# Is There Scale Invariance in $\mathcal{N}=1$ Supersymmetric Field Theories
$?$

Sibo Zheng
Department of Physics, Chongqing University, Chongqing 401331, P.R. China

Abstract

In two dimensions, it is well known that the scale invariance can be
considered as conformal invariance. However, there is no solid proof of this
equivalence in four or higher dimensions. we address this issue in the context
of $4d$ $\mathcal{N}=1$ SUSY theories. The SUSY version of dilatation current
for theories without conserved $R$ symmetry is constructed through the FZ-
multiplet. We discover that the scale-invariant SUSY theory is also conformal
when the real superfield in the dilatation current multiplet is conserved.
Otherwise, it is only scale-invariant, despite of the transformation of
improvement.

##  1 Introduction

In two dimensions, the equivalence between scale and conformal invariance has
been proved in terms of a “c theorem ” [1, 2]. At the classical level this
statement is argued to be still true in four-dimensional ( $4d$ ) QFT [3].
However, it is unclear at the quantum level generally, although some examples
( e.g, see [2, 4, 5, 6, 7]) with or without supersymmetry (SUSY) have been
proposed in the literature.

The difference between scale and conformal invariance in SUSY can be
understood from the viewpoint of structure of their groups [8]. The generators
for the group of scale invariance include a dilatation operator $\Delta$ and
those of super-Poincare group. The elements of the group for super-conformal
symmetry are bigger. In particular, the super-conformal group contains a
$R$-symmetric generator. So one might guess the role played by the $R$
symmetry in $4d$ SUSY theory is crucial for discriminating scale invariance
from conformal invariance.

Following the intuition, we address the connection between two symmetries in
the context of $4d$ $\mathcal{N}=1$ SUSY. The first task we should solve is
the realization of dilatation current of SUSY version. This is tied to two
elements. At first, the SUSY version of dilatation current is actually to
address the SUSY generalization of momentum-energy tensor $T_{\mu\nu}$ [10,
11]. There are a few such multiplets known as supercurrent multiplets which
admit the $T_{\mu\nu}$ as a component freedom. These supercurrent multiplets
are classified into Ferrara-Zumino (FZ)-multiplet [12], $\mathcal{R}$\-
multiplet [13] and $\mathcal{S}$\- multiplet [14], (see also [15, 16]). The
connection between scale and conformal invariance has been discussed in [9] in
terms of $\mathcal{R}$\- multiplet, which admits a conserved $R$ symmetry. In
this paper, we explore the FZ multiplet, which conversely doesn’t admit such
conserved $R$ symmetry.

The other element which is crucial for our discussion is the ambiguity in the
definition of $T_{\mu\nu}$. As this is the main source for incorporating the
scale and conformal invariance. In what follows, we simply review the
transformation of improvement due to the ambiguity in QFT, and then take care
of the improvement in its SUSY version.

The paper is organized as follows. In section 2, inspired by the construction
of dilatation current multiplet for $\mathcal{R}$-multiplet [9], we consider
the SUSY version of dilatation current and virial current multiplet in the
case of FZ multiplet. In section 3, we use the consistent constraints of
unitarity for scale-invariant SUSY theory and closure of SUSY algebra as the
main tool to explore the structure of the virial current multiplet. We find
that the the scale-invariant SUSY theory is also conformal when the real
superfield in the dilatation current multiplet is conserved. Otherwise, it is
only scale-invariant, despite of the transformation of improvement. Together
with the claim on conditions for the equivalence between these two symmetries
in $R$-symmetric case [9], we complete understanding their $4d$ SUSY version.

##  2 Supercurrent and Dilatation Current Multiplet

###  2.1 $4D$ version in QFT

Before we discuss the SUSY version of the dilatation current, let us recall
its definition in $4d$ quantum field theory (QFT). Given a QFT with scaling
invariance, there exists a conserved current, i.e, the dilatation current
$\Delta_{\mu}$, which is found to be,

$\displaystyle{}\Delta_{\mu}=x^{\nu}T_{\mu\nu}+\mathcal{O}_{\mu}$ (2.1)

Here $\mathcal{O}_{\mu}$ refers to the virial current that does not explicitly
depend on the spacetime coordinates. Scale invariance gives rise to an anomaly
for the derivative of the virial current,

$\displaystyle{}T=-\partial^{\mu}\mathcal{O}_{\mu}$ (2.2)

which shows that the virial current must also be conserved if QFT with scale
invariance is promoted to be conformal. In other words, the anomaly involved
the virial current in (2.2) is a character for scale invariance against
conformal invariance. As illustrated by Polchinski in Ref. [2], scale-
invariant QFT can be promoted to be a conformal one if and only if the virial
current permits the structure as

$\displaystyle{}\mathcal{O}_{\mu}=j_{\mu}+\partial^{\nu}L_{\mu\nu},$ (2.3)

with $L_{\mu\nu}$ is an anti-symmetric tensor and $j_{\mu}$ an conserved
current, $\partial^{\mu}j_{\mu}=0$.

There are two important issues which determine whether virial current is
allowed to have the structure as in (2.3). The first issue is the ambiguity in
the definition of the 4d energy-momentum tensor,

$\displaystyle{}T_{\mu\nu}$ $\displaystyle\rightarrow$
$\displaystyle~{}T_{\mu\nu}+(\partial_{\mu}\partial_{\nu}-\eta_{\mu\nu}\partial^{2})\varphi$
(2.4)

which transfers the ambiguity into the definition of virial current via (2.2).
The second issue involves in the SUSY version of (2.1). Inspired by the
connection between supersymmetric current $S_{\mu\alpha}$ and $T_{\mu\nu}$
from viewpoint of SUSY algebra, they can be embedded into super-multiplets
known as super-current multiplets. In what follows, we choose the super-
current multiplet which doesn’t allow the $R$ symmetry, i.e, the FZ-multiplet
$\mathcal{J}_{\mu}$ 111 Supercurrent multiplets without conserved $R$ symmetry
include $\mathcal{S}$\- and FZ-multiplet. Consider the fact that under certain
limits [14] the former reduces to either the later one or the $R$ multiplet,
we will study the FZ-multiplet.. We follow the conventions of Wess and Bagger
[17] 222The bi-spinor representation for vector field is taken as,
$\displaystyle
J_{\alpha\dot{\alpha}}=-2\sigma^{\mu}_{\alpha\dot{\alpha}}J_{\mu},~{}~{}~{}~{}~{}~{}and~{}~{}~{}~{}J_{\mu}=\frac{1}{4}\bar{\sigma}_{\mu}^{\dot{\alpha}\alpha}J_{\alpha\dot{\alpha}}$
, and present the explicit component expression for FZ-multiplet in the
appendix A. In the appendix, it is easy to see that the divergence of bottom
component
$j_{\mu}=\partial^{\alpha\dot{\alpha}}\mathcal{S}_{\alpha\dot{\alpha}}=i\left(\bar{D}^{2}\bar{X}-D^{2}X\right)$,
from which this global current is not conserved except in the case of
$\mathcal{R}$-multiplet.

###  2.2 Dilatation Current Multiplet

The SUSY version of (2.1) can be realized by a multiplet
$\mathbf{\Delta}_{\mu}$ defined as,

$\displaystyle{}\mathbf{\Delta}_{\mu}=x^{\nu}\left\\{-\frac{1}{8}\bar{\sigma}_{\mu}^{\dot{\alpha}\alpha}[D_{\alpha},\bar{D}_{\dot{\alpha}}]\mathcal{\mathcal{J}}_{\nu}+\frac{1}{16}\epsilon_{\nu\mu\rho\sigma}\left(\bar{\sigma}^{\dot{\alpha}\alpha}\right)^{\rho}\\{D_{\alpha},\bar{D}_{\dot{\alpha}}\\}\mathcal{J}^{\sigma}+\frac{1}{4}\eta_{\mu\nu}(D^{2}X+\bar{D}^{2}\bar{X})\right\\}+\mathcal{O}_{\mu}$

with $\Delta_{\mu}$ being the bottom component of real superfield
$\mathbf{\Delta}_{\mu}$ in (2.1), from which one can verify (2.1) by using the
component expression in the appendix A. The SUSY version of (2.2) can be
directly read from (2.2),

$\displaystyle{}\partial^{\mu}\mathcal{O}_{\mu}=\frac{3}{16}(\bar{D}^{2}\bar{X}+D^{2}X)$
(2.6)

In parrel to the previous discussion in the $4d$ version of QFT, it is crucial
to figure out the ambiguity of superfield $\mathcal{O}_{\mu}$ defined in (2.2)
before we proceed to discuss its structure. The constraint on supercurrent
$\mathcal{J}_{\mu}$,
$\bar{D}^{\dot{\alpha}}\mathcal{J}_{\alpha\dot{\alpha}}=D_{\alpha}X$ is not
affected by an improvement as [14],

$\displaystyle{}\mathcal{J}_{\alpha\dot{\alpha}}$ $\displaystyle\rightarrow$
$\displaystyle\mathcal{J}_{\alpha\dot{\alpha}}-2i\sigma^{\mu}_{\alpha\dot{\alpha}}\partial_{\mu}(Y-\bar{Y})$
$\displaystyle X$ $\displaystyle\rightarrow$
$\displaystyle~{}X-\frac{1}{2}\bar{D}^{2}\bar{Y}$ (2.7)

where $Y$ a chiral superfield. The improvement (2.2) shifts both the energy-
momentum tensor and the supersymmetry current simultaneously as,

$\displaystyle{}S_{\mu\alpha}$ $\displaystyle\rightarrow$ $\displaystyle
S_{\mu\alpha}+2i\left(\sigma_{\mu\nu}\right)^{\beta}_{\alpha}\partial^{\nu}Y\mid_{\theta^{\beta}}$
$\displaystyle T_{\mu\nu}$ $\displaystyle\rightarrow$
$\displaystyle~{}T_{\mu\nu}-(\partial_{\mu}\partial_{\nu}-\eta_{\mu\nu}\partial^{2})Re~{}Y\mid$
(2.8)

Substituting the second improvement in (2.2) into (2.6) leads to the SUSY
version of the improvement transformations,

$\displaystyle{}\partial^{\mu}\mathcal{O}_{\mu}\rightarrow\partial^{\mu}\mathcal{O}_{\mu}-\frac{3}{32}\left(D^{2}\bar{D}^{2}\bar{Y}+\bar{D}^{2}D^{2}Y\right)$
(2.9)

Here are a few comments in order. It is obvious that the transformations in
(2.2) don’t violate the conservations of supersymmetric current
$\partial^{\mu}S_{\mu\alpha}=0$ and energy-momentum tensor
$\partial^{\mu}T_{\mu\nu}=0$. Nevertheless, it modifies the trace part of
energy-momentum tensor as $T\rightarrow T-3\partial^{2}(ReY\mid)$. As we will
discuss later, this character potentially interpolates the scale invariant and
conformal invariant theories. That is to say, in a scale-invariant SUSY
theory, if there exists a well-defined $Y$ from UV to deep IR energy scale
such that trace part of energy-momentum tensor $T^{\prime}=0$ by the
improvement, then this SUSY theory is actually super-conformal. Otherwise, a
scale invariant SUSY theory is exactly allowed to exist, unless it doesn’t
satisfy the examination of unitary constraints (see below for more
discussions).

