Generalized Contagion Generalized Model of Contagion Principles of - PowerPoint PPT Presentation

Generalized Contagion Generalized Contagion Generalized Model of Contagion Principles of Complex Systems References Course 300, Fall, 2008 Prof. Peter Dodds Department of Mathematics & Statistics University of Vermont Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License . Frame 1/17

Generalized Outline Contagion Generalized Model of Contagion References Generalized Model of Contagion References Frame 2/17

Generalized Generalized contagion model Contagion Generalized Model of Contagion References Basic questions about contagion ◮ How many types of contagion are there? ◮ How can we categorize real-world contagions? ◮ Can we connect models of disease-like and social contagion? Frame 3/17

Generalized Some (of many) issues Contagion Generalized Model of Contagion References ◮ Disease models assume independence of infectious events. ◮ Threshold models only involve proportions: 3 / 10 ≡ 30 / 100. ◮ Threshold models ignore exact sequence of influences ◮ Threshold models assume immediate polling. ◮ Mean-field models neglect network structure ◮ Network effects only part of story: media, advertising, direct marketing. Frame 4/17

Generalized Generalized model—ingredients Contagion Generalized Model of Contagion References ◮ Incorporate memory of a contagious element [1, 2] ◮ Population of N individuals, each in state S, I, or R. ◮ Each individual randomly contacts another at each time step. ◮ φ t = fraction infected at time t = probability of contact with infected individual ◮ With probability p , contact with infective leads to an exposure. ◮ If exposed, individual receives a dose of size d drawn from distribution f . Otherwise d = 0. Frame 5/17

Generalized Generalized model—ingredients Contagion Generalized Model of Contagion S ⇒ I References ◮ Individuals ‘remember’ last T contacts: t � d i ( t ′ ) D t , i = t ′ = t − T + 1 ◮ Infection occurs if individual i ’s ‘threshold’ is exceeded: D t , i ≥ d ∗ i ◮ Threshold d ∗ i drawn from arbitrary distribution g at t = 0. Frame 6/17

Generalized Generalized model—ingredients Contagion Generalized Model of Contagion References I ⇒ R When D t , i < d ∗ i , individual i recovers to state R with probability r . R ⇒ S Once in state R, individuals become susceptible again with probability ρ . Frame 7/17

Generalized A visual explanation Contagion Generalized Model of Contagion References c 1 if D t,i < d ∗ i a φ t contact p receive S dose d > 0 infective 1 if D t,i ≥ d ∗ i 1 − p receive 1 − φ t no dose rρ if D t,i < d ∗ i ρ I r (1 − ρ ) if D t,i < d ∗ i b 1 − r if D t,i < d ∗ i d t − T d t − T +1 d t − 1 d t R 1 if D t,i ≥ d ∗ i � �� � � = D t,i 1 − ρ Frame 8/17

Generalized Generalized model Contagion Generalized Model of Contagion Important quantities: References   � ∞ k d d ∗ g ( d ∗ ) P �  where 1 ≤ k ≤ T . d j ≥ d ∗ P k =  0 j = 1 P k = Probability that the threshold of a randomly selected individual will be exceeded by k doses. e.g., P 1 = Probability that one dose will exceed the threshold of a random individual = Fraction of most vulnerable individuals. Frame 9/17

Generalized Generalized model—heterogeneity, r = 1 Contagion Generalized Model of Contagion Fixed point equation: References T � T � φ ∗ = � ( p φ ∗ ) k ( 1 − p φ ∗ ) T − k P k k k = 1 Expand around φ ∗ = 0 to find Spread from single seed if pP 1 T ≥ 1 ⇒ p c = 1 / ( TP 1 ) Frame 10/17

Generalized Heterogeneous case Contagion Generalized Model of Contagion References Example configuration: ◮ Dose sizes are lognormally distributed with mean 1 and variance 0.433. ◮ Memory span: T = 10. ◮ Thresholds are uniformly set at 1. d ∗ = 0 . 5 2. d ∗ = 1 . 6 3. d ∗ = 3 ◮ Spread of dose sizes matters, details are not important. Frame 11/17

