Large deviations for Poisson driven processes in epidemiology Peter Kratz joint work with Etienne Pardoux Aix Marseille Université CEMRACS Luminy. August 20, 2013 Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Overview Motivation 1 Deterministic compartmental models Long-term behavior Stochastic models Dynamically consistent finite difference schemes General models 2 Poisson models Law of large numbers Large deviations 3 Rate function Large deviations principle (LDP) Exit from domain Diffusion approximation 4 Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Overview Motivation 1 Deterministic compartmental models Long-term behavior Stochastic models Dynamically consistent finite difference schemes General models 2 Poisson models Law of large numbers Large deviations 3 Rate function Large deviations principle (LDP) Exit from domain Diffusion approximation 4 Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
A model with vaccination SIV model by Kribs-Zaleta and Velasco-Hernández (2000) S = # of susceptibles, I = # of infectives, V = # of vaccinated, N = S + I + V population size µ N β SI / N ✲ ✲ (birth rate) (infection rate) µ I ✲ S I µ S ✛ ✛ cI (death rate) (recovery rate) ❙ ✓ ✼ ♦ ❙ ✓ ❙ θ V ❙ ✓ ❙ (loss of vaccination) ❙ ✓ φ S σβ VI / N ( σ ∈ [ 0 , 1 ]) ❙ ❙ ✓ (vaccination rate) (infection of vaccinated) ❙ ✓ ❙ ✓ V ❙ ❙ ✇ ✓ µ V ✲ Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
ODE representation S ′ = µ N − β SI N − ( µ + φ ) S + cI + θ V l ′ = β ( S + σ V ) I (1) − ( µ + c ) I N V ′ = φ S − σβ VI N − ( µ + θ ) V Equation (1) has a unique solution satisfying 0 ≤ S , I , V ≤ S + I + V = N Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
ODE and equilibria We are interested in the long-term behavior of the model Does the disease become extinct or endemic? Find equilibria of the ODE (1) R 0 = basic reproduction number = “# of cases one case generates in its infectious period” a disease-free equilibrium ( I = 0) of (1) exists R 0 < 1 ⇒ the equilibrium is asymptotically stable ˜ R 0 = basic reproduction number without vaccination ˜ R 0 > 1 ⇒ the disease-free equilibrium is unstable R 0 < 1 < ˜ R 0 (and appropriate parameter choice) ⇒ two endemic equilibria ( I > 0) exist One is asymptotically stable, one is unstable Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Equilibria of the ODE Reduction of dimension s = S / N = 1 − i − v = proportion of susceptibles v = V / N = proportion of vaccinated 1 0.86 ❞ disease-free equilibrium 0.59 unstable endemic equilibrium ❞ 0.46 stable endemic equilibrium ❞ i = I / N proportion of infectives 0.18 0.31 1 0 Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Stochastic models Stochastic model corresponding to the deterministic model Replace the deterministic rates by (independent) non-homogenous Poisson processes An individual of type S becomes of type I at the jump time of the respective processes Jump rates are constant in-between jumps Example. Infection rate (at time t ): β S ( t ) I ( t ) N Questions What is the difference between the two processes for large N ? Can the stochastic process change between the domains of attraction of different stable equilibria (for large N )? When does this happen? For which population size N is it possible/probable? Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Alternative model with immigration We require a modification of the SIV-model in order to ensure that the process doesn’t get stuck at I = 0 Immigration of infectives at rate α > 0 (small) S ′ = µ N − β SI N − ( µ + φ + α ) S + cI + θ V l ′ = α N + β ( S + σ V ) I (2) − ( µ + c + α ) I N V ′ = φ S − σβ VI N − ( µ + θ + α ) V Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Equilibria with immigration For α ≈ 0 (but α > 0 sufficiently small) the equilibria and the regions of attraction remain similar The “disease-free” equilibrium satisfies I ≈ 0 (but I > 0) v = V / N = proportion of vaccinated 1 0.83 ❞ disease-free equilibrium 0.64 unstable endemic equilibrium ❞ 0.42 stable endemic equilibrium ❞ i = I / N proportion of infectives 0.01 0.14 0.34 1 Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Numerical solution of the ODE We require a numerical method for solving the ODE Anguelov et al. (2014): Non-standard finite difference scheme which is elementary stable The standard denominator h of the discrete derivatives is replaced by a more complex function φ ( h ) Nonlinear terms are approximated in a nonlocal way by using more than one point of the mesh The equilibria and their local stability is the same as for the ODE Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Overview Motivation 1 Deterministic compartmental models Long-term behavior Stochastic models Dynamically consistent finite difference schemes General models 2 Poisson models Law of large numbers Large deviations 3 Rate function Large deviations principle (LDP) Exit from domain Diffusion approximation 4 Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Poisson models � � t k Z N ( t ) := x + 1 � � N β j ( Z N ( s )) ds h j P j (3) N 0 j = 1 � t � � t b ( Z N ( s )) ds + 1 � � N β j ( Z N ( s )) ds = x + h j M j N 0 0 j d = number of compartments (susceptible individuals, ...) N = “natural size” of the population Z N i ( t ) = proportion of individuals in compartment i at time t A = domain of process (compact) P j ( j = 1 , . . . , k ): independent standard Poisson processes M j ( t ) = P j ( t ) − t : compensated Poisson processes h j ∈ Z d : jump directions β j : A → R + : jump intensities b ( x ) = � j h j β j ( x ) Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Law of large numbers Deterministic model � t � t k � φ ( t ) := x + b ( φ ( s )) ds = x + h j β j ( φ ( s )) ds (4) 0 0 j = 1 Theorem (Kurtz) x ∈ A, T > 0 , β j : R d → R + bounded and Lipschitz. There exist constants C 1 ( ǫ ) , C 2 > 0 (C 1 ( ǫ ) = Θ( 1 /ǫ ) as ǫ → 0 , C 2 independent of ǫ ) such that for N ∈ N , ǫ > 0 N | Z N ( t ) − φ ( t ) | ≥ ǫ log N ǫ 2 ) . � � ≤ C 1 ( ǫ ) exp ( − C 2 P sup t ∈ [ 0 , T ] In particular, Z N → φ almost surely uniformly on [ 0 , T ] . Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Overview Motivation 1 Deterministic compartmental models Long-term behavior Stochastic models Dynamically consistent finite difference schemes General models 2 Poisson models Law of large numbers Large deviations 3 Rate function Large deviations principle (LDP) Exit from domain Diffusion approximation 4 Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Rare events Recall (LLN): Z N → φ almost surely uniformly on [ 0 , T ] But: A (large) deviation of Z N from the ODE solution φ is nevertheless possible (even for large N , cf. Campillo and Lobry (2012)) Fix T > 0; D ([ 0 , T ]; A ) := { φ : [ 0 , T ] → A | φ càdlàg } ; Quantify P [ Z N ∈ G ] , P [ Z N ∈ F ] for G ⊂ D open, F ⊂ D closed ( N large) Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Legendre-Fenchel transform Legendre-Fenchel transform x ∈ A position, y ∈ R d direction of movement L ( x , y ) := sup θ ∈ R d ℓ ( θ, x , y ) for β j ( x )( e � θ, h j � − 1 ) � ℓ ( θ, x , y ) = � θ, y � − j � � L ( x , y ) ≥ L x , � j β j ( x ) h j = 0 L ( x , y ) < ∞ iff ∃ µ ∈ R k + s.t. y = � j µ j h j and µ j > 0 ⇒ β j ( x ) > 0 “Local measure” for the “energy” required for a movement from x in direction y Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
Rate function Rate function ( x ∈ A ) �� T 0 L (˜ φ ( t ) , ˜ for ˜ φ ( 0 ) = x and ˜ φ ′ ( t )) dt φ is abs. cont. I x , T (˜ φ ) := ∞ else I x , T ( φ ) = 0 iff φ solves (4) on [ 0 , T ] Interpretation of I x , T (˜ φ ) : the “energy” required for a deviation from φ Peter Kratz: Large deviations for Poisson driven processes in epidemiology Aix Marseille Université
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