short answer question

Mathematical Biosciences 238 (2012) 80–89

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Mathematical Biosciences

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Risk perception and effectiveness of uncoordinated behavioral responses in an emerging epidemic

Piero Poletti ⇑, Marco Ajelli, Stefano Merler Bruno Kessler Foundation, Trento, Italy

a r t i c l e i n f o

Article history: Received 13 April 2011 Received in revised form 16 April 2012 Accepted 19 April 2012 Available online 3 May 2012

Keywords: Risk perception Behavioral response Game theory Mathematical model Human behavior Influenza

0025-5564/$ - see front matter � 2012 Elsevier Inc. A http://dx.doi.org/10.1016/j.mbs.2012.04.003

⇑ Corresponding author. Address: Fondazione Brun I–38123 Trento, Italy. Tel.: +39 0461 314520; fax: +

E-mail addresses: [email protected] (P. Poletti), ajelli@fb (S. Merler).

a b s t r a c t

Beyond control measures imposed by public authorities, human behavioral changes can be triggered by uncoordinated responses driven by the risk perception of an emerging epidemic. In order to account for spontaneous social distancing, a model based on an evolutionary game theory framework is here pro- posed. Behavioral changes are modeled through an imitation process in which the convenience of differ- ent behaviors depends on the perceived prevalence of infections. Effects of misperception of risk induced by partial or incorrect information concerning the state of the epidemic are considered as well. Our find- ings highlight that, if the perceived risk associated to an epidemic is sufficiently large, then even a small reduction in the number of potentially infectious contacts (as a response to the epidemic) can remarkably affect the infection spread. In particular, the earlier the warning about the epidemic appears, the larger the possible reduction of the peak prevalence, and of the final epidemic size. Moreover, the epidemic spread is delayed if individuals’ perception of risk is based on a memory mechanism and the risk of infec- tion is initially overestimated. In conclusion, this analysis allows noteworthy inferences about the role of risk perception and the effectiveness of spontaneous behavioral changes during an emerging epidemic.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction larger uncoordinated responses have been detected for more severe

Contact patterns represent a key ingredient in epidemic model- ing of human to human infectious diseases [1–9]. Nonetheless, spontaneous behavioral response to the risk of infection is largely suspected to play a crucial role for determining the spread of epi- demics across human population [10–13]. In fact, during an epi- demic outbreak, individuals may change their behavior in order to reduce the risk of infection, especially if serious consequences are involved [14]. For instance, a population-based survey of peo- ple’s precautionary actions in response to a hypothetical influenza pandemic [15] reported that ‘‘more than 75% of respondents would avoid public transportation and 20–30% would try to stay indoors’’. Studies on behavioral response to the 2009 H1N1 influenza have highlighted an initial high level of anxiety about the pandemic [16] and different behavioral responses to the risk of infection [17–21]. In Australia, after the first pandemic wave, individuals ‘‘re- ported increasing handwashing (46%) and covering cough and sneezes (27%)’’ to reduce the risk of infection [20]. In the US, data collected on public response to H1N1 influenza from May 2009 to June 2009 suggest that ‘‘16–25% of Americans had avoided places where many people are gathered, like sporting events, malls, or public transportation and 20% had reduced contact with people outside [their] household as much as possible’’ [18]. Furthermore,

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o Kessler, via Sommarive 18, 39 0461 314591. k.eu (M. Ajelli), [email protected]

epidemics. As an example, the 2003 ‘‘SARS crisis resulted in an 80% reduction in travel to and from Hong Kong’’ [22]. These examples represent evidences that behavioral changes may appear in re- sponse to an emerging infectious disease, although it is still hard to quantify their impact on the epidemic spread.

The role of uncoordinated responses to risk perception is an increasingly relevant issue in disease transmission modeling for improving predictions about the spread of an emerging epidemic. Different models have been proposed to investigate the possible ef- fects of spontaneous human behavioral changes. However, it is still unclear which are the main determinants in both risk perception and the diffusion of human behavioral patterns leading to remark- able alterations in infection dynamics.

The aim of this manuscript is to assess which are the patterns of risk perception and information diffusion (here explicitly modeled) resulting in an effective alteration of the epidemic spread of a human–to–human infectious disease. The role of the key parame- ters regulating the mechanism of spontaneous self-protection is investigated through a detailed sensitivity analysis. The interplay between risk perception and disease transmission process is high- lighted and discussed.

1.1. Background

Several mathematical models have been proposed to investigate possible effects of behavioral changes on the dynamics of an epi- demic [23]. Belief systems may represent the primary barrier to

P. Poletti et al. / Mathematical Biosciences 238 (2012) 80–89 81

vaccination in high income countries, ‘‘producing clusters of unvac- cinated individuals and leading to increase in disease outbreak probability’’ [24]. In epochs characterized by a moderate prevalence of infection, high levels of uptake can be difficult to reach through voluntary vaccination programs as a consequence of perceived risk associated to vaccine side effects [25–29]. Moreover, media cover- age has the potential of triggering a vaccinating panic if the vaccine is imperfect [30] and the likelihood of eliminating an infectious dis- ease depends critically on the amount of information individuals have access to [26,31]. Besides, public subsides may have a perverse impact on behaviors adopted in the population by inducing more risky behavior and, in turn, by increasing the disease prevalence [32]. On the other hand, behavioral responses triggered by the per- ceived risk of experiencing the disease can stop the epidemic spread [33]. A spontaneous reduction of the risk of infection can be remarkably effective in reducing the spread of an emerging epi- demic [34] or, at least, in ‘‘delaying the epidemic until a vaccine be- comes widely available’’ [35]. Moreover, human behavioral changes can remarkably alter endemic equilibria [36], even making ‘‘impos- sible for a disease to establish itself in a population’’ [37].

