Active Control and Sustained Oscillations in actSIS Epidemic Dynamics
aa r X i v : . [ phy s i c s . s o c - ph ] J u l Active Control and Sustained Oscillationsin actSIS Epidemic Dynamics ⋆ Yunxiu Zhou * Simon A. Levin ** Naomi Ehrich Leonard *** ∗ Operations Research and Financial Engineering, Princeton University,Princeton, NJ 08544 USA (e-mail: [email protected]). ∗∗ Ecology and Evolutionary Biology, Princeton University, Princeton, NJ08544 USA (e-mail: [email protected]) ∗∗∗
Mechanical and Aerospace Engineering, Princeton University, Princeton,NJ 08544 USA (e-mail: [email protected])
Abstract:
An actively controlled Susceptible-Infected-Susceptible (actSIS) contagion model is presentedfor studying epidemic dynamics with continuous-time feedback control of infection rates. Our workis inspired by the observation that epidemics can be controlled through decentralized disease-controlstrategies such as quarantining, sheltering in place, social distancing, etc., where individuals activelymodify their contact rates with others in response to observations of infection levels in the population.Accounting for a time lag in observations and categorizing individuals into distinct sub-populationsbased on their risk profiles, we show that the actSIS model manifests qualitatively different features ascompared with the SIS model. In a homogeneous population of risk-averters, the endemic equilibriumis always reduced, although the transient infection level can exhibit overshoot or undershoot. In ahomogeneous population of risk-tolerating individuals, the system exhibits bistability, which can alsolead to reduced infection. For a heterogeneous population comprised of risk-tolerators and risk-averters,we prove conditions on model parameters for the existence of a Hopf bifurcation and sustained oscillationsin the infected population.
Keywords: active control, feedback, contagion models, epidemic processes, heterogeneity, oscillation.1. INTRODUCTIONDeterministic compartmental models in epidemiology have pro-vided valuable insights for the understanding of evolutionarydynamics of infectious disease spread in a host population(Kermack and McKendrick (1927)). These models have alsobeen widely applied to study various other spreading dynam-ics, including but not limited to information dissemination insocial networks (Jin et al. (2013)), sentiment contagion in hu-man society (Zhao et al. (2014)) and the propagation of sys-temic risks in financial market (Demiris et al. (2014)). Althoughthese deterministic models unavoidably ignore some importantdetails such as individual heterogeneity and network structure,they adequately capture the qualitative features of the spreadingdynamics, including transient system behavior and stability ofsolutions. The models have been shown to be close approxima-tions of certain Markov chain models of the underlying stochas-tic dynamics, see, e.g., Sahneh et al. (2013). The Susceptible-Infected-Susceptible (SIS) model has been widely studied andapplied in epidemiological modeling. While its assumption thatindividuals acquire no immunity after recovery may not be suit-able for certain diseases, it provides a worst case scenario, whichis valuable for a large class of contagious diseases in general.The SIS model in its simplest form assumes a constant infectionrate. However, it is well acknowledged that the rate can vary ⋆ The research was supported in part by Army Research Office grant W911NF-18-1-0325, Office of Naval Research grant N00014-19-1-2556, National ScienceFoundation grants CMMI-1635056, CNS-2027908, CCF-1917819, and C3.aiDigital Transformation Institute. over time due to the different control strategies taken by indi-viduals, which in turn affects the contagion dynamics. Indeed,understanding such models and their extensions is fundamentalto developing disease control strategies. A review of analysisand control of epidemics is provided in Nowzari et al. (2016).In this paper, we propose a model with feedback controlledinfection rates to account for active control strategies. Individ-uals modify their contact rates with others based on what theyobserve about the level of infection in the population. We modela time lag in the observations and distinguish individuals as risk-averters , risk-tolerators , and risk-ignorers . Risk-averters repre-sent those who change their contact rate with others in the oppo-site direction as the change in the observed infected populationlevel, e.g., those who could and did stay at home and practicedincreased social distancing during the COVID-19 pandemic asthey saw the infected population grow.
Risk-tolerators representthose whose contact rate with others changes in the same direc-tion as the change in the observed infected population level, e.g.,health care workers, delivery workers, and other essential work-ers, who were obliged to work during the COVID-19 pandemic.
