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The spread of awareness and its impact on epidemic outbreaks

Edited by Bryan Grenfell, Penn State University, Erie, PA, and accepted by the Editorial Board February 11, 2009 (received for review October 27, 2008)
Related Article
 In This Issue Apr 21, 2009
Abstract
When a disease breaks out in a human population, changes in behavior in response to the outbreak can alter the progression of the infectious agent. In particular, people aware of a disease in their proximity can take measures to reduce their susceptibility. Even if no centralized information is provided about the presence of a disease, such awareness can arise through firsthand observation and word of mouth. To understand the effects this can have on the spread of a disease, we formulate and analyze a mathematical model for the spread of awareness in a host population, and then link this to an epidemiological model by having more informed hosts reduce their susceptibility. We find that, in a wellmixed population, this can result in a lower size of the outbreak, but does not affect the epidemic threshold. If, however, the behavioral response is treated as a local effect arising in the proximity of an outbreak, it can completely stop a disease from spreading, although only if the infection rate is below a threshold. We show that the impact of locally spreading awareness is amplified if the social network of potential infection events and the network over which individuals communicate overlap, especially so if the networks have a high level of clustering. These findings suggest that care needs to be taken both in the interpretation of disease parameters, as well as in the prediction of the fate of future outbreaks.
Human reactions to the presence of disease abound, yet they have rarely been systematically investigated (1). Such reactions can range from avoiding social contact with infected individuals (social distancing) to wearing protective masks, vaccination, or more creative precautions. It has been shown, for instance, that local measles outbreaks are correlated with the demand for measles, mumps, and rubella vaccines (2). Similarly, the demand for condoms rises in areas where AIDS is prevalent (3), and condom use has been linked to the knowledge of someone who has died of AIDS (4).
Behavior that is responsive to the presence of a disease can potentially reduce the size of an epidemic outbreak. On closer inspection, it is not so much the presence of the disease itself that will prompt humans to change their behavior, as awareness of the presence of the disease. A change in behavior can be prompted without witnessing the disease first hand, but by being informed about it through others. This information in itself will spread through the population and have its own dynamic. For example, according to the Chinese Southern Weekend newspaper, the text message “There is a fatal flu in Guangzhou” was sent 126 million times in Guangzhou alone during the 2003 severe acute respiratory syndrome (SARS) outbreak (5), causing people to stay home or wear face masks when going outside. This figure stands in stark contrast to the comparatively low number of 5,327 cases recorded in the whole of China (6). It is not clear how much the individual behavioral responses contributed to containing the disease.
The spread of rumors has been described as “infection of the mind” (7) or “thought contagion” (8), and their spread is analogous to the spread of an infectious disease: information is passed on from carrier to carrier through a network of contacts. Therefore, when humans respond to the presence of a disease, we have a situation where an infectious agent and the information about the presence of this agent spread simultaneously, and will interact in their spread by a change in human behavior.
Here, we present a network model for the spread of awareness about a contagious disease. Awareness arises at the location of the disease and spreads among the population similarly to the way a disease would, an analogy that was suggested as early as 1964 (9). To capture the ephemeral nature of information, we implement an idea presented in ref. 10: as the information is passed from person to person, it loses its quality; in other words, firsthand information about a disease case will lead to a much more determined reaction than information that has passed through many people before arriving at a given individual.
Efforts to assess the potential for prevention of future outbreaks of contagious diseases have motivated previous studies on the effects of social distancing (11, 12) which, however, focused on behavioral changes imposed by a central organization on the population level. Attempts at extending this to incorporate individual behavioral reactions have focused on vaccination decisions and consequences thereof (13–16), dynamic rewiring of transmissive contacts (17), or incidencedependent reductions in contact rate (18).
In this study, we will investigate how the spread of awareness, prompted by firsthand contact with the disease, affects the spread of a disease. In this context, we understand awareness as the possession of information about the outbreak one is willing to act on as opposed to just generally knowing about the disease through media coverage or government programs without taking action. To study this, we have overlaid our model of information spread with a model for the spread of a contagious disease on two, not necessarily identical networks, with more informed individuals acting to reduce their susceptibility.