##  3 Constraints on Virial-current Multiplet

Now we proceed to uncover the structure of virial current superfield through
some consistent checks from SUSY algebra and unitarity of scale invariant
theories. The fact that the supercharge $Q_{\alpha}$ has scaling dimension
$1/2$ implies that,

$\displaystyle{}[Q_{\alpha},\Delta]=-\frac{i}{2}Q_{\alpha}=\int
d^{3}x[Q_{\alpha},\Delta_{0}]$ (3.1)

By using the component expression for FZ-multiplet in appendix A, one obtains,

$\displaystyle{}\int
d^{3}x\left(\mathcal{O}_{0\alpha}-\frac{i}{2\sqrt{2}}(\sigma_{0}\bar{\psi})_{\alpha}-i(\sigma^{\nu}_{0})_{\alpha}^{\beta}S_{\mu\beta}-\frac{i}{2}S_{0\alpha}\right)=0$
(3.2)

where $\psi=\frac{\sqrt{2}}{3}(\sigma^{\mu}\bar{S}_{\mu})$ in the case of FZ-
multiplet, and $Q_{\mu\alpha}\equiv[Q_{\alpha},\mathcal{O}_{\mu}]$ as in Ref.
[9]. Then (3.2) gives rise to,

$\displaystyle{}\mathcal{O}_{\mu\alpha}=\frac{i}{3}\sigma_{\mu\alpha\dot{\alpha}}\bar{\sigma}^{\nu\dot{\alpha}\beta}S_{\nu\beta}+(\sigma_{\mu}^{\nu})^{\beta\delta}\partial_{\nu}\gamma_{\beta\delta\alpha}+(\bar{\sigma}_{\mu}^{\nu})^{\dot{\beta}\dot{\delta}}\partial_{\nu}\gamma_{\dot{\beta}\dot{\delta}\alpha}$
(3.3)

where $\gamma_{\beta\delta\alpha}$ and
$\gamma_{\dot{\beta}\dot{\delta}\alpha}$ are local and gauge invariant
operators of dimension $5/2$.

In the case of conformal field theory, there exist well-known bounds on
dimension of local and gauge invariant operators [8]. Similar situation
happens in the case of non-conformal fixed points [18], in terms of which
operators $\gamma_{\beta\delta\alpha}$ and
$\gamma_{\dot{\beta}\dot{\delta}\alpha}$ (and higher spin operators) are found
to satisfy [9],

$\displaystyle{}(\sigma_{\mu}^{\nu})^{\beta\delta}\partial_{\nu}\gamma_{\beta\delta\alpha}=0,~{}~{}~{}(\bar{\sigma}_{\mu}^{\nu})^{\dot{\beta}\dot{\delta}}\partial_{\nu}\gamma_{\dot{\beta}\dot{\delta}\alpha}=0$
(3.4)

Thus, one finds,

$\displaystyle{}\mathcal{O}_{\mu\alpha}=\frac{i}{3}\sigma_{\mu\alpha\dot{\alpha}}\bar{\sigma}^{\nu\dot{\alpha}\beta}S_{\nu\beta}+(\sigma_{\mu}^{\nu})^{\alpha}_{\beta}\partial_{\nu}\gamma_{\beta}$
(3.5)

To proceed, we impose the closure of SUSY transformation to extract possible
information on $\gamma_{\beta}$ in (3.5) and its descents. It is
straightforward to impose the constraints,

$\displaystyle{}(\eta^{\beta}\xi^{\alpha}-\xi^{\beta}\eta^{\alpha})\delta_{\beta}\delta_{\alpha}\mathcal{O}_{\mu}$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle(\xi^{\alpha}\bar{\eta}_{\dot{\alpha}}\delta^{\dot{\alpha}}\delta_{\alpha}-\bar{\eta}_{\dot{\alpha}}\xi^{\alpha}\delta_{\alpha}\delta^{\dot{\alpha}})\mathcal{O}_{\mu}$
$\displaystyle=$ $\displaystyle
2i(\xi\sigma^{\nu}\bar{\eta})\partial_{\nu}\mathcal{O}_{\mu}$ (3.6)
$\displaystyle(\xi^{\alpha}\bar{\eta}_{\dot{\alpha}}\delta^{\dot{\alpha}}\delta_{\alpha}-\bar{\eta}_{\dot{\alpha}}\xi^{\alpha}\delta_{\alpha}\delta^{\dot{\alpha}})\gamma_{\beta}$
$\displaystyle=$ $\displaystyle
2i(\xi\sigma^{\nu}\bar{\eta})\partial_{\nu}\gamma_{\beta}$

which will give us some insights about the structure of $\gamma_{\beta}$. In
what follows, we follow a set of definitions of Ref.[9],

$\displaystyle{}\delta_{\alpha}\gamma_{\beta}$ $\displaystyle=$ $\displaystyle
i\epsilon_{\alpha\beta}\gamma-(\sigma^{\mu\nu})_{\alpha\beta}\gamma_{\mu\nu},$
$\displaystyle\delta_{\dot{\alpha}}\gamma_{\beta}$ $\displaystyle=$
$\displaystyle(\sigma^{\mu})_{\beta\dot{\alpha}}\gamma_{\mu}$ (3.7)

where $\gamma$, $\gamma_{\mu}$ and $\gamma_{\mu\nu}$ is gauge invariant
scalar, vector and anti-symmetric tensor operator, respectively.

In terms of the SUSY transformation (A), we obtain from the first constraint
in (3),

$\displaystyle{}i\partial^{\nu}\gamma_{\nu\mu}+\frac{1}{3}\partial_{\mu}\gamma=0$
(3.8)

Note that (3.8) coincides with what has been found in the case of
$\mathcal{R}$-multiplet. Thus, as discussed in [9], we arrive at the
conclusion that scalar $\gamma$ and tensor field $\gamma_{\mu\nu}$ both
vanish. In other words, $\gamma_{\alpha}$ is an anti-chiral superfield,

$\displaystyle{}D^{2}\mathcal{O}_{\mu}=\bar{D}^{2}\mathcal{O}_{\mu}=0$ (3.9)

Evaluating the second constraint in (3), one derives that,

$\displaystyle{}\partial_{\nu}\mathcal{O}_{\mu}=-\frac{2}{3}\eta_{\nu\mu}T-\frac{2}{3}\epsilon_{\mu\nu\rho\sigma}\partial^{\rho}j^{\sigma}-\frac{1}{4}\partial_{\nu}(\gamma_{\mu}+\bar{\gamma}_{\mu})+\frac{1}{4}\eta_{\nu\mu}\partial^{\rho}(\gamma_{\rho}+\bar{\gamma}_{\rho})-\frac{i}{4}\varepsilon_{\sigma\rho\mu\nu}\partial^{\sigma}(\gamma^{\rho}-\bar{\gamma}^{\rho})$

where we have used the anti-commutators of supercharges and supercurrent. From
(3) one finds the divergence of virial current,

$\displaystyle{}\partial^{\mu}\mathcal{O}_{\mu}=-\frac{1}{20}\partial^{\mu}(\gamma_{\mu}+\bar{\gamma}_{\mu})$
(3.11)

in term of the relation $T=-\partial^{\mu}\mathcal{O}_{\mu}$. Introduce
superfield $\Gamma_{\alpha}$ which accommodate $\gamma_{\alpha}$ as the bottom
component, $\Gamma_{\alpha}=\gamma_{\alpha}+\cdots$, we can write (3.11) in
the superfield expression

$\displaystyle{}\mathcal{O}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{1}{40}\bar{\sigma}_{\mu}^{\dot{\alpha}\alpha}\left(\bar{D}_{\dot{\alpha}}\Gamma_{\alpha}-D_{\alpha}\bar{\Gamma}_{\dot{\alpha}}\right)+J_{\mu}$
(3.12)

Following the fact that the anti-symmetric part involved $\gamma_{\mu}$ in (3)
doesn’t contribute to $\mathcal{O}_{\mu}$ in (3.12), one can accommodate this
part as,

$\displaystyle{}U_{\mu}=-i(\gamma_{\mu}-\bar{\gamma}_{\mu})+\mathcal{\hat{O}}_{\mu}$
(3.13)

where $U_{\mu}$ and $\mathcal{\hat{O}}_{\mu}$ is the vector freedom of real
superfield $U$ 333For illustration, superfield $U$ that can be decomposed into
a chiral and its anti-chiral superfield, can achieve the null contribution
from $U_{\mu}\sim\partial_{\mu}(A-A^{*})$.  and a primary operator
$\mathcal{\hat{O}}$. Using the last constraint in (3), one finds

$\displaystyle{}\Gamma_{\alpha}=\frac{i}{2}D_{\alpha}U+\frac{1}{2}\mathcal{\hat{O}}_{\alpha}$
(3.14)

from which (3.12) can be rewritten as

$\displaystyle{}\mathcal{O}_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\partial_{\mu}U+\frac{1}{8}\bar{\sigma}_{\mu}^{\dot{\alpha}\alpha}\left[D_{\alpha},\bar{D}_{\dot{\alpha}}\right]\mathcal{\hat{O}}$
(3.15)

where we have made a scaling of $\mathcal{O}_{\mu}$.

In summary, the virial current multiplet in a scale-invariant SUSY theory
satisfies

$\displaystyle{}0$ $\displaystyle=$ $\displaystyle
D^{2}\mathcal{O}_{\mu}=\bar{D}^{2}\mathcal{O}_{\mu}$
$\displaystyle\mathcal{O}_{\mu}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}~{}\partial_{\mu}U+\frac{1}{8}\bar{\sigma}_{\mu}^{\dot{\alpha}\alpha}\left[D_{\alpha},\bar{D}_{\dot{\alpha}}\right]\mathcal{\hat{O}}$
(3.16)

##  4 Scale Invariance vs Conformal Invariance

According to (2.9) the scale-invariant SUSY theory can be improved to be
conformal-invariant if and only if,

$\displaystyle{}\partial^{\mu}\mathcal{O}_{\mu}=\\{D^{2},\bar{D}^{2}\\}\hat{Y}$
(4.1)

with $\hat{Y}=Y+\bar{Y}$ a real superfield. Impose the first constraint of (3)
on its second one, one immediately finds that

$\displaystyle{}\partial^{\mu}\mathcal{O}_{\mu}=-\frac{1}{4}~{}\Box~{}U$ (4.2)

If $U$ is conserved, the scale-invariant SUSY theory is actually conformal-
invariant. Conversely, it is only scale-invariant, which is not affected by
the improvement (4.1) as we explain below.

Compare (4.2) with (4.1), one gets the intuition that scale-invariant SUSY
with non-conserved $U$ can be improved to be conformal-invariant if $U$
satisfies,

$\displaystyle{}\left(\Box~{}-\tilde{c}~{}\\{D^{2},\bar{D}^{2}\\}\right)U=0$
(4.3)

with an adjustable real coefficient $\tilde{c}$. Also the improvement suggests
that $U$ is proportional to $\hat{Y}$. This means $\hat{Y}$ should also
satisfy (4.3) and $D^{2}\hat{Y}=0$ simultaneously. Substituting the later into
(4.3) leads to,

$\displaystyle{}\Box~{}\hat{Y}=0~{}~{}~{}~{}~{}\Leftrightarrow~{}~{}~{}~{}~{}\Box~{}U=0$
(4.4)

In conclusion, when the virial current multiplet $\mathcal{O}_{\mu}$, defined
by the scale-invariant SUSY theory, doesn’t contain a conserved $U$, the
theory can not be improved to be conformal. Conversely, when such an $U$ is
conserved, the scale-invariant SUSY theory must also be conformal, despite of
the transition of improvement. Together with the claim on the equivalence
between these two symmetries in $R$-symmetric case [9], we complete
understanding their $4d$ SUSY version.

