# The influence of active agent motility on SIRS epidemiological dynamics

R. Kailasham  Department of Chemical Engineering, Carnegie Mellon University,
Pittsburgh, PA 15213, USA  Aditya S. Khair<EMAIL_ADDRESS>Department
of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA

###### Abstract

Active Brownian disks moving in two dimensions that exchange information about
their internal state stochastically are chosen to model epidemic spread in a
self-propelled population of agents under the susceptible-infected-recovered-
susceptible (SIRS) framework. The state of infection of an agent, or disk,
governs its self-propulsion speed; consequently, the activity of the agents in
the system varies in time. Two different protocols (one-to-one and one-to-
many) are considered for the transmission of disease from the infected to
susceptible populations. The effectiveness of the two protocols are
practically identical at high values of the infection transmission rate. The
one-to-many protocol, however, outperforms the one-to-one protocol at lower
values of the infection transmission rate. Salient features of the macroscopic
SIRS model are revisited, and compared to predictions from the agent-based
model. Lastly, the motility induced phase separation in a population of such
agents with a fluctuating fraction of active disks is found to be well-
described by theories governing phase separation in a mixture of active and
passive particles with a constant fraction of passive disks.

##  I  Introduction

Epidemiological models have been in use for more than a century to track the
spread of infectious diseases [1]. These models assume that the population is
compartmentalized into three populations: susceptible (S) individuals, who are
prone to the disease, infected (I) individuals who possess the disease and are
capable of spreading it, and individuals who have recovered (R) from the
disease and may or may not become prone to further infection [2]. The rates of
interconversion between the various populations are specified by
epidemiological constants, and the time-evolution of these populations is
governed by coupled ordinary differential equations (ODEs) [2]. Several
modifications to this susceptible-infected-recovered (SIR) model have been
made, to account for policy interventions such as vaccination drives, social
distancing protocols, and lockdown mandates [3]. SIR models and its variants
have been used to describe the spread of several diseases [4], such as: the
plague [1], COVID-19 [5, 6], varicella [7] and influenza [8]. These
macroscopic models assume a uniform, well-mixed population, such that the
epidemiological constants do not depend on the (heterogeneous) spatial density
of the various populations. A fine-grained, discrete treatment of epidemic
spread is agent-based modeling [9, 10, 11, 12] (ABM), which considers each
individual in the population as an autonomous entity, with the disease
transmission occurring if a susceptible individual comes within a contagion
radius of the infected individual. More recently, SIR modeling has been
coupled [13, 14, 15] with advances in the understanding of self-propelled or
active matter. Self-propelled entities such as light-activated Janus particles
undergo a transition from a homogeneous state to a clustered one, at
sufficiently high enough values of packing fraction and self-propulsion speed
[16, 17]. This experimentally observed phenomenon of motility induced phase
separation (MIPS) can be recapitulated in numerical simulations in which the
active particles translate with a constant self-propulsion speed, interact
sterically, and whose orientations evolve diffusively in time [18, 19]. It is
therefore possible, using such systems, to examine disease propagation in both
homogeneous and spatially heterogeneous configurations, by attaching to the
individual active agents an internal state (S, I or R). If the internal state
of the particle is coupled to its dynamics, such that the self-propulsion
speed of infected particles is different from that of susceptible or recovered
particles, then one would expect that MIPS in such a system would be affected.
We address this question in the present work, and also aim to draw a
connection between the agent-based, or microscopic, and macroscopic
descriptions of epidemic spread.

Norambuena  _et al._ [15] simulated a collection of active Brownian particles
as self-propelled agents moving in two dimensions in which a susceptible
particle gets infected as soon as it comes within a cut-off distance of an
infected particle. They considered a low particle density system, meaning that
the number of active agents per unit dimensionless area was approximately
$0.03$, and used a mean-free path approach to determine analytical expressions
for the rate of infection in their model. The internal state of the particle
does not affect its dynamics, and all particles move with the same self-
propulsion speed. Zhao  _et al._ [14] also endow an internal state to active
agents and examine how the effectiveness of the disease spread differs from
the macroscopic model that assumes a well-mixed population. Here too, a
particle’s self-propulsion speed is unaltered by its internal state. Useful
future directions of research identified in their paper include the study of
models in which the internal state is coupled to the motility of the particle,
and the simulation of systems at a large enough area fraction so that MIPS can
be observed. Forgács  _et al._ [13] study contagion dynamics using an agent-
based modeling approach, where the state of the particle determines its
motility, i.e., the infected particles move $50\%$ slower than susceptible or
recovered particles. Their simulation considers a collection of active
Brownian disks moving in two dimensions, the majority of which belong to a
large cluster that has undergone MIPS. The evolution of the disease for
various values of the epidemiological constants is examined, along with a
characterization of the spatial distribution of the susceptible particles
around the infected particles. The dynamics of active particles that exchange
information about their internal state has been studied in contexts outside of
epidemiological modeling as well [20, 21]. For example, Paoluzzi  _et al._
[20] examine MIPS in a mixture of motile and non-motile particles that can
change their identities upon collision. Quantifying phase separation arising
in agent-based modeling of epidemiological systems remains an open question,
particularly when the internal state is coupled to the motility of the agents,
causing the system to behave as a transient mixture of active and passive
particles. Also pertinent is establishing the connection (or lack thereof)
between the macroscopic and microscopic descriptions of disease spread in
populations, in the non-dilute limit. We address these questions in the
present work and find that: (a) the analytical theory defining the phase
boundary in a system of active and passive particles [22] can successfully be
adapted to describe phase separation in a collection where the activity of the
particles switches transiently, and (b) certain qualitative similarities can
be observed in the disease statistics predicted by the macroscopic and
microscopic models, although a direct mapping between the two is not found.

The remainder of this manuscript is organized as follows: Sec. II
recapitulates salient features of the macroscopic SIRS model, Sec. III
describes the numerical implementation of the microscopic model for two
protocols of the spread of infection: one in which each infected particle can
pass the disease on to exactly one susceptible particle within a contagion
radius, and another in which an infected particle can potentially spread the
disease to multiple susceptible particles within the contagion radius. A
comparison of the contagion dynamics predicted by the one-to-one and one-to-
many infection protocols are presented in Sec. IV. Sec. V presents a
comparison of the disease dynamics obtained using the macroscopic and
microscopic models. A discussion of phase separation in a mixture of active
and transiently passive disks is presented in Sec. VI, accompanied by a
description of the protocol used in the present work for identifying the
occurrence of phase separation in such systems, which could be used to analyze
MIPS in situations beyond the present study. We conclude in Sec. VII.

##  II  Macroscopic model revisited

We consider the SIRS model in this work, in which the immunity gained by the
recovered particles can be lost, causing them to become susceptible to the
infection again. This model is typically used to capture the contagion
dynamics of diseases like influenza [23, 24] and the Omicron variant of SARS-
CoV-2 [25], where the immunity gained is temporary. The population of each
compartment is denoted by $N_{S}$, $N_{I}$ and $N_{R}$, and evolve according
to the following coupled ODEs:

$\begin{split}\dfrac{dN_{S}}{dt}&=-\dfrac{\beta^{*}N_{I}N_{S}}{N}+\alpha^{*}N_{R},\\\\[5.0pt]
\dfrac{dN_{I}}{dt}&=\dfrac{\beta^{*}N_{I}N_{S}}{N}-\gamma^{*}N_{I},\\\\[5.0pt]
\dfrac{dN_{R}}{dt}&=\gamma^{*}N_{I}-\alpha^{*}N_{R}\end{split}$ (1)

where $\beta^{*}$ indicates the rate of infection or transmission of the
disease from an infected to a susceptible individual, $\gamma^{*}$ represents
the rate of recovery of an infected individual, and $\alpha^{*}$ represents
the rate of relapse or the loss of immunity of a recovered individual,
resulting in its conversion to a susceptible agent. We also define the
relative transmission rate $\omega^{*}\equiv\beta^{*}/\gamma^{*}$ and relative
relapse rate $\mu^{*}\equiv\alpha^{*}/\gamma^{*}$ for convenience. The
asterisks on the macroscopic (population-level) rate constants are used to
distinguish them from their counterparts in the microscopic model, which is
discussed later in the paper.

Depending on the context, the labels $\\{S,I,R\\}$ could refer to the type of
population being discussed, or the normalized value of that population, e.g.,
$S=N_{S}/N$. The governing equations [Eq. (1)] are subject to the initial
conditions $S_{0}\equiv S(t=0),I_{0},R_{0}$, and the normalization condition

$S+I+R=1.$ (2)

Two possible steady-state solutions to the system of equations (1) emerge: the
first corresponds to epidemic extinction, with
$S_{\infty}=1,I_{\infty}=0,R_{\infty}=0$, and the second is

$\begin{split}S_{\infty}&=\dfrac{1}{\omega^{*}}\\\\[5.0pt]
I_{\infty}&=\left(1-S_{\infty}\right)\left[\dfrac{\alpha^{*}}{\alpha^{*}+\gamma^{*}}\right]\\\\[5.0pt]
R_{\infty}&=\left(1-S_{\infty}\right)\left[\dfrac{\gamma^{*}}{\alpha^{*}+\gamma^{*}}\right]\end{split}$
(3)

Values of $\omega^{*}<1$ correspond to epidemic extinction, while
$\omega^{*}>1$ results in the endemic state given by eq. (3). Interestingly,
the steady-state susceptible population depends only on the rates of infection
and recovery, while the populations of the other-two species ($I$ and $R$)
depend on all the three rate constants.

Fig. 1 illustrates the effect of the rate constants on the time-evolution of
the infection, for different initial populations of the various populations.
The rate constants and the initial conditions uniquely determine the dynamics
of population evolution. The steady-state value of the various populations
($S_{\infty}$, for instance), however, are solely determined by the ratios
$\omega^{*}$ and $\mu^{*}$, and are independent of the initial conditions. In
the language of dynamical systems theory [26], eq. (3) is an attracting state
for $\omega^{*}>1$.

|
---|---
(a) | (b)
Figure 1: Population evolution as a function of time for two parameter sets
with $\omega^{*}=2$, $\mu^{*}=1.5$ and different values of the initial
populations of susceptible and infected individuals, i.e., (a) $S_{0}=0.7$,
$I_{0}=0.3$ and (b) $S_{0}=0.9$, $I_{0}=0.1$.

