TThe dynamics of two-stage contagion
Guy KatrielDepartment of Mathematics, ORT Braude College,Karmiel, Israel
Abstract
We explore simple models aimed at the study of social contagion, inwhich contagion proceeds through two stages. When coupled with de-mographic turnover, we show that two-stage contagion leads to nonlinearphenomena which are not present in the basic ‘classical’ models of math-ematical epidemiology. These include: bistability, critical transitions, en-dogenous oscillations, and excitability, suggesting that contagion modelswith stages could account for some aspects of the complex dynamics en-countered in social life. These phenomena, and the bifurcations involved,are studied by a combination of analytical and numerical means.
The notion of social contagion has been gaining increasing prominence, withaccumulating empirical evidence of its importance for many aspects of our lives,from political mobilization, spread of ideas and innovations, and psychologicalwell-being, to substance abuse, crime and violence, obesity, financial panics, andmass psychogenic illness [8, 12, 23, 37].Contagion phenomena have traditionally been mathematically modelled in thefield of infectious disease epidemiology [30, 39], and it is natural to apply thetools developed in this field to social contagion, as is indeed being done by re-searchers from diverse fields [6, 19, 34, 41, 44]. Models for various instancesof social contagion have been developed, including smoking [7], drug use [48],bulimia [20], political activism [29, 40], spread of rumors [16], crime [17], lan-guage dynamics [38], organized religions [24], diffusion of new products andtechnologies [2, 18] or ideas in a scientific community [3], among many moreexamples.It is important to address the various aspects in which social contagions differfrom biological contagions at the level of the individual, and the consequences ofthese differences for the dynamics of these phenomena at the population level.In elucidating this micro-macro link, mathematical modelling plays an impor-tant role, as it allows us to study the population-level patterns emerging fromdifferent contagion mechanisms, which are often far from intuitively obvious,and which may be quite different from those familiar from the study of classicalmodels for the transmission of infectious diseases.1 a r X i v : . [ phy s i c s . s o c - ph ] D ec ne important way in which social contagion mechanisms differ from those ofbiological contagion is that while infection with a pathogen is a discrete eventocurring upon contact of a susceptible individual with an infectious individual,social contagion may require transitions through several stages, with each suchtransition dependent on contact with ‘infectives’. This is a central tenet ofRogers’ Diffusion of Innovations theory [42], which studies the spread of innova-tions - including, for example, technologies, products, ideas, cultural practices,health-related behaviors, and more. Rogers’ theory posits that the adoption ofan innovation, at the individual level, involves a series of five stages in which theindividual comes to learn about the innovation, assess it, and finally adopt it.The transition rate from one stage to another is modulated by the interpersonalinfluence of other individuals in one’s social network, and hence depends on thenumber of people who have already adopted the innovation. As another exam-ple from the social sciences, Klandermans and Oegema [31] analyse the processof becoming a participant in a social movement as consisting of four stages:becoming part of the mobilization potential, becoming the target of mobiliza-tion attempts, becoming motivated to participate, and overcoming barriers toparticipation. A closely related idea is that of complex contagion [8], referringto social contagions which require sustained or repeated contacts with adoptersin order to spread.To incorporate the idea of stages of contagion into mathematical models, onecan divide the population into classes, each of which consists of individuals ata certain stage in the adoption process, with movement among the classes dueto contact with adopters. Models of this type have been proposed studied inseveral works [4, 10, 11, 13, 14, 22, 27, 28, 33], and in section 1.2 we will brieflydescribe and compare these with the model studied here.Our aim here is to make a detailed study of the dynamics of a two-stage conta-gion model, which, unlike in nearly all previous works, incorporates the processof demographic turnover - recruitment and departure of individuals from thepopulation. This may be due to births and deaths or, if considering a contagionspreading in particular institutional settings or age groups, individuals enteringand leaving the institution, or maturing into and out of the relevant age group.Our model is thus a two-stage analog of the classical SIR model with demo-graphic turnover [30, 39]. We will show that stages of contagion, in conjunctionwith demographic turnover, lead to new and sometimes surprising dynamicalphenomena, which are not present in the basic one-stage models familiar inmathematical epidemiology. The interesting behaviors we observe in our simplemodel suggest a possible generative mechanism for some of the complex phe-nomena observed in the social world: alternative stable states, discontinuoustransitions, critical mass effects, and periodic cycles.Since we wish to highlight the fact that stages of contagion lead to novel phe-nomena in the simplest of models, we resist the temptation to generalize byincorporating more mechanisms or more stages of contagion. The simplicityof the present models also provides the advantage that we can go quite far incharacterizing their dynamics analytically, in different parameter regimes, usingstandard tools of stability analysis of equilibria. Some aspects of the dynamics,however, will be studied numerically, and obtaining full mathematical proofsfor some of our conclusions from these simulations seems like a challenging task2or the future. Any phenomena present in our minimal model will also occur a fortiori in more elaborate models, for some parameter values, and such mod-els might give rise to further interesting dynamics not present in the two-stagemodel, and deserve to be studied further.In the remainder of this section we introduce the two models to be studied, one inwhich an individual’s adoption of an innovation is permanent and one in whichit is temporary, and make a comparison with previous models incorporatingtwo-stage contagion. The permanent adoption model will be studied in section2, and the temporary adoption model, which leads to a richer variety of possibledynamical behaviors, will be studied in section 3. In section 4 we will recapthe novel features that the two-stage contagion models have in comparison withtheir classical one-stage counterparts, and discuss their significance. We assume that the population is divided into three classes: class S consists of‘naive’ individuals who have not been exposed to the innovation, class S consistsof ‘informed’ individuals who have encountered the innovation but have not yetadopted it, and