Imperfect vaccination games and numerical procedures for MFG equilibriums

Transcription

1 Imperfect vaccination games and numerical procedures for MFG equilibriums Gabriel Turinici with L. Laguzet, F. Salvarani, G. Yahiaoui and E. Hubert CEREMADE, Université Paris Dauphine Institut Universitaire de France Workshop: PDE models for multi-agent phenomena InDAM, Rome Nov 28 th - Dec 2 nd, 2016 Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

2 Disclaimer: What follows is a THEORETICAL epidemiological investigation. It is not meant to be used directly for health-related decisions; if in need to take such a decision please seek professional medical advice. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

3 Outline 1 Vaccine scares, the need for MFG models 2 An analytical model: SIR-V 3 Taking into account the individual decisions Previous works Individual dynamics Definition of individual strategy Individual-societal equilibrium Taking into account a discount 4 Imperfect vaccination: efficacy, delays The model Optimal strategy and Nash equilibria Computing the equilibrium: background on -flows Numerical results Effects of failed vaccinations 5 SIR model with vital dynamics 6 Second order schemes for gradient flows and MFG 7 Conclusion Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

4 Vaccine scares: the need for MFG models Influenza A (H1N1) (flu) ( ) At 15/06/2010 flu (H1N1): deaths in 213 countries (WHO) France: 1334 severe forms (out of 7.7M-14.7M people infected) Countries Target coverage Effective rate of vaccination Germany 100 % 10% Belgium 100 % 6 % Spain 40 % < 4% France % 8.5 % Italy 40 % 1.4 % Previous vaccine scares (some have been disproved since): France: hepatitis B vaccines cause multiple sclerosis US: mercury additives are responsible for the rise in autism UK: the whooping cough (1970s), the measles-mumps-rubella (MMR) (1990s). Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

5 Further motivation: end of compulsory vaccination context in France: discussions on the end of general compulsory vaccination How is vaccination coverage evolving? Example: nobody vaccinates then an additional individual may vaccinate; if all vaccinate an additional invididual will not vaccinate. Will the vaccination coverage become unstable or chaotic? question: what are the determinants of individual vaccination hint: individual decisions sum up to give a global response; need a model. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

6 Outline 1 Vaccine scares, the need for MFG models 2 An analytical model: SIR-V 3 Taking into account the individual decisions Previous works Individual dynamics Definition of individual strategy Individual-societal equilibrium Taking into account a discount 4 Imperfect vaccination: efficacy, delays The model Optimal strategy and Nash equilibria Computing the equilibrium: background on -flows Numerical results Effects of failed vaccinations 5 SIR model with vital dynamics 6 Second order schemes for gradient flows and MFG 7 Conclusion Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

7 Influenza incidence historical data in France Sourse: Sentinelles network, INSERM/UPMC, Influenza A (H1N1) (flu) ( ) : see next. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

8 Influenza A (H1N1) (flu) ( ), France Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

9 The SIR-V model Global model βsidt γidt Susceptible Infected Recovered dv Vaccinated Figure: Graphical illustration of the SIR-V model. In this model all individuals are identical. β : probability of contamination, γ : recovery rates, dv (t) : measure of vaccination (!!.. cf. Bressan) I ds = βsidt dv (t) di(t) dt = βsi γi dr(t) dt = γi(t) S Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

10 Vaccination cost functional Cost for an infected person : r I. Cost of the vaccine (including side-effects) : r V. Global cost for the society (from the initial state X 0 = (S(0), I(0)) T ) : J(X 0, V ) = r I βsidt + r V dv (t) (1) 0 BUT this is not an individual perspective General tools: Abakuks, Andris, 1974 Optimal immunisation policies for epidemics,... viscosity solution (HJB version): LL & GT, Math. Biosci., 263: , Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

11 Outline 1 Vaccine scares, the need for MFG models 2 An analytical model: SIR-V 3 Taking into account the individual decisions Previous works Individual dynamics Definition of individual strategy Individual-societal equilibrium Taking into account a discount 4 Imperfect vaccination: efficacy, delays The model Optimal strategy and Nash equilibria Computing the equilibrium: background on -flows Numerical results Effects of failed vaccinations 5 SIR model with vital dynamics 6 Second order schemes for gradient flows and MFG 7 Conclusion Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

