The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension SIR epidemics with stages of infection Matthieu Simon (ULB) Joint work with Claude Lef` evre (ULB) Matrix Analytic Methods Conference 28-30 June 2016 Matthieu Simon (ULB) SIR epidemics with stages of infection 0
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Table of contents 1 The model 2 Martingales for the epidemic outcome 3 Contagion per infective 4 Semi-Markov extension Matthieu Simon (ULB) SIR epidemics with stages of infection 1
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Table of contents 1 The model 2 Martingales for the epidemic outcome 3 Contagion per infective 4 Semi-Markov extension Matthieu Simon (ULB) SIR epidemics with stages of infection 2
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension SIR models SIR models : spread of an epidemic amongst a closed and homogeneous population, according to the following scheme : S I R S : healthy individuals, but susceptible to be contaminated. I : infected individuals, who can infect the healthy ones (independently of each other). R : infectives whose infection period is finished. They take no longer part to the infection process (removed). Matthieu Simon (ULB) SIR epidemics with stages of infection 3
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension SIR models with stages We consider a SIR model with L stages of infection 1 , 2 , ..., L (e.g. for different degrees of infectiousness). p types of elimination ⋆ 1 , ⋆ 2 , ..., ⋆ p . (e.g. death or immunization). At the beginning : n susceptibles ans m j infectives in phase j . When contaminated, a susceptible begins in an initial stage given by α . Matthieu Simon (ULB) SIR epidemics with stages of infection 4
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Transitions between stages Contagion process When in stage j , an infective contaminates the s available susceptibles according to a Poisson process with parameter s β j n . Transitions for an infective For each infective, a Markov process { ϕ ( t ) } modulates the transitions between stages and the elimination time. � � Defined on ⋆ 1 , ⋆ 2 , ..., ⋆ p , 1 , 2 , ..., L and with generator 0 0 Q = . · · · A ❛ 1 ❛ 2 ❛ ♣ Here, t ∈ R + is the local time of an infection process. Matthieu Simon (ULB) SIR epidemics with stages of infection 5
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Epidemic outcome Let T be the ending time of the epidemic : T = inf { t ≥ 0 | I ( t ) = 0 } . We aim to determine the joint distribution of the statistics : S T : final size of the epidemic, R ( r ) : final number of eliminations of type r , T A T : cumulative total duration of all infection periods. Matthieu Simon (ULB) SIR epidemics with stages of infection 6
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Table of contents 1 The model 2 Martingales for the epidemic outcome 3 Contagion per infective 4 Semi-Markov extension Matthieu Simon (ULB) SIR epidemics with stages of infection 7
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Artificial time Time change : We follow the infectives one after the other. ⊲ Discrete time τ = 0 , 1 , 2 , ... S τ = number of susceptibles after τ infectives, R ( r ) = number of eliminations of type r after τ infectives, τ A τ = cumulative duration of the first τ infection periods. Initially, S 0 = n , A 0 = 0, R ( r ) = 0. 0 In this artificial time, the epidemic terminates at time ˜ T = inf { τ | τ + S τ = n + m } . By the characteristics of the model, T , R (1) T , ..., R ( p ) T ) d = ( S T , A T , R (1) T , ..., R ( p ) ( S ˜ T , A ˜ T ) . ˜ ˜ Matthieu Simon (ULB) SIR epidemics with stages of infection 8
