Network Science: Principles and Applications

Transcription

1 Network Science: Principles and Applications CS Fall 2016 Amarda Shehu,Fei Li [amarda, lifei](at)gmu.edu Department of Computer Science George Mason University Spreading Phenomena: Epidemic Modeling

2 Outline of Today s Class 1 Spreading Phenomena Epidemic Spreading - Why does it Matter? Epidemic Spreading and Networks 2 (Biological) Epidemic Modeling Homogeneous Mixing Hypothesis Population Models Susceptible-Infected (SI) Model Susceptible-Infected-Suspectible (SIS) Model Susceptible-Infected-Recovered (SIR) Model Summary 3 Network Epidemics SI Model on a Network Amarda Shehu,Fei Li () 2

3 Spreading Phenomena - Epidemic(s) Epi + demos upon + people Biological Airborne diseases (flu, SARS, etc.) Venereal diseases (HIV, etc.) Other infectious diseases including some cancers (HPV, etc.) Parasites (bedbugs, malaria, etc.) Digital Computer viruses, worms Mobile phone viruses Conceptual/Intellectual Diffusion of innovations Rumors Memes Business practices Amarda Shehu,Fei Li () Spreading Phenomena 3

4 Notable Biological Epidemic Outbreaks Amarda Shehu,Fei Li () Spreading Phenomena 4

5 Computer Viruses, Worms, Mobile Phone Viruses Amarda Shehu,Fei Li () Spreading Phenomena 5

6 Diffusion of Innovation Amarda Shehu,Fei Li () Spreading Phenomena 6

7 Information Spreading Amarda Shehu,Fei Li () Spreading Phenomena 7

8 Epidemic Spreading: Why does it Matter? From the Great Plague to SARS Amarda Shehu,Fei Li () Spreading Phenomena 8

9 14th Century - The Great Plague 4 years from France to Sweden Limited by the speed of human travel Amarda Shehu,Fei Li () Spreading Phenomena 9

10 21st Century - SARS Figure: Spread from one case in China in November 2002 to several continents in April Source: World Health Organization Amarda Shehu,Fei Li () Spreading Phenomena 10

11 Large Populations Provide the Fuel Separate, small population (hunter-gatherer society, wild animals) Connected, highly populated areas (cities) Human societies have crowd diseases consequences of large, interconnected populations (measles, tuberculosis, smallpox, influenza, common cold, etc.) Amarda Shehu,Fei Li () Spreading Phenomena 11

12 Epidemic Spreading - Why does it Matter? Amarda Shehu,Fei Li () Spreading Phenomena 12

13 Epidemic Spreading - Network Epidemic spreading always implies network structure! Spreading happens only when the carries of the diseases/virus/idea are connected to each other. Amarda Shehu,Fei Li () Spreading Phenomena 13

14 Types of Spreading Phenomena and Networks Amarda Shehu,Fei Li () Spreading Phenomena 14

15 Focus: Biological Epidemic Modeling Important questions to answer: What is the rate of infection? What strategies to employ to stop or slow it down? To our aid: Epidemic Models 1st-order approximation: population models with no information on how disease spreads on interaction/contact network 2nd-order approximation: network models 3rd-order approximation: network models with specific node dynamics Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 15

16 SIR model: flu, SARS, plague Classical Epidemic Models Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 16

17 Classical Epidemic Models SIS model: common cold Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 17

18 Simplest Model: SI SI model: devastating disease, no immunity, no treatment Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 18

19 Population Models Contact network not considered - homogeneous mixing assumed instead Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 19

20 Homogeneous Mixing Hypothesis The homogenous mixing hypothesis (also called fully-mixed or mass-action approximation) assumes that each individual has the same chance of coming into contact with an infected individual. This hypothesis eliminates the need to know the precise contact network on which the disease spreads, replacing it with the assumption that anyone can infect anyone else. Next: the dynamics of the SI, SIS and SIR models that help understand the basic building blocks of epidemic modeling. Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 20

