A Dynamical Framework for Modeling Fear of Infection and Frustration with Social Distancing in COVID-19 Spread
AA Dynamical Framework for Modeling Fear of Infection andFrustration with Social Distancing in COVID-19 Spread
Matthew D. Johnston and Bruce PellLawrence Technological University21000 W 10 Mile Rd.Southfield, MI 48075 [email protected]@ltu.edu
August 14, 2020
Abstract
In this paper, we introduce a novel modeling framework for incorporating fear of infectionand frustration with social distancing into disease dynamics. We show that the resultingSEIR behavior-perception model has three principal modes of qualitative behavior— no out-break , controlled outbreak , and uncontrolled outbreak . We also demonstrate that the modelcan produce transient and sustained waves of infection consistent with secondary outbreaks.We fit the model to cumulative COVID-19 case and mortality data from several regions.Our analysis suggests that regions which experience a significant decline after the first waveof infection, such as Canada and Israel, are more likely to contain secondary waves of infec-tion, whereas regions which only achieve moderate success in mitigating the disease’s spreadinitially, such as the United States, are likely to experience substantial secondary waves oruncontrolled outbreaks. Since being first detected in Wuhan, China, in December 2019, severe acute respiratory syndromecoronavirus 2 (SARS-CoV-2), and the resulting disease COVID-19, has spread rapidly around theglobe. Without a vaccine or generally effective treatment, the unprecedented international miti-gation effort has instead focused on reducing transmission through travel restrictions, mandatoryquarantines, work-from-home initiatives, school closures, and social distancing practices. Thesemeasures have in turn had detrimental effects on economic productivity, job stability, and overallquality of life, which in some regions has limited public willingness to abide by social distancingguidelines, even as cases have continued to rise. To date, COVID-19 has afflicted at least 20million people worldwide and resulted in over 700,000 deaths [1].Mathematical modeling has played a significant role in understanding the primary trans-mission pathways and epidemiological parameters of the COVID-19 pandemic. Studies haveestimated the basic reproductive number, a key measure of the transmissibility of a disease, ofSARS-CoV-2 [2, 3, 4, 5] and projected the disease’s spread under a wide variety of public policyintervention scenarios, including variances in social distancing policies, travel restrictions, andface mask utilization [6, 7, 8, 9, 10, 11]. Forecasting the extent of the spread of COVID-19,1 a r X i v : . [ q - b i o . P E ] A ug owever, has been complicated by many factors, including evidence of asymptomatic spread[12, 13], issues with parameter identifiability [14, 15], and the uncertain mechanisms by whichsocial behaviors have altered the spread of the disease to date [16, 17, 18].Several methods have been proposed in the research literature for incorporating and evaluat-ing the role of social perception and behavior changes in dynamical models of emerging infectiousdisease (see the excellent review paper [19]). The papers [20, 21, 22] incorporate social behav-ior changes into an SIR compartmental model by dividing the susceptible and infectious classesinto individuals who initiate behavior changes to reduce transmission and those who do not,and allowing behavior change to transmit through social contact like a contagion. The papers[23, 24, 25, 26, 27] instead incorporate social distancing behavior by allowing perception to evolveas an independent state-dependent variable and having it feedback directly into the transmissionrate. Other studies have focused on regional movement patterns, effectiveness of social distanc-ing, and the summer release of school children followed by their return to classes in the fall [28].The more recent study [29] specific to the study of COVID-19 incorporates public support forsocial distancing as a function of both infection level and economic losses.In this paper, we present a novel modeling framework for incorporating social perception andbehavior (3) into a compartmental SEIR model (1). Our model incorporates the effects of fearof infection ( P I ) and frustration with social distancing ( P ω ) on social distancing ( ω ), and theeffects of social distancing on disease dynamics by modifying the transmission rate ( β ). Analysisof the corresponding system of differential equations (4) suggests that fear of infection can be aneffective mitigator of disease spread but that a high level of frustration with social distancing canoverwhelm these efforts and result in an uncontrolled outbreak (see Figure 2). Our analysis alsosuggests that delays in social feedback can lead to transient and sustained waves of infection,even in populations where the disease’s spread is controlled (see Figure 3).We fit the model to cumulative COVID-19 case and mortality data across several regions:Canada, the United States, Israel, Michigan, California and Italy. (See Figures 4 and 5.) Ouranalysis suggests that, although the capacity for secondary waves of infection after controllingthe initial outbreak is widespread, the magnitude of the reduction in infection after the initialoutbreak is a strong indicator of a society’s ability to be able to mitigate the secondary waves.Regions which have significant reductions in infection after the initial outbreak, such as Canadaand Israel, are predicted to have modest and controllable secondary waves. Countries which hadonly modest reductions in infection levels after the initial outbreak, such as the United States,are predicted to have large secondary waves which threaten to become uncontrolled outbreaks.This paper is organized as follows. In Section 2 we develop the SEIR behavior-perceptionsystem (4), which incorporates social distancing, fear of infection, and frustration with socialdistancing as time-dependent variables capable of influencing the dynamics of disease spread. InSection 3, we analyze the SEIR behavior-perception system (4) and demonstrate the admissiblebehaviors in a variety of parameter regions, leading to controlled outbreaks, uncontrolled out-breaks, and sustained waves of infection. In Section 4, we fit the model to cumulative COVID-19case and mortality data from several regions to estimate key epidemiological and social per-ception and behavior parameters of the COVID-19 pandemic. In Section 5, we summarize ourresults and avenues for future work. In Appendix A, we present the mathematical details of thestability results found in Section 3. In this section, we introduce an SEIR behavior-perception feedback model of disease spread. Themodel consists of two parts: (a) an SEIR model for tracking the evolution of disease dynamics (1);2nd (b) a behavior-perception loop involving social distancing, fear of infection, and frustrationwith social distancing (3).
