1.

[17, Theorem 1] Consider system (4)-(6) under Assumption 1. If
$s(-D^{k}+B^{k})<0$, for each $k\in[3]$, then the DFE is exponentially stable,
with a domain of attraction containing $\mathcal{D}$.

Clearly, the conditions in Proposition 1 also imply asymptotic convergence to
the DFE. However, even when the strict inequalities in the aforementioned
proposition are weakened to allow equality, asymptotic convergence to the DFE
can still be achieved. The next proposition, which is a generalization of an
analogous result for the bivirus setting (see [27, Theorem 1]), formalizes the
same.

######  Proposition 2.

[35, Lemma 2] Consider system (4)-(6) under Assumption 1. If
$s(-D^{k}+B^{k})\leq 0$, for each $k\in[3]$, then the DFE is the unique
equilibrium of system (4)-(6). Moreover, it is asymptotically stable with the
domain of attraction $\mathcal{D}$.

Note that Proposition 1 and 2 provide guarantees on convergence to the DFE. It
is natural to ask what happens if, for some $k\in[m]$, one of the eigenvalues
of the matrix $-D^{k}+B^{k}$ has a positive real part. It turns out for each
$k\in[m]$, that violates the eigenvalue condition in Proposition 2, there
exists an equilibrium of the form
$(\textbf{0},\dots,\tilde{x}^{k},\dots,\textbf{0})$ in $\mathcal{D}$. The
following proposition formalizes this.

######  Proposition 3.

[9, Theorem 2.1] Consider system (4)-(6) under Assumptions 1 and 2. For each
$k\in[3]$, there exists a unique equilibrium
$(\textbf{0},\dots,\tilde{x}^{k},\dots,\textbf{0})$ in $\mathcal{D}$, with
$\textbf{0}\ll\tilde{x}^{k}\ll\textbf{1}$ if, and only if, $s(B^{k}-D^{k})>0$.

Indeed, it is straightforward to show that $\tilde{x}^{k}$ is the endemic
equilibrium for the single virus system (i.e., $m=1$) defined by the pair of
matrices ($D^{k},B^{k}$). Assuming $m=1$, analytic methods for computing the
endemic equilibrium333In the single-virus case, an endemic equilibrium, when
it exists, is unique have been provided in [51, Theorem 5].

Any non-zero equilibrium in $\mathcal{D}$ is referred to as an _endemic_
equilibrium. Endemic equilibria can be further classified as follows:
Equilibria of the form $(\textbf{0},\dots,\tilde{x}^{k},\dots,\textbf{0})$ are
referred to as the _boundary equilibria_. The equilibria of the form
$(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})$, where at least $\bar{x}^{i}$ and
$\bar{x}^{j}$ ($i,j\in[3],i\neq j$) are nonnegative vectors with at least one
positive entry in each of $\bar{x}^{i}$ and $\bar{x}^{j}$ are referred to as
_coexistence equilibria_. The coexistence equilibria can be further classified
as i) 2-coexistence equilibria, which are equilibria of the form
$(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})$ where $\bar{x}^{i}$ and $\bar{x}^{j}$
for some ($i,j\in[3],i\neq j$) are nonnegative vectors with at least one
positive entry in each of $\bar{x}^{i}$ and $\bar{x}^{j}$, and, for $k\neq
i,k\neq j$ there holds $\bar{x}^{k}=\textbf{0}$, and ii) 3-coexistence
equilibria, which are equilibria of the form
$(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})$ where, for each $i\in[3]$,
$\bar{x}^{i}$ is a nonnegative vector, that has at least one positive entry.
Indeed, such vectors are in fact strictly positive; see [17, Lemma 6]. That
is, at any endemic equilibrium, there is at least one individual in every
population node who is infected.

Let $J(x^{1},x^{2},x^{3})$ denote the Jacobian matrix of system (4)- (6) for
an arbitrary point in the state space. For future reference, it is easily
checked that $J(x^{1},x^{2},x^{3})$ is as given in (7) below.

(7) $\displaystyle J(x^{1},x^{2},x^{3})=$
$\displaystyle\scriptsize\begin{bmatrix}-D^{1}+(I-X^{1}-X^{2}-X^{3})B^{1}-\operatorname{diag}(B^{1}x^{1})&-\operatorname{diag}(B^{1}x^{1})&-\operatorname{diag}(B^{1}x^{1})\\\
-\operatorname{diag}(B^{2}x^{2})&-D^{2}+(I-X^{1}-X^{2}-X^{3})B^{2}-\operatorname{diag}(B^{2}x^{2})&-\operatorname{diag}(B^{2}x^{2})\\\
-\operatorname{diag}(B^{3}x^{3})&-\operatorname{diag}(B^{3}x^{3})&-D^{3}+(I-X^{1}-X^{2}-X^{3})B^{3}-\operatorname{diag}(B^{3}x^{3})\end{bmatrix}\normalsize.$

###  2.4 Preliminary lemmas

######  Lemma 3.

[27, 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, $\rho(-\Lambda^{-1}N)>1$.

We will also use the following variants of the Perron-Frobenius theorem for
irreducible matrices.

######  Lemma 4.

[31, Chapter 8.3] [53, 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 $\rho(A)<\rho(N)$.

######  Lemma 5.

[53, Lemma 2.3] Suppose that $M$ is an irreducible Metzler matrix. Then
$r=s(M)$ is a simple eigenvalue of $M$, and there exists a corresponding
eigenvector $\zeta\gg\textbf{0}$.

######  Lemma 6 (Restrictions on the trajectories of the tri-virus system).

Consider system (4)-(6) under Assumption 1. Suppose that the initial
conditions satisfy a) $x^{k}(0)\gg\textbf{0}$ for $k\in[3]$, and ii)
$(x^{1}(0),x^{2}(0),x^{3}(0))\in\mathcal{D}$. Further, suppose that matrix
$B^{k}$ for $k\in[3]$ is irreducible. For all finite $t>0$, $\textbf{0}\ll
x^{k}(t)\ll\textbf{1}$ for $k\in[3]$; and
$x^{1}(t)+x^{2}(t)+x^{3}(t)\ll\textbf{1}$.

The proof closely follows that of [62, Lemma 3.2] for the bivirus problem.

Proof: Define $z:=\textbf{1}-x^{1}-x^{2}-x^{3}$, and
$\hat{B}^{i}:=\operatorname{diag}(B^{i}x^{i})$. Hence, (4)-(6) can be
rewritten as:

$\displaystyle\dot{x}^{i}(t)$
$\displaystyle=-D^{i}x^{i}(t)+\hat{B}^{i}z(t),i=1,2,3$ (8)
$\displaystyle\dot{z}(t)$
$\displaystyle=D^{1}x^{1}(t)+D^{2}x^{2}(t)+D^{3}x^{3}(t)-[\hat{B}^{1}+\hat{B}^{2}+\hat{B}^{3}]z(t)$

Suppose that for some $\tau\in\mathbb{R}_{>0}$, and for some $i\in[n]$,
$z_{i}(\tau)=0$, which, since
$z_{i}(\tau)=1-x_{i}^{1}(\tau)-x_{i}^{2}(\tau)-x_{i}^{3}(\tau)$, implies that
either $x_{i}^{1}(\tau)\neq 0$ and/or $x_{i}^{2}(\tau)\neq 0$ and/or
$x_{i}^{3}(\tau)\neq 0$. Therefore, since a) $D^{1},D^{2},\text{and }D^{3}$
are positive diagonal matrices, and b) since, by assumption, $z_{i}(\tau)=0$,
it must be that $\dot{z}_{i}(\tau)>0$. This implies that $z_{i}(t)>0$ for all
$t>\tau$. This argument holds for all $i$, and hence we have that $z_{i}(t)>0$
for all $t$ and $i\in[n]$. As a result, we obtain $z(t)\gg\textbf{0}$ for all
$t>\tau$, which implies that $x^{1}(t)+x^{2}(t)+x^{3}(t)\ll\textbf{1}$. Hence,
we have $x^{k}(t)\ll\textbf{1}$ for $k\in[3]$.

Suppose that for some $t\in\mathbb{R}_{\geq 0}$, $\ell$ (where $\ell<n$)
entries in $x^{1}(t)$ equal zero, and let us label these entries
$i_{1},i_{2},\ldots,i_{\ell}$. Since, by assumption, the matrix $B^{1}$ is
irreducible, it follows from the property of irreducible matrices that there
is at least one entry in the vector $B^{1}x^{1}$ that is nonzero even though
the corresponding entry in the vector $x^{1}$ equals zero [27, Lemma 1]. Let
us assume that this occurs for the $i_{1}^{th}$ entry, i.e., $x^{1}_{i_{1}}=0$
yet $[B^{1}x^{1}]_{i_{1}}>0$. Observe that the evolution of the infection
level for virus 1 in node $i_{1}$ is as follows:

$\displaystyle\dot{x}^{1}_{i_{1}}(t)$
$\displaystyle=[-D^{1}x^{1}(t)]_{i_{1}}+[(I-X^{1}-X^{2}-X^{3})B^{1}x^{1}(t)]_{i_{1}}$
(9) $\displaystyle>0,$

where the inequality in (2.4) follows by noting that $x_{i_{1}}(t)=0$ by
assumption, and $[B^{1}x^{1}]_{i_{1}}>0$ as discussed above. This means that
there must exist some time instant $t^{\prime}$, with $t^{\prime}-t$ not too
large, such that $x^{1}(t^{\prime})$ has fewer than $\ell$ zero entries.
Repeating the argument for all the other zero entries in the vector
$x^{1}(t^{\prime})$ (and this can be done since the choice of node $i_{1}$ was
arbitrary), we have that $x^{1}(t)\gg\textbf{0}$ for $t>0$. Analogously, we
can prove that $x^{2}(t),x^{3}(t)\gg\textbf{0}$ for $t>0$.

Lemma 6 substantially limits the equilibria that can lie on the boundary of
the set $\\{x^{1},x^{2},x^{3}\in\mathbb{R}_{\geq 0}\mid
x^{1}+x^{2}+x^{3}\leq\textbf{1}\\}$.

It turns out that if $x=(x^{1},\dots,x^{3})\in\mathcal{D}$ is an equilibrium
of (4)-(6), then a certain dichotomy arises for $x^{k}$, $k=1,2,3$. The
following lemma formalizes said dichotomy.

######  Lemma 7.

[17, Lemma 6] Consider system (4)-(6) under Assumptions 1 and 2. If
$x=(x^{1},\dots,x^{3})\in\mathcal{D}$ is an equilibrium of (4)-(6), then, for
each $k\in[3]$, either $x^{k}=\textbf{0}$, or $\textbf{0}\ll
x^{k}\ll\textbf{1}$. Moreover, $\textstyle\sum_{k=1}^{3}x^{k}\ll\textbf{1}$.

###  2.5 Problem Statements

The focus of the present paper is to answer the following questions:

  1. i)

Is the tri-virus system monotone?

  2. ii)

Does the tri-virus system generically admit a finite number of equilibria?

  3. iii)

Can we identify a sufficient condition for local exponential convergence to a
boundary equilibrium?

  4. iv)

Can we identify a sufficient condition for the nonexistence of 3-coexistence
equilibria (resp. specific forms of 2-coexistence equilibria)?

  5. v)

Can we identify a sufficient condition for the existence of a 3-coexistence
equilibrium?

  6. vi)

Can we identify a sufficient condition for the existence and local
attractivity of a line of coexistence equilibria?

  7. vii)

Can we identify special case(s) where, irrespective of the non-zero initial
infection levels, the tri-virus dynamics converge to a plane of coexistence
equilibria?

##  3 Monotonicity (or lack thereof) of the tri-virus system

Note that the system defined by (4)-(6) is a (networked) nonlinear system. A
particular class of nonlinear systems is _monotone dynamical systems_ (MDS).
For a detailed overview of MDS, see, for instance, [47]. Before investigating
whether or not the tri-virus system is monotone, we detail the importance of
MDS in the context of the competitive bi-virus SIS networked model.

###  3.1 Monotone dynamical systems and competitive bivirus networked SIS
models

Notice that if $m=2$, then, under Assumption 2, system (3) is monotone; see
[62, Lemma 3.3] (and, for $k=1,2$, assuming $\delta_{i}^{k}=1$ for $i\in[n]$,
also [44, Theorem 18]). That is, setting $m=2$ for system (3), suppose that
$(x_{A}^{1}(0),x_{A}^{2}(0))$ and $(x_{B}^{1}(0),x_{B}^{2}(0))$ are two
initial conditions in $\textrm{int}(D)$ satisfying i)
$x_{A}^{1}(0)>x_{B}^{1}(0)$ and ii) $x_{A}^{2}(0)<x_{B}^{2}(0)$. Since the
bivirus system 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) $x_{A}^{2}(t)\ll x_{B}^{2}(t)$.
Additionally, it is also known that for almost all 444The term “almost all”
has a precise mathematical meaning: for all but a set of parameter values that
has measure zero. This set of exceptional values is defined by an algebraic or
semi-algebraic set. choices of $D^{i}$, $B^{i}$, $i=1,2$, system (3) has a
finite number of equilibria [62, Theorem 3.6]. Therefore, from [46, Theorems
2.5 and 2.6] (or [15, Theorem 9.4]), we know that for almost all initial
conditions in $\mathcal{D}$, system (3) with $m=2$ converges to a stable
equilibrium point, assuming such an equilibrium exists. There are initial
conditions for which convergence to a stable equilibrium does not occur. In
such cases, said initial condition is either a) in the stable manifold of an
unstable equilibrium, or b) in the stable manifold of a nonattractive limit
cycle. The set of initial conditions for which either a) or b) happens has
measure zero.