12 the vaccination is at birth only (probability p) ; at equilibrium vaccination occurs when the probability to be infected r V /r I. no time dependence; final state stationary. equilibrium ok if everybody does the same no eradication possible through voluntary vaccination, endemic state (coherent with literature) vaccine scare behavior : let other vaccinate (when perceived risk differs from that of majority) Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57 Taking into account the individual decisions: previous literature Bauch & Earn (PNAS 2004) consider the SIR-V model with births and deaths : ds = µ(1 p) βsidt µs di = (βsi γi)dt µi dr = γidt µr dv = µp β : probability of contamination, γ : recovery rates, dv (t) : measure of vaccination µ : the birth / death rate p the probability to vaccinate at birth

13 Taking into account the individual decisions: previous literature Francis (2004) : uses a SIR-V model to find an equilibrium. Results: They identify the vaccination region although the problem is not formulated as an optimization at the individual level. Galvani, Reluga & Chapman (PNAS 2006) consider a double SIR-V periodic model of flu with two age groups (break at 65yrs). Vaccination is separated from dynamics, once at the beginning of each season. Results : show that actual vaccine coverage is consistent with individual optimum; explain impact of age-targetted campaigns (children). Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

14 Taking into account the individual decisions: previous literature Bauch (2005) : time dependent vaccination rate p(t); probability to be infected is approximated by a rule of thumb (proportional to I(t)); the corresponding dynamics is a phenomenological proposal (m a parameter) : dp(t) = kp(1 p)(mr I I(t) r V ) dt B. Buonomo : vaccination as a feedback F. Fu : taking into account the topology (networks of acquaintances) Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

15 Individual dynamics βsidt γidt Susceptible Infected Recovered dv Vaccinated Global dynamics : continuous time deterministic ODE; is the master equation of the individual dynamics. rate βi rate γ Susceptible Infected Recovered... Vaccinated Individual dynamics: continuous time Markov jumps between Susceptible, Infected, Recovered and Vaccinated classes. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

16 Taking into account the individual decisions : individual strategy the simplest one, the pure strategy : vaccinate with certainty at some given instant τ [0, ] (τ = means no vaccination) In terms of probability to vaccinate a pure strategy is H( τ) with H( ) the Heaviside function. Here τ = 2.5. is it always optimal? NO does it give a stable equilibrium? NO Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

17 Taking into account the individual decisions : the cost of pure strategies Costs of pure strategies Π t vaccinating at t: J pure (Π 0 ) = r V, J pure (Π t ) = r I ϕ I (t) + r V (1 ϕ I (t)), J pure (Π ) = r I ϕ I ( ). more realistic mixed strategies (related game theory concepts in timing games cf. Drew & Tirole): the individual chooses a probability measure dϕ V over the space of all pure strategies The CDF function ϕ V (t) is increasing, right con tinuous with left limits (càdlàg) with ϕ V (0 ) = 0, ϕ V ( ) 1. In particular ϕ V (t) = probability to choose vaccination in the interval [0, t]. Overall cost of strategy ϕ V : J indi (ϕ V ; V ) = (1 ϕ V ( ))J pure (Π ) + 0 J pure (Π t )dϕ V (t) J indi (ϕ V ; V ) = r I ϕ I ( ) + ] 0 [r V r I ϕ I ( ) + (r I r V )ϕ I (t) dϕ V (t). Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

18 Discount and constraints discount factor D > 0 : use Φ I (t) = t 0 e Dτ dϕ I (τ), Φ I (0 ) = 0. J D pure(π t ) = r I Φ I (t) + e Dt r V (1 ϕ I (t)). Jpure(Π D ) = r I Φ I ( ). Cost of strategy ϕ V : Jindi D (ϕ V ) = r I Φ I ( )+ ] 0 [r I (Φ I (t) Φ I ( ))+r V e Dt (1 ϕ I (t)) dϕ V (t). constraints : if global vaccination is at maximal rate u max : V (t + t) V (t) u max t. umax t Probability of being vaccinated in [t, t + t]: S(t). A pure strategy is transformed into a queue (exponential). Individual constraint dϕ V (t) umax S(t) (1 ϕ V (t))dt. If everybody has the same ϕ V then dv (t) = dϕ V (t) 1 ϕ V (t) S(t) (when this operation makes sense). Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

