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##  Appendix A Corpus Description

Paper |  Sector |  Domain |  Purpose |  Methods |  Stakeholder
---|---|---|---|---|---
Frauenberger et al. (2012) (Frauenberger et al., 2012) |  Academia |  Education |  Scaffold the development of social skills for children |  Design sessions and workshops |  Children with and without Autism Spectrum Conditions
Duarte et al. (2014) (Duarte et al., 2014) |  Academia |  Assistive Living |  Assist therapists during interventions through gesture recognition |  Storytelling |  Children with Autism Spectrum Disorder
Šabanović et al. (2015) (Šabanović et al., 2015) |  Academia |  Assistive Living |  Provide companionship and therapeutic value |  Design sessions and workshops |  Independently living older adults experiencing co-occurring chronic mental and physical illness
Katan et al. (2015) (Katan et al., 2015) |  Academia |  Assistive Living |  Foster musical creativity |  Design sessions and workshops |  Musicians with learning and physical disabilities
Vines et al. (2015) (Vines et al., 2015) |  Academia |  Assistive Living |  Sustain peer-to-peer knowledge sharing platform |  Design sessions and workshops |  Older people
Azenkot et al. (2016) (Azenkot et al., 2016) |  Academia |  Assistive Living |  Interact with and guide a blind person through a building in an effective and socially acceptable way |  Design sessions and workshops |  People with visual impairment
Krishnaswamy (2017) (Krishnaswamy, 2017) |  Academia |  Assistive Living |  Increase independence of targeted stakeholder |  Surveys, Prototype Evaluation |  People with physical disabilities
Krishnaswamy, Moorthy, and Oates (2017) (Krishnaswamy et al., 2017) |  Academia |  Assistive Living |  Increase independence of targeted stakeholder |  Surveys, Prototype Evaluation |  People with reduced motor functionality
Lee et al. (2017) (Lee et al., 2017) |  Academia |  Assistive Living |  Establish social relations with people via robots |  Interviews, Design sessions and workshops |  Older adults, Clinical and caregiving staff
Rose and Björling (2017) (Rose and Björling, 2017) |  Academia |  Assistive Living |  Measure stress |  Design sessions and workshops, Prototype Evaluation |  Teens
Noothigattu et al. (2018) (Noothigattu et al., 2018) |  Academia |  Autonomous Vehicle |  Automate ethical decisions |  Crowdsource preferences |  None specified
Luhtala et al. (2018) (Luhtala et al., 2018) |  Government |  Public Services |  Predictive rescue work service |  Ethnography, Design sessions and workshops, Wizard of Oz, Prototype Evaluation |  Personnel of rescue departments (firefighters, operators, etc.)
Woodruff et al. (2018) (Woodruff et al., 2018) |  Industry |  Not Domain-Specific |  Explore how traditionally marginalized populations feel about algorithmic fairness |  Design sessions and workshops, Interviews |  Black or African American, Hispanic or Latinx, and low socioeconomic status participants in the United States
Woodward et al. (2018) (Woodward et al., 2018) |  Academia |  Assistive Living |  Improve recognition and accuracy |  Design sessions and workshops |  Older adults with depression, Therapists, Case workers
Chandrasekharan et al. (2019) (Chandrasekharan et al., 2019) |  Academia |  Online Moderation |  Automate decisions for content moderation |  Prototype Evaluation, Interviews |  Moderators of online communities
Herbig et al. (2019) (Herbig et al., 2019) |  Academia |  Translation |  Translate text |  Elicitation Study, Interviews |  Professional Translators
Lee et al. (2019) (Lee et al., 2019) |  Academia |  Social Welfare |  Allocate resources |  Preference Aggregation |  Donors, Volunteers, Recipient organizations, and Nonprofit organizations
Metatla, Oldfield, and Ahmed (2019) (Metatla et al., 2019) |  Academia |  Education |  Support inclusive education |  Focus Groups, Bodystorming, Design sessions and workshops |  Pupils with visual impairments
Hitron et al. (2019) (Hitron et al., 2019) |  Academia |  Education |  Recognize gestures to support learning |  Prototype Evaluation, Controlled Experiment |  Children
Kayacik et al. (2019) (Kayacik et al., 2019) |  Industry |  Arts & Entertainment |  Foster creativity for musicians |  Interviews, Design sessions and workshops, Prototype Evaluation |  Musicians
Bhargava et al. (2019) (Bhargava et al., 2019) |  Academia |  Online Moderation |  Filtering Content |  Survey, Prototype Evaluation |  Social media users
Grönvall and Lundberg (2019) (Grönvall and Lundberg, 2019) |  Academia |  Public Services |  Support parents with intellectual disabilities taking care of their small children |  Focus Groups, Design sessions and workshops |  Social workers
Walsh and Wronsky (2019) (Walsh and Wronsky, 2019) |  Academia |  Not Domain-Specific |  Create more inclusive co-design experiences for marginalized populations |  Wizard of Oz, Prototype Evaluation |  Marginalized populations
Brown et al. (2019) (Brown et al., 2019) |  Government |  Public Services |  Determine whether children should be placed in the child welfare system |  Design sessions and workshops |  Families involved in the child welfare system, Employees of child welfare agencies
Yadav et al. (2019) (Yadav et al., 2019) |  Academia |  Education |  Educate women in India on pregnancy-related issues |  Wizard of Oz |  Breastfeeding mothers from the slum areas of Delhi, India
Møller, Shklovski, and Hildebrandt (2020) (Holten Møller et al., 2020) |  Academia |  Public Services |  Support decision-making for job placement |  Design sessions and workshops |  Case workers
Alfrink and Tur (2020) (Alfrink et al., 2020) |  Academia |  Urban Planning |  Make decisions for electric vehicle (EV) charge points |  Prototype Evaluation |  Electric vehicle drivers
Candello et al. (2020) (Candello et al., 2020) |  Industry |  Education |  Teach children basic AI concepts |  Role-Playing, Design sessions and workshops |  Museum curators, guides, and educators
Kearns et al. (2020) (Kearns et al., 2020) |  Academia |  Healthcare |  Communicate health information |  Wizard of Oz, Design sessions and workshops |  Clinicians
Scott et al. (2022) (Scott et al., 2022) |  Academia |  Public Services |  Support job counselors in assessing job seekers as well as resource allocation (e.g., skills training, unemployment benefits) |  Design Sessions/Workshops |  Job seekers, practitioners
Mucha et al. (2020) (Mucha et al., 2020) |  Academia |  Multiple Domains |  Involve non-experts in complex technical discourses |  Workbook Sprint |  University students as non-experts
Saxena and Guha (2020) (Saxena and Guha, 2020) |  Academia |  Public Services |  Predict if a child needs state assistance |  Ethnography, Interviews |  Case workers in the Child-Welfare System
Skinner, Brown, and Walsh (2020) (Skinner et al., 2020) |  Academia |  Education |  Function as librarian in local libraries |  Design sessions and workshops |  Children of color in non-affluent neighborhoods of Baltimore City
Smith et al. (2020) (Smith et al., 2020) |  Academia, Nonprofit |  Online Moderation |  Predict quality of edits |  Interviews |  Wikipedia editors
Troiano, Wood, and Harteveld et al. (2020) (Troiano et al., 2020) |  Academia |  Intimacy |  Foster sexual interactions with humans |  Story Completion Method (SCM) |  Communities of fluent English writers (e.g., ”Fantasy Writers” on Reddit)
Yu et al. (2020) (Yu et al., 2020) |  Academia |  Criminal Justice |  Criminal defendants’ likelihood to re-offend |  Prototype Evaluation, Experimental Study, Interviews |  Criminal justice experts
Georgiou et al. (2020) (Georgiou et al., 2020) |  Academia |  Assistive Living |  Support stroke survivors in the home environment |  Sketching, Demoing, Focus Groups |  Stroke survivors
Halfaker and Geiger (2020) (Halfaker and Geiger, 2020) |  Academia, Nonprofit |  Online Moderation |  Predict quality of edits |  Aggregate stakeholder inputs on the design of system specs |  Wikipedia editors
Zhang, Bellamy, and Varshney (2020) (Zhang et al., 2020) |  Industry |  Not Domain-Specific |  Support decision-making |  Preference Aggregation |  None specified
Sendak et al. (2020) (Sendak et al., 2020) |  Academia |  Healthcare |  Diagnose conditions |  Immerse project team in existing workflow, Accept project proposal from stakeholders |  Front-line clinical professionals
Cheng et al. (2020) (Cheng et al., 2021) |  Academia |  Public Services |  Predict child maltreatment |  Preference Aggregation |  Social workers, Parents
Park and Lim (2020) (Park and Lim, 2020) |  Academia |  Communication |  Foster family-oriented sociality |  Design sessions and workshops |  People in family units
Vaziri et al. (2020) (Vaziri et al., [n. d.]) |  Academia |  Service Operation |  Collaborate with humans for service work |  Design sessions and workshops |  Service operators
Katell et al. (2020) (Katell et al., 2020) |  Academia |  Public Policy |  Re-center power with those most disparately affected by the harms of algorithmic systems |  Design sessions and workshops |  Non-profit organizations
Lin and Van Brummelen (2021) (Lin and Van Brummelen, 2021) |  Academia |  Education |  Integrate AI into core curriculum to leverage learners’ interests |  Design sessions and workshops |  K-12 teachers
Neto, Nicolau, and Paiva (2021) (Neto et al., 2021) |  Academia |  Education / Assistive Living |  Foster collaborative learning for visually impaired children |  Contextual inquiries, Participant observation, Interviews |  Visually impaired children
Jacobs et al. (2021) (Jacobs et al., 2021) |  Academia |  Healthcare |  Support antidepressant treatment decisionsa |  Interviews, Focus groups, Prototype evaluation |  Primary care physicians
Heitlinger et al. (2021) (Heitlinger et al., 2021) |  Academia |  Agriculture |  Foster decentralised non-extractive value exchange |  Role playing, Design sessions and workshops |  Urban agricultural communities
Subramonyam, Seifert, and Adar (2021) (Subramonyam et al., 2021) |  Academia |  Not Domain-Specific |  Identify desirable AI characteristics |  Data probes |  AI designers and engineers
Scurto, Caramiaux, and Bevilacqua (2021) (Scurto et al., 2021) |  Academia |  Arts & Entertainment |  Foster music performance and robotic art |  Art practices |  Musicians
Raz et al. (2021) (Raz et al., 2021) |  Academia |  Not Domain-Specific |  Demonstrate disparate accuracy rates in facial recognition |  Interactive demoing, Panel discussions, |  Community organizers
Lee et al. (2021) (Lee et al., 2021) |  Academia |  Labor Relations |  Manage the workforce |  Preference aggregation |  Shift workers
Schelenz et al. (2021) (Schelenz et al., 2021) |  Academia |  Online Moderation |  Foster diversity-aware social media platform |  Preference aggregation |  None specified
Bozic Yams and Aranda Muñoz (2021) (Bozic Yams and Aranda Muñoz, 2021) |  Academia |  Labor Relations |  Augment employee creativity and support their well-being at work |  Art practices, Design sessions and workshops |  Professionals with non-technical background
Li et al. (2021) (Li et al., 2021) |  Academia |  Software Engineering |  Enable natural language search queries and intuitive visualizations for code search |  Brainstorming sessions, Interviews, Prototype Evaluation, Design sessions and workshops |  Experts in software engineering
Flügge, Hildebrandt, and Møller (2021) (Ammitzbøll Flügge et al., 2019) |  Government |  Employment |  Support decision-making for job placement |  Design sessions and workshops, Interviews |  Case workers
Candello, Pichiliani, and Pinhanez (2021) (Candello et al., 2021) |  Industry |  Education |  Teach children basic AI concepts |  Role-Playing, Design sessions and workshops |  Museum curators, guides, and educators
Saxena et al. (2021) (Saxena et al., 2021) |  Academia |  Public Services |  Predict if a child needs state assistance |  Ethnography, Interviews |  Case workers in the Child-Welfare System
Zhang et al. (2022) (Zhang et al., 2022b) |  Academia, Industry |  Education |  Create interactive storytelling experiences |  Interviews, Design Sessions/Workshops |  Parents, children
Bäuerle et al. (2022) (Bäuerle et al., 2022) |  Industry |  Not Domain-Specific |  Not System-Specific |  Interviews, Design Sessions/Workshops, Prototype Evaluation |  Machine learning practitioners
Zhang et al. (2022) (Zhang et al., 2022a) |  Academia |  Labor Relations |  Manage, organize, coordinate, and evaluate workers |  Design Sessions/Workshops, Focus Groups |  Rideshare drivers
Park et al. (2022) (Park et al., 2022) |  Academia |  Labor Relations |  Evaluate employees’ work performance evaluation |  Design Sessions/Workshops |  Employees, employers/HR teams, AI/business experts
Stapleton et al. (2022) (Stapleton et al., 2022) |  Academia |  Public Services |  Inform workers’ decision-making for child welfare |  Design Sessions/Workshops |  Those who have been impacted by the child welfare system, those who work in the system
Suresh et al. (2022) (Suresh et al., 2022) |  Academia, Nonprofit |  Activism |  Retrieve and filter media coverage of feminicide |  Prototype Evaluation, Design Sessions/Workshops, Focus Groups, Surveys |  Activists
Miceli et al. (2022) (Miceli et al., [n. d.]) |  Academia, Nonprofit |  Not Domain-Specific |  Not System-Specific |  Interviews, Design Sessions/Workshops |  Data Workers
Storms et al. (2022) (Storms et al., [n. d.]) |  Academia |  Content Moderation |  Provide personalized content recommendation |  Prototype Evaluation, Design Sessions/Workshops |  News Audiences
Wolf and Blomberg (2019) (Wolf and Blomberg, [n. d.]) |  Industry |  Computer-Assisted Cooperative Work |  Support solution design |  Ethnography |  IT Architects
Park et al. (2022) (Park et al., [n. d.]) |  Academia |  Community Engagement |  Select volunteers for academic conferences |  Interviews |  Student Volunteers (SV)
Chen et al. (2022)(Chen et al., [n. d.]) |  Academia |  Content Moderation |  Provide personalized content recommendation |  Interviews |  Users and Practitioners
Lee et al. (2020) (Lee et al., [n. d.]) |  Academia |  Healthcare |  Identify salient features for stroke rehabilitation assessment |  Interviews, Prototype Evaluation |  Therapists
Benjamin et al. (2022) (Benjamin et al., [n. d.]) |  Academia |  Not Domain-Specific |  Not System-Specific |  Prototype Evaluation, Design Sessions/Workshops |  Non-ML Experts
Iqbal et al. (2021) (Iqbal et al., [n. d.]) |  Academia |  Computer-Assisted Cooperative Work |  Retrieve information from email records |  Interviews, Test Collection |  Potential Donors of Personal Records
Gu et al. (2021) (Gu et al., [n. d.]) |  Academia |  Healthcare |  Diagnose disease |  Prototype Evaluation |  Pathologists
Calacci and Pentland (2019) (Calacci and Pentland, [n. d.]) |  Academia |  Labor Relations |  Audit changes to platform workers’ pay |  Prototype Evaluation |  Preference Aggregation, Platform Workers
Puussaar et al. (2022) (Puussaar et al., [n. d.]) |  Academia |  Public Services |  Data-driven decision-making in local and national government |  Community Meeting, Design Workshops/Sessions, Ethnography |  Local Residents
Kaur et al. (2019) (Kaur et al., [n. d.]) |  Academia |  Employment |  Sort and screen job applications |  Speculative Design |  HR recruiters and job applicants
Nakao and Sugano (2020) (Nakao and Sugano, [n. d.]) |  Academia |  Accessibility |  Sound recognition |  Interviews, Design Workshops/Sessions |  Non-Expert Deaf, Hard-of-Hearing (DHH) People
Komatsu et al. (2020) (Komatsu et al., [n. d.]) |  Academia |  Computer-Assisted Cooperative Work |  Not System-Specific |  Interviews |  Journalists
Alvarado et al.(2020) (Alvarado et al., 2020) |  Academia |  Not Domain-Specific |  Not System-Specific |  Interviews, Diary Studies, Design Workshops/Sessions |  Not-specific
Seidelin et al. (2020) (Seidelin et al., 2020) |  Academia |  Not Domain-Specific |  Not System-Specific |  Interviews, Design Workshops/Sessions, Prototype Evaluation |  Domain Experts