21 Susceptible-Infected (SI) Model An individual is either suspectible (healthy) or infected (sick) S(t) : nr. of individuals suspectible at time t in a population of N individuals I (t) : number of infected individuals at time t At time t = 0, S(0) = N (all susceptible/healthy, so I (0) = 0) Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 21

22 Susceptible-Infected (SI) Model An individual is either suspectible (healthy) or infected (sick) S(t) : nr. of individuals suspectible at time t in a population of N individuals I (t) : number of infected individuals at time t At time t = 0, S(0) = N (all susceptible/healthy, so I (0) = 0) Assumptions: A typical individual has k contacts and that the likelihood that the disease will be transmitted from an infected to a susceptible individual in a unit of time is β Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 21

23 Susceptible-Infected (SI) Model An individual is either suspectible (healthy) or infected (sick) S(t) : nr. of individuals suspectible at time t in a population of N individuals I (t) : number of infected individuals at time t At time t = 0, S(0) = N (all susceptible/healthy, so I (0) = 0) Assumptions: A typical individual has k contacts and that the likelihood that the disease will be transmitted from an infected to a susceptible individual in a unit of time is β Question: If a single individual becomes infected at time t = 0 (i.e. I (0) = 1), how many individuals will be infected at some later time t? Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 21

24 Susceptible-Infected (SI) Model Within the homogenous mixing hypothesis, the probability that the infected person encounters a susceptible individual is S(t)/N So, an infected person comes into contact with k S(t)/N susceptible individuals in a unit time Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 22

25 Susceptible-Infected (SI) Model Within the homogenous mixing hypothesis, the probability that the infected person encounters a susceptible individual is S(t)/N So, an infected person comes into contact with k S(t)/N susceptible individuals in a unit time Since I (t) infected individuals are transmitting the pathogen, each at rate β, the average number of new infections di (t) during a timeframe dt is: S(t) I (t) β k N dt Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 22

26 Susceptible-Infected (SI) Model Within the homogenous mixing hypothesis, the probability that the infected person encounters a susceptible individual is S(t)/N So, an infected person comes into contact with k S(t)/N susceptible individuals in a unit time Since I (t) infected individuals are transmitting the pathogen, each at rate β, the average number of new infections di (t) during a timeframe dt is: S(t) I (t) β k N dt I (t) changes at the rate: di (t) S(t) I (t) dt = β k N Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 22

27 Susceptible-Infected (SI) Model Within the homogenous mixing hypothesis, the probability that the infected person encounters a susceptible individual is S(t)/N So, an infected person comes into contact with k S(t)/N susceptible individuals in a unit time Since I (t) infected individuals are transmitting the pathogen, each at rate β, the average number of new infections di (t) during a timeframe dt is: S(t) I (t) β k N dt I (t) changes at the rate: di (t) S(t) I (t) dt = β k N Let s use s(t) = S(t)/N and i(t) = I (t)/n to track the fraction of susceptible and infected sub-populations at time t; let s also drop the (t) variable from s(t) and i(t)for convenience and write: di dt = β k si = β k i (1 i) where the product β k is called the transmission rate or transmissability Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 22

28 Susceptible-Infected (SI) Model How do we solve di dt = β k s i = β k i (1 i)? Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 23

29 Susceptible-Infected (SI) Model How do we solve di dt = β k s i = β k i (1 i)? First, simple manipulation yields: di i + di 1 i = β k dt Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 23

30 Susceptible-Infected (SI) Model How do we solve di dt = β k s i = β k i (1 i)? First, simple manipulation yields: di i + di 1 i = β k dt Integrating both sides, we obtain: t 0 di(t) i(t) t 0 di(t) i(t) 1 = t 0 β k dt obtaining: ln(i) ln(i 0 ) [ln(i 1) ln(i 0 1)] = β k t... Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 23

