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29.05.2018 9_epidemics

file:///home/szwabin/Dropbox/Zajecia/Diffusion/Lectures/9_epidemics/9_epidemics.html 1/22

Diffusion processes on complex networks

Lecture 9 - epidemic processes on networks

Janusz SzwabińskiOutlook:

1. Introduction2. SI model on a network3. SIS model on a network4. SIR model on a network5. Contact networks6. Beyond the degree distribution7. Immunization8. Real-time forecast (attempts)

Further reading:

A.-L. Barabasi, Network ScienceR. Pastor-Satorras, C. Castellano, P. Van Mieghem and A. Vespignani, Epidemic processes incomplex networks, https://arxiv.org/abs/1408.2701 (https://arxiv.org/abs/1408.2701)

Introductionclassical models of epidemics do not incorporate the structure of the contact network that facilitatesthe spread of a pathogen

any individual can come into contact with any other individuals (homogenous mixinghypothesis)all individuals have comparable number of contacts (i.e. probability of getting infected)

both assumptions are falseindividuals can transmit a pathogen only to those they come into contact withpathogens spread on a complex contact network

contact networks are often scale-free, hence is not sufficient to characterize their topologythe failure of the basic hypotheses prompted a fundamental revision of the epidemic modelingframeworkPastor-Satorras and Vespignani, 2001

extension of basic models to complex networks

SI model on a networkIf a pathogen spreads on a network:

individuals with more links are more likely to be in contact with an infected individualthey are more likely to be infectedmathematical formalism must consider the degree of each node as an implicit variable

⟨k⟩

⟨k⟩

https://arxiv.org/abs/1408.2701



Degree block approximationnodes with the same degree are statistically equivalent

Let us denote with

the fraction of nodes with degree that are infected among all -degree nodes in the network. The totalfraction of infected nodes is then given by

Given the different node degrees, we write the SI model for each degree separately:

Here:

is the number of susceptible nodes with degree the infection rate is proportional to and the fraction of nodes with the degree that are notinfectedthe density function represents the fraction of infected neighbors of a susceptible node withdegree (in homogenous mixing we have )

coupled equations

=ikIk

Nk

k Nk k

i = ∑k

ik pk

= β(1 − )kdik

dtik Θk

(1 − )ik kβ k

Θk

k k = iΘk

kmax



Density function provides the fraction of infected nodes in the neighborhood of a susceptible node with the degree . We

must first determine it in order to calculate .

If there are no correlations between degrees in the network, the probability that a link points from a node withdegree to a node with degree is independent of . Hence the probability that a randomly chosen linkpoints to a node with degree is given by

It is sometimes called the excess degree, i.e. the probability that at the end of a randomly chosen link thereis a node with degree .

At least one link of each infected node is connected to another infected node, the one that transmitted theinfection. Therefore the number of links available for future transmissions is

Hence

In the absence of degree correlations the density function is indeed independent of . Differentiating thelast expression we obtain

Combining the last expression with the equation for the SI model itself, we get

To predict the early behavior of the epidemics, we consider the fact that for small the fraction of infectedindividuals is much smaller than one. Therefore we can neglect the corresponding terms in the equation for ,

The solution of the last equation is given by

where

If initially nodes are infected uniformly (i.e. for all ), then

We obtain:

Θ kik

k k′ kk′

=k′pk′

k∑k pk

k′pk′

⟨k⟩

k′

− 1k′

=Θk

( − 1)∑k′ k′ pk′ ik′

⟨k⟩

Θk k

=dΘ

dt∑

k

(k − 1)pk

⟨k⟩

dik

dt

= β (1 − )ΘdΘ

dt∑

k

( − k)k2 pk

⟨k⟩ik

tΘ

= β( − 1)ΘdΘ

dt

⟨ ⟩k2

⟨k⟩

Θ(t) = Cet/τ

τ =⟨k⟩

β (⟨ ⟩ − ⟨k⟩)k2

(t = 0) =ik i0 k

C = Θ(t = 0) = i0⟨k⟩ − 1

⟨k⟩



Let us come back now to the model,

At the beginning of the epidemics is small and the term can be neglected. Hence

Inserting the result for gives

where is the characteristic time for the spread of the pathogen,

Integrating the equation for we obtain the fraction of infected nodes with degree ,