19 Individual cost functional { ds = βsidt dv (t) di = (βsi γi)dt. vaccination at the society level with dv (t); it can have the form dv (t) = u G (t)dt. Society dynamics induces a cumulative probability of infection on [0, t] : ϕ I (t) solution of dϕ I (t) = βi(t)(1 ϕ I (t))dt, ϕ I (0) = 0. Individual decision: ϕ V (with constraints cf. above). Individual cost functional J indi (ϕ V ; V ) = r I ϕ I ( ) + ] 0 [r V r I ϕ I ( ) + (r I r V )ϕ I (t) dϕ V (t). Individual - global equilibrium condition (global strategy arise from individual decisions): dv (t) = S(t) 1 ϕ V (t) dϕ V (t) (when this operation makes sense) (Mean Field Games, cf. Lasry - Lions, Caines - Huang - Malhamé). related works: C. Gueant (graphs), O. Cardaliaguet (HJB), D. Gomes : hyp. of superlinear costs (here linear); pure strategy set is not compact. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

20 Theoretical results for discount=0 Analytic description of the Nash-MFG equilibrium (G.T., L. Laguzet 15) I 1 O 1 S Red region: vaccination in the societal model and in the individual model, green region : vaccination for the societal model but not for the individual, blue region: no vaccination in both models. Mean cost for an individual in the stable individual-global equilibrium is larger than the cost for the, non equilibrium, societal (non-individual) optimum state this is because in the green region, it is optimal for individuals to let other vaccinate. Conclusion: the stable strategy will be obtained even if it is more costly for everyone; the cost of anarchy in the model is strictly > 0. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

21 Equilibrium for D > 0, u max = Theoretical results, still analytical (GT, LL 2015) I Ω n Ω i (S, I ) Ω d S White region: no vaccination, Π gray region : delayed vaccination (new behavior, Π t ) dotted region: immediate vaccination, Π 0 Comparing Π 0 and Π (Francis) is not enough, more complicated dynamics than D = 0. More complicated regions for u max < (L. Laguzet, G. Yahiaoui, G. T ), uniqueness not proved. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

22 Application to Influenza A 2009/2010 vaccination The SIR model was fit to the observed vaccination coverage (J.P. Guthman et al. BEH 2010); the other parameters were chosen consistent with ranges from the literature (large CIs!). Time axis = weeks starting from W19Y2009; peak = W49Y2009 (30), vaccination peak = W51Y2009 (31-32), vaccination end W05Y2010 (40) Comparison of numbers of cases Real Model Comparison of cumulative numbers of cases Real Model Comparison of cumulative numbers of vaccination Real Modeled Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

23 Application to Influenza A 2009/2010 vaccination Infection risk 30 Vaccination intensity 30 Model based, optimal risk criterion Simpler criterion, model based Model free, data-driven criterion Figure: Time axis = weeks starting from W19Y2009; peak = W49Y2009 (30), vaccination peak = W51Y2009 (31-32), vaccination end W05Y2010 (40) The results indicate that the risk perception was inhomogeneous; first group to stop vaccination perceived relative risk r V /r I 5% 10%!! (week 32). The last group corresponds to r V /r I < 1% (week 40; model cannot be more precise with available data). Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

24 Summary for MFG for the SIR-V model Some remarks not a potential (i.e., benevolent planner) game (cost or anarchy...) some cost C(individual, global) appears (here J indi (ϕ V ; V ) ) together with a mapping relating societal and individual strategies; no symmetry for C. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

25 Outline 1 Vaccine scares, the need for MFG models 2 An analytical model: SIR-V 3 Taking into account the individual decisions Previous works Individual dynamics Definition of individual strategy Individual-societal equilibrium Taking into account a discount 4 Imperfect vaccination: efficacy, delays The model Optimal strategy and Nash equilibria Computing the equilibrium: background on -flows Numerical results Effects of failed vaccinations 5 SIR model with vital dynamics 6 Second order schemes for gradient flows and MFG 7 Conclusion Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

26 The individual model in DISCRETE TIME Failed Vaccination (F ) f λn (1 β n T In) β n T In Susceptible (S) β n T In Infected (I 0 ) 1 γ 0 T Infected (I 1 ) 1 γ 1 T... γ 0 T γ 1 T Recovered (1 f) λn (1 β n T In) α0β n T In α1β n T In αθ 1β n T In αθβ n T In (R) Vaccinated (not infected) (V 0 ) Partially Immunized (V 1 )... Partially Immunized (V Θ 1 ) Lost Immunity (V Θ ) Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