# Epidemic Propagation under Evolutionary Behavioral Dynamics:
Stability and Bifurcation Analysis ††thanks: This work was supported in part
by ARO grant W911NF-18-1-0325.

Abhisek Satapathi, Narendra Kumar Dhar, Ashish R. Hota, and Vaibhav Srivastava
A. Satapathi and A. R. Hota are with the Department of Electrical Engineering,
IIT Kharagpur, Kharagpur, West Bengal, India, 721302. Email:
abhisek.ee<EMAIL_ADDRESS>K. Dhar and V. Srivastava are
with the Department of Electrical and Computer Engineering, Michigan State
University, East Lansing, MI 48824. Email: {dharnare<EMAIL_ADDRESS>
###### Abstract

We consider the class of SIS epidemic models in which a large population of
individuals chooses whether to adopt protection or to remain unprotected as
the epidemic evolves. For a susceptible individual, adopting protection
reduces the probability of becoming infected but it comes with a cost that is
weighed with the instantaneous risk of becoming infected. An infected
individual adopting protection transmits a new infection with a smaller
probability compared to an unprotected infected individual. We focus on the
replicator evolutionary dynamics to model the evolution of protection
decisions by susceptible and infected subpopulations. We completely
characterize the existence and local stability of the equilibria of the
resulting coupled epidemic and replicator dynamics. We further show how the
stability of different equilibrium points gets exchanged as certain parameters
change. Finally, we investigate the system behavior under timescale separation
between the epidemic and the evolutionary dynamics.

##  I Introduction

Infectious diseases or epidemics spread through human society via social
interactions among infected and healthy individuals. There is a rich body of
work devoted to developing mathematical models that capture the dynamic
evolution of such epidemics [1, 2]. In the well-studied susceptible-infected-
susceptible (SIS) epidemic model, an individual belongs to one of two
infectious states: susceptible (healthy) and infected. A susceptible
individual becomes infected with a certain probability termed infection rate
if it comes in contact with an infected individual, while an infected
individual recovers with a certain recovery rate.

Individuals adopt protective measures, such as wearing masks or becoming
vaccinated, to avoid becoming infected, often in a strategic and decentralized
manner. Consequently, the interplay between epidemic dynamics and human
decision-making has been studied in the framework of game theory [3]; both
static or single-shot games to model vaccination decisions [4, 5] and dynamic
games that model evolution of protective decisions [6, 7, 8] have been
analyzed for SIS epidemics in recent past. Most of these works explore
structural properties of the equilibrium strategies and epidemic evolution
under those strategies.

However, the problem of a large group of strategic individuals finding or
learning equilibrium strategies is quite challenging [9, 10]. In settings with
a large population of agents, evolutionary dynamics have been proposed and
their convergence behavior to the set of equilibria is analyzed [11, 10]. This
class of learning dynamics involves repeated play or interaction of a static
game with the same set of payoff functions. Evolutionary learning with dynamic
payoff functions has also recently been explored by exploiting the concept of
passivity [12, 13]. In contrast with the above settings, epidemic games
exhibit a dynamically changing proportion of healthy and infected individuals
due to epidemic dynamics. A recent work [14] investigated this dynamic
evolution of infection dynamics and strategic decisions for a class of
susceptible-asymptomatic-infected-recovered (SAIR) epidemic setting in the
framework of dynamic population games [15].

Timescale separation has been used to study the interaction of both
evolutionary as well as epidemic dynamics with other dynamical systems. For
example, slow-fast coupled environmental and evolutionary game dynamics were
first studied in [16]. A recent paper [17] also studies slow-fast dynamics in
a setting related to environment and ecology. In the context of epidemics, a
recent work [18] explores slow-fast dynamics in the context of SIR epidemics
but does not consider a behavioral response to epidemics or any game-theoretic
considerations. SIS dynamics with risk-tolerator or risk-averter individuals
that respond differently as a function of the leaky-integrated infectious
population are studied in [19]; however, game-theoretic dynamics of such
behavioral responses have not been investigated. Coupled evolution of disease
and behavior has recently been investigated in [20, 21] for SIRS and SEIR
epidemic models, respectively.

In this paper, we build upon the above line of work and consider the class of
SIS epidemic models where a large population of individuals chooses whether to
adopt protection or to remain unprotected as the epidemic evolves. For a
susceptible individual, being protected reduces the probability of becoming
infected. Similarly, an infected individual who adopts protection transmits or
causes a new infection with a smaller probability compared to an infected
individual who is unprotected. For a susceptible individual, adopting
protection comes with a cost (such as discomfort due to wearing masks or
adhering to social distancing protocols) which is weighed with the
instantaneous risk of becoming infected; the latter depends on the current
epidemic prevalence and the proportion of individuals adopting protection.

In order to lay the foundations for a rigorous investigation of coupled
epidemic and behavioral dynamics, we focus on the replicator dynamics [11, 10]
to model the evolution of protection decisions by susceptible and infected
subpopulations. We completely characterize the equilibria (existence and local
stability) of the coupled epidemic and replicator dynamics and show how the
stability of different equilibrium points gets exchanged as certain parameters
change.

We further explore the behavior of the coupled dynamics under timescale
separation leading to a slow-fast dynamical system [22]. We focus on two
timescale separation scenarios. First, we analyze the case in which the
replicator dynamics are faster and the behavior converges to equilibrium
strategies quickly leading to a variable structure system as observed in prior
works (such as [8]). We then consider the case where the epidemic dynamics are
faster than the replicator dynamics. We numerically illustrate the dynamic
bifurcation behavior and the bifurcation delay phenomenon. Our results show
that the timescale separation of disease and behavioral dynamics plays a
significant role in disease evolution, and may even lead to inadequate and
counterproductive epidemic control measures if not properly accounted for.