Risk-ignorers represent those who do not actively modify theircontact rates.Our work contributes to the literature in the following ways.First, while active and passive spreading was first distinguishedin the social economics literature (see Hartmann et al. (2008)),our model rigorously demonstrates the differences betweenthem, and we prove new results on the dynamics of contagionwith active control. Further, our model serves as one form ofthe state-dependent approaches discussed in Rands (2010) thatffer evolutionarily grounded ways for studying social conta-gion in collective processes. Second, our feedback model usesa low-pass filter of the measured infected population level. Thismodels the observation delay and introduces an important ro-bustness to uncertainty. This is in contrast to contagion modelswhich feed back infected population level directly, as in Baker(2020); Franco (2020). Third, we prove a new and relatively sim-ple scenario under which sustained oscillations appear withinepidemiological frameworks without external forcing. Under-standing mechanisms that can lead to oscillations is criticallyimportant in the context of infectious disease spread (Lin et al.(1999); Dushoff et al. (2004); Camacho et al. (2011), Xu et al.(2020)). It is likewise of great interest in many other socio-economic processes, e,g., the rise and fall of business cycles(Mishchenko (2014)) and fluctuation of behavioral preferencesin social networks (Pais et al. (2012)). Fourth, in contrast tocontrol strategies that require global knowledge about the exactunderlying spreading dynamics, e.g., Nowzari et al. (2016), ourwork provides evidence for the promise of tunable decentralizedactive control strategies to manage the dynamics of epidemics.The paper is organized as follows. In Section 2, we reviewthe Susceptible-Infected-Susceptible (SIS) model. We introducethe actively controlled Susceptible-Infected-Susceptible (act-SIS) model in homogeneous populations of risk-tolerators and risk-averters respectively in Section 3, where we analyze theequilibrium solutions and stability conditions and show theirqualitatively different features as compared with the SIS model.We examine the actSIS model in a heterogeneous populationcomprised of risk-tolerators and risk-averters in Section 4, andprove conditions for a Hopf bifurcation with a stable limit cycle.We conclude in Section 5.2. BACKGROUND
Consider a disease spreading in a large, randomly-mixed pop-ulation, where individuals are divided into either susceptible(S) or infected (I) classes. Susceptible individuals get infectedat rate β while infected individuals recover at rate δ . Let p ( t ) be the fraction of infected individuals and s ( t ) the fraction ofsusceptible individuals at time t . The SIS model is ˙ s = − βsp + δp, ˙ p = βsp − δp. (1)The force of infection term βp in (1) describes the rate at whichsusceptible individuals get infected (Muench (1934)). It can bedecomposed into three terms: βp = ¯ β × α × p , where ¯ β is thetransmission rate of the disease and an intrinsic property of thedisease, and α is the effective number of contacts per unit time.Since s ( t ) = 1 − p ( t ) , (1) can be rewritten as ˙ p = β (1 − p ) p − δp. (2)The steady-state behavior of solutions to the well-mixed SISmodel (2) is characterized by the basic reproduction number R = β/δ , a key concept in epidemiology that defines theepidemic threshold of a particular infection (Diekmann et al.(1990)). If R > , the disease persists and a nonzero fraction ofthe population is infected at steady state: as t → ∞ , p ( t ) → − δ/β , the endemic equilibrium (EE). If R ≤ , the diseasedies out at steady state: as t → ∞ , p ( t ) → , the infectionfree equilibrium (IFE). At R = 1 , there is a transcriticalbifurcation. A natural extension of the homogeneous population setting isthe introduction of heterogeneities of various kinds. Charac-terizing population heterogeneity in terms of infection and/orrecovery rates has been discussed both in population subgroups(Anderson and May (1992)) and in networks (Hethcote and Yorke(1984), Pagliara and Leonard (2020)). In the following, we re-view the network SIS model that was originally introduced asthe multi-group SIS model in Lajmanovich and Yorke (1976).Consider a heterogeneous population of n sub-populations, eachlarge, well-mixed and homogeneous. Let susceptible individualsin sub-population i get infected through contact with infectedindividuals in sub-population j at rate β ij ≥ and infectedindividuals in sub-population i recover at rate δ i ≥ . Therate β ij can be decomposed as β ij = ¯ β × α ij where ¯ β is thetransmission rate and α ij represents the effective contact ratebetween sub-population i and j . The network SIS model is ˙ p i = (1 − p i ) n X i =1 β ij p j − δ i p i , (3)where p i ( t ) denotes the fraction of infected individuals in the i th sub-population, or equivalently, the probability that a typicalindividual in sub-population i is infected at time t . Let B = { β jk } and Γ = diag ( δ , . . . , δ N ) be the infection matrix andthe recovery matrix, respectively.For the network SIS model (3), the basic reproduction numberis R = ρ (cid:0) B Γ − (cid:1) , where ρ denotes the spectral radius. For R ≤ , solutions converge to the IFE as t → ∞ while for R > , solutions converge to the EE. See Lajmanovich and Yorke(1976), Fall et al. (2007), Mei et al. (2017) for details.3. HOMOGENEOUS POPULATIONOur proposal of the actively controlled Susceptible-Infected-Susceptible (actSIS) model is based on the following obser-vations. First, individuals conduct behavioral changes as theyacquire information of the epidemics which consequently affecttheir contact rates with others (Funk et al. (2009)). Such infor-mation, however, often involves estimations that are delayed andunavoidably omits details in the finer time scale. We are moti-vated in part by the model and study of change in susceptibilityafter first infection as presented Pagliara and Leonard (2020).Accordingly, we present the actSIS model for studying epidemicdynamics with continuous-time feedback control of infectionrates. We let the infection rate be the product of the intrinsictransmission rate ¯ β and an effective contact rate α ( · ) that isactively modified by individuals based on their observations ofthe system state. To account for the uncertainty and delay inmeasurements of the infection level in the population, we letthe feedback responses depend on the filtered state p s of theinfected fraction p , where p s tracks p and possibly some externalstimulus r ( t ) with a time constant τ s . The actSIS model is ˙ p = β (1 − p ) p − δp,τ s ˙ p s = − p s + p + r,β = ¯ βα ( p s ) . (4)Acknowledging that people conduct social-behavioral changesin a soft-threshold manner (Smaldino et al. (2018)), we con-sider sigmoidal-shaped functions for the feedback response α ( · ) .Similar feedback mechanisms in neuronal dynamics have beenhown to exhibit ultra-sensitivity and robustness to inputs andvariability (Sepulchre et al. (2019)). Let φ : [0 , → [0 , be amonotonically increasing saturating function φ ( p ; µ, ν ) = (cid:18) p · (1 − µ ) µ · (1 − p ) (cid:19) − ν ! − , with location parameter µ ∈ (0 , and slope parameter ν ∈ (0 , . For risk-tolerators we define α to vary directly with p s : α ( p s ) = φ ( p s ; µ T , γ T ) =: φ T ( p s ) . (5)For risk-averters we define α to vary inversely with p s : α ( p s ) = 1 − φ ( p s ; µ A , γ A ) =: 1 − φ A ( p s ) . (6)It is not surprising to find that the EE of the actSIS modelfor both risk-tolerators and risk-averters is upper bounded bythat of the SIS model, since the incorporation of feedbackresponses α ( · ) ∈ [0 , always decreases the effective infectionrates. However, the underlying structure resulting in these lowerendemic solutions differs among types of individuals, and theactSIS model shows qualitatively different dynamical featuresas compared to the SIS model.For homogeneous risk-tolerators , (4) undergoes a saddle nodebifurcation and exhibits bistability as illustrated in Fig. 1 andFig. 2(a). The saddle node bifurcation point is greater thanthe transcritical bifurcation point in the SIS model, which hasimplications for control design since it implies that it is moredifficult for the disease to spread in a population of risk-averters than in a population of risk-ignorers .For homogeneous risk-averters , (4) undergoes a transcriticalbifurcation, as in the SIS model, but with a reduced EE, asillustrated in Fig. 2(b). Further, the EE becomes a stable focusunder certain conditions, resulting in large overshoot and/orundershoot in the transient dynamics, as illustrated in Fig. 3.We devote the rest of this section to the detailed description andproof of these rich dynamics. For simplicity of exposition, weset δ = 1 , τ s = 10 throughout the rest of this paper. (a) IFE (b) EE Fig. 1.
Bi-stability of risk-tolerators . The IFE and EE are bothstable solutions for ¯ β > ¯ β c in the actSIS model for risk-tolerators . a) For p (0) = 0 . , the solution (green) ofactSIS ( risk-tolerators ) converges to the IFE, whereas thesolution (grey) of SIS ( risk-ignorers ) converges to the EE.b) For p (0) = 0 . , the solution (red) of actSIS and thesolution (grey) of SIS converge to the EE. Parameters forall simulations are ¯ β = 2 . , µ T = 0 . , ν T = 8 . For theseparameters, ¯ β c = 1 . < ¯ β . This particular form of a sigmoidal function over the unit interval wasproposed by Antweiler (2018). µ controls the value of p at which Φ( p ) = 1 / and ν controls how gradually or sharply the function grows. (a) Risk-tolerators (b) Risk-averters Fig. 2.