27 Individual dynamics: Dynamics: DISCRETE TIME controlled Markov chain P(M n+1 = S M n = S) = (1 λ n ) (1 β T n I n ) P(M n+1 = I 0 M n = S) = β T n I n P(M n+1 = V 0 M n = S) = (1 f ) λ n (1 β T n I n ) P(M n+1 = F M n = S) = f λ n (1 β T n I n ) P(M n+1 = R M n = I Ω ) = 1 P(M n+1 = R M n = I ω ) = γ T ω, ω = 0,..., Ω 1 P(M n+1 = I ω+1 M n = I ω ) = 1 γ T ω, ω = 0,..., Ω 1 P(M n+1 = I 0 M n = V θ ) = α θ β T n I n, θ = 0,..., Θ 1 P(M n+1 = I 0 M n = V θ ) = α θ β T n I n, θ = 0,..., Θ 1 P(M n+1 = I 0 M n = F ) = β T n I n. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

28 Individual cost The cost for an individual has three components: the cost r I of being infected before vaccination the cost r V of vaccination plus a possible cost of being infected while immunity is still building or after the persistence period the cost r V of failed vaccination plus a possible cost of being infected For an individual starting at M 0 = S, the total cost is ( ) J indi (ξ; V ) = r V P n<n {M n+1 = V 0, M n V 0 } M 0 = S ( ) +r V P n<n {M n+1 = F, M n F } M 0 = S ( ) +r I P n<n {M n+1 = I 0, M n I 0 } M 0 = S Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

29 Individual strategy is a probability density on {t 0,..., t N 1 } { }: ξ Σ N+1 = {(x 0,..., x N ) R N+1 x k 0, x x N = 1}. ξ n : probability that the individual vaccinates at time t n (if it was not infected before t n ); before the beginning of the epidemic he knows the time t n at which he will vaccinate (unless he is already infected by that time). Conditional rates λ n are derived from ξ. Mapping between λ = (λ n ) N 1 n=0 and ξ: ξ := N 1 n 1 (1 λ n ), ξ n := λ n (1 λ k ), n N 1 n=0 ξ n k=0, if ξ n + + ξ > 0 ξ n + + ξ n N 1 : λ n = 0, otherwise Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

30 The individual minimal cost The individual cannot change the overall societal vaccination, he can only choose his own vaccination strategy ξ. J indi (ξ; V ) = ξ, g V, g V, ξ R N+1. J indi (ξ; V ) to be minimized under the constraint ξ Σ N+1. Compatibility relationship between ξ and V when all individuals follow this vaccination policy: V n (the societal vaccination) = λ n (ξ)s n Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

31 Existence of equilibria Theorem (F. Salvarani, G.T., 16) The vaccination dynamics model admits a Nash equilibrium. Idea: existence: fixed point (Kakutani); uniqueness?? C ξ := g V with strategy ξ E(ξ): maximum gain obtained by an individual if he changes unilaterally his strategy (and everybody else remaining with the strategy ξ): E(ξ) = ξ, C ξ min η, C ξ 0 η Σ N+1 Σ N+1 : space of all possible strategies Equilibrium: it corresponds to a ξ such that E(ξ) = 0 (not really a potential game), difficult to exploit numerically. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

32 Computing the equilibrium: background on -flows (A, d) = metric space, F : A R {+ } a convex function. Evolution equation (with a sense to be defined latter) x (t) = F (x(t)). (2) JKO scheme (IE = implicit Euler): x k x(kτ) and d(x k, η) 2 x k+1 argmin η A + F (η). (3) 2τ When τ 0 the limit curve satisfies (2). Use for MFG? Not as such, because for instance, for F (η) = C(η, η), obtain a -flow converging to the minimum of F i.e. a benevolent planner perspective. Algebraic example (bi-linear, Hilbert) C(η, ξ) = η, Lξ JKO goes to the minimum of η, Lη = η,l+lt η 2, i.e., the anti-symmetric part of L (here the time) vanishes. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