class A consists of adopters. We choose the terminology of‘adoption of innovations’ for convenience; the models could just as well describepotential supporters of a social movement ( S ), actual supporters ( S ), andactivists ( A ), or many other examples of social contagion.The mechanisms involved in the models are: • Adopters randomly encounter other individuals, at a rate of β effectivecontacts per unit time. • Upon effective contact with an adopter, a naive individual adopts the in-novation with probability p (0 < p < • An informed individual adopts the innovation upon encounter with anadopter. • Demographic turnover (recruitment into and departure from the relevantpopulation, for example through births and deaths) occurs at per capitarate µ , so that µ − is the mean residence time of an individual in thepopulation ( e.g. , the life expectancy). Since recruitment and departurerates are assumed equal, we are assuming a constant population size. Itis also assumed that individuals entering the population are naive (class S ).Denoting the fraction of the population in each of the three classes at time t by S ( t ), S ( t ), A ( t ) (so that S ( t ) + S ( t ) + A ( t ) = 1), the above assumptions,represented also in the diagram of figure 1, translate, in the standard way [30,39], into the following differential equations: S (cid:48) = µ − βS A − µS , (1) S (cid:48) = (1 − p ) βS A − βS A − µS , (2)3igure 1: Diagram for the two-stage contagion model with permanent adoptionFigure 2: Diagram for the two-stage contagion model with temporary adoption A (cid:48) = β [ pS + S ] A − µA. (3)This is the permanent adoption model. In our second model, the temporaryadoption model, we assume that adopters abandon the innovation (or activistsin a social movement become ‘burned out’) at a constant per capita rate γ , sothat the mean duration of adoption is γ − . It is further assumed that those whohave abandoned the innovation will not re-adopt it. This adds an additionalclass of removed individuals R (see figure 2), and changes the model equationsto S (cid:48) = µ − βS A − µS , (4) S (cid:48) = (1 − p ) βS A − βS A − µS , (5) A (cid:48) = β [ pS + S ] A − ( γ + µ ) A, (6) R (cid:48) = γA − µR. (7)It should be stressed here that a crucial feature of the two-stage contagionmodels is that both transitions between stages depend on contagion. A modelin which the transition from stage S to stage A were a spontaneous one, thatis occuring at a constant per capita rate, would be equivalent to the standardSEIR model [30, 39], and would not generate any of the interesting behaviorsthat are our focus here. 4 .2 Previous work on two-stage contagion models We briefly survey previous work on two-stage contagion models of which we areaware, and compare the models which have been investigated with the modelstudied here.In the statistical physics literature, a two-stage contagion model was introducedin [27, 28] under the name Extended General Epidemic Process (EGEP). Likethe model considered here, this model includes two stages, with contagion lead-ing either directly to adoption ( A ) or to moving to a ‘weakened’ stage ( S ), whichupon further contagion can lead to adoption, and with permanent removal ofadopters to a recovered stage (temporary adoption). An equivalent model isintroduced in [9], as a way to describe co-infection with two pathogens, undersome symmetry assumptions. A similar model was introduced in [33], with theaim of describing innovations (with permanent adoption) and fads (with tem-porary adoption), and including an arbitrary number of stages, though withoutdirect movement of naive individuals into the adopter class. The models abovedo not include demographic turnover, so that there is no mechanism for the re-newal of the naive population, and they therefore generate transient epidemicsrather than endemic states (except for the permanent adoption model of [33], inwhich the entire population eventually adopts). Their investigation thus centerson the size of the epidemic, that is the total number of individuals who becomeinfected before it fades. Of prime interest here is the fact that, in contrast withthe one-stage epidemic model, in the two-stage model one has, under certainconditions, a discontinuous (‘first order’ in the language of statistical physics)dependence of the total size of the epidemic on the contact parameter. This hasled to interest in the statistical physics community, and several works investigatethe two-stage contagion process on lattices and random graphs [4, 10, 11, 13, 22].In [47], which develops models inspired by Rogers’ Diffusion of Innovation The-ory [42], mechanisms inducing renewal of the naive population are introduced,so that an endemic contagion becomes possible, as in this work. However, thereare also some signficant differences between the assumptions in [47] and in ourmodels, which we now detail.In the first model of [47], it is assumed that adoption is temporary, but theadopters who abandon the innovation return to the naive class so that theymay re-adopt (this is analogous to the SIS model in epidemiology). In addi-tion, informed individuals (class S ) move back to the naive stage (‘forgetting’)at a constant per capita rate. These two flows are the mechanism providingrenewal of the pool of naive individuals – the model does not incorporate de-mographic turnover. By contrast, in our model we do not allow individuals tomove back into the naive stage, and the renewal of the naive pool is providedby demographic turnover. This is not a trivial difference, as may be seen fromthe fact that the model of [47] reduces to a two-dimensional one, whereas ourtemporary adoption model is essentially three dimensional and cannot be re-duced to a two-dimensional one. The appropriateness of one set of assumptionsor the other (or some different combination of assumptions) is dependent on theprecise nature of the contagion phenomena involved, as well as on the relevanttime scale. One can construct a larger model including both models, but inorder to understand the specific contribution of each mechanism there is much5dvantage in investigating simple models highlighting that specific mechanism.The models of [47] also included the mechanisms of ‘spontaneous’ transition ofnaive individuals to the informed stage and of informed individuals to adoptionstage, at constant per capita rates, as a way to model the effects of mass-media. Indeed for some of the results in [47] it was assumed that the mass-mediaeffects are strong relative to the contagion effect. Here we treat the oppositeextreme of pure contagion, without adding the media effects, as in the previousworks on the EGEP model. Since contagion effects are nonlinear while mediaeffects are linear, some of the interesting nonlinear phenomena which follow fromthe contagion effects are muted or cancelled when media effects are sufficientlystrong. When media effects are present but sufficiently weak, behavior will besimilar to the pure-contagion case, by structural stability. On the other hand,the model of [47] does not include direct contagion of naive individuals into theadopter class. Here we do include