We consider the classic
SEIR model where the population is divided in four compartments: S - Susceptible to infection; E - Exposed to infection but not yet symptomatic; I - Actively infectious ; R - Removed from the infection [30]. This gives the following model:Susceptible( S ) Exposed( E ) Infectious( I ) Removed( R ) β λ γ (1)For simplicity, we assume that only actively infectious individuals can transmit the infection tosusceptible individuals, although we note that asymptomatic spread is suspected in COVID-19[12, 13]. We assume that removed individuals cannot become susceptible to the illness again. Itis currently unclear whether COVID-19 may be contracted multiple times but reinfection is notbelieved to be a significant factor in disease spread [31].The dynamics of the system (1) can be modeling by the following SEIR system : dSdt = − βN SIdEdt = βN SI − λEdIdt = λE − γIdRdt = γI (2)where the parameters are as in Table 3. Notice that the recovery/infectious period γ − is theexpected number of days until an individual is no longer infectious, regardless of whether thatis a result of recovery, death, quarantine, or another means.Although the SEIR system (2) cannot be solved explicitly, the dynamics are well-understood.Trajectories with non-negative initial conditions stay non-negative and satisfy the populationsize conservation equation N = S ( t ) + E ( t ) + I ( t ) + R ( t ). Trajectories asymptotically approachthe steady state ( ¯ S, ¯ E, ¯ I, ¯ R ) = ( N − R ∗ , , , R ∗ ) where R ∗ is the solution of N − R ∗ = e − βNγ R ∗ .The value of R ∗ is the extent of the disease since it corresponds to the number of people whocontracted the disease during its course.The critical parameter for determining whether a disease will spread or die in a population isthe basic reproductive number , R , which quantifies the expected number of secondary infectionsproduced by one active infection entering a fully susceptible population. When R > R < R = βγ , which can be calculatedusing the next-generation method [32, 33, 34]. It follows that there will be an outbreak in (2) if β > γ while the disease will dissipate without an outbreak if β < γ .3 .2 SEIR Model With Behavior-Perception Feedback The SEIR model (1) by itself does not account for the possibility that individuals may alter theirbehavior, and therefore the disease’s trajectory, in response to the disease’s spread. Since thereis no vaccine or generally effective treatment for the novel coronavirus SARS-CoV-2, mitigationefforts have necessarily focused on modifications to social behavior such as social distancing,hand washing, and face mask utilization.We extend the SEIR model (2) to incorporate the effects of social perception and behaviorchange over time. We introduce the following time-dependent variables: social distancing behav-ior (0 ≤ ω ≤ perceived fear of infection (0 ≤ P I ≤ perceived frustration with socialdistancing (0 ≤ P ω ≤
1) and assume the following network of dependencies:Spread ofdisease( λE ) Socialdistancing( ω )Fear ofinfection( P I ) Frustration withsocial distancing( P ω ) + + − + − (3)Each arrow indicates how the first quantity influences the second, with a positive label (+)indicating a positive influence and (-) indicating a negative influence.We now extend the SEIR system (2) to include social perceptions and behavior. We assumethat the rates of changes of ω , P I , and P ω are influenced by variables with arrows leading toit in (3) and that, in the absence of disease, ω , P I , and P ω will decay to zero. This gives thefollowing SEIR behavior-perception system : dSdt = − β (1 − ω ) N SIdEdt = β (1 − ω ) N SI − λEdIdt = λE − γIdRdt = γI dωdt = k ω ( f ( P I , P ω ) − ω ) dP I dt = k P I ( g ( λE ) − P I ) dP ω dt = k P ω ( h ( ω ) − P ω ) . (4)Notice that the social perception variables P I and P ω influence the social behavior variable ω which in turn influences the effective transmission rate β (1 − ω ), as outlined in (3). Notice alsothat we use the daily number of new active cases λE as the catalyst for social perception changerather than the number of current active infections I , as used in [29]. We believe λE correlatesbetter with publicly reported case incidence data than the current infection level I .We utilize the following functions f , g , and h : f ( P I , P ω ) = P I (1 − ω ∗ P ω ) (5) g ( λE ) = ( λE ) q M q + ( λE ) q (6) h ( ω ) = ω. (7)4ariable Units Description S ≥ E ≥ I ≥ R ≥ ω ∈ [0 ,
1] % Measure of social distancing behavior P I ∈ [0 ,
1] none Measure of socially perceived fear of infection P ω ∈ [0 ,
1] none Measure of socially perceived frustration with social distancing t ≥ β ≥ − Transmission rate λ − ≥ γ − ≥ k ω ≥ − Rate of social distancing behavior change k P I ≥ − Rate of socially perceived fear of infection change k P ω ≥ − Rate of socially perceived frustration with socially distancing change N ≥ N = S + E + I + R ) M ≥ q ≥ ω ∗ ∈ [0 ,
1] % Maximum reduction to social distancing due to frustrationTable 1:
Variables and parameters for the SEIR system (2) and the SEIR behavior-perception system(4). The parameters k ω , k P I , and k P ω control the rate of social perception and behavior change. Theparameter M > q ≥ ω ∗ controls the extent to which frustration withsocial distancing reduces social distancing behavior. Figure 1:
The relationship between new daily infections ( λE ) and social distancing ( ω ) in the SEIR behavior-perception model (4), assuming the social perception and behavior variables ω , P I , and P ω are at quasi-steadystate (8). On the left, we take M = 50, ω ∗ = 0, and vary q . As q increases, the response of social distancing tonew infections becomes sharper. On the right, we take M = 50, q = 5, and vary ω ∗ . As ω ∗ increases, the effectivelevel of social distancing decreases even when the level of new infections is high ( λE > M ). Note that enforcingthe bounds ω ∗ ∈ [0 ,
1] guarantees ω ( t ) ∈ [0 ,
1] for all t ≥ . ω value lower than0 .