##  II Coupled SIS Epidemic and Evolutionary Behavioral Model

In this section, we formally introduce the coupled evolution of the SIS
epidemic and protection adoption behavior in a homogeneous large-population
setting. Let the proportion of susceptible and infected individuals be $s(t)$
and $y(t)$, respectively. Both $s(t),y(t)\in[0,1]$ with $s(t)+y(t)=1$ for all
$t\geq 0$. Individuals choose whether to adopt protection against the epidemic
or not; these actions are denoted by $\mathtt{P}$ and $\mathtt{U}$,
respectively. Consequently, the population state at time $t$ is defined as
$x(t):=[x_{\mathtt{SU}}(t)\quad x_{\mathtt{SP}}(t)\quad
x_{\mathtt{IU}}(t)\quad x_{\mathtt{IP}}(t)]^{\intercal}\in[0,1]^{4}$, where
$x_{\mathtt{SU}}$ denotes the proportion of individuals who are susceptible
and choose to remain unprotected, $x_{\mathtt{IP}}$ denotes the proportion of
infected individuals who adopt protection, and so on. At time $t$, we have
$x_{\mathtt{SU}}(t)+x_{\mathtt{SP}}(t)=s(t)$,
$x_{\mathtt{IU}}(t)+x_{\mathtt{IP}}(t)=y(t)$, and
$\mathbb{1}^{\intercal}x(t)=1$ where $\mathbb{1}$ is a vector of appropriate
dimension with all entries $1$.

Individuals choose their action to maximize their payoffs which depends on the
population state $x(t)$ (which includes information regarding infection
prevalence since $x_{\mathtt{IU}}(t)+x_{\mathtt{IP}}(t)=y(t)$). For an
infected individual, there is no further risk of infection, and as a result,
we define its payoff to be constant parameters given by $-c_{\mathtt{IU}}$ if
it remains unprotected and $-c_{\mathtt{IP}}$ if it adopts protection,
respectively. In particular, $c_{\mathtt{IU}}>0$ captures the cost of breaking
isolation/quarantine protocols for an infected individual or could represent
worsening of the disease if an infected individual continues to be
irresponsible. Thus, we assume $c_{\mathtt{IU}}>c_{\mathtt{IP}}\geq 0$.

A susceptible individual trades off the cost of adopting protection, denoted
by $c_{P}>0$, and the expected cost of becoming infected. The latter is the
product of the loss upon infection $L>0$ and the instantaneous probability of
becoming infected which depends on its action as well as the population state.
Specifically, let $\beta_{u}$ and $\beta_{p}$ denote the probabilities of an
infected individual causing a new infection if it is unprotected and
protected, respectively. We assume that $\beta_{u}>\beta_{p}\geq 0$.
Similarly, let a protected susceptible individual be $\alpha\in(0,1)$ times
(less) likely to become infected compared to an unprotected susceptible
individual.

Thus, the payoff vector of individuals at population state $x$ is now defined
as

$F(x):=\begin{bmatrix}F_{\mathtt{SU}}(x)\\\ F_{\mathtt{SP}}(x)\\\
F_{\mathtt{IU}}(x)\\\
F_{\mathtt{IP}}(x)\end{bmatrix}=\begin{bmatrix}-L(\beta_{u}x_{\mathtt{IU}}+\beta_{p}x_{\mathtt{IP}})\\\
-c_{P}-L\alpha(\beta_{u}x_{\mathtt{IU}}+\beta_{p}x_{\mathtt{IP}})\\\
-c_{\mathtt{IU}}\\\ -c_{\mathtt{IP}}\end{bmatrix},$ (1)

where $F_{\mathtt{SU}}(x)$ denote the payoff for an individual who is
susceptible and unprotected at population state $x$, and so on. Susceptible
individuals who adopt protection pay a cost $c_{P}$ but experience a reduced
risk of becoming infected scaled by factor $\alpha$ as discussed above.

The population state $x$ is dynamic and time-varying, both due to the epidemic
dynamics as well as the evolutionary dynamics of strategically adopting
protective behavior. Since the proportion of susceptible and infected
individuals is time-varying, the framework of classical population games is
not applicable. We now introduce the coupled epidemic and evolutionary
dynamics. We first introduce some notation. Let $z_{S}(t)\in[0,1]$ denote the
fraction of susceptible nodes who remain unprotected, i.e.,
$x_{\mathtt{SU}}(t)=z_{S}(t)s(t)$ and $x_{\mathtt{SP}}(t)=(1-z_{S}(t))s(t)$.
Similarly, let $z_{I}(t)\in[0,1]$ denote the fraction of infected nodes who
remain unprotected.

Due to the presence of both unprotected and protected individuals with
different infection probabilities, the evolution of infected proportion is
given by

$\displaystyle\dot{y}(t)$ $\displaystyle=(x_{\mathtt{SU}}(t)+\alpha
x_{\mathtt{SP}}(t))(\beta_{u}x_{\mathtt{IU}}(t)+\beta_{p}x_{\mathtt{IP}}(t))-\gamma
y(t)$ $\displaystyle=\big{[}(1-y(t))(z_{S}(t)+\alpha(1-z_{S}(t)))$
$\displaystyle\qquad\qquad\cdot(\beta_{u}z_{I}(t)+\beta_{p}(1-z_{I}(t)))-\gamma\big{]}y(t)$
$\displaystyle=:f_{y}(y(t),z_{S}(t),z_{I}(t)).$ (2)

The above dynamics are analogous to the conventional scalar SIS epidemic
dynamics with effective infection rate
$\beta_{\textup{eff}}=(z_{S}(t)+\alpha(1-z_{S}(t)))(\beta_{u}z_{I}(t)+\beta_{p}(1-z_{I}(t)))$
which now depends on the efficacy of protection and the fractions that adopt
protection.111In particular, if $\alpha=1$ and $\beta_{p}=\beta_{u}=\beta$,
i.e., protection is not effective, the effective infection rate is $\beta$.

TABLE I: Existence and stability of equilibria of the coupled dynamics. Epidemic | Endemic | Equilibria
---|---|---
Parameters | Infection Level | $\mathbf{E1}:(0,1,0)$ | $\mathbf{E2}:(y^{*}_{u},1,0)$ | $\mathbf{E3}:(y^{*}_{\mathrm{int}},z^{*}_{S,int},0)$ | $\mathbf{E4}:(y^{*}_{p},0,0)$
$\gamma>\beta_{p}$ | $-$ | $\checkmark$, stable | $-$ | $-$ | $-$
$\alpha\beta_{p}<\gamma<\beta_{p}$ | $y^{*}_{u}<y^{*}_{\mathrm{int}}$ |  $\checkmark$, unstable | $\checkmark$, stable | $-$ | $-$
$y^{*}_{\mathrm{int}}<y^{*}_{u}$ |  $\checkmark$, unstable |  $\checkmark$, unstable | $\checkmark$, stable | $-$
$\gamma<\alpha\beta_{p}$ | $y^{*}_{u}<y^{*}_{\mathrm{int}}$ |  $\checkmark$, unstable | $\checkmark$, stable | $-$ |  $\checkmark$, unstable
$y^{*}_{p}<y^{*}_{\mathrm{int}}<y^{*}_{u}$ |  $\checkmark$, unstable |  $\checkmark$, unstable | $\checkmark$, stable |  $\checkmark$, unstable
$y^{*}_{\mathrm{int}}<y^{*}_{p}$ |  $\checkmark$, unstable |  $\checkmark$, unstable | $-$ | $\checkmark$, stable

We now define the evolutionary dynamics to capture the evolution of
unprotected fractions in both susceptible and infected subpopulations. We
focus on the class of replicator dynamics [10] in this work and assume that
susceptible nodes only replicate the strategies of other susceptible nodes
(likewise for infected nodes). Consequently, for susceptible nodes, we obtain

$\displaystyle\dot{z}_{S}(t)$
$\displaystyle={z}_{S}(t)(1-{z}_{S}(t))\big{[}[F]_{\mathtt{SU}}-[F]_{\mathtt{SP}}\big{]}$
$\displaystyle={z}_{S}(t)(1-{z}_{S}(t))\big{[}c_{P}-L(1-\alpha)(\beta_{u}z_{I}(t)$
$\displaystyle\qquad\qquad+\beta_{p}(1-z_{I}(t)))y(t)\big{]}$
$\displaystyle=:f_{S}(y(t),z_{S}(t),z_{I}(t)).$ (3)

Similarly, for infected nodes, we have

$\displaystyle\dot{z}_{I}(t)$
$\displaystyle={z}_{I}(t)\big{[}-c_{\mathtt{IU}}+(c_{\mathtt{IU}}{z}_{I}(t)+c_{\mathtt{IP}}(1-{z}_{I}(t)))\big{]}$
$\displaystyle={z}_{I}(t)(1-{z}_{I}(t))(c_{\mathtt{IP}}-c_{\mathtt{IU}})$
$\displaystyle=:f_{I}(y(t),z_{S}(t),z_{I}(t)).$ (4)

Thus, equations (2), (3), and (4) characterize the coupled evolution of the
epidemic and population states.

######  Lemma 1 (Invariant set)

For the coupled SIS epidemic and evolutionary behavior dynamics defined by
(2), (3) and (4), the set
$\\{(y,z_{S},z_{I})\;|\;(y,z_{S},z_{I})\in[0,1]^{3}\\}$ is invariant.

###### Proof:

For $y(t)\in\\{0,1\\}$, $\dot{y}$ is either zero or negative. Likewise, for
$z_{I},z_{S}\in\\{0,1\\}$, $\dot{z}_{I}=\dot{z}_{S}=0$. The result is
immediate from Nagumo’s theorem [23, Theorem 4.7]. ∎

##  III Equilibrium Characterization and Bifurcation Analysis

In this section, we first examine the equilibrium points of the above coupled
epidemic-replicator dynamics and their stability properties. We then examine
the bifurcations associated with the changes in the stability of these
equilibria.

###  III-A Equilibrium Characterization and Stability Analysis

First, we consider the evolution of $z_{I}$ in (4) which does not depend on
$y$ and $z_{S}$. There are two stationary points $z_{I}=0$ and $z_{I}=1$, and
it is easy to see that, for $c_{\mathtt{IU}}>c_{\mathtt{IP}}$, $z_{I}=1$ is
unstable and $z_{I}=0$ is exponentially stable with basin of attraction
$[0,1)$. It is also quite intuitive that if $c_{\mathtt{IP}}<c_{\mathtt{IU}}$,
infected nodes prefer to use protection, and the strategy for infected nodes
should converge to adopting protection.

Thus, in the remainder of this paper, we only focus on equilibria with
$z_{I}=0$. We begin with introducing a few variables that will be used to
define the equilibrium points.