To illustrate the roles played by $R$ symmetry, let us use the SUSY Wess-
Zumino model for example. Given the Kahler potential
$K(\Phi^{i},\bar{\Phi}^{i})$ and superpotential $W(\Phi_{i})$ for chiral
superfields $\Phi_{i}$, the FZ-multiplet and $X$ superfield is given by ,

$\displaystyle{}\mathcal{J}_{\alpha\dot{\alpha}}$ $\displaystyle=$
$\displaystyle
2g_{i\bar{i}}\left(D\Phi^{i}\right)\left(\bar{D}\bar{\Phi}^{\bar{i}}\right)-\frac{2}{3}[D_{\alpha},D_{\dot{\alpha}}]K,$
$\displaystyle X$ $\displaystyle=$ $\displaystyle 4W-\frac{1}{3}\bar{D}^{2}K$
(4.5)

If there is $R$ symmetry in SUSY Wess-Zumino models, $X$ can be written as the
specific form [14],

$\displaystyle
X=\bar{D}^{2}\left(\frac{1}{2}\sum_{i}R_{i}\Phi^{i}\partial_{i}K-\frac{1}{3}K\right)=-\frac{1}{2}\bar{D}^{2}\tilde{U},$
(4.6)

It is crucial to note that $U$ is identifies as $\tilde{U}$ that is indeed
decomposed of a chiral and its anti-chiral part. Such $U$ which is constrained
by $D^{2}U=\bar{D}^{2}U=0$ as in (3) trivially satisfies (4.4). Superficially,
the super-conformality in this type of $R$-symmetric Wess-Zumino model is
restored in terms of the transition of improvement. From the viewpoint of
virial current multiplet, this improvement is actually irrelevant.

$\bf{Acknowledgement}$
This work is supported in part by the Doctoral Fund of Ministry of Education
of China (No. 20110191120045).

##  Appendix A Communicators

The component expression for $\mathcal{S}_{\mu}$ that satisfies the constraint
$\bar{D}^{\dot{\alpha}}\mathcal{J}_{\alpha\dot{\alpha}}=D_{\alpha}X+\chi_{\alpha}$
is given by [14],

$\displaystyle{}\mathcal{J}_{\mu}$ $\displaystyle=$ $\displaystyle
j^{(S)}_{\mu}+\theta^{\alpha}(S_{\mu\alpha}-\frac{1}{\sqrt{2}}\sigma_{\mu}\bar{\psi})+\bar{\theta}(\bar{S}_{\mu}+\frac{1}{\sqrt{2}}\bar{\sigma}_{\mu}\psi)+\frac{i}{2}\theta^{2}\partial_{\mu}\phi^{{\dagger}}-\frac{i}{2}\bar{\theta}^{2}\partial_{\mu}\phi$
$\displaystyle+$
$\displaystyle(\theta\sigma^{\nu}\bar{\theta})\left(2T_{\nu\mu}-\eta_{\nu\mu}Z+\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\left(F^{(S)\rho\sigma}+\partial^{\rho}j^{(S)\sigma}\right)\right)$
$\displaystyle+$
$\displaystyle\theta^{2}\left(\frac{i}{2}\partial_{\rho}S_{\mu}\sigma^{\rho}-\frac{i}{2\sqrt{2}}\partial_{\rho}\bar{\psi}\bar{\sigma}^{\rho}\sigma_{\mu}\right)\bar{\theta}+\bar{\theta}^{2}\theta\left(-\frac{i}{2}\sigma^{\rho}\partial_{\rho}S_{\mu}+\frac{i}{2\sqrt{2}}\sigma_{\mu}\bar{\sigma}^{\rho}\partial_{\rho}\psi\right)$
$\displaystyle+$
$\displaystyle\theta^{2}\bar{\theta}^{2}\left(\frac{1}{2}\partial_{\mu}\partial^{\nu}j_{\nu}^{(S)}-\frac{1}{4}\partial^{2}j^{(S)}_{\mu}\right)$

with

$\displaystyle{}X$ $\displaystyle=$
$\displaystyle\phi+\sqrt{2}\theta\psi+\theta^{2}\left(Z+i\partial^{\rho}j^{(R)}_{\rho}\right)$
$\displaystyle\chi_{\alpha}$ $\displaystyle=$
$\displaystyle-i\lambda^{(S)}_{\alpha}+\left(D\delta_{\alpha}^{\beta}-2i(\sigma^{\rho}\bar{\sigma}^{\sigma})_{\alpha}^{~{}\beta}~{}F^{(S)}_{\rho\sigma}\right)\theta_{\beta}+\theta^{2}\sigma_{\nu\alpha\dot{\alpha}}\partial_{\nu}\bar{\lambda}^{(S)\dot{\alpha}}$
(A.2)

The component fields also satisfy two extra constraints,

$\displaystyle{}D=-4T^{\mu}_{\mu}+6Z,~{}~{}~{}~{}~{}~{}\lambda_{\alpha}^{(S)}=-2i\sigma^{\mu}\bar{S}_{\mu}+3i\sqrt{2}\psi.$
(A.3)

According to (A), one can derive the SUSY transformation of supercurrent
$S_{\mu\alpha}$,

$\displaystyle{}\delta_{\dot{\beta}}S_{\mu\alpha}$ $\displaystyle=$
$\displaystyle\sigma^{\nu}_{\alpha\dot{\beta}}\left(2T_{\nu\mu}-i\eta_{\nu\mu}\partial^{\rho}j^{(S)}_{\rho}+i\partial_{\nu}j^{(S)}_{\mu}-\frac{1}{2}\epsilon_{\nu\mu\rho\sigma}F^{(S)\rho\sigma}-\frac{1}{2}\epsilon_{\nu\mu\rho\sigma}\partial^{\rho}j^{(S)\sigma}\right)$
$\displaystyle\delta_{\beta}S_{\mu\alpha}$ $\displaystyle=$
$\displaystyle-2\varepsilon_{\lambda\beta}(\sigma_{\mu\rho})^{\lambda}_{\alpha}\partial^{\rho}\phi^{*}$
(A.4)

as well as their conjugators,

$\displaystyle{}\delta_{\beta}\bar{S}_{\mu\dot{\alpha}}$ $\displaystyle=$
$\displaystyle\sigma^{\nu}_{\beta\dot{\alpha}}\left(2T_{\nu\mu}+i\eta_{\nu\mu}\partial^{\rho}j^{(S)}_{\rho}-i\partial_{\nu}j^{(S)}_{\mu}-\frac{1}{2}\epsilon_{\nu\mu\rho\sigma}F^{(S)\rho\sigma}-\frac{1}{2}\epsilon_{\nu\mu\rho\sigma}\partial^{\rho}j^{(S)\sigma}\right)$
$\displaystyle\delta_{\dot{\beta}}\bar{S}_{\mu\dot{\alpha}}$ $\displaystyle=$
$\displaystyle-2\varepsilon_{\dot{\lambda}\dot{\beta}}(\bar{\sigma}_{\mu\rho})^{\dot{\lambda}}_{\dot{\alpha}}\partial^{\rho}\phi.$
(A.5)

We can obtain the corresponding relationes for the case of FZ multiplet by
taking the limit $\chi_{\alpha}=0$. The anti-commutators relationes (A) and
(A) modify as,

$\displaystyle{}\delta_{\dot{\beta}}S_{\mu\alpha}$ $\displaystyle=$
$\displaystyle\sigma^{\nu}_{\alpha\dot{\beta}}\left(2T_{\nu\mu}-i\eta_{\nu\mu}\partial^{\rho}j_{\rho}+i\partial_{\nu}j_{\mu}-\frac{1}{2}\epsilon_{\nu\mu\rho\sigma}\partial^{\rho}j^{\sigma}\right)$
$\displaystyle\delta_{\beta}S_{\mu\alpha}$ $\displaystyle=$
$\displaystyle-2\varepsilon_{\lambda\beta}(\sigma_{\mu\rho})^{\lambda}_{\alpha}\partial^{\rho}\phi^{*}$
(A.6)

and

$\displaystyle{}\delta_{\beta}\bar{S}_{\mu\dot{\alpha}}$ $\displaystyle=$
$\displaystyle\sigma^{\nu}_{\beta\dot{\alpha}}\left(2T_{\nu\mu}+i\eta_{\nu\mu}\partial^{\rho}j_{\rho}-i\partial_{\nu}j_{\mu}-\frac{1}{2}\epsilon_{\nu\mu\rho\sigma}\partial^{\rho}j^{\sigma}\right)$
$\displaystyle\delta_{\dot{\beta}}\bar{S}_{\mu\dot{\alpha}}$ $\displaystyle=$
$\displaystyle-2\varepsilon_{\dot{\lambda}\dot{\beta}}(\bar{\sigma}_{\mu\rho})^{\dot{\lambda}}_{\dot{\alpha}}\partial^{\rho}\phi.$
(A.7)

Here we have used the $D=0$ and $\lambda^{(S)}_{\alpha}=0$ in (A.3) to cancel
the dependence on $\psi$ and $Z$ fields, from which the conservation of
$\partial^{\mu}S_{\mu\alpha}=0$ is consistent with these anti-commutators.

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# A mean-field analysis of a network behavioural–epidemic model

Kathinka Frieswijk, Lorenzo Zino, Mengbin Ye, Alessandro Rizzo, and Ming Cao
K. Frieswijk, L. Zino, and M. Cao are with the Faculty of Science and
Engineering, University of Groningen, Groningen, the Netherlands
({k.frieswijk,lorenzo.zino,m.cao}@rug.nl). M. Ye is with the Centre for
Optimisation and Decision Science, Curtin University, Perth, Australia
(mengbin.ye@curtin.edu.au). A. Rizzo is with the Department of Electronics and
Telecommunications, Politecnico di Torino, Torino, Italy, and with the
Institute for Invention, Innovation, and Entrepreneurship, New York University
Tandon School of Engineering, Brooklyn NY, USA (alessandro.rizzo@polito.it).
This work was partially supported by the European Research Council (ERC-
CoG-771687), and the Western Australian Government (Premier’s Science
Fellowship Program).

###### Abstract

The spread of an epidemic disease and the population’s collective behavioural
response are deeply intertwined, influencing each other’s evolution. Such a
co-evolution typically has been overlooked in mathematical models, limiting
their real-world applicability. To address this gap, we propose and analyse a
behavioural–epidemic model, in which a susceptible–infected–susceptible
epidemic model and an evolutionary game-theoretic decision-making mechanism
concerning the use of self-protective measures are coupled. Through a mean-
field approach, we characterise the asymptotic behaviour of the system,
deriving conditions for global convergence to a disease-free equilibrium and
characterising the endemic equilibria of the system and their (local)
stability. Interestingly, for a certain range of the model parameters, we
prove global convergence to a limit cycle, characterised by periodic epidemic
outbreaks.