5, we can take ω ∗ > ω ( t ) value.
5e use a Hill function [35] on the level of new infections λE to control the perceived fear ofinfection (6) and note that this function become more “switch-like” around the threshold value M as q grows (see Figure 1). The parameters k ω , k P I , k P ω , q , M , and ω ∗ are as in Table 3.To further illustrate how social perception and behavior influences disease dynamics, we take ω , P I , and P ω at quasi-steady state in (4). This gives the following relationship for the level ofnew daily infections λE on the social distancing variable ω : ω = ( λE ) q M q + (1 + ω ∗ )( λE ) q . (8)Illustration of the effect of the parameters q , M , and ω ∗ on the relationship (8) is contained inFigure 1. In this section, we analyze the SEIR behavior-perception system (4) by considering reducedmodels which simulate the early-stage dynamics of the outbreak. We show that there are threedominants modes of behavior— no outbreak , controlled outbreak , and uncontrolled outbreak . Wealso show that the system (4) has the capacity for transient and sustained waves of infection,and investigate how this depends upon the delays in the social perception and behavior changes. We reduce the behavior-perception model (4) to consider the early-stage dynamics when nearlyeveryone in the population is susceptible to illness. To accomplish this, we set S = N and removethe equations for S and R . To further investigate the mechanisms which contribute to secondarywaves of infection, we consider three scenarios on the delays in social perception— no delays , onedelay , and two delays . In all cases, we state the model with frustration with social distancingincluded but can remove this variable by taking ω ∗ = 0. Model I: No delays.
We assume that the social perception and behavior variables in (4)operate on a significantly faster time scale than the disease dynamic variables. On the slow-timescale of the disease dynamics variables, this corresponds to taking ω , P I , and P ω in (4) at steadystate, which yields (8). Substituting (8) into (4) and removing S and R gives the following reduced no delay SEIR behavior-perception system : dEdt = β (cid:18) − ( λE ) q M q + (1 + ω ∗ )( λE ) q (cid:19) I − λEdIdt = λE − γI. (9)The behavior of this model is a reasonable approximation of that of (4) so long as the outbreakremains small ( S ≈ N ) and the social perceptions and behavior variables evolve on a significantlyfaster time-scale than the disease dynamics variables. Model II: One delay.
We assume that fear of infection evolves on a faster time scale thanthe remainder of the variables, which corresponds to taking P I at steady state in (4). After6emoving S and R , this gives the reduced one delay SEIR behavior-perception system : dEdt = β (1 − ω ) I − λEdIdt = λE − γI dωdt = k ω (cid:18) ( λE ) q M q + ( λE ) q (1 − ω ∗ P ω ) − ω (cid:19) dP ω dt = k P ω ( ω − P ω ) . (10)This model approximates (4) when the outbreak remains small ( S ≈ N ) and social perception ofthe disease evolves on a significantly faster time-scale than social behavior change or frustrationdue to social distancing. This is appropriate in societies where information is readily availabledue to media and other vectors of mass communication, but will or capability to change socialbehavior is limited. Model III: Two delays.
We remove S and R but do not assume any social perception orbehavior variables operate on a significantly faster time scale than the disease spread variables.This gives the reduced two delay SEIR behavior-perception system : dEdt = β (1 − ω ) I − λEdIdt = λE − γIdωdt = k ω ( P I (1 − ω ∗ P ω ) − ω ) dP I dt = k P I (cid:18) ( λE ) q M q + ( λE ) q − P I (cid:19) dP ω dt = k P ω ( ω − P ω ) (11)This model is the closest approximation of (4) since it only assumes that changes in S occur on aslower time scale than the remainder of the variables. The system (11) is appropriate for studyingthe early stages of an outbreak, diseases which are not easily transmitted, or diseases which aresignificant controlled through social intervention and never enter a mode of true outbreak. The primary epidemiological parameter for determining whether an outbreak will occur is the basic reproductive number R . This value corresponds to the expected number of secondaryinfections resulting from a single primary infection in a fully susceptible population (i.e. earlyin an outbreak when S ≈ N ). Consequently, when R < R > disease-free steady state ( ¯ E, ¯ I, ¯ ω, ¯ P I , ¯ P ω ) df = (0 , , , ,
0) of the reduced models (9), (10), and (11), and apply the next-generation method [32, 33, 34]. We will use this single steady state for all three reduced models byrestricting to the relevant model variables as required. For all three models, it can be computedthat the dominant eigenvalue of the next-generation matrix at the disease-free state is βγ so that R = βγ . It follows that the disease will die off if β < γ and we will have an outbreak if β > γ .The reduced models (9), (10), and (11) also have an endemic steady state , which is given by ( ¯ E, ¯ I, ¯ ω, ¯ P I , ¯ P ω ) end = (cid:32) Mλ (cid:18) β − γγ − ω ∗ ( β − γ ) (cid:19) q , Mγ (cid:18) β − γγ − ω ∗ ( β − γ ) (cid:19) q , β − γβ , β − γβ − ω ∗ ( β − γ ) , β − γβ (cid:33) . (12) Notice that the endemic steady state (12) is only physically meaningful for any of the threereduced models when 0 < β − γ < γω ∗ . Again, we can consider (12) to be the steady state for allthree reduced models by restricting to the appropriate model variables.7 a) q = 1 (b) q = 5 (c) q = 10 Figure 2:
Simulations of the active infection level of the SEIR behavior-perception system (4) with parametervalues λ = 0 . γ = 0 . M = 50, k ω = 0 . k P I = 0 . k P ω = 0 . ω ∗ = 0 .
5, and N = 1000000, and initialconditions S (0) = 999500, E (0) = 500, I (0) = 0, R (0) = 0, ω (0) = 0, P I (0) = 0, and P ω (0) = 0. Simulationswere conducted for β values from 0 .