In the literature, several models either assume a priori human re- sponse to the infection or they consider only behavioral response as driven by the diffusion of fear, which is modeled as a parallel infec- tion [16,37–41]. In this case, the ‘‘recovery from fear’’ occurs at a constant rate, regardless the current state of the epidemic and the behaviors adopted by the individuals. This assumption yields that ‘‘a population is always stable against invasion by a more risky behavioral form’’ [39]. However, as recently pointed out in [42], the attitude of individuals in reducing risky behaviors seems to de- pend on the current risk perception of the epidemic. An alternative and interesting approach based on evolutionary game theory [43– 45] has been proposed to investigate individuals’ choices in volun- tary vaccination programs [25,27–29,46–48], providing interesting and plausible explanations for the observed fluctuations in vaccine uptake [46,47]. The approach allows considering a symmetric mechanism regulating spontaneous behavioral changes. Different behaviors adopted in the population are represented by a given set of strategies. The adoption of different strategies is assumed to be driven by the perceived convenience of different behaviors, explicitly dependent on the epidemic dynamics.

The game theoretical approach has been already proposed in [49,50]. In [49], the authors discuss the conditions for observing multiple epidemic waves triggered by spontaneous behavioral changes while in [50] a similar approach has been applied for investigating the effect of human behavioral response during the 2009 H1N1 pandemic influenza. In both papers a simple model has been introduced, considering only the choices by susceptible individuals. In this manuscript, such model is extended in order to: (1) consider possible behavioral changes triggered by the per- ceived risk of infection occurring among infective agents; (2) inves- tigate the impact of different symptomaticity levels of infection on the risk perception and on the epidemic dynamics; (3) consider dif- ferent routes of misperception of the risk of infection.

1 In fact, while epidemic transmission can occur only through physical person-to- person contacts, it is fairly reasonable to consider that individuals can access the information required to decide whether or not to change behavior much more frequently by telephone, email, internet or, more in general, the media.

2. The model

2.1. Model assumptions

The proposed investigation is focused on possible behavioral re- sponses performed by individuals in order to avoid or to reduce the risk of infection. In particular, in the model, individuals are sup- posed to be able to reduce their susceptibility to the infection. Such defensive response takes into account both reduction in physical contacts and, more in general, all self-prophylaxis measures which can reduce the transmission probability during potentially infec-

tious contacts. For instance, a reduction of contacts can be achieved through the avoidance of crowded environments or by limiting travels. A reduction of transmission probability can be achieved by increasing wariness in usual activities e.g., washing hands fre- quently or respecting cough/respiratory etiquette, or by using face masks.

Actually, only susceptible individuals are exposed to the risk of infection. On the other hand, in principle, asymptomatic infective individuals and recovered individuals who did not experience symptoms have no reason to behave differently from susceptibles, as they are equally concerned about the risk of experiencing the disease. Thus, we assume that self-protective behavior possibly adopted by asymptomatic infective individuals similarly results in a reduction of the force of infection.

In epidemic modeling other spontaneous responses to the dis- ease can be considered. For instance, changes in contact patterns can appear in individuals that suffer symptoms as a consequence of their sickness (workplaces and school attendance can drastically reduce during a serious infection outbreak [51,52]). Such defensive response depends on the severity of symptoms and it appears regardless of the current state of the epidemic in the population and of the behavior of other individuals while it does not depend on the dynamics of the perceived risk of infection. We assume that all symptomatic infected individuals enact the same defensive re- sponse. Thus, a different transmission rate for symptomatic indi- viduals is considered. Undeniably, the understanding of behavioral changes adopted by infective symptomatic individuals represents a major task in epidemic modeling. However, the inves- tigation of this phenomenon is beyond the scope of this study, which is mainly focused on the dynamic aspects of behavioral changes that depend on risk perception.

These assumptions can be summarized as follows. Behavioral changes caused by symptoms involve only symptomatic infected individuals and they are here accounted for by simply considering a different transmission rate for symptomatic infective individuals. On the contrary, behavioral changes triggered by the concern of experiencing the disease involve susceptible individuals, infected asymptomatic individuals and recovered individuals who did not experience symptoms.

Cast in the language of evolutionary game theory, the dynamics of self-protection can be modeled as a suitable dynamic game, where behaviors adopted by individuals correspond to strategies with given expected payoffs. The spread of different behaviors in response to the perceived risk of infection is modeled through an imitation dynamics [44–46,49,53]. The latter is a learning process, in which the convenience of strategies is assessed through per- sonal encounters, comparing the payoff of different strategies: individuals change strategy as they become aware that their payoff can increase by adopting another behavior.

The diffusion of the pathogen across the population is modeled by an SIR scheme: differences between symptomatic and asymp- tomatic individuals and between individuals that spontaneously reduce or not their risk of infection are considered as well.

The model of the infection dynamics is the result of the combi- nation of two contributions: the disease transmission dynamics (equations related to this process will be denoted by the super- script tr) and the imitation dynamics (equations related to this pro- cess will be denoted by the superscript im). Since the type of contacts that are relevant for the transmission of the pathogen may be different from those contacts relevant for the diffusion of information and behaviors, the disease transmission process and the imitation process are modeled in two different time scales.1

82 P. Poletti et al. / Mathematical Biosciences 238 (2012) 80–89

Therefore, we introduce s as the time unit of spontaneous behavioral changes and t ¼ as (with a 2 R) as the time unit of the transmission dynamics.

2.2. Transmission process

The transmission model is kept as simple as possible. The host population is assumed to be divided into five classes, namely, sus- ceptible S, infective symptomatic IS and asymptomatic IA, recov- ered individuals that when infective were symptomatic RS and recovered individuals that when infective were asymptomatic RA . Here we refer to these latter classes of individuals as ‘‘recovered symptomatics’’ and ‘‘recovered asymptomatics’’, respectively. Fur- thermore, susceptible, asymptomatic infective and asymptomatic recovered individuals are divided in two subclasses each: individ- uals adopting a ‘‘normal’’ behavior (Sn; IAn ; RAn ) and individuals adopting an ‘‘altered’’ one (Sa; IAa ; RAa ). Specifically, individuals adopting an altered behavior represent individuals that reduce their number of potentially infectious contacts by a factor q with 0 6 q 6 1 as a response to the risk of infection. From now on, let us denote as bn and ba the two different behaviors adopted by Sn; IAn ; RAn and Sa; IAa ; RAa respectively. Sn; Sa; IS; IAn ; IAa ; RS; RAn ; RAa rep- resent fractions of the population instead of absolute numbers.