the higher the degree of a node, the more likely it becomes infected, because

Θ(t) = i0⟨k⟩ − 1

⟨k⟩et/τ

= β(1 − )kdik

dtik Θk

ik β kik Θk

≃ βkdik

dtΘk

Θ

≃ βkdik

dti0

⟨k⟩ − 1

⟨k⟩et/τ SI

τ SI

=τ SI ⟨k⟩

β (⟨ ⟩ − ⟨k⟩)k2

ik k

= [1 + ( − 1)]ik i0k (⟨k⟩ − 1)

⟨ ⟩ − ⟨k⟩k2et/τ SI

(t) = g(t) + kf(t)ik



characteristic time depends not only on , but also on the network's degree distribution through

Characteristic times

Random network

Scale-free network with

and are finite is finite as well

spreading dynamics is similar to the behavior predicted for a random network, but with an altered

Scale-free network with

the spread of a pathogen is instantaneous!!!vanishing reflects the role of hubs in epidemic phenomena

the hubs are first to be infected

Summary

the degree heterogeneity affects only the characteristic time

τ ⟨k⟩⟨ ⟩k2

⟨ ⟩ = ⟨k⟩ (⟨k⟩ + 1)k2

=τ SIER

1

β⟨k⟩

γ ≥ 3

⟨k⟩ ⟨ ⟩k2

τ SI

τ SI

γ < 3⟨ ⟩ = ∞lim

N→∞k2

→ 0τ SI

τ SI



SIS model on a networkIn this case we have

The derivation of the density function is similar to the SI model, with one important difference:

in the SI model, if a node is infected, then at least one of its neighbors must be infected at most of its neighbors are susceptible

in the SIS model, the previously infected neighbor can become susceptible againall links of a node can be available to spread the disease

Hence

Again, keeping only first order terms we obtain

Multiplying the equation with and summing over we get

This equation again has the solution

with

A global outbreak is possible if , which yields the condition for an outbreak

and the epidemic threshold

= β(1 − )k − μdik

dtik Θk ik

Θk

→k − 1

k

=Θk

∑k′ k′pk′ ik′

⟨k⟩

= βkΘ − μdik

dtik

k /⟨k⟩pk k

= (β − μ)ΘdΘ

dt

⟨ ⟩k2

⟨k⟩

Θ(t) = Cet/τ SIS

=τ SIS ⟨k⟩

β⟨ ⟩ − ⟨k⟩μk2

> 0τ SIS

λ = >β

μ

⟨k⟩

⟨ ⟩k2

=λc

⟨k⟩

⟨ ⟩k2



Characteristic time

Random networks

The condition yields the following epidemic threshold

is always finitenon-zero epidemic threshold (always)

if , the pathogen will spread until it reaches an endemic state

if , the pathogen dies out

Scale-free networks

if is finite, spreading dynamics similar to that on a random networkif diverges

the epidemic threshold expected to vanish (direct consequence of hubs)even viruses that are hard to pass from individual to individual can spread successfully

⟨ ⟩ = ⟨k⟩ (⟨k⟩ + 1)k2

=τ SISER

1

β (⟨k⟩ + 1) − μ

τ SISER

=λc

1

⟨k⟩ + 1

⟨k⟩

λ = >β

μ λc

λ < λc

⟨ ⟩k2

⟨ ⟩k2



SIR model on a networkIn the SIR model the density of infected nodes with degree follows the equation

Again, keeping only first order terms (i.e. ignoring and in the parenthesis above) we obtain