$y^{*}_{u}:=1-\frac{\gamma}{\beta_{p}},\quad
y^{*}_{\mathrm{int}}:=\frac{c_{P}}{L(1-\alpha)\beta_{p}},\quad
y^{*}_{p}:=1-\frac{\gamma}{\alpha\beta_{p}},\\\ \text{and
}z^{*}_{S,\mathrm{int}}:=\frac{1}{1-\alpha}\left[\frac{\gamma}{\beta_{p}(1-y^{*}_{\mathrm{int}})}-\alpha\right].$

We now define all possible equilibria of the coupled SIS epidemic and
evolutionary behavior dynamics (2–4) corresponding to $z_{I}=0$:

$\mathbf{E0}=(0,0,0),\quad\mathbf{E1}=(0,1,0),\quad\mathbf{E2}=(y^{*}_{u},1,0),\\\
\mathbf{E3}=(y^{*}_{\mathrm{int}},z^{*}_{S,\mathrm{int}},0),\text{
and}\quad\mathbf{E4}=(y_{p}^{*},0,0).$

At $\mathbf{E0}$ everyone adopts protection and there is no infection. At
$\mathbf{E1}$, there is no infection, and susceptible individuals do not adopt
protection. $\mathbf{E2}$ is an endemic equilibrium, i.e., a fraction of the
population is infected, at which susceptible individuals continue to remain
unprotected. $\mathbf{E3}$ is an endemic equilibrium at which some susceptible
individuals adopt protection. Finally, $\mathbf{E4}$ is an endemic equilibrium
at which all susceptible individuals adopt protection. The existence and
stability of these equilibria are established below and summarized in Table I.

######  Proposition 1 (Equilibria and Stability)

For the equilibrium points of the coupled SIS epidemic and evolutionary
behavioral dynamics (2–4) corresponding to $z_{I}=0$, the following statements
hold:

  1. (i.

$\mathbf{E0}$ exists for all parameter regimes, and is unstable;

  2. (ii.

$\mathbf{E1}$ exists for all parameter regimes, is stable if
$\beta_{p}<\gamma$, and is unstable, otherwise;

  3. (iii.

$\mathbf{E2}$ exists only when $\beta_{p}>\gamma$, is stable when
$y^{*}_{u}<y^{*}_{\mathrm{int}}$, and is unstable otherwise;

  4. (iv.

$\mathbf{E3}$ exists only when $y^{*}_{p}<y^{*}_{\mathrm{int}}<y^{*}_{u}$, and
is stable;

  5. (v.

$\mathbf{E4}$ exists only when $\gamma<\alpha\beta_{p}$, is stable when
$y^{*}_{p}>y^{*}_{\mathrm{int}}$, and is unstable otherwise.

The proof is presented in the Appendix.

###  III-B Bifurcation Analysis

We now numerically explore the bifurcations associated with the transition of
stability among the equilibria. For the numerical illustration we choose the
parameter values in equations (2), (3), and (4) as summarized below:

$c_{P}$ | $\alpha$ | $\beta_{u}$ | $c_{\mathtt{IU}}$ | $L$ | $\beta_{p}$ | $c_{\mathtt{IP}}$
---|---|---|---|---|---|---
$1$ | $0.5$ | $0.3$ | $2$ | $80$ | $0.15$ | $1$

We adopt the recovery rate $\gamma$ as a bifurcation parameter and use the
numerical continuation package MATCONT [24] to compute the bifurcation
diagram. The bifurcation diagram is shown in Fig. 1.

Figure 1: Bifurcation diagram of the equilibria of the coupled SIS epidemic
and evolutionary behavioral dynamics. (solid blue: stable branch and dashed
red: unstable branch of equilibria)

For $\gamma\to 0^{+}$, $\mathbf{E4}$ is the stable equilibria, while
$\mathbf{E0}$, $\mathbf{E1}$, and $\mathbf{E2}$ are unstable. As the value of
$\gamma$ is increased at point T3 in Fig. 1, $\mathbf{E4}$ exchanges stability
to $\mathbf{E3}$ in a transcritical bifurcation. Note that the unstable branch
of $\mathbf{E3}$ is not visible since it is associated with negative values of
$z_{S}$. As the value of $\gamma$ is increased, the fraction of susceptible
population adopting the protection decreases and at T2, $\mathbf{E3}$
exchanges stability to $\mathbf{E2}$ in another transcritical bifurcation.
Again, the unstable branch of $\mathbf{E3}$ at T2 corresponds to $z_{s}>1$ and
is not visible in the bifurcation diagram. Upon further increasing $\gamma$,
the fraction of the infected population continues to decrease, and at T1,
$\mathbf{E2}$ exchanges stability with the disease-free equilibria
$\mathbf{E1}$. The unstable branch of $\mathbf{E2}$ at T1 corresponds to
negative values of $y$ and is not visible.

Another transcritical bifurcation takes place at T0
$(\gamma=\alpha\beta_{p})$, where $\mathbf{E0}$ and $\mathbf{E4}$ cross. For
$\gamma<\alpha\beta_{p}$ (resp., $\gamma>\alpha\beta_{p}$), $\mathbf{E0}$ has
two (resp., one) eigenvalues in the right-half plane. As $\mathbf{E4}$
approaches T0 from $y>0$, it has one eigenvalue in the right-half plane, while
for $y<0$ near T0, $\mathbf{E4}$ is stable. Thus, the transcritical
bifurcation at T0 corresponds to the exchange of the stability of stable and
unstable eigenvalues of two unstable equilibrium points.

Figure 2: Trajectories for equilibrium points $\mathbf{E1}$, $\mathbf{E2}$,
$\mathbf{E3}$, and $\mathbf{E4}$. The top panels shows trajectory for
$\gamma=0.2$ and $\gamma=0.13$ at which $\mathbf{E1}$ and $\mathbf{E2}$ are
stable, respectively. The bottom panels shows trajectory for $\gamma=0.1$ and
$\gamma=0.05$ at which $\mathbf{E3}$ and $\mathbf{E4}$ are stable,
respectively.

Fig. 2 illustrates the behavior of trajectories for equilibrium points
$\mathbf{E1}$, $\mathbf{E2}$, $\mathbf{E3}$, and $\mathbf{E4}$. For
$\mathbf{E1}$, $\mathbf{E2}$ and $\mathbf{E4}$, the trajectories evolve such
that they first approach $z_{S}=0$, i.e., every susceptible individual adopts
protection and subsequently, they approach $z_{S}=0$ or $1$ depending on the
equilibria. In contrast, trajectories for $\mathbf{E3}$ are highly oscillatory
which we investigate further in the next section.

##  IV Coupled Epidemic-Behavioral Dynamics under Timescale Separation

We now study the coupled epidemic-behavioral dynamics (2), (3), and (4) under
timescale separation to obtain further insights into their behavior.

###  IV-A Replicator dynamics being faster than epidemic dynamics

In this regime, the coupled dynamics is modeled as a slow-fast system:

$\displaystyle\begin{split}\dot{y}(t)&=f_{y}(y(t),z_{S}(t),z_{I}(t))\\\
\epsilon\dot{z}_{S}(t)&=f_{S}(y(t),z_{S}(t),z_{I}(t))\\\
\epsilon\dot{z}_{I}(t)&=f_{I}(y(t),z_{S}(t),z_{I}(t)),\end{split}$ (5)

where $\epsilon\in(0,1]$ is a timescale separation variable [22].

At a given epidemic prevalence $y$, we characterize the (stable) equilibria of
the fast system involving the replicator dynamics with states $(z_{S},z_{I})$.
For reasons discussed earlier, we focus on equilibria with $z_{I}=0$. It is
now easy to see that if $y\neq y^{*}_{\mathrm{int}}$, there are two
equilibrium points: $(0,0)$ and $(1,0)$. If $y=y^{*}_{\mathrm{int}}$, then
$(z_{S},0)$ is an equilibrium point of the fast system for any
$z_{S}\in[0,1]$. Following analogous arguments as the proof of Proposition 1,
it follows that $(0,0)$ is locally stable for the fast system when
$y>y^{*}_{\mathrm{int}}$ and $(1,0)$ is locally stable for the fast system
when $y<y^{*}_{\mathrm{int}}$. Consequently, we obtain the following reduced
dynamics for the slow system:

$\displaystyle\dot{y}(t)=\begin{cases}\big{[}(1-y(t))\beta_{p}-\gamma\big{]}y(t),\quad&\text{if
}y<y^{*}_{\mathrm{int}},\\\
\big{[}(1-y(t))\alpha\beta_{p}-\gamma\big{]}y(t),\quad&\text{if
}y>y^{*}_{\mathrm{int}},\end{cases}$ (6)

which approximates the coupled dynamics (5) in the limit $\epsilon\to 0$. We
now establish the limiting behavior of the above dynamics (6).

######  Proposition 2 (Trajectories under fast behavioral response)

For the epidemic dynamics (6) with $y(0)\neq 0$, the following statements hold

  1. (i.

if $y^{*}_{u}<0$, then $y(t)$ monotonically decreases and converges to the
origin;

  2. (ii.

if $0<y^{*}_{u}<y^{*}_{\mathrm{int}}$, then $y(t)$ monotonically converges to
$y_{u}^{*}$;

  3. (iii.

if $y^{*}_{p}<y^{*}_{\mathrm{int}}<y^{*}_{u}$, then $y(t)$ converges to
$y^{*}_{\mathrm{int}}$, and the convergence may not be monotonic;

  4. (iv.

if $y^{*}_{\mathrm{int}}<y^{*}_{p}$, then $y(t)$ monotonically converges to
$y_{p}^{*}$.

The proof is presented in the Appendix.

###### Remark 1

The dynamics in (6) potentially represent a class of non-pharmaceutical
interventions where authorities impose social distancing measures that reduce
the infection rate by a factor $\alpha$ when the infection prevalence exceeds
a threshold. Thus, the result in Proposition 2 is of potential independent
interest. In prior work [8], an analogous result was obtained in the special
case where $\alpha=0$.

###  IV-B Epidemic dynamics being faster than behavior

When epidemic dynamics evolves faster than the adaptation of protective
behavior, we arrive at the dynamics

$\displaystyle\begin{split}\epsilon\dot{y}(t)&=f_{y}(y(t),z_{S}(t),z_{I}(t))\\\
\dot{z}_{S}(t)&=f_{S}(y(t),z_{S}(t),z_{I}(t))\\\
\dot{z}_{I}(t)&=f_{I}(y(t),z_{S}(t),z_{I}(t)),\end{split}$ (7)

where $\epsilon\in(0,1]$ as before. Let
$\beta_{\textup{eff}}(z_{S},z_{I}):=(z_{S}+\alpha(1-z_{S}))(\beta_{u}z_{I}+\beta_{p}(1-z_{I}))$
denote the effective infection rate as a function of player strategies. Then,
in the limit $\epsilon\to 0$, the infection prevalence converges to the stable
equilibrium point of the fast system given by

$y^{*}(z_{S},z_{I})=\begin{cases}0,\quad&\text{if
}\beta_{\textup{eff}}(z_{S},z_{I})<\gamma,\\\
1-\frac{\gamma}{\beta_{\textup{eff}}(z_{S},z_{I})},\quad&\text{if
}\beta_{\textup{eff}}(z_{S},z_{I})>\gamma.\end{cases}$ (8)

Consequently, we obtain the following reduced dynamics for the slow system:

$\displaystyle\begin{split}\dot{z}_{S}(t)&=f_{S}(y^{*}(z_{S}(t),z_{I}(t)),z_{S}(t),z_{I}(t))\\\
\dot{z}_{I}(t)&=f_{I}(y^{*}(z_{S}(t),z_{I}(t)),z_{S}(t),z_{I}(t)),\end{split}$
(9)

which approximates the behavioral dynamics when $\epsilon\to 0$.