##  I Introduction

Mathematical models of epidemic spreading on networks have been of increasing
interest to the systems and control community [1, 2, 3, 4]. Since 2020, the
COVID-19 pandemic has given an extra impetus to such an interest [5, 6]. In
particular, the ongoing pandemic has highlighted the key role of human
behavioural response in shaping the course of an epidemic outbreak and how
such a response is deeply intertwined with the epidemic spreading process.
Some efforts have been made to incorporate human behaviour into epidemic
models [7, 8, 9, 10, 11, 12], in particular, by adding an alert state, in
which individuals take self-protective measures based on factors such as the
awareness of the infection prevalence [9, 12], communication with neighbours
[10], awareness campaigns [11, 12], or by incorporating opinion dynamics
mechanisms [13, 14]. While these models proved useful in capturing some key
aspects of real-world epidemics, their inherent oversimplification of the
evolving nature of human behaviour limits their practical applicability.

Recently, evolutionary game theory has emerged as a powerful framework to
develop realistic behavioural–epidemic models [15, 16, 17, 18, 19, 20]. In
[21], a novel game-theoretic paradigm was proposed, in which human decision
making and epidemics co-evolve on a two-layered network, with the decision
making influenced by a range of factors such as social influence,
interventions, risk perception, and immediate and accumulated costs of using
protection. However, except for the approximation of the epidemic threshold,
[21] relies only on numerical simulations, which suggest that the
behavioural–epidemic model can reproduce a wide range of behaviours, including
eradication of the disease, convergence to endemic equilibria, or periodic
oscillations and multiple epidemic waves.

In this letter, we expand on [21] to provide an analytical treatment of the
long-term behaviour of a game-theoretical behavioural–epidemic model. To this
aim, we propose a continuous-time implementation of the framework proposed in
[21], combined with a susceptible–infected–susceptible epidemic model. Through
a mean-field approach [22], we derive analytical results on the asymptotic
behaviour of the system. After having established the epidemic threshold, we
analyse the behaviour of the system below and above such a threshold. Below
the threshold, we prove global convergence to a disease-free equilibrium
(DFE). Above the threshold, we characterise the endemic equilibria (EEs) of
the system and their local stability properties. Furthermore, we derive
conditions under which the system undergoes periodic oscillations with
multiple waves, converging to a limit cycle. Finally, numerical simulations
suggest that the locally exponentially stable equilibria are also globally
stable, paving the way for future research towards extending our theoretical
findings.

##  II Model

_Notation:_ The set of real, real nonnegative, and strictly positive real
numbers is denoted by $\mathbb{R}$, $\mathbb{R}_{\geq 0}$, and
$\mathbb{R}_{>0}$, respectively. We say that an event $E$ is triggered by a
_Poisson clock_ with (possibly time-varying) rate $q_{E}(t)$, if $\lim_{\Delta
t\searrow 0}{\mathbb{P}\big{[}E\text{ occurs during }(t,t+\Delta
t)\big{]}}/{\Delta t}=q_{E}(t)$.

###  II-A Population and Network Model

We consider a population of $n$ individuals $\mathcal{V}=\\{1,\ldots,n\\}$.
Each individual $i\in\mathcal{V}$ is characterised by a two-dimensional state
$(x_{i}(t),y_{i}(t))$, reflecting their _behavioural state_
$x_{i}(t)\in\\{0,1\\}$ and _health state_ $y_{i}(t)\in\\{S,I\\}$, at time
$t\in\mathbb{R}_{\geq 0}$. In particular, an individual $i\in\mathcal{V}$
either chooses to use self-protective measures ($x_{i}(t)=1$) at time $t$,
thereby preventing any possible contraction of the disease, or to not employ
them ($x_{i}(t)=0$); simultaneously, the individual can have two different
health states: $y_{i}(t)=I$ if $i$ is infected, and $y_{i}(t)=S$ if $i$ is
healthy and susceptible to the infection.

Each individual is represented by a node in a two-layer temporal network
$\mathcal{G}(t)=(\mathcal{V},\mathcal{E}_{\text{I}},\mathcal{E}_{\text{C}}(t))$,
illustrated in Fig. 1a. The influence layer captures social influence on the
individual’s decision-making process through the (possibly directed) link set
$\mathcal{E}_{\text{I}}$, whereby node $j$ is an (out)-neighbour of $i$
($(i,j)\in\mathcal{E}_{\text{I}}$) if and only if (iff) $j$ can influence
$i$’s behaviour. The set of neighbours of $i$ is denoted by
$\mathcal{N}_{i}:=\\{j\in\mathcal{V}\ :\ (i,j)\in\mathcal{E}_{\text{I}}\\}$,
with size $d_{i}:=|\mathcal{N}_{i}|$. Since the spreading of a disease
typically evolves much faster than social ties do, we assume that the
influence layer is time-invariant.

Disease transmission from an infectious to a susceptible individual occurs
through interactions in close physical proximity, henceforth denoted by
_contacts_ , modelled by the contact layer $\mathcal{E}_{\text{C}}(t)$, where
$\\{i,j\\}\in\mathcal{E}_{\text{C}}(t)$ iff $i$ and $j$ have a contact at time
$t\in\mathbb{R}_{\geq 0}$. We assume that contacts are generated according to
a continuous-time activity-driven network [23], in which each individual
$i\in\mathcal{V}$ is assigned an _activity rate_ $a_{i}\in\mathbb{R}_{>0}$,
which captures the level of physical activity of individual $i$. Then, $i$
activates if triggered by a Poisson clock with rate $a_{i}$ and, once active,
generates a contact with another individual, selected uniformly at random from
$\mathcal{V}\setminus\\{i\\}$.

$\mathcal{E}_{\text{I}}$$\mathcal{E}_{\text{C}}(t)$

(a) Two-layer network

$I$$S$$q_{SI}$$\mu$$0$$1$$q_{01}$$q_{10}$ (b) Co-evolving dynamics

Figure 1: Illustration of the network model and dynamics.

###  II-B Behavioural–Epidemic Model

In the behavioural–epidemic framework proposed in [21], each individual
$i\in\mathcal{V}$ decides whether to adopt self-protective measures according
to an evolutionary game-theoretic mechanism [24], depending on social
influence, risk perception, costs for adopting self-protective measures,
frustration, and government policy interventions. Here, we propose a
simplified decision-making mechanism in which the last two factors are
omitted. Such a simplification allows the reduction of the number of
parameters involved in the system, simplifying its analysis and the
presentation of the results, without restricting the broad range of possible
emergent behaviours, as we shall demonstrate in this letter.

To capture these factors, we introduce the payoff function

$\displaystyle\pi_{1}^{(i)}(t)$
$\displaystyle=\displaystyle\frac{1}{d_{i}}\sum\nolimits_{j\in\mathcal{N}_{i}}x_{j}(t)+\zeta\bar{y}(t)\,,$
(1a) which captures the payoff for adopting self-protective measures
($x_{i}=1$), where $\bar{y}(t):=\tfrac{1}{n}\big{|}\\{i\in\mathcal{V}\ :\
y_{i}(t)=I\\}\big{|}$ denotes the infection prevalence at time $t$; and
$\displaystyle\pi_{0}^{(i)}(t)$
$\displaystyle=\displaystyle\frac{1}{d_{i}}\sum\nolimits_{j\in\mathcal{N}_{i}}\big{(}1-x_{j}(t)\big{)}+c\,,$
(1b)

which captures the payoff associated with not adopting self-protections. The
first term, present in both formulae, represents social influence: the more
neighbours of $i$ adopt a certain action, the higher the payoff for the
corresponding action. The term $\zeta\bar{y}(t)$, with
$\zeta\in\mathbb{R}_{\geq 0}$, increases the payoff for adopting self-
protections as the infection prevalence grows, capturing the risk perception.
Here, we assume that people react in a linear fashion in response to the
information they receive on the infection prevalence $\zeta\bar{y}(t)$, but
more complex and nonlinear terms may be considered. Finally, the constant
$c\in\mathbb{R}_{\geq 0}$ represents the psychological, social, and economical
cost per unit-time associated with the adoption of self-protections, thereby
increasing the payoff for not adopting self-protections.

Individuals change their behaviour following a stochastic implementation of
the classical imitation dynamics mechanism, which is often used in
evolutionary game theory [24, 25], in which they imitate their peers triggered
by Poisson clocks with rate equal to their corresponding payoff functions.
Specifically, an individual $i$ who is not adopting self-protective measures
at time $t$ (i.e. $x_{i}(t)=0$) will adopt them if triggered by a Poisson
clock with rate

$\displaystyle q_{01}^{(i)}(t)$
$\displaystyle=\frac{1}{d_{i}}\sum\nolimits_{j\in\mathcal{N}_{i}}x_{j}(t)\pi_{1}^{(j)}(t)\,,$
(2a) and an individual $i$ who is adopting them (i.e. $x_{i}(t)=1$) will stop
if triggered by a Poisson clock with rate $\displaystyle q_{10}^{(i)}(t)$
$\displaystyle=\frac{1}{d_{i}}\sum\nolimits_{j\in\mathcal{N}_{i}}\big{(}1-x_{j}(t)\big{)}\pi_{0}^{(j)}(t)\,.$
(2b)

Simultaneously, if a susceptible individual $i$ ($y_{i}(t)=S$) who does not
use protective measures ($x_{i}(t)=0$) has a physical encounter with an
infected individual $k$ ($y_{k}(t)=I$), then $i$ becomes infected with per-
contact infection probability $\lambda\in(0,1]$. We assume that self-
protective measures are $100\%$ effective in preventing contagion. Hence, if
individual $i$ employs protections at time $t$ ($x_{i}(t)=1$), then they
cannot be infected at time $t$. Following [26], we compute that if $i$ is
susceptible at time $t$ ($y_{i}(t)=S$), then $i$ will become infected if
triggered by a Poisson clock with rate

$q_{SI}^{(i)}(t)=\dfrac{\lambda(1-x_{i}(t))}{n-1}\bigg{(}na_{i}\bar{y}(t)+\sum_{j\in\mathcal{V}:y_{j}(t)=I}a_{j}\bigg{)}\,,$
(3)

where the first term in the parentheses accounts for the contact initiated by
$i$ with infected individuals, and the second accounts for contacts initiated
by infected individuals who interact with $i$. If the disease can be
transmitted only in one direction, then only the corresponding term should be
considered in Eq. (3). Note that if individual $i$ employs protection at time
$t$, then $q_{SI}^{(i)}(t)=0$. An infected individual $i$
$\left(y_{i}(t)=I\right)$ spontaneously recovers, if triggered by a Poisson
clock with node-independent and time-invariant rate $\mu\in\mathbb{R}_{>0}$.
All the state transitions and rates are shown in Fig. 1b.

##  III Mean-Field Dynamics

The evolution of the state of each individual $(x_{i}(t),y_{i}(t))$,
$i\in\mathcal{V}$, is determined by independent Poisson clocks. Hence, the
state of the system follows a Markov process on a state space with size
growing exponentially with the population size $n$, making its direct analysis
unfeasible. We employ a mean-field relaxation of the stochastic process to
derive analytical insight, following the $n$-intertwined mean-field approach
described in [22]. Specifically, we define and study for each individual
$i\in\mathcal{V}$ the probabilities of adopting protective behaviours
$p_{x}^{(i)}(t):=\mathbb{P}\left[x_{i}(t)=1\right]$ and of being infected
$p_{y}^{(i)}(t):=\mathbb{P}\left[y_{i}(t)=I\right]$, which evolve according to

$\displaystyle\dot{p}_{x}^{(i)}$
$\displaystyle=(1-p_{x}^{(i)})q_{01}^{(i)}-p_{x}^{(i)}q_{10}^{(i)}\,,$ (4a)
$\displaystyle\dot{p}_{y}^{(i)}$
$\displaystyle=(1-p_{y}^{(i)})q_{SI}^{(i)}-p_{y}^{(i)}\mu\,.$ (4b)

Also, we introduce the macroscopic variables

$x(t):=\dfrac{1}{n}\sum\nolimits_{i\in\mathcal{V}}p_{x}^{(i)}(t)\,,\quad
y(t):=\dfrac{1}{n}\sum\nolimits_{i\in\mathcal{V}}p_{y}^{(i)}(t)\,,$ (5)

which are the average probability that a randomly selected individual is
adopting protections and is infected at time $t$, respectively. Let
$\bar{x}(t):=\frac{1}{n}\sum_{i\in\mathcal{V}}x_{i}(t)$ denote the fraction of
adopters of self-protection in the population at time $t$. In the limit of
large-scale populations, $n\to\infty$, the central limit theorem ensures that
$\bar{x}(t)$ and $\bar{y}(t)$ converge to $x(t)$ and $y(t)$, respectively.
Hence, the macroscopic variables in Eq. (5) approximate with arbitrary
accuracy the fraction of adopters of self-protective measures and the epidemic
prevalence, for any finite-time horizon [27].