33 Explicit numerical schemes on metric spaces On the contrary a (semi-) explicit numerical scheme (Explicit Euler) would yield x k η 2 x k+1 argmin η A + η, Lx k. (4) 2τ and the dynamics x (t) = Lx(t), i.e., x(t) = e Lt x 0. Explicit numerical schemes are needed. Typical situation C(ξ I, ξ G ) (individual and global arguments). When individual is at ξk I the corresponding global is at ξg k (here ξi k ) and the next individual strategy is ξk+1 I d(ξk I argmin, η)2 η A + C(η, ξk G ). (5) 2τ (cf. G. Carlier and co-workers on best reply ; P. Cardialiaguet et al. on fictitious play ). Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

34 Explicit numerical schemes on metric spaces Question: explicit τ 0 has a meaning? JKO strategy: variational interpolation, then compactness by equi-continuity and uniform boundedness. Bounds (assumptions: Lipschitz in the global argument; global cost has relatively compact level sets): C(ξ k+1, ξ k+1 ) C(ξ k+1, ξ k ) + Ld(ξ k+1, ξ k ) C(ξ k+1, ξ k ) + d(ξ k+1, ξ k ) 2 + τl 2 /2 2τ C(ξ k, ξ k ) + τl 2 /2... C(ξ 0, ξ 0 ) + TL 2 /2. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

35 The algorithm How to compute the equilibrium? Step 1: Choose a step τ > 0 and a starting distribution ξ 1. Set iteration count k = 1. Step 2: Compute dist(η, ξ k ) 2 ξ k+1 argmin η ΣN+1 + η, C ξk. (6) 2τ Step 3: If E(ξ k+1 ) is smaller than a given tolerance then stop and exit, otherwise set k k + 1 and go back to Step 1. Remark: similar to an explicit -flow (depends on which distance is chosen). related to best reply and fictitious play learning methods in game theory. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

36 Short persistence, perfect efficacy Total simulation time T = 1 (one year), number of time instants: N = (three times a day) Recovery rate γ ω = γ = 365/3.2 (mean recovery time 3.2 days, Ω = 20) Reproduction number R 0 = 1.35, β = γr 0 ; Initial proportion of susceptibles S 0 = 0.94 and infected I 0 = ; Relative costs r I = 1, r V = β min = γ/s 0 and β(t) = β for t t β 2 := 1/2 (6 months) and then β(t) = β min for t > t β 2 = 1/2 Persistence of the vaccine: t 1 = 5/365, t 2 = 1/12 (one month, Θ = 93) and α θ = 1 1 [t1,t 2 ]. Vaccine efficacy: 100% (i.e., failure rate f = 0). Step τ = 0.1, 1000 iterations Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

37 9 Short persistence, perfect efficacy # Solution strategy ξ MFG supported at several time instants between 0.25 and 0.43, with 68% of nonvaccinating individuals time C Cost C ξ MFG of the optimal converged strategy ξ MFG, red line: cost of the non-vaccinating pure strategy (C ξ MFG ) N time Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

38 S Itotal S # Itotal time time Evolution of the susceptible class S n (left) and of the total infected class I n (right) Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

39 E(9) flowtime Decrease of the incentive to change strategy E(ξ k ) (not monotonic because not a -flow for E( ). Cost of the solution ξ MFG, C ξ MFG : M(ξ MFG ) = Cost of anarchy > 0 for instance for Π 0 (vaccination with certainty at time t = 0.0 unless infected by that time) has cost M(Π 0 ) = Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

40 9 Long persistence, perfect efficacy # Optimal converged strategy ξ MFG. The weight of the non-vaccinating pure strategy (i.e., corresponding to time t = ) is 91% time C9 # time Cost C ξ MFG of the optimal converged strategy ξ MFG. The thin horizontal line corresponds to the cost of the non-vaccinating pure strategy (C ξ MFG ) N+1. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

41 Effects of the failed vaccination rate on the strategy Individual vaccination policy with respect to the failed vaccination rate of the vaccine. 8 f 1 ξ failure rate (f ) r V = (the other parameters are unchanged) If f = 0.85 the probability of being infected is 14.38%. % of vaccination (1 ξ ) Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

42 Imperfect vaccination Continuous time-version: S (t) = βs(t)i(t) [ U (t) I (t) = β S(t) + F (t) + ] + 0 A(θ)V (t, θ)dθ I(t) γi(t) t V (t, θ) + θ V (t, θ) = βa(θ)v (t, θ)i(t) F (t) = fu (t) βf (t)i(t). Initial and boundary conditions S(0) = S 0, I(0) = I 0, t 0 : S(t) 0, F (0 ) = 0, V (0, θ) = 0, θ 0, V (t, 0) = (1 f )U (t), Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