Fig. 2 illustrates a plot of the steady-state populations as a function of the
rate constants. Fig. 2 (a) illustrates the case of varying the infection rate
fixed values of the recovery and relapse rate. Low values of the relative
transmission rate, i.e., $\omega^{*}<1$ result in epidemic extinction, in
which there are no infected particles in the long-time limit. As
$\omega^{*}\geq 1$, however, the steady-state population of susceptibles
declines with an increase in the infection rate. The transition of the steady-
state numbers from the low $\omega^{*}$ branch to the high $\omega^{*}$ branch
appears to follow a transcritical bifurcation [26]. Holding the infection and
recovery rates constant, while varying the relapse rate, as shown in Fig. 2
(b), has no effect on the population of susceptibles. There are more recovered
than infected individuals for values of $\mu^{*}<1$, while the balance is
reversed as the value of $\mu^{*}$ crosses unity. Lastly, Fig. 2 (c) examines
the consequences of varying the recovery rate as the infection and relapse
rates are held constant. At both small and large values of the recovery rate,
the recovered population is nearly zero. This is because for $\omega^{*}<1$
the population is solely made of susceptible individuals, while the infected
individuals dominate the population for $\omega^{*}\gg 1$.

|  |
---|---|---
(a) | (b) | (c)
Figure 2: Steady-state population as a function of (a) relative transmission
rate at constant values of recovery and relapse rates, (b) relative relapse
rate at constant values of infection and recovery rates, and (c) relative
transmission rate at constant values of infection and relapse rates. Data in
Figs. 1 and 2 have been generated by solving the coupled ODEs given by eq.
(1), using a code shared on Mathworks File Exchange [27].

##  III  Numerical simulations of microscopic model for SIRS dynamics in
active Brownian particles

We simulate a system of $N$ active Brownian particles (ABPs) of unit radius
each ($d=2r=2$) moving in a periodic square box of side $L$. The position of
the particles evolves in time according to

$\dot{\bm{r}}_{i}=\bm{v}^{\text{HM}}_{i}+U_{0}\bm{e}_{i}$ (4)

where $U_{0}$ is the self-propulsion speed of the disks. The first term on the
RHS of eq. (4) models a short-ranged steric repulsion that operates only when
the particles are in contact, and is implemented numerically through the
Heyes-Melrose algorithm [28, 29] as described in ref. 30. The particles move
in the direction $\bm{e}_{i}\equiv\left[\cos\theta_{i},\sin\theta_{i}\right]$,
where $\theta_{i}$ denotes the particle’s orientation measured with respect to
the positive $x-$axis. The particle orientation evolves in time according to a
rotational diffusive process, such that

$\left<\dot{\theta}_{i}(t)\dot{\theta}_{i}(t^{\prime})\right>=2D_{\text{r}}\delta(t-t^{\prime})$
(5)

where $D_{\text{r}}$ denotes the rotational diffusion constant. The positions
and orientations of the particles are updated using a forward Euler algorithm
with a timestep $\Delta t$. We next describe the update rules to simulate the
spreading of diseases in this microscopic, agent-based model, which largely
follows the algorithm outlined by Forgács  _et al._ [13].

Each particle has an associated internal state, i.e., it could be a
susceptible ($S$), infected ($I$), or recovered ($R$) agent. A census of the
number of particles in each sub-category is taken at the beginning of each
timestep. The allowed transitions are $S\to I$, $I\to R$ and $R\to S$, with
the rate constants associated with the transitions given by $\beta,\gamma$ and
$\alpha$, respectively. The protocol governing the spread of the infection is
as follows: a loop is run over all the $N_{I}$ infected particles in the box
at a given time instant, and the number of susceptible and infected particles
within a cut-off radius $r_{\text{c}}$ of an infected particle $I^{(n)}$ is
recorded as $N_{S}^{(n)}$ and $N_{I}^{(n)}$, respectively. Note that the index
$n$ runs over all the infected disks in the system at a given time instant. By
this definition, the smallest allowable value for $N_{S}^{(n)}$ is zero, while
that for $N_{I}^{(n)}$ is unity. The probability of infection is calculated as
$p_{\text{inf}}=\beta N_{I}^{(n)}\Delta t$. The probability of recovery is
given by $p_{\text{rec}}=\gamma\Delta t$. The values of the three
probabilities $\\{p_{\text{inf}},p_{\text{rec}},p_{\text{rel}}\\}$ are capped
at unity. If any of these probabilities exceed unity during a timestep, it is
reset to unity.

We consider two protocols by which the infection might spread:

Protocol A: one-to-one

For each value of the index $n$ during the loop over $N_{I}$, if the outcome
of a binomial trial with probability $p_{\text{inf}}$ is non-zero, then $one$
susceptible particle from $N_{S}^{(n)}$ is picked at random for conversion to
an infected particle. Similarly, if the outcome of a binomial trial with
probability $p_{\text{rec}}$ is non-zero, and there are multiple infected
particles within a cut-off radius $r_{\text{c}}$ of an infected particle
$I^{(n)}$, then the state of $one$ infected particle changes to recovered. An
infected particle in this protocol can spread the disease to only one other
particle in a timestep. Having multiple infected particles in the vicinity of
a susceptible particle only increases the probability of infection.

Protocol B: one-to-many

For each value of the index $n$ during the loop over $N_{I}$, the number of
susceptible particles that get infected is decided by drawing a sample from a
binomial distribution of probability $p_{\text{inf}}$, $N_{S}^{(n)}$ number of
times, and calculating $S^{+}$, the total number of successful outcomes. A
total of $S^{+}$ particles from amongst $N_{S}^{(n)}$ are then randomly
selected to be infected. Once the loop over all $N_{I}$ is completed, the
number of infected particles that recover is decided by drawing a sample from
a binomial distribution of probability $p_{\text{rec}}$, $N_{I}$ number of
times, and calculating $I^{+}$, the total number of successful outcomes. A
total of $I^{+}$ particles from amongst $N_{I}$ are then randomly selected to
undergo recovery. In this protocol, therefore, an infected particle could
potentially transmit the disease to multiple susceptible particles in its
neighborhood.

For both the protocols discussed above, the relapse of recovered particles
into the susceptible category is governed by a binomial process of probability
$p_{\text{rel}}=\alpha\Delta t$, and is invoked once the loop over all $N_{I}$
is completed. Furthermore, the infected particles become immobile ($U_{0}=0$),
while both the susceptible and recovered particles retain their activity. The
infected particles regain their mobility upon recovery. The instantaneous
fraction of active disks in the system is therefore given by
$x_{A}\equiv\left(N_{S}+N_{R}\right)/N$.

The population and system size in all cases are chosen to be $N=1600$ and
$L=100$, corresponding to a number density of $\rho=N/L^{2}=0.16$, and an area
fraction of $\phi_{0}=\rho\pi d^{2}/4\approx 0.5$. The self-propulsion speed
is fixed at $U_{0}=0.1$. A discrete timestep width of $\Delta t=0.1$ is used
in all the simulations, such that the displacement of a disk over a single
timestep is smaller than its diameter. The contagion radius is chosen to be
$r_{\text{c}}=3$ in all the simulations, unless specified otherwise. The
motility of the system is quantified using the Péclet number, $\text{Pe}\equiv
3U_{0}/dD_{\text{r}}$. The persistence length $\ell\equiv U_{0}/D_{\text{r}}$
of the active particles must be smaller than the box dimensions, to minimize
finite-size effects [31, 32]. We adjust the rotational diffusion constant at
each Pe so that this condition is met. The effects of thermal noise on the
translational motion are ignored in the present work, but may be included by
adding a white-noise process to the RHS of eq. (4). In Section IV that
compares the contagion dynamics predicted by the two protocols (A and B), and
in Section V that explores the connection between the microscopic and
macroscopic models, the numerical results are obtained from simulations that
are $n=2\times 10^{4}$ steps long. In Section VI that charts the phase
behavior of a collection of active and transiently passive disks, simulations
of at least $\mathcal{O}(10^{5})$ steps are used. The total length of the
simulation is simply the product of the number of simulation steps and the
discrete timestep, and is denoted by $t_{\text{sim}}=n\Delta t$.

##  IV  Comparison between protocols A and B

In this section, we compare the dynamics of infection spread, and the steady-
state statistics for numerical simulations performed using the two protocols
(A and B) discussed above. Given the broad span of the parameter space, we
restrict ourselves to studying two values of the relative transmission rate
$\omega\equiv\beta/\gamma$, and perform a scan across a range of relative
relapse rates, $\mu\equiv\alpha/\gamma$, by holding the recovery rate constant
at $\gamma=0.1$, and varying $\alpha$ and $\beta$ appropriately. We divide the
parameter space into four quadrants as shown in Fig. 3 and examine the
simulation results accordingly.

Figure 3: Schematic of parameter space explored for the comparison between
protocol A and protocol B.

Fig. 4 (a) explores the first and second quadrants of the parameter space and
illustrates that, at a higher value of the relative relapse rate, $\mu=2$, we
notice two different behaviors, based on the value of the relative
transmission rate. In Quadrant I, both the relative transmission and the
relapse rates are high, implying that the disease spread is more probable, as
is the replenishment of the numbers of the susceptible population. In this
quadrant, we observe that the higher transmission rate plays a more dominant
role, leading to a faster spread of the disease, and a rapid decrease in the
number of susceptibles. Furthermore, it is immaterial if the infection spreads
via the one-to-one or the one-to-many route, as they both result in a nearly
identical prediction for the steady-state susceptible population. In Quadrant
II, the relapse rate is higher than the infection rate. The steady-state in
such a case is decided by the protocol for disease spread. Stipulating a one-
to-one spreading protocol causes the infection to die out faster, so that the
population is entirely composed of susceptibles in the long time limit.
Allowing for a one-to-many spreading protocol for the infection compensates
for the low value of the transmissibility in this quadrant, resulting in a
faster spread of the disease, and a lower value of the steady-state
susceptible population as compared to the one-to-one protocol. Fig. 4 (b)
explores the third and fourth quadrants of the parameter space, for the
smallest value of the relative relapse rate considered in this section,
$\mu=0.1$. Over the time window considered in the figure, the numbers
predicted by the one-to-many protocol are comparable to or lower than that
predicted by the one-to-one protocol, indicating, unsurprisingly, that the
former is more efficient in spreading the disease. This effectiveness of the
one-to-many protocol, however, is manifest only transiently, as the steady-
state values of the susceptible population are independent of the relative
transmissibility, and the route for disease spread.

|
---|---
(a) | (b)
Figure 4: Time-evolution of the susceptible population predicted by the two
protocols at (a) high and (b) low values of the relative relapse rate
$\mu\equiv\alpha/\gamma$, for relative transmission rates of $\omega=0.5$ and
$\omega=2$.

