Classic epidemic models (full mixing) Epidemic models over networks Epidemic models over networks Argimiro Arratia & R. Ferrer-i-Cancho Universitat Polit` ecnica de Catalunya Complex and Social Networks (20 20 -202 1 ) Master in Innovation and Research in Informatics (MIRI) Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
Classic epidemic models (full mixing) Epidemic models over networks Official website: www.cs.upc.edu/~csn/ Contact: ◮ Ramon Ferrer-i-Cancho, rferrericancho@cs.upc.edu, http://www.cs.upc.edu/~rferrericancho/ ◮ Argimiro Arratia, argimiro@cs.upc.edu, http://www.cs.upc.edu/~argimiro/ Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
Classic epidemic models (full mixing) Epidemic models over networks Epidemic models Epidemic models attempt to capture the dynamics in the spreading of a disease (or idea, computer virus, product adoption). Central questions they try to answer are: ◮ How do contagions spread in populations? ◮ Will a disease become an epidemic? ◮ Who are the best people to vaccinate? ◮ Will a given YouTube video go viral? ◮ What individuals should we market to for maximizing product penetration? Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model In today’s lecture Classic epidemic models (full mixing) The SI model The SIR model The SIS model Epidemic models over networks Homogeneous models A general network model for SIS Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model Full mixing in classic epidemiological models Full mixing assumption In classic epidemiology, it is assumed that every individual has an equal chance of coming into contact with every other individual in the population Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model Full mixing in classic epidemiological models Full mixing assumption In classic epidemiology, it is assumed that every individual has an equal chance of coming into contact with every other individual in the population Dropping this assumption by making use of an underlying contact network leads to the more realistic models over networks (second half of the lecture)! Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model The SI model ( fully mixing susceptible – infected ) Notation (following [Newman, 2010]) ◮ Let S ( t ) be the number of individuals who are susceptible to sickness at time t ◮ Let X ( t ) be the number of individuals who are infected at time t 1 ◮ Total population size is n ◮ Contact with infected individuals causes a susceptible person to become infected ◮ An infected never recovers and stays infected and infectious to others 1 Well, really S and X are random variables and we want to capture number of infected and susceptible in expectation . Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model The SI model In the SI model, individuals can be in one of two states: ◮ infective (I), or ◮ susceptible (S) S I ✲ The parameters of the SI model are ◮ β infection rate: probability of contagion after contact per unit time Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model The SI model Dynamics dX dt = β SX dS dt = − β SX and n n where ◮ S / n is the probability of meeting a susceptible person at random per unit time ◮ XS / n is the average number of susceptible people that infected nodes meet per unit time ◮ β XS / n is the average number of susceptible people that become infected from all infecteds per unit time Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model The SI model Logistic growth equation and curve Define s = S / n and x = X / n , since S + X = n or equivalently s + x = 1, we get: dx dt = β ( 1 − x ) x The solution to the differential equation (known as the “ logistic growth equation ”) leads to the logistic growth curve x 0 e β t x ( t ) = 1 − x 0 + x 0 e β t where x ( 0 ) = x 0 Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model The SI model Logistic growth equation and curve Logistic growth curve 1.0 0.8 Fraction of infected x 0.6 x 0 e β t x ( t ) = 1 − x 0 + x 0 e β t 0.4 0.2 0.0 2 4 6 8 10 t Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model Solving the logistic growth equation I dx dt = β ( 1 − x ) x � x � t 1 ⇐ ⇒ ( 1 − x ) x dx = β dt x 0 0 � x � x 1 1 ⇐ ⇒ ( 1 − x ) dx + x dx = β t − β 0 x 0 x 0 � x � x 1 1 ⇐ ⇒ ( 1 − x ) dx + x dx = β t x 0 x 0 ⇒ ln 1 − x 0 1 − x + ln x ⇐ = β t x 0 ⇒ ln ( 1 − x 0 ) x ⇐ = β t ( 1 − x ) x 0 Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model Solving the logistic growth equation II ln ( 1 − x 0 ) x = β t ( 1 − x ) x 0 ⇒ ( 1 − x 0 ) x = e β t ⇐ ( 1 − x ) x 0 1 − x = x 0 e β t x ⇐ ⇒ 1 − x 0 ⇒ 1 − x = 1 − x 0 ⇐ x 0 e β t x x 0 e β t + 1 = 1 − x 0 + x 0 e β t ⇒ 1 x = 1 − x 0 ⇐ x 0 e β t x 0 e β t ⇐ ⇒ x = 1 − x 0 + x 0 e β t Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model The SIR model Allowing recovery and immunity In the SIR model, individuals can be in one of two states: ◮ infective (I), or ◮ susceptible (S), or ◮ recovered (R) S I R ✲ ✲ The parameters of the SIR model are ◮ β infection rate: probability of contagion after contact per unit time ◮ γ recovery rate: probability of recovery from infection per unit time Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model The SIR model Dynamics ds dx dr dt = − β sx dt = β sx − γ x dt = γ x The solution to this system (with s + x + r = 1) is not analytically tractable, but solutions look like the following: Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model The SIR model I A threshold phenomenon Now we are interested in considering the fraction of the population that will get sick (i.e. size of the epidemic), basically captured by r ( t ) as t → ∞ Substituting dt = dr γ x from the third equation into ds = − β sxdt and solving for s (assuming r 0 = 0), we obtain that s ( t ) = s 0 e − β γ r and so dr dt = γ ( 1 − r − s 0 e − β γ r ) Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model The SIR model II A threshold phenomenon As t → ∞ , we get that r ( t ) stabilizes and so dr dt = 0, thus: r = 1 − s 0 e − β γ r Assume that s 0 ≈ 1, since typically we start with a small nr. of infected individuals and we are considering large populations, and so r = 1 − e − β γ r Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model The SIR model III A threshold phenomenon 1.0 1.0 0.8 0.8 y = r y = 1 − e ( − 1.5r ) epidemic size in the limit 0.6 0.6 y = 1 − e ( − 1r ) 0.4 0.4 y = 1 − e ( − 0.5r ) 0.2 0.2 0.0 0.0 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 β γ r Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
The SI model Classic epidemic models (full mixing) The SIR model Epidemic models over networks The SIS model The SIR model IV A threshold phenomenon 1.0 0.8 ◮ if β γ � 1 then no epidemic occurs epidemic size in the limit 0.6 ◮ if β γ > 1 then epidemic occurs 0.4 ◮ β = γ is the epidemic transition 0.2 0.0 0 1 2 3 4 β γ Argimiro Arratia & R. Ferrer-i-Cancho Epidemic models over networks
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