Long-time asymptotic behavior for nonlinear size-structured population models - Applications to prion diseases -

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1 1 / 23 Long-time asymptotic behavior for nonlinear size-structured population models - Applications to prion diseases - Pierre Gabriel INRIA Lyon, France In collaboration with Marie Doumic and Vincent Calvez SIAM PDE, San Diego, November 211

2 2 / 23 Prion diseases ï Transmissible Spongiform Encephalopathies are infectious, fatal and neurodegenerative diseases ï Examples: madcow disease (BSE), Kreuzfeld Jakob disease, scrapie disease ï The pathogenic agent, known as prion, is a protein ï This protein has the ability to aggregate under an abnormal form into polymers

3 3 / 23 polymer of size x µôxõ βôxõ τôxõ λ δ normal protein abnormal protein Figure: Prion proliferation

4 4 / 23 Prion model V ÔtÕ: quantity of monomers at time t, uôt,xõ: quantity of polymers of size x at time t. ³² ³± d V ÔtÕ λ δv ÔtÕ V ÔtÕ τôxõuôt,xõ dx, dt uôt,xõ V ÔtÕ τôxõuôt,xõ µôxõuôt,xõ FuÔt,xÕ, t x where FuÔxÕ : 2 βôyõuôyõκôx,yõdy βôxõuôxõ. x

5 4 / 23 Prion model V ÔtÕ: quantity of monomers at time t, uôt,xõ: quantity of polymers of size x at time t. ³² ³± d V ÔtÕ λ δv ÔtÕ V ÔtÕ τôxõuôt,xõ dx, dt uôt,xõ V ÔtÕ τôxõuôt,xõ µôxõuôt,xõ t x FuÔt,xÕ, where FuÔxÕ : 2 βôyõuôyõκôx,yõdy βôxõuôxõ. x

6 4 / 23 Prion model V ÔtÕ: quantity of monomers at time t, uôt,xõ: quantity of polymers of size x at time t. ³² ³± d V ÔtÕ λ δv ÔtÕ V ÔtÕ τôxõuôt,xõ dx, dt uôt,xõ V ÔtÕ τôxõuôt,xõ µôxõuôt,xõ t x FuÔt,xÕ, where FuÔxÕ : 2 βôyõuôyõκôx,yõdy βôxõuôxõ. x Assumptions on κôx,yõ: ï y κôx,yõdx 1

7 4 / 23 Prion model V ÔtÕ: quantity of monomers at time t, uôt,xõ: quantity of polymers of size x at time t. ³² ³± d V ÔtÕ λ δv ÔtÕ V ÔtÕ τôxõuôt,xõ dx, dt uôt,xõ V ÔtÕ τôxõuôt,xõ µôxõuôt,xõ t x FuÔt,xÕ, where FuÔxÕ : 2 βôyõuôyõκôx,yõdy βôxõuôxõ. x Assumptions on κôx,yõ: ï y κôx,yõdx 1 FuÔxÕdx βôxõuôxõ dx

8 4 / 23 Prion model V ÔtÕ: quantity of monomers at time t, uôt,xõ: quantity of polymers of size x at time t. ³² ³± d V ÔtÕ λ δv ÔtÕ V ÔtÕ τôxõuôt,xõ dx, dt uôt,xõ V ÔtÕ τôxõuôt,xõ µôxõuôt,xõ t x FuÔt,xÕ, where FuÔxÕ : 2 βôyõuôyõκôx,yõdy βôxõuôxõ. x Assumptions on κôx,yõ: ï ï y y κôx,yõdx 1 xκôx,yõdx y 2 FuÔxÕdx βôxõuôxõ dx

9 4 / 23 Prion model V ÔtÕ: quantity of monomers at time t, uôt,xõ: quantity of polymers of size x at time t. ³² ³± d V ÔtÕ λ δv ÔtÕ V ÔtÕ τôxõuôt,xõ dx, dt uôt,xõ V ÔtÕ τôxõuôt,xõ µôxõuôt,xõ t x FuÔt,xÕ, where FuÔxÕ : 2 βôyõuôyõκôx,yõdy βôxõuôxõ. x Assumptions on κôx,yõ: ï ï y y κôx,yõdx 1 xκôx,yõdx y 2 FuÔxÕdx βôxõuôxõ dx xfuôxõdx mass conservation

10 5 / 23 Outline The linear growth-fragmentation equation Steady states of the prion equation Long-time behavior for prion-type equations

11 6 / 23 Outline The linear growth-fragmentation equation Steady states of the prion equation Long-time behavior for prion-type equations

12 7 / 23 The linear growth-fragmentation equation t uôt,xõ x τôxõuôt,xõ FuÔt,xÕ

13 7 / 23 The linear growth-fragmentation equation t uôt,xõ x τôxõuôt,xõ FuÔt,xÕ Dual equation: ϕôt,xõ τôxõ t x ϕôt,xõ F ϕôt,xõ x where F ϕôxõ 2βÔxÕ κôy,xõϕôyõdy βôxõϕôxõ.