From the above analysis, it is clear that the sharpest contrast between the
steady-state outcomes predicted by the two protocols occurs in Quadrant II.
Given the fluctuations in the values of the susceptible population, the
average steady-state value, $S_{\infty}$, is estimated by computing the mean
of the last $20\%$ of the time series. Fig. 5 illustrates the steady-state
susceptible population as a function of the relative relapse rate, for two
different values of the relative transmission rate. The effectiveness of the
one-to-many protocol in governing the spread of infection is most evident from
the low transmission regime as identified in Fig. 5 (a). For $\mu>0.2$, the
$S_{\infty}$ resulting from the one-to-many protocol is significantly lower
than that predicted by the one-to-one protocol, indicating a spread of the
disease amongst a larger fraction of the population. In the high transmission
regime, however, the protocol for disease propagation has a less pronounced
effect on the steady-state statistics, as evinced by Fig. 5 (b). Having
compared the outcomes from the two protocols, we now present results obtained
with Protocol A only for the rest of the paper.

|
---|---
(a) | (b)
Figure 5: Steady-state susceptible population as a function of the relative
relapse rate $\mu$ for (a) high and (b) low values of the relative
transmission rate $\omega$. Error bars represent standard deviation of data
obtained from the time-averaging procedure used to estimate the mean steady-
state population. Where invisible, error bars are smaller than symbol size.

##  V  Connection between microscopic and macroscopic models

A major distinction between the microscopic agent-based model (ABM) and the
macroscopic population-based model for disease spread is the specification of
a contagion radius $r_{\text{c}}$ for the former, which makes the spread of
the infection depend not only on the number of infected and susceptible
individuals at a given time, but also on their locations. We illustrate below
a few salient features of the steady-state numbers predicted by the
microscopic model, and how they compare to the macroscopic model predictions.

|
---|---
(a) | (b)
|
(c) | (d)
Figure 6: Steady-state susceptible population as a function of (a) initial
fraction of infected individuals, (b) relative relapse rate at fixed values of
infection rate, (c) relative transmissibility at fixed values of relapse rate,
and (d) contagion radius at fixed values of the epidemiological constants. All
results obtained using Protocol A (one-to-one) for disease spread.

Fig. 6 (a) illustrates that the steady-state population for the microscopic
model is independent of the initial fraction of the infected population
$I_{0}$, at a fixed value of the contagion radius ($r_{\text{c}}=3$), for
various values of the relative transmission and relapse rates. The
independence from initial conditions, over the range examined in this figure,
is a trait shared by the microscopic and macroscopic models.

Fig. 6 (b) shows that the steady-state susceptible population decreases as a
function of the ratio of the relative relapse rate ($\mu$), for fixed values
of $\omega$ and the contagion radius ($r_{\text{c}}=3$). This marks a crucial
departure from the macroscopic model (see fig. 2 (b)) in which $S_{\infty}$ is
solely a function of the relative transmission rate.

The variation of the steady-state population as a function of the relative
transmission rate $\omega$ is shown in Fig. 6 (c), for fixed values of $\mu$
and contagion radius ($r_{\text{c}}=3$). The susceptible population is
independent of the relative transmission rate for small values of the latter.
Beyond a threshold value of the relative transmission rate, however, the
susceptible population decreases with $\omega$. The crossover value depends on
the ratio $\alpha/\gamma$, unlike in the macroscopic model where the
transition occurs at $\omega^{*}=1$ and is independent of the relapse rate.
While the existence of a bifurcation in the macroscopic model predictions (a
system of coupled ODEs) is unsurprising [26], it is remarkable that an
evidence of bifurcation is also seen in the agent-based model. This also
indicates that the stochastic update rule for the various compartments is
faithful to the contagion dynamics as predicted by the ordinary differential
equations of the macroscopic model.

Lastly, Fig. 6 (d) illustrates the dependence of $S_{\infty}$ on the contagion
radius, for various values of the epidemiological rate constants. For all the
cases examined in the figure, the steady-state susceptible population
decreases as a function of $r_{\text{c}}$.

We briefly revisit the comparison between protocols A and B before concluding
this section. The results reported in Section IV all come from numerical
simulations with a contagion radius of $r_{\text{c}}=3$. Fig. 7 illustrates
the effect of using a larger value of the contagion radius, $r_{\text{c}}=10$,
on the steady-state numbers obtained using Protocol A, over a range of
relative relapse rates and a fixed relative transmission rate of $\omega=2$.
Keeping the contagion radius fixed at $r_{\text{c}}=10$, using a lower value
of $\omega$, or a different protocol results in a nearly identical plot. We
note that at a large value of the contagion radius, there is essentially no
distinction between the steady-state predictions of the two protocols. The
long-time population of susceptibles is practically zero and the particles are
split between the infected and recovered categories. This indicates that when
an infected particles has an abundance of susceptible neighbors to transmit
the diseases, it is immaterial if the disease spreads via the one-to-one or
one-to-many route, and there are negligible susceptible individuals remaining
in the long time limit.

In the limit of a large contagion radius, each particle can “see” all the
other particles in the box, and one could therefore expect that the effect of
spatial heterogeneity is reduced, bringing the microscopic model predictions
closer to that obtained from the macroscopic model. Probing this line of
thought, we note a qualitative similarity between Fig. 7 and the macroscopic
model results given by Fig. 2 (b), in that the steady state numbers are
independent of the relative relapse rate. A distinction between the
microscopic and macroscopic model predictions is that while the former
predicts a vanishing of the susceptible population across the range of the
relative relapse rates considered, the latter predicts a finite non-zero value
for the steady-state susceptible population.

Figure 7: Steady state population as a function of the relative relapse rate, as obtained using Protocol A. |
---|---
(a) | (b)
Figure 8: Steady-state statistics obtained using Protocol A at (a)
$r_{\text{c}}=3$ and (b) $r_{\text{c}}=10$.

We note another similarity between the macroscopic model result in Fig. 2 (a)
and the microscopic model results in Fig. 8. Keeping the relative relapse rate
$\mu$ fixed, increasing the relative transmission rate $\omega$ drives the
spread of infection from a state of extinction ($S_{\infty}=1$) to one in
which the fraction of susceptible individuals has reduced considerably. As
noted in the discussion of Fig. 6 (c), the relative transmissibility at which
the transition away from the epidemic extinction state occurs depends on the
value of $\mu$ in the microscopic model. For the macroscopic model, however,
the location of this transition is fixed at the analytically determinable
value of $\omega^{*}=1$, and is independent of the relative relapse rate
$\mu$. Fig. 8 illustrates the effect of the contagion radius on the location
of this bifurcation: higher values of $r_{\text{c}}$ push the transition to
lower values of $\omega$. This makes intuitive sense: a smaller infection rate
is required when the infected particles can see a larger number of the
susceptible population, resulting in a more effective spread of the disease.

We conclude this section by noting that the precise mapping (if indeed one
exists) between the rate constants used in the ABM and those appearing in the
ordinary differential equations at the population-level remains unknown, we
predict that the contagion radius $r_{\text{c}}$ and the transmission protocol
would crucially affect this relationship.

##  VI  Phase separation in motility-modified SIRS model

A collection of self-propelled (or active) particles interacting sterically
undergo a motility induced phase separation (MIPS) at large enough values of
the Péclet number and the area fraction $\phi_{0}$ of the particles. In this
transition, the particles go from being in a gas-like, single-phase to a
phase-separated state consisting of a dense large cluster that dynamically
exchanges particles with the surrounding dilute phase [see refs. 18, 19 for an
extensive review of the topic]. The boundary separating the homogeneous state
from the phase-separated one in the Pe-$\phi_{0}$ plane has been determined
through direct numerical simulations [33, 34, 35] and analytical theory [36,
33, 37, 38, 39].

The effect of the presence of passive particles - ones that move
translationally under the effect of Brownian noise or not at all - on the
phase separation behavior of an active-passive mixture has also received
interest [22, 40, 41, 42, 43]. Stenhammar  _et al._ [22] studied such a
mixture with a total area fraction of $\phi_{0}$, of which a number fraction
$x_{A}$ is active. They derive the following analytical expression for the
phase boundary in the $\text{Pe}-x_{A}$ plane:

$x_{\text{A}}=\dfrac{3\pi^{2}\kappa}{4\phi_{0}\text{Pe}}$ (6)

using the kinetic model introduced by Redner  _et al._ [33]. Here $\kappa$ is
an empirical fitting parameter [33] that represents the average total number
of particles that are lost from a phase-separated cluster in an escape event.
The boundary obtained using $\kappa=4.05$ is seen to accurately demarcate the
homogeneous and demixed states in the phase diagram generated from numerical
simulations at $\phi_{0}=0.6$. A value of $\kappa=4.5$ accurately predicts the
phase boundary in the Pe-$\phi_{0}$ plane for a system composed solely of
active disks ($x_{A}=1$). Takatori and Brady [40] derive an expression for the
phase boundary in an active-passive mixture using an alternative approach that
relies on the concept of active swim pressure in a collection of self-
propelled swimmers. They obtain an agreement with the predictions of
Stenhammar  _et al._ [22] without any fitting parameters.

Figure 9: Normalized size of the largest cluster as a function of time, for the case of $N=1600$ active disks in which all the epidemiological constants have been set to zero and there is no contagion dynamics. Snapshots of the system at the three indicated time-instances are given in Fig. 10. |  |
---|---|---
(a) | (b) | (c)
Figure 10: Snapshots of the system in Fig. 9, recorded at (a) $t=1000$, (b)
$t=5527$, and $t=8000$, respectively. Figure 11: Cluster size evolution as a
function of time for two representative systems. Parameters needed for the
algorithm that detects motility induced phase separation are identified here,
and their numerical values provided in Table 1.