By defining bS and bA as the transmission rates for symptomatic and asymptomatic infective individuals respectively, 1=c the aver- age length of the infectivity period (corresponding here to the gen- eration time) and p the probability of developing symptoms, the epidemic flow (summarized in Fig. 1) can be modeled as follows:

_Strn ðtÞ ¼ �kSn; _Stra ðtÞ ¼ �qkSa; _ItrS ðtÞ ¼ p½kSn þ qkSa�� cIS; _ItrAnðtÞ ¼ ð1 � pÞkSn � cIAn; _ItrAaðtÞ ¼ ð1 � pÞqkSa � cIAa ; _RtrS ðtÞ ¼ cIS; _RtrAnðtÞ ¼ cIAn ; _RtrAaðtÞ ¼ cIAa ;

8>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>:

ð1Þ

where k is the force of infection, which is modeled as follows:

k ¼ bSIS þ bAIAn þ qbAIAa :

The force of infection k is the result of three contributions: first, the transmission associated to symptomatic infected individuals (bS IS); second, the transmission associated to asymptomatic in- fected individuals adopting the normal behavior (bA IAn ); third, the (reduced) transmission associated to asymptomatic infected indi- viduals adopting the altered behavior (qbA IAa ). Moreover, in the first equation of system (1), the susceptibility of individuals adopting the altered behavior (i.e. Sa) is reduced by a factor q as well.

Fig. 1. Epidemic flow between epidemiological classes due to the disease transmission process only.

2.3. Risk perception and convenience of responsiveness

Human behavior is assumed to be driven by the evaluation of prospective outcomes deriving from alternative decisions and cost-benefit considerations, depending on the perceived risk of infection.

In this model, the perceived prevalence represents the informa- tion about the risk of infection, it accounts for the number of symp- tomatic cases occurred over a certain (past) period of time and it is modeled through an exponentially fading memory mechanism (similar to those used in [47,54,55]) as follows:

dM dt ðtÞ¼ p½kðtÞSnðtÞþ qkðtÞSaðtÞ�� mIðtÞ

where m weighs the decay of the perceived risk due to new symp- tomatic cases. Actually, 1=m can be read as the average duration of the memory of symptomatic cases. When m ¼ c and Mð0Þ¼ ISð0Þ, the perceived prevalence corresponds exactly to the prevalence of symptomatic infections in the population (i.e. MðtÞ¼ ISðtÞ 8t).

The convenience of the two different strategies is defined by their corresponding payoff function. All individuals pay a cost for the risk of infection, which we assume to depend linearly on the perceived prevalence (MðsÞ) and to be higher for bn than for ba. Be- sides, individuals playing ba pay an extra, fixed cost. Therefore, the payoffs associated with bn and ba result respectively:

PnðsÞ¼�mnMðsÞ; PaðsÞ¼�k � maMðsÞ

with mn > ma. mn and ma can be read as parameters related to the risk of developing symptoms induced by the two different behav- iors bn and ba, while k represents the cost of any self-imposable pro- phylactic measure. Thus the altered behavior gives the advantage of reducing the risk of infection, but the extra cost associated to the al- tered behavior implies that the normal behavior is the most conve- nient one when the perceived prevalence M is small (or in absence of disease).

2.4. Imitation process

We assume that individuals can change strategy after having compared, through encounters with other individuals, the payoffs of the two different strategies, at a rate proportional to the differ- ence between the corresponding payoff functions (DP ¼ Pn � Pa), with proportionality constant /.

In a two strategy game, the imitation dynamics can be de- scribed as follows:

_x ¼ ~xxð1 � xÞ/DP; ð2Þ

where x and 1 � x are the fractions of population performing the two different strategies and ~x is the rate at which individuals meet each other. The encounters useful for changing strategy are those involving individuals playing a different strategy. As a matter of fact, DP sets the sign of _x, i.e. the direction of the behavioral changes.

However, Eq. (2) has the property that a strategy can spread in the population if and only if it is already played by some individu- als (as x ¼ 0 and x ¼ 1 are equilibria for Eq. (2)). In order to avoid this undesirable effect, we consider also that individuals, rarely (at a rate ~l � 1), change strategy regardless the payoff balance, acting an irrational exploration of the set of available strategies [45,49].

Therefore, Eq. (2) becomes

_x ¼ xxð1 � xÞDP þ ~lð1 � xÞ� ~lx;

where x :¼ ~x/. The imitation process drives only changes of strategy while it

does not involve any transition among epidemiological classes.

2 In fact, the population of players is composed by individuals belonging to classes S; IA; RA .

3 Indeed, individuals adopting the altered behavior leave the susceptible class fo entering the symptomatic infective class at a lower rate than susceptibles adopting the normal behavior.

P. Poletti et al. / Mathematical Biosciences 238 (2012) 80–89 83

As a consequence, encounters between individuals belonging to different epidemiological classes, e.g. susceptible and infective individuals, have to be carefully considered: in fact, neither suscep- tibles can become infectives nor infectives can become suscepti- bles through imitation.

