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Spread of epidemics on random graphs Viet Chi TRAN - Université Lille 1 - France December 11, 2015 1 Course 2 : Convergence of a random individual-based model to Volz’ equations 2 SIR on a configuration model graph FConfiguration model (CM) : Bollobas (80), Molloy Reed (95), Durett (07), van der Hofstad (in prep.) F Description of an SIR epidemics spreading on a configuration model graph : I Infinite system of denumberable equations, Ball and Neal (2008)). I 5 ODEs, Volz (2008), Miller (2011). I Recently: Barbour Reinert (2014), Janson Luczak Winridge (2014) F Individuals are separated into 3 classes : I Susceptibles St Infectious It I Removed Rt I 3 Stochastic model for a finite graph with N vertices F Only the edges between the I and R individuals are observed. The degree of each individual is known. F To each I individual is associated an exponential random clock with rate α to determine its removal. F To each open edge (directed to S), we associate a random exponential clock with rate β. F When it rings, the edge of an S is chosen at random. We determine whether its remaining edges are linked with S, I or R-type individuals. (a) (b) (c) I S I I I I R I R R I I I S I 4 Edge-based quantities F The idea of Volz is to use network-centric quantities (such as the number of edges from I to S) rather than node-centric quantities. F St , It , Rt , St , It , Rt , di , di (St )... µ finite measure on N and f bounded or > 0 function: P hµ, f i = k ∈N f (k )µ(k ). 5 Edge-based quantities F The idea of Volz is to use network-centric quantities (such as the number of edges from I to S) rather than node-centric quantities. F St , It , Rt , St , It , Rt , di , di (St )... µ finite measure on N and f bounded or > 0 function: P hµ, f i = k ∈N f (k )µ(k ). F We introduce the following measures: X X µSt (dk ) = δdu (dk ) µSI δdu (St ) (dk ) t (dk ) = u∈St µSR (dk ) t = X u∈It δdu (St ) (dk ) u∈Rt 5 Edge-based quantities F The idea of Volz is to use network-centric quantities (such as the number of edges from I to S) rather than node-centric quantities. F St , It , Rt , St , It , Rt , di , di (St )... µ finite measure on N and f bounded or > 0 function: P hµ, f i = k ∈N f (k )µ(k ). F We introduce the following measures: X X µSt (dk ) = δdu (dk ) µSI δdu (St ) (dk ) t (dk ) = u∈St µSR (dk ) t = X u∈It δdu (St ) (dk ) u∈Rt This sums up the evolution of the epidemic (but does not allow the reconstruction of the complicated graph on which the illness propagates). X It = Card(It ) = hµSI NtSI = hµSI du (St ) t , 1i, t , ki = u∈It 5 Dynamics F Global force of infection: βNtSI . − F Choice of a given susceptible of degree k : k /NtS− . So that the rate of infection of a given susceptible of degree k is: βkptI− . F The probability that its k − 1 remaining edges are linked to I or R is: ` m j p(j, `, m|k − 1, t) = NtSI −1 − NtSR − k −1 NtS −1 NtSS − 1j+`+m=k −1 1j<NtSI 1`≤NtSR − − − 6 Dynamics F Global force of infection: βNtSI . − F Choice of a given susceptible of degree k : k /NtS− . So that the rate of infection of a given susceptible of degree k is: βkptI− . F The probability that its k − 1 remaining edges are linked to I or R is: ` m j p(j, `, m|k − 1, t) = NtSR NtSI −1 − − k −1 NtS −1 NtSS − 1j+`+m=k −1 1j<NtSI 1`≤NtSR − − − SR F To modify the degree distributions µSI t− (idem for µt− ): We draw a sequence e = (eu )u∈It− of integers. • eu is the number of edges to the infectious individual u at t− . • not all sequences are admissible. The probability of drawing the sequence e is Q du ρU (e | j, µSI T− ) = u∈It− NtSI − j+1 eu 1{P eu =j+1, e is admissible} . 