Multiplying this equation with

and summing over we get

The solution is

with

Outbreak is possible if . This yields the condition for a global outbreak as

This allows us to write the epidemic threshold for the SIR model in the form

k

= β(1 − − )k − μdik

dtik rk Θk ik

ik rk

≃ βk − μdik

dtΘk ik

(k − 1)pk

⟨k⟩

k

= (β − μ)ΘdΘ

dt

⟨ ⟩ − ⟨k⟩k2

⟨k⟩

Θ(t) = Cet/τ SIR

=τ SIR ⟨k⟩

β⟨ ⟩ − ⟨k⟩(β + μ)k2

> 0τ SIR

λ = >β

μ

⟨k⟩

⟨ ⟩ − ⟨k⟩k2

=λc

1

− 1⟨ ⟩k2

⟨k⟩



Contact networks

Sexual contact network in Sweden

Romantic links in a high school



Air transportation network



Face-to-face network in a school



Beyond the degree distributionour results indicate that the speed with which a pathogen spreads, depends on the degreedistribution of the relevant contact networkindeed, affects both the characteristic time and the epidemic threshold however, real networks have a number of characteristics that are not captured by alone:

temporal structure

bursty contact patterns

degree correlationslink weightscommunities

⟨ ⟩k2 τ λc

pk



Immunization

Immunization is the process by which an individual's immune system becomes fortifiedagainst an agent (known as the immunogen).

When this system is exposed to molecules that are foreign to the body, it will orchestrate animmune response, and it will also develop the ability to quickly respond to a subsequentencounter because of immunological memory. Therefore, by exposing an animal to animmunogen in a controlled way, its body can learn to protect itself (active immunization).

The most important elements of the immune system that are improved by immunization arethe T cells, B cells, and the antibodies B cells produce. Memory B cells and memory T cellsare responsible for a swift response to a second encounter with a foreign molecule. Passiveimmunization is direct introduction of these elements into the body, instead of production ofthese elements by the body itself.

Goals:

to protect the immunized individual from an infectionto reduce the speed with which a pathogen spreads in a population

Consider a situation when a randomly selected fraction of individuals is immunized in a population. Let usassume that the pathogen follows the SIS model.

The immunized nodes are invisible to the pathogen, and only the remaining fraction of nodes cancontact and spread the disease. As a consequence the effective degree of each susceptible node changes,

Hence

Random networksFor sufficiently high the spreading rate could fall below the epidemic threshold. The immunization rate

necessary to achieve this can be calculated from

Hence

Thus, immunization may indeed lead to dying out of the pathogen.

Heterogenous networks

If the pathogen spreads on a network with , then

g

(1 − g)

⟨k⟩ → ⟨k⟩(1 − g)

λ = → = λ(1 − g)β

μλ′

g λ′

gc

=(1 − )βgc

μ

1

⟨k⟩ + 1

= 1 −gc

μ

β

1

⟨k⟩ + 1

⟨ ⟩k2

(1 − ) =β

μgc

⟨k⟩

⟨ ⟩k2



Hence

For a scale-free network with we have

virtually all nodes have to be immunized to stop the epidemicconsistent with data

measles requires 95% of population to be immunizeddigital viruses - random strategies require almost 100% of antivirus software installations

Strategies for scale-free networksimmunization of hubs

smaller variance, i.e. network fragmentationproblem - we usually do not know the hubs!!!

selective immunizationfriendship paradox - on average the neighbors of a node have a higher degree than thenode itselfto immunize the population:

1. Choose randomly a fraction of nodes (Group 0).2. Select randomly a link for each node in Group 0 Group 13. Immunize individuals from Group 1

requires no information about the global structure of the network

= 1 −gc

μ

β

⟨k⟩

⟨ ⟩k2

γ < 3⟨k2

gc

⟩ → ∞

→ 1

⟨ ⟩k2

→



Real-time forecastduring much of its history humanity has been helpless when faced with a pandemiclacking drugs and vaccines, infectious diseases repeatedly swept through continents, decimatingthe world's populationthe first vaccine was tested only in 1796the systematic development of vaccines and cures against new pathogens became possible only inthe 1990swe still have effective vaccines only against a small number of pathogenstransmission-reducing and quarantine-based measures remain the main tools of healthprofessionals in combatting new pathogensfor the combination of vaccines, treatments and quarantine-based measures to be effective, weneed to predict when and where the pathogen emerges next, allowing local health officials to bestdeploy their resourcesthe real-time prediction of an epidemic outbreak is a very recent developmentthe ground was set by

