Large Deviations in Epidemiology : Malaria modelling

Transcription

1 Kouegou Kamen Boris Supervised by : Etienne Pardoux Ecole de printemps de la Chaire MMB - Aussois May 13, 218

2 Figure: Internet... - Mosquitoes are the deadliest animals for Humans ( 745 pers./an) - Guess who is second?

3 Structure Simplified Malaria model : SIRS-SI 1 Simplified Malaria model : SIRS-SI SIRS-SI Malaria Model 2 3 Large deviations for our process Extinction time probability Lower Bound 4

4 We consider an SIRS-SI Malaria model including both Human and mosquito populations: S h susceptibles Human..., S v susceptibles vectors (mosquitoes)..., Ψ h S h Ψ h I h Ψ h R h Ψ h H I λ h S vv h γ h I h S h I h R h ρ h R h Ψ vv S v λ vs v I hh I v Ψ vs v Ψ vi v H = S h + I h + R h, V = S v + I v are assumed to be constants.

5 Dynamic of the system : In Human s comparments, ds h dt di h dt dr h dt where ( λ h = λ V ) σ vσ hv h H = σ vv + σ hh βhv on humans I v = Ψ hh + ρ hr h λ h V I v = λ h Sh (γh + Ψh)Ih V = γ hi h (ρ h + Ψ h)r h Sh ΨhSh is seen as the infectious force of mosquitoes σ v denotes the maximum number of time a mosquito can bite a human per unit time σ h denotes the maximum number of bites a human can have per unit time β hv denotes the transmission probability from an infected mosquito to a susceptible human assuming contact

6 and in mosquito s one ds v dt di v dt I h = Ψ vv λ v Sv ΨvSv H I h = λ v Sv ΨvIv H where ( λ v = λ V ) σ vσ hh v H = βvh is the infectious force of humans on mosquitoes σ vv + σ hh β vh denotes the probability of transmition from an infected human ta a susceptible mosquito assuming contact Proportions : Pose s h = S h/h..., s v = S v/v..., V/H = m >> 1 and denote. a v(m) = λ v = σvσh β vh, a h(m) = λ h = σvσhm β vh σ vm + σ h σ vm + σ h

7 Renormalized deterministic system : di h = a h(m)i v(1 i h r h) γ hi h Ψ hi h dt dr (1) h = γ hi h ρ hr h Ψ hr h dt di v = a v(m)i h(1 i v) Ψ vi v dt with s h = 1 i h r h and s v = 1 i v (1) is well posed for any given initial condition on his domain : { } (2) D := (i h, r h, i v) R 3 + : i h + r h 1, i v 1 if a h(m)a v(m) < Ψ v(γ h + Ψ h) : z fde = (,, ) stable. if a h(m)a v(m) > Ψ v(γ h + Ψ h): and z end = (i h, γ h ρ h + Ψ h i h, z fde = (,, ) unstable 1 ) stable. 1 + Ψv a v(m)i h Basic reproductive number: R = a h(m)a v(m)/ψ v(γ h + Ψ h) average number of new cases induced by one cases.

8 Renormalized Transition rates : For x = (x 1, x 2, x 3) β 1(x) = a h(m)x 3(1 x 1 x 2), β 2(x) = γ hx 1, β 3(x) = Ψ hx 1 β 4(x) = ρ hx 2 β 5(x) = Ψ hx 2 β 6(x) = a v(m)x 1(1 x 3) β 7(x) = Ψ vx 3 We note Zt H,V = (i H t, rt H, i V t ). Thus Infected and recovered humans are described respectively by and i H t r H t = i H + 1 H 1 H = r H + 1 H Samely define i V t 1 H Nβ1 (Z H,V β3 (Z H,V β2 (Z H,V β5 (Z H,V Q 1(ds, du) 1 H Q 3(ds, du) Q 2(ds, du) 1 H Q 2(ds, du) β2 (Z H,V β4 (Z H,V Q 2(ds, du) Q 2(ds, du)

9 General form of the process : Zt H,V = z + 1 k h j(m) H Hβj (Z H,V Proposition: (Large Number Law : m is fixed ) Q j(ds, du), (k = 7) For any given initial condition z, the Renormalized stochastic process Zt H,V (m) converges to the deterministic one Z t(m) a.s uniformly on [, T] and z. That is lim sup Zt H,V (m) Z H,V + t(m) = a.s t T Proposition: (Large Number Law : m goes to infinity) For any given initial condition z, the Renormalized stochastic process Zt H,V (m) converges to the deterministic one Zt a.s uniformly on [, T] and z. where Z is the same deterministic process as Z t including the fact that lim ah(m) = lim σ vσ hm β vh = a h, lim m + m + σ vm + σ av(m) = h m + lim m + σ vσ h σ vm + σ h =

