quancol . ........ . . . ... ... ... ... ... ... ... - PowerPoint PPT Presentation

quan�col . ........ . . . ... ... ... ... ... ... ... www.quanticol.eu Introduction to Mean-Field Luca Bortolussi Nicolas Gast Dipartimento di Matematica e Geoscienze Università degli studi di Trieste CNR/ISTI, Pisa luca@dmi.units.it SFM, Bertinoro, June 20-24, 2016 SFM 1 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Predicting Pandemics www.quanticol.eu hex Worldwide model of H1N1 2009 influenza virus. SFM 2 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Motivational overview www.quanticol.eu � Population models are the natural way to describe large systems of interacting agents, common in systems biology, epidemiology, ecology, computer performance, CAS, . . . � Such models are typically described as a stochastic process in discrete space and continuous time. f 1 ( f 2 ( � These stochastic f 3 ( f 4 ( models are very hard to analyse, in particular Z ! for large populations. � Mean field theorems, when they can be applied, allow us to replace for large populations the stochastic process with a small system of ODE, that can be easily solved numerically. SFM 3 / 59

Introduction 1 Basics of Population Models 2 Continuous Time Markov Chains Basics of Mean-Field Approximation 3 Mean-Field of Stochastic Process Algebras Fast Simulation and Approximate Stochastic Verification 4 Example: bike-sharing 5