###  3.2 The tri-virus system is not monotone

It is natural to ask how well the notion of MDS applies to multi-competitive
networked SIS epidemics; this subsection aims to answer this question
conclusively. To this end, we construct a graph associated with the Jacobian
(7) of system (4)-(6), say $\bar{G}$. The construction follows the outline
provided in [48]. More specifically, the graph $\bar{G}$ has $3n$ nodes. The
edges of $\bar{G}$ are based on the entries in the Jacobian matrix
$J(x^{1},x^{2},x^{3})$ in (7). Specifically, if
$[J(x^{1},x^{2},x^{3})]_{ij}<0$ for $i\neq j$, then we draw an edge labelled
with “-” sign; if $[J(x^{1},x^{2},x^{3})]_{ij}>0$ for $i\neq j$, then we draw
an edge labelled with “+” sign. Thus, $\bar{G}$ is a signed graph. Note that
$\bar{G}$ has no self-loops. As an aside, also observe that since
$x^{k}(t)\geq 0$ for $k\in[3]$ and $t\in\mathbb{R}_{+}$, it is immediate that
the sign of the elements in $J(x^{1},x^{2},x^{3})$ do not change with the
argument so that $\bar{G}$ is the same for all points in the interior of
$\mathcal{D}$.

We also need the following concept from graph theory. A signed graph is
considered consistent if every undirected cycle in the graph has a net
positive sign, i.e., it has an even number of “-” signs [48]. We have the
following result.

######  Theorem 1.

System (4)-(6) is not monotone.

###### Proof.

Note that the Jacobian $J(x^{1},x^{2},x^{3})$ is a block matrix, with all
blocks along the off-diagonal being negative diagonal matrices. Pick any node
$i$, where $i\in\\{1,2,\ldots,n\\}$. Observe that, since all blocks along the
off-diagonal of $J(x^{1},x^{2},x^{3})$ are negative diagonal matrices, it is
clear that there exists an edge from node $i$ to node $i+n$, an edge from node
$i+n$ to node $i+2n$, and an edge from node $i+2n$ to node $i$. Furthermore,
each of these edges has a “-” sign. Hence, a loop starting from node $i$,
traversing through nodes $i+n$, $i+2n$, and back to node $i$ is a 3-length
cycle with an odd number of negative signs. Therefore, from [48, page 62], the
signed graph $\bar{G}$ is not consistent. Consequently, from [48, page 63], it
follows that the system (4)-(6) is not monotone.

The conclusion of Theorem 1 offers several important and interesting insights,
including key differences with bi-virus systems (m = 2). The fact that a
bivirus system is monotone coupled with the fact that for almost all choices
of $D^{k}$, $B^{k}$, $k=1,2$, the bivirus system has a finite number of
equilibria allows one to draw general conclusions on the limiting behavior of
bivirus dynamical systems. One can thus focus on the characterization of
equilibria given network parameters $D^{i}$, $B^{i}$, such as the number of
equilibria and their regions of attraction; it is known that bivirus systems
can have multiple stable endemic equilibria [1]. A consequence of Theorem 1 is
that one cannot draw upon the rich literature on monotone dynamical systems
(see[46]) to study the limiting behavior of system (4)-(6). In general, for
non-monotone systems, no dynamical behavior, including chaos, can be
definitively ruled out [48]. Indeed, and in contrast to the bivirus system,
the basic convergence properties of the trivirus system to endemic equilibria
are not well understood.

Another possible consequence of the lack of monotonicity is as follows: It is
known that setting $D^{k}=I$ for $k\in[2]$ has no bearing on either the
location of equilibria of system (3) with $m=2$ nor on their (local) stability
properties [62, Lemma 3.7]. That is, consider two bivirus systems, namely
$\mathcal{S}$ and $\hat{\mathcal{S}}$, where $\mathcal{S}$ is defined by
$(B^{1},D^{1},B^{2},D^{2})$ and $\hat{\mathcal{S}}$ is defined by
$(\hat{B}^{1}=(D^{1})^{-1}B^{1},\hat{D}^{1}=I,\hat{B}^{2}=(D^{2})^{-1}B^{2},\hat{D}^{2}=I)$.
Then, the location of equilibria is the same for both bivirus systems.
Moreover, local stability of an equilibrium in bivirus system $\mathcal{S}$
implies, and is implied by, that in bivirus system $\hat{\mathcal{S}}$. For
system (3) with $m=3$, by extending the arguments from [62, Lemma 3.7], it is
straightforward to show that the _location_ of the equilibria is the same for
an arbitrary system defined by $D^{k},B^{k}$, $k\in[3]$ and the system defined
by $\hat{D}^{k}=I$, and $\hat{B}^{k}=(D^{k})^{-1}B^{k}$. However, since the
tri-virus system is not monotone, the arguments for the stability of
equilibria in the proof of [62, Lemma 3.7] cannot be adapted. Hence, for the
tri-virus case, when the healing rates for all nodes with respect to all
viruses are ‘scaled’ in the manner above to become unity, the preservation of
stability properties remains an open question.

As a matter of independent interest, in the next section, we ask whether for
almost all choices of $D^{k}$, $B^{k}$, $k=1,2,3$, the trivirus system has a
finite number of equilibria.

##  4 Finiteness of equilibria for generic trivirus networks

In this section, we show that for generic tri-virus networks, i.e., for almost
all choices of $D^{k},B^{k},k=1,2,3$, the number of equilibria is finite, and
the associated Jacobian matrices are nonsingular. For bi-virus networks, an
argument for the corresponding result based on algebraic geometry ideas was
provided in [62, Theorem 3.6]. That argument becomes much more intricate for
tri-virus systems. So an alternative proof is provided, based on a topological
tool, termed the Parametric Transversality Theorem, see [25, p.145] and [13,
p.68]. This tool has the advantage that it will apply to variations of the
tri-virus dynamics, where the right side of (1) is replaced by
$-f(x_{i}^{k})+(1-\sum x_{i}^{k})g(x^{1},x^{2},x^{3})$ for general nonlinear
functions $f$ and $g$, such as those arising where feedback control is
present, see, e.g., [63], or more refined models are constructed, see, e.g.,
[60, 7].

######  Proposition 4.

For generic parameters $D^{i}$, $B^{i}$, with $i=1,2,3$, the trivirus system
(4)-(6) has a finite number of equilibria which are all nondegenerate, i.e.
the associated Jacobian matrices are nonsingular.

The proof of this proposition is contained in the appendix, using as it does
very different tools to the rest of the paper. The first component of the
proof establishes the nondegeneracy claim. Further, we prove a slight
modification, viz. that for any fixed set of $B^{k}$, almost all choices of
the $D^{k}$ result in a finite number of equilibria. Proving that for fixed
$D^{k}$, almost all choices of $B^{k}$ result in a finite number of equilibria
is achieved with completely analogous calculations, but these calculations are
not provided. Together, these two observations establish the proposition. Note
that one cannot replace the ‘almost all’ requirement with an ‘all’
requirement. As for the bivirus problem, there can exist special values of the
parameters for which there is a continuum of equilibria; see Sections 8 and 9
for details.

One of the practical ramifications of Proposition 4 is recorded in the
following remark.

######  Remark 1.

The fact that, for almost all choices of infection and recovery rates, the
tri-virus networked SIS model has a finite number of equilibria means that
assuming chaos and limit cycles are known to be not present, then, even in the
absence of interventions for controlling the spread, the range of outcomes
that health administration officials need to prepare for are limited (although
not necessarily few).

##  5 Persistence of one or more viruses

In this section, we first show that for low-dimensional systems existence of
certain kinds of equilibria can be ruled out for almost all choices of system
parameters. Further, we identify a sufficient condition for local exponential
convergence to a boundary equilibrium.

###  5.1 The special case of low dimension systems

When $n=1$ or $n=2$, certain types of equilibria are generically impossible,
as we now show.

######  Proposition 5.

Consider system (4)-(6) and suppose that $n=1$. Suppose that
$(\tilde{x}^{1},\tilde{x}^{2},\tilde{x}^{3})$ is an equilibrium. Provided no
two values of $(D^{k})^{-1}B^{k}$ (with $k=1,2,3$) are identical, at most one
of the $\tilde{x}^{k}$ is nonzero at an equilibrium.

Proof: At any equilibrium, there holds (recall that the $\tilde{x}^{k}$ are
scalar, so that $\tilde{X}^{k}=\tilde{x}^{k})$

$(-D^{k}+(1-\tilde{x}^{1}-\tilde{x}^{2}-\tilde{x}^{3})B^{k})\tilde{x}^{k}=0$

Note that if all $B^{i}$ are nonzero, there cannot hold
$(1-\tilde{x}^{1}-\tilde{x}^{2}-\tilde{x}^{3})=0,$ else a contradiction is
easily obtained.

It is immediate that either

$(1-\tilde{x}^{1}-\tilde{x}^{2}-\tilde{x}^{3})^{-1}=(D^{k})^{-1})B^{k}\quad\mbox{or}\quad\tilde{x}^{k}=0$

The first alternative corresponds to $\tilde{x}^{k}\neq 0$, and if, for
example, $\tilde{x}^{1}\neq 0,\tilde{x}^{2}\neq 0$ then the parameters are
constrained by $(D^{1})^{-1}B^{1}=(D^{2})^{-1}B^{2}$.

For the case $n=2$, we shall appeal to the following lemma.

######  Lemma 8.

Consider three positive matrices $B^{i},i=1,2,3$ of dimension $2\times 2$. For
generic values of the matrix entries, there cannot exist a diagonal matrix,
$\Lambda$ say, such that each of $\Lambda B^{i}$ has a unity eigenvalue.

Proof: Denote the diagonal entries of $\Lambda$ by $\lambda_{1},\lambda_{2}$,
The eigenvalue condition on $\Lambda B^{i}$ is det$(I-\Lambda B^{i})=0$, or

$(\beta^{i}_{11}\beta^{i}_{22}-\beta^{i}_{12}\beta^{i}_{21})\lambda_{1}\lambda_{2}-\beta^{i}_{11}\lambda_{1}-\beta^{i}_{22}\lambda_{2}+1=0$

For fixed $B^{i}$, the set of $\lambda_{i}$ satisfying this equation is
evidently defined by a rectangular hyperbola. It is clear that for generic
$B^{i}$, there cannot be a common point of intersection between three such
hyperbolae. Algebraic geometry gives methods for calculating the actual
semialgebraic set of $B^{i}$ for which a common point of intersection exists,
but this is irrelevant for our purposes. For almost all $B^{i}$, no such
intersection exists.

Now we can state the following.

######  Proposition 6.

Consider system (4)-(6) and suppose that $n=2$. Suppose that
$(\tilde{x}^{1},\tilde{x}^{2},\tilde{x}^{3})$ is an equilibrium. Then for
almost all values of the parameters $D^{k},B^{k}$ ($k=1,2,3$), at least one
$\tilde{x}^{k}$ must be equal to 0.

Proof: The equilibrium equations yield

$(-D^{k}+(I-\tilde{X}^{1}-\tilde{X}^{2}-\tilde{X}^{3})B^{k})\tilde{x}^{k}=0$

Set $\Lambda=I-\tilde{X}^{1}-\tilde{X}^{2}-\tilde{X}^{3}$ and observe that the
equilibrium equations can be rewritten as

$\tilde{x}^{k}=\Lambda[(D^{k})^{-1}B^{k}]\tilde{x}^{k}$

Hence either $\tilde{x}^{k}$ is an eigenvector of $\Lambda[(D^{k})^{-1}B^{k}]$
or it is zero. The preceding lemma applies (with $(D^{k})^{-1}B^{k}$ replacing
$B^{k}$ in the lemma statement) and the conclusion of the proposition is
immediate.

The proposition indicates that for generic $D^{i}$, $B^{i}$, and with $n=2$,
the only equilibria of the trivirus system are the healthy equilibrium, those
defined by the three associated single virus systems, and those defined by the
three associated bivirus systems. Note that for $n=2$, all these equilibria
are analytically computable, [62].

It turns out that we cannot have multiple 3-coexistence equilibria that differ
in the coordinate vector corresponding to only one of the viruses; we
formalize the same in the following lemma.

######  Lemma 9.

Consider system (4)-(6) under Assumptions 1 and 2. If
$(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})$ and
$(\bar{x}^{1},\bar{x}^{2},\hat{x}^{3})$ are endemic equilibria (i.e.,
$\textbf{0}\ll(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})\ll\textbf{1}$ and
$\textbf{0}\ll\hat{x}^{3}\ll\textbf{1}$) of (4)-(6), then
$\bar{x}^{3}=\hat{x}^{3}$.

Proof: Suppose that $(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})$ and
$(\bar{x}^{1},\bar{x}^{2},\hat{x}^{3})$ are endemic equilibria of (4)-(6),
then by writing the equilibrium version of equation (4), we obtain:

(10) 0
$\displaystyle=-D^{1}\bar{x}^{1}+((I-\bar{X}^{1}-\bar{X}^{2}-\bar{X}^{3}))B^{1}\bar{x}^{1}$
(11) 0
$\displaystyle=-D^{1}\bar{x}^{1}+((I-\bar{X}^{1}-\bar{X}^{2}-\hat{X}^{3}))B^{1}\bar{x}^{1}$

From (10) and (11), it is clear that

$\displaystyle\bar{x}^{1}$
$\displaystyle=(D^{1})^{-1}((I-\bar{X}^{1}-\bar{X}^{2}-\bar{X}^{3}))B^{1}\bar{x}^{1}$
$\displaystyle=(D^{1})^{-1}((I-\bar{X}^{1}-\bar{X}^{2}-\hat{X}^{3}))B^{1}\bar{x}^{1},$

which, since i) $\bar{x}^{1}\gg\textbf{0}$ and ii) by Assumption 2 it follows
that for each $i\in[n]$, $\sum_{j=1}^{n}B^{1}_{ij}>0$, which implies that
$\bar{X}^{3}=\hat{X}^{3}$, i.e., $\bar{x}^{3}=\hat{x}^{3}$.

###  5.2 Stability of boundary equilibria

In this subsection, we establish that the local stability (resp. instability)
of the boundary equilibrium corresponding to virus 1 is dependent on whether
(or not) the state matrices, obtained by linearizing the dynamics of viruses
$2$ and $3$ around the boundary equilibrium of virus $1$, are Hurwitz.

######  Theorem 2.

Consider system (4)-(6) under Assumptions 1 and 2. The boundary equilibrium
$(\tilde{x}^{1},\textbf{0},\textbf{0})$ is locally exponentially stable if,
and only if, each of the following conditions is satisfied:

  1. i)

$\rho((I-\tilde{X}^{1})(D^{2})^{-1}B^{2})<1$; and

  2. ii)

$\rho((I-\tilde{X}^{1})(D^{3})^{-1}B^{3})<1$.