For instance, when susceptibles adopting strategy bn (Sn) com- pare their payoff with infectives adopting strategy ba (IAa ), neither Sn nor IAa can increase. Indeed if bn is more convenient than ba then some infectives adopting ba will migrate to the class of infectives adopting bn (namely IAn ). Similarly, if ba is more convenient than bn then some susceptibles adopting bn will migrate to the class of susceptibles adopting ba (namely Sa ). The same argument holds for encounters between susceptible and removed individuals as well. Thus the resulting imitation dynamics for susceptible individ- uals is modeled as follows:

_Simn ðsÞ ¼ x½SnSa þ SaðIAn þ RAnÞ�DP þ ~lSa � ~lSn if DP > 0 _Simn ðsÞ ¼ x½SnSa þ SnðIAa þ RAaÞ�DP þ ~lSa � ~lSn if DP < 0 _Sima ðsÞ ¼ �x½SnSa þ SaðIAn þ RAnÞ�DP � ~lSa þ ~lSn if DP > 0 _Sima ðsÞ ¼ �x½SnSa þ SnðIAa þ RAaÞ�DP � ~lSa þ ~lSn if DP < 0:

8>>>>>< >>>>>:

Similar equations can be defined for all other classes involved in the imitation process (namely, IAn ; IAa ; RAn ; RAa ). On the contrary, as neither IS nor RS are assumed to perform any self-protection mea- sure induced by the risk of experiencing symptoms, we assume _IimS ðsÞ¼ _RimS ðsÞ¼ 0 8s. In fact, as reinfection is not considered in this study, neither IS nor RS can take advantage by adopting an al- tered behavior.

2.5. Resulting coupled model

As previously mentioned, behavioral changes and epidemic transitions are modeled in two different time scales (equations re- lated to the imitation process are described in s instead of t). Hence, their contribution in the coupled dynamics are weighted properly by the factor a ¼ t=s. For instance, the resulting dynamics of susceptibles individuals can be written as

_SnðtÞ ¼ _Strn þ 1 a

_Simn ; _SaðtÞ ¼ _Stra þ 1a

_Sima :

(

Therefore the full system becomes:

_SnðtÞ ¼ �kSn þ 1a½xSn SaDPþxSaðIAn þRAnÞDPHðDPÞ � xSnðIAa þRAaÞDPHð�DPÞþ ~lSa � ~lSn�

_SaðtÞ ¼ �qkSa þ 1a½xSn SaDP�xSaðIAn þRAnÞDPHðDPÞ þ xSnðIAa þRAaÞDPHð�DPÞ� ~lSa þ ~lSn�

_ISðtÞ ¼ p½kSn þqkSa��cIS _IAnðtÞ ¼ ð1�pÞkSn �cIAn þ 1a½xIAn IAa DPþxIAaðSn þRAnÞDPHðDPÞ

� xIAnðSa þRAaÞDPHð�DPÞþ ~lIAa � ~lIAn� _IAaðtÞ ¼ ð1�pÞkSa �cIAa þ 1a½xIAn IAa DP�xIAaðSn þRAnÞDPHðDPÞ

þ xIAnðSa þRAaÞDPHð�DPÞ� ~lIAa þ ~lIAn� _RSðtÞ ¼ cIS _RAnðtÞ ¼ cIAn þ 1a½xRAn RAa DPþxRAaðSn þIAnÞDPHðDPÞ

� xRAnðSa þIAaÞDPHð�DPÞþ ~lRAa � ~lRAn� _RAaðtÞ ¼ cIAa þ 1a½xRAn RAa DP�xRAaðSn þIAnÞDPHðDPÞ

þ xRAnðSa þIAaÞDPHð�DPÞ� ~lRAa þ ~lRAn�;

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð3Þ

where H is the Heaviside function i.e., HðDPÞ¼ 1 if DP P 0 and HðDPÞ¼ 0 if DP < 0

2.6. Approximated model

By introducing the variables S ¼ Sa þ Sn; IA ¼ IAa þ IAn RA ¼ RAa þRAn corresponding to the fraction of susceptible, asymptomatic infected and asymptomatic recovered individuals in the popula- tion, and by defining the variable x ¼ðSn þ IAn þ RAnÞ=ðS þ IA þ RAÞ, corresponding to the fraction of the population of players2 adopt- ing the normal behavior, the following result holds:

Proposition 1. If

Sn Sn þ Sa

¼ IAn

IAn þ IAa ¼

RAn RAn þ RAa

; ð4Þ

then

Sx ¼ Sn; IA x ¼ IAn; RAx ¼ RAn and system (3) becomes:

_S ¼�k½x þ qð1 � xÞ�S; _IA ¼ð1 � pÞk½x þ qð1 � xÞ�S � cIA; _IS ¼ pk½x þ qð1 � xÞ�S � cIS; _RA ¼ cIA; _RS ¼ cIS; _M ¼ pk½x þ qð1 � xÞ�S � mM; _x ¼ xð1 � xÞ½pS=ð1 � RS � ISÞ�½kðq � 1Þ�; þq½xð1 � xÞð1 � IS � RSÞð1 � mMðtÞÞþ lð1 � 2xÞ�;

8>>>>>>>>>>>>>>< >>>>>>>>>>>>>>:

ð5Þ

where m ¼ðmn � maÞ=k; q ¼ kx=a; l ¼ ~l=xk and k ¼ bSIS þ bA IAx þqbAIAð1 � xÞ.

The former part of the equation for x (i.e., xð1 � xÞ½pS= ð1 � RS � ISÞ�½kðq � 1Þ�) comes from rewriting system (1) under assumption (4). This part can be read as a ‘‘natural’’ selection pro- cess embedded in the transmission dynamics that favors individu- als adopting the altered behavior.3 The latter (i.e., q½xð1 � xÞ ð1 � IS � RSÞð1 � mMÞþ lð1 � 2xÞ�) accounts for spontaneous changes in individual behaviors due to the imitation process and the irrational exploration of strategies. Specifically, xð1 � xÞ ð1 � IS � RSÞ is the fraction of useful encounters for having a switch of strategy due to imitation; 1 � mM represents the balance between the payoff associated with the two possible behaviors; m defines a threshold determining which behavior would represent the most convenient choice; the term lð1 � 2xÞ represents the possibility that individuals, rarely (at a rate l � 1) change strategy independently by the payoff values, by enacting an irrational exploration of strate- gies [45,49,56]; q essentially represents the speed of spontaneous behavioral changes with respect to pathogen transmission dynamics. For large values of q (and small values of l) the sign of _x essentially depends on the balance of payoff between the two possible behav- iors (1 � mM). When the perceived prevalence M is over 1=m (here- after denoted as prevalence threshold or risk threshold), the altered behavior is perceived as the most convenient, and thus x decreases.