We summarize the common metrics used in the literature to quantify motility-
induced phase separation, before describing the algorithm introduced in the
present work to identify systems that have undergone phase separation. The
fraction of particles in the largest cluster $N_{\text{LC}}/N$ is a popular
metric [31, 44, 30] to track the approach to MIPS. When the majority of the
particles in the system belong to a single cluster, it is taken to be an
indication of MIPS. Large fluctuations in the number of particles within a
subregion are taken to be a sign of inhomogeneity and the onset of MIPS [45,
46]. Another signature for the occurrence of MIPS is the appearance of a
bimodality in the probability distribution of the local area fraction [47, 48,
49]. As a system evolves from a homogeneous state to a phase separated one,
the size of the domains (calculated from either the static structure factor or
the density correlation function) grows with time as a power law [33, 34, 50,
51] $\mathcal{L}(t)\sim t^{1/3}$. These metrics have also been used to analyze
systems containing a mixture of active and passive particles [22], in which
the relative populations of the two species remain constant in time.

In the present work, however, the fraction of active and passive disks
$fluctuate$ in time due to the disease spreading, and we seek to identify an
appropriate metric for the identification of MIPS in such systems. To that
end, Fig. 9 illustrates the time evolution of $N_{\text{LC}}/N$ in a system
consisting entirely of susceptible particles, with the epidemiological
constants set to zero. There is no spread of infection in such a system, and
the fraction of active disks is therefore unity at all times. For the values
of Pe and the area fraction considered in Fig. 9, the system undergoes MIPS,
as evinced by the snapshots recorded at the various time instances. Our goal
is to define metrics for the identification of MIPS based on this reference
time series, for application to other systems in our work in which the
fraction of active disks fluctuate in time. Firstly, we note that the
normalized size of the largest cluster reaches a steady state value of
$N_{\text{LC}}/N\approx 0.83$ following an initial transient. Secondly, after
$t\approx 5520$, there are no significant dips in the value of
$N_{\text{LC}}/N$, and the fluctuations in this quantity are minimal. We use
these two observations to devise a methodology (Fig. 11 and Fig. 12) for
ascertaining if a system has undergone MIPS or not, given the time-series of
the largest cluster. In case a system has phase separated, this algorithm also
estimates the time needed for MIPS. The various parameters needed by the
algorithm are $\\{M_{\text{cut}},M_{\text{min}},\tau,\sigma\\}$, and a brief
explanation is as follows. If the fractional size of the largest cluster
$\mu\equiv N_{\text{LC}}/N$ does not exceed $M_{\text{cut}}$ at any point in
its time series, then we consider that MIPS has not occurred. The algorithm
searches for a chunk of data in the time series in the interval
$\left[t_{\text{s}},t_{\text{s}}+\tau\right]$, such that each data point in
the interval exceeds $M_{\text{min}}$. If such a chunk is not found in the
input data series, the algorithm concludes that MIPS has not occurred.
Provided such a data chunk is found, we then test if the standard deviation of
the data series (normalized by the total number of particles), is smaller than
$\sigma$. If this requirement is met, the algorithm concludes that MIPS has
occurred and returns $t_{\text{s}}$, the starting point of the data chunk, as
the time at which MIPS has occurred. If the standard deviation of the data
series (normalized by the total number of particles) exceeds $\sigma$, then we
conclude that MIPS has not occurred. The parameter values used in this
algorithm are listed in Table 1. The time to MIPS as estimated by the
algorithm for the timeseries indicated in Figs. 9 and 11 are 3730 and 5527,
respectively.

Figure 12: Flowchart illustrating the algorithm for ascertaining if a system
has undergone MIPS, and to evaluate the time needed for phase separation in
case it has.

Fig. 13 explores the phase behavior of the system in the $\text{Pe}-x_{A}$
plane, obtained from simulations using a wide range of the epidemiological
constants (see Table 2 for values). At the steady-state value of the fraction
of active disks for any given value of the Péclet number, we denote if MIPS
has occurred or not using the algorithm described above. The boundary between
the homogeneous states and the MIPS states is well described by eq. 6 when a
value of $\kappa=1.875$ is used. Snapshots of the system at a fixed value of
Pe and varying fractions of the active disks are given in Fig. 14 A-D. We
notice that there is no preference for disks with identical internal states to
cluster together. A more quantitative analysis would involve the calculation
of the pair correlation function for the various populations.

Table 1:  Parameters used in the algorithm (Fig. 12) to determine if phase separation has occurred or not. Parameter | Value used in present work
---|---
$M_{\text{cut}}$ | $0.8$
$M_{\text{min}}$ | $0.6$
$\tau$ | $0.1\,t_{\text{sim}}$
$\sigma$ | $0.1$
Table 2:  Data points in the $\text{Pe}-x_{A}$ phase plane (Fig. 13) and the epidemiological constants used in the simulations for obtaining them. Pe |  $x_{A}$ |  $\beta$ |  $\gamma$ |  $\alpha$
---|---|---|---|---
|  $1.0$ |  0 |  0 |  0
|  $0.84$ |  0.1 |  0.1 |  0.05
|  $0.62$ |  0.05 |  0.1 |  0.2
50 |  $0.53$ |  0.2 |  0.15 |  0.2
|  $0.38$ |  0.2 |  0.1 |  0.2
|  $0.22$ |  0.2 |  0.05 |  0.2
|  $0.004$ |  0.2 |  0.001 |  0.2
|  $1.0$ |  0.05 |  0.1 |  0.05
|  $0.82$ |  0.1 |  0.1 |  0.05
|  $0.63$ |  0.2 |  0.2 |  0.2
60 |  $0.55$ |  0.2 |  0.15 |  0.2
|  $0.42$ |  0.2 |  0.1 |  0.2
|  $0.23$ |  0.2 |  0.05 |  0.2
|  $0.03$ |  0.2 |  0.005 |  0.2
|  $1.0$ |  0 |  0 |  0
|  $0.78$ |  0.1 |  0.1 |  0.05
|  $0.59$ |  0.2 |  0.2 |  0.2
75 |  $0.51$ |  0.05 |  0.1 |  0.2
|  $0.37$ |  0.2 |  0.1 |  0.2
|  $0.22$ |  0.2 |  0.05 |  0.2
|  $0.006$ |  0.2 |  0.001 |  0.2
|  $1.0$ |  0.05 |  0.1 |  0.05
|  $0.76$ |  0.1 |  0.1 |  0.05
|  $0.56$ |  0.2 |  0.15 |  0.2
90 |  $0.41$ |  0.2 |  0.1 |  0.2
|  $0.21$ |  0.2 |  0.05 |  0.2
|  $0.03$ |  0.2 |  0.005 |  0.2
|  $1.0$ |  0 |  0 |  0
|  $0.78$ |  0.1 |  0.1 |  0.05
|  $0.59$ |  0.2 |  0.2 |  0.2
100 |  $0.53$ |  0.05 |  0.1 |  0.2
|  $0.39$ |  0.2 |  0.1 |  0.2
|  $0.23$ |  0.2 |  0.05 |  0.2
|  $0.008$ |  0.2 |  0.001 |  0.2

Figure 13: Phase separation in a mixture of active and passive disks. Open
circles indicate a homogeneous phase, while closed diamonds represent a phase-
separated system. Dash-dotted redline represents eq. (6) with $\kappa=1.875$.
Snapshots of the system at the locations A,B,C,D are given in Fig. 14. Figure
14: Representative snapshots of the system for $\text{Pe}=75$ at the active-
disk fraction indicated in Fig. 13 and recorded at the halfway point of the
total simulation. The blue disks denote susceptible agents, while red and
green disks represent infected and recovered individuals, respectively. Figure
15: Time taken for motility induced phase separation, as a function of the
active disk fraction.

In Fig. 15, the time taken for motility induced phase separation, as
identified using the algorithm described in Fig. 12, is plotted as a function
of the fraction of active disks in the system, for a range of Péclet numbers.
We observed no definitive trend, implying that we cannot conclude if the
presence of transiently immobile disks helps to aid or suppress phase
separation in an active-passive mixture. Forgács  _et al._ [13] observe that
the presence of quenched disorder, or immobile obstacles, in an active matter
system causes the formation of numerous small clusters in addition to the
large cluster that is characteristic of MIPS. Additionally, MIPS is a re-
entrant phenomenon [37, 44], meaning that increasing the Péclet number in an
already phase separated system can cause the system to go back to being in a
homogeneous phase.

##  VII  Conclusions

We performed an agent-based modeling of disease spread according to the SIRS
model using a collection of active Brownian particles moving in two dimensions
whose internal state encodes their state of infection. Two protocols for
infection were considered, and their efficacies for the spread of the disease
were analyzed for various combinations of the epidemiological constants. The
coupling of the particle’s internal state to its motility causes the
population to behave as a collection of particles in which the fraction of
active disks is time-dependent. We developed an algorithm to determine the
occurrence of motility induced phase separation in such systems with transient
activity, and find that it is well-described by the theories for phase
separation in active-passive mixtures where the fraction of active disks
remains constant in time. Although a direct mapping between the agent-based
(microscopic) and macroscopic model is not found, several common features
between the contagion dynamics predicted by the two models are noted. We see
evidence for a transcritical bifurcation in the microscopic model where the
agents are modeled as active Brownian particles. The use of active Brownian
disks permits a tractable method to tune the density distribution of the
system by changing the Péclet number. Humans in general, however, do not move
at a constant self-propulsion speed with randomly varying orientations.
Simulating the dynamics of individuals in a crowd has typically relied on the
use of social forces that describe the interaction between the individual
members [52, 53]. The use of such pedestrian models to describe the motion of
individual agents could permit the extension of the present work to model
epidemic spread in human populations. Another interesting exercise for future
work could be the effect of the type of motility on the nature and location of
the bifurcation point, i.e., would a system of agents modeled using social
forces exhibit a different kind of bifurcation when the steady-state numbers
are plotted as a function of the relative rate of transmission.

##  Acknowledgements

A. K. and R. K. acknowledge support from the Donors of the American Chemical
Society Petroleum Research Fund (ACS-PRF 61858-ND9).