In the following, we will introduce the model and, in a first approximation, cast it into a system of ordinary differential equations under the assumption of random mixing of individuals within the population. This will allow us to show how awareness can reduce the number of individuals infected during an epidemic, while the threshold for disease invasion, and thus the potential for outbreaks, remains unchanged. Subsequently, we will consider a full spatial version of the same model. We will see that if the assumption of random mixing is lifted and the local nature of the interaction taken into account, locally spreading awareness can prevent a disease from breaking out, and how social network structure and overlap between the networks have an effect on this interaction.
The Model
We associate with each individual X in the population of size N a level of awareness indicated by an index i which denotes the number of passages the information has undergone before arriving at the given individual, i.e., X_{0} will stand for an individual with firsthand information and X_{i} for one with information that has passed through i other individuals before arriving at the given individual. The two transitions governing the information dynamics are information transmission (X_{i} + X_{j>(i+1)} → X_{i} + X_{i+1}) and fading of awareness (X_{i} → X_{i+1}). As the quality of information decreases at each transmission event while it is also gradually lost within each individual, information eventually disappears from the population if it is not refreshed.
We link this model to an epidemiological susceptibleinfectedrecovered (SIR) model (19), assigning each individual a diseaserelated state of susceptible (denoted S_{i}, the subscript i again representing the level of information), infected (I_{i}) or recovered (R_{i}) with the usual transitions of infection and recovery.
To capture the impact of individual actions, we make transmission of the disease dependent on the quality of the information available to a given susceptibility. The susceptibility of individuals in states S_{i} increases with i as (1 − ρ^{i}), 0 < ρ < 1. The decay constant ρ therefore governs how much the tendency to act is reduced with decreasing quality of information. The total amount of awareness in the susceptible part of the population g(ρ,{S_{i}(t)}) at any given time t can then be calculated as
We assume that information can be generated de novo if the disease is present, so we link generation of new information to a transition through which awareness about the disease is generated in infected individuals at rate ω. As the parameter ω thus reflects the likelihood per unit time of an infected individual to find out about their infection, it distinguishes between diseases with obvious and readily interpreted symptoms and cases where, for instance, the infection is contagious but asymptomatic, or where infection does not necessarily entail awareness about its nature (e.g., SARS, which may be mistaken for common flu). All the transitions and their respective rates are summarized in Table 1. There, and in the following, we denote with a hat per contact as opposed to populationlevel rates such that
MeanField Analysis
In the meanfield approximation, individual variables are replaced by population aggregates. By assuming random mixing and therefore ignoring any spatial structure within the population, we can describe the model system fully considering only the number of individuals in each possible state.
In the meanfield version of our model of information spread, the population is compartmentalized according to level of awareness, and the information dynamics for the part of the population at awareness level i is governed by
where
At any moment, awareness is then somehow distributed in the population, and this distribution changes over time according to the model dynamics. If new, and thus highquality information is introduced once in a population in which no or only lowquality information is available, this will initially spread to increase the total amount of information in the population, given by
By linking the model of information spread with the SIR model of the spreading disease, we obtain the full set of differential equations describing the interaction between the two processes (see SI Appendix). Now, a mutual feedback between information and disease emerges: higher prevalence of the disease entails more highly informed individuals, which in turn disseminate more information into the susceptible population, thereby impeding the further spread of the disease.
We can obtain a clearer picture of this interaction by summing the equations over the information states. In that case, the meanfield equations reduce to a form similar to the SIR equations, where β′(ρ,{S_{i}(t)}) = β · [1 − g(ρ,{S_{i}(t)})] reflects the current level of awareness within the susceptible population and can be interpreted as the effective rate of infection as part of the population is shielded by its awareness and the corresponding behavioral response. Since β′(ρ,{S_{i}(t)}) depends on the distribution of the S_{i}(t), this system is not closed, but it is still useful for understanding the behavior at the start of an outbreak. If at any time all susceptibles were maximally aware (S = S_{0}), β′ would be 0 and the disease would not spread at all, a situation that will never arise in the model because susceptibles can at best obtain S_{1} status if they are informed by infecteds with firsthand information (I_{0}). If, however, at any instant nobody is aware (S → S_{∞}), β′ becomes equal to β, and the model reduces to the conventional SIR model (see, e.g., refs. 19 and 20) with infection rate β and recovery rate γ. In the conventional SIR model, the epidemic threshold is at R_{0} = β/γ = 1, meaning that an initially low number of infecteds will increase if β > γ to cause an epidemic, whereas the disease will die out if β < γ.