If $\rho((I{-}\tilde{X}^{1})(D^{2})^{-1}B^{2})>1$ or if
$\rho((I{-}\tilde{X}^{1})(D^{3})^{{-}1}B^{3})>1$, then
$(\tilde{x}^{1},\textbf{0},\textbf{0})$ is unstable.

The proof follows the strategy outlined for the $m=2$ case in [62, Theorem
3.10].
Proof: Consider the equilibrium point $(\tilde{x}^{1},\textbf{0},\textbf{0})$,
and note that the Jacobian evaluated at this point is as follows:

(12) $\displaystyle J(\tilde{x}^{1},\textbf{0},\textbf{0})=$
$\displaystyle\scriptsize\begin{bmatrix}-D^{1}+(I-\tilde{X}^{1})B^{1}-\hat{B}^{1}&{-}\hat{B}^{1}&{-}\hat{B}^{1}\\\
\textbf{0}&{-}D^{2}{+}(I{-}\tilde{X}^{1})B^{2}&\textbf{0}\\\
\textbf{0}&\textbf{0}&{-}D^{3}{+}(I{-}\tilde{X}^{1})B^{3}\end{bmatrix}\normalsize,$

where $\hat{B}^{i}=\operatorname{diag}(B^{i}\tilde{x}^{i})$, for $i=1,2,3$.
Observe that the matrix $J(\tilde{x}^{1},\textbf{0},\textbf{0})$ is block
upper triangular. Hence, it is Hurwitz if, and only if, the blocks along the
diagonal are Hurwitz. We will now show that this condition is fulfilled as a
consequence of the assumptions of Theorem 2.

Since $(\tilde{x}^{1},\textbf{0},\textbf{0})$ is an equilibrium point of
system (4)-(6), by considering the equilibrium version of equation (4), we
have the following:

(13) $\displaystyle(-D^{1}+(I-\tilde{X}^{1})B^{1})\tilde{x}^{1}=\textbf{0}.$

By Assumption 1, we have that $D^{1}$ is positive diagonal and $B^{1}$ is
nonnegative. Furthermore, by assumption, we know that $B^{1}$ is irreducible.
Moreover, from [17, Lemma 6], it follows that $(I-\tilde{X}^{1})$ is positive
diagonal, implying that $(I-\tilde{X}^{1})B^{1}$ is nonnegative irreducible.
Thus, we can conclude that the matrix $(-D^{1}+(I-\tilde{X}^{1})B^{1})$ is
irreducible Metzler. From [17, Lemma 6] we know that
$\textbf{0}\ll\tilde{x}^{1}$. Hence, by applying [53, Lemma 2.3] to (13), it
must be that $\tilde{x}^{1}$ is, up to scaling, the only eigenvector of
$(-D^{1}+(I-\tilde{X}^{1})B^{1})$ with all entries being strictly positive.
Furthermore, $\tilde{x}^{1}$ is the eigenvector that is associated with, and
only with, $s(-D^{1}+(I-\tilde{X}^{1})B^{1})$. Therefore,
$s(-D^{1}+(I-\tilde{X}^{1})B^{1})=0$.
Define $Q:=D^{1}-(I-\tilde{X}^{1})B^{1}$, and note that $Q$ is an M-matrix.
Since $s(-Q)=0$ and $B^{1}$ is irreducible, $Q$ is a singular irreducible
M-matrix. Observe that $\hat{B}^{1}$ is a nonnegative matrix and because
$B^{1}$ is irreducible and $\tilde{x}^{1}\gg\textbf{0}$, it must be that at
least one element in $\hat{B}^{1}$ is strictly positive. Therefore, from [40,
Lemma 4.22], it follows that $Q+\hat{B}^{1}$ is an irreducible non-singular
M-matrix, which from [40, Section 4.3, page 167] implies that $-Q-\hat{B}^{1}$
is Hurwitz. Therefore, we have that
$s(-D^{1}+(I-\tilde{X}^{1})B^{1}-\hat{B}^{1})<0$.

By assumption, $\rho((I-\tilde{X}^{1})(D^{2})^{-1}B^{2})<1$ and
$\rho((I-\tilde{X}^{1})(D^{3})^{-1}B^{3})<1$. Therefore, by noting that
$D^{2}$ (resp. $D^{3}$) are positive diagonal matrices and $B^{2}$ (resp.
$B^{3}$) are nonnegative irreducible matrices, from [27, Proposition 1] it
follows that $s(-D^{2}+(I-\tilde{X}^{1})B^{2})<0$ (resp.
$s(-D^{3}+(I-\tilde{X}^{1})B^{3})<0$). Since each diagonal block of
$J(\tilde{x}^{1},\textbf{0},\textbf{0})$ is Hurwitz, it is clear that
$J(\tilde{x}^{1},\textbf{0},\textbf{0})$ is Hurwitz. Local exponential
stability of $(\tilde{x}^{1},\textbf{0},\textbf{0})$, then, follows from [21,
Theorem 4.15 and Corollary 4.3].

The proof of necessity follows by first noting that if either condition in
statement i) or that in statement ii) is violated, then, since at least one of
the blocks along the diagonal of $J(\tilde{x}^{1},\textbf{0},\textbf{0})$ is
not Hurwitz, the matrix $J(\tilde{x}^{1},\textbf{0},\textbf{0})$ is not
Hurwitz. Then, by invoking the necessary part of [21, Theorem 4.15 and
Corollary 4.3], the result follows.

The claim for instability can be proved by noting that if either of the
eigenvalue conditions is violated, then, since
$J(\tilde{x}^{1},\textbf{0},\textbf{0})$ is block diagonal, the matrix
$J(\tilde{x}^{1},\textbf{0},\textbf{0})$ is not Hurwitz. The result follows
from [21, Theorem 4.7, item ii)].

Analogous results for the boundary equilibria
$(\textbf{0},\tilde{x}^{2},\textbf{0})$ and
$(\textbf{0},\textbf{0},\tilde{x}^{3})$ can be similarly obtained.

##  6 Nonexistence of various coexistence equilibria

In this section, we establish conditions on the system parameters that
guarantee the nonexistence of different kinds of coexistence equilibria.
Specifically, we identify a condition for the nonexistence of a 3-coexistence
(resp. 2-coexistence) equilibria. We draw on the stability properties
established in the previous section.

######  Theorem 3 (Nonexistence of 3-coexistence equilibria).

Consider system (4)-(6) under Assumption 1. Suppose that matrix $B^{k}$ for
$k\in[3]$ is irreducible. Suppose that $\rho((D^{k})^{-1}B^{k})>1$ for
$k\in[3]$. Let $\tilde{x}^{1}$, $\tilde{x}^{2}$, and $\tilde{x}^{3}$ denote
the single-virus endemic equilibrium corresponding to virus 1, 2, and 3,
respectively. If $(D^{3})^{-1}B^{3}>(D^{2})^{-1}B^{2}>(D^{1})^{-1}B^{1}$, then

  1. i)

the equilibrium point $(\textbf{0},\textbf{0},\textbf{0})$ is unstable;

  2. ii)

the equilibrium point $(\tilde{x}^{1},\textbf{0},\textbf{0})$ is unstable;

  3. iii)

the equilibrium point $(\textbf{0},\tilde{x}^{2},\textbf{0})$ is unstable;

  4. iv)

the equilibrium point $(\textbf{0},\textbf{0},\tilde{x}^{3})$ is locally
exponentially stable;

  5. v)

there does not exist a 3-coexistence equilibrium.

Proof: The proof strategy is quite similar to those in [17, Theorem 6] and
[62, Corollary 3.10].

The proof of statements i)-iii) are quite similar to that of statements i) and
ii) in Theorem 4. Proof of statement i): Consider the equilibrium point
$(\textbf{0},\textbf{0},\textbf{0})$, and observe that the Jacobian computed
at this point is as follows:

(14) $\displaystyle J(\textbf{0},\textbf{0},\textbf{0})=$
$\displaystyle\footnotesize\begin{bmatrix}-D^{1}+B^{1}&\textbf{0}&\textbf{0}\\\
\textbf{0}&-D^{2}+B^{2}&\textbf{0}\\\
\textbf{0}&\textbf{0}&-D^{3}+B^{3}\end{bmatrix}\normalsize,$

Since by assumption, $B^{k}$ is non-negative irreducible and
$\rho((D^{k})^{-1}B^{k})>1$ for $k\in[3]$, it follows from Lemma 3 that
$s(-D^{k}+B^{k})>0$ for $k\in[3]$, which further implies that
$s(J(\textbf{0},\textbf{0},\textbf{0}))>0$. Instability of the equilibrium
point $(\textbf{0},\textbf{0},\textbf{0})$, then, follows from [21, Theorem
4.7, item ii)].
Proof of statement ii): Consider the equilibrium point
$(\tilde{x}^{1},\textbf{0},\textbf{0})$, and observe that the Jacobian
computed at this point is as follows:

(15) $\displaystyle J(\tilde{x}^{1},\textbf{0},\textbf{0})=$
$\displaystyle\scriptsize\begin{bmatrix}-D^{1}+(I-\tilde{X}^{1})B^{1}-\hat{B}^{1}&-\hat{B}^{1}&-\hat{B}^{1}\\\
\textbf{0}&-D^{2}+(I-\tilde{X}^{1})B^{2}&\textbf{0}\\\
\textbf{0}&\textbf{0}&-D^{3}+(I-\tilde{X}^{1})B^{3}\end{bmatrix}\normalsize,$

Note that since $\tilde{x}^{1}$ is the single-virus endemic equilibrium for
virus 1, by the definition of an equilibrium point, we have the following:

$[-D^{1}+(I-\tilde{X}^{1})B^{1}]\tilde{x}^{1}=\textbf{0}.$

Observe that $-D^{1}+(I-\tilde{X}^{1})B^{1}$ is an irreducible Metzler matrix,
and, since $\tilde{x}^{1}\gg\textbf{0}$, from Lemma 5 we have that
$s(-D^{1}+(I-\tilde{X}^{1})B^{1})=0$. Consequently, from Lemma 3,
$\rho((I-\tilde{X}^{1})(D^{1})^{-1}B^{1})=1$. By assumption
$(D^{2})^{-1}B^{2}>(D^{1})^{-1}B^{1}$, which implies
$(I-\tilde{X}^{1})(D^{1})^{-1}B^{1}<(I-\tilde{X}^{1})(D^{2})^{-1}B^{2}$.
Therefore, since the matrices $(I-\tilde{X}^{1})(D^{1})^{-1}B^{1}$ and
$(I-\tilde{X}^{1})(D^{2})^{-1}B^{2}$ are nonnegative, from Lemma 4 item iv),
we have that $\rho((I-\tilde{X}^{1})(D^{2})^{-1}B^{2})>1$. Consequently, from
Lemma 3, it must be that $s(-D^{2}+(I-\tilde{X}^{1})B^{2})>0$, which implies
that $s(J(\tilde{x}^{1},\textbf{0},\textbf{0}))>0$, and hence the equilibrium
point $(\tilde{x}^{1},\textbf{0},\textbf{0})$ is unstable.
The proof of statement iii) is analogous to that of statement ii), and is,
therefore, omitted.

Proof of statement iv): Consider the equilibrium point
$(\textbf{0},\textbf{0},\tilde{x}^{3})$, and observe that the Jacobian is as
follows:

(16) $\displaystyle J(\textbf{0},\textbf{0},\tilde{x}^{3})=$
$\displaystyle\scriptsize\begin{bmatrix}-D^{1}+(I-\tilde{X}^{3})B^{1}&\textbf{0}&\textbf{0}\\\
\textbf{0}&-D^{2}+(I-\tilde{X}^{3})B^{2}&\textbf{0}\\\
-\hat{B}^{3}&-\hat{B}^{3}&-D^{3}+(I-\tilde{X}^{3})B^{3}-\hat{B}^{3}\end{bmatrix}\normalsize,$

(Recall that $\hat{B}^{3}$ is shorthand for
$\operatorname{diag}(B^{3}\tilde{x}^{3}))$. By following similar arguments as
in the proof of Theorem 2 (for establishing that the matrix
$-D^{1}+(I-\tilde{X}^{1})B^{1}-\hat{B}^{1}$ is Hurwitz), we can show that the
matrix $-D^{3}+(I-\tilde{X}^{3})B^{3}-\hat{B}^{3}$ is Hurwitz, i.e.,
$s(-D^{3}+(I-\tilde{X}^{3})B^{3}-\hat{B}^{3})<0$. Since $\tilde{x}^{3}$ is the
single-virus endemic equilibrium corresponding to virus 3, we have that

$\rho((I-\tilde{X}^{3})(D^{3})^{-1}B^{3})=1.$

Since, by assumption, $(D^{3})^{-1}B^{3}>(D^{2})^{-1}B^{2}>(D^{1})^{-1}B^{1}$,
it is clear that $(D^{3})^{-1}B^{3}>(D^{2})^{-1}B^{2}$ and
$(D^{3})^{-1}B^{3}>(D^{1})^{-1}B^{1}$. Hence, it follows that a)
$(I-\tilde{X}^{3})(D^{3})^{-1}B^{3}>(I-\tilde{X}^{3})(D^{2})^{-1}B^{2}$, and
b) $(I-\tilde{X}^{3})(D^{3})^{-1}B^{3}>(I-\tilde{X}^{3})(D^{1})^{-1}B^{1}$.
Therefore, from Lemma 4 item iv), it must be that a)
$\rho((I-\tilde{X}^{3})(D^{2})^{-1}B^{2})<1$, and b)
$\rho((I-\tilde{X}^{3})(D^{1})^{-1}B^{1})<1$, which from Lemma 3 further
implies that a) $s(-D^{2}+(I-\tilde{X}^{3})B^{2})<0$, and b)
$s(-D^{3}+(I-\tilde{X}^{3})B^{1})<0$. As a consequence,
$s(J(\textbf{0},\textbf{0},\tilde{x}^{3}))<0$, which, from [21, Theorem 4.7],
leads us to conclude that the equilibrium point
$(\textbf{0},\textbf{0},\tilde{x}^{3})$ is locally exponentially stable.
Proof of statement v): We now show that under the conditions of the Theorem, a
3-coexistence equilibrium (i.e., of the form
$\bar{x}=(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})$ with all $\bar{x}^{i}>0$ )
cannot exist. Suppose that, by way of contradiction, there exists a
3-coexistence equilibrium point
$\bar{x}=(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})$. Therefore, from Lemma 9 it
must be that
$\textbf{0}\ll(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})\ll\textbf{1}$, and
$\bar{x}^{1}+\bar{x}^{2}+\bar{x}^{3}\ll\textbf{1}$. Furthermore, by the
definition of an equilibrium point, we also have the following:

(17) 0
$\displaystyle=(-D^{1}+(I-\bar{X}^{1}-\bar{X}^{2}-\bar{X}^{3})B^{1})\bar{x}^{1},$
(18) 0
$\displaystyle=(-D^{2}+(I-\bar{X}^{1}-\bar{X}^{2}-\bar{X}^{3})B^{2})\bar{x}^{2},$
(19) 0
$\displaystyle=(-D^{3}+(I-\bar{X}^{1}-\bar{X}^{2}-\bar{X}^{3})B^{3})\bar{x}^{3}.$

Since $\tilde{x}^{k}$ is the single-virus endemic equilibrium for virus $k$,
$k\in[3]$, we also have the following:

(20) 0
$\displaystyle=\Big{(}-D^{1}+\big{(}I-\tilde{X}^{1}\big{)}B^{1}\Big{)}\tilde{x}^{1}(t),$
(21) 0
$\displaystyle=\Big{(}-D^{2}+\big{(}I-\tilde{X}^{2}\big{)}B^{2}\Big{)}\tilde{x}^{2}(t)$
(22) 0
$\displaystyle=\Big{(}-D^{3}+\big{(}I-\tilde{X}^{3}\big{)}B^{3}\Big{)}\tilde{x}^{3}(t)$

Define
$\kappa:=\max_{i\in[n]}[\bar{x}^{1}+\bar{x}^{2}+\bar{x}^{3}]_{i}/\tilde{x}^{3}_{i}$,
which implies that
$\bar{x}^{1}+\bar{x}^{2}+\bar{x}^{3}\leq\kappa\tilde{x}^{3}$. Let $j$ be the
maximizing value of $i$ in the definition of $\kappa$. Assume by way of
contradiction that $\kappa\geq 1$. Consider

$\displaystyle(\bar{x}^{1}_{j}+\bar{x}^{2}_{j}+\bar{x}^{3}_{j})$
$\displaystyle=(1-\bar{x}^{1}_{j}-\bar{x}^{2}_{j}-\bar{x}^{3}_{j})((D^{1})^{-1}B^{1}\bar{x}^{1}+(D^{2})^{-1}B^{2}\bar{x}^{2}+(D^{3})^{-1}B^{3}\bar{x}^{3})_{j}$
(23)
$\displaystyle<(1-\bar{x}^{1}_{j}-\bar{x}^{2}_{j}-\bar{x}^{3}_{j})((D^{3})^{-1}B^{3}\bar{x}^{1}+(D^{3})^{-1}B^{3}\bar{x}^{2}+(D^{3})^{-1}B^{3}\bar{x}^{3})_{j}$
$\displaystyle=(1-\bar{x}^{1}_{j}-\bar{x}^{2}_{j}-\bar{x}^{3}_{j})((D^{3})^{-1}B^{3}(\bar{x}^{1}+\bar{x}^{2}+\bar{x}^{3}))_{j}$
(24)
$\displaystyle\leq(1-\bar{x}^{1}_{j}-\bar{x}^{2}_{j}-\bar{x}^{3}_{j})((D^{3})^{-1}B^{3}\kappa\tilde{x}^{3})_{j}$
(25)
$\displaystyle=\kappa(1-\bar{x}^{1}_{j}-\bar{x}^{2}_{j}-\bar{x}^{3}_{j})(1-\tilde{x}^{3}_{j})^{-1}\tilde{x}^{3}_{j}$
(26)
$\displaystyle=(1-\bar{x}^{1}_{j}-\bar{x}^{2}_{j}-\bar{x}^{3}_{j})(1-\tilde{x}^{3}_{j})^{-1}(\bar{x}^{1}_{j}+\bar{x}^{2}_{j}+\bar{x}^{3}_{j})$
(27) $\displaystyle<(\bar{x}^{1}_{j}+\bar{x}^{2}_{j}+\bar{x}^{3}_{j}),$

where (23) follows from the assumption that
$(D^{3})^{-1}B^{3}>(D^{2})^{-1}B^{2}$ and
$(D^{3})^{-1}B^{3}>(D^{1})^{-1}B^{1}$ leads to
$(D^{3})^{-1}B^{3}y>(D^{2})^{-1}B^{2}y$ and
$(D^{3})^{-1}B^{3}y>(D^{1})^{-1}B^{1}y$ for any $y\gg\textbf{0}$. The
inequality (24) is due to
$(\bar{x}^{1}+\bar{x}^{2}+\bar{x}^{3})\leq\kappa\tilde{x}^{3}$; (25) is
immediate from (22); and (26) stems from the fact that
$\bar{x}^{1}_{j}+\bar{x}^{2}_{j}+\bar{x}^{3}_{j}=\kappa\tilde{x}^{3}_{j}$. The
inequality (27) follows by noting that since by assumption $\kappa\geq 1$, it
must be that
$(1-\tilde{x}^{3}_{j})>(1-\bar{x}^{1}_{j}-\bar{x}^{2}_{j}-\bar{x}^{3}_{j})$.
Observe that (27) is a contradiction, thus implying that $\kappa<1$. Thus,
$\bar{x}^{1}+\bar{x}^{2}+\bar{x}^{3}\ll\tilde{x}^{3}$.

Note that since $\tilde{x}^{3}\gg\textbf{0}$, and $\bar{x}^{3}\gg\textbf{0}$,
it follows from (19) and (22) that
$\rho((I-\bar{X}^{1}-\bar{X}^{2}-\bar{X}^{3})(D^{3})^{-1}B^{3})=1$, and
$\rho(\big{(}I-\tilde{X}^{3}\big{)}(D^{3})^{-1}B^{3})=1$. Since
$\bar{x}^{1}+\bar{x}^{2}+\bar{x}^{3}\ll\tilde{x}^{3}$, it must be that
$(I-\bar{X}^{1}-\bar{X}^{2}-\bar{X}^{3})>(I-\tilde{X}^{3})$, which from Lemma
4 item iv) implies that
$\rho((I-\bar{X}^{1}-\bar{X}^{2}-\bar{X}^{3})(D^{3})^{-1}B^{3})>\rho((I-\tilde{X}^{3})(D^{3})^{-1}B^{3})$,
which is a contradiction of the assumption that there exists a 3-coexistence
equilibrium. Consequently, there does not exist a 3-coexistence equilibrium.

Theorem 3 identifies a sufficient condition for the non-existence of a
3-coexistence equilibrium. For the bivirus system, a condition based on the
row sum of matrices $B^{k}$, for $k\in[2]$, guarantees the non-existence of
any 2-coexistence equilibria; see [62, Corollary 3.10, item ii)]. It is
unknown if an analogous condition (but for the nonexistence of any
3-coexistence equilibria) can be obtained for the tri-virus system; a rigorous
investigation is left for future work. It turns out that a condition less
restrictive than that in Theorem 3 precludes the possibility of the existence
of certain 2-coexistence equilibria, as formalized in the following theorem.

######  Theorem 4.

Consider system (4)-(6) under Assumptions 1 and 2. Suppose that
$\rho((D^{k})^{-1}B^{k})>1$ for $k\in[3]$.

  1. i)

The equilibrium point $(\textbf{0},\textbf{0},\textbf{0})$ is unstable.

  2. ii)

If $(D^{2})^{-1}B^{2}>(D^{1})^{-1}B^{1}$ and
$(D^{3})^{-1}B^{3}>(D^{1})^{-1}B^{1}$, then

    1. a)

the equilibrium point $(\tilde{x}^{1},\textbf{0},\textbf{0})$ is unstable; and

    2. b)

there does not exist a 2-coexistence equilibrium of the form a)
$(\bar{x}^{1},\bar{x}^{2},\textbf{0})$, or b)
$(\bar{x}^{1},\textbf{0},\bar{x}^{3})$, where
$\textbf{0}\ll\bar{x}^{1}\ll\textbf{1}$,
$\textbf{0}\ll\bar{x}^{2}\ll\textbf{1}$, and
$\textbf{0}\ll\bar{x}^{3}\ll\textbf{1}$.

Proof: The proof of statement i) (resp. statement ii a)) is the same as that
of statement i) (resp. statement ii)) in Theorem 3.

Proof of statement ii) b): We now show that, under the conditions of the
Theorem, a 2-coexistence equilibrium of the form
$(\bar{x}^{1},\bar{x}^{2},\textbf{0})$, where
$\textbf{0}\ll\bar{x}^{1}\ll\textbf{1}$ and
$\textbf{0}\ll\bar{x}^{2}\ll\textbf{1}$, cannot exist. Suppose that, by way of
contradiction, there exists a 2-coexistence equilibrium of the form
$(\bar{x}^{1},\bar{x}^{2},\textbf{0})$ with
$\textbf{0}\ll\bar{x}^{1}\ll\textbf{1}$ and
$\textbf{0}\ll\bar{x}^{2}\ll\textbf{1}$. By taking recourse to the equilibrium
version of equations (4)-(6), we have the following:

(28) 0
$\displaystyle=\Big{(}\big{(}I-(\bar{X}^{1}+\bar{X}^{2})\big{)}B^{1}-D^{1}\Big{)}\bar{x}^{1},$
(29) 0
$\displaystyle=\Big{(}\big{(}I-(\bar{X}^{1}+\bar{X}^{2})\big{)}B^{2}-D^{2}\Big{)}\bar{x}^{2},$

By a suitable rearrangement of terms in (28) and (29), and by noting that the
matrices $I-(\bar{X}^{1}+\bar{X}^{2})\big{)}B^{1}$ and
$I-(\bar{X}^{1}+\bar{X}^{2})\big{)}B^{2}$ are irreducible, from Lemma 4 item
i), we have the following:

(30) $\displaystyle\rho(I-(\bar{X}^{1}+\bar{X}^{2})\big{)}(D^{1})^{-1}B^{1})$
$\displaystyle=1$ (31)
$\displaystyle\rho(I-(\bar{X}^{1}+\bar{X}^{2})\big{)}(D^{2})^{-1}B^{2})$
$\displaystyle=1$

Since by assumption $(D^{2})^{-1}B^{2}>(D^{1})^{-1}B^{1}$, it must be that
$(I-(\bar{X}^{1}+\bar{X}^{2})\big{)}(D^{2})^{-1}B^{2}>(I-(\bar{X}^{1}+\bar{X}^{2})\big{)}(D^{1})^{-1}B^{1}$,
which from Lemma 4 item iv) further implies that
$\rho((I-(\bar{X}^{1}+\bar{X}^{2})\big{)}(D^{2})^{-1}B^{2})>\rho((I-(\bar{X}^{1}+\bar{X}^{2})\big{)}(D^{1})^{-1}B^{1})$,
which contradicts the conclusions from (30) and (31). Thus, there cannot exist
a 2-coexistence equilibrium of the form $(\bar{x}^{1},\bar{x}^{2},\textbf{0})$
with $\textbf{0}\ll\bar{x}^{1}\ll\textbf{1}$ and
$\textbf{0}\ll\bar{x}^{2}\ll\textbf{1}$. The nonexistence of a 2-coexistence
equilibrium of the form $(\bar{x}^{1},\textbf{0},\bar{x}^{3})$ with
$\textbf{0}\ll\bar{x}^{1}\ll\textbf{1}$ and
$\textbf{0}\ll\bar{x}^{3}\ll\textbf{1}$ can be shown analogously, by
leveraging the assumption that $(D^{3})^{-1}B^{3}>(D^{1})^{-1}B^{1}$.

Note that Theorem 4 conclusively rules out the existence of a 2-coexistence
equilibrium of the form $(\bar{x}^{1},\bar{x}^{2},\textbf{0})$, and of the
form $(\bar{x}^{1},\textbf{0},\bar{x}^{3})$. It, however, does not preclude
the possibility of the existence of 2-coexistence equilibria of the form
$(\textbf{0},\bar{x}^{2},\bar{x}^{3})$. Therefore, Theorem 4 does not set out
a condition for the nonexistence of any 2-coexistence equilibrium in a tri-
virus system.

Observe that the condition in Theorem 3 implies (but is not implied by) the
condition in Theorem 4, and, consequently, Theorem 3 also rules out the
existence of a 2-coexistence equilibrium of the form
$(\bar{x}^{1},\bar{x}^{2},\textbf{0})$, and of the form
$(\bar{x}^{1},\textbf{0},\bar{x}^{3})$.

##  7 Existence of coexistence equilibria

Section 6 focuses on the nonexistence of certain equilibria. In this section,
we provide conditions for the existence of a 2-coexistence equilibrium and
identify circumstances under which the existence of one or more 3-coexistence
equilibria can be inferred. Define, for each $i\in[3]$,
$\bar{X}^{i}:=\operatorname{diag}{(\bar{x}^{i})}$, where $\bar{x}^{i}$ is as
defined in Section 2. The following proposition guarantees the existence of a
2-coexistence equilibrium.

######  Proposition 7.

Consider system (4)-(6) under Assumptions 1 and 2. Suppose that
$\rho((D^{k})^{-1}B^{k})>1$ for $k\in[3]$. There exists at least one
2-coexistence equilibrium, i.e., $(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})$ where
$\textbf{0}\ll\bar{x}^{i},\bar{x}^{j}\ll\textbf{1}$ for some $i,j\in[3],i\neq
j$, and, for $\ell\neq i,\ell\neq j$, $\bar{x}^{\ell}=\textbf{0}$.