It is worth of noticing that if p ¼ 0 then ISðtÞ¼ RSðtÞ¼ MðtÞ ¼ 0 8t and no spontaneous behavioral changes occur, but for the small initial contribution coming from rare irrational explorations. On the other hand, if p ¼ 1 all infections are symptomatics and IAðtÞ¼ RAðtÞ¼ 0 8t, thus only susceptibles are involved in sponta- neous behavioral response to the epidemic. Moreover, if p ¼ 1 and m ¼ c, model (5) is similar to the model introduced in [49].

By assuming that condition (4) holds, Proposition 1 shows that model (3) is equivalent to model (5). The mathematical proof that

r

84 P. Poletti et al. / Mathematical Biosciences 238 (2012) 80–89

conditions of Proposition 1 are at least approximately satisfied in a global sense is still an open problem. Nonetheless, by comparing the trajectories produced by simulating the full model (3) with the trajectories produced by simulating its approximated version (5) in a wide range of parameters, we show that model (5) can be considered a numerically validated approximation of model (3) (see Supplementary materials for details). This considered, for the sake of simplicity, in the following sections only the approxi- mated model (5) is analyzed.

2.7. Reproductive number

The basic reproductive number R0 is essentially the average number of secondary infections that result from a single infectious individual in a fully susceptible population [57]. It can be com- puted by using next generation technique [58]. In this case, the resulting basic reproductive number is

R0 ¼ð1 � pÞ bA c ½x þ qð1 � xÞ�2 þ p

bS c ½x þ qð1 � xÞ� ð6Þ

which can be interpreted as a combination of two basic reproduc- tive numbers:

Rn0 ¼ð1 � pÞ bA c þ p

bS c ;

the reproductive number for a population where all individuals are adopting the normal behavior (x ¼ 1) and

Ra0 ¼ q 2ð1 � pÞ

bA c þ qp

bS c ;

the reproductive number for a population where all individuals are adopting the altered behavior (i.e., x ¼ 0). As 0 6 q 6 1 and 0 6 p 6 1, then Ra0 6 R

n 0 . The reduction in the number of contagious

contacts accumulates when both susceptibles and asymptomatic infective individual are adopting the altered behavior; this leads to the term q2 in Ra0.

Eq. (6) highlights that R0 depends on the fraction of individuals in the population who are adopting either normal or altered behav- ior. If xð0Þ¼ 1 and q ¼ 0, the system (5) reduces to a classical SIR model (with the distinction on symptomatic and asymptomatic individuals) driven by Rn0; on the contrary, if xð0Þ¼ 0 and q ¼ 0, it reduces to a classical SIR model driven by Ra0 . The smaller q is, the smaller Ra0 and the larger the effect of self-protection when individuals are adopting the altered behavior.

Two important assumptions characterize this model: (i) all indi- viduals are able to adopt a self-protection strategy 4; (ii) only two possible behaviors are available and it is assumed that all individuals adopting the altered behavior enact the same reduction in the num- ber of potentially infectious contacts. Nonetheless, as the payoffs are linear and a population of players is considered, such assumptions are analogous to consider a population of individuals that can choose among the infinite set of mixed strategies defined by the linear con- vex combination of these two pure strategies. Moreover, assuming that all individuals are reducing contagious contacts by q, it is similar to assume that only a fraction 1 � ~x acts a reduction of a factor ~q ¼ðq � ~xÞ=ð1 � ~xÞ or that only a fraction f of individuals is able to re- duce their contacts by acting a reduction of a factor ~q ¼ðq þ f � 1Þ=f . For instance, a reduction by a factor q ¼ 0:85 in contacts adopted by all individuals has the same effect on Ra0 of a reduction by a factor q ¼ 0:8 adopted only by 75% of individuals.

4 However, self-protection against the risk of infection is not enacted by IS and RS .

2.8. Model parametrization

In this study the spread of a ‘‘generic’’ influenza-like infection is simulated. Therefore, parameters that merely characterize the dis- ease transmission process are taken from reliable estimates avail- able for the 2009 H1N1 pandemic influenza. Specifically, the basic reproductive number is assumed to be 1:4 [59–63] and for the gen- eration time 2:8 days are assumed [61,62]. In general, the trans- mission rate for symptomatic and asymptomatic individuals can be different, as a consequence of a larger infectivity of symptom- atic infections or as a consequence of changes in usual habits asso- ciated with sickness. However, in order to investigate the interplay between risk perception and the epidemic diffusion, the specific choice of the value of the transmission rates is not crucial and hereafter we made the simplest assumption bS ¼ bA. The impact of different behavioral responses on the epidemic is investigated, by varying one-by-one the parameters, starting from a baseline configuration (see Table 1).

The scenario proposed as the baseline represents the simple case where all infections are symptomatic (p ¼ 1), the perceived prevalence M is exactly the prevalence of symptomatic infections IS (c ¼ m) and at the beginning of the epidemic the perceived risk of infection is zero (Mð0Þ¼ ISð0Þ¼ 0). The reduction of the number of potentially infectious contacts q has been derived from an esti- mation of the possible reduction of transmissibility potential due to behavioral responses during the 2009 H1N1 pandemic influenza in Italy [50]. Plausible values for q and 1=m have been assumed in order to consider a representative situation in which: (a) respon- siveness becomes more convenient than the normal behavior when the prevalence becomes larger than the 1% of the popula- tion; (b) the delay between the time at which responsiveness be- comes convenient and the time at which a relevant fraction of the population becomes responsive is about 5 days. Among all the simulations proposed, the rate of irrational exploration l has been fixed to a value already used in [49]. The initial conditions considered in the baseline scenario are: Sð0Þ¼ 1 � 10�3; ISð0Þ¼ 10�3; xð0Þ¼ 1 � 10�6; IAð0Þ¼ RAð0Þ¼ RSð0Þ¼ Mð0Þ¼ 0.