In the rest of this letter, we will make the following simplifying assumption.

######  Assumption 1.

We assume that a) the influence layer is complete, i.e.
$\mathcal{N}_{i}=\mathcal{V},\,\forall\,i\in\mathcal{V}$; b) individuals have
homogeneous activity, i.e. $a_{i}=\alpha\in\mathbb{R}_{>0}$,
$\forall\,i\in\mathcal{V}$; and c) individuals have the same initial
probability of adopting protections, i.e.
$p_{x}^{(i)}(0)=p_{x}(0)\in[0,1],\forall\,i\in\mathcal{V}$.

Under item a) of Assumption 1, Eq. (1) reduces to $\pi_{1}(t)=x(t)+\zeta y(t)$
and $\pi_{0}(t)=1-x(t)+c$, where we have dropped the index $i$ since the
payoffs are uniform across the population. Under Assumption 1, we derive a
planar system that governs the mean-field evolution of the macroscopic
variables (proof in Appendix -A) and rigorously analyse it.

######  Proposition 1.

In the limit of large-scale populations $n\to\infty$ and under Assumption 1,
the two macroscopic quantities in Eq. (5) evolve according to the following
planar system:

$\begin{array}[]{lll}\dot{x}&=&x(1-x)(2x+\zeta y-1-c)\,,\\\
\dot{y}&=&2\alpha\lambda y(1-x)(1-y)-\mu y\,.\end{array}$ (6)

The following result guarantees that Eq. (6) is always well-defined, i.e. that
the variables $x$ and $y$, which represent fractions of the population, remain
within $[0,1]\times[0,1]$.

######  Lemma 1.

The domain $[0,1]\times[0,1]$ is positively invariant for Eq. (6).

###### Proof.

The domain $[0,1]\times[0,1]$ is compact and convex and the vector field in
Eq. (6) is Lipschitz-continuous. Hence, Nagumo’s Theorem can be applied (see
[28]). We are left with checking the direction of the vector field at the
boundaries of the domain. We observe that $\dot{x}=0$ for $x=0$ and $x=1$,
while $\dot{y}=0$ for $y=0$ and $\dot{y}<0$ for $y=1$, implying that any
trajectory such that $(x(0),y(0))\in[0,1]\times[0,1]$ has
$(x(t),y(t))\in[0,1]\times[0,1]$ for any $t\geq 0$. ∎

In the following, we will make some realistic assumptions on the model
parameters. In particular, we want to guarantee that the use of self-
protective measures is always preferred when the entire population is
infected, while their use is disfavoured in the absence of a disease. To
guarantee this, we need to enforce in Eq. (1) that, for any $x\in[0,1]$,
$\pi_{1}(t)<\pi_{0}(t)$, if $y=0$, and $\pi_{0}(t)<\pi_{1}(t)$, if $y=1$.
These conditions are satisfied by making the following assumption.

######  Assumption 2.

We assume that $c>1$ and $\zeta>c+1$.

(a) $\zeta=5$

(b) $\zeta=8$

(c) $\zeta=9.5$

Figure 2: Simulations of Eq. (6) for different values of the risk perception
parameter (in the captions) for Example 1. Stable equilibria, saddle points
and unstable equilibria are marked with a black, black-white and white
asterisk, respectively.

##  IV Main Results

We study the asymptotic behaviour and the equilibria characteristics of the
behavioural–epidemic model using the mean-field system in Eq. (6). The
following lemma characterises the equilibria of the system in Eq. (6) and
their local stability properties. Its proof can be found in Appendix -B.

######  Lemma 2.

Under Assumption 2, Eq. (6) has at most five equilibria: three on the boundary
of $[0,1]\times[0,1]$, two in the interior. The three equilibria on the
boundary are:

  1. i)

the DFE $(0,0)$, which is locally asymptotically stable if
$\lambda\leq\tfrac{\mu}{2\alpha}$ (with exponential stability if strict
inequality holds), and a saddle point if $\lambda>\tfrac{\mu}{2\alpha}$;

  2. ii)

the DFE $(1,0)$, which is a saddle point;

  3. iii)

the protection-free EE $(0,1-\tfrac{\mu}{2\alpha\lambda})$, which exists iff
$\lambda>\tfrac{\mu}{2\alpha}$. When it exists, it is locally asymptotically
stable if $\zeta\leq\tfrac{2\alpha\lambda(1+c)}{2\alpha\lambda-\mu}$ (with
exponential stability if strict inequality holds) and a saddle point if
$\zeta>\frac{2\alpha\lambda(1+c)}{2\alpha\lambda-\mu}$.

Next, define

$\scalebox{0.89}{\mbox{$\displaystyle\beta_{\pm}:=\dfrac{1}{4}\left[c+3-\zeta\pm\sqrt{(c+3-\zeta)^{2}+8\left[\zeta\left(1-\tfrac{\mu}{2\alpha\lambda}\right)-1-c\right]}\right]$}}.$

The two EEs in the interior are:

  1. iv)

$\displaystyle\big{(}\beta_{+},1-\tfrac{\mu}{2\alpha\lambda(1-\beta_{+})}\big{)}$
, which exists iff  $\displaystyle\lambda>\tfrac{\mu}{2\alpha(1-\beta_{+})}$
and one of the following conditions is satisfied: a)  $\displaystyle
c-1+\tfrac{2\mu}{\alpha\lambda}+\sqrt{\tfrac{4\mu}{\alpha\lambda}\left(c-1+\tfrac{\mu}{\alpha\lambda}\right)}\leq\zeta<c+3$
, where necessarily  $\displaystyle c<\tfrac{4\alpha\lambda}{\mu}-3$ ; or
$\displaystyle\zeta\geq c+3$ and
$\displaystyle\zeta>\tfrac{2\alpha\lambda(1+c)}{2\alpha\lambda-\mu}$ . If it
exists, it is locally exponentially stable if
$\tfrac{4\alpha\lambda}{\zeta}(1-\beta_{+})^{2}<\mu<2(1-\beta_{+})(\alpha\lambda-\beta_{+})$,
and unstable if $\mu<\tfrac{4\alpha\lambda}{\zeta}(1-\beta_{+})^{2}$ or
$\mu>2(1-\beta_{+})(\alpha\lambda-\beta_{+})$.

  2. v)

$\displaystyle\big{(}\beta_{-},1-\tfrac{\mu}{2\alpha\lambda(1-\beta_{-})}\big{)}$
, which exists iff  $\displaystyle\lambda>\tfrac{\mu}{2\alpha(1-\beta_{-})}$
and  $\displaystyle
c-1+\tfrac{2\mu}{\alpha\lambda}+\sqrt{\tfrac{4\mu}{\alpha\lambda}\left(c-1+\tfrac{\mu}{\alpha\lambda}\right)}\leq\zeta<\min\left\\{c+3,\tfrac{2\alpha\lambda(c+1)}{2\alpha\lambda-\mu}\right\\}$
, where necessarily  $\displaystyle c<\tfrac{4\alpha\lambda}{\mu}-3$ . If it
exists, it is locally exponentially stable if
$\tfrac{4\alpha\lambda}{\zeta}(1-\beta_{-})^{2}<\mu<2(1-\beta_{-})(\alpha\lambda-\beta_{-})$,
and unstable if $\mu<\tfrac{4\alpha\lambda}{\zeta}(1-\beta_{-})^{2}$ or
$\mu>2(1-\beta_{-})(\alpha\lambda-\beta_{-})$.

Lemma 2 leads to the establishment of the epidemic threshold for the system in
Eq. (6), i.e. the conditions under which the system converges to one of the
DFEs.

######  Theorem 1.

Assume that Assumption 2 holds and $\lambda\leq\tfrac{\mu}{2\alpha}$. Then, if
$x(0)<1$, the system in Eq. (6) converges to the DFE $(0,0)$; otherwise, it
converges to the DFE $(1,0)$.

###### Proof.

Let $f(y)=2\alpha\lambda y(1-y)-\mu y$. By [2, Lemma 4.1], the solution of
$\dot{z}=f(z)$ converges to $z=0$ if $\lambda\leq\tfrac{\mu}{2\alpha}$. As a
consequence of the nonnegativity of $y(t)$ (Lemma 1) and the fact that
$\dot{y}\leq f(y)$, $y(t)$ converges to $0$. If $x(0)=1$, convergence to
$(1,0)$ is straightforward, since $x=1$ is an invariant manifold. Otherwise,
since $y(t)\to 0$ and $c>1$, there exists a time $\bar{t}>0$ such that
$y(t)\leq\frac{c-1}{2\zeta}$, for any $t\geq\bar{t}$. From Eq. (6), we observe
that for any  $\displaystyle t>\bar{t}$ ,
$\scalebox{1.0}{\mbox{$\displaystyle\dot{x}=x(1-x)(2x+\zeta
y-1-c)<-x(1-x)(c-1)$}}/2$, which yields the claim. ∎

Theorem 1 fully characterises the behaviour of the system when the disease is
not highly infectious, i.e. $\lambda\leq\tfrac{\mu}{2\alpha}$. In the
following, we will consider the opposite scenario
$\lambda>\tfrac{\mu}{2\alpha}$. We also assume
$c\geq\tfrac{4\alpha\lambda}{\mu}-3$, ensuring that the unique endemic
equilibrium only exists for a high enough level of risk perception $\zeta$.
Under an upper bound of $c$, which depends on the other model parameters,
Lemma 2 reduces to the following proposition, which clearly illustrates the
role of the risk perception $\zeta$. The proof is reported in Appendix -C.