14 7 / 23 The linear growth-fragmentation equation t uôt,xõ x τôxõuôt,xõ FuÔt,xÕ Dual equation: ϕôt,xõ τôxõ t x ϕôt,xõ F ϕôt,xõ x where F ϕôxõ 2βÔxÕ κôy,xõϕôyõdy βôxõϕôxõ. General Relative Entropy (Michel, Mischler, Perthame 24): ϕôt,xõvôt,xõh uôt,xõ dx vôt,xõ

15 7 / 23 The linear growth-fragmentation equation t uôt,xõ x τôxõuôt,xõ FuÔt,xÕ Dual equation: ϕôt,xõ τôxõ t x ϕôt,xõ F ϕôt,xõ x where F ϕôxõ 2βÔxÕ κôy,xõϕôyõdy βôxõϕôxõ. General Relative Entropy (Michel, Mischler, Perthame 24): d dt ϕôt,xõvôt,xõh H ½ uôt,xõ vôt,xõ uôt,xõ dx vôt,xõ uôt,xõ uôt,yõ βôyõκôy,xõϕôt,xõvôt,yõ H H vôt,xõ vôt,yõ uôt,xõ vôt,xõ uôt,yõ Å dxdy for H convex vôt,yõ

16 8 / 23 Eigenproblem and convergence ΛUÔxÕ x τôxõuôxõ F UÔxÕ, ΛφÔxÕ τôxõ x φôxõ F φôxõ,

17 8 / 23 Eigenproblem and convergence ΛUÔxÕ x τôxõuôxõ F UÔxÕ, ΛφÔxÕ τôxõ x φôxõ F φôxõ, UÔxÕ, φôxõ, U φu 1.

18 8 / 23 Eigenproblem and convergence ΛUÔxÕ x τôxõuôxõ F UÔxÕ, UÔxÕe Λt ΛφÔxÕ τôxõ x φôxõ F φôxõ, φôxõe Λt UÔxÕ, φôxõ, U φu 1.

19 8 / 23 Eigenproblem and convergence ΛUÔxÕ x τôxõuôxõ F UÔxÕ, UÔxÕe Λt ΛφÔxÕ τôxõ x φôxõ F φôxõ, φôxõe Λt UÔxÕ, φôxõ, U φu 1. Any solution uôt, xõ to the growth-fragmentation equation satisfies uôt,xõe Λt t ρ UÔxÕ with ρ uôt,xõφôxõdx. [Perthame, Ryzhik ; Michel, Mischler, Perthame ; Laurençot, Perthame ; Cáceres, Cañizo, Mischler]

20 9 / 23 Assumptions To avoid concentration at x ï β τ È L1 locôr Õ,

21 9 / 23 Assumptions To avoid concentration at x ï β τ È L1 locôr Õ, ï C, γ s.t. ³² ³± x x γ κôz,yõdz min 1,C y and x γ τôxõ È L1 locôr Õ.

22 9 / 23 Assumptions To avoid concentration at x ï β τ È L1 locôr Õ, ï C, γ s.t. ³² ³± x x γ κôz,yõdz min 1,C y and x γ τôxõ È L1 locôr Õ. To avoid formation of infinitely long polymers ï xβôxõ lim x τôxõ.

23 1 / 23 Existence and uniqueness Theorem (Doumic, G.) Under the previous assumptions and some technical ones, there exists a unique solution ÔΛ,U,φÕ È R L 1 ÔR Õ W 1, loc Ô, Õ to the eigenvalue problem. Moreover we have: Λ, x α τ U È L p ÔR Õ, α γ, p È Ö1,, x α τ U È W 1,1 ÔR Õ, α, and k s.t. φ 1 x k È L ÔR Õ.

24 11 / 23 Outline The linear growth-fragmentation equation Steady states of the prion equation Long-time behavior for prion-type equations

25 12 / 23 Prion equation ³² ³± d V ÔtÕ λ δv ÔtÕ V ÔtÕ τôxõuôt,xõ dx, dt uôt,xõ V ÔtÕ τôxõuôt,xõ µôxõuôt,xõ t x FuÔt,xÕ.

26 12 / 23 Steady states ³² ³± λ δv V τ ÔxÕuÔxÕ dx, V x τôxõuôxõ µuôxõ FuÔxÕ.