6 Renormalization F We are interested in increasing the number of vertices N without rescaling the degree distribution. µN,S , µN,SI , µN,SR . F We now consider µ(N),S , µ(N),SI and µ(N),SR where for ex: (N),S µt (dk ) = (N),SI (idem for µ0 1 N,S µ (dk ) N t with (N),S lim µ0 N→+∞ (N),SR with N̄0SI > ε and µ0 = µ̄S0 in MF (N) with N̄0SR > ε) 7 Renormalization F We are interested in increasing the number of vertices N without rescaling the degree distribution. µN,S , µN,SI , µN,SR . F We now consider µ(N),S , µ(N),SI and µ(N),SR where for ex: (N),S µt (dk ) = (N),SI (idem for µ0 1 N,S µ (dk ) N t with (N),S lim µ0 N→+∞ (N),SR with N̄0SI > ε and µ0 = µ̄S0 in MF (N) with N̄0SR > ε) F 3 SDE: (N),SI (N),SI (N),SI,f (N),SI,f hµt , f i = hµ0 , f i + At + Mt , where M (N),SI,f is a square integrable martingale started from 0 and with previsible quadratic variation in 1/N. 7 (N),IS,f F At (N),IS,f At : Z =− t (N),SI αhµs , f ids 0 + Z tX (N),I (N),S µs (k ) 0 k ∈N × X e∈U X βkps ρU (e|j + psN (j, `, m|k − 1, t) j+`+1≤k 1, µn,SI ) s f (m) + X f (du − eu ) − f (du ) ds, u∈IsN 8 (N),IS,f F At : t Z (N),IS,f =− At (N),SI αhµs , f ids 0 + Z tX (N),I (N),S µs (k ) 0 k ∈N X × X βkps psN (j, `, m|k − 1, t) j+`+1≤k ρU (e|j + 1, µn,SI ) s e∈U f (m) + X f (du − eu ) − f (du ) ds, u∈IsN Th: Under appropriate moment conditions, (N),S (N),SI (N),SR (µt , µt , µt SR (µ̄St , µ̄SI , µ̄ ) t∈R+ t t hµ̄SI t ,fi = hµ̄SI 0 ,fi )t∈R+ converge to a deterministic limit Z t − αhµ̄SI s , f ids 0 Z + t X 0 k ∈N∗ X βk p̄sI j+`+m=k −1 × f (m) + (j + 1) X k 0 ∈N∗ j, `, m (p̄sI )j (p̄sR )` (p̄sS )m k −1 0 k 0 µ̄SI s (k ) f (k 0 − 1) − f (k 0 ) µ̄Ss (k )ds SI hµ̄s , k i 8 Deterministic limit F Limit equations: µ̄St (k ) = µ̄S0 (k )θtk , θt = e−β Rt 0 p̄sI ds hµ̄SI t , f i = ... Z t hµ̄SR , f i = αhµ̄SI t s , f ids 0 + Z tX 0 k ∈N βk p̄sI (k − 1)p̄sR X k 0 ∈N 0 k 0 µSR s (k ) S µ̄s (k )ds f (k 0 − 1) − f (k 0 ) SR N̄s F This allows us to recover Volz’equations: • Choosing f ≡ 1 gives S̄t , Īt , • Choosing f (k ) = k gives N̄ S , N̄ SI , N̄ SR , from which we can deduce p̄I = N̄ SI /N̄ S ... 9 Volz’equations P Prop: let g(z) = k ∈N µ̄S0 (k )z k be the generating function of µ̄S0 . Z t θt = exp − β psI ds 0 S̄t =g(θt ), Īt = Ī0 + Z t β p̄sI θs g 0 (θs ) − αĪs ds 0 N̄ SI p̄tI = t S = p̄0I + N̄t p̄tS = N̄tSS N̄tS Z t g 00 (θs ) I I I − β p̄ (1 − p̄ ) − α p̄ ds. s s s g 0 (θs ) 0 Z t g 00 (θs ) = p̄0S + β p̄sI p̄sS 1 − θs 0 ds. g (θs ) 0 β p̄sI p̄sS θs 2 Recall the limit for mixing models: d S̄t = − β S̄t Īt , dt d Īt = β S̄t Īt − αĪt . dt 10 Volz’equations P Prop: let g(z) = k ∈N µ̄S0 (k )z k be the generating function of µ̄S0 . Z t θt = exp − β psI ds 0 S̄t =g(θt ), Īt = Ī0 + Z t β × N̄sSI − αĪs ds 0 N̄ SI p̄tI = t S = p̄0I + N̄t p̄tS = N̄tSS N̄tS Z t g 00 (θs ) I I I − β p̄ (1 − p̄ ) − α p̄ ds. s s s g 0 (θs ) 0 Z t g 00 (θs ) = p̄0S + β p̄sI p̄sS 1 − θs 0 ds. g (θs ) 0 β p̄sI p̄sS θs 2 Recall the limit for mixing models: d S̄t = − β S̄t Īt , dt d Īt = β S̄t Īt − αĪt . dt Here: d S̄t =g 0 (θt )θ̇t = −βg 0 (θt )θt p̄tI = −β N̄tS p̄tI = −β N̄tSI . dt 10 Sketch of the proof (N),SI (N),S , 1 + k 5 i < +∞, Assumption: supN∈N∗ hµ0 , 1 + k 5 i + hµ0 F Tightness: topology on MF (N). Roelly’s criterion. Aldous-Rebolledo criterion. |>ε ≤ε − A(N),SI,f P |Aτ(N),SI,f σN N P |hM (N),SI,f iτN − hM (N),SI,f iσN | > ε ≤ ε. F Convergence of the generators. • The identification of the limit is OK on [0, T ] IF T < τεN where (N),SI τεN = inf{t ≥ 0, Nt < ε}. F Uniqueness: • Gronwall’s lemma gives that solutions of the limiting equation have same mass and same moments of order 1 and 2. • Uniqueness of the generating function of µ̄IS which solves a transport equation. 11 600 400 0 200 Sizes of classes 800 1000 Degree distribution of the “initial condition” 0 1 2 3 4 5 6 Time Prop: For ε > 0, when N → +∞, the degree distribution when after [εN] infections converges to: 1 X pk (1 − z ε )k δk 1−ε k ≥0 where z ε is the solution of 1 − ε = f (1 − z), f being the generating function of the original degree distribution. 12