31 Susceptible-Infected (SI) Model How do we solve di dt = β k s i = β k i (1 i)? First, simple manipulation yields: di i + di 1 i = β k dt Integrating both sides, we obtain: t 0 di(t) i(t) t 0 di(t) i(t) 1 = t 0 β k dt obtaining: ln(i) ln(i 0 ) [ln(i 1) ln(i 0 1)] = β k t... So: i = i 0e β k t 1 i 0 +i 0 e β k t Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 23

32 Susceptible-Infected (SI) Model Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 24

33 Susceptible-Infected (SI) Model: Summary At early times, the fraction of infected individuals grows exponentially With time, an infected individuals encounters fewer and fewer susceptible individuals. Hence, the growth of i slows for large t, ending epidemic when all are infected, i.e., when i(t ) = 1 and s(t ) = 0 The characteristic time required to reach an 1/e fraction (about 36%) of all susceptible individuals is: τ = 1 β k Think of τ as the inverse of the speed with which the pathogen spreads through the population Increasing either the density of links k or β enhances the speed of the pathogen and reduces the characteristic time Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 25

34 Susceptible-Infected-Suspectible (SIS) Model Most pathogens are eventually defeated by our immune system SIS model allows infected individuals to recover at rate µ, becoming susceptible again So, the equation describing the dynamics of the SIS model is now: di dt = β k i(1 i) µi µi captures the rate at which the population recovers from the disease Solving for i gives: i = (1 µ β k ) Ce(β k µ)t 1+Ce (β k µ)t where the initial condition i 0 = 0 gives C = i 0/(1 i 0 µ/(β k )) While in the SI model eventually everyone becomes infected, in SIS the epidemic has two possible outcomes: exponential outbreak endemic state Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 26

35 Susceptible-Infected-Suspectible (SIS) Model Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 27

36 Susceptible-Infected-Suspectible (SIS) Model Stationary/Endemic State (µ < β k ) For low recovery rate, the fraction of infected individuals, i, follows a logistic curve similar to the one observed for the SI model Yet, not everyone is infected, but i reaches a constant i( ) < 1 value; at any moment, only a finite fraction of the population is infected The number of newly infected individuals equals the number of individuals who recover (the infected fraction of the population) does not change with time; di/dt = 0 = : i( ) = 1 µ β k Disease-free State (µ > β k ) For a sufficiently high recovery rate, the exponent is negative Therefore, i decreases exponentially with time, indicating that an initial infection will die out exponentially This is because in this state the number of individuals cured per unit time exceeds the number of newly infected individuals With time, the pathogen disappears from the population Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 28

37 Susceptible-Infected-Suspectible (SIS) Model: Summary The SIS model predicts that some pathogens will persist in the population, while some will die out shortly Let the characteristic time of a pathogen be: τ = 1 µ(r 0 1) where R 0 = β k µ R 0 is the reproductive number, representing the average number of susceptible individuals infected by an infected individual during its infectious period in a fully susceptible population So, R 0 is the number of new infections each infected individual causes under ideal circumstances If R 0 > 1, τ > 0, so the epidemic is in the endemic state (if each infected individual infects more than one healthy person, the pathogen is poised to spread and persist); the higher R 0, the faster is the spreading process If R 0 < 1, τ < 0, so the epidemic dies out (if each infected individual infects less than one additional person, the pathogen cannot persist in the population) Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 29

38 Susceptible-Infected-Suspectible (SIS) Model The reproductive number is one of the first parameters epidemiologists estimate for a new pathogen, gauging the severity of the problem they face. The high R 0 of some of these pathogens underlies the dangers they pose. For example, each individual infected with measles causes over a dozen subsequent infections. Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 30

39 Susceptible-Infected-Recovered (SIR) Model Model needs to allow individuals to develop immunity and so be removed from the population rather than return to the suspectible state These individuals do not count anymore from the perspective of the pathogen, as they cannot be infected, nor can they infect others SIR model can capture this dynamics Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 31