###### Remark 2

The slow timescale dynamics (9) sets the value of
$\beta_{\textup{eff}}(z_{S}(t),z_{I}(t))$ for the faster $y$ dynamics in (7).
The class of problems in which the bifurcation parameter of a differential
equation is varied on a slower timescale are referred to as dynamic
bifurcation problems [22]. It follows from the theory of dynamic bifurcations
that when $\beta_{\textup{eff}}(z_{S}(t),z_{I}(t))$ is not in the neighborhood
of the bifurcation point $\beta_{\textup{eff}}(z_{S}(t),z_{I}(t))=\gamma$, the
system trajectories track the stable branches of the bifurcation diagram.
System trajectories may not track the stable branch of the bifurcation diagram
in a neighborhood of bifurcation point, which leads to the so-called
_bifurcation delay_ [22].

###  IV-C Numerical Results

We now provide further insights into the dynamics under timescale separation,
which exhibits large oscillations and the bifurcation delay phenomenon via
numerical simulations.

We first investigate the behavior of system (5) for
$\epsilon\in\\{0.01,0.1,1\\}$. The same model parameters as in Section III-B
are selected and $\gamma$ is selected as $0.1$ so that $\mathbf{E3}$ is the
stable equilibrium point. Fig. 3 (right panel) shows the time-evolution of $y$
and $z_{S}$. More oscillatory behavior is observed as $\epsilon$ becomes
smaller, i.e., the behavioral dynamics becomes faster. To understand this
behavior, we focus on $z_{S}$ dynamics (3) with

$\Delta F=c_{P}-L(1-\alpha)(\beta_{u}z_{I}(t)+\beta_{p}(1-z_{I}(t)))y(t)$

as a dynamic parameter. To this end, we illustrate $z_{S}$ trajectories in
$z_{S}-\Delta F$ plane in Fig. 3 (left panel).

Recall that if $\Delta F$ is a positive (resp. negative) constant, then
$z_{S}=1$ (resp. $z_{S}=0$) is a stable equilibrium point. Accordingly,
$z_{S}=0$ and $z_{S}=1$ are marked blue and red in Fig. 3 (left panel), when
they are stable and unstable, respectively.

Since behavioral dynamics is fast, $y$ is quasi-stationary and $z_{I}$ very
quickly converges to zero. In Fig. 3 (left panel), the initial fraction of the
infected population is sufficiently high such that $\Delta F<0$, then the fast
behavioral dynamics quickly converge to $z_{S}=0$ (the bottom solid blue
line), i.e., every susceptible individual adopts protection. This results in a
decrease in the fraction of the infected population and increases $\Delta F$.
As $\Delta F$ becomes positive, $z_{S}=0$ becomes unstable and $z_{S}$ quickly
jumps to $z_{S}=1$ (the top solid blue line), and a similar process repeats
which again drives $\Delta F$ to negative values. This process leads to the
highly oscillatory behavior seen in Fig. 3. Eventually, trajectories converge
such that $\Delta F=0$, and in this regime $z_{S}$ settles to an equilibrium
value in the interval $(0,1)$.

Figure 3: Trajectories of the coupled epidemic-behavioral dynamics when the
behavioral dynamics evolve on a faster timescale. The left panel shows
trajectories in $z_{S}-\Delta F$ plane. For $\Delta F<0$ (resp. $\Delta F>0$)
$z_{S}=0$ (resp. $z_{S}=1$) is the stable equilibrium point and is shown in
solid blue line. The slower epidemic dynamics drive $\Delta F$ to change sign
leading to an abrupt switch in the equilibrium value of $z_{S}$, which in turn
causes $\Delta F$ to again switch sign. This process repeats and results in
oscillatory trajectories. Finally, the trajectories reach an equilibrium point
at which $\Delta F=0$. The right panel shows the oscillatory trajectories. The
oscillations increase as $\epsilon$ is decreased.

We now investigate the behavior of system (7) for
$\epsilon\in\\{0.0001,0.01,1\\}$. The same model parameters as above are
selected. The time evolution of $y$ and $z_{S}$ trajectories is shown in Fig.
4 (right panels). To better understand these trajectories, we investigate them
in the $y-\beta_{\textup{eff}}$ plane in Fig. 4, left panel), where
$\beta_{\textup{eff}}(z_{S},z_{I})=(z_{S}+\alpha(1-z_{S}))(\beta_{u}z_{I}+\beta_{p}(1-z_{I}))$
is a function of $z_{I}$ and $z_{S}$. The solid blue and red lines show the
bifurcation diagram of the faster SIS epidemic dynamics with
$\beta_{\textup{eff}}$ as the bifurcation parameter.

In Fig. 4(a), the initial condition is selected such that the initial value of
$\beta_{\textup{eff}}$ corresponds to endemic equilibria being stable. For
small $\epsilon$ values, the trajectories very quickly converge to the stable
branch of the bifurcation diagram and traverse along with the bifurcation
diagram until they reach the stable equilibria $\mathbf{E3}$ of the coupled
dynamics. This behavior is consistent with the theory of dynamic bifurcations
[22]. For moderate values of $\epsilon$, similar behavior is observed but
trajectories overshoot the stable branch and subsequently return to it. For
$\epsilon=1$, we observe oscillatory behavior similar to the case in which the
behavioral dynamics are faster.

In Fig. 4(b), the initial value of $\beta_{\textup{eff}}$ corresponds to
disease-free equilibria being stable. For small $\epsilon$, the trajectories
converge very quickly to the stable branch of the bifurcation diagram for the
SIS epidemic dynamics and track the stable branch until they reach the
bifurcation point. Subsequently, they continue on the unstable branch for some
time before switching to the stable branch. This behavior is referred to as
_bifurcation delay_ [22] and is attributed to the fact that at the bifurcation
point, the SIS epidemic dynamics become slower than the otherwise slower
behavioral dynamics. Similar behavior is observed for moderate $\epsilon$,
while for $\epsilon=1$ we observe oscillatory behavior as earlier.

(a)

(b)

Figure 4: Trajectories of the coupled epidemic-behavioral dynamics when the
epidemic dynamics evolve on a faster timescale. The left panels shows
trajectories in $y-\beta_{\textup{eff}}$ plane. The right panels show the
time-evolution of $y$ and $z_{S}$ trajectories. Left panel (a) shows for small
$\epsilon$, trajectories quickly converge to the stable branch of the
bifurcation diagram and then move along the bifurcation diagram to reach the
stable equilibria. Left panel (b) shows for small $\epsilon$, trajectories
quickly converge to the stable branch of the bifurcation diagram and then move
along on the bifurcation diagram until they reach the bifurcation point. In a
neighborhood of the bifurcation point, trajectories do not move along the
stable branch of the bifurcation diagram but they return to the stable branch
after some delay. Oscillatory behavior is observed for larger values of
$\epsilon$. The right panel shows the time evolution of $y$ and $z_{S}$.

##  V Conclusions

We proposed and analyzed a novel model that captures the interaction of
epidemic propagation dynamics with human protection adoption behavioral
dynamics. For the proposed coupled epidemic-behavioral dynamics, we completely
characterized the equilibrium points, their stability properties, and the
associated bifurcations. The set of equilibria corresponds to a disease-free
population or an endemic, and complete, partial, or no adoption of protection
by susceptible population. We showed that the stability of these equilibria is
a function of infection and recovery rate as well as the utility functions of
the individuals in the population. We investigated the transient behavior of
the model under two timescale scenarios: (i) faster behavioral dynamics, (ii)
faster epidemic dynamics. In the former case, we showed that the coupled
dynamics may lead to highly oscillatory behavior and may take a long time to
converge to the stable equilibria. In the latter case, we illustrated the
bifurcation delay phenomenon in the onset of endemic.

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Proof of Proposition 1: It can be verified that $\mathbf{E0}$ and
$\mathbf{E1}$ are always equilibria of the coupled dynamics. The Jacobian
matrix at $\mathbf{E0}$ and $\mathbf{E1}$ are

$J_{\mathbf{E0}}=\left[\begin{smallmatrix}\alpha\beta_{p}-\gamma&0&0\\\
0&c_{P}&0\\\
0&0&c_{\mathtt{IP}}-c_{\mathtt{IU}}\end{smallmatrix}\right],\text{ and
}J_{\mathbf{E1}}=\left[\begin{smallmatrix}\beta_{p}-\gamma&0&0\\\
0&-c_{P}&0\\\ 0&0&c_{\mathtt{IP}}-c_{\mathtt{IU}}\end{smallmatrix}\right].$

$J_{\mathbf{E0}}$ has a positive eigenvalue if $c_{P}>0$. Thus, for any
nonzero cost of adopting protection, $\mathbf{E0}$ is not a stable equilibrium
point of the coupled dynamics. Likewise, $\mathbf{E1}$ is stable if
$\beta_{p}<\gamma$, and unstable, otherwise.