43 Imperfect vaccination Continuous time-version: Analytic expressions V (t, θ) := (1 f )U (t θ) exp F (t) = f t 0 U (t θ) exp Model simpler form: S (t) = βs(t)i(t) U (t) I (t) I(t) [ = β + S(t) + U (t θ) 0 (fe t [ t t θ βi(τ)a(τ t + θ)dτ ], [ t t θ βi(τ)dτ ] dθ. t θ βi(τ)dτ + (1 f )A(θ)e t t θ βi(τ)a(τ t+θ)dτ ) dθ ] γ. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

44 Outline 1 Vaccine scares, the need for MFG models 2 An analytical model: SIR-V 3 Taking into account the individual decisions Previous works Individual dynamics Definition of individual strategy Individual-societal equilibrium Taking into account a discount 4 Imperfect vaccination: efficacy, delays The model Optimal strategy and Nash equilibria Computing the equilibrium: background on -flows Numerical results Effects of failed vaccinations 5 SIR model with vital dynamics 6 Second order schemes for gradient flows and MFG 7 Conclusion Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

45 SIR model with vital dynamics SIR model with birth/death and newborn vaccination: S (t) = µ(1 u) βsi µs (7) I (t) = βsi γi µi (8) R (t) = γi µr (9) V (t) = µ u µv, (10) where we used the cummulative vaccination function u(t) = t 0 p(τ)dτ which belongs to the set: C = {u : [0, [ R u(0) = 0, b a 0 : u(a) u(b) a b, u(b) u(a)}. (11) Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

46 SIR model with vital dynamics (in coll. with E. Hubert) SIR model with birth/death and newborn vaccination: Susceptible βi (t) t Infected γ t Recovered (S) (I) (R) µ t µ t µ t Vaccinated µ t Death without Death after (V ) infection (D 1 ) infection (D 2 ) Figure: Individual model is a continuous time non-homogeneous Markov chain. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

47 Infected SIR model with vital dynamics T in years (S 1, I 1 ) Susceptible Figure: Evolution of the susceptibles and infected classes in the time-independent model with parameters : µ = 1/80, R 0 = 15, γ/µ = 1000, β = R 0 (γ + µ), r = 1.0e 3, T = 400(years) and initial parameters S 0 = 7%, I 0 = 1.0e 4. For these parameters we have p csne = (convergently stable Nash equilibrium), S = 6.6%, I = 6.6e 08, (S(T ) = 6.6%, and I(T ) = 6.36e 34). Several hundreds years are required in order for the long time limit to be reached. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

48 SIR model with vital dynamics MFG- Nash equilibrium = fixed point of T : u C T (u) P(C) (P(C) = the ensemble of subsets of C) T (u) = {v C p v : p (τ) (r V r I φ u I (τ)) q (τ) (r V r I φ u I (τ)), q P, τ 0} (12). Theorem (E. Hubert, G.T. 2016) The SIR model with newborn vaccination admits at least one Nash equilibrium. Theorem (Kakutani-Fan-Glicksberg) Let L be a locally convex topological linear space and C a compact convex set in L. Let R(C) be the family of all closed convex (non empty) subsets of C. Then, for any upper semi-continuous point to set correspondence T from C to R(C), there exists a point x 0 C such that x 0 T (x 0 ). Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

49 SIR model with vital dynamics Numerics: iterative. For p n u n, minimize: min p(t) [0,1] { (p (t) p n (t)) 2 2τ + p (t) (r (t))} V r I φ un I. (13) The next iterate p n+1 is the solution of the minimization problem (13): )} p n+1 (t) = proj [0,1] {p n (t) τ (r V r I φ un I (t). Rk: 100k iterations needed. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

50 I SIR model with vital dynamics (S 0,I 0 ) (S(t),I(t)) (S 1,I 1 ) S Figure: We consider µ = 1/80, R 0 = 15, γ = 1000µ, β = R 0 (γ + µ), r = 0.001, T = 20, S 0 = 7%, I 0 = 1.0e 4. The long-term, time independent, convergently stable Nash equilibrium is attained at a level of p csne = 1 1/(R 0 (1 r)) = The evolution of S and I for the equilibrium (compare with figure 5). Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