this mechanism (controlled by the parameter p ), as in the EGEP model, which is in fact essential for the generation of someof the more interesting phenomena, such as the periodic oscillations.A further element which is introduced in the first model of [47] is the possibilityof a nonlinear dependence of the per-capita transition rate from the informedclass ( S ) to the adopter class on the number of adopters, that is replacing theterm βAS by a term of the form g ( A ) S , where g is a nonlinear function. It isshown that choosing g ( A ) to be quadratic leads to bistability. In our models weshow that bistability occurs even without this additional mechanism, and wewill not introduce such nonlinear dependence.The second model of [47] introduces a time delay to model the intermediateevaluation stage, as well as demographic turnover, though not the direct con-tagion of naive individuals into the adoption stage. The time delay makes thestability analysis of equilibria considerably more difficult, but an interestingconsequence of this delay is that it gives rise to periodic oscillations for someparameter values. Note that in our model with temporary adoption, we willshow that periodic oscillations arise even in the absence of a delay. We will notconsider delays in this work.To conclude this short survey, we mention the work [14], which contains rigorousresults on a stochastic two-stage contagion model on a lattice with return ofadopters to the naive stage. In this work we restrict ourselves to mean-fielddeterministic models. In this section we analyze the dynamics of the permanent adoption model. Itwill be useful to define the dimensionless contact parameter δ = βµ , which plays a role similar to the basic reproductive number R in epidemio-logical models: since an adopter spends an average time µ − in the adoptercompartment before leaving the population, δ is the average total number of6ffective encounters that an adopter has.We note that since S + S + A = 1, we can eliminate one of the variables, say S , by substituting S = 1 − S − A, (8)and reformulate the model equations (1)–(3) as a two-dimensional system S (cid:48) = f ( S , A ) . = µ − βS A − µS , (9) A (cid:48) = g ( S , A ) . = β [1 − (1 − p ) S − A ] A − µA. (10)This defines a flow in the invariant region of the phase plane given by S ≥ , A ≥ , S + A ≤ . We can use the Bendixon-Dulac criterion [45] to verify that this system doesnot have limit cycles. Indeed ∂∂S (cid:18) A f ( S , A ) (cid:19) + ∂∂A (cid:18) A g ( S , A ) (cid:19) = − β − µA < . Since this is a two dimensional system in a bounded region, the Poincar´e-Bendixon theorem [45] implies that
Proposition 1.
Every trajectory of the system (9),(10) approaches an equilib-rium point as t → ∞ . We therefore now study the equilibria of the model, which correspond to solu-tions of the algebraic equations µ − βS A − µS = 0 , (11) β [1 − (1 − p ) S − A ] A − µA = 0 . (12)From (11) we have S = 1 δA + 1 . (13)From (12) we have A = 0 or (1 − p ) S + A = 1 − δ − . (14)If A = 0 then we obtain the contagion-free equilibrium E : ( S , S , A ) = (1 , , . Assuming A (cid:54) = 0, and substituting (13) into (14) gives1 − pδA + 1 + A = 1 − δ − , which may be rewritten as a quadratic equation and solved to give A , = 12 · (cid:104) − δ − ± (cid:112) − − p ) δ − (cid:105) . (15)7hese solutions will correspond to endemic equilibria if and only if they arereal and positive. We will denote by E the equilibrium corresponding to A = A , with the other components given by (13),(8), and by E the equilibriumcorresponding to A . We now determine the conditions on the parametersunder which these equilibria exist. A , A are real if and only if δ ≥ − p ) . (16)Assuming now that (16) holds, we have A > ⇔ (cid:112) − − p ) δ − > δ − − ⇔ δ > p < δ ≤ . (17)Noting that p ≥ − p ), we get A > ⇔ δ ≥ (cid:40) − p ) when p ≤ p when p > . For the equilibrium E , we have, assuming (16) holds, A > ⇔ (cid:112) − − p ) δ − < − δ − ⇔ < δ < p , (18)so that A > ⇔ p <
12 and 4(1 − p ) < δ < p . To determine the stability of the equilibria, we linearize (9) around an equilib-rium, obtaining the Jacobian matrix [45] J = (cid:18) − βA − µ − βS − β (1 − p ) A β [1 − (1 − p ) S ] − βA − µ (cid:19) For the contagion-free equilibrium E we have J = (cid:18) − µ − β βp − µ (cid:19) , with eigenvalues − µ, βp − µ , and we conclude that E will be stable when bothof these eigenvalues are negative, that is when δp <
1, and unstable if δp > E , E we have, using (14) J = (cid:18) − βA − µ − β − p (1 − A − δ − ) − β (1 − p ) A − βA (cid:19) so that tr ( J ) = − βA − µ < , det( J ) = β A [2 A + 2 δ − − , hence the equilibrium is stable if and only if det( J ) >
0, that is iff2( A + δ − ) > ± (cid:112) − − p ) δ − > , which holds if and only if the + sign is taken. We conclude that E is stablewhenever it exists, while E is always unstable whenever it exists.We summarize the results of the preceeding analysis. Proposition 2. (I) If p < then: • For δ < − p ) there are no endemic equilibria, and the contagion-freeequilibrium E is stable. • For − p ) < δ < p there are two endemic equilibria E , E , with E stable and E unstable, and the contagion-free equilibrium E is stable. • For δ ≥ p there is a unique endemic equilibrium E , which is stable, andthe contagion-free equilibrium E is unstable.(II) If p ≥ then: • For δ ≤ p there is no endemic equilibrium, and the contagion-free equilib-rium E is stable. • For δ > p there is a unique endemic equlibrium E , and the contagion-freeequilibrium E is unstable. We now discuss the interesting dynamical consequences of the preceding results.When p ≥ (see figure 3, right), the behavior is similar to that familiar from astandard one-stage contagion model – the SI model with demographic turnover:if contact rate is low ( δ < p ) then contagion cannot spread (the contagion-freeequilibrium is stable, and there is no endemic equilibrium), and as δ crosses the invasion threshold δ = p the contagion-free equilbrium loses stability and anendemic equilibrium is born, so that contagion is established. This transitionis a continuous one: for values of δ slightly above the threshold, the fraction ofadopters A is small, and it increases as δ increases.Things are more interesting in the case p < (see figure 3, left), since in thiscase we have a critical transition [43] at the endemicity threshold δ = 4(1 − p ), inwhich two endemic equilibria E , E appear (‘out of the blue’), so that contagioncan establish at the level A corresponding to the stable equilibrium E . At theendemicity threshold δ = 4(1 − p ) we have A = 12 · (cid:20) − − p ) (cid:21) > δ = p (which is larger) is crossed. This means that for values of δ between thesetwo thresholds we have bistability - contagion may establish, or not, depending9n whether the initial conditions belong to the basin of attraction of E or of E . This is demonstrated in figure 4, in which we show the solution A ( t ) forparameter values p = 0 . , δ = 5, for two initial conditions: when the initialfraction of adopters is 4% the contagion is extinguished, while for an initialfraction 5% contagion is established, with equilibrium value of A = 56 .