Proof: By assumption, we have that $\rho((D^{k})^{-1}B^{k})>1$ for $k\in[3]$.
Therefore, from Proposition 3 it follows that there exist single-virus endemic
equilibria (also referred to as boundary equilibria) corresponding to virus 1,
2, and 3, namely $\tilde{x}^{1}$, $\tilde{x}^{2}$, and $\tilde{x}^{3}$,
respectively. Observe that each of these equilibria could be either stable or
unstable, which implies that there must be either i) a pair which are both
stable or ii) a pair that are both unstable. We consider the two cases
separately.
Case i): Assume, without loss of generality, that the two stable boundary
equilibria are $(\tilde{x}^{1},{\bf{0}},{\bf{0}})$ and
$({\bf{0}},\tilde{x}^{2},{\bf{0}})$. Observe that for the bivirus system,
defined by neglecting $x^{3}$, the boundary equilibria are
$(\tilde{x}^{1},{\bf{0}})$ and $({\bf{0}},\tilde{x}^{2})$, the existence of
which, to reiterate, is guaranteed from the assumption that $\rho(B^{k})>1$
for $k=1,2$. The bivirus system is known to be monotone [62, Lemma 3.3].
Therefore, from [46, Theorem 2.8] and [46, Proposition 2.9], it follows that
there is necessarily an unstable equilibrium $(\bar{x}^{1},\bar{x}^{2})$,
where $\textbf{0}\ll(\bar{x}^{1},\bar{x}^{2})\ll\textbf{1}$. Corresponding to
this equilibrium of the bivirus system is the equilibrium
$(\bar{x}^{1},\bar{x}^{2},{\bf{0}})$ of the trivirus system.
Case ii): Assume, without loss of generality, that the two unstable boundary
equilibria are $(\tilde{x}^{1},{\bf{0}},{\bf{0}})$ and
$({\bf{0}},\tilde{x}^{2},{\bf{0}})$. Since the bivirus system (defined by
neglecting $x^{3}$) is monotone. Since for monotone systems between any two
unstable boundary equilibria, there must exist a stable equilibrium [46],
there must be an equilibrium point $(\bar{x}^{1},\bar{x}^{2})$. Moreover,
since from [62, Lemma 3.1], it is known that, for the bivirus system, besides
the DFE, only the equilibrium points $(\tilde{x}^{1},{\bf{0}})$ and
$({\bf{0}},\tilde{x}^{2})$ can be on the boundary of the surface defined by
the set $\mathcal{D}^{k}$, $k=1,2$; it follows that the equilibrium point
$(\bar{x}^{1},\bar{x}^{2})$ must lie in the interior of $\mathcal{D}^{k}$,
$k=1,2$, which further implies that
$\textbf{0}\ll(\bar{x}^{1},\bar{x}^{2})\ll\textbf{1}$, with
$(\bar{x}^{1},\bar{x}^{2},{\bf{0}})$ again an equilibrium of the trivirus
system.

Hence, it follows that in either case, there exists a 2-coexistence
equilibrium for system (4)-(6).

We now turn our attention to identifying a sufficient condition for the
existence of a 3-coexistence equilibria. To this end, we will borrow some
results on systems invariant in $\mathbb{R}^{n}_{+}$ with ultimately bounded
trajectories, see [16]; a tri-virus system indeed meets these requirements.

Specifically, and following the terminology of [16], we classify equilibria as
saturated or unsaturated and derive a counting lower bound on the possible
equilibria of tri-virus systems. Observe that for a boundary equilibrium or a
2-coexistence equilibrium, the associated Jacobian is block-triangular; see
(7). We say that an equilibrium is saturated if the diagonal block
corresponding to the zero entries of the equilibrium is Hurwitz and
unsaturated otherwise [16]. A boundary equilibrium of (4)-(6) 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 (12). However, a 2-coexistence equilibrium
might be unstable but still saturated because we only consider one diagonal
block of the Jacobian to determine if it is saturated, while stability is
determined by the remainder of the Jacobian truncated to remove this diagonal
block; for a 2-coexistence equilibrium, this may or may not be Hurwitz. In
contrast, a 3-coexistence equilibrium, with no diagonal block in the Jacobian
matrix corresponding to zero entries of the equilibrium, is always saturated.

The index of an equilibrium, as per the convention in [16], is defined as the
sign, i.e., $\pm 1$, of the negative of the determinant of the Jacobian at the
point. (Note that all such Jacobians are nonsingular for generic parameter
values, as established in Proposition 4). We will require the following
results from [16]: (a) if a solution beginning in the interior converges to a
limit as $t\to\infty$, that limit must be a saturated fixed point, and (b) the
sum of the indices over saturated equilibria is necessarily $+1$. We will now
show that there exists at least one saturated equilibrium, and if the only
such saturated equilibrium is a boundary equilibrium, it must be stable.

######  Theorem 5.

Consider system (4)-(6) under Assumption 1, and assume, in the light of
genericity of the parameters, that all equilibria are nondegenerate. Suppose
that, for each $k\in[3]$, $\rho((D^{k})^{-1}B^{k})>1$. There exists at least
one of the following:

  1. i)

a stable boundary equilibrium;

  2. ii)

a saturated 2-coexistence equilibrium; or

  3. iii)

a 3-coexistence equilibrium.

Proof: We first show that the assumptions of the Theorem guarantee the
existence of boundary equilibria and at least one 2-coexistence equilibrium.
By assumption, we have that $\rho((D^{k})^{-1}B^{k})>1$ for $k\in[3]$.
Therefore, from Proposition 3 it follows that there exist single-virus endemic
equilibria (also referred to as boundary equilibria) corresponding to virus 1,
2, and 3, namely $\tilde{x}^{1}$, $\tilde{x}^{2}$, and $\tilde{x}^{3}$,
respectively. Moreover, from Proposition 7, the existence of the said boundary
equilibria guarantees the existence of at least one 2-coexistence equilibrium.
Observe that Lemma 1 guarantees that, for each $k\in[3]$,
$x^{k}(0)>\textbf{0}$ implies that $x^{k}(t)>\textbf{0}$ for all
$t\in\mathbb{R}_{\geq 0}$, and that the set $\mathcal{D}$ (which is compact)
is forward invariant. Therefore, from [16, Theorem 2], it follows that system
(4)-(6) 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,
due to Lemma 7, of the form $(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})$, where
$\textbf{0}\ll(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})\ll\textbf{1}$, thus
implying that $(\bar{x}^{1},\bar{x}^{2},\bar{x}^{3})$ is a 3-coexistence
equilibrium.
Case 2: Suppose there are no fixed points in the interior of $\mathcal{D}$.
This implies that a saturated fixed point must be on the boundary of
$\mathcal{D}$ [16]. Therefore, at least one of the following should be true:
a) at least one of the single-virus boundary equilibrium is saturated; or b)
at least one of the 2-coexistence equilibria is saturated. The following is an
immediate consequence of Theorem 5, and [16, Theorem 1].

######  Corollary 1.

Consider system (4)-(6) under Assumption 1 and with generic parameter values
ensuring that all equilibria are isolated. Suppose that, for each $k\in[3]$,
$\rho((D^{k})^{-1}B^{k})>1$. Let $(\tilde{x}^{1},\textbf{0},\textbf{0})$,
$(\textbf{0},\tilde{x}^{2},\textbf{0})$ and
$(\textbf{0},\textbf{0},\tilde{x}^{3})$ denote the boundary equilibria
corresponding to viruses 1,2 and 3, respectively. Suppose further for such
equilibria that, for each pair $(i,j)$ such that $i,j\in[3]$ and $i\neq j$,
$\rho((I{-}\tilde{X}^{i})(D^{j})^{-1}B^{j})>1$. Further, let
$(\bar{x}^{i},\bar{x}^{j},\bar{x}^{k})$ be a 2-coexistence equilibrium, i.e.,
such that for some $i,j\in[3]$, $\bar{x}^{i},\bar{x}^{j}>0$, and for some
$k\in[3]$, $\bar{x}^{k}=\textbf{0}$. Suppose further that, for every such
triplet $(i,j,k)$, $s(-D^{k}+(I-\bar{X}^{i}-\bar{X}^{j})B^{k})\geq 0$. Then
there exists at least one 3-coexistence equilibrium. In fact, there must be an
odd number of 3-coexistence equilibrium.

Proof: By assumption, we have that $\rho((D^{k})^{-1}B^{k})>1$ for $k\in[3]$.
Therefore, from Proposition 3 it follows that there exist boundary equilibria
corresponding to virus 1, 2, and 3, namely $\tilde{x}^{1}$, $\tilde{x}^{2}$,
and $\tilde{x}^{3}$, respectively. By assumption, for each pair $(i,j)$ such
that $i,j\in[3]$ and $i\neq j$,
$\rho((I{-}\tilde{X}^{i})(D^{j})^{-1}B^{j})>1$. Therefore, from Theorem 2,
each of the boundary equilibrium points,
$(\tilde{x}^{1},\textbf{0},\textbf{0})$,
$(\textbf{0},\tilde{x}^{2},\textbf{0})$ and
$(\textbf{0},\textbf{0},\tilde{x}^{3})$, is unstable, and, thus, unsaturated.
Given the existence of the boundary equilibria, from Proposition 7, the
existence of at least one equilibrium (i.e., 2-coexistence equilibrium),
$(\bar{x}^{i},\bar{x}^{j},\bar{x}^{k})$ such that for some $i,j\in[3]$,
$\bar{x}^{i},\bar{x}^{j}>0$, and for some $k\in[3]$, $\bar{x}^{k}=\textbf{0}$,
is guaranteed. By assumption, for every such triplet $(i,j,k)$,
$s(-D^{k}+(I-\bar{X}^{i}-\bar{X}^{j})B^{k})\geq 0$, which implies that no
2-coexistence equilibrium is saturated. Therefore, from Theorem 5, it must be
that there exists an equilibrium point $(\bar{x}^{i},\bar{x}^{j},\bar{x}^{k})$
such that $\bar{x}^{i},\bar{x}^{j},\bar{x}^{k}>\textbf{0}$, i.e., a
3-coexistence equilibrium.

##  8 Existence and attractivity of a line of coexistence equilibria for
nongeneric tri-virus networks

Sections 5-7 pertain to the existence or nonexistence of various endemic
equilibria. Note that in those sections, the discussion centers around the
(non)existence (and, in Section 5 also stability (or lack thereof)) of _an_
equilibrium point. In this section, however, we will identify a scenario that
guarantees the existence and local exponential attractivity of a line of
coexistence equilibria. As for the bivirus case [62], special values of the
system parameters are needed.

Throughout this section, we will assume that $D^{k}=I$ for $k=1,2,3$ on the
one hand, recall, from the discussion in Section 3.2, that whether one can
make such an assumption without loss of generality is an open question. On the
other hand, said assumption makes it easier to describe the phenomenon as we
will see in the sequel.

Let $z$ denote the single-virus endemic equilibrium corresponding to virus 1,
with $Z=\operatorname{diag}(z)$. Therefore, since, by assumption, $D^{1}=I$,
the vector $z$ fulfils the following:

(32) $(-I+(I-Z)B^{1})z=\textbf{0}.$

Furthermore, since $z$ is an endemic equilibrium, from [17, Lemma 6] it
follows that $\textbf{0}\ll z\ll\textbf{1}$. Let $C$ be any nonnegative
irreducible matrix for which $z$ is also an eigenvector corresponding to
eigenvalue one. That is $Cz=z$. Therefore, from [53, Theorem 2.7], it follows
that $\rho(C)=1$, and that the vector $z$, up to a scaling, is the unique
eigenvector of $C$ with all entries being strictly positive. Define

(33) $B^{2}:=(I-Z)^{-1}C.$

We have the following result.

######  Theorem 6.

Consider system (4)-(6) under Assumption 1. Suppose that $D^{k}=I$ for
$k\in[3]$. Suppose that $B^{1}$ and $B^{3}$ are arbitrary nonnegative
irreducible matrices, and the vector $z$ and matrix $B^{2}$ are as defined in
(32) and (33), respectively. Then, a set of equilibrium points of the trivirus
equations is given by $(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$ for all
$\beta_{1}\in[0,1]$. Furthermore,

  1. i)

if $s(-I+(I-Z)B^{3})<0$, then the equilibrium set
$(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$, with $\beta_{1}\in[0,1]$, is locally
exponentially attractive.

  2. ii)

if $s(-I+(I-Z)B^{3})>0$, then the equilibrium set
$(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$, with $\beta_{1}\in[0,1]$, is
unstable.

Proof: We first show that, for all $\beta_{1}\in[0,1]$, the point
$(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$ fulfils the equilibrium version of
equations (4)-(6). To this end, observe that the right hand side of (4)-(6)
evaluated at $(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$ yields:

$\displaystyle(-I+(I-\beta_{1}Z-(1-\beta_{1})Z)B^{1})\beta_{1}z$ (34)
$\displaystyle=(-I+(I-Z)B^{1})\beta_{1}z=0,$

where (34) follows by noting that $\beta_{1}$ is a scalar, and $z$ is the
single-virus endemic equilibrium corresponding to virus 1. Similarly,

$\displaystyle(-I+(I-\beta_{1}Z-(1-\beta_{1})Z)B^{2})(1-\beta_{1})z$
$\displaystyle=(-I+(I-Z)B^{2})(1-\beta_{1})z$
$\displaystyle=(-I+(I-Z)(I-Z)^{-1}C)(1-\beta_{1})z$ (35)
$\displaystyle=(-I+C)(1-\beta_{1})z=0,$

where (35) follows by noting that $Cz=z$. Thus, from (34) and (35), it is
clear that, for every $\beta_{1}\in[0,1]$,
$(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$ is an equilibrium point of system
(4)-(6); i.e. there is a set of equilibrium points
$(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$ with $\beta_{1}\in[0,1]$.

Next, observe that, for any $\beta_{1}\in[0,1]$, the Jacobian evaluated at
$(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$ is as given in (36).