40 Models Summary SI, SIS, and SIR agree on the early stages: when the nr. of infected individuals is small, the disease spreads freely and the nr. of infected individuals increases exponentially The outcomes are different for large times: In the SI model, everyone becomes infected; the SIS model either reaches an endemic state, in which a finite fraction of individuals are always infected, or the infection dies out; in the SIR model everyone is removed at the end. The reproductive number predicts the long-term fate of an epidemic: for R 0 > 1, the pathogen persists in the population, while, for R 0 < 1, it dies out naturally The models discussed so far have ignored the fact that that an individual comes into contact only with its network-based neighbors in the pertinent contact network We assumed homogenous mixing instead, which means that an infected individual can infect any other individual, meaning that an infected individual typically infects only k other individuals, ignoring variations in node degrees To accurately predict the dynamics of an epidemic, we need to consider the precise role the contact network plays in epidemic phenomena Amarda Shehu,Fei Li () (Biological) Epidemic Modeling 32

41 Network Epidemics Both assumptions in honogeneous mixing hypothesis are false: 1. any individual can come into contact with any other individual... False 2. all individuals have a comparable number of contacts k... False Instead, an individual can transmit a pathogen only to those with whom they come in contact = pathogen spreads in contact network Furthermore, these networks are often scale-free, hence k is not sufficient to characterize their topology Figure: Bubonic plague outbreak Amarda Shehu,Fei Li () Network Epidemics 33

42 Fundamental Revision of Epidemic Modeling Framework In 2001, Romualdo Pastor-Sorras and Alessandro Vespignani extended the basic epidemic models to incorporate in a self-consistent fashion the topological characteristics of the underlying contact network. Formalism developed by them now deemed network epidemics. Amarda Shehu,Fei Li () Network Epidemics 34

43 SI Model on a Network Must consider the degree of each node as an implicit variable Degree block approximation distinguishes nodes based on their degree and assumes that nodes with the same degree are statistically equivalent Amarda Shehu,Fei Li () Network Epidemics 35

44 SI Model on a Network Now we can write a separate rate equation for each degree block/compartment Denote with i k = I k N the fraction of nodes with degree k that are infected among all k N k degree-k nodes in the network Amarda Shehu,Fei Li () Network Epidemics 36

45 SI Model on a Network Now we can write a separate rate equation for each degree block/compartment Denote with i k = I k N the fraction of nodes with degree k that are infected among all k N k degree-k nodes in the network Total fraction of infected nodes is then: i = k p ki k Amarda Shehu,Fei Li () Network Epidemics 36

46 SI Model on a Network Now we can write a separate rate equation for each degree block/compartment Denote with i k = I k N the fraction of nodes with degree k that are infected among all k N k degree-k nodes in the network Total fraction of infected nodes is then: i = k p ki k Now we can write the SI model for each degree k separately: di k dt = β(1 i k)kθ k Amarda Shehu,Fei Li () Network Epidemics 36

47 SI Model on a Network Now we can write a separate rate equation for each degree block/compartment Denote with i k = I k N the fraction of nodes with degree k that are infected among all k N k degree-k nodes in the network Total fraction of infected nodes is then: i = k p ki k Now we can write the SI model for each degree k separately: di k dt = β(1 i k)kθ k The equation has the same structure as in the SI model with no contact network; the infection rate is proportional to β, and the fraction of degree-k nodes that are not yet infected, which is (1 i k ) Amarda Shehu,Fei Li () Network Epidemics 36

48 SI Model on a Network Now we can write a separate rate equation for each degree block/compartment Denote with i k = I k N the fraction of nodes with degree k that are infected among all k N k degree-k nodes in the network Total fraction of infected nodes is then: i = k p ki k Now we can write the SI model for each degree k separately: di k dt = β(1 i k)kθ k The equation has the same structure as in the SI model with no contact network; the infection rate is proportional to β, and the fraction of degree-k nodes that are not yet infected, which is (1 i k ) There are key differences: k replaced with actual degree k θ k, the fraction of infected neighbors of a susceptible node Not one equation, but k max equations (each degree block has its own dynamics) Amarda Shehu,Fei Li () Network Epidemics 36