######  Proposition 2.

Let $\lambda>\tfrac{\mu}{2\alpha}$ and $\tfrac{4\alpha\lambda}{\mu}-3\leq
c<\tfrac{32\alpha\lambda}{5\mu}-3$. Under Assumption 2, the following hold:

  1. i)

if $\zeta<\tfrac{2\alpha\lambda(1+c)}{2\alpha\lambda-\mu}$, then Eq. (6) has
three equilibria: the DFEs $(0,0)$ and $(1,0)$, which are saddle points, and
the (locally) exponentially stable EE $(0,1-\tfrac{\mu}{2\alpha\lambda})$;

  2. ii)

if $\zeta>\tfrac{2\alpha\lambda(1+c)}{2\alpha\lambda-\mu}$, then Eq. (6) has
four equilibria: three saddle points—the DFEs $(0,0)$, $(1,0)$, and the EE
$(0,1-\tfrac{\mu}{2\alpha\lambda})$—and the interior EE
$(\beta_{+},1-\tfrac{\mu}{2\alpha\lambda(1-\beta_{+})})$, which is (locally)
exponentially stable iff $\zeta$ satisfies all of the following three
conditions: a)
$\displaystyle\zeta>c-1+\tfrac{25\mu}{8\alpha\lambda}+\tfrac{5}{2}\sqrt{\tfrac{\mu}{\alpha\lambda}(c-1+\tfrac{25\mu}{16\alpha\lambda})}$
, b)
$\displaystyle\zeta>\tfrac{\alpha\lambda}{2\alpha\lambda-\mu}((c+1)[1-\sqrt{(\alpha\lambda-1)^{2}+2\mu}]+\alpha\lambda(c-3)+2\mu)$
, and c)
$\displaystyle\zeta<\tfrac{\alpha\lambda}{2\alpha\lambda-\mu}((c+1)\big{[}1+\sqrt{(\alpha\lambda-1)^{2}+2\mu}]+\alpha\lambda(c-3)+2\mu)$
.

Proposition 2 focuses on local stability and instability of endemic
equilibria. In the following result we establish sufficient conditions under
which sustained oscillations with periodic epidemic waves occur, with proof in
Appendix -D.

######  Theorem 2.

Let Assumption 2 hold, $\lambda>\tfrac{\mu}{2\alpha}$, and
$\tfrac{4\alpha\lambda}{\mu}-3\leq c<\tfrac{32\alpha\lambda}{5\mu}-3$.
Furthermore, assume
$\displaystyle\zeta>\tfrac{2\alpha\lambda(1+c)}{2\alpha\lambda-\mu}$ ,
$\displaystyle\zeta>c-1+\tfrac{25\mu}{8\alpha\lambda}+\tfrac{5}{2}\sqrt{\tfrac{\mu}{\alpha\lambda}(c-1+\tfrac{25\mu}{16\alpha\lambda})}$
, and
$\scalebox{0.92}{\mbox{$\displaystyle\zeta$}}>\tfrac{\alpha\lambda}{2\alpha\lambda-\mu}[(c+1)[1+\sqrt{(\alpha\lambda-1)^{2}+2\mu}]+\alpha\lambda(c-3)+2\mu]$.
If the initial condition $x(0),y(0)$ is in the interior of the domain
$[0,1]\times[0,1]$, then the system in Eq. (6) converges to a periodic
solution, within the domain $[0,1]\times[0,1-\frac{\mu}{2\lambda\alpha}]$.

We conclude the section by presenting an example.

######  Example 1.

Let $c=3$, $\alpha=3$, $\lambda=0.5$, and $\mu=1$. In Fig. 2a, we set the risk
perception to $\zeta=5$, which satisfies the conditions in item i) of
Proposition 2. Hence, the only (locally) stable equilibrium of Eq. (6) is the
EE $(0,1-\tfrac{\mu}{2\alpha\lambda})$. Simulations suggest that all
trajectories converge to it. In Fig. 2b, we increase the risk perception to
$\zeta=8$, which satisfies all the conditions in item ii) of Proposition 2.
Hence, the interior EE is locally stable: all trajectories converge to it,
suggesting globally stability. Finally, we set $\zeta=9.5$, which satisfies
the conditions of Theorem 2. Consistently, all the trajectories in Fig. 2c
converge to a limit cycle.

##  V Conclusion

We studied a behavioural–epidemic model in which human behaviour and epidemics
co-evolve in a mutually influencing manner. Employing a mean-field approach,
we painted an extensive picture of the system behaviour, including a stability
analysis of the equilibria and the expression of the epidemic threshold.
Furthermore, we explored the role of risk perception in the occurrence of
periodic oscillations and established conditions for global convergence to
such a periodic solution.

Our promising results pave the way for several avenues of future research.
First, the numerical findings suggest that our local stability results might
be extended towards obtaining global results. Second, interventions may be
incorporated, towards designing control policies to favour a collective
behavioural response and mitigate an epidemic outbreak.Third, our theoretical
analysis relies on the simplifying Assumption 1. Efforts should be placed
towards extending our theoretical findings to more general scenarios,
including non-trivial directed networks. Finally, further factors should be
incorporated into the model, including limited effectiveness of self-
protections, accumulation of socio-economic fatigue, and nonlinear terms to
capture more complex risk perception (as in [21]). This will be key for real-
world applications.

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###  -A Proof of Proposition 1

In the limit $n\to\infty$, Eq. (4a) reduces to
$\dot{p}_{x}^{(i)}=(1-p_{x}^{(i)})x(x+\zeta y)-p_{x}^{(i)}(1-x)(1-x+c)$.
Similarly, using item b) of Assumption 1 and Eq. (3), we write Eq. (4b) as
$\dot{p}_{y}^{(i)}=\lambda(1-p_{x}^{(i)})(1-p_{y}^{(i)})2\alpha\frac{n}{n-1}y-\mu
p_{y}^{(i)}$. Next, observe that, under item c) of Assumption 1,
$p_{x}^{(i)}(t)$ is the same $\forall\,i\in\mathcal{V}$, so we drop the index
$i$ and write $p_{x}^{(i)}(t)=\bar{x}(t)=x(t)$, where the last equality holds
for $n\to\infty$. Finally, we combine the expressions above and Eq. (5), to
derive Eq. (6).

###  -B Proof of Lemma 2

Solving $\dot{y}=0$ yields $y=0$ or $(1-x)(1-y)=\tfrac{\mu}{2\alpha\lambda}$,
where for the latter, $\tfrac{\mu}{2\alpha\lambda}>0$ necessarily requires
$x<1$ and $y<1$ as conditions. Here, $(1-x)(1-y)=\tfrac{\mu}{2\alpha\lambda}$
with $x<1$ and $y<1$ can be written as $y=1-\tfrac{\mu}{2\alpha\lambda(1-x)}$.
If $y=0$, then the solutions to $0=\dot{x}=x(1-x)(2x-1-c)$ in the domain
$[0,1]$ are given by $x=0$ and $x=1$. Thus, the only DFEs are $(0,0)$ and
$(1,0)$. Next, let us consider equilibria $(x,y)$ with
$y=1-\tfrac{\mu}{2\alpha\lambda(1-x)}\in(0,1)$, $x\in[0,1)$, and let
$\mu<2\alpha\lambda(1-x)$ for existence of the equilibria. Substituting
$y=1-\tfrac{\mu}{2\alpha\lambda(1-x)}$ in $\dot{x}=0$ yields
$x(1-x)(2x+\zeta(1-\tfrac{\mu}{2\alpha\lambda(1-x)})-1-c)=0$, which for $x<1$
has solutions $x=0$, and
$\displaystyle\beta_{\pm}=\tfrac{1}{4}(c+3-\zeta\pm\sqrt{(c+3-\zeta)^{2}+8[\zeta(1-\tfrac{\mu}{2\alpha\lambda})-1-c]})$
. The EEs are given by $(0,1-\tfrac{\mu}{2\alpha\lambda})$ and
$(\beta_{\pm},1-\tfrac{\mu}{2\alpha\lambda(1-\beta_{\pm})})$. First, we
investigate for which values of $\zeta$ we have
$(c+3-\zeta)^{2}+8[\zeta\left(1-\tfrac{\mu}{2\alpha\lambda}\right)-1-c]=\zeta^{2}+2\zeta(1-c-\tfrac{2\lambda}{\alpha\lambda})+(c-1)^{2}\geq
0$, which is a necessary condition for $\beta_{\pm}\in\mathbb{R}$. Note that
the roots of
$\displaystyle\zeta^{2}+2\zeta(1-c-\tfrac{2\lambda}{\alpha\lambda})+(c-1)^{2}$
are
$\zeta_{\pm}=c-1+\tfrac{2\mu}{\alpha\lambda}\pm\sqrt{\tfrac{4\mu}{\alpha\lambda}(c-1+\tfrac{\mu}{\alpha\lambda})}$.
Since $\zeta_{-}<c-1$, $\beta_{\pm}\in\mathbb{R}$ iff $\zeta\geq\zeta_{+}$.
Next, we study for which values of $\zeta$ and $c$ we have
$\beta_{\pm}\in(0,1)$, while assuming that the previously identified
conditions necessary for $\beta_{\pm}\in\mathbb{R}$ hold. We start with
$\beta_{+}$. If $\zeta<c+3$, then $\beta_{+}>0$. If $\zeta\geq c+3$, then
$\beta_{+}>0$ iff $\zeta>\tfrac{2\alpha\lambda(1+c)}{2\alpha\lambda-\mu}$, for
which $\zeta(1-\tfrac{\mu}{2\alpha\lambda})-1-c>0$. Observe that $\beta_{+}<1$
iff
$\sqrt{(c+3-\zeta)^{2}+8[\zeta(1-\tfrac{\mu}{2\alpha\lambda})-1-c]}<\zeta+1-c$,
which is always satisfied. Hence, $\beta_{+}\in(0,1)$ if $\zeta\geq c+3$ and
$\zeta>\tfrac{2\alpha\lambda(1+c)}{2\alpha\lambda-\mu}$, or if
$\displaystyle\zeta<c+3$ . Combining this with the conditions for
$\beta_{+}\in\mathbb{R}$, yields the regions of $\zeta$ for which
$\beta_{+}\in(0,1)$. Note here that if
$\zeta>\tfrac{2\alpha\lambda(1+c)}{2\alpha\lambda-\mu}$, then
$\beta_{+}\in\mathbb{R}$. The condition on $c$ in a) results from the fact
that, for the region to exist, the lower bound must have a lower value than
the upper bound. Likewise, we consider $\beta_{-}$. Observe that $\beta_{-}>0$
iff $\zeta<\min\\{c+3,\tfrac{2\alpha\lambda(c+1)}{2\alpha\lambda-\mu}\\}$ and
$\beta_{-}<1$ iff
$\sqrt{(c+3-\zeta)^{2}+8[\zeta(1-\tfrac{\mu}{2\alpha\lambda})-1-c]}>c-\zeta-1$,
which is satisfied. Combining the above with the conditions ensuring that
$\beta_{-}\in\mathbb{R}$, gives the regions of $\zeta$ for which
$\beta_{-}\in(0,1)$.