27 12 / 23 Steady states ³² ³± λ δv V τ ÔxÕuÔxÕ dx, V x τôxõuôxõ µuôxõ FuÔxÕ. ï disease-free steady state: ū, V λ δ

28 12 / 23 Steady states ³² ³± λ δv V τ ÔxÕuÔxÕ dx, V x τôxõuôxõ µuôxõ FuÔxÕ. ï disease-free steady state: ū, V λ δ ï disease steady state: u, u and V

29 12 / 23 Steady states ³² ³± λ δv V τ ÔxÕuÔxÕ dx, V x τôxõuôxõ µuôxõ FuÔxÕ. ï disease-free steady state: ū, V λ δ ï disease steady state: u, u and V µu ÔxÕ V x τôxõu ÔxÕ Fu ÔxÕ

30 12 / 23 Steady states ³² ³± λ δv V τ ÔxÕuÔxÕ dx, V x τôxõuôxõ µuôxõ FuÔxÕ. ï disease-free steady state: ū, V λ δ ï disease steady state: u, u and V µu ÔxÕ V x τôxõu ÔxÕ Fu ÔxÕ

31 12 / 23 Steady states ³² ³± λ δv V τ ÔxÕuÔxÕ dx, V x τôxõuôxõ µuôxõ FuÔxÕ. ï disease-free steady state: ū, V λ δ ï disease steady state: u, u and V µu ÔxÕ V x τôxõu ÔxÕ Fu ÔxÕ ΛÔV Õ µ, u ÔxÕ ρ UÔV ;xõ,

32 12 / 23 Steady states ³² ³± λ δv V τ ÔxÕuÔxÕ dx, V x τôxõuôxõ µuôxõ FuÔxÕ. ï disease-free steady state: ū, V λ δ ï disease steady state: u, u and V µu ÔxÕ V x τôxõu ÔxÕ Fu ÔxÕ ΛÔV Õ µ, u ÔxÕ ρ UÔV ;xõ, ρ λßv δ τôxõuôv ;xõdx

33 12 / 23 Steady states ³² ³± λ δv V τ ÔxÕuÔxÕ dx, V x τôxõuôxõ µuôxõ FuÔxÕ. ï disease-free steady state: ū, V λ δ ï disease steady state: u, u and V µu ÔxÕ V x τôxõu ÔxÕ Fu ÔxÕ ΛÔV Õ µ, u ÔxÕ ρ UÔV ;xõ, ρ λßv δ τôxõuôv ;xõdx so V V.

34 12 / 23 Steady states ³² ³± λ δv V τ ÔxÕuÔxÕ dx, V x τôxõuôxõ µuôxõ FuÔxÕ. ï disease-free steady state: ū, V λ δ ï disease steady state: u, u and V µu ÔxÕ V x τôxõu ÔxÕ Fu ÔxÕ ΛÔV Õ µ, u ÔxÕ ρ UÔV ;xõ, ρ λßv δ τôxõuôv ;xõdx so V V.

35 13 / 23 Powerlaw coefficients If τôxõ τx ν and βôxõ βx γ, then the conditions which ensure the existence of eigenelements reduce to k : γ 1 ν.

36 13 / 23 Powerlaw coefficients If τôxõ τx ν and βôxõ βx γ, then the conditions which ensure the existence of eigenelements reduce to k : γ 1 ν. If additionally κôx,yõ 1 y κ x y, then we can compute explicitly ΛÔV Õ ΛÔ1ÕV γk 1 and UÔV;xÕ V k UÔ1;V k xõ.

37 13 / 23 Powerlaw coefficients If τôxõ τx ν and βôxõ βx γ, then the conditions which ensure the existence of eigenelements reduce to k : γ 1 ν. If additionally κôx,yõ 1 y κ x y, then we can compute explicitly ΛÔV Õ ΛÔ1ÕV γk 1 and UÔV;xÕ V k UÔ1;V k xõ. So there is a unique possible disease steady state given by V µ kγ 1. ΛÔ1Õ

38 14 / 23 Existence of multiple disease steady states Theorem (Calvez, Doumic, G.) For L or L, if βôxõ xl βx γ and τôxõ xl τx ν, then lim ΛÔV Õ lim βôxõ. V L xl

39 14 / 23 Existence of multiple disease steady states Theorem (Calvez, Doumic, G.) For L or L, if βôxõ xl βx γ and τôxõ xl τx ν, then lim ΛÔV Õ lim βôxõ. V L xl Consequence: if lim βôxõ lim βôxõ x x then, for µ small enough, there exist at least two values of V such that ΛÔV Õ µ.