49 Early-time Behavior of i k How do we halt spread in liew of a vaccine or treatment? Amarda Shehu,Fei Li () Network Epidemics 37

50 Early-time Behavior of i k How do we halt spread in liew of a vaccine or treatment? Need an accurate estimate of nr. of individuals infected in early stages of epidemic Amarda Shehu,Fei Li () Network Epidemics 37

51 Early-time Behavior of i k How do we halt spread in liew of a vaccine or treatment? Need an accurate estimate of nr. of individuals infected in early stages of epidemic At small t, i k is small, and the higher-order term βi k kθ k can be neglected, writing: βkθ k di k dt Amarda Shehu,Fei Li () Network Epidemics 37

52 Early-time Behavior of i k How do we halt spread in liew of a vaccine or treatment? Need an accurate estimate of nr. of individuals infected in early stages of epidemic At small t, i k is small, and the higher-order term βi k kθ k can be neglected, writing: βkθ k di k dt For a network lacking degree correlations, θ k is independent of k (Barabasi 10.13): di k dt βki o k 1 k et/τ SI where τ SI is the characteristic time for the spread of pathogen: τ SI k = β( k 2 k ) Amarda Shehu,Fei Li () Network Epidemics 37

53 Early-time Behavior of i k How do we halt spread in liew of a vaccine or treatment? Need an accurate estimate of nr. of individuals infected in early stages of epidemic At small t, i k is small, and the higher-order term βi k kθ k can be neglected, writing: βkθ k di k dt For a network lacking degree correlations, θ k is independent of k (Barabasi 10.13): di k dt βki o k 1 k et/τ SI where τ SI is the characteristic time for the spread of pathogen: τ SI k = β( k 2 k ) Integrating, we obtain the fraction of infected nodes with degree k: i k = i 0 (1 + k( k 1) k 2 k (et/τ SI 1)) So, the higher the degree of a node, the more likely that it becomes infected Amarda Shehu,Fei Li () Network Epidemics 37

54 Early-time Behavior of i k How do we halt spread in liew of a vaccine or treatment? Need an accurate estimate of nr. of individuals infected in early stages of epidemic At small t, i k is small, and the higher-order term βi k kθ k can be neglected, writing: βkθ k di k dt For a network lacking degree correlations, θ k is independent of k (Barabasi 10.13): di k dt βki o k 1 k et/τ SI where τ SI is the characteristic time for the spread of pathogen: τ SI k = β( k 2 k ) Integrating, we obtain the fraction of infected nodes with degree k: i k = i 0 (1 + k( k 1) k 2 k (et/τ SI 1)) So, the higher the degree of a node, the more likely that it becomes infected The total fraction of infected nodes grows with time as: i = k max 0 i k p k dk = i 0 (1 + k 2 k k 2 k (et/τ SI 1)) Amarda Shehu,Fei Li () Network Epidemics 37

55 Fraction of Infected Nodes in SI model Figure: In an Erdos-Renyi network with average degree k = 2. Amarda Shehu,Fei Li () Network Epidemics 38

56 Random Network Since k 2 = k ( k + 1), obtaining: τer SI = 1 β k Summary Scale-free Network with γ 3 Both k and k 2 are finite, so τ SI is also finite, and spreading dynamics is similar to what is predicted for a random network but with an altered τ SI Scale-free Network with γ < 3 For γ < 3 in the N limit, k 2, which predicts that τ SI 0 The spread of a pathogen on a scale-free network is instantaneous (most unexpected prediction of network epidemics) Inhomogeneous networks A network does not need to be strictly scale-free. As long as k 2 > k ( k + 1), τ SI is reduced; so, heterogenous network enhance the speed of any pathogen Using the degree block concept, the mathematical formalisms for the rate of infection can be extended simillarly for the SIS and SIR models Amarda Shehu,Fei Li () Network Epidemics 39