(36) $\displaystyle
J(\beta_{1}z,(1{-}\beta_{1})z,\textbf{0})=\scriptsize\begin{bmatrix}-I+(I-Z)B^{1}-\operatorname{diag}(B^{1}\beta_{1}z)&-\operatorname{diag}(B^{1}\beta_{1}z)&-\operatorname{diag}(B^{1}\beta_{1}z)\\\
-\operatorname{diag}(B^{2}(1-\beta_{1})z)&-I+(I-Z)B^{2}-\operatorname{diag}(B^{2}(1-\beta_{1})z)&-\operatorname{diag}(B^{2}(1-\beta_{1})z)\\\
\textbf{0}&\textbf{0}&-I+(I-Z)B^{3}\end{bmatrix}\normalsize$

Hence, we can rewrite $J(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$ as

(37) $\displaystyle J(\beta_{1}z,(1-\beta_{1})z,\textbf{0})=$
$\displaystyle\scriptsize\begin{bmatrix}\bar{J}(\beta_{1}z,(1{-}\beta_{1})z,\textbf{0})&&\hat{J}\\\
\mathbf{0}&&{-}I{+}(I{-}Z)B^{3}\end{bmatrix},$

where

$\displaystyle\bar{J}(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$ (38)
$\displaystyle=\scriptsize\begin{bmatrix}{-}I{+}(I{-}Z)B^{1}{-}\operatorname{diag}(B^{1}\beta_{1}z)&-\operatorname{diag}(B^{1}\beta_{1}z)\\\
-\operatorname{diag}(B^{2}(1-\beta_{1})z)&{-}I{+}(I{-}Z)B^{2}{-}\operatorname{diag}(B^{2}(1{-}\beta_{1})z)\end{bmatrix},$

while
$\hat{J}=\scriptsize\begin{bmatrix}-\operatorname{diag}(B^{1}\beta_{1}z)\\\
-\operatorname{diag}(B^{2}(1-\beta_{1})z)\end{bmatrix}$.

Note that the matrix $J(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$ is block upper
triangular. Hence, it is clear that
$s(J(\beta_{1}z,(1-\beta_{1})z,\textbf{0}))=\max\\{s(\bar{J}(\beta_{1}z,(1-\beta_{1})z,\textbf{0}),s(-I+(I-Z)B^{3})\\}$.

Proof of statement i): Consider the matrix
$\bar{J}(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$. Define
$P:=\begin{bmatrix}I_{n}&&\textbf{0}\\\ \textbf{0}&&-I_{n}\end{bmatrix}$.
Therefore,

$\displaystyle P\bar{J}((\beta_{1}z,(1-\beta_{1})z,\textbf{0})P$ (39)
$\displaystyle=\scriptsize\begin{bmatrix}{-}I{+}(I{-}Z)B^{1}{-}\operatorname{diag}(B^{1}\beta_{1}z)&\operatorname{diag}(B^{1}\beta_{1}z)\\\
\operatorname{diag}(B^{2}(1-\beta_{1})z)&{-}I{+}(I{-}Z)B^{2}{-}\operatorname{diag}(B^{2}(1{-}\beta_{1})z)\end{bmatrix}.$

Note that $P\bar{J}((\beta_{1}z,(1-\beta_{1})z,\textbf{0})P$ is an irreducible
Metzler matrix. Hence, by considering the element-wise positive vector
$\begin{bmatrix}z^{\top}&&z^{\top}\end{bmatrix}^{\top}$, and by invoking the
equilibrium version of the equation of the single-virus system corresponding
to virus 1, it follows that
$P\bar{J}((\beta_{1}z,(1-\beta_{1})z,\textbf{0})Pz=\textbf{0}$. Therefore,
from [53, Lemma 2.3], it follows that
$s(P\bar{J}(\beta_{1}z,(1-\beta_{1})z,\textbf{0})P)=0$, which implies
$s(\bar{J}(\beta_{1}z,(1-\beta_{1})z,\textbf{0})=0$. Consequently, since, by
assumption, $s(-I+(I-Z)B^{3})<0$, we obtain
$s(J(\beta_{1}z,(1-\beta_{1})z,\textbf{0}))=0$. Furthermore, since
$s(P\bar{J}(\beta_{1}z,(1-\beta_{1})z,\textbf{0})P)=0$, then since
$P\bar{J}(\beta_{1}z,(1-\beta_{1})z,\textbf{0})P$ is an irreducible Metzler
matrix, by [53, Theorem 2.7] we have that the matrix
$\bar{J}(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$ has exactly one eigenvalue at
the origin, and all other eigenvalues have negative real parts. Therefore,
since $s(-I+(I-Z)B^{3})<0$, the matrix
$J(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$ has exactly one eigenvalue at the
origin, and all other eigenvalues have negative real parts. Hence, the
trivirus equations associated with the line of equilibria define a one-
dimensional center manifold along which the Jacobian is singular. Therefore,
from [21] it follows that the set of equilibrium points
$(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$, with $\beta_{1}\in[0,1]$, is locally
exponentially attractive.

Proof of statement ii): By assumption, $s(-I+(I-Z)B^{3})>0$. Hence,
$s(J(\beta_{1}z,(1-\beta_{1})z,\textbf{0}))>0$. Therefore, from [21, Theorem
4.7, statement ii)] it follows that the set of equilibrium points
$(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$, with $\beta_{1}\in[0,1]$, is
unstable.

Observe that the key idea behind Theorem 6 is to fix the matrix $B^{1}$, and
then choose $B^{2}$ (as given in (33)) so as to obtain a locally exponentially
attractive (resp. unstable) equilibrium set, namely
$(\beta_{1}z,(1-\beta_{1})z,\textbf{0})$, with $\beta_{1}\in[0,1]$. It is
straightforward to note that, using the other two possible pairs, namely
$(B^{1},B^{3})$ and $(B^{2},B^{3})$, the same idea can be applied so as to
secure corresponding locally exponentially attractive (or, in case of
condition ii) in Theorem 6 being, up to a suitable adjustment of notation,
fulfilled, unstable) equilibrium sets.

The result in Theorem 6 does not contradict the claim for finiteness of
equilibria presented in Proposition 4. Note that the entries in matrices
$D^{i}$, $B^{i}$, $i=1,2,3$, are either a priori fixed to a specific value or
they are not. In the case where those are not fixed to a specific value, we
refer to those as free parameters, in the sense that these are allowed to take
any value in $\mathbb{R}_{+}$. The dimension of the space of free parameters
equals the number of free parameters in the tri-virus system. Each choice of
free parameters results in a realization of system (4)-(6). The set of choices
of free parameters that fall within the special case identified by Theorem 6
has measure zero.

######  Remark 2.

Theorem 6 admits a non-zero initial infection level in the network for virus
3. If one were to assume that $x_{i}^{3}(0)=0$ for all $i\in[n]$, then the
trivirus system of equations (i.e., (4)-(6)) collapses into the bivirus
equation set (i.e., equation (3) where $m=2$). Under such a setting, Theorem 6
coincides with [62, Proposition 3.9], which, in turn, subsumes [27, Theorems 6
and 7]; both with respect to i) admitting a larger class of parameters than
[27, Theorems 6 and 7] (and, assuming the setup in [35] is restricted to the
bi-virus case, [35, Corollaries 2 and 3]), and ii) providing guarantees for
local exponential attractivity.

##  9 Plane of coexistence equilibria for nongeneric trivirus networks

The focus of Section 8 was on the existence and attractivity (resp.
instability) of a line of coexistence equilibria. In this section, we identify
two scenarios that permit the existence of a plane of coexistence equilibria
(more strictly, the equilibria of interest are defined by the intersection of
a plane with the positive orthant), and, in the case of one of those
scenarios, we provide conditions for global convergence to the associated
plane of coexistence equilibria.

We consider a case where three identical copies of a virus are spreading over
the same graph as formalized next.

######  Assumption 3.

We suppose that

  1. (i)

All three viruses are spreading over the same graph.

  2. (ii)

For all $i\in[n]$ $\delta_{i}^{1}=\delta_{i}^{2}=\delta_{i}^{3}>0$.

  3. (iii)

For all $i=j\in[n]$ and $(i,j)\in\mathcal{E}$,
$\beta_{ij}^{1}=\beta_{ij}^{2}=\beta_{ij}^{3}$.

Note that for the special case identified in Assumption 3, assuming that the
setting in [35] is restricted to the tri-virus case, the existence of a plane
of coexistence equilibrium has been secured by [35, Corollary 3]. However,
[35, Corollary 3] does not provide guarantees for even local (let alone
global) convergence to the said plane. To address this shortcoming, first,
consider the system

(40) $\dot{x}(t)=(-D+(I-\operatorname{diag}(\tilde{x}))B)x(t),$

where $\tilde{x}$ is the unique endemic equilibrium of the single virus SIS
system associated with $(D,B)$, and with $B$ irreducible. The matrix
$Q:=D-(I-\operatorname{diag}(\tilde{x}))B$ is a singular irreducible
$M$-matrix, with a simple eigenvalue at $0$, and all other eigenvalues have a
positive real part. Associated with this simple zero eigenvalue is the right
eigenvector $\tilde{x}\gg{\bf 0}$ and a left eigenvector
$\tilde{u}^{\top}\gg{\bf 0}^{\top}$. We assume that $\tilde{u}^{\top}$ is
normalized to satisfy $\tilde{u}^{\top}\tilde{x}=1$. Moreover, there exists a
positive diagonal matrix $P$ such that $\bar{Q}:=PQ+Q^{\top}P\succeq 0$. In
fact, if we choose
$P=\operatorname{diag}(\tilde{u}_{1}/\tilde{x}_{1},\ldots,\tilde{u}_{n}/\tilde{x}_{n})$,
then it can be verified that $\bar{Q}$ is an irreducible $M$-matrix of rank
$n-1$, with the nullvector $\tilde{x}$ associated with the simple eigenvalue
at $0$, see [40, Section 4.3.4].

######  Theorem 7.

Consider system (4)-(6) under Assumptions 1, 2 and 3. Further, suppose that
$\rho(D^{-1}B)>1$. Then

  1. i)

For all initial conditions satisfying $x^{1}(0)>{\bf 0}_{n}$, $x^{2}(0)>{\bf
0}_{n}$, and $x^{3}(0)>{\bf 0}_{n}$, we have that
$\lim_{t\to\infty}(x^{1}(t),x^{2}(t),x^{3}(t))\in\mathcal{E}$ with
exponentially fast convergence rate, where

$\mathcal{E}=\\{(x^{1},x^{2},x^{3})|\alpha_{1}x^{1}+\alpha_{2}x^{2}+\alpha_{3}x^{3}=\tilde{x},\textstyle\sum_{i=1}^{3}\alpha_{i}=1\\},$

and $\tilde{x}$ is the unique endemic equilibrium of the single virus SIS
dynamics defined by $(D,B)$.

  2. ii)

Every point on the connected set $\mathcal{E}$ is a coexistence equilibrium.

Proof: Proof of statement i): The initial conditions guarantee that, at some
finite time $t$, we have $x^{1}(t)\gg{\bf 0}_{n}$, $x^{2}(t)\gg{\bf 0}_{n}$,
and $x^{3}(t)\gg{\bf 0}_{n}$. Define $z=x^{1}+x^{2}+x^{3}$, and
$Z=X^{1}+X^{2}+X^{3}$. Therefore, together with Assumption 3, it follows that

$\displaystyle\dot{z}$
$\displaystyle=\Big{[}-D+\big{(}I-(X^{1}(t)+X^{2}(t)+X^{3}(t))\big{)}B\Big{]}\times$
(41)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}(x^{1}(t)+x^{2}(t)+x^{3}(t))$
(42) $\displaystyle=\big{[}-D+(I-Z(t))B\big{]}z(t).$

Since $z(t)\gg{\bf 0}_{n}$, and $\rho(D^{-1}B)>1$ by hypothesis, it follows
from [22, Theorem 2] that $\lim_{t\to\infty}z(t)=\tilde{x}$ exponentially
fast. In fact, $\tilde{x}$ is the exponentially stable equilibrium of (42),
with domain of attraction $z(0)\in[0,1]^{n}\setminus{\bf 0}$. It follows that,
for all $z(0)\in[0,1]^{n}\setminus{\bf 0}$, $\|\tilde{x}-z(t)\|\leq ae^{-bt}$
for some positive constants $a,b$, with $\|\cdot\|$ being the Euclidean norm.

The dynamics for virus $i$, where $i\in[3]$, can be written as

$\dot{x}^{i}(t)=-[D+(I-\operatorname{diag}(\tilde{x}))B]x^{i}(t)+(\operatorname{diag}(\tilde{x})-Z(t))Bx^{i}(t).$

Without loss of generality, we consider virus $1$ and drop the superscript.
That is, we study the system

(43) $\dot{x}(t)=-Qx(t)+(\operatorname{diag}(\tilde{x})-Z(t))Bx(t),$

where $Z(t)$ is treated as an external time-varying input, and
$Q=D-(I-\operatorname{diag}(\tilde{x}))B$ is an irreducible singular
$M$-matrix, as detailed below (40). Define the oblique projection matrix
$R=I-\tilde{x}\tilde{u}^{\top}$. Define also $\zeta=Rx$, and note that
$\tilde{u}^{\top}\zeta=0$ and $\zeta={\bf 0}\Leftrightarrow x=\alpha\tilde{x}$
for some $\alpha\in\mathbb{R}$. That is, $\zeta$ is always orthogonal to
$\tilde{u}$, and $\zeta$ is the zero vector precisely when $x$ is in the span
of $\tilde{x}$.

Observe that $\dot{\zeta}(t)=R\dot{x}(t)$. Substituting in the right of (43)
for $\dot{x}(t)$, we obtain

(44) $\dot{\zeta}(t)=-Q\zeta(t)+R(\operatorname{diag}(\tilde{x})-Z(t))Bx(t),$

by exploiting the fact that $QR=Q=RQ$.

Consider the Lyapunov-like function

(45) $V=\zeta(t)^{\top}P\zeta(t),$

with $P$ defined below (40). It is positive definite in $\zeta$.
Differentiating $V$ with respect to time yields

$\displaystyle\dot{V}$
$\displaystyle=-2\zeta(t)^{\top}PQ\zeta(t)+2\zeta(t)^{\top}PR(\operatorname{diag}(\tilde{x})-Z(t))Bx(t)$
(46)
$\displaystyle=-\zeta(t)^{\top}\bar{Q}\zeta(t)+2\zeta(t)^{\top}PR(\operatorname{diag}(\tilde{x})-Z(t))Bx(t).$

Due to submultiplicativity of matrix norms, $\|\zeta(t)\|\leq\|R\|\|x(t)\|$.
Hence, since $\|x(t)\|$ is bounded, (9) yields

(47) $\displaystyle\dot{V}$
$\displaystyle\leq-\zeta(t)^{\top}\bar{Q}\zeta(t)+\kappa\|\tilde{x}-z(t)\|\leq-\zeta(t)^{\top}\bar{Q}\zeta(t)+\bar{a}e^{-bt},$

where $\kappa$ and $\bar{a}$ are positive constants, and the second inequality
is since $\|\tilde{x}-z(t)\|\leq ae^{-bt}$.