We now analyze equilibrium points where infection is endemic. It can be
verified that $\mathbf{E2}$ exists only when $\beta_{p}>\gamma$. The Jacobian
matrix at $\mathbf{E2}$ is

$J_{\mathbf{E2}}=\left[\begin{smallmatrix}d_{1}&d_{2}&d_{3}\\\
0&-[c_{P}-L(1-\alpha)\beta_{p}y^{*}_{u}]&0\\\
0&0&c_{\mathtt{IP}}-c_{\mathtt{IU}}\end{smallmatrix}\right],$

where $d_{1}=(1-2y^{*}_{u})\beta_{p}-\gamma$,
$d_{2}=y^{*}_{u}(1-y^{*}_{u})(1-\alpha)\beta_{p}>0$ and
$d_{3}=y^{*}_{u}(1-y^{*}_{u})(\beta_{u}-\beta_{p})>0$. Thus, $J_{\mathbf{E2}}$
is an upper triangular matrix. Furthermore, the first diagonal entry is

$\displaystyle(1-2y^{*}_{u})\beta_{p}-\gamma$
$\displaystyle=\gamma-\beta_{p}<0,$

in the regime where $\mathbf{E2}$ exists. Therefore, $\mathbf{E2}$ is stable
when

$\displaystyle c_{P}$ $\displaystyle>L(1-\alpha)\beta_{p}y^{*}_{u}\iff
y^{*}_{u}<y^{*}_{\mathrm{int}}.$

It can be verified that
$\mathbf{E3}=(y^{*}_{\mathrm{int}},z^{*}_{S,\mathrm{int}},0)$ is an
equilibrium point. We now derive conditions that ensure
$y^{*}_{\mathrm{int}},z^{*}_{S,\mathrm{int}}\in[0,1]$. We now examine the
conditions under which $y^{*}_{\mathrm{int}}\in(0,1)$ and
$z^{*}_{S,\mathrm{int}}\in(0,1)$. By definition, $y^{*}_{\mathrm{int}}>0$. We
now observe that

$\displaystyle z^{*}_{S,\mathrm{int}}>0$
$\displaystyle\iff\frac{\gamma}{\alpha\beta_{p}}>1-y^{*}_{\mathrm{int}}\iff
y^{*}_{\mathrm{int}}>1-\frac{\gamma}{\alpha\beta_{p}}$ $\displaystyle\text{and
}z^{*}_{S,\mathrm{int}}<1$
$\displaystyle\iff\frac{\gamma}{\beta_{p}}<1-y^{*}_{\mathrm{int}}\iff
y^{*}_{\mathrm{int}}<1-\frac{\gamma}{\beta_{p}}.$

Thus, $\mathbf{E3}$ exists when $y^{*}_{p}<y^{*}_{\mathrm{int}}<y^{*}_{u}$.
Note that the third row of the Jacobian of the dynamics at $\mathbf{E3}$,
$J_{\mathbf{E3}}$, would be $[0\quad 0\quad c_{\mathtt{IP}}-c_{\mathtt{IU}}]$
as before, and as a result, $c_{\mathtt{IP}}-c_{\mathtt{IU}}<0$ would be an
eigenvalue. Thus, we focus on the $2\times 2$ sub-matrix containing the first
two rows and columns of the Jacobian matrix which simplifies to

$\displaystyle\hat{J}_{\mathbf{E3}}=\left[\begin{smallmatrix}\frac{-\gamma
y^{*}_{\mathrm{int}}}{1-y^{*}_{\mathrm{int}}}&d_{4}\\\
-z^{*}_{S,\mathrm{int}}(1-z^{*}_{S,\mathrm{int}})L(1-\alpha)\beta_{p}&0\\\
\end{smallmatrix}\right],$

where $d_{4}=y^{*}_{\mathrm{int}}(1-y^{*}_{\mathrm{int}})(1-\alpha)\beta_{p}$.
For the above matrix, the sum of the eigenvalues is negative and the
determinant is positive, and as a result, all three eigenvalues of
$J_{\mathbf{E3}}$ are negative. Therefore, $\mathbf{E3}$, when it exists, is a
stable equilibrium of the coupled dynamics.

It can be verified that $\mathbf{E4}$ exists when $y^{*}_{p}\in(0,1)$ or
equivalently, when $\gamma<\alpha\beta_{p}$. The Jacobian matrix at
$\mathbf{E4}$ is

$J_{\mathbf{E4}}=\left[\begin{smallmatrix}(1-2y^{*}_{p})\alpha\beta_{p}-\gamma&d_{5}&d_{6}\\\
0&d_{7}&0\\\ 0&0&c_{\mathtt{IP}}-c_{\mathtt{IU}}\end{smallmatrix}\right],$
(10)

where $d_{5}=y^{*}_{p}(1-y^{*}_{p})(1-\alpha)\beta_{p}>0$,
$d_{6}=y^{*}_{p}(1-y^{*}_{p})\alpha(\beta_{u}-\beta_{p})>0$ and
$d_{7}=c_{P}-L(1-\alpha)\beta_{p}y^{*}_{p}$. Thus, $J_{\mathbf{E4}}$ is a
diagonal matrix with the first diagonal entry

$\displaystyle(1-2y^{*}_{p})\alpha\beta_{p}-\gamma$
$\displaystyle=\gamma-\alpha\beta_{p}<0,$

in the regime where $\mathbf{E4}$ exists. Therefore, $\mathbf{E4}$ is stable
when

$\displaystyle c_{P}$ $\displaystyle<L(1-\alpha)\beta_{p}y^{*}_{p}\iff
y^{*}_{p}>y^{*}_{\mathrm{int}}.$

This concludes the proof.

Proof of Proposition 2: Recall that since $\alpha\in(0,1)$, we have
$y^{*}_{p}<y^{*}_{u}$. It follows from [1] that the dynamics

$\dot{y}(t)=[(1-y(t))\beta-\gamma]y(t)$

converges to $0$ if $\gamma\geq\beta$ and converges to
$y^{*}=1-\frac{\gamma}{\beta}$, otherwise. Furthermore, it is easy to see that
in case of the latter, $|y(t)-y^{*}|$ is monotonically decreasing.222For
$V(y)=(y-y^{*})^{2}$, we have $\dot{V}(y)=-2\beta yV(y)$. Note that when
$y<y^{*}_{\mathrm{int}}$ (respectively, $y>y^{*}_{\mathrm{int}}$), (6) is
analogous to the above dynamics with $\beta=\beta_{p}$ (respectively,
$\beta=\alpha\beta_{p}$). We now analyze the four cases stated above.

Case 1: $y^{*}_{u}<0$. In this case, we have
$\alpha\beta_{p}<\beta_{p}<\gamma$ and as a result, $y=0$ is the only
equilibrium both when $y>y^{*}_{\mathrm{int}}$ and $y<y^{*}_{\mathrm{int}}$.

Case 2: $0<y^{*}_{u}<y^{*}_{\mathrm{int}}$. Since $y^{*}_{p}<y^{*}_{u}$,
$y(t)$ is monotonically decreasing when $y(t)>y^{*}_{\mathrm{int}}$.
Similarly, $y(t)$ is monotonically decreasing when
$y_{u}^{*}<y(t)<y^{*}_{\mathrm{int}}$ and monotonically increasing for
$y(t)<y_{u}^{*}$. Thus, the claim follows.

Case 3: $y^{*}_{p}<y^{*}_{\mathrm{int}}<y^{*}_{u}$. In this case, $y(t)$ is
monotonically decreasing when $y(t)>y^{*}_{\mathrm{int}}$ as
$\max(0,y^{*}_{p})<y^{*}_{\mathrm{int}}$, and monotonically increasing when
$y(t)<y^{*}_{\mathrm{int}}$ since the corresponding stable equilibrium
$y^{*}_{u}>y^{*}_{\mathrm{int}}$. Thus, $y^{*}_{\mathrm{int}}$ acts as a
sliding surface for the dynamics (6). $y(t)$ may switch between two subcases
in (6) leading to an oscillatory behavior as will be illustrated numerically.

Case 4: $y^{*}_{p}>y^{*}_{\mathrm{int}}$. In this case, $y(t)$ is
monotonically increasing when $y(t)<y^{*}_{\mathrm{int}}$ since
$y^{*}_{u}>y^{*}_{p}>y^{*}_{\mathrm{int}}$. Monotonic behavior for
$y(t)>y^{*}_{\mathrm{int}}$ follows from analogous arguments as Case 2.

This concludes the proof.


SCIPP 20/01

Low Energy Effective Theory for Axion Strings

Michael Dine(a)

(a)Santa Cruz Institute for Particle Physics and
Department of Physics, University of California at Santa Cruz
Santa Cruz, CA 95064.

We consider aspects of the low energy (classical) effective field theory
description of global cosmic strings. In the non-relativistic limit, we study
the extent to which these are described by the combination of the Nambu-Gotto
and Kalb-Ramond actions. While in a formal sense this is the case for
infinitely long strings, for more realistic situations, the collective
coordinates of the system do not provide a complete description for more than
brief intervals; as time evolves it becomes necessary to include low momentum
goldstone excitations as well.

##  1 Introduction

There has been ongoing discussion for almost four decades of the possible role
of axion strings in axion dark matter production (see, e.g., [1, 2, 3, 4, 5,
6]. There are a range of analyses, some analytical, many numerical
simulations, with the latter quite large and sophisticated in recent years[7,
8, 9, 10, 11, 12, 13]. In some cases, these simulations appear to point to a
parameterically enhanced production of axion dark matter due to cosmic
strings; in other cases, not so much so. There is a large parameter in the
problem, the infrared sensitive value of the string tension divided by
$f_{a}^{2}$. Assuming that the Hubble constant provides the infrared cutoff,
this parameter is typically taken to be:

$\displaystyle\xi\equiv\log(f_{a}/H_{qcd})\sim 70.$ (1.1)

If there is of order one string per horizon, and if most of the energy in the
axion field surrounding axion strings were converted into very low momentum
axions, there would be an enhancement of the dark matter axion density by such
a factor over naive, textbook[14] considerations. This could have significant
consequences for cosmology and for the ability of experiments such as ADMX to
detect the dark matter. In [15], qualitative arguments were put forth that
there is no such enhancement. This work did enumerate a number of challenges
for analytic dark matter estimates at the order one level, which simulations
have the potential to resolve.

In this paper, we will look more closely at the effective theory of axion
strings, asking whether one might expect enhancements of axion emission at low
momentum by powers of $\xi$. In effect, we are revisiting the older literature
on these objects. Crucial will be the role of the infrared in axion
production. Simulations, when sufficiently accurate, generally consider all
degrees of freedom simultaneously. They do not exploit low energy actions for
collective coordinates and low momentum axions of the type we will consider
here. It would be interesting to perform simulations in the low energy
effective theory, and/or to match the results of simulations to these
theories.

As opposed to Nielsen-Olesen, gauge strings[16], strings associated with the
breaking of a global symmetry have some problematic features. In particular,
if one tries to define a tension, one finds that it is infrared divergent. As
with all infrared divergences, this is a signal, as stressed in [15], that one
must include an enlarged set of degrees of freedom in describing the
system111It is perhaps worth recalling how this is realized in QED. There, the
three point function is infrared divergent. It cannot be used in an effective
field theory by itself to compute scattering amplitudes; it is necessary to
consider emission of extremely low energy photons as well.. In the case of
global cosmic strings, this means that one can’t consider the system of string
collective coordinates by themselves. Instead, it is necessary to consider an
effective action involving these coordinates and axions with momentum below
some cutoff scale. For an infinitely long string, the cutoff should be chosen
so as to avoid problems of causality. Implementing such a cutoff might be
complicated. For a closed string of circumference of order $R$, $R$ itself
cuts off infrared divergences, and ideally one should choose the (momentum
space) cutoff large compared to $R^{-1}$. For long, parallel string anti-
string segments separated by a distance $b$, $b$ acts as a cutoff. Such
configurations are likely more representative than the infinite string of the
actual cosmological situation. In this note, we will look at the effective
action for this system, particularly in the non-relativistic limit in which
the description of solitons becomes simple. We will see that for a long
straight string, with little excitation of non-zero modes, the Nambu-Goto
action, supplemented with the Kalb-Ramond coupling[14] of the axion to the
string (NGKR), in a certain formal sense, captures the dynamics. But for
closed strings or string-antistring segments the NGKR description, by itself,
fails to correctly capture the dynamics of the system. The problem is that
while, at one instant, the system might be well described by its collective
coordinates without axion excitations, this will typically not be true a short
time later. This is in sharp contrast to infrared insensitive systems like
magnetic monopoles and gauge strings in four dimensions. Said another way, for
most of its history, the string, as an entity, is not the simple object
considered in many analyses. Note, here, we are speaking of the classical (as
opposed to quantum) theory of these strings. This will be our focus throughout
this paper.