01 to 0 . .
01 taking three different values of q : (a) q = 1; (b) q = 5; and (c) q = 10. The results are divided into three classes according to Table 2: (i) no outbreak,0 < β < .
1, (green); (ii) controlled outbreak, 0 . ≤ β < . . ≤ β ≤ . q does not affect whether there is controlled or uncontrolled outbreakbut does control the distinction between trajectories in the respective regions.Model Behavior Parameterrange Disease-freesteady state Endemicsteady state Oscillationsno delays(9) No outbreak β < γ stable DNE noneControlled outbreak 0 < β − γ < γω ∗ unstable stableUncontrolled outbreak β − γ > γω ∗ unstable DNEone delay(10) No outbreak β < γ stable DNE transientControlled outbreak 0 < β − γ < γω ∗ unstable stableUncontrolled outbreak β − γ > γω ∗ unstable DNEtwo delays(11) No outbreak β < γ stable DNE sustainedControlled outbreak 0 < β − γ < γω ∗ unstable varies ∗ Uncontrolled outbreak β − γ > γω ∗ unstable DNE Table 2:
Summary of the analysis of the disease-free and endemic steady states of the reduced SEIR behavior-perception systems (9), (10), and (11). The analysis predicts three distinct modes of epidemic behavior: (a) nooutbreak if β < γ ; (b) controlled outbreak if 0 < β − γ < γω ∗ ; and (c) uncontrolled outbreak if β − γ > γω ∗ .The distinction between these three cases on system (4) is illustrated in Figure 2. The analysis also suggeststhat delays in social perceptions feeding back into social behavior are required in order to have secondary wavesof infection. In particular, for the two delay model, the endemic steady state may lose stability even when theoutbreak is controlled, which yields limit cycles. These sustained waves of infection are demonstrated in Figure3. The mathematical analysis and numerically derived boundaries of stability are contained in Appendix A. The endemic steady state (12) corresponds to a controlled outbreak where the the fear ofinfection keeps the spread of the disease from growing uncontrolled (see Figure 2). Notice,8owever, that if ω ∗ is sufficiently high, the endemic steady state is not physically relevant andthe system exhibits a full outbreak similar to as though no social interventions had taken place.This suggests that frustration with social distancing can undo the gains in mitigating diseasespread given by fear of infection and the resultant social distancing.To further understand the behavior of the reduced models (9), (10), and (11), and thereforethe full SEIR behavior-perception model (4), we conduct linear stability analysis on the disease-free and endemic steady states of the reduced model (see Appendix A for mathematical details).A summary of the analysis is contained in Table 2. Simulations of the full SEIR behavior-perception model (4) are contained in Figure 2 for various values of the transmission rate β andsharpness parameter q .These analyses suggest three distinct possibilities for the dynamics:(i) No outbreak when β < γ (green in Figure 2). In this case, the overall level of infection E ( t ) + I ( t ) tends monotonically to zero. This corresponds to a disease which is not virulentenough to maintain presence in a population.(ii) Controlled outbreak when 0 < β − γ < γω ∗ (blue in Figure 2). In this case, the infectionnears the endemic steady state (12). This corresponds to a population which, throughsocial perception and behavior feedback, is able to tolerate a certain amount of ambientinfection without ever fully ridding it from the population.(iii) Uncontrolled outbreak when β − γ > γω ∗ (red in Figure 2). In this case, the infectionruns through the population relatively unimpeded and forms a characteristic infection peak.This corresponds to a population for which the frustration of social distancing is simply toogreat to allow the level of social behavior change required to mitigate the disease’s spread.Note that the parameter bounds for these three cases can be stated in terms of the basicreproductive number R = βγ . Defining R crit = ω ∗ ω ∗ , we have that the system (4) exhibits nooutbreak if 0 < R <
1, a controlled outbreak if 1 < R < R crit , and an uncontrolled outbreakif R crit < R . Intuitively, diseases with high reproductive numbers require less frustration withsocial distancing in order to revert to full outbreaks. In contrast, outbreaks of diseases with lowbasic reproductive numbers and high levels of frustration can still be controlled since the diseasedoes not spread as fast. For diseases with high reproductive numbers, it is crucial that mitigationstrategies are implemented in such a way that frustrations with disease mitigation strategies donot feedback to cause a decrease in social distancing. Significant discussion and public policy planning has centered around the possibility of secondarywaves of infection. It is widely feared that after initial success in mitigating the spread of COVID-19, social perceptions and behaviors will change in response to falling infection numbers andthat this will lead to a second wave of infection, and indeed this is suspected to be occurring inseveral regions around the world. Given the costs associated with social restrictions, predictingand assessing the timing and scale of potential second waves of infection is an area of significantresearch and concern.To analyze the possibility of secondary waves of infection in the SEIR behavior-perceptionsystem (4), we further analyze the stability of the endemic steady state (12). The mathematicalanalysis is contained in Appendix A and the results are summarized in Table 2. We see thatno oscillations are possible for the reduced no delay system (9), transient but not sustainedoscillations (i.e. limit cycles) are possible in certain parameter regions for the one delay system(10), and both transient and sustained oscillations are possible in certain parameter regions for9 a) q = 1 (No oscillations) (b) q = 5 (transient oscillations) (c) q = 10 (sustained oscillations) Figure 3:
Simulations of the SEIR behavior-perception system (4) with the parameter values β = 0 . λ = 0 . γ = 0 . M = 100, k ω = 0 . k P I = 0 . k P ω = 0 . ω ∗ = 0 .