40 15 / 23 Constant case: τ ÔxÕ τ, βôxõ βx, κôx,y Õ 1 y µ ΛÔV Õ V Globally Stable V Unstable V Stability results: Engler, Prüss, Pujo-Menjouet, Webb, Zacher 26

41 15 / 23 Constant case: τ ÔxÕ τ, βôxõ βx, κôx,y Õ 1 y µ ΛÔV Õ Globally Stable V V Unstable.5.1 Globally Stable V Stability results: Engler, Prüss, Pujo-Menjouet, Webb, Zacher 26

42 Case lim xβôxõ lim xβôxõ µ ΛÔV Õ V 1 V2.5 V Stable.5 Unstable Stable V Stability results: Calvez, Deslys, Laurent, Lenuzza, Mouthon, Oelz, Perthame / 23

43 Case lim xβôxõ lim xβôxõ µ ΛÔV Õ V 1 V2.5 Stable.5 V Unstable Stable V Stability results: Calvez, Deslys, Laurent, Lenuzza, Mouthon, Oelz, Perthame / 23

44 Case lim xβôxõ lim xβôxõ µ ΛÔV Õ V 1 V2.5 Stable.5 Unstable V Stable V Stability results: Calvez, Deslys, Laurent, Lenuzza, Mouthon, Oelz, Perthame / 23

45 17 / 23 Outline The linear growth-fragmentation equation Steady states of the prion equation Long-time behavior for prion-type equations

46 18 / 23 A prion-type equation with particular solutions Consider the nonlinear growth-fragmentation equation Å uôt,xõ f x p u xuôt,xõ Å g x q u uôt, xõ FuÔt, xõ (1) t x with βôxõ βx γ and κôx,yõ 1 y κ x y.

47 18 / 23 A prion-type equation with particular solutions Consider the nonlinear growth-fragmentation equation Å uôt,xõ f x p u xuôt,xõ Å g x q u uôt, xõ FuÔt, xõ (1) t x with βôxõ βx γ and κôx,yõ 1 y κ x y. If ÔW,QÕ is a solution to the ODE system, W γw f ÔW pßγ QÕ W, Q Q W gôw qßγ QÕ, then Ôt,xÕ QÔtÕUÔW ÔtÕ;xÕ is a solution to equation (1).

48 19 / 23 Long-time behavior Å uôt,xõ f x p u xuôt,xõ µuôt,xõ t x FuÔt,xÕ

49 19 / 23 Long-time behavior Å uôt,xõ f x p u xuôt,xõ µuôt,xõ t x FuÔt,xÕ ï trivial steady state: u,

50 19 / 23 Long-time behavior Å uôt,xõ f x p u xuôt,xõ µuôt,xõ t x FuÔt,xÕ ï trivial steady state: u, ï nontrivial steady states: u ÔxÕ Q UÔW ;xõ where W µ and Q is such that f x p u Å µ.

51 19 / 23 Long-time behavior Å uôt,xõ f x p u xuôt,xõ µuôt,xõ t x FuÔt,xÕ ï trivial steady state: u, ï nontrivial steady states: u ÔxÕ Q UÔW ;xõ where W µ and Q is such that f x p u Å µ.

52 19 / 23 Long-time behavior Å uôt,xõ f x p u xuôt,xõ µuôt,xõ t x FuÔt,xÕ ï trivial steady state: u, ï nontrivial steady states: u ÔxÕ Q UÔW ;xõ where W µ and Q is such that f x p u Å µ. Theorem (G.) ï If limsupf ÔI Õ µ, then any solution converges to a steady state. I ï The trivial steady state is: ï stable if f ÔÕ µ, ï unstable if f ÔÕ µ. ï A positive steady state u is: ï stable if f ½ x p uôxõdx, ï unstable if f ½ x p uôxõdx.

53 2 / 23 Steady states and stability f ÔI Õ stable µ unstable I

54 21 / 23 Long-time behavior Å uôt,xõ f x p u xuôt,xõ Å g x q u uôt, xõ t x FuÔt, xõ Can we prove the same kind of behavior for general functions g?

55 21 / 23 Long-time behavior Å uôt,xõ f x p u xuôt,xõ Å g x q u uôt, xõ t x FuÔt, xõ Can we prove the same kind of behavior for general functions g? No Theorem (G.) There exist situations with increasing functions f and g where periodic solutions appear. More precisely there exist solutions of the form with Q and W periodic functions. uôt,xõ QÔtÕUÔW ÔtÕ;xÕ

56 22 / 23 Periodic oscillations uô, xõ x

57 23 / 23 A generalized prion equation ³² ³± Å d V ÔtÕ λ δv ÔtÕ V ÔtÕf x p u dt uôt,xõ V ÔtÕf t xuôt,xõ dx, Å x p u xuôt,xõ µuôt,xõ x FuÔt,xÕ, introduced by Greer, van den Driessche, Wang and Webb (27).

58 23 / 23 A generalized prion equation ³² ³± Å d V ÔtÕ λ δv ÔtÕ V ÔtÕf x p u dt uôt,xõ V ÔtÕf t xuôt,xõ dx, Å x p u xuôt,xõ µuôt,xõ x FuÔt,xÕ, introduced by Greer, van den Driessche, Wang and Webb (27). Theorem (G.) There exist functions f and parameters λ, δ, µ, γ and p for which this equation admits periodic solutions.