Bifurcation diagrams of the actSIS model. a) For ho-mogeneous risk-tolerators , the actSIS model undergoes asaddle-node bifurcation at bifurcation point ¯ β c > . b) Forhomogeneous risk-averters, the actSIS model undergoes atranscritical bifurcation at ¯ β = 1 as in the SIS model. TheEE for the actSIS models are upper-bounded by that ofthe SIS model in both cases. The bifurcation diagrams ofthe actSIS models are drawn in blue and the SIS in grey.Solid curves denote stable solutions and dashed curvesdenote unstable solutions. Parameters for these plots are µ T = 0 . , ν T = 8 , µ A = 0 . , ν A = 8 . The bifurcationdiagrams: Fig.1 and Fig. 4 (a), are produced with the MAT-LAB package matcont (Dhooge et al. (2003)).
Theorem 3.1 (Steady-state behavior of the actSIS model withhomogeneous risk-tolerators) . Consider the actSIS dynamicsfor a homogeneous population of risk-tolerators given by (4) with r ( t ) = 0 and α ( · ) = φ T ( · ) . Then the following hold:(i) There are two types of equilibrium solutions:(a) IFE: p = p s = 0 . It is always stable;(b) EE: p = p s = p ∗ satisfying φ T ( p ∗ )(1 − p ∗ ) = δ ¯ β .There are zero, one, or two such solutions. A solutionis stable if Φ T ′ ( p ∗ ) · (1 − p ∗ ) < δ ¯ β .(ii) The system undergoes a saddle node bifurcation at thebifurcation point ( ¯ β c , p ∗ ) with stable upper branch (stableEE) and unstable lower branch (unstable EE). The epi-demic threshold ¯ β c /δ is always larger than that of the SISmodel, i.e., ¯ β c /δ > .(iii) For ¯ β > ¯ β c the system exhibits bistability of the IFE andthe EE. For ¯ β ≤ ¯ β c the IFE is the only stable equilibrium.(iv) As ¯ β c < ¯ β → ∞ , p ∗ → − δ/ ¯ β , the EE of SIS.Proof. (i) The equilibrium solutions are straightforward tocompute. For stability, we compute the Jacobian as J T = (cid:20) ¯ βφ T ( p s )(1 − p ) − δ ¯ β (1 − p ) pφ T ′ ( p s ) τ s − τ s (cid:21) . (7)For the IFE, (7) reduces to J T (cid:12)(cid:12) IFE = (cid:20) − δ τ s − τ s (cid:21) , (8)implying that the IFE is always stable. For the EE, J T (cid:12)(cid:12) EE = " − δ p ∗ − p ∗ ¯ β (1 − p ∗ ) p ∗ φ T ′ ( p ∗ ) τ s − τ s . (9)Stability requires the equilibrium solution to satisfy δ − p ∗ − ¯ β (1 − p ∗ ) φ T ′ ( p ∗ ) > . Since p ∗ ∈ (0 , , it is equivalentto requiring that φ T ′ ( p ∗ ) · (1 − p ∗ ) < δ ¯ β .(ii) We start by computing the critical value ¯ β c . Since φ T ( · ) isa monotonically increasing function taking values between and 1, g ( p ) := φ T ( p )(1 − p ) takes value between 0 and1 and it first increases from 0 (since g (0) = 0 ) and thendecreases to 0 ( g (1) = 0 ). Depending on parameter values( µ T , ν T , ¯ β, δ ), the EE has either zero, one or two solutions.Let h ( p ) := φ T ′ ( p ) · (1 − p ) . It first increases and thendecreases for p ∈ [0 , . Denoting ˆ p := argmax p g ( p ) , onecan check that h ( p ) intersects with g ( p ) at three points: , ˆ p and . This implies that when the EE has two solutions(when δ/β < g (ˆ p ) ), the smaller solution is unstable andthe larger one is stable. ¯ β c can be solved analytically bysolving ¯ β c = δ/g (ˆ p ) .To prove the existence of a saddle-node bifurcation, weuse the classification of equilibria for a two-dimensionalsystem presented in Section 4.2.5 in Izhikevich (2007).Specifically, we show that one of the eigenvalues of theJacobian at ( ¯ β c , p ∗ ) becomes zero. Using equations ¯ β c = δ/g (ˆ p ) and g ′ (ˆ p ) = 0 , one can easily verify the deter-minant of the Jacobian at ( ¯ β c , p ∗ ) equals zero thus theexistence of a saddle node bifurcation.Since g (ˆ p ) ∈ (0 , , we have ¯ β c /δ is always larger thanthe epidemic threshold in the SIS model.(iii) This follows directly from (i) and (ii).(iv) Comparing the equations satisfied by the EE solutionsfor the SIS and the homogeneous actSIS ( risk-tolerators )model, we observe that φ T ( p ∗ ) and p ∗ increase with ¯ β . As φ T ( p ∗ ) approaches , p ∗ approaches − δ/ ¯ β . (a) Phase Portrait (b) Undershoot and overshoot Fig. 3.