25, and N = 1000000 and initial condition S (0) = 999500, E (0) = 500, I (0) = 0, R (0) = 0, ω (0) = 0, P I (0) = 0, and P ω (0) = 0. Simulations are run forthree cases on the parameter q : (a) q = 1; (b) q = 5; and (c) q = 10. We can see that, as q becomes larger, thedynamics changes from quasi-stability (Figure (a)), to transient oscillations (Figure (b)), to sustained oscillations(Figure (c)). This suggests that the sharpness in the feedback from the infection level to social distancing behaviorplays a key role in creating secondary waves of infection. the two-delay system (11). This analysis suggests that delays in the social feedback and behaviorvariables play a key role in the generating the capacity for secondary waves of infection. Thedistinctions between no oscillations, transient oscillations, and sustained oscillations for the fullSEIR behavior-perception model (4) are illustrated in Figure 3. In this section, we fit the SEIR behavior-perception system (4) to COVID-19 case incidence andmortality data. Data is taken from the database hosted by the Center for Systems Science andEngineering (CSSE) at Johns Hopkins University [36]. Model parameters and initial conditionsfor ω , P I and P ω were estimated through nonlinear least-squares curve fitting to the cumulativereported case and mortality data and their corresponding rates of change. We used the built-inMATLAB functions fmincon and multistart to minimize the following objective function: Obj ( θ ) = T (cid:88) i =1 (cid:16) c ( θ, t i ) − ˆ c i ) + ( d ( θ, t i ) − ˆ d i ) (cid:17) + T (cid:88) i = T − (cid:0) r c ( θ, t i ) − ˆ r ci ) + ( r d ( θ, t i ) − ˆ r di ) (cid:1) , where c ( θ, t i ) and d ( θ, t i ) are the estimated cumulative cases and death cases from the modelwith parameter set θ at the calendar date t i , respectively. ˆ c i and ˆ d i correspond to the actualreported cumulative cases and death cases at calendar date t i . The second term consists of therates of change of cumulative cases and death cases. That is, c ( θ, t i ) and d ( θ, t i ) are the rates ofchange of cumulative cases and death cases and ˆ r ci and ˆ r di are the actual rates of change fromthe data. T is the total number of data points. We chose to include fitting to the number ofdeaths to promote constraining the model in the fitting process.To test the model, we have selected six regions which have experienced different profiles inhow the epidemic has spread: Canada, Italy, the United States, Israel, Michigan, and California.10 a) Canada (b) United States (c) Israel Figure 4:
Model fits of the SEIR behavior-perception system model (4) to data sets of varying dynamics: (a)Canada, (b) United States and (c) Israel. Fitted Model parameters are shown in Table 3. Top row: cumulativecases and active cases (scaled according to the left and right axis, respectively). Bottom row: social distancing,perception of fear of infection and frustration with social distancing ( ω , P I and P ω ). The basic reproductivenumbers for Canada, US and Israel are 1.96, 3.78 and 1.53, respectively. In Canada and Italy, after a substantial initial outbreak, the outbreak has been largely contained.In the United States, there was a large outbreak in March and April, a moderate dip in May andearly June, and then a larger outbreak in July. In Israel, there was a small outbreak in Marchand April, a period of near eradication in May and June, and then a large outbreak in July. InMichigan, there was an initial outbreak that was largely contained, until early July where newcases of infection started to increase. California, much like the United States experienced a initialoutbreak followed by a moderate dip in the month of April, and then entered the beginning ofa much larger outbreak. The results are contained in Figures 4 and 5 and parameter values arecontained in Table 3.Simulated model fits indicate that the SEIR behavior-perception model supports differentepidemic profiles as fear of infection, social distancing and frustration with social distancingare allowed to dynamically change between 0 and 1. These profiles include controlled diseaseincidence with near eradication (Canada in Figure 4), loss of disease control and management(United States in Figure 4), and disease resurgence in the form of a secondary wave (Israel inFigure 4).Figure 4 shows model fits to reported case data for Canada, the United States and Israel.The model fit for Canadian reported case data shows an initial outbreak in early March which isultimately controlled over the following four months. The model predicts stable levels of high fearof infection while social distancing and frustration with social distancing both tend to sufficientlevels for disease control and near eradication. On the other hand, the United States showsan initial outbreak which is temporarily controlled until late May. Fear of infection quicklysaturates to 1, while frustration with social distancing slowly increases over time. After theinitial increase in social distancing, the increasing levels of frustration lead to a steady decreasein social distancing levels which become insufficient for control or eradication and thus active11 a) Michigan (b) California (c) Italy
Figure 5:
Model fits of the SEIR behavior-perception system model (4) to data sets of varying dynamics: (a)Michigan, (b) California and (c) Italy. Fitted Model parameters are shown in Table 3. Top row: cumulativecases and active cases (scaled according to the left and right axis, respectively). Bottom row: social distancing,perception of fear of infection and frustration with social distancing ( ω , P I and P ω ). The basic reproductivenumbers for Michigan, California and Italy are 2.2, 3.62 and 2.51, respectively. Parameter Canada United States Israel Michigan California Italy β λ γ k ω k P I k P ω q M ω ∗ N S (0) 37589621 328198408 9151800 9986769 39509766 60359707 E (0) 200 200 200 200 200 200 I (0) 179 1392 0 31 34 93 R (0) 0 0 0 0 0 0 ω (0) 0 0 0.187 0.177 0.144 0.121 P I (0) 0.2 0.2 0.183 0.165 0.089 0.078 P ω (0) 0.019 0.2 0 0.2 0.166 0.109Table 3: Parameter and initial values used to create model simulations in Figure 4 and 5. cases of infection increase. Interestingly, Israel shows an initial outbreak followed by a periodof near eradication of the disease. However, fear of infection and social distancing decrease to12nsufficient levels too quickly after the first wave of infections and leads to a secondary outbreak.Figure 5 shows model fits of the SEIR behavior-perception system for Michigan, Californiaand Italy. Michigan shows an initial outbreak that is relatively under control. However, themodel predicts a period starting after mid-July where social distancing, fear of infection andfrustration with social distancing begin to oscillate, but not to the extend seen in the Israelmodel fit from Figure 4. California experiences an initial outbreak that is only temporarilymanaged before increasing exponentially. After the initial outbreak, the rise in active cases isaccompanied by high fear of infection and an increasing level of frustration with social distancing.This leads to a reduction in social distancing. Italy dynamically behaves similar to what is seenin Michigan and Canada. That is, after the initial outbreak, high fear of infection drives a periodof stable social distancing and frustration with social distancing.