10 a sequence X n define on a space X isatisfies a LDP with good rate function I, * I is lower semicontinuous with level set {x X I(x) a} compact. ** For O, F X respectively open and closed sets lim inf n 1 log P [Xn O] inf n I(x), x O Consider the general form of processes defined on D([, T], R d ) Z N,z t = z + 1 N k h j 1 lim sup log P [Xn F] inf n n I(x) x F Nβj (s,z N,x Q j(ds, du) Good rate Function : Let note for all φ D([, T], R d ) L(t, φ, φ) = sup θ, φ k t β j(t, φ)(e θ,hj 1) θ R d then the Cramer s Transform expression is T L(t, φ, I T(φ) = φ)dt if φ is absolutely continuous + if not

11 For a Poisson random variable of parameter λ >, Log-Laplace transform : Λ(θ) = λ(e θ 1) Legendre transform : Shwartz and Weiss rate function : Î T(φ) = T f (y, λ) := Λ (y) = y log y λ y + λ inf c C t(φ) k f (c j, β j(t, φ)) { where C t(φ) = c R k + φ. t = k hjcj and } cj > only if βj(t, φ) > Interchange integration and infimum : Î T(φ) = T inf c C T (φ) C T(φ) = {c Lebesgue integrable define on [, T] s.t L(c(t), φ t)dt c(t) C t(φ)}. Of course, Î T(φ) = I T(φ).

12 Theorem (under Assumptions...) Z N,z satisfies a LDP with good rate function I T on D([, T]; R d +), locally uniformly in the initial condition z Proposition: ( Malaria model : m fixed) Z H,V satisfies a LDP with good rate function I T on the space D([, T]; D). That is for any O, F D([, T]; D) open and closed sets respectively,, 1 lim inf H,V + H log P [ Z H,V O ] inf IT(φ) φ O 1 lim sup H,V + H log P [ Z H,V F ] inf IT(φ) φ F

13 Freidlin-Wentzell Theory: (Exit time Problem) For our SIRS-SI model, Remember if R > 1, two equilibriums states : z fde(unstable) and z end(stable). O z end := D {i h = i v = } is the attractive domain of z end. If z O z end. * The law of large number says that for H, V are large, Z H,V z end. ** Large Deviations quantifies the rare event of going far from the limit z end Let denote then τ H,V z How long could take the process to exit O z end???? τ H,V z e HĒ for large H Proposition: (Under assumptions...) and it follows that = inf{t > Z H,V t / O z end } [ lim P e (Ē δ)h < τz H,V < e (Ē+δ)H] = 1 δ > H,V + 1 lim H,V + H log E [ τz H,V ] = Ē

14 where (3) Ē = inf { E(z end, z) : z O z end } with (4) E(z end, z) = inf { E(z end, z, T) : T > } and (5) E(z end, z, T) = inf { I T (φ) : φ D([, T]; D), φ() = z end, φ(t) = z } (3), (4) and (5) form a Bolza type minimisation Problem in optimal control theory. subject to the dynamic φ t = Ē = min z,t,φ,c 7 h jc j t. T L(c t, φ t)dt Pontryagin minimum principle is helpfull to have some intiution on the optimal trajectory.

15 The lower bound : Simplified Malaria model : SIRS-SI 1 lim N + N log P [ Z N,z O ] inf IT(φ). φ O Process : Z N,z t = z + 1 N k h j LLN limit : Z t = z + Nβj (s,z N,z k h j β j (s, Z s) ds Q j(ds, du), if Q N j = 1 N M.A.P with intensity ds Ndu, QN = (Q N 1,..., Q N k ). then Z N,z t = z + k h j βj (s,z N,z and if we consider the map defined on M k by : η = (η 1,..., η k) (6) Φ z t (η) := x + k h j βj(s,φ z η j(ds, du) Q N j (ds, du), we have Φ z t (Q N ) = Z N,z and Z t = Φ x t (Λ 2 ) where Λ 2 = Leb Leb

16 Well known : Q N satisfies a LDP on M k with rate function say J T If φ z was a continuous map : CONTRACTION PRINCIPLE At least one φ in O is absolutely continuous. Note that if φ is absolutely continuous then φ t = Φ z t (η φ ) where η φ has density h j(s, u) = cj(s) β j(s, φ) 1 (,β j (s,φ)](u) + 1 [βj (s,φ), β](u), c C T(φ). and β = sup t β(t, φ) QUASICONTINUITY : For every L, R we have N large enough and δ small enough [ ( P Z N,xN φ T > L ; d T, β Q N, η φ) ] δ e NR where d T, β (η, ν) := k sup t,u New rate function form : Ĩ T(φ) = inf ηj ([, t] [, u]) νj ([, t] [, u]). { } J T(η) : η (Φ z ) 1 (φ)

17 A More realistic model we had in mind: (µ 1h, µ 2h N h ) (µ 1h, µ 2h N h ) (δ h, µ 1h, µ 2h N h ) (µ 1h, µ 2h N h ) (Λ h, Ψ h N h ) α h ν S h E h γ h I h ρ h R h h Ψv α v S v E v I v ν v (µ 1v, µ 2v Nv) (µ 1v, µ 2v Nv) (µ 1v, µ 2v Nv) Populations sizes are not constant, the domain is no longer compact but at least convex. Need more for Wentzell freidlin theory to work. Coercivity of z E(z, z)???