The EE as a stable focus. (a) Phase portrait of the actSISmodel for a homogeneous population of risk-averters . Greyarrows depict the vector fields. The initial conditions andend points of the simulations are plotted as circles andsquares respectively. Starting from low initial values, p exhibits a rapid increase followed by a decrease to the EE.Starting from high initial values, p exhibits a rapid decreaseto nearly zero before an increase to the equilibrium state.(b) For p (0) = p s (0) = 0 . , the transient state p under-shoots to near zero for a long time and then overshoots. Theparameters are ¯ β = 2 . , µ A = 0 . , ν A = 8 . Theorem 3.2 (Steady-state behavior of the actSIS model withhomogeneous risk-averters) . Consider the actSIS dynamics fora homogeneous population of risk-averters given by (4) with r ( t ) = 0 and α ( · ) = 1 − φ A ( · ) . Then the following hold:(i) There are two equilibrium solutions:(a) IFE: p s = p = 0 . It is stable if ¯ β < δ ;(b) EE: p s = p = p ∗ satisfying (1 − Φ A ( p ∗ ))(1 − p ∗ ) = δ/ ¯ β . It is always stable if it exists, which iswhen ¯ β > δ .(ii) The EE is a stable focus if b > ( a − c ) / (4 ac ) , where a = δp ∗ − p ∗ , b = − p ∗ − φ A ( p ∗ ) φ A ′ ( p ∗ ) , c = τ s . (iii) The EE is always upper bounded by − δ/ ¯ β , the SIS EE.Proof. (i) Since − Φ A ( · ) is monotone decreasing, there ex-ists at most one EE solution when δ/ ¯ β < . The Jacobianof (4) is J = (cid:20) ¯ β (1 − Φ A ( p s ))(1 − p ) − δ − ¯ β (1 − p ) pφ A ′ ( p s ) τ s − τ s (cid:21) , (10)which simplifies to J | IFE = (cid:20) ¯ β − δ τ s − τ s (cid:21) , (11)for the IFE, and J | EE = " − δp ∗ − p ∗ − ¯ β (1 − p ∗ ) p ∗ φ A ′ ( p ∗ ) τ s − τ s (12)for the EE. Therefore, the IFE is stable when ¯ β < δ . Forthe EE, stability requires δp ∗ − p ∗ + ¯ β (1 − p ∗ ) p ∗ Φ A ′ ( p ∗ ) > .Since Φ A ′ ( p ∗ ) is always non-negative, the stability condi-tions always holds when such a solution exists.(ii) We prove when EE is a stable focus, by proving when (12)has a pair of complex-conjugate eigenvalues with negativereal part. We use the classification of equilibria for a two-dimensional system according to the trace and determinantof the Jacobian (Izhikevich (2007)). Denoting A := J | EE and using the definitions of a, b and c , we have Tr( A ) = − ( a + c ) and det( A ) = ac (1 + b ) . Therefore the condition b > ( a − c ) ac , guarantees that Tr( A ) − A ) < , andtogether with Tr( A ) < , that the EE is a stable focus.(iii) As ¯ β increases, the right hand side of (1 − Φ A ( p ∗ ))(1 − p ∗ ) = δ/ ¯ β decreases. As a result, p ∗ increases with ¯ β . Onthe other hand, − φ A ( p ∗ ) decreases away from thus theEE are always bounded above by − δ/ ¯ β .4. HETEROGENEOUS POPULATIONWe introduce the heterogeneous network actSIS model. Thetransmission rates, recovery rates and feedback responses canall be distinct. However, we restrict to the following set upas a first step in exploring the role of heterogeneity. Let thepopulation comprise two homogeneous sub-populations, risk-tolerators and risk-averters , which differ only in their feedbackresponses to the infection. We assume the disease transmissionoccurs across sub-populations but not within. For the generaliza-tion of the network SIS model (3) to the network actSIS model,this translates as n = 2 , ¯ β i = ¯ β , δ i = δ , β ii = 0 , α ( · ) = α ( · ) for all risk-tolerators and α ( · ) = α ( · ) for all risk-averters . Let p ( t ) ( p ( t ) ) denote the fraction of risk-tolerators ( risk-averters )that are infected at time t . Let β = ¯ βα ( p s ) = ¯ βφ T ( p s ) bethe effective infection rate from risk-averters to risk-tolerators ,and β = ¯ βα ( p s ) = ¯ β (cid:0) − φ A ( p s ) (cid:1) from risk-tolerators to risk-averters . The heterogeneous network actSIS model in thecase of these two subpopulations is ˙ p = (1 − p ) β · p − δp , ˙ p = (1 − p ) β · p − δp ,τ s ˙ p s = − p s + p ,τ s ˙ p s = − p s + p . (13)The equilibrium solutions satisfy p ∗ = p ∗ s , p ∗ = p ∗ s and ∗ − p ∗ · α ( p ∗ ) p ∗ = p ∗ − p ∗ · α ( p ∗ ) p ∗ = δ ¯ β . (14)As shown in Fig. 4(a), there are model parameters for whichsystem (13) undergoes a Hopf bifurcation with a stable limitcycle. As ¯ β increases from zero, system (13) undergoes a saddle-node bifurcation from a single stable IFE to bistability of the EEand the IFE. As ¯ β increases further across the Hopf bifurcationpoint ¯ β ∗ , the system exhibits a stable limit cycle about the EE.For completeness, we present the following theorem (Theorem3.4.2 in Guckenheimer and Holmes (1983)), which we use toprove the existence of stable limit cycles for (13). (a) Hopf bifurcation of system (13) β - λ (b) Verification of (H2) Fig. 4.
Hopf bifurcation. (a) Bifurcation diagram for p withbifurcation parameter ¯ β . For small positive values of ¯ β , theIFE is the only stable solution. As ¯ β increases a saddle-node bifurcation leads to bistability of the IFE and EE. As ¯ β continues to increase there is a Hopf bifurcation (reddot) and stable oscillations about the EE. The blue solidcurves depict stable solutions, while dashed curves depictunstable solutions. (b) The real and imaginary parts of theeigenvalues of the Jacobian D p f at ( p ∗ , ¯ β ∗ ) are plottedin solid and dashed lines. At around ¯ β ∗ = 7 . , a pair ofcomplex eigenvalues crosses the real line ( x -axis) with anonzero derivative, verifying (H2). Parameters are δ =1 , τ s = 10 , µ T = 0 . , ν T = 0 . , µ A = 0 . , ν A = 20 . Theorem 4.1 (Guckenheimer and Holmes) . Suppose that theheterogeneous actSIS model (13) expressed as ˙ p = f ( p , ¯ β ) , p = ( p , p , p s , p s ) , ¯ β ∈ R , has an equilibrium at (cid:0) p ∗ , ¯ β ∗ (cid:1) and the following properties are satisfied: • ( H1 ) The Jacobian D p f | ( p ∗ , ¯ β ∗ ) has a simple pair of pureimaginary eigenvalues λ (cid:0) ¯ β ∗ (cid:1) and λ (cid:0) ¯ β ∗ (cid:1) and no othereigenvalues with zero real parts, • (H2) dd ¯ β (Re λ ( ¯ β )) (cid:12)(cid:12)(cid:12) ( ¯ β = ¯ β ∗ ) = 0 .Then the dynamics undergo a Hopf bifurcation at (cid:0) p ∗ , ¯ β ∗ (cid:1) resulting in periodic solutions. The stability of the periodicsolutions is given by the sign of the first Lyapunov coefficientof the dynamics ℓ | ( p ∗ , ¯ β ∗ ) . If ℓ < , then these solutionsare stable limit cycles and the Hopf bifurcation is supercritical,while if ℓ > , the periodic solutions are repelling. We show in Proposition 4.2 conditions on the model parame-ters that guarantee that the non-hyperbolicity condition (H1) issatisfied for the heterogeneous actSIS model (13). We verifynumerically that condition (H2) is satisfied and ℓ < . Proposition 4.2.
Denote m = (1 − p ∗ )(1 − p ∗ ) , q = p ∗ β ∗ /p ∗ , s = p ∗ β ∗ /p ∗ , v = p ∗ β ′∗ /p ∗ , w = p ∗ β ′∗ /p ∗ , β ∗ = ¯ βα ( p ∗ ) ,β ∗ = ¯ βα ( p ∗ )) . Then, for system (13) , the non-hyperbolicitycondition (H1) in Theorem is satisfied if c > , a = 0 , where a := s + q + 2 τ s ,ac := 2 τ s (1 − m ) sq + 1 τ s ( s + q ) − m ( vβ ∗ + wβ ∗ ) . Proof.