We have introduced an SEIR behavior-perception system (4) for modeling the feedback of socialfear of infection and frustration with social distancing on the dynamics of disease spread. Wehave shown the following: (1) including fear of infection leads to a controlled outbreak the levelof which depends on each society’s tolerance for infection; (2) delays in the fear of infection canlead to secondary and sustained waves of infection ; and (3) frustration with social distancing canovercome the fear of infection to lead to an uncontrolled outbreak . Where analytically possible,we have provided parameter ranges where the relevant behaviors occur.We have also fit the model to cumulative COVID-19 case data from several regions: Canada,the United States, Israel, Michigan, California and Italy. (See Figures 4 and 5.) The fits obtainedvalidate our SEIR behavior-perception model as capable of capturing the emergence of social-feedback-driven secondary waves of infection. Our analysis furthermore suggests that regionswhich experience significant reductions in new infection levels following their initial outbreak,such as Israel and Canada, are likely to be able to mitigate secondary waves of infection. Regionswhich experiences only moderate reductions in new infection levels, such as the United States,are likely to experience more dramatic secondary waves and are at increase risk for entering intoan uncontrolled outbreak.Using the full model (4), we estimated the basic reproductive number for Canada, UnitedStates, Israel, Michigan, California and Italy, to be 1.96, 3.78, 1.53, 2.2, 3.62 and 2.51 respectively.These estimations are inline with other studies [4, 5]. In addition, the time varying effectivereproductive number is now implicitly a function of the social behavior in the population, becauseof the incorporation of behavior-perception feedback variables, ω , P I and P ω . This effectivereproductive number may provide further real-time and future insight into not only the diseasedynamics, but the social dynamics that drive the spread of the disease.The reduced model (9) produces a threshold number, R crit > R > R crit the solutions become unbounded. This unbounded behavior manifests itself within the full model(4) as the uncontrolled outbreak which forms a characteristic peak often seen in disease outbreaks.However, unlike in the reduced model, the full model has bounded solutions, because of the finitesusceptible population. Thus herd immunity ultimately reduces the disease burden to zero andthe disease dies out. These dynamics are illustrated in Figure 2.The model introduced in this paper presents several immediate opportunities for further work.1. The model (4) assumes the same critical infection level M for both the ascending anddescending phases of disease dynamics. In practice, however, we have seen some countriesquick to lockdown and slow to open up, while other countries are slow to lockdown and quick13o open up, which suggests a different critical value M in the ascending and descendingphase of an outbreak.2. Currently, our estimates for the social distancing variable ω only come through diseaseprevalence. Throughout the COVID-19 pandemic, however, data on social mobility hasbeen provided by several sources, including Google, Apple, and the United States De-partment of Transportation [37, 38]. Further insight in the social mechanisms underlyingdisease dynamics might be gained by fitting ω to this data as well.3. Numerical simulations such as those in Figure 3 suggest that the reduced SEIR behavior-perception model (11) undergoes a Hopf bifurcation as the endemic steady state (12) losesstability. Numerical results also suggest that trajectories of the reduced models (10) and(11) are bounded in the controlled outbreak scenario but unbounded for the uncontrolledoutbreak scenario. These results are currently unproved. References [1]
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Math. Ann. , (1895), 273284.16 Stability analysis of SEIR behavior-perception system
In this Appendix, we provide the mathematical stability analysis for the disease-free and endemicsteady states of the reduced feedback systems (9), (10), and (11). We make use of the trace-determinant condition for planar systems [39] and the Routh-Hurwitz criterion for non-planarsystems [40, 41]. The Routh-Hurwitz conditions states that a given polynomial has all rootswith negative real part if and only if the first column entries of the Routh-Hurwitz table are allpositive. Consequently, if the Routh-Hurwitz table of the characteristic polynomial of a matrixhas any first column entry which is negative, then the matrix has an eigenvalue with positivereal part.
A.1 No delays
Assuming that ω , P I , and P ω (if considered) equilibriate immediately, (11) reduces to the fol-lowing direct systems: No frustration With frustration dEdt = β (cid:18) − ( λE ) q M q + ( λE ) q (cid:19) I − λEdIdt = λE − γI. dEdt = β (cid:18) − ( λE ) q M q + (1 + ω ∗ )( λE ) q (cid:19) I − λEdIdt = λE − γI. (13) In addition to the disease-free steady state ( ¯ E, ¯ I ) df = (0 , E, ¯ I ) end = (cid:32) Mλ (cid:18) β − γγ − ω ∗ ( β − γ ) (cid:19) q , Mγ (cid:18) β − γγ − ω ∗ ( β − γ ) (cid:19) q (cid:33) (14)which is only physical meaningful if 0 < β − γ < γω ∗ . We will consider the “With frustration”system in (13) and reduced to the “No frustration” case by taking ω ∗ = 0.We have the following result. Theorem A.1.