5% ofthe population. Thus under essentially the same conditions - that is the sameparameter values and only slightly different initial conditions, the system mayachieve radically different outcomes.The critical transition displayed by this model has important implications withregard to changes in outcomes under variation of its parameters. If we assumethat initially we are near the contagion-free equilibrium E , with δ < − p ),and slowly increase the contact parameter δ ( e.g. by increasing the contact rate β ) then, as the endemicity threshold δ = 4(1 − p ) is crossed we will still be in thebasin of attraction of the E , so that the population will remain contagion-free.This will continue until the invasion threshold δ = p is reached, at which point E loses stability, and then we will observe an even more dramatic jump to thevalue A corresponding to δ = p , that is to A = 12 · (cid:104) − p + (cid:112) − − p ) p (cid:105) . On the other hand, if initially δ > p and we are near the endemic equilibrium E + and slowly decrease δ , then we will remain near the endemic equilibriumeven as δ drops below the invasion threshold and the contagion-free equilibriumbecomes stable. This will continue until the endemicity threshold δ = 4(1 − p ) isreached, at which point the endemic equilibria disappear and we will jump fromthe value given by (20) to a contagion-free state. We thus have the phenomenonof hysteresis in which discontinuous transitions occur at different values of theparameter, depending on whether it is increased or decreased.In the extreme case p = 0 (no direct adoption by naive individuals), there isno invasion threshold and the contagion-free equilibrium is stable for all δ . Inthis two endemic equilibria are born when δ = 4, and we have bistability for all δ > E on the contact parameter δ . A = A is given by (15),and by (8),(13), we have S = 1 − (cid:112) − − p ) δ − − p ) , S = δ − − p (1 − (cid:112) − − p ) δ − )2(1 − p ) . While it is immediate that S is monotone decreasing and A is monotone in-creasing as a function δ , the fraction S of individuals at the intermediate stageis not monotone in δ , as shown in figure 5. Indeed it is easy to calculate thatthe fraction S is maximized at δ = p +1 , attaining the value S = − p .10igure 3: Fraction of adopters A at endemic equilibria, as a function of δ , for p = 0 . p = 0 . E , whichis stable, and the blue (dashed) line in the left part is the unstable equilibrium E .Figure 4: Fraction of adopters A ( t ), for p = 0 . µ = 0 . β = 0 .
25 (so δ = 5),for initial conditions: A (0) = A , S (0) = 1 − A , S (0) = 0, A = 0 . , A =0 . S at the stable equilbirium E , as afunction of δ , for p = 0 . p = 0 . The temporary adoption model
We now move to the analysis of the two-stage contagion model with temporaryadoption, which displays a richer repertoire of dynamic behaviors. As for theprevious model, it will be useful to define the non-dimensional parameter δ = βγ + µ , which can again be interpreted as the mean number of effective contacts thatan adopter makes during the period of adoption, so that it will be called thecontact parameter. Equilibria of the model are given by solutions, with non-negative components,of the equations obtained by equating the derivatives in (4)-(7) to zero: µ − βS A − µS = 0 , (21)(1 − p ) βS A − βS A − µS = 0 , (22) βA [ pS + S ] − ( γ + µ ) A = 0 , (23) γA − µR = 0 . (24)We now analyze the solutions of this algebraic system.From (21),(22),(24) we have S = 1 δ ( γµ + 1) A + 1 , S = (1 − p ) · δ ( γµ + 1) A ( δ ( γµ + 1) A + 1) , R = γµ · A. (25)From (23) we have that either A = 0 or pS + S = δ − . (26)In the case A = 0 we obtain the contagion-free equilibrium E : A = 0 , S = 1 , S = 0 , R = 0 . (27)In the case A (cid:54) = 0 we have, substituting (25) into (26), p · γµ + 1) A + δ − + (1 − p ) · ( γµ + 1) A (( γµ + 1) A + δ − ) = 1 , which is equivalent to a quadratic equation with solutions A , = 12 · (cid:18) γµ (cid:19) − (cid:104) − δ − ± (cid:112) − − p ) δ − (cid:105) . (28)12he expression (28) is identical to the expression (15) for the permanent adop-tion model, apart from the multiplicative factor (cid:16) γµ (cid:17) − , (indeed when γ = 0the temporary adoption model degenerates to the permanent adoption model).Therefore the analysis of the conditions for existence of equilibria is the samein both cases, and we obtain: Proposition 3. (I) If p < then: • For δ < − p ) there are no endemic equilibria, • For − p ) ≤ δ < p there are two endemic equilibria E , E (whichcoincide when δ = 4(1 − p ) ). • For δ ≥ p there is a unique endemic equilibrium E .(II) If p ≥ then: • For δ ≤ p there is no endemic equilibrium, • For δ > p there is a unique endemic equlibrium E . However, the conditions for stability of the equilibria, which to a large extentdetermine the dynamics of the model, are quite different from those for thepermanent adoption model, and we turn to these next.