Let $\lambda_{2}$ denote the smallest strictly positive eigenvalue of
$\bar{Q}$. The Courant-Fischer min-max theorem [65, Theorem 8.9] yields

(48)
$\frac{\zeta^{\top}\bar{Q}\zeta}{\zeta^{\top}\zeta}\geq\min_{\begin{subarray}{c}v^{\top}\tilde{u}=0\\\
v^{\top}v=1\end{subarray}}v^{\top}\bar{Q}v=\lambda_{2},$

for all $\zeta\neq{\bf 0}$ perpendicular to $\tilde{u}$. Since
$\zeta^{\top}\tilde{u}=0$ holds for all $\zeta\in\mathbb{R}$ by definition,
(48) holds for all $\zeta\neq{\bf 0}$. Next, and recalling the definition of
$P$ below (40), it follows that

(49)
$\underline{p}\zeta^{\top}\zeta\leq\zeta^{\top}P\zeta\leq\bar{p}\zeta^{\top}\zeta,$

for all $\zeta$, where
$\underline{p}=\min_{i\in[n]}\tilde{u}_{i}/\tilde{x}_{i}$ and
$\bar{p}=\max_{i\in[n]}\tilde{u}_{i}/\tilde{x}_{i}$.

From (47) it follows that
$\dot{V}\leq-\lambda_{2}\zeta(t)^{\top}\zeta+\bar{a}e^{-bt}$, which further
implies that
$\dot{V}\leq-\bar{\lambda}\bar{p}\zeta(t)^{\top}\zeta+\bar{a}e^{-bt}$, where
$\bar{\lambda}=\lambda_{2}/\bar{p}$. Consequently, from (49), it follows that
$\dot{V}\leq-\bar{\lambda}V+\bar{a}e^{-bt}$. Thus, and recalling the
definition of $\zeta$, it follows that $\lim_{t\to\infty}x(t)=\alpha\tilde{x}$
exponentially fast, where $\alpha\in(0,1)$ because $x(t)\in(0,1)^{n}$ for all
$t\geq\tau$, for some positive $\tau$.

This analysis holds not only for virus $1$ but also virus $2$ and virus $3$.
In other words, $\lim_{t\to\infty}x^{i}(t)=\alpha_{i}\tilde{x}$ for some
$\alpha_{i}\in(0,1)$, for all $i\in[3]$. Recall that
$\lim_{t\to\infty}z(t)=\tilde{x}$, and we immediately conclude that
$\sum_{i=1}^{n}\alpha_{i}=1$, thus proving statement i).

Proof of statement ii): We now prove that every point in $\mathcal{E}$ is an
equilibrium (coexistence follows trivially by definition). Consider an
arbitrary point $(x^{1},x^{2},x^{3})$ in $\mathcal{E}$. From (4), we have

(50) $\displaystyle\dot{x}^{1}(t)$
$\displaystyle=(-D+(I-X^{1}-X^{2}-X^{3})B)x^{1}$ (51)
$\displaystyle=(-D+(I-\operatorname{diag}(\tilde{x}))B)\alpha_{1}\tilde{x}={\bf
0}.$

By the same arguments, it follows that $\dot{x}^{2}(t)={\bf 0}$ and
$\dot{x}^{3}(t)={\bf 0}$ at the point $(x^{1},x^{2},x^{3})$ in $\mathcal{E}$.
In other words, $(x^{1},x^{2},x^{3})$ is an equilibrium of the system system
(4)-(6). Since this holds for any arbitrary point in $\mathcal{E}$, the proof
of statement ii) is complete.

We identify yet another special case of the tri-virus model parameters that
admits the existence of a plane of 3-coexistence equilibrium. To this end, let
$B^{1}$ be some arbitrary nonnegative irreducible matrix with $z$ being as
defined in (32), and $B^{2}$ being as defined in (33). Let $\hat{C}\neq C$ be
a nonnegative irreducible matrix such that $z$ is the eigenvector
corresponding to eigenvalue unity, that is, $\hat{C}z=z$. Define

(52) $B^{3}:=(I-Z)^{-1}\hat{C}.$

We have the following result.

######  Proposition 8.

Consider system (4)-(6) under Assumptions 1 and 2. Suppose that $B^{1}$ is an
arbitrary nonnegative irreducible matrix and the vector $z$ together with
matrices $B^{2}$ and $B^{3}$ are as defined in (32), and (33) and (52),
respectively. Then, a set of equilibrium points of the trivirus equations is
given by $(\alpha_{1}z,\alpha_{2}z,\alpha_{3}z)$, for all $\alpha_{i}\geq 0$,
$i=1,2,3$, and $\sum_{i=1}^{3}\alpha_{i}=~{}1$.

Proof: Assume, without loss of generality, that $D^{k}=I$ for $k\in[3]$.
Observe, then, that the right hand side of (4)-(6) evaluated at
$(\alpha_{1}z,\alpha_{2}z,\alpha_{3}z)$ yields:

$\displaystyle(-I+(I-\alpha_{1}Z-\alpha_{2}Z-\alpha_{3}Z)B^{1})\alpha_{1}z$
(53) $\displaystyle=(-I+(I-Z)B^{1})\alpha_{1}z=0,$

where (53) follows by noting that by assumption, $\alpha_{1}$ is a nonnegative
scalar, and $\sum_{i=1}^{3}\alpha_{i}=1$, and $z$ is the single-virus endemic
equilibrium corresponding to virus 1. Similarly,

$\displaystyle(-I+(I-\alpha_{1}Z-\alpha_{2}Z-\alpha_{3}Z)B^{2})\alpha_{2}z$
$\displaystyle=(-I+(I-Z)B^{2})\alpha_{2}z$
$\displaystyle=(-I+(I-Z)(I-Z)^{-1}C)\alpha_{2}z$ (54)
$\displaystyle=(-I+C)\alpha_{2}z=0,$

where (54) follows by noting that $Cz=z$. Finally,

$\displaystyle(-I+(I-\alpha_{1}Z-\alpha_{2}Z-\alpha_{3}Z)B^{3})\alpha_{3}z$
$\displaystyle=(-I+(I-Z)B^{3})\alpha_{3}z$
$\displaystyle=(-I+(I-Z)(I-Z)^{-1}\hat{C})\alpha_{3}z$ (55)
$\displaystyle=(-I+\hat{C})\alpha_{3}z=0,$

where (55) follows by noting that $\hat{C}z=z$. Thus, from (53), (54) and
(55), it is clear that, for every $\alpha_{i}\geq 0$ with $i=1,2,3$, such that
$\sum_{i=1}^{3}\alpha_{i}=1$, $(\alpha_{1}z,\alpha_{2}z,\alpha_{3}z)$ is an
equilibrium point of system (4)-(6).

Note that Proposition 8 and, assuming the setting in [35] is limited to the
tri-virus case, [35, Corollaries 2 and 3] identify special cases that admit
the existence of a plane of coexistence equilibria. Since Proposition 8 covers
a larger class of parameters than that in [35, Corollaries 2 and 3],
Proposition 8 subsumes [35, Corollaries 2 and 3].

Proposition 8 and Theorem 7 compare in the following sense: On the one hand,
note that if $(I-Z)B^{1}=C=\hat{C}$, then, in view of (33) and (52), it
follows that $B^{1}=B^{2}=B^{3}$, which coincides with the setting of Theorem
7. Thus, Proposition 8 covers a larger class of parameters than Theorem 7,
and, therefore, with respect to the existence of a set of 3-coexistence
equilibria, is more general. Conversely, Proposition 8 provides no stability
guarantees for the said equilibrium set, whereas Theorem 7 establishes global
stability of the equilibrium set for the smaller class of parameters that it
covers.

With respect to the existence of a continuum of equilibria, Proposition 8 is
stronger than Theorem 6 because it adds a larger class of $i=1,2,3$, $D^{i}$,
$B^{i}$ matrices. That said, note that Theorem 6 admits arbitrary $B^{3}$,
whereas Proposition 8 insists on $B^{3}$ obeying a specific functional form,
as given in (52). In conclusion, neither subsumes the other.

##  10 Simulations

We now present a set of simulations that highlight the key theoretical results
of our paper. We choose $D^{i}=I$ for $i=1,2,3$, with the following $B^{i}$
matrices, where $\hat{\beta}^{k}_{ij}$ are constants that are changed
depending on the simulation example being presented.

$B^{1}=\scriptsize\begin{bmatrix}0&0&0&1.5\\\ 1.5&0&0&0\\\ 0&1.5&0&0\\\
0&0&1.5&0\end{bmatrix},\\\
B^{2}=\begin{bmatrix}0&1.5+\hat{\beta}^{2}_{12}&0&0\\\ 0&0&1.5&0\\\
0&0&0&1.5\\\ 1.5&0&0&0\end{bmatrix},$
$B^{3}=\scriptsize\begin{bmatrix}1&0&0.5+\hat{\beta}^{3}_{13}&0\\\
0&1+\hat{\beta}^{3}_{22}&0.5&0\\\ 0&0.5&0&1\\\
0.3+\hat{\beta}^{3}_{31}&0&1.2&0\end{bmatrix}.$

Except as otherwise stated, initial conditions are obtained according to the
following procedure. First, for $i\in[n]$ and $s\in[4]$, we sample a value
$p_{i}^{s}$ from a uniform distribution $(0,1)$. Then, for $i\in[n]$ and
$k\in[3]$ we set $x_{i}^{k}(0)=p_{ij}^{k}/\sum_{s=1}^{4}p_{ij}^{s}$, which
ensures that the initial conditions are in $\mathcal{D}$ but otherwise
randomized.

Example 1: We set $\hat{\beta}^{3}_{13}=0$, $\hat{\beta}^{2}_{12}=-0.1$,
$\hat{\beta}^{3}_{22}=-0.1$, and $\hat{\beta}^{3}_{31}=0.1$. In this example,
and following the notation of Section 2.3 and Theorem 2, we obtain
$\rho((I-\tilde{X}^{1})(D^{2})^{-1}B^{2})=0.9829$ and
$\rho((I-\tilde{X}^{1})(D^{3})^{-1}B^{3})=0.99624$. Thus,
$(\tilde{x}^{1},\bf{0},\bf{0})$ is locally exponentially stable, and Fig. 1a
shows convergence to $(\tilde{x}^{1},\bf{0},\bf{0})$. It can be computed that
$\rho((I-\tilde{X}^{2})(D^{1})^{-1}B^{1})=1.0174$ and
$\rho((I-\tilde{X}^{2})(D^{3})^{-1}B^{3})=1.0127$ and similarly,
$\rho((I-\tilde{X}^{3})(D^{1})^{-1}B^{1})=1.003$ and
$\rho((I-\tilde{X}^{3})(D^{2})^{-1}B^{2})=0.9863$. Hence, in line with Theorem
2, both $({\bf 0},\tilde{x}^{2},{\bf 0})$ and $({\bf 0},{\bf
0},\tilde{x}^{3})$ are unstable.

Example 2: We set $\hat{\beta}^{3}_{13}=0.05$,
$\hat{\beta}^{2}_{12}=\hat{\beta}^{3}_{22}=\hat{\beta}^{3}_{31}=0$. In this
example, and following the notation of Theorem 6, we have a line of equilibria
$(\beta_{1}z,(1-\beta_{1})z,{\bf 0})$, with $z=\frac{1}{3}\bf{1}$ and
$\beta_{1}\in[0,1]$. Since $\rho((I-Z)B^{3})=1.0043$, in line with statement
ii) in Theorem 6, this line of equilibria is unstable; see Figure 1b. We can
compute that $({\bf 0},{\bf 0},\tilde{x}^{3})$ is locally exponentially
stable, following reasoning analogous to that in Theorem 2. Note, however,
that even though the line of equilibria $(\beta_{1}z,(1-\beta_{1})z,{\bf 0})$
is unstable, it still has an attractive manifold (of measure zero in
$\mathcal{D}$); trajectories on this manifold will converge to the line. As
established in [62, Proposition 3.9], this manifold includes points
$(x^{1},x^{2},{\bf 0})$, where $x^{1}$ and $x^{2}$ are in a small open
neighbourhood around $\beta_{1}z$ and $(1-\beta_{1})z$, respectively.

Example 3: We set $\hat{\beta}^{3}_{13}=-0.1$,
$\hat{\beta}^{2}_{12}=\hat{\beta}^{3}_{22}=\hat{\beta}^{3}_{31}=0$. Similar to
Example 2, we have a line of equilibria $(\beta_{1}z,(1-\beta_{1})z,{\bf 0})$,
with $z=\frac{1}{3}\bf{1}$. Now, however, $\rho((I-Z)B^{3})=0.9911$, and this
line of equilibria is locally exponentially attractive according to statement
i) in Theorem 6; see Figure 1c. The boundary equilibrium $({\bf 0},{\bf
0},\tilde{x}^{3})$ is unstable, following reasoning analogous to that in
Theorem 2.

Example 4: We set
$\hat{\beta}^{3}_{13}=\hat{\beta}^{2}_{12}=\hat{\beta}^{3}_{22}=\hat{\beta}^{3}_{31}=0$.
The aforementioned choice of $B^{i}$ satisfies Proposition 8, and hence there
is a plane of equilibria of the form $(\alpha_{1}z,\alpha_{2}z,\alpha_{3}z)$,
with $\sum_{i=1}^{3}\alpha_{i}=1$ and $z=\frac{1}{3}\bf{1}$. Although we do
not have a theoretical result for convergence to this plane, Fig. 1d and 1e
illustrate convergence to this plane of equilibria from different initial
conditions; the particular equilibrium reached depends on the choice of
initial conditions.

We now consider a second set of simulations with $n=5$ nodes. We again set
$D^{i}=I$ for $i=1,2,3$, but we now study a different set of $B^{i}$ matrices
to demonstrate other relevant theoretical results of the paper. Let

(56) $B=\begin{bmatrix}1&0&2&0&0.5\\\ 0.5&2&0&0&0\\\ 0&5&0.1&0&0\\\
0.1&0&0&0.2&0\\\ 0&0&0&0.1&0.9\end{bmatrix}.$

Our $B^{i}$ matrices will be perturbations of $B$, which we describe in the
respective simulation examples. To facilitate this, let $e_{i}$ be the $i$th
basis vector of $\mathbb{R}^{5}$, i.e., $e_{i}$ has $1$ in its $i$th entry,
and all other entries are $0$. The initial conditions are selected following
the same random procedure described above.