Consider the reduced direct SEIR behavior-perception models (13) . The followingbehaviors are possible:1. If β < γ then there is only the disease-free steady state and it is asymptotically stable.2. If < β − γ < γω ∗ then the disease-free steady state is unstable, and the endemic steadystate is positive and asymptotically stable. Furthermore, trajectories near the endemicsteady state may not exhibit oscillatory behavior.3. If β − γ > γω ∗ then there is only the disease-free steady state and it is unstable, and solutionsbecome unbounded.Proof. We consider the Jacobian of the system (13) evaluated at the disease-free steady stateand the endemic steady state (14). After simplifying, we have J df = (cid:20) − λ βλ − γ (cid:21) and J end = (cid:34) − λ (cid:16) ( β − γ )( γ − ω ∗ ( β − γ )) q − γβγβ (cid:17) γλ − γ (cid:35) . (15)17e have tr( J df ) = − ( λ + γ ) < J df ) = − λ ( β − γ ). It follows that disease-free steadystate is a stable node if β < γ (no outbreak) and a saddle if β > γ (controlled or uncontrolledoutbreak). For the endemic steady state, we havedet( J end ) = ( β − γ )( γ − ω ∗ ( β − γ )) λqβ tr( J end ) = − ( β − γ )( γ − ω ∗ ( β − γ )) λq + γβ ( γ + λ ) γβ . Since we are only concerned with the endemic steady state when 0 < β − γ < γω ∗ , we havethat det( J end ) > J end ) <
0. It follows that the endemic steady state (14) of (13) isasymptotically stable whenever it exists.In order for trajectories near the endemic steady state (14) to oscillate, we require thattr( J end ) − J end ) <
0. To analyze this condition, we computetr( J end ) − J end ) = [ λ ( β − γ )( γ − ω ∗ ( β − γ )) q + γβ ( λ − γ )] + 4 λγ β γ β > . (16)It follows that the linearized system may not permit oscillations around the endemic steady state(14) so that (13) does not permit the oscillatory behavior.To prove solutions become unbounded if β − γ > γω ∗ (uncontrolled outbreak) we note thatthis condition implies that ( λE ) q > > ( β − γ ) M q γ − ω ∗ ( β − γ )for E >
0. It follows that ( λE ) q M q + (1 + ω ∗ )( λE ) q < β − γβ so that β (cid:18) − ( λE ) q M q + (1 + ω ∗ )( λE ) q (cid:19) > γ. Now consider the quantity E + I . From (9) we have that ddt ( E + I ) = (cid:20) β (cid:18) − ( λE ) q M q + (1 + ω ∗ )( λE ) q (cid:19) − γ (cid:21) I > . It follows that, provided
E >
0, then we have that E + I is continually increasing for all time.Now suppose that there is a least upper bound E lim + I lim > E ( t )+ I ( t ) ≤ E lim + I lim .It follows that E ( t ) + I ( t ) approaches a single point on this set and by continuity of (9), thispoint must be a steady state. When β − γ > γω ∗ , however, there is only the disease-free steadystate and this does not lie on this set. It follows that there is not a least upper bound E lim + I lim to E ( t ) + I ( t ). Consequently, it follows thatlim t →∞ E ( t ) + I ( t ) = ∞ . That is, the overall level of infection in the population is unbounded.18 .2 One Delay
Assuming that fear of infection ( P I ) operates on a significantly faster timescale than the remain-der of the variables, we can set dP I dt = 0 in (11) to get the reduced one delay SEIR behavior-perception systems : No frustration With frustration dEdt = β (1 − ω ) I − λEdIdt = λE − γIdωdt = k ω (cid:18) ( λE ) q M q + ( λE ) q − ω (cid:19) dEdt = β (1 − ω ) I − λEdIdt = λE − γIdωdt = k ω (cid:18) ( λE ) q M q + ( λE ) q (1 − ω ∗ P ω ) − ω (cid:19) dP ω dt = k P ω ( ω − P ω ) . (17) The system (17) has the disease-free steady state ( ¯ E, ¯ I, ¯ ω, ¯ P ω ) df = (0 , , ,
0) and the endemicsteady state( ¯ E, ¯ I, ¯ ω, ¯ P ω ) end = (cid:32) Mλ (cid:18) β − γγ − ω ∗ ( β − γ ) (cid:19) q , Mγ (cid:18) β − γγ − ω ∗ ( β − γ ) (cid:19) q , β − γβ , β − γβ (cid:33) (18)which again is only physical meaningful if 0 < β − γ < γω ∗ . Once again, we will consider onlythe “with frustration” case and limit to the “no frustration” case by taking ω ∗ = 0.We have the following result. Theorem A.2.