To investigate stability of the equilibria, we examine the linearization of thesystem around an equilibrium [45]. In fact since R does not appear in the firstthree equations, and is determined by the other variables by R = 1 − S − S − A ,it suffices to consider (4)-(6). Linearization of this system around an equilibrium( S , S , A ) gives the Jacobian matrix J = − βA − µ − βS (1 − p ) βA − βA − µ (1 − p ) βS − βS pβA βA β [ pS + S ] − γ − µ . Beginning with the contagion-free equilibrium (27), we have J = − µ − β − µ (1 − p ) β βp − γ − µ With eigenvalues λ = λ = − µ and λ = βp − γ − µ , so that we obtain Proposition 4.
The contagion-free equilibrium E is stable if and only if δ < p . Moving to the endemic equilibria, we compute the characteristic polynomial of J , and substitute the expressions (25) for S , S at the equilibrium, to obtain P ( λ ) = λ + a λ + a λ + a , (29)13here a = βµ (cid:18) (1 − δ − )( βA + µ ) + (1 − p ) µ (cid:18) µβA + µ − (cid:19)(cid:19) (30) a = ( βA + µ ) + ( p − δ − ) βµ (31) a = 2( βA + µ ) . (32)By the Routh-Hurwitz stability criterion [45], the equilibrium is stable when: a > , a > , a a > a . The condition a > a >
0, wesubstitute (28) into (30) and obtain (with a + sign for A and a − sign for A ): a = 12 · βµ δ (cid:16)(cid:112) − − p ) δ − ± (1 − δ − ) (cid:17) (cid:112) − − p ) δ − , (33)whence a > ⇔ (cid:112) − − p ) δ − ± (1 − δ − ) > . For A = A (+ sign), the last condition always holds, while for A = A it isequivalent to (cid:112) − − p ) δ − > − δ − ⇔ δ > p , but under the last condition A is the unique endemic equilibrium, so we con-clude that the condition a > A = A .We have therefore shown that Proposition 5.
The equilibrium E , when it exists, is unstable. By the above we have that A = A will be stable if and only if a a > a , andwe proceed to check when this condition holds. Substituting A = A as givenby (28) into (30)-(32) we calculate a a − a = 14 · µ δ (cid:104) (cid:112) − − p ) δ − (cid:105) + βµ δ ( p − δ − ) (cid:104) (cid:112) − − p ) δ − (cid:105) − · βµ δ (cid:16)(cid:112) − − p ) δ − + (1 − δ − ) (cid:17) (cid:112) − − p ) δ − , so that a a > a is equivalent to µγ + µ δ (cid:104) (cid:112) − − p ) δ − (cid:105) > − p ) · (cid:16) ( δ −
2) + δ (cid:112) − − p ) δ − (cid:17) . (34)We note that if p ≥ then (34) automatically holds, since the left hand side ispositive and the right hand side is non-positive. If p < then (34) is equivalentto: γµ < F ( δ ) . = δ − p ) · (cid:104) (cid:112) − − p ) δ − (cid:105) ( δ −
2) + δ (cid:112) − − p ) δ − − . (35)We therefore conclude that 14 roposition 6. (i) If p ≥ then the endemic equilibrium E is stable.(ii) If p < then the endemic equilibrium E is stable if γµ < F ( δ ) and unstableif γµ > F ( δ ) , where F ( δ ) is defined in (35). Since the function F ( δ ) plays an important role, we wish to understand theshape of its graph, assuming p < . This function is defined for δ > − p ),and we have F (4(1 − p )) = (cid:18) − p + 1 (cid:19) − , lim δ →∞ F ( δ ) = + ∞ . Differentiating F , we find that F (cid:48) ( δ ) > ⇔
12 ( δ − (cid:112) − − p ) δ − > − p − − pδ − δ , which, using some elementary algebra, gives Proposition 7. (i) If p < then F ( δ ) is monotone increasing for all δ ≥ − p ) .(ii) If ≤ p < then F ( δ ) is decreasing for − p ) ≤ δ < √ p andincreasing for δ > √ p . We now synthesize our previous result, to obtain a picture of the dependence ofthe model dynamics on the parameters.We first note that when p ≥ , the behavior is rather simple: for δ < p wehave only the contagion-free equilibrium E , which is stable. At δ = p thecontagion-free equilibrium loses stability and a stable endemic equilibrium E arises in a continuous transition. This behavior is similar to that observed inthe standard SIR model with demographic turnover [30, 39].In the case p < we observe new phenomena. To understand these, we dividethe plane of parameters (cid:16) δ, γµ (cid:17) into five regions, each of which corresponds todifferent properties of the equilibria, as determined in section 3.1. These regionsare shown in figure 6 for the case p = 0 .
15 and p = 0 . δ -axis). The qualitative difference inappearance between the two cases stems from the fact that the function F ( δ ),which defines the boundary γµ = F ( δ ) between regions II and V and betweenregion III and IV is monotone increasing for when p < and has a minimumwhen p > (proposition 7).We now study the dynamics for each of these regions in turn, using both theanalytical results regarding the equilibria and their stability obtained above andnumerical simulations. 15igure 6: Phase diagram for p = 0 .
15 (left) and for p = 0 . I : No contagion In region I , defined by the condition δ < − p ) , the contagion-free equilibrium E is stable, and there exist no endemic equilibria(propositions 3,4). When the parameters are in this region the contagion willnot spread. II : Bistability In region II , given by the conditions4(1 − p ) < δ < p , γµ < F ( δ ) , the contagion-free equilibrium E is stable, but there exist also two endemicequilibria E , E , with E stable and E unstable (propositions 3,5,6). Thus inthis region we have bistability - the contagion may persist or not, depending onthe initial conditions. This is demonstrated in figure 7 (left), in which, whenthe initial fraction of adopters is 3% the contagion dies out, while if the initialfraction of adopters is 4% the contagion persists and approaches an endemicequilibrium.Let us note that in this region the coefficients of the characteristic polynomial(29) of the linearization at E satisfy a < a > a < a > E is a saddle with aone-dimensional unstable manifold. 16igure 7: Dynamics for parameter values in region II . Left: Solution A ( t )for parameters p = 0 . , µ = 0 . , δ = 3 . , γµ = 5. Initial conditions are A (0) = A , S (0) = 1 − A , S = R = 0, with A = 0 . A = 0 . A (0) = A , S (0) = 1 − A , S = R = 0 with A = 0 . , . , . , . , .
