Epidemical Models

University of Athens, Pervasive Computing Research Group

p-comp@di.uoa.grChristos Anagnostopoulos

Epidemics

Viruses and the dynamics of epidemics are simple. They spread widely and resiliently, without central control

Gossip in social interactions spreads well, without central control

– Build computer systems by analogy with epidemics. Gain fault-tolerance without increased design complexity [Demers et al. '87]

Information Dissemination Problem

One node x possesses a piece of information.

How to spread it to all n nodes quickly and reliably?

Broadcast Tree

Nodes organize themselves into broadcast tree.Messages are sent along edges of tree.

Not very fault-tolerant. And how is the tree constructed?

Uniform Gossip

In each round, each node calls a uniformly selected node and forwards all information it has.

Dynamics•O(logn) to reach everyone•O(logn) to reach an individual•Fault-tolerant

Probability ~ n-1

What if message were an alarm?Locality and Topology is often important!

Flooding

Each node forwards information to its neighbors only, in a round-robin fashion.

Dynamics•O(d) to reach an individual •O(sqrt(n)) to reach everyone•Not so fault-tolerant•Message is re-circulated

Probability = 1

Exponentially worse than uniform gossip

Spatial Gossip

Spatial Gossip with exponent p. Each round, each node x calls node y (at distance d) with:Probability ~ d-p

Spatial gossip achieves dissemination time polynomial in log(d) With high probability, node x receives information from node y within O(log1+ed)

rounds.

Spatial gossip has desirable features of flooding and uniform gossip.Is there an inherent tradeoff between speed and locality?

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Epidemical Model Dynamics

– A population of individuals, M– States (Infected, Susceptible, Immune, Recovered, Exposed)– State transition probabilities (infection / birth rate β, recovery / cure rate δ)– Epidemical threshold (endemic, pandemic, epidemic), λ = β/δ– Fraction of infected individuals, p(t)

Network Topology (homogeneous / scale-free networks) Semantics (selectivity attribute) Reliability or completeness, R: The proportion of infected nodes out of the

population Efficiency: Let L be the imposed load to the network denoting the number

of contacts established per node i.e., number of messages on average each node sends to its neighbors in order to propagate an epidemic.

– The efficiency E of the whole process is represented by the number of nodes which get infected per unit load, i.e., E = R/L

Epidemical Model

States: Susceptible - Exposed – Infected – Recovered (SEIR)

– There is a period of time during which the individual has been infected but is not yet infectious

States: Susceptible – Infected - Carrier / Recovered (SICR)

– An infectious individual is never completely recovered and continues to carry the infection, whilst not suffering the disease themselves (e.g., tuberculosis)

S I RE

S I R

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SIR: Epidemical Model

A population of individuals, M States (Infected, Susceptible, Immune, Recovered, Exposed) State transition probabilities (infection rate / birth rate β, recovery / cure rate δ) Epidemical threshold (endemic, pandemic, epidemic), λ = β/δ Fraction of infected individuals, p(t)

The epidemic stops when the number of susceptible individuals, S(t), drops.

S I R

time (t)

p(t)

β δ

dS(t)/dt = -βS(t)I(t)

dI(t)/dt = βS(t)I(t)-δI(t)

dR(t)/dt=δI(t)

S(t) I(t)

R(t)

SIS: Epidemical Model States: Infected – Susceptible – Infected (SIS) State transition probabilities: infection rate β, recovery rate δ

Susceptible and infected individuals get equilibrated (e.g., common cold)

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δLet p(t)=I(t)/(S(t)+I(t)),

dp(t)/dt = βp(t)(1-p(t))-δp(t)

The solution is:

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t)(e)p(p)(p

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Network Topologies and Epidemiology

Erdos-Renyi (ER): Degree distribution follows a Poison distribution: Vertices connected with probability p

kVk ppk

VkobPr

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The probability of having a node connected to another spatially contiguous node is the same as that of having the node connected to a far node: there is no local neighbourhood

The epidemic threshold tends to increase as the average degree of the vertices is lowered.

The epidemic threshold is higher compared toboth the ER and the homogenous graphs [Kephart, 91]

Average degree <k> = 3

Spatial model: Regular lattice with local neighborhood connection

Community of level = 3

2D lattice: Infection is quadratic

Network Topologies and Epidemiology

Small World: Any pair of nodes can be connected through a small number of intermediate nodes (rewiring with p)

Small value of the average path length and clustering coefficient grater than ER, lesser than lattice

Clustering coefficient indicates if neighboring nodes tend to connect to the same neighbors

Degree of k-distribution

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Small World: Exponential Networks, i.e., Prob(k) is exponentially bounded.

