This interpretation suggests a relatively simple way to compute the whole distribution of the PR. In principle once the matrices of the Laplacian operator and the potential operator are known, the ψ (and henceforth the set of PR values) could be computed by inverting these operators. This simple operation is unfeasible when the size of the matrix is of the order of tens of billions of pages as in the WWW. Here we adopt a different approach based on a matrix expansion that can be also extended to study the
time evolution. The idea is to rewrite the above equation by using the common Taylor expansion, while starting from the equation (4.5):
ψ= I − V−1∇2D−1
V−1F. (4.7)
We now expand the expression in brackets by writing I− V−1∇2D−1 = ∞ X n=0 V−1∇2Dn (4.8) provided all the eigenvalues λh of V−1∇2D
have |λh| < 1.
This allows to invert only the diagonal matrix V (that can be done easily by taking the inverse of the elements on the diagonal). The expression above can be rewritten as
ψ= kO− αAT−1F0 = (I − αB)−1 kO−1F0 (4.9) where F0= αF, kO is a matrix whose elements are all zero apart on the diagonal where they are given by the outdegree of vertices, AT is the transpose of the adjacent matrix and B = kO−1
AT.
Equation (4.9) closely resembles the original equation for PR, with the important caveat that we are now working with a wave function ψ. In this case, the expansion:
(I − αB)−1= ∞ X
n=0
(αB)n (4.10)
converges and we can calculate with the desired precision ψ and so the associated PR. The results of this matrix expansion are in good agreement with the solution obtained by traditional methods. One can increase as desired the order of the expansion with a computational cost that increases only linearly with the order.
Behavioral Changes
It’s not that I’m afraid to die, I just don’t want to be there when it happens. W. Allen Contents
5.1 Fear of the sick . . . 90 5.2 Self-reinforcing fear . . . 93 5.3 Mass-media effect . . . 99
Human behavior has long been recognized as one of the key points in understanding epidemic spreading (106), leading to a concerted effort to include social complexity in epidemiological models. Age structure (107), air line travel (108; 109) and commuting are now incorporated in all realistic models (21). However, much remains to be done. The recent H1N1 pandemic has demonstrated the feasibility and reliability of epidemic forecasting in real time (16; 110), but it has also brought to light the limitations under- lying the state of the art of epidemic models (17). In particular, it has become clear that societal reactions can have an important impact on epidemic spreading (111). These reactions can be classified into different classes. In the first, changes are imposed by authorities through the closure of schools, churches, public offices and bans of public gatherings (112; 113). In the second, individuals decide to modify their behavior due to concern about a disease, by avoiding social contacts with infected individuals and
crowded spaces, reducing traveling or preventing children from attending school. In both cases we will have a modification of the spreading process due to the reduction of contacts in the population. In general, the result of these measures is very important. A reduction of the epidemic outbreak and a delay of the epidemic peaks are possible outcome. For these reasons social distancing policies are crucial measures during serious epidemic spreading.
Many studies have been done in order to evaluate the impact and role of organized public health measures in real epidemics (112; 114; 115). Instead just a few, recent, attempts have considered spontaneous social distancing phenomena. In some approaches individ- ual behaviors were modeled by introducing different contact rates (normal or altered) in response to the state of disease (116; 117), in others new compartments representing individual states or levels of self-imposed isolation were proposed (118) while in others the spreading of awareness was coupled with the disease (119). However, there is no consensus on how spontaneous social distancing is related to the perceived current state of the epidemic. The definition of a general model is still an open issue.
In this study we propose a general framework to model the spreading of awareness and social distancing in a single population. We modify the classical SIR model (79) by introducing a new compartment, SF, that represents susceptible people aware of an infectious disease. These people decide to reduce their number of contacts as a way of trying to reduce the likelihood of becoming infected. We modeled the spread of epidemic awareness within the population considering different mechanisms. We related the awareness to the state of the epidemic at a given time, to the number of scared people (through “fear contagion” (120)) and to the information that spreads by the media (121). In this Chapter we present a complete survey of these different processes, their implementation and the analysis of their main features.