## References

  * Kermack and McKendrick [1927] W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. London, Ser. A 115, 700 (1927).
  * Anderson and May [1991] R. M. Anderson and R. M. May,  _Infectious Diseases of Humans_ (Oxford Science Publications, Oxford, 1991).
  * Cabrera  _et al._ [2021] M. Cabrera, F. Córdova-Lepe, J. P. Gutiérrez-Jara, and K. Vogt-Geisse, An SIR-type epidemiological model that integrates social distancing as a dynamic law based on point prevalence and socio-behavioral factors, Sci. Rep. 11, 10170 (2021).
  * Bjørnstad  _et al._ [2020] O. N. Bjørnstad, K. Shea, M. Krzywinski, and N. Altman, The SEIRS model for infectious disease dynamics, Nature methods 17, 557 (2020).
  * Wu  _et al._ [2022] Y. Wu, Y. Sun, and M. Lin, SQEIR: An epidemic virus spread analysis and prediction model, Computers and Electrical Engineering 102, 108230 (2022).
  * Liu  _et al._ [2023] T. Liu, J. Huang, Z. He, Y. Zhang, N. Yan, C. J. Zhang, and W. K. Ming, A real-world data validation of the value of early-stage SIR modelling to public health, Scientific Reports 13, 9164 (2023).
  * Giraldo and Palacio [2008] J. O. Giraldo and D. H. Palacio, Deterministic SIR (Susceptible-Infected-Removed) models applied to varicella outbreaks, Epidemiology and Infection 136, 679 (2008).
  * Osthus  _et al._ [2017] D. Osthus, K. S. Hickmann, P. C. Caragea, D. Higdon, and S. Y. Del Valle, Forecasting seasonal influenza with a state-space SIR model, Annals of Applied Statistics 11, 202 (2017).
  * Peruani and Sibona [2008] F. Peruani and G. J. Sibona, Dynamics and steady states in excitable mobile agent systems, Phys. Rev. Lett. 100, 168103 (2008).
  * Peruani and Sibona [2019] F. Peruani and G. J. Sibona, Reaction processes among self-propelled particles, Soft Matter 15, 497 (2019).
  * González and Herrmann [2004] M. C. González and H. J. Herrmann, Scaling of the propagation of epidemics in a system of mobile agents, Physica A: Statistical Mechanics and its Applications 340, 741 (2004).
  * Rodríguez  _et al._ [2019] J. P. Rodríguez, F. Ghanbarnejad, and V. M. Eguíluz, Particle velocity controls phase transitions in contagion dynamics, Sci. Rep. 9, 6463 (2019).
  * Forgács  _et al._ [2022] P. Forgács, A. Libál, C. Reichhardt, N. Hengartner, and C. J. Reichhardt, Using active matter to introduce spatial heterogeneity to the susceptible infected recovered model of epidemic spreading, Sci. Rep. 12, 11229 (2022).
  * Zhao  _et al._ [2022] Y. Zhao, C. Huepe, and P. Romanczuk, Contagion dynamics in self-organized systems of self-propelled agents, Sci. Rep. 12, 2588 (2022).
  * Norambuena  _et al._ [2020] A. Norambuena, F. J. Valencia, and F. Guzmán-Lastra, Understanding contagion dynamics through microscopic processes in active Brownian particles, Sci. Rep. 10, 20845 (2020).
  * Buttinoni  _et al._ [2013] I. Buttinoni, J. Bialké, F. Kümmel, H. Löwen, C. Bechinger, and T. Speck, Dynamical clustering and phase separation in suspensions of self-propelled colloidal particles, Phys. Rev. Lett. 110, 238301 (2013).
  * Bialké  _et al._ [2015] J. Bialké, T. Speck, and H. Löwen, Active colloidal suspensions: Clustering and phase behavior, J Non-Cryst Solids 407, 367 (2015).
  * Cates and Tailleur [2015] M. E. Cates and J. Tailleur, Motility-induced phase separation, Annual Review of Condensed Matter Physics 6, 219 (2015).
  * Zöttl and Stark [2023] A. Zöttl and H. Stark, Modeling Active Colloids: From Active Brownian Particles to Hydrodynamic and Chemical Fields, Ann. Rev. Cond. Matt. Phys. 14, 109 (2023).
  * Paoluzzi  _et al._ [2020] M. Paoluzzi, M. Leoni, and M. C. Marchetti, Information and motility exchange in collectives of active particles, Soft Matter 16, 6317 (2020).
  * Zhong  _et al._ [2023] W. Zhong, Y. Deng, and D. Xiong, Burstiness and information spreading in active particle systems, Soft Matter 19, 2962 (2023).
  * Stenhammar  _et al._ [2015] J. Stenhammar, R. Wittkowski, D. Marenduzzo, and M. E. Cates, Activity-induced phase separation and self-assembly in mixtures of active and passive particles, Phys. Rev. Lett. 114, 018301 (2015).
  * Edlund  _et al._ [2011] S. Edlund, J. Kaufman, J. Lessler, J. Douglas, M. Bromberg, Z. Kaufman, R. Bassal, G. Chodick, R. Marom, V. Shalev, Y. Mesika, R. Ram, and A. Leventhal, Comparing three basic models for seasonal influenza, Epidemics 3, 135 (2011).
  * Bucyibaruta  _et al._ [2023] G. Bucyibaruta, C. B. Dean, and M. Torabi, A discrete-time susceptible-infectious-recovered-susceptible model for the analysis of influenza data, Infect. Dis. Model. 8, 471 (2023).
  * Nill [2023] F. Nill, Endemic oscillations for SARS-CoV-2 Omicron—A SIRS model analysis, Chaos, Solitons and Fractals 173, 113678 (2023).
  * Strogatz [2015] S. Strogatz, Nonlinear dynamics and chaos (CRC Press, 2015) p. 532, 2nd ed.
  * Valentini [2020] G. Valentini, SIR Epidemic Spread Model (2020), https://www.mathworks.com/matlabcentral/fileexchange/75100-sir-epidemic-spread-model.
  * Heyes and Melrose [1993] D. M. Heyes and J. R. Melrose, Brownian dynamics simulations of model hard-sphere suspensions, J. Non-Newton. Fluid Mech 46, 1 (1993).
  * Foss and Brady [2000] D. R. Foss and J. F. Brady, Brownian Dynamics simulation of hard-sphere colloidal dispersions, J. Rheol. 44, 629 (2000).
  * Kailasham and Khair [2023] R. Kailasham and A. S. Khair, Effect of speed fluctuations on the collective dynamics of active disks, Soft Matter 19, 7764 (2023).
  * Patch  _et al._ [2017] A. Patch, D. Yllanes, and M. C. Marchetti, Kinetics of motility-induced phase separation and swim pressure, Phys. Rev. E 95, 012601 (2017).
  * Patch  _et al._ [2018] A. Patch, D. M. Sussman, D. Yllanes, and M. C. Marchetti, Curvature-dependent tension and tangential flows at the interface of motility-induced phases, Soft Matter 14, 7435 (2018).
  * Redner  _et al._ [2013] G. S. Redner, M. F. Hagan, and A. Baskaran, Structure and dynamics of a phase-separating active colloidal fluid, Phys. Rev. Lett. 110, 055701 (2013).
  * Stenhammar  _et al._ [2014] J. Stenhammar, D. Marenduzzo, R. J. Allen, and M. E. Cates, Phase behaviour of active Brownian particles: The role of dimensionality, Soft Matter 10, 1489 (2014).
  * Levis  _et al._ [2017] D. Levis, J. Codina, and I. Pagonabarraga, Active Brownian equation of state: Metastability and phase coexistence, Soft Matter 13, 8113 (2017).
  * Stenhammar  _et al._ [2013] J. Stenhammar, A. Tiribocchi, R. J. Allen, D. Marenduzzo, and M. E. Cates, Continuum theory of phase separation kinetics for active brownian particles, Phys. Rev. Lett. 111, 145702 (2013).
  * Bialké  _et al._ [2013] J. Bialké, H. Löwen, and T. Speck, Microscopic theory for the phase separation of self-propelled repulsive disks, Europhys. Lett. 103, 30008 (2013).
  * Takatori and Brady [2015a] S. C. Takatori and J. F. Brady, Towards a thermodynamics of active matter, Phys. Rev. E 91, 032117 (2015a).
  * Worlitzer  _et al._ [2021] V. M. Worlitzer, G. Ariel, A. Be’er, H. Stark, M. Bär, and S. Heidenreich, Motility-induced clustering and meso-scale turbulence in active polar fluids, New J. Phys. 23, 033012 (2021).
  * Takatori and Brady [2015b] S. C. Takatori and J. F. Brady, A theory for the phase behavior of mixtures of active particles, Soft Matter 11, 7920 (2015b).
  * Rogel Rodriguez  _et al._ [2020] D. Rogel Rodriguez, F. Alarcon, R. Martinez, J. Ramírez, and C. Valeriani, Phase behaviour and dynamical features of a two-dimensional binary mixture of active/passive spherical particles, Soft Matter 16, 1162 (2020).
  * Agrawal and Mahapatra [2021] N. K. Agrawal and P. S. Mahapatra, Alignment-mediated segregation in an active-passive mixture, Phys. Rev. E 104, 044610 (2021).
  * Zhang  _et al._ [2022] J. Zhang, T. Huang, G. Xu, and Y. Chen, Density fluctuations of two-dimensional active-passive mixtures, Commun. Theor. Phys. 74, 075601 (2022).
  * Su  _et al._ [2023] J. Su, M. Feng, Y. Du, H. Jiang, and Z. Hou, Motility-induced phase separation is reentrant, Commun. Phys. 6, 58 (2023).
  * Fily and Marchetti [2012] Y. Fily and M. C. Marchetti, Athermal phase separation of self-propelled particles with no alignment, Phys. Rev. Lett. 108, 235702 (2012).
  * Desgranges  _et al._ [2023] C. Desgranges, M. E. Tuckerman, M. Ferrari, P. M. Chaikin, S. Sacanna, and J. Delhommelle, Microswimmers under the spotlight : interplay between agents with different levels of activity, Soft Matter 19, 7334 (2023).
  * Gonnella  _et al._ [2015] G. Gonnella, D. Marenduzzo, A. Suma, and A. Tiribocchi, Motility-induced phase separation and coarsening in active matter, C. R. Phys 16, 316 (2015).
  * Knezevic and Stark [2022] M. Knezevic and H. Stark, Collective motion of active particles exhibiting non-reciprocal orientational interactions, Sci. Rep. 12, 19437 (2022).
  * Theeyancheri  _et al._ [2024] L. Theeyancheri, S. Chaki, T. Bhattacharjee, and R. Chakrabarti, Dynamic Clustering of Active Rings, Phys. Rev. Res 6, L012038 (2024).
  * Caporusso  _et al._ [2022] C. B. Caporusso, L. F. Cugliandolo, P. Digregorio, G. Gonnella, D. Levis, and A. Suma, Dynamics of Motility-Induced clusters: coarsening beyond Ostwald ripening, Phys. Rev. Lett. 131, 068201 (2022).
  * Caporusso  _et al._ [2020] C. B. Caporusso, P. Digregorio, D. Levis, L. F. Cugliandolo, and G. Gonnella, Motility-Induced Microphase and Macrophase Separation in a Two-Dimensional Active Brownian Particle System, Phys. Rev. Lett. 125, 178004 (2020).
  * Teknomo [2006] K. Teknomo, Application of microscopic pedestrian simulation model, Transportation Research Part F: Traffic Psychology and Behaviour 9, 15 (2006).
  * Miller [2007] J. C. Miller, Epidemic size and probability in populations with heterogeneous infectivity and susceptibility, Phys. Rev. E 76, 010101(R) (2007).