The rest of this paper is organized as follows. In the next section, we
briefly review well known aspects of the collective coordinate method for
solitons, both particles and strings. For systems such as monopoles and
Nielsen-Olsen strings, we explain the sense in which the collective
coordinates provide a complete description of the system, in uniform motion,
for all times. We consider the collective coordinate analysis for an
infinitely long, straight global string in section 3. We derive, formally, the
non-relativistic Nambu-Gotto action for the collective coordinates, with its
infrared divergent tension, and in section 4 we will see that the Kalb-Ramond
coupling does formally describe the axion couplings to the collective modes.
We then consider, in section 5, the description of closed string systems. We
note that, related to the infrared divergence of the infinite string, there is
not a unique definition of collective coordinates. It is then not surprising,
as we elucidate, that the NGKR description of such systems is inadequate. In
section 6, we summarize and describe our expectations for the spectrum of
axion radiation emitted by the sorts of closed string systems one might expect
to encounter in early universe cosmology and in simulations of such
cosmologies.

##  2 Collective Coordinate Review

The utility of the collective coordinate method follows from elementary
considerations. Consider, for example, a kink in a translationally invariant
theory in two dimensions, described by a classical solution, $\Phi_{cl}(x)$

$\displaystyle\Phi(x,t)=\Phi_{cl}(x-X(t))+\delta\Phi.$ (2.1)

$\delta\Phi$ describes the fluctuations about the classical solution. We can
obtain an action for the degrees of freedom $X$ and $\delta\Phi$ by
substituting 2.1 into the field theory action. There is a kinetic term for $X$
of the form

$\displaystyle{\cal L}(X,\delta\Phi)={1\over 2}M\dot{X}^{2}+\dots{}$ (2.2)

and higher order terms in $\dot{X}$. Because of translational invariance,
there are no terms involving $X$ without derivatives. In addition, precisely
because $\Phi_{cl}$ is a solution of the classical equations, when we work out
the action for $X$, there are no terms in the action linear in $\delta\Phi$
for a kink in uniform motion. So if one has a soliton in uniform motion at
time $t=t_{0}$, say, with $\Delta\Phi=0$, $\delta\Phi$ remains zero for all
time.

The same is true for other collective coordinates associated with symmetries
of solutions, such as translational and dyonic excitations of monopoles.
Infinite gauge strings in four dimensions have two collective coordinates, the
transverse coordinates of the string solution, $\vec{X}_{\bot}$. Calling $z$
the coordinate along the string, and writing the field in the presence of the
string as

$\displaystyle\Phi(\vec{x}_{\perp},z,t)=\Phi(\vec{x}_{\perp}-\vec{X}(z,t))+\delta\Phi,$
(2.3)

the action for these collective coordinates has the form:

$\displaystyle\int d^{2}\sigma{\cal L}(X,\delta\Phi)=\int d^{2}\sigma{1\over
2}T\left({\partial\vec{X}\over\partial\sigma_{\alpha}}\right)^{2}+\dots{}$
(2.4)

Here $\sigma=(z,t)$; this is the non-relativistic limit of the usual Nambu-
Gotto action. Again, there is no term linear in $\delta\Phi$. So, for a string
executing uniform motion, starting with $\delta\Phi=0$ at some time,
$\delta\Phi$ remains zero. All of this follows from the translational
invariance of the theory. The Nambu-Gotto description remains good if the
string is curved, with curvature small compared to the characteristic scales
of the system.

This simple observation is the origin of the power of the collective
coordinate method for such systems. It will not be the case for strings
arising from the breaking of global symmetries in four dimensions (or vortices
in $2+1$ dimensions). This is precisely because of the infrared divergences.
The effects which cut off the divergence, such as finite circumference, will
spoil the collective coordinate story, in the sense that the system as
described by the collective coordinates alone does not obey the equations of
motion. This is closely tied to the need to include other degrees of freedom
in the effective action. The infrared divergence yields a dependence on the
breaking scale of the symmetry. There will be a potential for the collective
coordinates and coupling of the collective coordinates to $\delta\Phi$.

##  3 The Effective Action For the Collective Coordinates for a Long Global
String

Consider a theory of a complex scalar, $\Phi$, with action symmetric under
$\Phi\rightarrow e^{i\alpha}\Phi$:

$\displaystyle S=\int
d^{4}x\left(|\partial_{\mu}\Phi|^{2}+m^{2}|\Phi|^{2}-{\lambda\over
2}|\Phi|^{4}\right).$ (3.1)

For this action,

$\displaystyle\langle|\Phi|\rangle\equiv f_{a}\equiv v={m^{2}\over\lambda}.$
(3.2)

This action admits static string solutions. Taking the string to lie along the
$z$ axis, with coordinates $(x,y,z)\equiv(\vec{x}_{\bot},z)=(\rho,\theta,z)$,

$\displaystyle\Phi(t,z,\vec{x}_{\bot})=\Phi_{cl}(\vec{x}_{\bot})=f(\rho)e^{i\theta}.$
(3.3)

The function $f(\rho)$ has the properties:

$\displaystyle f(\rho)\rightarrow 0~{}{\rm as}~{}\rho\rightarrow
0;~{}~{}~{}f(\rho)\rightarrow f_{a}(1-{1\over f_{a}\rho^{2}})~{}{\rm
as}~{}\rho\rightarrow\infty.$ (3.4)

We first obtain the effective action for the collective coordinates for the
string. Note, first that if we calculate the tension, thought of as the energy
per unit length of the string, the result is infrared divergent:

$\displaystyle T=\int
d^{2}x_{\bot}\left(|\vec{\nabla}\Phi|^{2}+V(\Phi)-V_{0}\right)\approx 2\pi
f_{a}^{2}\int d\rho{1\over\rho^{2}}=2\pi\log(\Lambda/\mu).$ (3.5)

Here $\Lambda$ is an ultraviolet cutoff, of order the core size; we’ll take
$\Lambda=f_{a}$. $\mu$ is an infrared cutoff. If we have a closed string with
radius $R$, $\mu=R^{-1}$; if we have a parallel string and antistring
separated by a distance $b$, $\mu=b^{-1}$.

We can see more directly that $T$ is the tension of the string by allowing
slow variation in the transverse directions, introducing collective
coordinates $X^{i}(z,t),~{}i=1,2$. We also allow variation of the phase of
$\Phi$ from $i\theta$, i.e. a spatially and time-dependent axion field:

$\displaystyle\Phi(t,z,\vec{x}_{\bot})=\Phi_{cl}(\vec{X}(t,z)-\vec{x}_{\bot})e^{i\delta
a(t,z,\vec{x}_{\bot})}.$ (3.6)

Now plug this expression into the terms in the action:

$\displaystyle S=\int
dtdzd^{2}x_{\bot}\left(|\partial_{t}\Phi|^{2}-|\partial_{z}\Phi|^{2}\right)$
(3.7) $~{}~{}~{}\approx\int
dtdz\left((\partial_{t}X^{i})^{2}-(\partial_{z}X^{i})^{2}\right)\int
d^{2}x_{\bot}|\vec{\nabla}\Phi_{cl}(x_{\bot})|^{2}$ $~{}~{}~{}~{}~{}+\int
d^{4}xf_{a}^{2}\left(\partial_{t}X^{i}\partial_{i}~{}\theta\partial_{t}\delta
a-\partial_{z}X^{i}\partial_{i}~{}\theta\partial_{z}\delta a\right).$
$~{}~{}~{}~{}\equiv I_{transverse}+I_{\delta a}$

This is valid for distances from the string small compared to the wavelength
of the disturbance. In this regime, the variation of the coordinate is roughly
constant, and the light transit time can be neglected. The first term can be
rewritten as

$\displaystyle
I_{transverse}=T\int{dzdt}\left(\left({\partial\vec{X}_{\bot}(t,z)\over\partial
t}\right)^{2}-\left({\partial\vec{X}_{\bot}(t,z)\over\partial
z}\right)^{2}\right)$ (3.8)

with $T$ given by our earlier expression. The infrared divergence appears
again. In the second
term,$\partial_{i}\theta={\epsilon_{ij}x^{j}\over|\vec{x}_{\bot}|^{2}}.$ so

$\displaystyle I_{\delta a}=\int
d^{4}x\epsilon_{ij}{x^{j}\over|\vec{x}_{\bot}|^{2}}f_{a}^{2}\left(\partial_{t}X^{i}\partial_{t}\delta
a-\partial_{z}X^{i}~{}\theta\partial_{z}\delta a\right).$ (3.9)

Equation 3.8 is, again, the non-relativistic limit of the Nambu-Goto action,
but with an infrared divergent tension. Before considering how this divergence
may be tamed, we can ask about the coupling to $\delta a$. As we now
demonstrate (and was remarked in [15]), equation 3.9 is of the form of the
Kalb-Ramond action[14].

##  4 Appearance of the KR term for a Long, Slowly Moving String

In this section, we observe formally that for an infinitely long, straight
string, the KR action does describe the coupling of the string collective
coordinates to axion radiation for a slowly moving string. The $a$
configuration defining the string changes as $X^{i}$, the string collective
coordinates, change. If the collective coordinates change slowly enough, we
might expect $a$ to be, at each instant, nearly in its lowest energy (the
$\nabla^{2}a=0$) configuration. The slight differences will correspond to
radiation. This is encoded in our collective coordinate analysis, and we will
see it is reproduced by the KR action.

###  4.1 The Circulating Axion Field

We first express the axion field surrounding a static string in terms of the
antisymmetric tensor field. Consider an infinite static string along the $z$
direction, i.e. $X^{0}=t$, $X^{z}=z$. The KR action leads to the equation:

$\displaystyle\Box B_{0z}=f_{KR}.$ (4.1)

(We will fix the constant, $f_{KR}$ in a moment.) So, taking $B_{0z}$ to be
$z$ and $t$ independent,

$\displaystyle B_{0z}=G(\vec{x}_{\bot})f_{KR}$ (4.2)

where $G(\vec{x}_{\bot})$ is the two dimensional Green’s function,

$\displaystyle G(\vec{x}_{\bot})={1\over 2\pi}\log(|\vec{x}_{\bot}|).$ (4.3)

From this, we can construct

$\displaystyle H_{0zi}=\partial_{i}B_{0z}=\epsilon_{0zij}\partial_{j}a$ (4.4)

so

$\displaystyle\partial_{j}a=\epsilon_{ij}\partial_{i}f_{KR}G=\epsilon_{ij}\partial_{j}f_{KR}{x^{i}\over|\vec{x}_{\bot}|^{2}}$
(4.5)

which is $f_{KR}\partial_{j}\theta$, the polar angle in cylindrical
coordinates. So the static configuration is as expected. Note that this is the
minimal energy configuration for fixed $B$ ($a$), with the boundary condition
of unit winding.