Then, we study local stability. First, we consider the DFE $(0,0)$.
Linearising Eq. (6) about $(0,0)$, we find that the eigenvalues of the
Jacobian matrix are $2\alpha\lambda-\mu$ and $-(c+1)<0$, where the former is
negative iff $2\alpha\lambda<\mu$. Thus, the equilibrium is locally
exponentially stable (LES) if $2\alpha\lambda<\mu$, and a saddle point if
$2\alpha\lambda>\mu$. The case $2\alpha\lambda=\mu$ is studied separately,
yielding asymptotic stability. Computations are omitted due to space
constraints. Next, we consider the DFE $(1,0)$. In a similar way, we linearize
the system about the equilibrium and we find the eigenvalues of the Jacobian
matrix, which are given by $c-1>0$ and $-\mu<0$, implying that the DFE $(1,0)$
is a saddle point. Now consider the EE $(0,1-\tfrac{\mu}{2\alpha\lambda})$,
with $\mu<2\alpha\lambda$, so it exists. Linearising the system about this
equilibrium, we obtain a Jacobian matrix with eigenvalues
$\zeta(1-\tfrac{\mu}{2\alpha\lambda})-1-c$ and $\mu-2\alpha\lambda<0$, so the
equilibrium is LES if $\zeta<\tfrac{2\alpha\lambda(1+c)}{2\alpha\lambda-\mu}$,
and a saddle point if the opposite inequality holds. If the equality holds,
Eq. (6) is studied directly (computations omitted), yielding asymptotic
stability. Finally, consider the interior EE with $x=\beta_{+}$. Let
$\beta_{+}\in(0,1)$ and $\mu<2\alpha\lambda(1-\beta_{+})$ for existence.
Linearising Eq. (6) about the EE yields the Jacobian matrix

$A=\begin{bmatrix}2\beta_{+}(1-\beta_{+})&\zeta\beta_{+}(1-\beta_{+})\\\
-\tfrac{\mu}{1-\beta_{+}}(1-\tfrac{\mu}{2\alpha\lambda(1-\beta_{+})})&\mu-2\alpha\lambda(1-\beta_{+})\end{bmatrix}.$

By the determinant-trace method, the EE with $x=\beta_{+}$ is LES if
$\tfrac{4\alpha\lambda}{\zeta}(1-\beta_{+})^{2}<\mu<2(1-\beta_{+})(\alpha\lambda-\beta_{+})$
and unstable if at least one of the opposite inequalities holds. By replacing
$\beta_{+}$ with $\beta_{-}$ in the argument above, we complete the proof by
obtaining the conditions for the other interior EE.

###  -C Proof of Proposition 2

Item i) follows from Lemma 2. We now prove ii). Under the parameter
constraints imposed by the hypothesis of the proposition, it follows from
Lemma 2 that Eq. (6) has three equilibria on the boundary: $(0,0)$, $(1,0)$
and $(0,1-\tfrac{\mu}{2\alpha\lambda})$, all of which are saddle points.
Moreover, the interior EE with $x=\beta_{-}$ does not exist. For existence of
the interior EE with $x=\beta_{+}$, we need
$\lambda>\tfrac{\mu}{2\alpha(1-\beta_{+})}$, which is equivalent to
$\displaystyle\sqrt{(c+3-\zeta)^{2}+8[\zeta(1-\tfrac{\mu}{2\alpha\lambda})-1-c]}<\zeta+1-c-\tfrac{2\mu}{\alpha\lambda}$
, where the right-hand side (RHS) is positive for
$\lambda>\tfrac{\mu}{2\alpha}$ and $\zeta>c+3$. Note that
$c+3\leq\tfrac{2\alpha\lambda(1+c)}{2\alpha\lambda-\mu}$ iff
$c\geq\tfrac{4\alpha\lambda}{\mu}-3$. Squaring both sides, algebraic
simplifications yield the equivalent expression
$\displaystyle\tfrac{\mu}{\alpha\lambda}+c-1>0$ , which holds.

The interior EE is LES iff
$\tfrac{4\alpha\lambda}{\zeta}(1-\beta_{+})^{2}<\mu<2(1-\beta_{+})(\alpha\lambda-\beta_{+})$
(Lemma 2). The condition $\mu<2(1-\beta_{+})(\alpha\lambda-\beta_{+})$ is
equivalent to
$\scalebox{0.85}{\mbox{$\displaystyle\tfrac{1}{2}[\alpha\lambda+1+\tfrac{1}{2}(\zeta-c-3)]\sqrt{(c+3-\zeta)^{2}+8\left[\zeta\left(1-\tfrac{\mu}{2\alpha\lambda}\right)-1-c\right]}$}}<$
$\displaystyle
2\alpha\lambda-\mu+\tfrac{1}{4}(\zeta-c-3)^{2}+\tfrac{1}{2}(\zeta-c-3)(\alpha\lambda+1)+[\zeta(1-\tfrac{\mu}{2\alpha\lambda})-1$
$\displaystyle-c].$ For $\lambda>\tfrac{\mu}{2\alpha}$ and
$\zeta>\max\\{c+3,\tfrac{2\alpha\lambda(1+c)}{2\alpha\lambda-\mu}\\}$, both
the left-hand side (LHS) and RHS of the equation above are positive. After
squaring both sides and some straightforward rewriting, we obtain the
equivalent condition
$(2\alpha\lambda-\mu)\zeta^{2}-2(\alpha\lambda)^{2}[c-3+\tfrac{1}{\alpha\lambda}(c+1+2\mu)]\zeta+2(\alpha\lambda)^{2}(c^{2}+2\alpha\lambda(1-c)-2\mu-1)<0$,
where the roots of the polynomial are given by
$\displaystyle\bar{\zeta}_{\pm}=\tfrac{\alpha\lambda}{2\alpha\lambda-\mu}((c+1)[1\pm\sqrt{(\alpha\lambda-1)^{2}+2\mu}]+\alpha\lambda(c-3)+2\mu)$
. The leading term of the polynomial on the LHS is positive, so the condition
is satisfied iff $\bar{\zeta}_{-}<\zeta<\bar{\zeta}_{+}$.

Next, $\mu>\tfrac{4\alpha\lambda}{\zeta}(1-\beta_{+})^{2}$ is equivalent to
$\displaystyle(\zeta+1-c)\sqrt{(c+3-\zeta)^{2}+8[\zeta(1-\tfrac{\mu}{2\alpha\lambda})-1-c]}>\zeta^{2}+(c-1)^{2}$
$\scalebox{0.85}{\mbox{$\displaystyle+2\zeta(1-c-\tfrac{5\mu}{4\alpha\lambda})$}},$
where the roots of the RHS are
$\tilde{\zeta}_{\pm}=c-1+\tfrac{5\mu}{4\alpha\lambda}\pm\sqrt{\tfrac{5\mu}{2\alpha\lambda}(c-1+\tfrac{5\mu}{8\alpha\lambda})}$.
For $c<\tfrac{32\alpha\lambda}{5\mu}-3$, we have $c+3>\tilde{\zeta}_{+}$, so
the RHS is positive. Squaring both sides and rewriting yields the equivalent
expression
$-\zeta^{3}+[2(c-1)+\tfrac{25\mu}{4\alpha\lambda}]\zeta^{2}-(c-1)^{2}\zeta<0$,
where the roots of the LHS are $\zeta=0$ and
$\breve{\zeta}_{\pm}=c-1+\tfrac{25\mu}{8\alpha\lambda}\pm\frac{5}{2}\sqrt{\tfrac{\mu}{\alpha\lambda}(c-1+\tfrac{25\mu}{16\alpha\lambda})}.$
Since $c>1$ implies $\breve{\zeta}_{-}<c-1$, we need $\zeta>\breve{\zeta}_{+}$
for stability.

###  -D Proof of Theorem 2

Under the conditions of Theorem 2, Eq. (6) has three saddle points on the
boundary: $(0,0)$, $(1,0)$ and $(0,1-\tfrac{\mu}{2\alpha\lambda})$, and the
unique fully unstable interior EE
$(\beta_{+},1-\tfrac{\mu}{2\alpha\lambda(1-\beta_{+})})$. Also, $\dot{y}<0$ if
$y\neq 0$ and $\tfrac{\mu}{2\alpha\lambda}>(1-x)(1-y)$. For all
$y>1-\tfrac{\mu}{2\alpha\lambda}$, we have
$\tfrac{\mu}{2\alpha\lambda}>1-y\geq(1-x)(1-y)$, so $\dot{y}<0$. Furthermore,
$\dot{y}=-\mu xy\leq 0$ at $y=1-\tfrac{\mu}{2\alpha\lambda}$, which implies
that the region $[0,1]\times[0,1-\tfrac{\mu}{2\alpha\lambda}]$ is attractive
and invariant.

We study now the behaviour of the system near the boundaries of
$[0,1]\times[0,1-\tfrac{\mu}{2\alpha\lambda}]$, and examine whether it is
possible to reach the boundary of the domain. Consider the boundary $x=1$. Let
us assume that the trajectory reaches $x=1-\varepsilon$, with $\varepsilon>0$
arbitrarily small at time $t_{0}$. From Eq. (6), we observe that
$\dot{x}(t_{0})=\varepsilon(1-\varepsilon)(1+\zeta y-c-2\varepsilon)$ when
$x(t_{0})=1-\varepsilon$. Hence, $\dot{x}$ can be positive only if
$y>\tfrac{c-1}{\zeta}$. We consider the regions
$\mathcal{R}_{\varepsilon}:=[1-\varepsilon,1]\times[\tfrac{c-1}{\zeta},1-\tfrac{\mu}{2\alpha\lambda}]$
and
$\displaystyle\mathcal{S}_{\varepsilon}:=[1-\varepsilon,1]\times[0,\tfrac{c-1}{\zeta}]$
, where $\dot{x}$ can be positive only in $\mathcal{R}_{\varepsilon}$, while
it is negative in $\mathcal{S}_{\varepsilon}$. Furthermore, we derive the
following uniform bound for any pair $(x,y)\in\mathcal{R}_{\varepsilon}$:
$\dot{y}=-\mu y+2\alpha\lambda y\varepsilon(1-y)<-\mu\tfrac{c-1}{\zeta}<0$,
for $\varepsilon$ sufficiently small. Similarly, we observe that
$\dot{x}=x(1-x)(2x+\zeta y-1-c)\leq k(1-x)$, for some constant $k>0$. These
bounds yield a strict bound on the distance between the trajectory and $x=1$
before the trajectory exits the region $\mathcal{R}_{\varepsilon}$ from the
bottom and enters $\mathcal{S}_{\varepsilon}$. We define $u(t)=1-x(t)$. The
uniform bound on $\dot{x}$ in $\mathcal{R}_{\varepsilon}$ is equivalent to
$-\dot{u}\leq-k(-u)$, so by the Gronwall-Bellman inequality [29],
$-u(t)\leq-u(t_{0})e^{-k(t-t_{0})}$, which is equivalent to $x(t)\leq
1-\varepsilon e^{-k(t-t_{0})}$, for any $t\geq t_{0}$. Here, we used the fact
that $x(t_{0})=1-\varepsilon$. Then, the uniform bound on $\dot{y}$ in
$\mathcal{R}_{\varepsilon}$ is used to derive a bound on the time needed for
the trajectory to exit $\mathcal{R}_{\varepsilon}$. Specifically, since the
length along the $y$-axis of $\mathcal{R}_{\varepsilon}$ is equal to
$(1-\tfrac{\mu}{2\alpha\lambda}-\tfrac{c-1}{\zeta})$, and the time-derivative
of the trajectory along the $y$-component is negative and greater in modulus
than $\mu\tfrac{c-1}{\zeta}$, then there necessarily exists a time
$\tilde{t}\leq\zeta(1-\tfrac{\mu}{2\alpha\lambda}-\tfrac{c-1}{\zeta})/(\mu(c-1))$
such that $y(t_{0}+\tilde{t})<\tfrac{c-1}{\zeta}$ and $x(t_{0}+\tilde{t})\leq
1-\varepsilon^{\prime}$, with $\varepsilon^{\prime}=\varepsilon
e^{-k\tilde{t}}$. Hence, the trajectory will exit from
$\mathcal{R}_{\varepsilon}$ and will enter $\mathcal{S}_{\varepsilon}$, in
which $\dot{x}<0$ and $\dot{y}<0$. This establishes that the trajectory cannot
further approach the boundary $x=1$, nor re-enter $\mathcal{R}_{\varepsilon}$
from the boundary between $\mathcal{S}_{\varepsilon}$ and
$\mathcal{R}_{\varepsilon}$. Thus, there exists a constant
$\varepsilon^{\prime}>0$ such that
$[0,1-\varepsilon^{\prime}]\times[0,1-\tfrac{\mu}{2\alpha\lambda}]$ is
positively invariant for Eq. (6).