3. Results

3.1. Baseline scenario

Let us consider now the impact of spontaneous behavioral changes on the epidemic dynamics for the baseline scenario de- scribed in the previous section. The resulting dynamics of the Sys- tem (5) is displayed in Fig. 2a and Fig. 2b. After an initial growth of the epidemic, the perceived prevalence reaches the prevalence threshold 1=m and the altered behavior becomes more convenient. Then, after few days, the altered behavior becomes widely adopted in the population and the epidemic growth rate remarkably de- creases. As the prevalence decreases below the threshold, the pop- ulation starts to adopt the normal behavior again, producing an heavy tail in the infection dynamics.

Two key parameters characterize the timing of the behavioral response: m and q. The first one describes how the perceived prev- alence M is weighted in the payoff functions i.e., in the balance of the cost associated to the risk of infection and the cost of a self-pro- tection strategy. As a matter of fact, 1=m defines the threshold for the perceived prevalence above which individuals reducing con- tacts have a larger payoff. The larger m is, the earlier the altered behavior is perceived as the most convenient choice. On the other hand, q represents the speed of the imitation process with respect to the disease transmission time–scale. As a matter of fact, q en- tails the delay (embedded in the imitation dynamics) between the time at which a strategy becomes more convenient and the

Table 1 Epidemiological and behavioral parameters.

Parameter Interpretation Investigated range Baseline value References

1=c Average length of the infectivity - 2:8 [61,62] period (days)

bS Transmission rate for symptomatic - 0:5 a

individuals (days�1) bA Transmission rate for asymptomatic - bS when p – 1

b

individuals (days�1) p Probability of developing symptoms ½0; 1� 1 c q Reduction factor in the number ½0; 1� 0:85 [50]

of potentially infectious contacts 1=m Prevalence (or risk) threshold ½0; 0:05� 0:01 Assumed q Speed of the behavioral ½10�1; 103� 10 Assumed

changes (days�1) l Rate of irrational exploration - 10�8 [49] 1=m Average memory length ½1; 30� 1=c c

a The baseline configuration of epidemiological parameters is chosen in order to represent (when all individuals adopt the normal behavior) a simple SIR model with a basic reproductive number R0 ¼ 1:4 [59–63].

b If p ¼ 1, all infected individuals are symptomatic, thus there are no asymptomatic individuals transmitting the disease. When the impact of different symptomaticity levels of infection (p – 1) on the risk perception is considered, we assume that the transmission rate for asymptomatic and symptomatic individuals is the same.

c For the sake of simplicity, in the baseline scenario all infections are symptomatic (p ¼ 1) and the perceived prevalence is the prevalence of symptomatic infections (m ¼ c).

P. Poletti et al. / Mathematical Biosciences 238 (2012) 80–89 85

time at which the majority of population adopts this strategy. For instance, when 1=m < Isð0Þ, the altered behavior is more conve- nient since the beginning of the epidemic. Nonetheless, if x – 0, the altered behavior takes some time to spread in the population as well (see Fig. 2d). We define the time between when the per- ceived prevalence crosses the threshold 1=m and when more than 50% of the population has changed strategy as the imitation delay. The relation between the imitation delay and q is depicted in Fig. 2c. Our analysis highlights that the larger is q, the earlier the most convenient strategy spreads in the population. It is worth noticing that, if q ¼ 10 (as assumed in the baseline scenario), the imitation delay is about 4 days. This delay is compliant to that ob- served in the simulated dynamics of the daily prevalence of symp- tomatic infections for the baseline scenario (see Fig. 2a). The

Fig. 2. (a) Daily prevalence of symptomatic infection in the case of no responsiveness of t line). Other parameters are as in the baseline scenario reported in Table 1. The horizontal about 4 days after the perceived prevalence MðtÞ¼ ISðtÞ crosses the threshold 1=m pro dynamics of 1 � x (blue line, scaled on the left) and the effective reproductive number ov Other parameters are as in the baseline scenario. For each value of q the imitation de prevalence threshold 1=m and when more than 50% of individuals have adopted the alte delay. This, in turn, determines plausible values of q. For instance, if q ¼ 10 (as assumed (a). (d) Daily prevalence of symptomatic infection in the case of no responsiveness of th line), but for the prevalence threshold 1=m ¼ 10�3 . This example shows that, althoug responsiveness takes about 4 days to spread in the population. (For interpretation of the this article.)

computation of the imitation delay is robust under different assumptions (e.g., for different values of m), as shown in Fig. 2d.

In sum, the time at which the transition between two possible behaviors occurs is driven by m, while the duration of this transi- tion is driven by q. Therefore, the risk threshold (1/m) and the speed of the behavioral changes q are the main parameters deter- mining the responsiveness of the population to an epidemic outbreak.

Thanks to the above considerations, it is possible to define which are the ranges of values to be worth exploring by consider- ing the meaning of different model parameters.