N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{1-\xi}L(t)V(t)-w(t)l_{R}(t)L(t)-\theta\varphi(t)^{\phi}.$

Thus, the free-entry conditions in this case are:

$(1-\xi)\eta N(t)^{\zeta}\varphi(t)^{\xi}l_{R}(t)^{-\xi}V(t)=w(t)$ (100)

and

$\xi\eta
N(t)^{\zeta}\varphi(t)^{\xi-1}l_{R}(t)^{1-\xi}L(t)V(t)=\theta\phi\varphi(t)^{\phi-1}.$
(101)

Next, we derive the BGP growth rate in this case. Similarly, we still have
(31) and (32). Now, from (101), we rewrite it as

$\zeta\frac{\dot{N}(t)}{N(t)}+(\xi-1)\frac{\dot{\varphi}(t)}{\varphi(t)}+(1-\xi)\frac{\dot{l}_{R}(t)}{l_{R}(t)}+n+\frac{\dot{V}(t)}{V(t)}=(\phi-1)\frac{\dot{\varphi}(t)}{\varphi(t)}$

In BGP, this equation can be further derived as

$\zeta\frac{\dot{N}(t)}{N(t)}+(\xi-\phi)\frac{\dot{\varphi}(t)}{\varphi(t)}+2n=0.$
(102)

Combining (102) with (32), we derive the BGP growth rate as

$\frac{\dot{N}(t)}{N(t)}=\left[\frac{\xi+\phi}{\phi(1-\zeta)-\xi}\right]n.$

Meanwhile, the BGP growth rate of data provision is

$\frac{\dot{\varphi}(t)}{\varphi(t)}=-\frac{n}{\xi}+\frac{1-\zeta}{\xi}\frac{\dot{N}(t)}{N(t)}=\left[\frac{2-\zeta}{\phi(1-\zeta)-\xi}\right]n>0.$

The labor allocations in this case have the same form as those shown in
Proposition 3.2, except that the BGP growth rate is different here. Q.E.D.

In (3), we require that the growth rate of data provision should not exceed
the growth rate of consumption. As a result, when $2-\zeta>\xi+\phi$, the
constraint binds. In this case, free-entry condition of data fails since firms
always require excessive data. We can then calculate the growth rates from the
free-entry condition of labor (19) as

$g^{*}=g^{*}_{\varphi}=\frac{n}{1-\zeta-\xi},$

which means the constraint (3) affects the BGP growth rates in this case.
Also, following Jones and Tonetti (2020), the innovation possibility frontier
in this paper is set as

$\dot{N}_{t}=\frac{1}{\chi}L_{E}(t),$

which can be viewed as setting the knowledge spillover effect $\zeta$ to be 0,
while data are set as different characters. Then, the condition
$\xi+\phi<2-\zeta$ is likely to be satisfied when we take proper values of
$\xi$ and $\phi$. As a result, we can derive similar conclusion under this
condition: When firms own data, they are inclined to use up all the data they
have.


# AudioSlots: A slot-centric generative model for audio separation

###### Abstract

In a range of recent works, object-centric architectures have been shown to be
suitable for unsupervised scene decomposition in the vision domain. Inspired
by these methods we present AudioSlots, a slot-centric generative model for
blind source separation in the audio domain. AudioSlots is built using
permutation-equivariant encoder and decoder networks. The encoder network
based on the Transformer architecture learns to map a mixed audio spectrogram
to an unordered set of independent source embeddings. The spatial broadcast
decoder network learns to generate the source spectrograms from the source
embeddings. We train the model in an end-to-end manner using a permutation
invariant loss function. Our results on Libri2Mix speech separation constitute
a proof of concept that this approach shows promise. We discuss the results
and limitations of our approach in detail, and further outline potential ways
to overcome the limitations and directions for future work.

Index Terms—  Speech separation, object-centric representation

##  1 Introduction

Recently there has been a lot of research into neural network based
architectures that operate on set-structured data and architectures that learn
to map from unstructured inputs to set-structured output spaces. In
particular, in the vision domain, slot-centric or object-centric architectures
underpin recent advances in object detection [1] and unsupervised object
discovery [2, 3].

These object-centric architectures have an inbuilt inductive bias of
permutation equivariance, making them a natural fit for the task of audio
separation. In this paper we apply the core ideas from these architectures to
the problem of sound separation: the task of separating audio sources from
mixed audio signals without access to privileged knowledge about the sources
or the mixing process. Sound separation is inherently a set-based problem, as
the ordering of the sources is arbitrary.

We frame sound separation as a permutation-invariant conditional generative
modeling problem: we learn a mapping from a mixed audio spectrogram to an
unordered set of independent source spectrograms. Our method, AudioSlots,
separates audio into individual latent variables per source, which are then
decoded into individual source spectrograms. It is built using permutation-
equivariant encoder and decoder functions based on the Transformer
architecture [4] and thus invariant to the ordering of the source latent
variables (“slots”).

To evaluate the promise of such an architecture, we train AudioSlots using a
matching-based loss to generate separate sources from the mixed audio-signal.
We demonstrate our method on a simple two-speaker speech separation task from
Libri2Mix [5].

While our results primarily constitute a proof of concept for this idea, we
find that sound separation with slot-centric generative models shows promise,
but comes with certain challenges: the presented version of our model
struggles to generate high-frequency details, relies on heuristics for
stitching independently predicted audio chunks, and still requires ground-
truth reference audio sources for training. We are optimistic that these
challenges can be overcome in future work, for which we outline possible
directions in this paper.

Fig. 1: Architecture overview. The input waveform is first cropped and
transformed into a spectrogram. Then the neural network encodes the
spectrogram to a set of permutation invariant source embeddings $s_{1...n}$,
these embeddings are decoded to generate a set of individual source
spectrograms. The whole pipeline is supervised with the groundtruth source
spectrograms using a matching based permutation invariant loss function.

##  2 Related Work

Our work explores a novel set-based generative modeling approach for sound
separation. In the following, we provide a brief overview on recent prior
learning-based approaches for sound separation as well as on set-based (or
slot-centric) neural network architectures used in other domains.

Sound separation. A variety of neural network methods have been proposed for
supervised sound separation, differing in terms of their sound reconstruction
mechanisms and their overall architectures. Mask-based methods reconstruct
sound by predicting separation masks that are applied to an analysis/synthesis
basis representation of the input audio signal (such as STFT or learned
basis), e.g. [6, 7, 8, 9, 10, 11, 12, 13]. Alternatively, direct
reconstruction methods estimate the source signals or their spectra without
explicitly estimating masks [14, 15].

Many generic architectures have been proposed for sound separation, including
recurrent networks [16], convolutional networks [9], U-nets [14], attention
networks [17], and their combinations. These address the arbitrary permutation
of output sources using a permutation-invariant loss during training [7, 8].

Some methods have gone further to address permutation invariance at the
architecture level by producing un-ordered representations corresponding to
each source. Deep clustering and deep attractor networks [6, 18, 7], employ a
permutation-equivariant architecture, which operates in an attention-like way
over embeddings of each time-frequency bin.

Our approach produces an embedding for each source, using a slot-based
attention mechanism, and decodes it to directly estimate the source
spectrogram, using a NeRF-like [19] method. This differs from mask-based
methods in using direct prediction, and the method of direct prediction using
NeRF architecture is novel. The slot-based attention mechanism is different
from previous attention networks for sound separation, and is more closely
related to deep clustering. However our attention method works on higher level
spectrogram regions rather than individual time-frequency bins, and uses
general-purpose attention mechanisms instead of simple affinity-based methods.

Recent unsupervised approaches, such as mixture invariant training (MixIT)
[20], us only audio mixtures for training. While we only explore supervised
sound separation using ground-truth isolated reference sources in our work, a
training setup like in MixIT is orthogonal to our approach and would be
interesting to explore in future work.

Slot-centric neural networks. Neural networks that operate on unordered sets
of features have been studied for some time [21, 22, 23]. Most closely related
to our work are approaches that generate an unordered set of outputs
conditioned on some input data [24, 25, 1] and methods that use a set of
latent variables (“slots”) to model permutation-invariant aspects of the input
[2, 3], such as objects in visual scenes. We refer to the latter as slot-
centric neural networks. In the context of vision, this class of models forms
the basis for many modern scene understanding approaches, including object-
detection [1], panoptic segmentation [26], and unsupervised object discovery
[2, 3, 27]. In our work, we demonstrate that slot-centric generative models
similarly hold promise for compositional, permutation-invariant tasks in audio
processing, specifically for separating individual audio sources from a
mixture.

##  3 Method

We present a generalized permutation invariant training framework for
supervised sound separation. Unlike previous methods, we approach the source-
separation task from a generative perspective. The main objective of our
method is to project the input audio into a set of embeddings each
representing a different source in the input. These embeddings are then used
to generate the magnitude spectrograms of individual sources in a permutation
invariant manner. The whole pipeline is supervised with the ground-truth
source spectrograms using a permutation-invariant loss. In rest of this
section we elaborate on different steps in our training pipeline.

