Epidemic spreading on preferred degree adaptive networks Shivakumar Jolad, Wenjia Liu, R. K. P. Zia and Beate Schmittmann Department of Physics, Virginia Tech, Blacksburg-VA SESAPS-2011 NSF-DMR Materials Theory

Motivation Most epidemic models are based on regular lattices, Erdös-Renyi or scale free networks, with active nodes and static links.  Active nodes: Links influence the nodes (epidemics, opinions, Ising model)  Active links: network topology changes Preferred degree networks: Nodes (individuals) have a preferred number of links (social connection). Make more realistic networks. Static epidemic model: active nodes, static links Adaptive model: active nodes and active links.  Feedback between Dynamics on the network and Dynamics of the network. Two community Coupled networks: Epidemic propagation from one community to other. 2

Preferred degree networks 3

Degree distribution w Two tailed exponential (Laplace) distribution All have same preferred degree 4

SIS disease dynamics 5

SIS dynamics on static preferred degree network Absorbing state Active State 6 Heterogeneous mean field theory

Adaptive behavior 7

Reckless Typical Nosophobic (highly scared) Reckless Typical Nosophobic Minimum required for survival 8

Infection phase diagram : Adaptive SIS 9 Non-adaptive Reckless Typical Nosophobic Adaptive SIS model with varying preferred degree doesn't change the epidemic threshold, but changes the level of infection in the active state.

Mean field analysis Mean field predictions closely match simulations 10 Infection probability connected: mean field

11 Heterogeneous preferred degree Two community network (extroverts and introverts) different preferred degree and coupling  Simplest network with heterogeneous preference Questions:  How does disease spread across different communities?  Are extroverts more prone to contagious diseases?  How does disease depend on the interaction/coupling between the communities?

Two community network 12 external (cross) degree internal degree Decision to make internal (within) or external (across) link Introverts Extroverts

Results- uncoupled communities 13 Extroverts Introverts

Coupled communities 14 Extroverts Introverts

Summary Behavior SIS model with active nodes and active links. Feedback between dynamics on the network and dynamics of the network. Adaptive SIS model with varying preferred degree doesn't change the epidemic threshold, but changes the level of infection in the active state. Generalizations to more realistic problems:  Population with different preferred degrees:  Epidemic on a realistic model network with different degree distribution  Population with mixed behavior: 15

Collaborators 16

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Classification of Adaptive behavior Belief based Prevalence Based Local Information taken from the social or spatial neighborhood only. Local information, prevalence based. People know who is infected in their network (Small pox, Cold). Global behavioral choice based on publicly available information like media, websites etc. Behaviors linked to global information directly linked to disease prevalence. ( Flu, SARS etc). Funk et al, “Modeling the inuence of human behavior on the spread of infectious diseases: a review”, J. R. Soc. Interface, 7 (50) 1247 (2010) 18

19 Inflexible and reckless 10 max ~ 25 small Effect on Network

Make new friends, break old ties (establish/cut contacts ) according to some preference Preferences can be dynamic. Preference can vary from person to person Preferred degree networks 20

Outline Recap of networks Dynamics on the networks: Epidemic spreading in population Dynamics of the network: Adaptive behavior of people Model and Simulation Summary 21

SIS phase diagram- basic picture cc disease spreads disease dies I Phase transition from ‘absorbing state’ to an ‘active state’ 22

Erdös-Rényi and Scale Free Networks From Linked: The New Science of Networks - Albert-Laszlo Barabasi 23

Realistic Social Contact Networks (a) T. Platini, S. Eubank, Private communication (b) C. L. Barrett et al, “Generation and analysis of large synthetic social contact networks”, Winter Simulation Conference: Austin, Texas Social contact network: (a) Hawaii (b) LA, NYC, Seattle 24

Identical preferred degree w Two tailed exponential (Laplace) distribution Small world network Let us follow KISS (Keep It Simple, Silly! ) first 25

Effect on Network Fearless Fearless Fearful Nosophobic 26

Contact process on networks 27

Monte-Carlo simulations 28

29 SIS phase diagram near transition point, comparison with mean field theories.

Noisy perception 30 Constant noise strength Variable noise strength