Intriguingly, in this version of our model, the epidemic threshold does not change compared with the conventional SIR model if we start with a fully uninformed and susceptible population. In that case, awareness arises only through the process of information generation, coupled to the parameter ω and the number of infected I. This becomes relevant only once sufficiently many carry the disease, and only then is β′ reduced with respect to β. During the initial stages of the outbreak, however, β′ ≈ β, and the number of infected will always increase initially if β > γ. Only if a certain level of awareness were already present at the time t_{0} of the beginning of the outbreak, the threshold would be reduced to R_{0} = β′(ρ,{S_{i}(t_{0})})/γ.
Even with an unchanged epidemic threshold, the outbreak ceases to grow once S(t) = Nγ/β′(ρ,{S_{i}(t)}), which can be at a significantly lower level than the usual peak at S = Nγ/β, and, similarly, the final size of the epidemic can be much lower than without the effect of spreading awareness (see SI Appendix).
IndividualBased Analysis
The analysis presented in the previous section regarded the system at the population level under the assumption of random mixing, such that both the pathogen and the different levels of awareness were each distributed homogeneously within the population. In individualbased network models, however, each member of the population is embedded into a network of contacts and can infect others only over the connections of that network. In real social networks, mixing is far from random, and the number of connections each individual forms is limited and can vary significantly (21).
In a conventional SIR model, the infection events originating from a given infected individual are realized independently with identical probability T, and the average number of secondary individuals infected by a randomly chosen individual that has been infected is given by (20, 22–24)
where
Here, we will first consider the case where disease spreads locally, but information is disseminated globally, as in the case where awareness is triggered by information broadcast through the media. If the spread of information is welldescribed by the meanfield approximation presented above, we can assume that information quality is independent and identically distributed within susceptible contacts of infected individuals. In that case, the probability of infection at time t over a given link chosen at random is
where p_{i}(t) is the probability of the susceptible at risk of infection to possess information having gone through i hands at time t, and T_{i} is the probability of infection of that neighbor. If the distribution of awareness is already present at the time t_{0} of the beginning of an outbreak the basic reproductive number is reduced to
In the limit of random mixing of disease contacts, this reduces to
A completely different picture emerges if awareness, just like the disease, is not just globally present but spreads locally from individual to individual during the initial stages of the outbreak. Before we look at the full picture, let us assume for the moment that information transmission is only occurring between infected individuals informing their susceptible contacts, but that the information is not passed on any further, which could be regarded as analogous to singlestep contact tracing. In that case, the impact of awareness depends on the number of edges emanating from each node that are common to both networks. If we let (kc) denote the common degree, that is the number of contacts for possible disease transmission that are also information contacts, and (kd) the degree for contacts of disease transmission only, the reduced basic reproductive number is given by
where T_{∞} =
While the full expression for this critical
Even if ρ is large, information needs to be both generated and spread at a sufficiently high rate to have an effect on the disease outbreak. Given either the rate of information generation ω, or the rate of information spread
The full effect of the interaction between the two spreading processes comes into play when we let the information propagate independently without limiting the number of steps it can spread. In that case, there is a chance for an infected individual to have its susceptible contacts informed through others, and T′ can be further reduced. However, there remains a limit to the effect as the first upper bound on
A way to push that limit toward higher values of
The reduction in the basic reproductive number and its limits are clearly a consequence of the contactbased view, and they did not appear in the meanfield analysis. In fact, the meanfield limit of the full expression for
Network Overlap
The singlestep analysis presented in the previous section allowed for the two networks to be different, in that contacts of infectious individuals on the disease network that were not connected to the same individual on the information network were assumed to be completely unprotected. However, if awareness is allowed to spread for more than just one step, such missing links can partially be compensated for if there are other paths connecting an infected individual and its susceptible neighbor, i.e., if information links are clustered around the disease links.