5.1
Fear of the sick
The first model we considered, is a generalization of the SIR model, on a single pop- ulation, that includes a new compartment of susceptibles: SF 1. Individuals in this compartment are more careful about their contacts, thus reducing the contagion rate
β → rββ with (0 ≤ rβ < 1). Normal susceptible people reach the SF compartment (be- come feared) after interacting with infected people in compartment I. In this case, fear is generated by the presence of infected persons in the community. This process can be considered as a parallel contagion process. By analogy we defined the reproductive number for the fear as:
RF = βµF
F. (5.1)
People can recover from fear and return into the susceptible compartment by interacting with recovered people, R, and other susceptibles, S, with a constant rate µF. People stop being afraid after seeing that not many people are affected by the disease and that the ones that were infected are now recovered. The full epidemic model is then described by the following set of equations:
dtS(t) = −βS(t)I(t) N − βFS(t)I(t) N + µFSF(t) S(t) + R(t) N , dtSF(t) = −rββSF(t)I(t)N + βFS(t)I(t)N − µFSF(t) S(t) + R(t) N , dtI(t) = −µI(t) + βS(t)I(t)
N + rββSF(t)I(t)N , dtR(t) = µI(t).
It is important to stress that we considered a process in which: X
i
dtXi(t) = 0 for ∀ t and Xi ∈ S, SF, I, R , (5.2) meaning that the total number of individuals in the population does not change. In diseases like flu, the time scale of the spreading is very small on respect to the average life time of a person. This allowed us to ignore birth or death processes. The dynamics take place with a fix number of individuals. The flows between compartments balance each others. For each negative term there is another one, equal, but with a positive sign. To explain the equations we can just consider negative terms. In particular: in the first equation in the (5.2) the first term takes into account individuals in the susceptible compartment S that interacting with infected individual become sick:
The second term takes into account individuals in the susceptible compartment S that interacting with infected individuals become scared by disease:
S + I βF
−−→ SF + I. (5.4)
The first term of the second equation takes into account individuals in the compartment SF that interacting with infected individuals get sick:
SF + I rββ
−−→ 2I. (5.5)
This happen with a rate rββ < β because, as we said, people aware of the disease reduce their contacts. The last term in the second equation takes into account people in the compartment SF that interacting with healthy individuals, S, and recovered ones, R, stop to be aware and move back in the compartment S:
SF + S µF
−−→ 2S, (5.6)
and
SF + R−−→ S + R.µF (5.7)
The first term in the third equation takes into account the spontaneous recovery of sick individuals:
I −→ R.µ (5.8)
When the disease spreads much faster than public awareness, the model reduces to the classical SIR, with basic reproductive number, R0 = β/µ. In this limit, the early time of the compartment SF is given by (assuming SF
t=0≡ 0): SF(t) ∼ βF
µ(R0− 1) + µF e
µ(R0−1)t− e−µFt . (5.9) It is clear a transition between two regimes. For
µ(R0− 1) > µF, (5.10)
the rate of increase of the fear is governed by R0, otherwise fear dies out: the rate of fear production it is not enough to sustain it. However, when panic spreads faster than the disease, RF RSIR0
everyone quickly becomes scared and our model reduces to an SIR model with a reduced reproductive rate RF
0 = rββ/µ, dominated by the character- istics of the SF compartment. We explored numerically the intermediate regime between
these two limits. For small values of RF the presence of fear does not significantly affect the timing of the disease, as showed in Figure (5.1). It simply produces a mild reduction on epidemic size. Around the 20% for rβ = 0 and RF = 5 (see Figure (5.2)).
Increasing the value of RF results in two different scenarios:
1. rββ/µ > 1 the reduction of epidemic size is bounded to the value of an SIR model with β → βrβ;
2. rββ/µ < 1 fear completely stops the progression of the disease.
After the end of the epidemic, the system enters in the so called disease-free equilibrium. In the phase space this is describe by:
(S∞, S∞F, I∞, R∞) = (1 − R∞, 0, 0, R∞). (5.11) From the equations above, it is easy to show that fear disappears exponentially:
SF(t) ∼ e−µFt. (5.12)
There is not possibility of an endemic state of fear. Fear can only be produced by the presence of infected people, as soon the infection dies, scared people can recover from fear by interacting with all the susceptible and recovered becoming susceptible themselves.