First of all, if the perceived risk is always lower than the thresh- old (MðtÞ < 1=m for any time t), then no behavioral changes (in re- sponse to risk perception) can appear in the population. If m ¼ c

he population (q ¼ 1, bold gray line), and in the baseline scenario (q ¼ 0:85, bold red gray line represents the prevalence threshold 1=m. The behavioral response appears ducing a lower increase in the prevalence of infection. (b) Baseline scenario: the er time (dark green line, scaled on the right). (c) Imitation delay as a function of q.

lay is computed as the time between when the perceived prevalence crosses the red strategy (x < 0:5). The gray region represents a plausible range for the imitation in the baseline scenario) the imitation delay is 4 days, compliant to that observed in e population (q ¼ 1, bold gray line), and in the baseline scenario (q ¼ 0:85, bold red h the altered behaviors is (almost) always the most convenient strategy, human references to colour in this figure legend, the reader is referred to the web version of

Fig. 3. (a) Daily prevalence of symptomatic infections (a1), final epidemic size (a2), daily peak prevalence of symptomatic infections (a3) and peak day (a4) as obtained for different values of the prevalence threshold 1=m. Other parameters are as in the baseline scenario (see Table 1). (b) As (a) but for different values of the speed of behavioral changes q. (c) As (a) but for different values of the reduction factor q. (d) As (a) but for different values of the probability of developing symptoms p.

86 P. Poletti et al. / Mathematical Biosciences 238 (2012) 80–89

and Mð0Þ¼ ISð0Þ¼ 0, then MðtÞ¼ ISðtÞ. Thus, in order to observe behavioral changes in the population during the epidemic, the prevalence threshold 1=m must be lower than the maximum value of ISðtÞ expected in absence of any behavioral response. Moreover, it is possible to define a plausible range for q, by considering a cor- responding plausible range for the imitation delay (gray region in Fig. 2c). As q is a reduction factor and p is a probability, they are both explored in the range ½0; 1�. Finally, the investigated range for 1=m has been considered in order to represent an average dura- tion of memory from 1 day to 30 days. Investigated parameters’ ranges are displayed in Table 1.

From a mathematical point of view, this happens when 1=m is larger than ¼ 1 � 1

Rn0 þ 1

Rn0 log 1

Rn0 i.e., the largest possible daily peak prevalence, obtained when all

individuals adopt the normal behavior throughout the whole course of the epidemic.

3.2. Effectiveness of human self-protection

The effectiveness of human self-protection is analyzed in terms of: (i) final epidemic size (defined as the total number of infections at the end of the epidemic); (ii) daily peak prevalence; (iii) peak day.

As mentioned above, a major responsiveness of the population to an infection corresponds to a small prevalence threshold (large values of m) and to a low imitation delay (large values of q). As the responsiveness of population increases, a larger reduction in the fi- nal epidemic size and in the daily peak prevalence is observed (see Fig. 3a and Fig. 3b). The responsiveness of the population is related to when the behavioral response becomes effective. In fact, if the prevalence threshold or the imitation delay are too large the hu- man response never takes place and the epidemic spreads follow- ing the dynamics of an SIR model driven by Rn0 . If an epidemic is not

perceived sufficiently severe to trigger a behavioral response of the population,5 this corresponds to an unreachable level of the preva- lence threshold.

The size of reduction in contagious contacts associated to the al- tered behavior has a strong impact on the epidemic dynamics. As q decreases, a larger reduction of the risk of infection is enacted by individuals adopting the altered behavior. This leads to a decrease in both the final epidemic size and the daily peak prevalence (see Fig. 3c). The peak day is remarkably anticipated as q decreases; however, the burden for health care centers at the epidemic peak decreases as well (see the dynamics of the daily prevalence over time in Fig. 3c). Moreover, Fig. 3c shows that, as q becomes large, for small value of q the daily peak prevalence corresponds to the threshold value 1=m.

Three interesting aspects raise from our analysis:

(i) A small reduction in the number of contagious contacts enacted by the population can remarkably alter the spread of the epidemic;

(ii) For small values of q, multiple epidemic waves can occur. (iii) There exists a threshold for q such that smaller values do

not determine a larger impact of behavioral changes on the final epidemic size, the daily peak prevalence and the peak day;

5

Ip

Fig. 4. (a) Daily prevalence of symptomatic infections (a1), final epidemic size (a2), daily peak prevalence of symptomatic infections (a3) and peak day (a4) as obtained for different values of the average memory length 1=m. Other parameters are as in the baseline scenario (see Table 1). (b) As (a) but for different values of Mð0Þ, assuming 1=m ¼ 30 and m ¼ 1. (c) As (a) but for different values of the alarm times T. (d) As (a) but for different frequencies of information update: the perceived risk is updated every f days.

P. Poletti et al. / Mathematical Biosciences 238 (2012) 80–89 87

Indeed, a reduction of 100% in the number of potentially infectious contacts (corresponding to q ¼ 0 i.e., assuming total isolation) pro- duces the same effects obtained by considering a reduction of 25% (q ¼ 0:75). The possibility of observing multiple epidemic waves in- creases together with the human responsiveness to the epidemic i.e., for large values of q or m. A discussion on this topic can be found in the Supplementary materials.

When considering the possibility of asymptomatic infections (p < 1, but still assuming bS ¼ bA ), asymptomatic infectives can adopt the altered behavior as well. In fact, we remind that, since asymptomatic infectives are not aware of their infection, they have no reason to behave differently from susceptibles. As the fraction of asymptomatic infections increases (i.e., as p decreases), the frac- tion of individuals adopting a self-protection strategy increases as well. On the other hand, a larger symptomaticity of the disease re- sults in a larger number of observable infections and, in turn, a lar- ger perceived prevalence and risk. Our numerical simulations show that, in this trade off, the latter phenomenon seems to prevail. In- deed, spontaneous behavioral changes have a larger impact on epi- demic dynamics when the probability of developing symptoms is large (see Fig. 3d).

3.3. Risk perception and information diffusion

Hereafter we consider different assumptions on the risk percep- tion, on the information diffusion and, more in general, on the effect of the misperception of the risk of infection. So far, we have assumed that the perceived prevalence M at time t is exactly the symptomatic prevalence IS at time t (i.e., m ¼ c and Mð0Þ¼ ISð0Þ¼ 0). Several alternative assumptions can be considered.