Ultimately, the influence of spreading awareness on a disease outbreak depends on how much the individuals at the front of the growing epidemic are aware of its presence. Although the impact of heterogeneities in the degree distribution [including socalled scalefree network topologies (26)] can be captured in the factors D_{(kc)} and D_{(kd)}, other properties of the two networks and their relation to each other can have a strong impact on the containment of the disease. Going back to a deterministic description of the system, we can get some insight into the relevant processes and their dependence on network structure and overlap by considering the dynamics of the populationlevel variables in terms of pairs (27). Denoting the number of pairs of a given type on the disease network with […]^{d}, the equation for the number of infected individuals contains a term
which can be rewritten as
Here, p_{i}^{SI} = [S_{i}I]^{d}/[SI]^{d} represents the probability that the S member in a randomly chosen SI pair on the disease network to be at information level i, such that
Although it is not practical to derive an analytical expression for the behavior of p_{i}^{SI} in terms of network structure and overlap, we can measure it on simulated networks. In Fig. 3, one sees that if the information network is connected randomly and independently of the disease network, we obtain the meanfield situation where p_{i}^{SI} = p_{i}^{S}. If both networks are connected randomly but coincide, information is distributed more effectively and we observe a mild departure from equality of p_{i}^{SI} and p_{i}^{S}. A much more pronounced effect, however, can be observed if the two networks are triangular lattices, which contain many clusters, or triangles of connections. In that case, information is distributed much more effectively if the two networks coincide, resulting in significant correlation between risk of infection and information level, such that that much less total information is needed to protect the part of the population most at risk. Fig. 4 illustrates this effect, showing a snapshot of a simulated disease outbreak with awareness spreading on an triangular lattice completely overlapping with the disease network. Clouds of information have already formed around infected individuals, strongly limiting the further spread of the disease.
Discussion
On a social network, spreading awareness of a contagious disease in conjunction with a reduction in susceptibility does not only lower the incidence of that disease, but in some cases can even prevent that disease from growing into an epidemic. This is the case even if the awareness is not triggered by central information, but instead based on information that is passed on from person to person. However, beyond a critical infection rate, spreading awareness can slow down the spread of a disease and lower the final incidence, but it cannot completely stop it from reaching epidemic proportions and taking over large parts of the population. Only if the disease is easily recognized and information spreads rapidly, while at the same time there is a strong tendency toward protective behavior, awareness of a disease outbreak can bring the infection rate of a disease down significantly. If all of these factors work together, rapid drops in the transmissibility of a disease, as have been observed, for example, in the 2003 outbreak of SARS in Hong Kong (28), might be rooted in processes similar to the ones here presented.
Social network structure is found to play a significant role in the way spreading awareness and a contagious disease interact. The relative clustering of the information network around infectious individuals determines how effectively spreading awareness can constrain an epidemic outbreak. This effectiveness is significantly lowered when the network of disease spread differs from the communication network. This could be of relevance in the case of sexually transmitted disease, where a strong heterogeneity in the relevant network has been observed (29) and highly sexually active individuals are of crucial importance, yet do not necessarily find themselves in the same parts of the communication network as potential infectious contacts (e.g., sex workers might not communicate frequently with their customers). However, contact tracing programs work exactly to bring the two networks to match and can be seen as a special case of overlapping networks with just one step of information transmission.
Because the presence of a disease can change human behavior, care should be taken when trying to predict disease progression from behavioral observations in populations where the disease is not present (30, 31) or from observations on a different disease (32). Our model suggests how the interaction of social network structure with the properties of the disease induces a change in behavior in individuals and our results show how this could feed back to alter the disease dynamics.
Up to now, the effects of social distancing have predominantly been studied from a viewpoint of centrally controlled action. We argue that it is of equal importance to consider the selfinitiated reactions of individuals in the presence of a contagious disease. The model we analyzed here differs from the previous studies of the effect of social distancing in that we treat it as a local effect within the population which depends on the awareness of the social proximity of a disease. The importance of this is particularly relevant but not limited to cases like SARS in China where initially no information was made available by the governing bodies. Therefore, we think this can provide a valuable contribution to the ongoing discussion about the impact to be expected from social distancing in disease outbreaks to come.
Acknowledgments
This work was supported by UK Engineering and Physical Sciences Research Council Grant EP/D002249/1.
Footnotes
 ^{1}To whom correspondence should be addressed. Email: s.funk{at}rhul.ac.uk

Author contributions: S.F., E.G., C.W., and V.A.A.J. designed research; S.F., E.G., and V.A.A.J. performed research; S.F. analyzed data; and S.F. and V.A.A.J. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission. B.G. is a guest editor invited by the Editorial Board.

This article contains supporting information online at www.pnas.org/cgi/content/full/0808904106/DCSupplemental.
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