###  4.2 Axion Emission From the String

Now consider axion emission from a moving string. Consider the term in the
action

$\displaystyle\int d^{4}x{\cal L}\sim\int
d^{4}x\left(H_{zij}^{2}-H_{0ij}^{2}\right).$ (4.6)

Noting, for example, that

$\displaystyle
H_{zij}(t,z,\vec{x}_{\bot})=\partial_{0}a+\epsilon_{ij}\partial_{i}G^{(2)}(\vec{x}_{\bot}){\partial
X^{j}\over\partial t}$ (4.7)

yields for the action a sum of terms, ${\cal L}={\cal L}_{X^{i}}+{\cal
L}_{a-X^{i}}$, where:

$\displaystyle{\cal L}_{X^{i}}=\int d^{4}x{\cal L}=\int
dz~{}dtx\left(({dX^{i}\over dt})^{2}-({dX^{i}\over dz})^{2}\right)\int
d^{2}x_{\bot}\left({1\over x_{\bot}^{2}}\right)$ (4.8)

and

$\displaystyle{\cal L}_{a-X^{i}}=\int
dzdt~{}d^{2}x_{\bot}\left(\partial_{0}a{dX^{i}\over
dt}-\partial_{z}a{dX^{i}\over dz}\right)\epsilon_{ij}{x^{j}\over
x_{\bot}^{2}}.$ (4.9)

The first expression is identical to the action for the transverse
fluctuations of the string found in the field theory; the second is the axion-
string collective coordinate coupling found there.

So formally, the collective coordinates and the axion perturbation of the
solitonic string in field theory are described by the Nambu-Gotto-Kalb-Ramond
(NGKR) action. We say formally both because of the infrared divergent tension
and because of issues of causality: the axion field changes everywhere in
response to transverse string displacements. These issues should be addressed
by systems of closed strings, such as long, parallel strings and antistrings
or circular strings, where string separations and/or radii will act as an
infrared cutoff. We consider these issues in the next section. As we
anticipated earlier, the fact that these configurations are not static
solutions of the equations of motion limits the applicability of the
collective coordinate description.

##  5 Systems of Closed Strings

In [15], two limits of string motion were considered, referred to as
“adiabatic” and “sudden”. These were considered in a rather heuristic way. The
adiabatic limit corresponds to slow motion of the string, and, in this limit,
as the strings moves and the axion field surrounding them changes, the energy
change in the axion field is compensated by the change in the kinetic energy
of the collective coordinate. The sudden limit corresponds to fast motion of
the string. In this case, most of the energy stored in the axion field is
converted into freely moving axions. If this picture is correct, than in the
case of adiabatic motion, the system should be well described by its
collective coordinates; this is not true of the sudden case. In this section
we will understand this picture in more detail in terms of the equations of
motion of the system.

Because of the infrared issues associated with the axion field, equation 3.6
is not really sensible for an infinite string; it also implies an
instantaneous change in the axion field throughout space. For a closed system,
e.g. a closed, roughly circular string, or alternatively for a long, parallel
string and antistring, this potentially makes more sense, as the infrared
divergences cancel. In this case, however, we have to consider what we mean by
$\Phi_{cl}$, since there are not static solutions of the equations of motion
of this type. We might take this to be, for example, the minimal energy static
(instantaneously) configuration with specified location of the string core.
Then we can make the substitution of equation 3.6 and obtain an action for
these coordinates. But the question of what configuration our collective
coordinates describe does not have a unique answer. We have no reason to
expect that as the system evolves, even if it starts in one such
configuration, say, again, the axion configuration of lowest energy, it will
remain in the corresponding lowest energy configuration.

For a long string-antistring pair, separated by a distance $b$ and moving
slowly, the field configuration of lowest energy, far from the string core, is
just the superposition of two static solutions. The corresponding field
configuration is, far from the string core:

$\displaystyle\Phi=f_{a}e^{i{a(x,t)+\delta a\over f_{a}}}$ (5.1)

where

$\displaystyle a(x,t)=a_{0}(x_{\bot}-{b(t)\hat{y}\over
2})-a_{0}(x_{\bot}+{b(t)\hat{y}\over 2}).$ (5.2)

Here $a_{0}(\vec{x}_{0})$ is the axion configuration around a single string.
We now show that with initial conditions

$\displaystyle b(0)=b_{0};~{}\dot{b}(0)=0$ (5.3)

this is almost a solution of the equations of motion, for short times.

Substituting $\Phi$ from equation 5.1 into the field theory action, yields an
action for $b$:

$\displaystyle L\approx f_{a}^{2}\ell(\dot{b}^{2}\log(f_{a}b)+\log(f_{a}b)).$
(5.4)

as well as a coupling of the collective mode, $b$, to the axion, $\delta a$:

$\displaystyle\int d^{3}x~{}dt~{}\left(\dot{b}{\partial a\over\partial
b}\partial_{t}\delta a-{\partial a\over\partial x^{i}}{\partial\delta
a\over\partial x^{i}}\right)$ (5.5)

We can integrate by parts in the first term with respect to time. Provided
that $\dot{b}$ is small, the source for $\delta a$ is then proportional to
$\ddot{b}={1\over b\log(f_{a}b)}$. The energy stored in $\delta a$ is
suppressed, at early times, by $1/\over\log(f_{a}b)$ relative to the kinetic
energy in $b$. But we can make a stronger statement. The system can be
described as approximately adiabatic[15]. Note that the source for the mode

$\displaystyle\delta a=C(t){\partial a\over\partial b}$ (5.6)

vanishes at early times as a consequence of the equation of motion for $b$. To
see this, note that, the kinetic term, after integration by parts, neglecting
$\dot{b}$, yields:

$\displaystyle\ddot{b}C\int d^{2}x_{\bot}({\partial a\over\partial b})^{2}$
(5.7)

which is just $C$ times the second derivative term in the equation of motion
for $b$, while from the integration over the $\vec{\nabla}_{\bot}^{2}$ term
one has:

$\displaystyle C{\partial\over\partial b}\int
d^{2}x_{\bot}(\vec{\nabla}a)^{2}=C{\partial V(b)\over\partial b}.$ (5.8)

So, provided $\dot{b}\approx 0$, the field of equation 5.1 is almost a
solution of the equations of motion. But as $\dot{b}$ grows, $C$ is
significantly sourced. In particular, the $\dot{b}^{2}$ terms dominate once
$\dot{b}>\log(f_{a}b)^{-1/2}.$ At this point the adiabatic approximation is
breaking down.

This is quite general. For circular motion, one instead studies a classical
solution with a circular core of radius $R$. One can again start with a static
configuration, minimizing the axion field energy as a function of $R$. Again
the mode:

$\displaystyle\delta a=C(t){\partial a\over\partial R}$ (5.9)

is not sourced initially as a consequence of the equations of motion for $R$.
Now, however, the system quickly becomes relativistic, and $C$ is driven away
from zero. Correspondingly, much of the field energy is converted into
radiation, rather than kinetic energy of $R$.

Loosely speaking, treatments of radiation utilizing the Kalb-Ramond action
require a careful treatment of “back reaction”. Once one enters the non-
adiabatic regime, most of the energy of the field is converted to radiation
rather than kinetic energy of $b$. To fully account for this requires solving
the full set of equations. It is reassuring, at least, that both regimes can
be analyzed, for the string-antistring case, in a non-relativistic limit.
Circular strings become relativistic more quickly, and these back-reaction
effects will be pronounced early on.

We conclude from this that initially for a long parallel string and
antistring, if the system starts at rest, the bulk of the energy in the axion
field is converted into kinetic energy of $b$. Little radiation is produced.
As $b$ decreases and the velocity becomes relativistic, production of axions
increases. Note, in particular, that one produces the mode $C$. This
corresponds to the fact that, as the velocity increases, this motion is less
and less an approximate solution of the equations of motion. All of this is
consistent with the expectation that the energy in the axion field at a length
scale $b(t)$ is converted into radiation of wavelength $b(t)$ except,
possibly, at early stages of the motion, where long wavelength axions are
suppressed if the velocity, $\dot{b}$, is small.

##  6 Conclusions

Global strings are peculiar objects. In isolation, their tensions are infrared
divergent. This indicates that they are not really objects at all, in that
they cannot, except possibly for brief periods, be described simply in terms
of their collective coordinates. We have seen this explicitly for long,
parallel strings and antistrings, and for closed strings. For these systems,
the infrared divergences cancel, but static configurations are not solutions
of the equations of motion. We have given a plausible definition of the
collective coordinates for such systems. But as these systems move, even if
initially the collective coordinates provide a good description, the
description quickly becomes incomplete, with a substantial (typically order
one or larger) fraction of the energy in excitations. In this sense, the
Nambu-Gato-Kalb-Ramond description is inadequate. As signaled by the infrared
divergence, it is necessary to consider degrees of freedom beyond the string
collective coordinates.

As we have recalled, in [15], at a heuristic level two limiting cases of axion
radiation by strings were considered for a long, parallel string and
antistring. One, referred to as adiabatic, involved slow motion of strings,
with the axion field adjusting at each instant to the configuration of the
string core. In this limit, it was argued, the energy of the axion field would
largely be converted to kinetic energy of the collective coordinate $b$. In
this paper, we have seen how this is realized explicitly at the level of the
equations of the effective field theory, in the limit that $\dot{b}$ is small.
A second limit, referred to as sudden, involved rapid motion of the strings,
with most of the energy in the axion field of the strings converted to axions.
Again, we have seen this at the level of the field theory equations when
$\dot{b}$ is not small. In the small $\dot{b}$, adiabatic case, few axions are
produced, In the large $\dot{b}$ limit, most of the energy in the axion field
surrounding the strings is converted to axions, but these have a spread in
wavelength, with typical wavelengths at time $t$ of order the string
separation at time $t$. From this, we see that in the sudden limit, the system
is not well described by the collective coordinates alone. We also see that
there is no reason to expect a lowlogarithmic enhancement of the low energy
axion density; rather the energy should be distributed uniformly (on a log
scale) over wave numbers between the Hubble scale and the scale $f_{a}$. These
observations support the picture, put forward in [15], that cosmic strings are
unlikely to yield a parameterically enhanced contribution to the population of
low momentum axions, and thus to an enhanced axion dark matter density. Recent
simulations with substantial reach in $\xi$[13] appear consistent with these
expectations.

Acknowledgements: I am grateful for collaboration and conversations with
Nicolas Fernandez, Akshay Ghalsasiand Hiren Patel. This work was supported in
part by the U.S. Department of Energy grant number DE-FG02-04ER41286.

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