the development of the epidemic modeling framework in the 1980sthe 2003 SARS epidemic, which resulted in worldwide reporting guidelines about ongoingoutbreaks

availability of data pertaining to a pandemic offers real-time input to modeling effortsthe 2009 H1N1 outbreak was the first was the first pandemic whose spread was predictedin real time

the emergence of any new pathogen raises several key questions:Where did the pathogen originate?Where do we expect new cases?When will the epidemic arrive at various densely populated areas?How many infections are to be expected?What can we do to slow its spread?How can we eradicate it?

GLEAM

In [1]:

from IPython.display import HTML



In [2]:

%%HTML

<video width=800 controls>

<source src="GLEAMviz.mp4" type="video/mp4">

</video>

0:00 / 2:03



Global Epidemic and Mobility (GLEAM) computational modelhttp://www.gleamviz.org/ (http://www.gleamviz.org/)desktop application available

The population layer: census data and population densities

population data from the websites of the Gridded Population of the World and the Global Urban-Rural Mapping projects, which are run by the Socioeconomic Data and Application Center (SEDAC)of Columbia University

the world is divided into a grid of cells and assigned an estimated population valueGLEAM uses cells that are approximately 25 x 25 km, dividing the globe into over 250,000populated cells

coordinates of each cell are knowncoordinates of all commercial airports in the World Airport Network known as well

by considering the distance between the cells and airports, each cell assigned to a ‘local’airportover 3,300 subpopulations, each centered at a local transportation hub

The mobility layer: commuting and flight patterns

Airport network

12 different flight networks (one for each month) derived from the worldwide booking datasets fromthe Official Airline Guide (OAG) database

more than 3,800 commercial airports in about 230 countries4,000,000 connections representing the estimated bookings between any two of theseairports for each month

http://www.gleamviz.org/



airport network data reveals significant variations in both the number of destinations per airport andin the number of passengers per connection

some airports with lots of connections and large volumes (hubs)many airports with few connections and low volumes

Commuting network

information obtained from the national statistics offices of more than 40 countries in five continentsmore than 78 000 administrative regionsdue to different semantics and organizational structures with varying degrees of detail, the datahave been standardized before being integratedover five million commuting connections between GLEAM’s geographic subpopulations, capturingthe irregular network structure that affects the local diffusion of infections between neighboringsubpopulations

Disease dynamics layer

the infection dynamics is simulated according to the characteristics of the disease coupled with anyprevention and intervention measuresdisease characteristics:

incubation timesthe proportion of asymptomatic yet infectious individualsmortality ratesimmunity

'compartmental model'each individual fits, at any given point in time, within a certain ‘compartment’ thatcorresponds with a particular disease-related statecompartments are connected by paths that define how individuals may pass from one stateto another



H1N1 pandemics (2009)



GLEAMviz predictions

Peak Timeit corresponds to the week when most individuals are infected in a particular countrypredicting the peak time helps health officials decide the timing and the quantity of thevaccines or treatments they distributethe peak time depends on the arrival time of the first infection and the demographic and themobility characteristics of each countrythe observed peak time fell within the prediction interval for 87% of the countriesin the remaining cases the difference between the real and the predicted peak was at mosttwo weeks

Early PeakGLEAM predicted H1N1 epidemic will peak out in November, rather than in January orFebruary (the typical peak time of influenza-like viruses)this unexpected prediction turned out to be correct, confirming the model’s predictivepowerthe early peak time was a consequence of the fact that H1N1 originated in Mexico, ratherthan South Asia (where many flu viruses come from)

it took the virus less time to arrive to the northern hemisphereThe Impact of Vaccination

several countries implemented vaccination campaigns to accelerate the decline of thepandemicthe simulations indicated that these mass vaccination campaigns had only negligibleimpact on the course of the epidemicthe reason is that the timing of these campaigns was guided by the expectation of aJanuary peak time, prompting the deployment of the vaccines after the November 2009peak, too late to have a strong effect

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