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Useful relations in the artificial time Suppose that the τ -th infective begins in stage j . Then S τ − 1 ( k ) � S τ � � = 1 j ( k ; u ) , k u =1 A τ = A τ − 1 + D j , = R ( r ) R ( r ) τ − 1 + 1 j , r , τ 1 j ( k ) = I (a fixed group of k susceptibles escape from the infective) 1 j ( r ) = I (the infective will become an eliminated of type r ) D j = infection duration of the infective. Matthieu Simon (ULB) SIR epidemics with stages of infection 9
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Martingales for the epidemic outcome With the preceding relations, one can show that for each k = 0 , 1 , ..., n , θ ≥ 0 and ③ ∈ R p , the process � S τ � p � e − θ A τ z r R ( r ) , τ ≥ m = m 1 + · · · + m L τ q ( k , θ, ③ ) τ k r =1 is a martingale, provided that L � q ( k , θ, ③ ) = α j q j ( k , θ, ③ ) , j =1 p � 1 j ( k ) e − θ D j z r 1 j ( r ) . q j ( k , θ, ③ ) = E r =1 Matthieu Simon (ULB) SIR epidemics with stages of infection 10
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Joint distribution of S T , A T and R ( r ) T Applying the optional stopping theorem on this martingale for ˜ T = inf { τ | τ + S τ = n + m } , after having considered the effect of the initial infectives : Proposition For 0 ≤ k ≤ n, θ ≥ 0 and ③ ∈ R p : � S T � R � z r R ( r ) e − θ A T q ( k , θ, ③ ) S T E T k r =1 � n � L � q ( k , θ, ③ ) n q j ( k , θ, ③ ) m j . = k j =1 Matthieu Simon (ULB) SIR epidemics with stages of infection 11
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Some consequences of the preceding formula A triangular system to determine the distribution of S T : � s � � n � � n � L q ( k ) s P ( S T = s ) = q ( k ) n q j ( k ) m j k k j =1 s = k , � n P ( S T = s ) = 1 s =0 where q j ( k ) ≡ q j ( k , 0 , 0 ). The moments of A T and R ( r ) : T L � � � � � E [ A T ] = m j E D j + n − E [ S T ] E [ D α ] , j =1 � � L � � � R ( r ) = m j q (0 , 0 , ❡ r ) + n − E [ S T ] q j (0 , 0 , ❡ r ) . E T j =1 Matthieu Simon (ULB) SIR epidemics with stages of infection 12
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Table of contents 1 The model 2 Martingales for the epidemic outcome 3 Contagion per infective 4 Semi-Markov extension Matthieu Simon (ULB) SIR epidemics with stages of infection 13
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Contagion per infective To obtain the epidemic outcome, we only need the parameters p � 1 j ( k ) e − θ D j z r 1 j ( r ) . q j ( k , θ, ③ ) = E r =1 We only need to analyse the behaviour of a unique infective facing k susceptibles, who are immediately removed when infected. Let N ( k , t ) be the number of infections generated by this single infective up to time t ( t is the local time of the infectious period). Matthieu Simon (ULB) SIR epidemics with stages of infection 14
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension contagion process { ( N ( k ; t ) , ϕ ( t )) | t ∈ R + } is a Markov process with state space { ⋆ 1 , ..., ⋆ p , [(0 , 1) , ..., (0 , L )] , . . . , [( k , 1) , ..., ( k , L )] } , and its generator is 0 0 ❛ 1 · · · A 0 ( k ) A 1 ( k ) 0 · · · 0 ❛ ♣ ❛ 1 · · · 0 A 0 ( k − 1) A 1 ( k − 1) · · · 0 ❛ ♣ , ❛ 1 · · · 0 0 A 0 ( k − 2) · · · 0 ❛ ♣ . . . . . . . . . . . . . . . . . . ❛ 1 · · · 0 0 0 · · · A 1 (1) ❛ ♣ ❛ 1 · · · 0 0 0 · · · A ❛ ♣ where A 1 ( h ) = h n B and A 0 ( h ) = A − A 1 ( h ). Matthieu Simon (ULB) SIR epidemics with stages of infection 15
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Formula for the coefficients By using the structure of this last generator, one can show that Proposition For 1 ≤ j ≤ L, p � � � − 1 q j ( k , θ, ③ ) = ❡ j θ I − A 0 ( k ) z r ❛ r . r =1 The same formula holds for q ( k , θ, ③ ) except that α is substituted for ❡ j . Matthieu Simon (ULB) SIR epidemics with stages of infection 16
The model Martingales for the epidemic outcome Contagion per infective Semi-Markov extension Table of contents 1 The model 2 Martingales for the epidemic outcome 3 Contagion per infective 4 Semi-Markov extension Matthieu Simon (ULB) SIR epidemics with stages of infection 17
Recommend
More recommend