Example 5: We set $B^{i}=B$ for all $i=1,2,3$, i.e., we have three identical
copies of a virus spreading over the same graph. According to Theorem 7, there
is a plane of coexistence equilibria, which in this case is defined by the
vector $\tilde{x}=[0.691,0.610,0.758,0.078,0.051]^{\top}$. Fig. 1f and 1g show
convergence to two different equilibria in this plane of equilibria from two
different initial conditions.

Example 6: We set $B^{1}=B$, $B^{2}=B+0.5e_{1}e_{4}^{\top}$, and
$B^{3}=B^{2}+0.1e_{5}e_{1}^{\top}$. This tri-virus system satisfies the
inequality conditions of Theorem 3; it follows that there does not exist a
$3$-coexistence equilibrium, while the boundary equilibrium $({\bf 0},{\bf
0},\tilde{x}^{3})$ is locally exponentially stable. Fig. 1h shows the tri-
virus system converging to the $({\bf 0},{\bf 0},\tilde{x}^{3})$ equilibrium.

Example 7: We set $B^{1}=B$, $B^{2}=B+2e_{1}e_{4}^{\top}$, and
$B^{3}=B+0.1e_{5}e_{1}^{\top}$. This system satisfies the inequality
conditions of Theorem 4 but not those of Theorem 3. In Fig. 2a, we observe
convergence to a $2$-coexistence equilibrium. Consistent with Theorem 4, this
$2$-coexistence equilibrium is such that only virus $2$ and virus $3$ exist in
each node of the network, i.e., it has the form $({\bf
0},\bar{x}^{2},\bar{x}^{3})$.

Example 8: We set $B^{1}=B+0.7e_{3}e_{2}^{\top}$,
$B^{2}=B+2e_{1}e_{4}^{\top}$, and $B^{3}=B+0.1e_{5}e_{1}^{\top}$. This system
satisfies the conditions of Corollary 1. The inequalities concerning the
boundary equilibria are easily checked. The inequalities involving the
$2$-coexistence equilibria are checked by verifying that there are precisely
$3$ such equilibria, one unique equilibrium corresponding to each of virus
$1$, virus $2$, and virus $3$ being extinct. This uniqueness can be
established by a rapid simulation procedure that exploits the fact that
bivirus systems are MDS and involves simulation of just two different initial
conditions [62, Corollary 3.14]. Based on this knowledge, Corollary 1
establishes that there is at least one $3$-coexistence equilibrium, and it is
saturated. Indeed, Fig. 2b shows convergence to a $3$-coexistence equilibrium.

Example 9: In each of the examples above, except for Example 5, the
simulations correspond to theoretical results that establish local exponential
convergence (either to an isolated equilibrium or a line of equilibria).
Extensive additional simulations with many other randomized initial conditions
suggest that, for these examples, convergence to the equilibrium of interest
(or line/plane of equilibria) occurs for all initial conditions in the
interior of $\mathcal{D}$, thereby suggesting that they (i.e., the equilibrium
of interest, or line/plane of equilibria), are globally stable. We conclude
here with a final example, which serves as a cautionary note highlighting that
tri-virus systems can have multiple attractive equilibria with different
regions of attraction with nonzero measure in $\mathcal{D}$. First, define the
following matrices:

$\displaystyle B_{11}$ $\displaystyle=\begin{bmatrix}1.6&1\\\
1&1.6\end{bmatrix},B_{12}=\begin{bmatrix}2.1&0.156\\\
3.0659&1.1\end{bmatrix},$ $\displaystyle B_{21}$
$\displaystyle=\begin{bmatrix}1.7&1\\\
1.2&0.5\end{bmatrix},B_{22}=\begin{bmatrix}1.6&1\\\
1.2&0\end{bmatrix},C=0.001\times{\bf 11}^{\top}.$

We consider $n=4$, and set $D^{i}=I$ for all $i=1,2,3$, and

$B^{1}=\begin{bmatrix}B_{11}&C\\\
C&B_{21}\end{bmatrix},B^{2}=\begin{bmatrix}B_{12}&C\\\
C&B_{22}\end{bmatrix},B^{3}=\begin{bmatrix}B_{22}&C\\\ C&B_{11}\end{bmatrix}.$

Fig. 2c and 2d show convergence to two different $2$-coexistence equilibria
from different initial conditions in the interior of $\mathcal{D}$; we
verified that the Jacobian matrix at both these equilibria are Hurwitz, i.e.,
both equilibria are locally exponentially stable. In Fig. 2c, convergence
occurs to a $2$-coexistence equilibrium where virus $2$ is extinct, whereas in
Fig. 2d, the $2$-coexistence equilibrium is such that virus $1$ is extinct.

(a) Example 1

(b) Example 2

(c) Example 3

(d) Example 4, First Initial Condition

(e) Example 4, Second Initial Condition

(f) Example 5, First Initial Condition

(g) Example 5, Second Initial Condition

(h) Example 6

Figure 1: Trajectories of the simulated trivirus system (3), for different
simulation parameters detailed in Section 10.

(a) Example 7

(b) Example 8

(c) Example 9, First Initial Condition

(d) Example 9, Second Initial Condition

Figure 2: Trajectories of the simulated trivirus system (3), for different
simulation parameters detailed in Section 10.

##  11 Conclusion

This paper analyzed a competitive tri-virus SIS model. Specifically, we
established that the tri-virus system is not monotone, unlike the bi-virus
system. Subsequently, using the Parametric Transversality Theorem, we showed
that, generically, the tri-virus system has a finite number of equilibria and
that the Jacobian matrices associated with each equilibrium are nonsingular.
We then identified a sufficient condition for local exponential convergence to
a boundary equilibrium. Thereafter, we identified a sufficient condition for
the nonexistence of various kinds of coexistence equilibria, followed by the
provision of a sufficient condition for the existence of a 3-coexistence
equilibrium (resp. 2-coexistence equilibrium). We identified i) a setting that
permits the existence and local exponential attractivity of a line of
coexistence equilibria; and ii) two settings that permit the existence of,
and, in one case, global convergence to, a plane of coexistence equilibria.
All of the aforementioned settings have measure zero.

There are several directions of future research that could be pursued.
Recalling the discussion in Section 3.2, a straightforward problem is
establishing whether or not the stability properties of a trivirus system
defined by $\\{D^{i},B^{i},i=1,2,3\\}$ are the same as those defined by
$\\{I,(D^{i})^{-1}B^{i},i=1,2,3\\}$. Second, given that the tri-virus system
is not monotone, a comprehensive view of the model that, in particular,
focuses on its limiting behavior would be extremely relevant since, to provide
the said view, novel tools may be required. On a related note, investigating
scenarios that admit the existence (resp. exclude the possibility of
existence) of limit cycles would provide further insights. Third, while the
present paper identifies conditions for the existence of a 3-coexistence
equilibrium point, no claims are made regarding its uniqueness or its
stability (or lack thereof). Hence, one line of investigation could, perhaps
by leveraging Morse-Smale inequalities [45], focus on providing bounds on the
number of 3-coexistence equilibria that a tri-virus system is permitted.
Fourth, assuming that one of the competing viruses is benign as compared to
the other two, devising (feedback) control strategies that can drive the virus
dynamics to the boundary equilibrium of the benign virus could provide
policymakers with another tool for disease mitigation.

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## Appendix A: Proof of Proposition 4

The use below of the Parametric Transversality Theorem, the principal tool in
the proof, requires the preliminary calculation of certain Jacobians. We will
consider the case where the $B^{i}$ are fixed, to demonstrate that for almost
all allowed $D^{i}$, equilibria are nondegenerate (i.e. the associated
Jacobians are nonsingular). Given the bounded nature of the set of interest,
viz. $\\{\bf{0}\leq x^{k}\leq\bf{1}\\}\cap\\{x^{1}+x^{2}+x^{3}\leq{\bf{1}}\\}$
and continuity of the Jacobian, finiteness of the number of zeros follows
straightforwardly from the nondegeneracy conclusion.

It is standard that there is a finite number of equilibria (viz. 1 or 2)),
when two of the $x^{i}$ are zero, the equilibria being expressible using those
of a single virus network. If precisely one of the $x^{i}$ is zero, the
equilibria can be defined using a bivirus system, and again, the finiteness
for generic $D^{k}$ is known from [62]. It, therefore, remains to consider
coexistence equilibria with all $x^{k}$ nonzero separately, and in this case,
as shown in Lemma 7 we can assume $x^{k}\gg 0,k=1,2,3$.

Without loss of generality, we will assume there is some fixed positive
$\bar{d}<1$ such that all diagonal entries of the $D^{i}$ lie in
$(\bar{d},\bar{d}^{-1})$. Let $\mathcal{D}$ denote the manifold defined by the
set of allowed $D^{i}$ consistent with the choice of $\bar{d}$.

Let $\mathcal{X}$ denote the manifold $\\{\bf{0}\ll
x^{k}\ll{\bf{1}}\\}\cap\\{x^{1}+x^{2}+x^{3}\ll{\bf{1}}\\}$. Consider for any
$x=[(x^{1})^{\top},(x^{2})^{\top},(x^{3})^{\top}]^{\top}\in\mathcal{X}$ and
$\delta=\mbox{vec}[D^{1},D^{2},D^{3}]\in\mathcal{D}$ the map

$\displaystyle f_{\delta}:$
$\displaystyle\mathcal{X}\times\mathcal{D}\to\mathcal{Y},$
$\displaystyle(x,\delta)$ $\displaystyle\mapsto
y=\begin{bmatrix}[-D^{1}+(I_{n}-X^{1}-X^{2}-X^{3})B^{1}]x^{1}\\\
[-D^{2}+(I_{n}-X^{1}-X^{2}-X^{3})B^{2}]x^{2}\\\
[-D^{3}+(I_{n}-X^{1}-X^{2}-X^{3})B^{3}]x^{3}\end{bmatrix}$

We now claim:

######  Lemma 10.

With notation as above, the matrix $\frac{\partial
f_{\delta}(x,\delta)}{\partial(x,\delta)}$ has full row rank at any
coexistence equilibrium.

Proof: We first observe that

(57) $\displaystyle\frac{\partial f_{\delta}(x,\delta)}{\partial(x,\delta)}$
$\displaystyle=\frac{\partial
f_{\delta}(x^{1},x^{2},x^{3},D^{1},D^{2},D^{3})}{\partial(x^{1},x^{2},x^{3},D^{1},D^{2},D^{3})}$
$\displaystyle=\begin{bmatrix}\left(\frac{\partial
f_{\delta}(x^{1},x^{2},x^{3},D^{1},D^{2},D^{3})}{\partial(x^{1},x^{2},x^{3})}\right)^{\top}\\\
\left(\frac{\partial
f_{\delta}(x^{1},x^{2},x^{3},D^{1},D^{2},D^{3})}{\partial(D^{1},D^{2},D^{3})}\right)^{\top}\\\
\end{bmatrix}^{\top}$

Observe next, using the diagonal nature of the $D^{i}$, that there holds (with
$e_{j}$ denoting the $j$-th unit vector)

(58) $\displaystyle\frac{\partial
f_{\delta}(x^{1},x^{2},x^{3},D^{1},D^{2},D^{3})}{\partial\delta^{1}_{j}}$
$\displaystyle=x^{1}_{j}e_{j}$ $\displaystyle\frac{\partial
f_{\delta}(x^{1},x^{2},x^{3},D^{1},D^{2},D^{3})}{\partial\delta^{2}_{j}}$
$\displaystyle=x^{2}_{j}e_{j+n}$ $\displaystyle\frac{\partial
f_{\delta}(x^{1},x^{2},x^{3},D^{1},D^{2},D^{3})}{\partial\delta^{3}_{j}}$
$\displaystyle=x^{3}_{j}e_{j+2n}$

It follows that

(59) $\displaystyle\frac{\partial
f_{\delta}(x^{1},x^{2},x^{3},D^{1},D^{2},D^{3})}{\partial\delta}$
$\displaystyle=\frac{\partial
f_{\delta}(x^{1},x^{2},x^{3},D^{1},D^{2},D^{3})}{\partial(D^{1},D^{2},D^{3})}$
$\displaystyle=\begin{bmatrix}X^{1}&0&0\\\ 0&X^{2}&0\\\
0&0&X^{3}\end{bmatrix}$

At any equilibrium in $\mathcal{X}$, this matrix has full row rank, as then
does the Jacobian matrix $\frac{\partial
f_{\delta}(x,\delta)}{\partial(x,\delta)}$ in (57) of which it is a submatrix.

Let $\mathcal{Z}$ be the submanifold of $\mathcal{Y}$ consisting of the single
point ${\bf{0_{3n}}}$. It is, in fact, the image of those points in
$\mathcal{X}\times\mathcal{D}$ defined by
$f_{\delta}(x,\delta)={\bf{0_{3n}}}$. Hence at such a point, $x^{k}\gg 0$ or
$X^{k}$ is nonsingular. Hence the Jacobian of $f_{\delta}$ with respect to
$(x,\delta)$ has full rank at such a point, which means that the map
$f_{\delta}$ is transversal to $\mathcal{Z}$. Then by the Parametric
Transversality Theorem [25, see p.145], [13, see p.68], it follows that for
almost all choices $\bar{\delta}$ of $\delta$ , the mapping
$f_{\bar{\delta}}:\mathcal{X}\to\mathcal{Y},f_{\bar{\delta}}(x)=f_{\delta}(x,\bar{\delta})$
will be transversal to $\mathcal{Z}$, i.e. the Jacobian $\frac{\partial
f_{\delta}(x,\bar{\delta})}{\partial x}$ will be of full rank $3n$ at the
zeros of $f_{\bar{\delta}}$. Since this matrix is square and of size $3n\times
3n$, this says that every zero of $f(x,\bar{\delta})$ is nondegenerate and as
a consequence, isolated. As already noted, since the zeros occur in a bounded
set, they are finite in number. The choices of $\delta$ for which they are not
finite in number, if indeed such choices exist, define a set of measure zero.