1. Broad scale network (power-low /exp. Distr.)2. Single scale network (Watz-Strogatz, WS)

Average path length logarithmically increases with population

Local structure (neighboring nodes share many common neighbors)

Each node has at least same node degree

3. Scale free network (Barabasi-Albert, BA) Incremental growth, preferential

connectivity, i.e., newly added nodes tend to get connected with existing nodes that already have a high number of connections (e.g., hubs)

Network Topologies and Epidemiology

BA model assumes connectivity distribution with γ = 1 2k~kobPr

Each node has a statistically significant probability of having a very large number of connections compared to the average connectivity <k>

Examples of scale-free networks

1. Social networks, including collaboration networks.2. Protein-Protein interaction networks.3. Sexual partners in humans, which affects the dispersal of sexually transmitted disease4. Computer networks, including the WWW (γ=1)

Small World: The existence of small number of highly connected nodes

A relatively small average distance from any two nodes

Random Network Scale-free network

Small world Scale-free

Epidemiology in Random Networks

The number of infected nodes does not significantly affect the probability of infection

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δLet p(t)=I(t)/(S(t)+I(t)),

dp(t)/dt = β(1-p(t))-δp(t)

The prevalence at equilibrium is:

1-δ1-β

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ER network with 900 nodes.

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Epidemiology in Random Networks

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Survival probability: The fraction of the infected nodes decreases.

Epidemical threshold λ < 1The higher the nodes connectivity the

lower the threshold

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Epidemiology in Small Word Networks

Survival probability: An outbreak will occur whatever the value of λ. The prevalence probability is lower than that in a Homogeneous network.

For λ > 0, a fraction of susceptible nodes will be infected showing the absence of the epidemical threshold.

me~tp 1

0 < γ <= 1: No epidemical threshold. 1 < γ < 2:Threshold reappears with a vanishing

slope. The connectivity distribution becomes bounded as in WS, ER model.

γ >=2: The usual critical behavior at the threshold The prevalence probability is independent from the

network size The outbreak is worst when the starting node is not

necessarily the most connected but the one in the best location (it has many highly connected neighbors).

The starting node can affect the survival probability in the short term, but the system tends to the same state (system is not chaotic [Yorke et. al, 2000])

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Efficiency in Epidemiology

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Let a be the average number of contacts with which infectious nodes become uninterested in disseminating a rumor / diffuse an epidemic (stifle). At this point, a node realizes that the update / rumor has lost its novelty and becomes uninterested in diffusing it. The tradeoff is to maximizing the number of updated nodes and minimizing the number of contacts.

The efficiency depends on the rate at which nodes lose interest in further spreading of the epidemic and the network topology.

Strategy: After a given a, a node stops spreading in order to achieve high efficiency. This is based on the assumption that someone else WILL disseminate that information across the network.

Achtung!: The logic behind the strategy tempts selfish nodes to remain idle when it comes to relaying information!

dS(t)/dt = -βS(t)I(t)

dI(t)/dt = βS(t)I(t)-(1/a)I(t)(I(t)+U(t))

Epidemical Threshold: An eigenvalue approach

Let pi,t: be probability that a node i be infected at time t. j denotes a node in the neighborhood of the node i. Then,

pi,t = 1 - δ pi,t-1 – (1 - pi,t-1) Πj(1-βpj,t-1) =>

pi,t = 1 - δ pi,t-1 – (1 - pi,t-1) (1-βΣjpj,t-1) =>

pi,t = 1 - δ pi,t-1 – (1 - pi,t-1-βΣjpj,t-1) =>

pi,t = - δ pi,t-1 + pi,t-1+βΣjpj,t-1 =>

pi,t = (1- δ) pi,t-1 +β Σjpj,t-1

Converting to matrix notation, we have (Pt = [p1,t , …, pM,t ]):

Pt=((1-δ)I + βΑ).Pt-1 => Pt = St.P0, with S = (1-δ)I + βΑ

Hence, if mi,S and mi,A are the eigenvalues of the S and A matrices then, mi,S = 1 – δ + β. mi,A

Using spectral decomposition, S = Σi mi,Sfi,S fTi,S, i=1, … ,M and then,

St = Σi(mti,Sfi,S fT

i,S) => Pt = Σi(mti,Sfi,S fT

i,S).P0

For an infection to die off and not an epidemic, the should go to zero at larger t, thus, the maximum eigenvalue m1,A< 1, i.e.,

(1 – δ + βm1,A ) < 1 => (βm1,A ) < δ =>

λ = 1/ m1,A = β/δ (Epidemical threshold) [Faloutsos et., al 2001]

ReferencesD. Demers, D. Greene, C. Hauser, W. Irish, J. Larson, S. Shenker, H. Sturgis,

D. Swinehart, D. Terry, Epidemic Algorithms for Replicated Database Maintenance, In Proceedings of the ACM Principles of Distributed Computing, 1-12, 1987.

D. Kempe, J. Kleinberg, A. Demers, Spatial gossip and resource location protocols, J. ACM, 51(6), 943-967, 2004.

Barabási, Albert-László, Reka, Albert, Emergence of scaling in random networks, Science, 286, 509-512, 1999.

Yamir Moreno, Maziar Nekovee, Alessandro Vespigianni, Efficiency and reliability of epidemic data dissemination in complex networks, Physical Review E, vol. 69,DOI: 10.1103/PhyRevE.69.055101, 2004.

P.T. Eugster, R. Guerraoui, A.-M. Kermarrec, L. Massoulie, From Epidemics to Distributed Computing, IEEE Computer, 2004

Z Chen, Spatial-temporal modeling of malware propagation in networks, IEEE Transactions on Neural Networks, 2005