Preprocessing: Given a mixture waveform during training we first randomly crop
a 0.5-second audio clip. Then following [28] we transform the clip into a
spectrogram using a short-time Fourier transform with window size 512 and hop
size 125. Then the absolute values of the complex spectrograms are non-
linearly scaled by exponentiating them by power of 0.3 to emphasize low
values. These scaled absolute values are passed as the input to the next step.

Encoding: To infer source embeddings, the input spectrogram is first encoded
using a ResNet-34 network [29] to a $32\times 8$ grid of encoded “image”
features $z$. We use a reduced stride in the ResNet root block to retain a
higher spatial resolution. Next a transformer with 4 layers maps $z$ to source
embeddings $s_{1...n}$ where $n$ is the number of sources and $s_{i}\in
R^{d}$. Unlike the original formulation [4], in each transformer layer we
first perform a self-attention operation between query vectors $q$ and then
perform cross-attention between the outputs of the self-attention step and
$z$. We use same variables as both the key and value in self-attention and
cross-attention steps. The initial queries $q$ of dimensionality $4096$ are
learned via backpropagation, similar to DETR [1].

Decoding: We use a spatial broadcast decoder [30] to generate individual
source spectrograms from $s_{1...n}$. First each embedding $s_{i}$ are copied
across a 2D grid to create tensor $g_{i}$ of shape $F\times T\times d$ where
$F,T$ are the frequency bins and timesteps of the output spectrogram. Then
positional embeddings with fourier features [31] are appended to $g_{i}$
making its shape $F\times T\times(d+e)$ where $e$ is the size of each fourier
feature embeddings. Subsequently a fully connected network with shared
parameters is applied across all the vectors in $g_{i}$ to arrive at a set of
spectrograms each with shape $F\times T$. Note that this decoder is similar to
a NeRF model [19], which similarly takes positional codes as input and learns
a fully-connected network to produce coordinate-dependent outputs. In our
case, we directly produce spectrogram values, arranged on a 2D image grid, as
outputs and further condition the network on the respective latent source
embeddding $s_{i}$ for each generated spectrogram.

Objective: The predicted spectrograms are generated in no pre-determined
ordering as the source separation problem is permutation invariant. Therefore
we need to match estimated spectrograms with ground-truth spectrograms to
calculate the loss of the network. Among all the possible matches between
ground-truth and estimated spectrograms we seek to find the optimal assignment
with minimum reconstruction error. We suggest to use the Hungarian matching
algorithm to solve this assignment problem because of its speed, accuracy and
ability to handle large numbers of items efficiently (although we only use
datasets with a small number of sources in our experiments). It works by
assigning costs or weights to each possible pair and then selects pairs which
minimize the total cost or weight associated with them. Finally the mean
squared error between matched ground-truth and estimated spectrograms is
minimized as the training objective to optimize the network parameters.

Source Separation: During testing, given an input mixture waveform we first
break it into multiple non-overlapping waveforms of length 0.5-seconds. We add
zero-padding at the end in case the input waveform length is not evenly
divisible into chunks of 0.5s. These waveforms are preprocessed as mentioned
above and are passed as inputs to the network, trained using the above
pipeline to estimate absolute values of the spectrograms of individual
sources. The estimated spectrograms are first rescaled by exponentiating them
by power of 1/0.3 to invert the scaling done during preprocessing. These
rescaled estimates are used to calculate the masks in the oracle method to
create complex spectrogram estimates of individual sources from the input.
Given input spectrogram $I$ the output source spectrograms are computed as
$m_{i}*I$ where $m_{i}$ is the mask corresponding to the $i^{th}$ source
estimated using the spectrograms predicted by the neural network. These
complex spectrograms are inverted to waveforms using an inverse short-time
Fourier transform (STFT) and then stitched together, resolving matching using
the best match with the ground-truth signal for simplicity.

Training: We train using Adam [32] for 300k steps with a batch size of 64, a
learning rate of 2e-4, 2500 warmup steps and a cosine decay schedule.

##  4 Experiments

Fig. 2:  Comparison between absolute value of the individual source
spectrograms of Groundtruth, Autoencoder estimates and AudioSlots (Ours)
estimates. The input spectrogram (top) is a mixture and rest of the rows show
the spectrograms of the individual sources. The input and groundtruth
spectrograms are preprocessed using the steps mentioned in Sec 3. Notice that
our method is able to reconstruct harmonics fairly well, however struggles
with estimating the high-frequency features (see highlighted example regions).
Table 1:  Results. We use the estimated absolute spectrograms as masks over
the input complex spectrogram to produce complex spectrograms of the
individual sources [33]. In the table below we present SI-SNR[dB] and SI-
SNRi[dB] values with IBM and Wiener-like masking. Higher is better. Notice
that there is only a minimal drop in performance between Autoencoder and
AudioSlots showing that our pipeline is able to learn to separate speech well.

| SI-SNR | SI-SNRi | SI-SNR | SI-SNRi
---|---|---|---|---
Masking Type | IBM | IBM | Wiener | Wiener
Oracle | 12.07 | 12.07 | 12.32 | 12.33
Autoencoder | 10.19 | 10.20 | 10.15 | 10.15
AudioSlots | 09.50 | 09.50 | 09.96 | 09.97
Oracle (1-sec) | 13.15 | 13.16 | 13.46 | 13.47
AudioSlots (1-sec) | 09.66 | 09.67 | 10.20 | 10.20

We evaluate the performance of our method on speech separation using the
Libri2Mix [5] dataset. We use the anechoic version of the dataset. Each
instance in the dataset is sampled at 16kHz and 10 seconds long. Libri2Mix
contains contains utterances from both male and female speakers drawn from
LibriSpeech [34]. The train-360-clean split of the dataset contains 364 hours
of mixtures and the sources are drawn without replacement.

As mentioned above we use the masking to estimate the complex spectrograms of
the individual sources using the input spectrogram and the network
predictions. There are various masking functions that can be used [33]. In our
experiments we use the ideal binary mask (IBM) [35] and Wiener filter like
mask as mask functions which are defined as:

$\text{IBM:}~{}m_{i}=\begin{cases}1&i=\mathrm{argmax}(m)\\\
0&otherwise\end{cases}$ $\text{“Wiener
like”:}~{}m_{i}=\frac{(m_{i})^{2}}{\Sigma_{i=1}^{n}(m_{i})^{2}}$

Metrics: We measure the separation performance using scale-invariant signal-
to-noise ratio (SI-SNR) [36] and SI-SNR improvement (SI-SNRi). Let $y$ denote
the target and $\hat{y}$ denote the estimate obtained by our method. Then SI-
SNR measures the fidelity between $y$ and $\hat{y}$ within an arbitrary scale
by rescaling the target:

$\text{SI-SNR}(y,\hat{y})=10\log_{10}\frac{||\alpha y||^{2}}{||\alpha
y-\hat{y}||^{2}}$

where $\alpha=\mathrm{argmin}_{a}||ay-\hat{y}||^{2}=y^{T}\hat{y}/||y||^{2}$.
The SI-SNRi is the difference between the SI-SNR of each source estimate after
processing and the SI-SNR obtained using the input mixture as the estimate for
each source. During evaluation we first match the targets and estimates to
maximize SI-SNR and then average the resulting SI-SNR and SI-SNRi scores.

Results: In Table 1 we compare our performance with an autoencoder variant of
our method (which receives ground-truth reference sources as input) and the
performance of separation obtained using the (preprocessed) ground-truth
signals. The metrics computed using the ground-truth signals represent the
maximum values that can be obtained with our (lossy) preprocessing. In the
autoencoder variant we train the network to reconstruct the individual source
signals with $n=1$. We then use the individual estimates as the masks over the
complex spectrogram of the mixture. Since the spectrograms contain high-
frequency features, this would help us understand the ability of our
architecture to faithfully represent these features. We also present an
ablation by increasing the crop length in the preprocessing step to 1 second.

The difference in performance between the autoencoder variant and the
separation model is only $0.18\pm 0.01$. This indicates that our method,
AudioSlots, is able learn speech separation well, closely approaching the
performance of the baseline which receives fully-separated sources as input.

Still, there is substantial headroom for improvement, both in terms of our
model as well as our overall pipeline: prior masking-based approaches [11, 12,
13] already solve Libri2Mix speaker separation to an impressive degree,
achieving significantly higher SI-SNR values than we report here. Very
recently, diffusion-based approaches have also shown competitive performance
[37]. This gap is in part due to our lossy preprocessing pipeline: for
example, computing STFTs on pre-chunked audio (done here for simplicity)
introduces border artifacts which even reduces our ground-truth SI-SNR scores
below what other models can achieve. We further zero-pad all audio signals to
the same length for simplicity. Also, since we create masks from the generated
spectrograms, we are also bound by the limitations of mask-based methods, e.g.
that masked spectral content cannot be regenerated.

Limitations: Our experimental comparison highlights the main limitation of our
method, which is reconstruction fidelity: both the autoencoder baseline and
our AudioSlots model encode the signal using a latent bottleneck
representation and tend to discard certain high-frequency details (among
others). This is also qualitatively visible in Figure 2. The comparison
further shows that the crop length affects the separation performance: while
this can be partially explained by boundary artifacts due to our simplistic
cropping approach, it also hints towards sensitivity of AudioSlot’s ability to
both separate and reconstruct audio spectrograms on chunk length.

Discussion: Our results show promise for addressing audio separation using
slot-centric generative models that represent the audio using a set of source-
specific latent variables. This is a significant departure from earlier
methods that directly operate on the input audio using e.g. a masking-based
approach [20]. Learning source-specific latent variables further has the
benefit that these decomposed latent variables can likely be used not just for
generation, but also for recognition tasks, similar to how slots in object-
centric computer vision models serve as a basis for object detection and
segmentation.

We are optimistic that the current limitations of our approach can be overcome
in future work:

  * •

To address the issue of reconstruction fidelity (blurry reconstructions for
high-frequency features), it is likely that moving away from a deterministic
feedforward decoder to e.g. an autoregressive decoding approach, as in AudioLM
[38], or an iterative diffusion-based decoder, as in [39], can bridge the gap
to high-fidelty generation.