First of all, to determine which behavior is the most convenient to adopt, individuals can take into account infections occurred over a (past) period of time. This case corresponds to assume 1=m > 1=c. As a matter of fact, assuming 1=m > 1=c results in a larger per- ceived risk of infection associated to every single new infection. Therefore, it is not surprising that a longer memory duration leads to a larger diffusion of the altered behavior, which results in decreasing the daily peak prevalence and the final epidemic size and in delaying the epidemic peak (see Fig. 4a).

The role of memory becomes more relevant when the popula- tion overestimates the risk of infection during the early phases of the epidemic. For instance, such kind of overestimation may hap- pen as a consequence of the concern raised after a mass media campaign (as could have happened during 2009 H1N1 pandemic [17,50,64]). An initial overestimation of the risk of infection can re- sult in an initial perceived risk above the threshold, even when the threshold is too large for supporting the altered behavior as the most convenient one. If the number of new cases produced by the epidemic is not sufficient to support the altered behavior as more convenient, the effect of misperception can vanish. On the other hand, when the initial prevalence misperception is supported by a long lasting memory, the altered behavior can result as the most convenient one for a relevant period of time. This essentially leads to an initial reduced growth rate of the epidemic, which de- lays the disease spread (see Fig. 4b). Moreover, if the initial overes- timation of risk is supported for a sufficiently large period of time, both daily peak prevalence and final epidemic size decrease as well.

The misperception of risk can also occur when the population becomes aware of a new epidemic outbreak after a period of time

Fig. 5. The influence of different key features of human behavioral responses on: (a) final epidemic size, (b) daily peak prevalence and (c) peak day. Different parameters have been investigated in the ranges defined in Table 1. Colors represent the responsiveness of the population: warmer colors correspond to higher responsive population. Gray boxes have been used where no specific relations were observed. The dashed line represents the baseline scenario, the red line represents a fully non responsive population (i.e., it corresponds to a ‘‘simple’’ SIR model driven by Rn0 ) and the blue line represents a fully responsive population (i.e., it corresponds to a ‘‘simple’’ SIR model driven by Ra0 . (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

88 P. Poletti et al. / Mathematical Biosciences 238 (2012) 80–89

since the emergence of the epidemic. This phenomenon can be investigated by assuming that the perceived prevalence is initially equal to zero for a period of time T. Numerical simulations show that the larger the delay is, the lower the effectiveness of human response becomes. In particular, daily peak prevalence and final epidemic size increase as T increases (see Fig. 4c) and, if the ‘‘alert’’ takes place too late no relevant effects on the outbreak can be de- tected (see Fig. 4c).

Finally, regardless of misperceptions or misjudgments of the risk of infection, it is fairy reasonable to assume that individuals may acquire information about the status of the epidemic only once in a while (e.g., once in f days), rather than in real time.6

Our investigation highlights that less frequent information does not remarkably reduce the effectiveness of the behavioral response of the population, unless very rare information is considered (see Fig. 4d).

Fig. 5 summarizes the main results presented in the manuscript.

4. Discussion

Human response to the perceived risk of infection can remark- ably affect the epidemic spread both qualitatively and quantita-

6 However, driven by the balance based on the last information acquired by the individuals, behavioral changes may occur continuously over time.

tively. Nonetheless, human behavior cannot be merely considered as an independent background process of the infection dynamics. The introduced model provides a promising approach, based on evolutionary game theory, to investigate the complex interaction between human behavior and disease transmission process. The model proposed in this manuscript extends the model already introduced in [49,50] in order to consider also (i) behav- ioral changes triggered by risk perception occurring in the infective agents; (ii) the impact of different symptomaticity levels of infec- tion on the risk perception and on the epidemic dynamics; (iii) the role of misperception during an emerging epidemic. The model is fairy general to be applied for describing any kind of epidemic outbreak (e.g. due to influenza, smallpox, SARS, etc.).

Key features of a human self-protection are captured as well as how and when behavioral responses are effective. Possible effects of spontaneous behavioral changes on epidemic spread highlighted by our analysis are compliant with results obtained in previous works through different assumptions. Specifically, if behavioral changes are fast enough, they can have a remarkable effect in reducing the daily prevalence of infection [41] and the final epi- demic size [16,38]. Moreover, for suitable parameter configura- tions, the epidemic dynamics becomes quite rich and can account for multiple epidemic waves, as shown in [40,49,65].

In addition, our analysis describes which are the main determi- nants of human behavior and risk perception leading to remark- able alterations of the dynamics of an emerging infectious disease. Our main findings can be summarized as follows. First, if the perceived risk associated to an epidemic is sufficiently large even a small decrease in the number of potentially infectious con- tacts can remarkably reduce the impact of an epidemic. Second, the disease spread results highly sensitive to how rapidly people adopt a self-reduction in their contact activity rates. However, when the mechanism regulating the spread of information about the disease is sufficiently fast, spontaneous social distancing is always effec- tive, especially when the disease is characterized by a large sym- ptomaticity. On the contrary, mechanisms of misperception of the risk of infection may essentially delay the epidemic spread.

In conclusion, accounting for spontaneous behavioral changes would be helpful for giving insights to public health policy makers, for planning control strategies and for better estimating the burden for health care centers over time. However, at the current stage, the proposed model could hardly be used for real time predictions. In order to gain a major consciousness on how such mechanisms work, further analysis may be devoted to the investigation of ‘‘real’’ epidemics, only rarely addressed in practice [50].

Acknowledgments

The authors thank Giuseppe Jurman (Bruno Kessler Foundation) for his useful comments that contributed to improve the presenta- tion of this manuscript. Additionally, the authors would like to thank two anonymous reviewers for their helpful comments and suggestions. This work has been partially funded by the EC-ICT contract No. 231807 (EPIWORK).

Appendix A. Supplementary data

Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.mbs.2012.04.003.

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