quan�col . ........ . . . ... ... ... ... ... ... ... Example: SIRS epidemic model www.quanticol.eu S I R SFM 5 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Example: SIRS epidemic model www.quanticol.eu S I R SFM 6 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Example: SIRS epidemic model www.quanticol.eu S I R SFM 7 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Example: SIRS epidemic model www.quanticol.eu S I R SFM 8 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Example: SIRS epidemic model www.quanticol.eu 1.0 S I R 0.8 S 0.6 occupancy 0.4 I R 0.2 0.0 0 5 10 15 20 Time SFM 9 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Population Models www.quanticol.eu � Assumption: agents are individually indistinguishable and homogeneously mixed. � State: we just need to count how many agents are in each different state. � Dynamics: what are the interactions among agents, how many interact and how they change state. SFM 10 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Population Models www.quanticol.eu A population CTMC model is a tuple X = ( X , D , T , x 0 ) , where: 1. X — vector of variables counting how many individuals in each state. 2. D — (countable) state space. 3. x 0 ∈ D — initial state . 4. η i ∈ T — global transitions . They can be visualised as chemical reaction/ rewriting rules: r 1 X 1 + . . . r n X n − → s 1 X 1 + . . . s n X n . Formally, they are pairs η i = ( v , r ( X )) 4.1 v ∈ R n , v = s − r — update vector (state changes from X to X + v ) 4.2 r : D → R ≥ 0 — rate function. SFM 11 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Example: SIRS epidemics www.quanticol.eu � Three variables: X S , X I , X R . � State space: D = { ( n 1 , n 2 , n 3 ) | n 1 + n 2 + n 3 = N } ⊂ { 0 , . . . , N } 3 . S � Transitions: � S + I − → I + I X I η inf = (( − 1 , 1 , 0 ) , sk I N X S ) I R � I − → R η rec = (( 0 , − 1 , 1 ) , k R X I ) � R − → S η susc = (( 1 , 0 , − 1 ) , k S X R ) SFM 12 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Exponential Distribution www.quanticol.eu Definition A random variable T : (Ω , S ) → [ 0 , ∞ ] is Exp ( λ ) iff � Cdf is P ( T < t ) = 1 − e − λ t � Density is f T ( t ) = λ e − λ t , t ≥ 0. � ∞ P ( T > t ) dt = 1 The expected value of T is E ( T ) = λ . 0 Memoryless Property T ∼ Exp ( λ ) if and only if the following memoryless property holds: P ( T > s + t | T > s ) = P ( T > t ) for all s , t ≥ 0 . Instantaneous firing probability An exponential distribution with rate λ models the firing time of an event who has probability of firing between time t and t + dt equal to λ dt . SFM 13 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... CTMC: definition www.quanticol.eu S -valued Continuous Time Markov Chain � Let S be finite or countable. � A CTMC on a state space S is a labelled graph, where labels are the rates of exponential distributions. � In each state, there is a race condition between the different exiting edges: the fastest is traversed. � The CTMC has the memoryless property: the future depends only on the current state. Formally A Continuous Time Markov Chain is a right-continuous continuous-time random process (with cadlag sampling paths) satisfying the memoryless condition: for each n , t i and s i : P ( X t n = s n | X t 0 = s 0 , . . . , X t n − 1 = s n − 1 ) = P ( X t n = s n | X t n − 1 = s n − 1 ) . SFM 14 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... CTMC: infinitesimal generator www.quanticol.eu Formally A Continuous Time Markov Chain is a right-continuous continuous-time random process (with cadlag sampling paths) satisfying the memoryless condition: for each n , t i and s i : P ( X t n = s n | X t 0 = s 0 , . . . , X t n − 1 = s n − 1 ) = P ( X t n = s n | X t n − 1 = s n − 1 ) . Q -matrix A Q -matrix is the | S | × | S | matrix such that: 1. q ij ≥ 0, i � = j is the rate of the exponential distribution giving the time needed to go from state s i to state s j 2. q ii = − � j � = i q ij is the opposite of the exit rate from state i . Therefore, each row of the Q -matrix sums up to zero. SFM 15 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... A simple example: single agent infection www.quanticol.eu S = { S , I , R } S 1 0.1   − 1 1 0 2 Q = 0 − 2 2   I R 0 . 1 0 − 0 . 1 SFM 16 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... CTMC for population models: the SIRS example www.quanticol.eu 2k i k s k r 2k r 2k i k s 2k s k s k r k i 3k r 2k s 3k s 2k r k r SFM 17 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Master Equation www.quanticol.eu The equation for the time evolution of the probability mass for CTMC is known as Kolmogorov equation. In the context of Population Processes is often know as master equation. There is one equation per state x ∈ D , for the probability mass P ( x , t ) , which considers the inflow and outflow of probability at time t . dP ( x , t ) � � = r η ( x − v η ) P ( x − v η , t ) − r η ( x ) P ( x , t ) dt η ∈T η ∈T These differential equations, for finite state spaces, can be solved by numerical integration or by using specialised methods for CTMC (uniformization). Finite state projections can be used for infinite state spaces. The cost is polynomial in the size of the state space. SFM 18 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Example: SIRS model www.quanticol.eu dP ([ 1 , 1 , 1 ] , t ) / dt = 2 k r P ([ 1 , 2 , 0 ] , t ) + 2 k s P ([ 0 , 1 , 2 ] , t ) − k i P ([ 1 , 1 , 1 ] , t ) − 2k i k s k r k r P ([ 1 , 1 , 1 ] , t ) − k s P ([ 1 , 1 , 1 ] , t ) 2k r 2k i k s dP ([ 3 , 0 , 0 ] , t ) / dt = k s P ([ 2 , 0 , 1 ] , t ) 2k s k s k r k i 3k r 2k s dP ([ 0 , 3 , 0 ] , t ) / dt = 3k s 2k r k r 2 k i P ([ 1 , 2 , 0 ] , t ) − 3 k r P ([ 0 , 3 , 0 ] , t ) SFM 19 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Stochastic Simulation www.quanticol.eu An alternative to solve the master equation is to generate sample trajectories of the CTMC, and then extract statistical information from them. The most famous simulation algorithm is due to Doob-Gillespe. It is based on the fact that a CTMC can be factorized in two independent processes. � The time T x spent in a state (holding time) x is exponentially distributed with rate r 0 ( x ) = � η r η ( x ) (exit rate). � The probability of taking transition η (jump chain) is independent of T and is equal to r η ( x ) / r 0 ( x ) . SFM 20 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Stochastic Simulation www.quanticol.eu The Doob-Gillespie algorithm samples the time spent in a state and the next state according to the previous characterization. Let x be the current state (initially x 0 ) and t the current time (initially, t = 0). While t < t final 1. Sample dt ∼ T x (using dt = − log U / r 0 ( x ) , U uniform in [ 0 , 1 ] ) and update time to t + dt 2. Choose a transition η with probability r η ( x ) / r 0 ( x ) and update the state to x + v η . Complexity is proportional on the number of reactions and on the steps to be done till the final time is reached, which on average are bounded by t final · max x r 0 ( x ) . SFM 21 / 59

quan�col . ........ . . . ... ... ... ... ... ... ... Example: SIRS model www.quanticol.eu 1.0 S I R 0.8 0.6 occupancy 0.4 0.2 0.0 0 5 10 15 20 Time SFM 22 / 59