For a four-dimensional system to satisfy (H1)the eigenvalues of the Jacobian D p f | ( p ∗ , ¯ β ∗ ) must satisfy (cid:0) λ + c (cid:1) (cid:0) λ + aλ + b (cid:1) = 0 , (15)for some a = 0 , b ∈ R ,and c > . We compute the Jacobian D p f | ( p ∗ , ¯ β ∗ ) = − β ∗ p ∗ − δ (1 − p ∗ ) β ∗ − p ∗ ) p ∗ β ′∗ (1 − p ∗ ) β ∗ − β ∗ p ∗ − δ (1 − p ∗ ) p ∗ β ′∗ τ s − τ s τ s − τ s which has eigenvalues that satisfy (cid:18) τ s + λ (cid:19) ( s + λ ) ( q + λ ) − m (cid:18) τ s + λ (cid:19) sq − m (cid:18) τ s + λ (cid:19) vβ ∗ − m (cid:18) τ s + λ (cid:19) wβ ∗ − mvw = 0 . (16)Matching coefficients of (15) and (16), we derive the ex-pressions for a , b , and c , in terms of the model parameters( ¯ β, µ T , γ T , µ A , γ A ) for (13) to satisfy the first part of (H1). Wehave λ (cid:0) ¯ β ∗ (cid:1) = √ c i , a pure imaginary eigenvalue. a = 0 guaran-tees there are no other eigenvalues with zero real part.A proof of conditions guaranteeing (H2) of Theorem 4.1 is thesubject of ongoing work. Fig 4(b) shows numerically that (H2) issatisfied for the parameters selected. We also checked that ℓ < . An illustration of the sustained oscillations corresponding tothe stable limit cycle of (13) is depicted in Fig 5.5. FINAL REMARKSThe actSIS model incorporates two novel mechanisms as com-pared with the SIS model: a feedback mechanism for the ef-fective infection rates, and a time scale separation between thestate of the system and the state used in the feedback law. Thequalitative differences we have shown for the actSIS model aredue to both of the mechanisms. We have observed sustainedoscillations even when some of the individuals are risk-ignorers and under relaxed assumptions on interconnections. We will ex-amine the broader set of possibilities in future work and considerapplications in other biological and socio-ecological processes.REFERENCESAnderson, R.M. and May, R.M. (1992). Infectious diseases ofhumans: dynamics and control . Oxford University Press.Antweiler, W. (2018).
A sigmoid-logit proba-bility function for the (0,1) domain . URL http://wernerantweiler.ca/blog.php?item=2018-11-03 .Baker, R. (2020). Reactive social distancing in a SIR model ofepidemics such as COVID-19. arXiv:2003.08285v1 .Camacho, A., Ballesteros, S., Graham, A.L., Carrat, F., Rat-mann, O., and Cazelles, B. (2011). Explaining rapid reinfec-tions in multiple-wave influenza outbreaks: Tristan da cunha
10 20 30 40 50 60 7000.20.40.60.8 (a) Sustained oscillations (b) β evolution (c) β evolution Fig. 5.
Stable limit cycles of the heterogeneous actSIS model. a) When ¯ β > ¯ β ∗ , system (13) exhibits stable oscillations. States p and p initially decrease quickly due to different mechanisms: p decreases because p s (0) < µ T , which drives β down and p decreases because p s (0) > µ A , which drives β down. Thelow p leads to a decrease in p s , which in turn drives up β ,thus increasing p . The increase in p leads to an increase in p s , which in turn drives up p . The process repeats resulting insustained oscillations. b-c): The time evolution of the effectiveinfection rates. Parameters used for the simulation: ¯ β = 8 . , µ T = 0 . , ν T = 0 . , µ A = 0 . , ν A = 20 . Initial conditions: p (0) = 0 . , p (0) = 0 . , p s (0) = 0 . , p s (0) = 0 . . Proceedings of the RoyalSociety B: Biological Sciences , 278(1725), 3635–3643.Demiris, N., Kypraios, T., and Smith, L.V. (2014). On theepidemic of financial crises.
Journal of the Royal StatisticalSociety. Series A (Statistics in Society) , 697–723.Dhooge, A., Govaerts, W., and Kuznetsov, Y.A. (2003). MAT-CONT: a MATLAB package for numerical bifurcation anal-ysis of ODEs.
ACM Transactions on Mathematical Software(TOMS) , 29(2), 141–164.Diekmann, O., Heesterbeek, J.A.P., and Metz, J.A. (1990). Onthe definition and the computation of the basic reproductionratio R0 in models for infectious diseases in heterogeneouspopulations.