Consider the reduced direct SEIR behavior-perception models (13) . The followingbehaviors are possible:1. If β < γ then there is only the disease-free steady state and it is asymptotically stable.2. If < β − γ < γω ∗ then the disease-free steady state is unstable, and the endemic steady stateis positive and asymptotically stable. Furthermore, trajectories near the endemic steadystate may or may not exhibit oscillatory behavior depending on the parameter values.3. If β − γ > γω ∗ then there is only the disease-free steady state and it is unstable.Proof. The Jacobian of (10) evaluated at the disease-free steady state is J df = − λ β λ − γ − k ω
00 0 k P ω − k P ω . (19)The diagonal structure of (19) means the eigenvalues can be analysed by considering the decom-posed matrices (cid:20) − λ βλ − γ (cid:21) and (cid:20) − k ω k P ω − k P ω (cid:21) . (20)The right matrix in (20) has the eigenvalues λ , = − k ω , − k P ω < β < γ and a saddle if β > γ .19he Jacobian of (10) evaluated at the endemic steady state ( ¯ E, ¯ I, ¯ ω, ¯ P I ) (18) is J end = − λ γ − βMγ (cid:16) β − γγ − ω ∗ ( β − γ ) (cid:17) q λ − γ λk ω ( β − γ ) − q ( γ − ω ∗ ( β − γ ))
1+ 1 q βM ( β − ω ∗ ( β − γ )) − k ω − ω ∗ ( β − γ ) k ω β − ω ∗ ( β − γ ) k P ω − k P ω (21)It is not feasible to determine the eigenvalues of (21) directly. The Routh-Hurwitz criterion,however, can be applied to the characteristic polynomial of (21). The first two entries of theRouth-Hurwitz table are 1 and λ + γ + k P ω + k ω , which are trivially positive. The third and fourthentries can be expanded and factored into terms which are either positive of contain factors of β − γ of β − ω ∗ ( β − γ ) (see supplemental computational material). We notice that the endemiccondition 0 < β − γ < γω ∗ implies that β − ω ∗ ( β − γ ) > γ − ω ∗ ( β − γ ) > γ − ω ∗ (cid:16) γω ∗ (cid:17) > k P ω k ω λ ( β − γ )( γ − ω ∗ ( β − γ )) qβ − ω ∗ ( β − γ )This is positive, so that all the first-column entries of the Routh-Hurwitz table are positive. Itfollows that all of the eigenvalues of (15) have negative real part so that the endemic steadystate of (17) is locally asymptotically stable. It follows that the system with one delay is notconsistent with sustained oscillations around the endemic steady state (18).It is challenging to find explicit conditions on the parameters which guarantee (21) has com-plex eigenvalues. It can be checked numerically, however, that this is possible. One example setof parameters is β = 0 . λ = 0 . γ = 0 . q = 10, M = 100, k ω = 0 . k P ω = 0 . ω ∗ = 0.Substituted in (21), this produces the eigenvalues: λ = − . λ = − . λ , = − . ± . i . Itfollows that the one-delay models (17) permit transient waves of infection. A.3 Two Delays
Now consider the full two delay model, with and without frustration with social distancing:
No frustration With frustration dEdt = β (1 − ω ) I − λEdIdt = λE − γIdωdt = k ω ( P I − ω ) dP I dt = k P I (cid:18) ( λE ) q M q + ( λE ) q − P I (cid:19) dEdt = β (1 − ω ) I − λEdIdt = λE − γIdωdt = k ω ( P I (1 − ω ∗ P ω ) − ω ) dP I dt = k P I (cid:18) ( λE ) q M q + ( λE ) q − P I (cid:19) dP ω dt = k P ω ( ω − P ω ) . (22) The system (22) has the disease-free steady state ( ¯ E, ¯ I, ¯ ω, ¯ P I , ¯ P ω ) df = (0 , , , , ( ¯ E, ¯ I, ¯ ω, ¯ P I , ¯ P ω ) end = (cid:32) Mλ (cid:18) β − γγ − ω ∗ ( β − γ ) (cid:19) q , Mγ (cid:18) β − γγ − ω ∗ ( β − γ ) (cid:19) q , β − γβ , β − γβ − ω ∗ ( β − γ ) , β − γβ (cid:33) (23) < β − γ < γω ∗ . We consider the “with frustration”case with the understanding that we can limit to the “no frustration” case by taking ω ∗ = 0.We have the following result. Theorem A.3.
Consider the reduced direct SEIR behavior-perception models (13) . The followingbehaviors are possible:1. If β < γ then there is only the disease-free steady state and it is asymptotically stable.2. If < β − γ < γω ∗ then the disease-free steady state is unstable, and the endemic steadystate is positive. Trajectories near the endemic steady state may converge exponentiallytoward it, exhibit damped oscillations, or converge toward nearby limit cycles, dependingon the parameter values.3. If β − γ > γω ∗ then there is only the disease-free steady state and it is unstable.Proof. The Jacobian of (22) evaluated at the disease-free steady state is − λ β λ − γ − k ω k ω
00 0 0 − k P I
00 0 k P ω − k P ω (24)The diagonal structure of (24) means the eigenvalues can be analysed by considering the decom-posed matrices (cid:20) − λ βλ − γ (cid:21) and − k ω k ω − k P I k P ω − k P ω . (25)The right matrix in (25) has the eigenvalues λ , , = − k ω , − k P I , − k P ω < β < γ and a saddle if β > γ .The Jacobian evaluated at the endemic steady state ( ¯ E, ¯ I, ¯ ω, ¯ P I , ¯ P ω ) end (23) is − λ γ − Mβγ (cid:16) β − γγ − ω ∗ ( β − γ ) (cid:17) q λ − γ − k ω k ω (cid:16) β − ω ∗ ( β − γ ) β (cid:17) − k ω ω ∗ ( β − γ ) β − ω ∗ ( β − γ ) λk PI q ( β − γ ) − q ( γ − ω ∗ ( β − γ )
1+ 1 q M ( β − ω ∗ ( β − γ )) − k P I
00 0 k P ω − k P ω (26)The eigenvalues of (26) cannot be reasonably computed directly. The Routh-Hurwitz condition,however, can be utilized to determine if the matrix permits eigenvalues with positive real part.The first two column entries of the Routh-Hurwitz table are 1 and λ + γ + k P I + k P ω + k ω , bothof which are positive. The third entry can be factored as − n ( q, · q + n ( q, d ( q,
0) (27)where n ( q,
1) = k P I λk ω ( β − γ )( γ − ω ∗ ( β − γ ))) n ( q,
0) = γ ( k P I + k ω + k P ω ( λ + γ + k P ω + k ω )( λ + k P I + γ )( β − ω ∗ ( β − γ )) + βγk P ω k ω ( k P ω + k ω ) d ( q,
0) = γ ( γ + k P I + k P ω + k ω + λ )( β − ω ∗ ( β − γ ))21or the endemic condition 0 < β − γ < γω ∗ we have β − γ > , γ − ω ∗ ( β − γ ) > , and β − ω ∗ ( β − γ ) > n ( q, > n ( q, >
0, and d ( q, >
0. It follows that (27) can be made negative bychoosing q > n ( q, n ( q, . In particular, for any values of the parameters β, λ, γ, k P I , and k ω we can choose q sufficientlylarge so that the first column of the Routh-Hurwitz table has a sign change, so that the endemicsteady state ( ¯ E, ¯ I, ¯ ω, ¯ P I , ¯ P ω ) end (23) is unstable. The value of qq