05. This demonstrates the bistability of contagion-free and the endemic equilibrium E , when parameters are in region II. Theendemic equilibria are shown in blue. III : Endemic equilibrium or bistability of equilbriumand limit cycle
In region
III , given by the conditions δ > p , γµ < F ( δ ) , the contagion-free equilibrium E is unstable, and there exists a unique endemicequilibrium E , which is stable (propositions 3,6). Thus in this region contagionwill not fade out, and it can persist at the stable endemic equilibrium. Thesimulations show that this is indeed the case for most parameter values in region III , as illustrated in figure 8. However we also find a small range of parametervalues in region
III , near its boundary with region IV , for which a stable limitcycle coexists with the stable equilibrium E . For these parameter values wehave bistabilty of the endemic equilibrium and a limit cycle, so that contagioncan persist either at a constant or at periodically varying prevalence, dependingon initial conditions. This phenomenon will be explained in section 3.3.3, whenwe discuss the Hopf bifurcation at the boundary of regions III and IV . IV : Endogenous oscillations In region IV , given by the conditions δ > p , γµ > F ( δ ) , III . Left: Solution A ( t ) for parameters p = 0 . , µ = 0 . , δ = 8 , γµ = 10. Right: A trajectory inthe phase space, for the same parameter values.the contagion-free equilibrium E is unstable, and there exists a unique endemicequilibrim E , but it too is unstable (propositions 3,6). Indeed in this region wehave that the coefficients of the characteristic polynomial (29) of the lineariza-tion at E satisfy: a > , a > , a a < a . The fact that a a < a impliesthat at least one eigenvalue of the linearization has a positive real part. a > E is two-dimensional.In view of the fact that the contagion-free equilibrium is unstable, we know thatthe contagion cannot fade out. On the other hand, since both equilibria E , E are unstable, the system cannot stabilize at an equilibrium, and we conclude thatthe contagion must persist in a non-stationary regime. Simulations show thatcontagion persists as a limit cycle, leading to sustained oscillations, representingrepeated epidemic cycles. This is demonstrated in figure 9. In this example δ = 8, the population turnover time is µ − = 20 years and the mean duration ofadoption is γ − = 1 year, and periodic oscillations have a period of around 29years are obtained, with the fraction of adopters varying between less than 0 . δ = 7 (by reducing β ), keeping the same populationturnover time and mean duration of adoption. The period of oscillations is nowapproximately 128 years.It should be noted that although the stability analysis of the equilibrium pointsshowed that contagion must persist in a non-stationary state, it does not followautomatically that this must be a periodic one - indeed it is known that athree dimensional system can also exhibit quasi-periodic and chaotic behaviors.However, our simulations for various parameter values in region IV have notrevealed any non-stationary dynamics other than a limit cycle. Verifying this18igure 9: Dynamics for parameter values in region IV. Left: Solution A ( t ) forparameters p = 0 . , µ = 0 . , δ = 8 , γµ = 20. Right: A trajectory in the phasespace, for the same parameter values.Figure 10: Solution A ( t ) , S ( t ) , S ( t ) for parameters p = 0 . , µ = 0 . , δ =7 , γµ = 20.mathematically appears to be a challenging problem. In region V , given by the conditions4(1 − p ) < δ < p , γµ > F ( δ ) , the contagion-free equilibrium E is stable, and there exist two endemic equilib-ria E , E , both of which are unstable (propositions 3,5,6). Thus in this regionthe contagion cannot persist in the form of an endemic equilibrium. A priorione could think that contagion might persist in the form of endogenous oscilla-tions (as is the case for region IV), but the simulations show that this is not thecase, and in fact generic trajectories converge to the contagion-free equilibriumas t → ∞ . However, here we observe a different phenomenon (see figure 11):19igure 11: Dynamics for parameter values in region V. Left: Solution A ( t ) forparameters p = 0 . , µ = 0 . , δ = 5 , γµ = 20. Right: A trajectory in the phasespace, for the same parameter values.trajectories in phase space, starting at points near the contagion-free equilib-rium, make a large excursion away from this equilibrium and then back to it.This means that contagion spreads as a large epidemic, and then disappears.This is quite different from the behavior in region I , where contagion disappearswithout spreading. This phenomenon is known as excitability , and it is familiarin the field of neuroscience [26]. To understand the underlying reason for it, weneed to look at the unstable manifold of the equilibrium E . Note that, as is thecase for region II , since the coefficients of the characteristic polynomial of thelinearization at E satisfy a < , a > E ), see figure 12, we observe thatits two ends connect E to the stable contagion-free equilibrium E , forminga heteroclinic cycle. This cycle attracts nearby trajectories and is responsiblefor the excitability phenonmenon. As we will see further on, the heterocliniccycle above can be understood as a ‘residue’ of the limit cycle which exists whenparameters are in region IV , arising from it through a homoclinic bifurcation. Having characterized the model dynamics in each of the five regions of theparameter plane, it is interesting to understand the transitions that occur whenthe boundaries from one region to another are crossed. Each such crossingcorresponds to a bifurcation involving one of the equilibrium points. We willexamine the six transitions that can occur when the parameters vary along ageneric curve in the parameter plane. To illustrate this, we take p = 0 .