J. of Mathematical Biology , 28(4), 365–382.Dushoff, J., Plotkin, J.B., Levin, S.A., and Earn, D.J. (2004).Dynamical resonance can account for seasonality of influenzaepidemics.
Proc. National Academy of Sciences , 101(48),16915–16916.Fall, A., Iggidr, A., Sallet, G., and Tewa, J.J. (2007). Epi-demiological models and lyapunov functions.
MathematicalModelling of Natural Phenomena , 2(1), 62–83.Franco, E. (2020). A feedback SIR (fSIR) model highlights ad-vantages and limitations of infection-based social distancing. sociarXiv:2004.13216v1s . Funk, S., Gilad, E., Watkins, C., and Jansen, V.A. (2009). Thespread of awareness and its impact on epidemic outbreaks.
Proc. National Academy of Sciences , 106(16), 6872–6877.Guckenheimer, J. and Holmes, P. (1983).
Nonlinear Oscilla-tions, Dynamical Systems, and Bifurcations of Vector Fields .Springer-Verlag.Hartmann, W.R., Manchanda, P., Nair, H., Bothner, M., Dodds,P., Godes, D., Hosanagar, K., and Tucker, C. (2008). Model-ing social interactions: Identification, empirical methods andpolicy implications.
Marketing Letters , 19(3-4), 287–304.Hethcote, H.W. and Yorke, J.A. (1984).
Lecture Notes inBiomathematics . Springer.Izhikevich, E.M. (2007).
Dynamical Systems in Neuroscience .MIT press.Jin, F., Dougherty, E., Saraf, P., Cao, Y., and Ramakrishnan,N. (2013). Epidemiological modeling of news and rumorson twitter. In
Proceedings of the 7th Workshop on SocialNetwork Mining and Analysis , 1–9.Kermack, W.O. and McKendrick, A.G. (1927). A contributionto the mathematical theory of epidemics.
Proceedings of theRoyal Society of London. Series A , 115(772), 700–721.Lajmanovich, A. and Yorke, J.A. (1976). A deterministic modelfor gonorrhea in a nonhomogeneous population.
Mathemati-cal Biosciences , 28(3-4), 221–236.Lin, J., Andreasen, V., and Levin, S.A. (1999). Dynamics ofinfluenza A drift: the linear three-strain model.
MathematicalBiosciences , 162, 33–51.Mei, W., Mohagheghi, S., Zampieri, S., and Bullo, F. (2017).On the dynamics of deterministic epidemic propagation overnetworks.
Annual Reviews in Control , 44, 116–128.Mishchenko, Y. (2014). Oscillations in rational economies.
PloS One , 9(2), e87820.Muench, H. (1934). Derivation of rates from summation databy the catalytic curve.
Journal of the American StatisticalAssociation , 29(185), 25–38.Nowzari, C., Preciado, V.M., and Pappas, G.J. (2016). Analysisand control of epidemics: A survey of spreading processes oncomplex networks.
IEEE Control Syst. Mag. , 36(1), 26–46.Pagliara, R. and Leonard, N.E. (2020). Adaptive susceptibilityand heterogeneity in contagion models on networks.
IEEETransactions on Automatic Control . doi:10.1109/TAC.2020.2985300.Pais, D., Caicedo-Nunez, C.H., and Leonard, N.E. (2012). Hopfbifurcations and limit cycles in evolutionary network dynam-ics.
SIAM Journal on Applied Dynamical Systems , 11(4),1754–1784.Rands, S.A. (2010). Group movement ’initiation’ and state-dependent decision-making.
Behavioural Processes , 84(3),668–670.Sahneh, F.D., Scoglio, C., and Van Mieghem, P. (2013). Gen-eralized epidemic mean-field model for spreading processesover multilayer complex networks.
IEEE/ACM Transactionson Networking , 21(5), 1609–1620.Sepulchre, R., Drion, G., and Franci, A. (2019). Control acrossscales by positive and negative feedback.
Annual Review ofControl, Robotics, and Autonomous Systems , 2, 89–113.Smaldino, P.E., Aplin, L.M., and Farine, D.R. (2018). Sig-moidal acquisition curves are good indicators of conformisttransmission.
Scientific Reports , 8(1), 1–10.Xu, B., Cai, J., He, D., Chowell, G., and Xu, B. (2020). Mech-anistic modelling of multiple waves in an influenza epidemicor pandemic.
Journal of Theoretical Biology , 486, 110070.hao, L., Wang, J., Huang, R., Cui, H., Qiu, X., and Wang, X.(2014). Sentiment contagion in complex networks.