  * •

At present, AudioSlots assumes supervision in the form of ground-truth
sources. An extension to fully-unsupervised training on raw, mixed audio would
be desirable. To this end, we explored replacing the Transformer in AudioSlots
with a Slot Attention [3] module which has an inductive bias towards
decomposition that allows it to be trained unsupervised in the context of
visual scene decomposition. In initial experiments we found, however, that
this inductive bias might not suffice for decomposing audio spectrograms in a
fully-unsupervised fashion. A supervised version of AudioSlots with a Slot
Attention module, however, performed similar to the version with a Transformer
module in initial experiments, highlighting that further exploration is still
promising for future work.

  * •

We think that the limitation of processing individual chunks in isolation,
which requires post-hoc stitching, can be overcome by using a sequential
extension of the model, where slots of the past time step are used as
initialization for the next time step as in Slot Attention for Video [40]. We
leave this for future work.

##  5 Conclusion

We present AudioSlots, a slot-centric generative architecture for audio
spectrograms. We demonstrate a proof of concept that AudioSlots holds promise
for addressing the task of audio source separation using structured generative
models. While our current implementation of AudioSlots has several
limitations, including low reconstruction fidelity for high-frequency features
and requiring separated audio sources as supervision, we are optimistic that
these can be overcome and outline several possible directions for future work.

## References

  * [1] Nicolas Carion, Francisco Massa, Gabriel Synnaeve, Nicolas Usunier, Alexander Kirillov, and Sergey Zagoruyko,  “End-to-end object detection with transformers,”  in ECCV, 2020.
  * [2] Klaus Greff, Raphaël Lopez Kaufman, Rishabh Kabra, Nick Watters, Christopher Burgess, Daniel Zoran, Loic Matthey, Matthew Botvinick, and Alexander Lerchner,  “Multi-object representation learning with iterative variational inference,”  in ICML, 2019.
  * [3] Francesco Locatello, Dirk Weissenborn, Thomas Unterthiner, Aravindh Mahendran, Georg Heigold, Jakob Uszkoreit, Alexey Dosovitskiy, and Thomas Kipf,  “Object-centric learning with slot attention,”  NeurIPS, 2020.
  * [4] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Lukasz Kaiser, and Illia Polosukhin,  “Attention is all you need,”  NeurIPS, 2017.
  * [5] Joris Cosentino, Manuel Pariente, Samuele Cornell, Antoine Deleforge, and Emmanuel Vincent,  “Librimix: An open-source dataset for generalizable speech separation,”  arXiv preprint arXiv:2005.11262, 2020.
  * [6] John R Hershey, Zhuo Chen, Jonathan Le Roux, and Shinji Watanabe,  “Deep clustering: Discriminative embeddings for segmentation and separation,”  in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2016.
  * [7] Yusuf Isik, Jonathan Le Roux, Zhuo Chen, Shinji Watanabe, and John R Hershey,  “Single-channel multi-speaker separation using deep clustering,”  arXiv preprint arXiv:1607.02173, 2016.
  * [8] Morten Kolbæk, Dong Yu, Zheng-Hua Tan, and Jesper Jensen,  “Multitalker speech separation with utterance-level permutation invariant training of deep recurrent neural networks,”  IEEE/ACM Transactions on Audio, Speech, and Language Processing, 2017.
  * [9] Yi Luo and Nima Mesgarani,  “Conv-tasnet: Surpassing ideal time–frequency magnitude masking for speech separation,”  IEEE/ACM Transactions on Audio, Speech, and Language Processing, 2019.
  * [10] Ethan Manilow, Gordon Wichern, Prem Seetharaman, and Jonathan Le Roux,  “Cutting music source separation some Slakh: A dataset to study the impact of training data quality and quantity,”  in Proc. IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA), 2019.
  * [11] Yi Luo, Zhuo Chen, and Takuya Yoshioka,  “Dual-path rnn: efficient long sequence modeling for time-domain single-channel speech separation,”  in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2020.
  * [12] Cem Subakan, Mirco Ravanelli, Samuele Cornell, Mirko Bronzi, and Jianyuan Zhong,  “Attention is all you need in speech separation,”  in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2021.
  * [13] Kai Li, Runxuan Yang, and Xiaolin Hu,  “An efficient encoder-decoder architecture with top-down attention for speech separation,”  arXiv preprint arXiv:2209.15200, 2022.
  * [14] Marco Tagliasacchi, Yunpeng Li, Karolis Misiunas, and Dominik Roblek,  “SEANet: A multi-modal speech enhancement network,”  in Proc. Interspeech, 2020.
  * [15] Zhong-Qiu Wang, Samuele Cornell, Shukjae Choi, Younglo Lee, Byeong-Yeol Kim, and Shinji Watanabe,  “Tf-gridnet: Making time-frequency domain models great again for monaural speaker separation,”  arXiv preprint arXiv:2209.03952, 2022.
  * [16] Felix Weninger, John R Hershey, Jonathan Le Roux, and Björn Schuller,  “Discriminatively trained recurrent neural networks for single-channel speech separation,”  in 2014 IEEE global conference on signal and information processing (GlobalSIP). IEEE, 2014, pp. 577–581.
  * [17] Yuma Koizumi, Shigeki Karita, Scott Wisdom, Hakan Erdogan, John R Hershey, Llion Jones, and Michiel Bacchiani,  “Df-conformer: Integrated architecture of conv-tasnet and conformer using linear complexity self-attention for speech enhancement,”  in 2021 IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA). IEEE, 2021, pp. 161–165.
  * [18] Zhuo Chen, Yi Luo, and Nima Mesgarani,  “Deep attractor network for single-microphone speaker separation,”  in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). IEEE, 2017.
  * [19] Ben Mildenhall, Pratul P Srinivasan, Matthew Tancik, Jonathan T Barron, Ravi Ramamoorthi, and Ren Ng,  “Nerf: Representing scenes as neural radiance fields for view synthesis,”  in ECCV, 2020.
  * [20] Scott Wisdom, Efthymios Tzinis, Hakan Erdogan, Ron Weiss, Kevin Wilson, and John Hershey,  “Unsupervised sound separation using mixture invariant training,”  NeurIPS, 2020.
  * [21] Oriol Vinyals, Samy Bengio, and Manjunath Kudlur,  “Order matters: Sequence to sequence for sets,”  arXiv preprint arXiv:1511.06391, 2015.
  * [22] Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnabas Poczos, Russ R Salakhutdinov, and Alexander J Smola,  “Deep sets,”  NeurIPS, 2017.
  * [23] Juho Lee, Yoonho Lee, Jungtaek Kim, Adam Kosiorek, Seungjin Choi, and Yee Whye Teh,  “Set transformer: A framework for attention-based permutation-invariant neural networks,”  in ICML, 2019.
  * [24] Yan Zhang, Jonathon Hare, and Adam Prugel-Bennett,  “Deep set prediction networks,”  NeurIPS, 2019.
  * [25] Adam R Kosiorek, Hyunjik Kim, and Danilo J Rezende,  “Conditional set generation with transformers,”  arXiv preprint arXiv:2006.16841, 2020.
  * [26] Yi Zhou, Hui Zhang, Hana Lee, Shuyang Sun, Pingjun Li, Yangguang Zhu, ByungIn Yoo, Xiaojuan Qi, and Jae-Joon Han,  “Slot-vps: Object-centric representation learning for video panoptic segmentation,”  in CVPR, 2022.
  * [27] Pradyumna Reddy, Paul Guerrero, and Niloy J Mitra,  “Search for concepts: Discovering visual concepts using direct optimization,”  arXiv preprint arXiv:2210.14808, 2022.
  * [28] Hu Xu, Juncheng Li, Alexei Baevski, Michael Auli, Wojciech Galuba, Florian Metze, Christoph Feichtenhofer, et al.,  “Masked autoencoders that listen,”  arXiv preprint arXiv:2207.06405, 2022.
  * [29] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun,  “Deep residual learning for image recognition,”  in CVPR, 2016.
  * [30] Nicholas Watters, Loic Matthey, Christopher P Burgess, and Alexander Lerchner,  “Spatial broadcast decoder: A simple architecture for learning disentangled representations in vaes,”  arXiv preprint arXiv:1901.07017, 2019.
  * [31] Matthew Tancik, Pratul Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan Barron, and Ren Ng,  “Fourier features let networks learn high frequency functions in low dimensional domains,”  NeurIPS, 2020.
  * [32] Diederik P Kingma and Jimmy Ba,  “Adam: A method for stochastic optimization,”  arXiv preprint arXiv:1412.6980, 2014.
  * [33] Hakan Erdogan, John R. Hershey, Shinji Watanabe, and Jonathan Le Roux,  “Phase-sensitive and recognition-boosted speech separation using deep recurrent neural networks,”  in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2015.
  * [34] Vassil Panayotov, Guoguo Chen, Daniel Povey, and Sanjeev Khudanpur,  “Librispeech: an asr corpus based on public domain audio books,”  in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2015.
  * [35] Yipeng Li and DeLiang Wang,  “On the optimality of ideal binary time–frequency masks,”  Speech Communication, 2009.
  * [36] Jonathan Le Roux, Scott Wisdom, Hakan Erdogan, and John R Hershey,  “Sdr–half-baked or well done?,”  in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2019.
  * [37] Shahar Lutati, Eliya Nachmani, and Lior Wolf,  “Separate and diffuse: Using a pretrained diffusion model for improving source separation,”  arXiv preprint arXiv:2301.10752, 2023.
  * [38] Zalán Borsos, Raphaël Marinier, Damien Vincent, Eugene Kharitonov, Olivier Pietquin, Matt Sharifi, Olivier Teboul, David Grangier, Marco Tagliasacchi, and Neil Zeghidour,  “Audiolm: a language modeling approach to audio generation,”  arXiv preprint arXiv:2209.03143, 2022.
  * [39] Curtis Hawthorne, Ian Simon, Adam Roberts, Neil Zeghidour, Josh Gardner, Ethan Manilow, and Jesse Engel,  “Multi-instrument music synthesis with spectrogram diffusion,”  arXiv preprint arXiv:2206.05408, 2022.
  * [40] Thomas Kipf, Gamaleldin F. Elsayed, Aravindh Mahendran, Austin Stone, Sara Sabour, Georg Heigold, Rico Jonschkowski, Alexey Dosovitskiy, and Klaus Greff,  “Conditional Object-Centric Learning from Video,”  in ICLR, 2022.