15 (thecorresponding phase diagram is in figure 6, left), choose three horizontal linesin the ( δ, γµ )-plane, ( γµ = 4 , ,
20) and examine the bifurcations along theselines, as they cross regions. The A -values of the equilibria, as well the range of A -values for the limit cycles, are plotted in figures 13,14,16, using the numerical20igure 12: Heteroclinic loop involving the stable contagion-free equilibrium E and the unstable endemic equilibrium E , for parameters p = 0 . , µ = 0 . , δ =5 , γµ = 20 (equlibrium E is also plotted, but not involved).continuation package MATCONT [15]. The investigation to follow will revealthat there are also global bifurcations which occur at interior points of some ofthe regions, and not only at their boundaries. I → II and I → III : fold bifurcations
When the line δ = 4(1 − p ) (the endemicity threshold) is crossed from left toright, we have a fold (also known as limit-point) bifurcation [36] in which thetwo endemic equilbria E , E appear. In the case of crossing from region I toregion II (that is when the point of crossing satisfies γµ < F ( δ ), as is the case infigure 13), the equilibrium E is stable and E is a saddle with a one-dimensionalunstable manifold - this is also known as a saddle-node bifurcation. In the caseof crossing from I to III (as is the case in figure 14 and 16) both E and E aresaddles, with E having a two-dimensional unstable manifold and E having aone-dimensional unstable manifold. II → III : transcritical bifurcation
When we cross the invasion threshold δ = p from region II to region III , atranscritical bifurcation ocurrs whereby the unstable endemic equilbirium E merges with the stable contagion-free equilibrium E and then disappears (its A component becomes negative), and E becomes unstable.21igure 13: Left: Equilibria in dependence on δ , when p = 0 . γµ = 4Figure 14: Left: Equilibria and limit cycles in dependence on δ , when p = 0 . γµ = 20. Right: A closer look at values of the parameter δ near the Hopfbifurcation point. IV → III : Hopf bifurcation
When we cross from region
III to region IV , along the curve γµ = F ( δ ), theunique endemic equilibrium E loses stability as two eigenvalues of the lineariza-tion around E move from the left to right-hand side of the complex plane, lead-ing to the birth of a limit cycle through a Hopf bifurcation [36]. For the case γµ = 20, this occurs at δ = 8 .
14, and the bifurcating limit cycle can be observedin figure 14. By taking a closer look at the neighborhood of the bifurcationpoint (figure 14,right), we find that the bifurcation is of subcritical type: as δ increases beyond the critical value δ = 8 .
14, a branch of unstable limit cyclesis born out of the equilibrium point (which changes from unstable to stable).At δ = 8 . IV .This means that for 8 . < δ < .
184 - parameter values for which we are inregion
III , there exist both a stable endemic equilibrium and a stable limitcycle, and in addition there is an unstable limit cycle - thus we have bistabiltyof periodic and stationary behavior. It also implies that the transition from astable endemic equilibrium to periodic behavior as δ decreases, so that we movefrom region III to region IV , occurs in a discontinuous manner - when E losesstability the stable limit cycle which characterizes the dynamics is a large one.22 = 8 δ = 7 . δ = 6 . δ = 6 δ = 5 δ = 4Figure 15: Saddle-node homoclinic bifurcation as δ decreases, for p = 0 . γµ = 20. IV → V : Saddle-node homoclinic bifurcation Referring to figure 14 (left) we see that when δ is reduced beyond the value δ = p = . = 6 . IV to region V , the limitcycle disappears, and we would like to understand the type of bifurcation in-volved. We recall (section 3.2.5) that when parameter values are in region V the dynamics is characterized by excitability, stemming from the a hetroclinicloop connecting the unstable equilibrium E to the stable contagion-free equi-librium E . The transition from the limit cycle to the heteroclinic loop occursthrough a saddle-node homoclinic bifurcation [36], see figure 15: as δ approaches . from above, part of the limit cycle approaches closer to the contagion-freeequilibrium E , and when δ = . the limit cycle touches E , thus forming ahomoclinic (note that this implies that as δ approaches the critical value, theperiod of the limit cycle approaches infinity). As soon as δ < . , the unstableendemic equilibrium E bifurcates out of E , which now becomes stable, andthe heteroclinic loop E → E → E is formed (note that the other unstableequilibrium E is not involved in this bifurcation). The limit cycle has thusbeen replaced by a heteroclinic loop, so that oscillations have been replaced byexcitability. V → II : Hopf bifurcation of an unstable limit cycle When we move from region V to region II across the boundary defined by thecurve γµ = F ( δ ), the endemic equilibrium E becomes stable, as two eigenvalues23igure 16: Left: Equilibria and limit cycles in dependence on δ , when p = 0 . γµ = 10. Right: A closer look at values of the parameter δ near the Hopfbifurcation point.Figure 17: For p = 0 . , γµ = 10, unstable manifold of E before ( δ = 4 .
9, left)and after ( δ = 5 .
1, right) the homoclinic bifurcation.of its linearization cross from the right to the left halves of the complex plane.This is accompanied, as expected, by a Hopf bifurcation, in which an unstable limit cycle emerges from E . This can be seen in Figure 16, where we take γµ = 10 and the transition from region V to region II occurs at δ = 4 . δ = 4 . δ = 4 . E . For values of δ below the bifurcation value δ = 4 .