Generalized Heterogeneous case—Three universal Contagion classes Generalized Model of Contagion References I. Epidemic threshold II. Vanishing critical mass III. Critical mass 1 0.8 φ * 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 p p p ◮ Epidemic threshold: P 1 > P 2 / 2, p c = 1 / ( TP 1 ) < 1 ◮ Vanishing critical mass: P 1 < P 2 / 2, p c = 1 / ( TP 1 ) < 1 ◮ Pure critical mass: P 1 < P 2 / 2, p c = 1 / ( TP 1 ) > 1 Frame 12/17

Calculations—Fixed points for r < 1, d ∗ = 2, Generalized Contagion and T = 3 Generalized Model of Contagion References T � T � . Eq: φ ∗ = Γ( p , φ ∗ ; r ) + � ( p φ ∗ ) i ( 1 − p φ ∗ ) T − i . F .P i i = d ∗ ∞ ( 1 − r )( p φ ) 2 ( 1 − p φ ) 2 + � ( 1 − r ) m ( p φ ) 2 ( 1 − p φ ) 2 × Γ( p , φ ∗ ; r ) = m = 1 � � χ m − 1 + χ m − 2 + 2 p φ ( 1 − p φ ) χ m − 3 + p φ ( 1 − p φ ) 2 χ m − 4 [ m / 3 ] � m − 2 k � � ( 1 − p φ ∗ ) m − k ( p φ ∗ ) k . where χ m ( p , φ ∗ ) = k k = 0 Frame 13/17

Generalized SIS model Contagion Generalized Model of Contagion References Now allow r < 1: 1 0.8 φ * 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 p II-III transition generalizes: p c = 1 / [ P 1 ( T + τ )] (I-II transition less pleasant analytically) Frame 14/17

Generalized More complicated models Contagion Generalized Model of Contagion References 1 φ * 0.5 0 0 0.5 1 0 0.5 1 0 0.5 1 p ➤ Due to heterogeneity in individual thresholds. ➤ Same model classification holds: I, II, and III. Frame 15/17

Generalized Hysteresis in vanishing critical mass models Contagion Generalized Model of Contagion References 1 φ * 0.5 0 0 0.5 1 p Frame 16/17

Generalized Generalized model—heterogeneity, r ≤ 1 Contagion Generalized Model of Contagion References II-III transition generalizes: p c = 1 / [ P 1 ( T + τ )] where τ = 1 / r = expected recovery time Frame 17/17

Generalized Discussion Contagion Generalized Model of Contagion ◮ Memory is crucial ingredient. References ◮ Three universal classes of contagion processes: I. Epidemic Threshold II. Vanishing Critical Mass III. Critical Mass ◮ Dramatic changes in behavior possible. ◮ To change kind of model: ‘adjust’ memory, recovery, fraction of vulnerable individuals ( T , r , ρ , P 1 , and/or P 2 ). ◮ To change behavior given model: ‘adjust’ probability of exposure ( p ) and/or initial number infected ( φ 0 ). Frame 18/17

Generalized Discussion Contagion Generalized Model of Contagion ◮ If pP 1 ( T + τ ) ≥ 1, contagion can spread from single References seed. ◮ Key quantity: p c = 1 / [ P 1 ( T + τ )] ◮ Depends only on: 1. System Memory ( T + τ ). 2. Fraction of highly vulnerable individuals ( P 1 ). ◮ Details unimportant (Universality): Many threshold and dose distributions give same P k . ◮ Most vulnerable/gullible population may be more important than small group of super-spreaders or influentials. Frame 19/17

Generalized Future work/questions Contagion Generalized Model of Contagion References ◮ Do any real diseases work like this? ◮ Examine model’s behavior on networks ◮ Media/advertising + social networks model ◮ Classify real-world contagions Frame 20/17

Generalized References I Contagion Generalized Model of Contagion References P . S. Dodds and D. J. Watts. Universal behavior in a generalized model of contagion. Phys. Rev. Lett. , 92:218701, 2004. pdf ( ⊞ ) P . S. Dodds and D. J. Watts. A generalized model of social and biological contagion. J. Theor. Biol. , 232:587–604, 2005. pdf ( ⊞ ) Frame 21/17