We investigate the effects of risk perception in a simple model of epidemic spreading. We assume that the perception of the risk of being infected depends on the fraction of neighbors that are ill. The effect of this factor is to decrease the infectivity, that therefore becomes a dynamical component of the model. We study the problem in the mean-field approximation and by numerical simulations for regular, random and scale-free networks. We show that for homogeneous and random networks, there is always a value of perception that stops the epidemics. In the ``worst-case'' scenario of a scale-free network with diverging input connectivity, a linear perception cannot stop the epidemics; however we show that a non-linear increase of the perception risk may lead to the extinction of the disease. This transition is discontinuous, and is not predicted by the mean-field analysis.
Deep Dive into Risk perception in epidemic modeling.
We investigate the effects of risk perception in a simple model of epidemic spreading. We assume that the perception of the risk of being infected depends on the fraction of neighbors that are ill. The effect of this factor is to decrease the infectivity, that therefore becomes a dynamical component of the model. We study the problem in the mean-field approximation and by numerical simulations for regular, random and scale-free networks. We show that for homogeneous and random networks, there is always a value of perception that stops the epidemics. In the ``worst-case’’ scenario of a scale-free network with diverging input connectivity, a linear perception cannot stop the epidemics; however we show that a non-linear increase of the perception risk may lead to the extinction of the disease. This transition is discontinuous, and is not predicted by the mean-field analysis.
In spring 2006, the potential threat of bird flu dominated headlines in UK newspapers. On 26 March 2006 The Sun has called it "the day we all dreaded", while the Guardian says avian flu is "almost certain to spread to wild birds across the UK". The Daily Telegraph adds that the most likely human victims will be poultry farmers, who will be bankrupted. But the Mirror calls for calm, saying people have a better chance of winning the lottery than catching the virus. Interestingly, given a certain amount of clustering of wealth residents and correlation between wealth and readers preference, this would translate into a differently informed neighborhood. When the epidemic is over its peak or other news has just peaked or media has "cried wolf" too many times over unfounded health scares, there is a quick drop in the attention to that disease (something similar is reported nowadays for HIV). In other parts of the world, for example Indonesia, a country with 18000 islands, people reacted differently to the bird flu epidemics. Despite awareness campaigns in the media and even door-to-door visits in some of the islands, many Indonesians remained oblivious to the dangers of being in contact with diseased birds, and aware of the need to inform the authorities and implement a cull. Note that awareness campaigns, such as during the SARS epidemics, are expensive and may result in culling, reductions in commerce, travels and tourism. The media hype over epidemics threat has a close sim- * Electronic address: franco.bagnoli@unifi.it ilarity in how worried or fatalist, resilient, skeptical or cheeky may be friends and neighborhood. Therefore, the individual perception of the risk of becoming infected is a key factor influencing the spreading of an epidemics and, toward realistic inference, epidemiological models should incorporate such parameter [1].
In order to investigate the effect of risk perception in influencing the spreading of a disease, let us start from simple, yet meaningful models, such SIS or SIR ones. These models are defined on a network where individuals or groups of individuals corresponds to the nodes and links represent social contacts and relationships among them. Most of classical studies used either a regular lattice, or a random one. Both of those choices are characterized by a well defined value of the mean connectivity k , and small variance k 2 -k 2 . As shown by Watts and Strogatz [2], the simple rewiring of a small fraction of links in an otherwise regular lattice results in a sudden lowering of the diameter of the graph, without affecting the average connectivity or the degree of clustering. This small world effect manifests itself in a dramatic shortage of the distance between any two individuals, almost without affecting the local perception of the network of contacts. The consequences for epidemics spreading are important: just a few long-distance connections may promote the spreading of a disease in rural areas, whereby an epidemic would otherwise diffuse very slowly.
However, the investigations of social networks have shown that they are quite different from regular and random graphs [3,4]. The probability distribution of contacts often exhibits a power-law behavior (P (k) ∝ k -γ ), with an exponent γ between two and three [5,6]. This distribution is characterized by a relatively large number of highly connected hubs, which are presumably responsible for epidemics spreading. Moreover, such distributions have a diverging second moment k 2 for γ ≤ 3 and a diverging average connectivity k for γ ≤ 2.
The influence of the connectivity on the spreading dynamics is well outlined by a simple mean-field analysis. Let us consider for the moment a tree with fixed connectivity k. In a SIS model with immediate recovery dynamics, a single infected individual may infect up to k neighbors [7], each one with probability τ . The temporal behavior of the mean fraction c of infected individuals is given by
where c ≡ c(t), c ′ ≡ c(t + 1) and the sum runs over the number s of infected individuals. The basic reproductive ratio R 0 [8] is simply given by R 0 = kτ , so that the epidemic threshold R 0 = 1 corresponds to τ c = 1/k. This means that for a fixed connectivity, only diseases with an infectivity less than 1/k do not spread.
In heterogeneous networks (nodes with different connectivity) the mean field analysis, reported in Section III, gives
In summary, the result is that on very non homogeneous networks, with diverging second moment k 2 (and even worse on those with diverging average connectivity k ), a disease will always spread regardless of its intrinsic morbidity [9].
This result can be modified by the assortativity degree of the network and by the presence of loops, not considered in tyhe mean-field analysis. In networks with assortative connections (hubs are preferentially connected to other hubs), it may happen that epidemics spread for any finite infectivity even when the second moment is not
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