A similar argument guarantees that any trajectory that starts from the
interior is bounded away from the boundaries $y=0$ and $x=0$. Since
convergence to the boundaries is impossible, the boundary equilibria points
cannot be reached if the initial conditions of the system are in the interior
of the domain $(0,1)\times(0,1)$. Finally, we consider the open set
$(x,y)\in(0,1)\times(0,1-\tfrac{\mu}{2\alpha\lambda})$. Since the unique
interior EE is an unstable point under the above conditions, there does not
exist a homoclinic orbit. It follows directly from the generalised Poincaré-
Bendixson theorem [30] that every non-empty compact $\omega$-limit set of an
orbit is periodic.


#  Learning Effective NeRFs and SDFs Representations with
3D Generative Adversarial Networks for 3D Object Generation:
Technical Report for ICCV 2023 OmniObject3D Challenge

Zheyuan Yang, Yibo Liu, Guile Wu, Tongtong Cao, Yuan Ren, Yang Liu, Bingbing
Liu
Huawei Noah’s Ark Lab
{andrew.yang1, yibo.liu3, guile.wu, caotongtong, yuan.ren3, yang.liu9,
<EMAIL_ADDRESS>
###### Abstract

In this technical report, we present a solution for 3D object generation of
ICCV 2023 OmniObject3D Challenge. In recent years, 3D object generation has
made great process and achieved promising results, but it remains a
challenging task due to the difficulty of generating complex, textured and
high-fidelity results. To resolve this problem, we study learning effective
NeRFs and SDFs representations with 3D Generative Adversarial Networks (GANs)
for 3D object generation. Specifically, inspired by recent works, we use the
efficient geometry-aware 3D GANs as the backbone incorporating with label
embedding and color mapping, which enables to train the model on different
taxonomies simultaneously. Then, through a decoder, we aggregate the resulting
features to generate Neural Radiance Fields (NeRFs) based representations for
rendering high-fidelity synthetic images. Meanwhile, we optimize Signed
Distance Functions (SDFs) to effectively represent objects with 3D meshes.
Besides, we observe that this model can be effectively trained with only a few
images of each object from a variety of classes, instead of using a great
number of images per object or training one model per class. With this
pipeline, we can optimize an effective model for 3D object generation. This
solution is one of the final top-3-place solutions in the ICCV 2023
OmniObject3D Challenge.

Figure 1: An overview of the framework of our solution for 3D object
generation.

##  1 Introduction

3D object generation aims at generating meaningful 3D surfaces and synthesis
images of 3D objects given random inputs [23, 7]. Inspired by the great
success of 2D object generation [9, 4], there have been many studies [12, 5]
extending 2D methods to 3D generation. Recent 3D generation approaches focus
more on effective 3D representation learning [18, 15] and advanced generation
strategy [10, 2, 21]. Despite the great progress and promising results in
recent years, 3D object generation is still a challenging task due to the
difficulty of generating complex, textured and high-fidelity results.

To devise a solution for 3D object generation in the ICCV 2023 OmniObject3D
Challenge, we simultaneously consider effective 3D representation learning and
advanced generation strategy. In terms of 3D representation learning, textured
mesh [3] and Neural Radiance Fields (NeRFs) [14] embrace the great potential
to effectively encode 3D object features. In particular, textured mesh
representations can support individual texture map generation, which allows us
to easily replace surface textures, while NeRFs-based representations excel at
rendering high-fidelity images. Furthermore, when considering effective shape
representation, there are a variety of options, ranging from explicit
representations like point-based [20], mesh-based [1] and voxel-based [22]
approaches to implicit ones such as the Signed Distance Functions (SDFs) [17,
11] and occupancy function [13]. In the light of these cutting-edge
techniques, we mainly focus on exploring effective NeRFs representations for
3D object generation while still optimizing SDFs representations to
effectively represent objects with 3D meshes.

Moving on to advanced generation strategy, there are usually different
options, such as latent generation [21] and direct generation [16]. In
particular, latent generation methods aim to generate the representation of
the desired output in the latent space and decode the latent code back to the
3D space, while direct generation focus on directly generating the desired 3D
representation without utilizing the latent space. Besides, from the
perspective of generation process, generation strategies can be categorized
into Generative Adversarial Networks (GANs) [10, 2], Variational AutoEncoder
(VAE) [19], flow-based [6], Diffusion Models (DMs) [21], _etc_. In our
solution, we employ the efficient geometry-aware 3D GANs [2] as the backbone
for 3D object generation.

The overall framework of our solution is depicted in Fig. 1. Specifically, we
aim to learn effective NeRFs and SDFs representations with 3D Generative
Adversarial Networks (GANs) for 3D object generation. Inspired by the recent
success of efficient geometry-aware 3D GANs, _i.e_., EG3D [2, 18], we employ
it as the backbone and incorporate label embedding and color mapping [18] to
enable model training on different taxonomies simultaneously. Next, we
generate NeRFs-based representations for rendering high-fidelity synthetic
images and SDFs to effectively represent objects with 3D meshes through a
decoder. In addition, we empirically find that we can effectively train this
model with only a few images per object from various classes, rather than
training with a great number of images per object or optimizing one model per
class. With this pipeline, we can optimize an effective model for 3D object
generation. This solution is one of the final top-3-place solutions in the
ICCV 2023 OmniObject3D Challenge.

##  2 Technical Solution

As aforementioned, to realize 3D object generation for ICCV 2023 OmniObject3D
Challenge, we explore 3D GANs to learn effective NeRFs representations for
synthetic images and SDFs representations for 3D shapes.

### Motivation of Using GANs instead of DMs.

Although diffusion models have recently gained an increasing popularity in 3D
object generation, most existing works [16, 21] have only been verified to
work under specific category conditions (e.g., chairs [16] or avatars [21]).
Besides, since the OmniObject3D dataset [23] is a collection of objects from
hundreds of taxonomies, following the pipeline of existing DMs would
necessitate the use of hundreds of pre-trained models, which is impractical.
Hence, we employ a method that enables simultaneous training on all the
taxonomies in the OmniObject3D dataset. Moreover, given the success of
StyleGAN2 [10] in representing NeRFs in latent space, we have chosen latent
generation for our pipeline.

### Solution Pipeline.

The framework of our solution is depicted in Fig. 1. Overall, our solution
adopts a GAN-based generation process. Specifically, with EG3D [2] as the
backbone, firstly, we generate a one-dimensional latent code, $\mathbf{z}$
($1\times 512$), which is obtained by randomly sampling 512 values from a
standard normal distribution. Meanwhile, we randomly generate an integer,
indicating the object label, within the range of the total category number of
OmniObject3D. The integer is then transformed into a label embedding by an
embedding layer. This enables the model to be trained on different taxonomies
simultaneously. Next, we forward the label embedding and latent code
$\mathbf{z}$ into a mapping network and generate an intermediate latent code
$\mathbf{w}$ ($1\times 15\times 512$) with promoted dimensions. Here, the
first 14 dimensions of $\mathbf{w}$ ($\mathbf{w}_{1}$ in the figure) are used
as the input to the object generator ($\mathbf{G}_{1}$ in the figure) while
the last dimension of $\mathbf{w}$ ($\mathbf{w}_{2}$ in the figure) are
employed as the input to the color generator ($\mathbf{G}_{2}$ in the figure).
$\mathbf{G}_{1}$ and $\mathbf{G}_{2}$ generate the tri-plane representation
($3\times 32\times 64\times 64$) and the color tensor ($3\times 10$),
respectively. Following [18], we use the color generator which allows
convenient texture conversion. After that, we determine queries, $\mathbf{Q}$,
based on the randomly generated camera pose. The tri-plane representation is
then taken as keys, $\mathbf{K}$, and transmitted into the tri-plane decoder
together with $\mathbf{Q}$. Then, we use a decoder to generate the SDF of the
object, which can be sampled to create a 3D mesh. This decoder also produces a
feature as the input to a view direction mapper, from which the output is
multiplied by the color tensor, $\mathbf{V}$ (the values), in the attention
module. Finally, the attention module produces the 2D synthesis image and a
discriminator [2] which only functions during training is used for adversarial
training.

### Model Training.

Since we are simultaneous optimizing NeRFs and SDFs representations, we ensure
the generator can initially generate a unit sphere by initializing our model
with SDF-pretraining as [18]. Then, the generator is optimized using the Adam
optimizer with a learning rate of 0.0025, while the discriminator [2] uses a
learning rate of 0.002. The batch size is set to 32 and the model is trained
for 300,000 iterations. Additionally, we also use adaptive discriminator
augmentation [8] to improve model generalization. Overall, the model training
objective is formulated as:

$\alpha_{0}\mathcal{L}_{path}+\alpha_{1}\mathcal{L}_{e}+\alpha_{2}\mathcal{L}_{v}+\alpha_{3}\mathcal{L}_{sdf},$
(1)

where $\mathcal{L}_{path}$ denotes the Path length regulation loss [10],
$\mathcal{L}_{e}$ and $\mathcal{L}_{v}$ denote the entropy loss and total
variation loss [10], respectively, $\mathcal{L}_{sdf}$ represents the SDF
eikonal loss [18], and $\alpha_{i}$ are weighting parameters. We set
$\alpha_{0}{=}2$, $\alpha_{1}{=}0.05$, $\alpha_{2}{=}0.5$ and
$\alpha_{3}{=}0.1$.

Figure 2: Visualization results of the synthesis images rendered from the
NeRFs generated by our solution.

##  3 Experiments

### Dataset.

We train our model from scratch for 3D object generation on the training
dataset provided by the ICCV 2023 OmniObject3D Challenge. The complete
OmniObject3D dataset [23] contains 216 classes from a wide range of daily
categories and approximately 6k objects in total where each object has 100
images in different views.

### Efficient Model Training.

Instead of training one model for each class, we train a single model with
various types of objects. The model takes each image along with its camera
pose, focal length and object type. Besides, we empirically find that we can
effectively train this model with only a few images per object from various
classes, so we select the first 8 images of each object for model training.
This significantly improves the model training efficiency.

Figure 3: Visualization results of the synthesis images rendered from the NeRFs generated by our solution without additional label embedding. Method | FID
---|---
$\Delta$= Ours w/ label embedding - | -4.8
Ours w/o label embedding
Table 1: Improvement in terms of FID of our solution after using the
additional label embedding.

### Results.

Fig. 2 shows some visualization results of synthesis images generated by our
solution on OmniObject3D. It can be seen that our approach is capable of
generating high-fidelity images from different views for 3D object generation.
Besides, Fig. 3 shows results of our solution without the additional label
embedding. Comparing Fig. 2 with Fig. 3, we can see that without the
additional label embedding, the quality of the synthesis images is
significantly worse than the original solution. Furthermore, Table. 1 shows
that using the additional label embedding can improve the solution by 4.86 in
terms of FID. This verifies that the additional label embedding is beneficial
for improving model generalization.

##  4 Conclusion

In this technical report, we present a solution for 3D object detection of
ICCV 2023 OmniObject3D Challenge. The key insight of our solution is to learn
effective NeRFs and SDFs Representations with 3D generative adversarial
networks for 3D object generation. Overall, this solution is one of the final
top-3-place solutions in the ICCV 2023 OmniObject3D Challenge.

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