953 (figure 17, left), this unstable manifoldforms a homoclinic loop with E . At the bifurcation the unstable limit cyclecollides with the unstable equilibrium E , forming a homoclinic orbit, whichthen vanishes, and the two parts of the unstable manifold of E now connect to E and to E (figure 17, right). 24 Discussion
We have shown that simple two stage contagion models with demographicturnover generate interesting nonlinear effects, which do not arise in their ‘clas-sical’ one-stage counterparts (SI and SIR models [30, 39]). We now summarizethese effects in a non-technical way, so as to highlight the qualitative conclusionsthat can be drawn regarding conditions under which different types of dynam-ics are obtained, and consider their significance for the behavior of contagion atthe population level. We expect that the broad features described below will berobust, in the sense that they will hold also under various modifications of themodel.(1) When the probability of adoption on first encounter is sufficiently high ( p ≥ ), the behavior of the two-stage contagion models is much like that of a one-stage model: there exists an invasion threshold δ = p such that when contagionis weak ( i.e. the contact parameter δ satisfies δ ≤ p ) the contagion cannotpersist, while for δ above this threshold the contagion persists and approaches anendemic equilibrium. Moreover, the transition from non-contagion to contagionis continuous, in the sense that when the contact parameter is slightly abovethe threshold, the extent of contagion ( A ) at equilibrium will be small.When the probability of adoption on first encounter is sufficiently low ( p < ),new qualitative features emerge. In this case the dynamical behavior of themodel depends on two factors: the contact parameter δ , and the duration ofthe adoption period relative to the average residence time of an individual inthe population. The different phenomena which occur, depending on these twofactors, are summarized below.(2) When the contact paramater is below the endemicity threshold ( δ < − p )),contagion will not spread in the population.(3) When the contact parameter is above the invasion threshold ( δ > p ), weobserve different behaviors, depending on the duration of adoption:(i) If the mean duration of adoption ( γ − ) is sufficiently long relative to the meanresidence time of individuals in the population ( µ − ) – including the extremecase γ = 0 which corresponds to the model with permanent adoption, the con-tagion becomes established at a constant level (a stable endemic equilibrium),regardless of the initial number of adopters.(ii) If the mean duration of adoption is sufficiently short (that is γµ is sufficientlylarge), sustained periodic oscillations occur - corresponding to cyclic epidemicsof contagion. These oscillations are an emergent phenomenon at the populationlevel, arising from interactions among individuals, each of which displays nocyclic behavior - recall that in this model each individual who abandons theinnovation does not re-adopt it. We therefore have a mechanism for the endoge-nous generation of periodic fads and fashions. We note that a quite differentmechanism capable of generating periodic fashions, based on ‘snobs’ and ‘follow-ers’, is modelled in [1, 5]. We note that endogenous oscillations do not occur inthe basic models of mathematical epidemiology, and some special mechanismsare required to generate such oscillations, the most prominent being temporaryimmunity with delay [25, 49]. Our results show that two-stage contagion is an-25ther mechanism which induces periodic oscillations, and it appears that it isamong the simplest mechanisms producing this effect.(iii) In a narrow intermediate range of values of the mean duration of adoption,endemic equilibrium and periodic oscillations are both stable, so that contagionwill be maintained either at a constant level or in the form of cyclic epidemics,depending on initial conditions.(4) When the contact parameter is above the endemicity threshold but below theinvasion threshold (4(1 − p ) < δ < p ), we observe different behaviors, dependingon the duration of adoption:(i) If the mean duration of adoption ( γ − ) is sufficiently long relative to themean residence time of individuals in the population ( µ − ) (including in thelimit γ = 0, corresponding to permanent adoption), we have bistability of thecontagion-free and endemic equilibria, (‘alternative stable states’, [43]), givingrise to a ‘critical mass’ threshold, so that contagion will only ‘catch’ if suffi-ciently many individuals adopt it at the start. An important implication of thisbistability is that eradication of an established contagion will require reducingthe contact parameter to a much lower value than the invasion threshold. Forexample, if p = 0 .
1, then the invasion threshold is δ = p = 10, but eradicatingan existing endemic contagion will require reducing δ below δ = 4(1 − p ) = 3 . δ crossesthe invasion threshold from below, a jump from no contagion to a high levelof contagion will occur. Similarly, if a cotagion is already established and thecontact parameter is reduced until it reaches the endemicity threshold, a largecontagion will disappear without warning. This type of ‘critical transition’ or‘regime shift’ phenomenon [43] can provide an explanation for rapid opinionshifts and dramatic behavioral changes which can arise under minor changes inexternal conditions [35, 46]. Under this explanation, the discontinuous transi-tion is a collective effect arising from the interactions among individuals - thebehavior of individuals changes only in a gradual way as the contact parametervaries.The bistability and hysteresis effects described above do not occur in the basic‘one stage’ epidemiological models, in which transition from the contagion-freestate to endemicity is continuous. However such effects do occur in some moreelaborate models, and are known under the term ‘backward bifurcation’. Var-ious epidemiological mechanisms are known to induce backward bifurcations,e.g. exogenous reinfection of latently infected individuals, imperfect vaccina-tion, and risk structure [21]. Our results show that contagion with stages isanother mechanism which generates backward bifurcation.(ii) If the mean duration of adoption is sufficently short (that is γµ is sufficentlylarge), then contagion cannot become endemic, but we observe the phenomenonof excitability: starting with a small fraction of initial adopters, a large epidemicdevelops before the contagion fades. This contrasts with one-stage contagionmodels, in which, below the invasion threshold, the number of adopters will al-ways decrease, whatever its initial value. Thus the two-stage model can accountfor large contagion epidemics which do not become endemic, despite the renewalof the susceptible population provided by the demographic turover.26eveloping the mathematical theory of social contagion requires classifying rele-vant mechanisms at the micro-level, and exploring their dynamical consequencesat the population level using mathematical modelling. Simple models, like theone considered here, show that a combination of basic mechanisms (here, two-stage contagion and demographic turnover) can give rise to rich phenomena,suggestive of some of the complexities found in social systems. As always, thefact that a mathematical model can produce a phenomenon which is reminis-cent of a real-world one is far from proof that the mechanisms described by themodel are those responsible for the real effect, but it does constitute a proof-of-principle that the mechanisms involved are capable of producing the effect [32].It would be of great interest to attempt a direct validation of a two-stage (ormulti-stage) contagion model by fitting it to empirical data. References [1] Rafa(cid:32)l Apriasz, Tyll Krueger, Grzegorz Marcjasz, and Katarzyna Sznajd-Weron. The hunt opinion model - an agent based approach to recurringfashion cycles.
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