51 SIR model with vital dynamics 1 probability of vaccination p(t) (T=20) p(t) (T=80) pcsne Figure: The solution p(t). Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

52 Outline 1 Vaccine scares, the need for MFG models 2 An analytical model: SIR-V 3 Taking into account the individual decisions Previous works Individual dynamics Definition of individual strategy Individual-societal equilibrium Taking into account a discount 4 Imperfect vaccination: efficacy, delays The model Optimal strategy and Nash equilibria Computing the equilibrium: background on -flows Numerical results Effects of failed vaccinations 5 SIR model with vital dynamics 6 Second order schemes for gradient flows and MFG 7 Conclusion Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

53 Second order schemes for gradient flows and MFG What about higher orders in τ? ODE: x (t) = f (x(t)) Implicit Euler x k+1 = x k + τf k+1, first order, corresponds to JKO Crank-Nicholson or modified midpoint x k+1 = x k + τ f k+1+f k 2, second order, It corresponds to the Variational Implicit Midpoint (VIM) scheme (G.T., G. Legendre 2016): d(x x k+1 argmin k,y) 2 x A 2τ + 2F ( x k+y 2 ) (with x+y 2 being the midpoint of the geodesic from x to y. ) Variational Implicit Euler : (EVIE) formulation : do a τ/2 IE (= JKO) step and then extrapolate on the geodesic. Rk: of course, additional smoothness of F is needed to have order 2. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

54 W 2 error estimated order Second order schemes for gradient flows and MFG Numerical results for ρ log(ρ): order JKO ref VIM order VIM ref VIM error JKO ref EVIE 10-5 error EVIE ref EVIE error VIM ref EVIE error EVIE ref VIM error VIM ref VIM error JKO ref VIM time step = number of time steps Figure: Left: error of the JKO scheme (dotted line) and of the VIM and EVIE schemes (solid lines) Right: numerical estimation of the order of convergence of the JKO scheme (dotted line) and of the VIM and EVIE schemes (solid lines) Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

55 Second order schemes for gradient flows and MFG What about 2nd order for (explicit) MFG? Recent work by L. Laguzet, 3 high order schemes inspired from Heun, RK3, RK4: The (standard, Hilbert space) Heun: p 1 = x k + τf (t k, x k ), x k+1 = x k + τ 2 The variational (metric space) Heun scheme [ ] f (t k, x k ) + f (t k+1, p 1 ). { d(η, ξk ) ξ 2 } k+1 argmin + C(η, ξ k ), η A 2τ { d(η, ξk ) 2 ξ k+1 argmin + 1 η A 2τ 2 C(η, ξ k) + 1 } 2 C(η, ξ k+1 ). Two minimizations are required in order to obtain ξ k+1. Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

56 Outline 1 Vaccine scares, the need for MFG models 2 An analytical model: SIR-V 3 Taking into account the individual decisions Previous works Individual dynamics Definition of individual strategy Individual-societal equilibrium Taking into account a discount 4 Imperfect vaccination: efficacy, delays The model Optimal strategy and Nash equilibria Computing the equilibrium: background on -flows Numerical results Effects of failed vaccinations 5 SIR model with vital dynamics 6 Second order schemes for gradient flows and MFG 7 Conclusion Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57

57 Final remarks Bibliography L. Laguzet & G. T., Bull. Math. Biol., 2015, 77(10) : L. Laguzet, G. T., G. Yahiaoui Equilibrium in an individual - societal SIR vaccination model in presence of discounting and finite vaccination capacity, doi = / , (2016) F. Salvarani, G. T., Individual vaccination equilibrium for imperfect vaccine efficacy and limited persistence, submitted (2016) (HAL) E. Hubert, G. T., Nash-MFG equilibrium in a SIR model with time dependent newborn vaccination, submitted (2016) (HAL) G. Legendre, G.T. Second-order in time schemes for gradient flows in Wasserstein and geodesic metric spaces (submitted, 16, HAL) Extensions a model with limited information may be more realistic structure by age groups geographical heterogeneity additional stochastic dynamics uniqueness, metric space framework Gabriel Turinici (CEREMADE & IUF) Vaccination Nash-MFG Indam, Rome, / 57