The Hamilton-Jacobi-Bellman equation

6 7 8 9

t α*(t)

α_max

α_min

α_opt

0 10 20 30 40 50 60 70 80

0 1 2 3 4 5 6 7 8 9

t α*(t)

α_min α_max

α_opt

0 10 20 30 40 50 60 70 80

0 1 2 3 4 5 6 7 8 9

t α*(t)

α_max

α_min

0 10 20 30 40 50 60 70 80

0 1 2 3 4 5 6 7 8 9

t α*(t)

α_max

α_min

Figure 5.2: Piecewise optimal controls obtained for different values of ∆twithr^pα^q ₁⁶_α a decreasing convex function. The final time isT 80, the space dimension isn 3,and the coefficients areτ1 0.01, τ2 10, β20.1 andβ30.9.The time step varies: ∆t0.8 (top left), ∆t0.4 (top right), ∆t0.2 (bottom left) and

∆t0.1 (bottom right).

5.2.2 The Hamilton-Jacobi-Bellman equation

We assume that there is no influence of αon the transport term by consideringr 1. In this case we know that there exists an optimal controlα for Problem (5.9). Define cx,t^pα^pqq as the payoff for the dynamical system

$

&

%

U9^ps^qM^pα^ps^qqU^ps^q, t^¤s^¤T, U^pst^qx.

(5.28)

Then, thevalue functionv^px, t^qdefined by

v^px, t^q: sup

α^pqPA

c_x,t^pα^pqq satisfies the Hamilton-Jacobi-Bellman (HJB) equation

v_t^px, t^q max

A proof of this result can be found in [48] for instance, it is also explained how the solution to Equa-tion (5.29) allows to compute the optimal controlα thanks to the dynamic programming method.

Here we solve numerically Equation (5.29) forn3.We consider coefficients satisfying τ2 ^¡2τ1 and β3 ^¥β2 so that Proposition 5.1.4 ensures that there exists α which maximises the Perron eigenvalue, and thatλ1^pα^q increases from λ1^p0^q0 to λ1^pα^qand then decreases to limα^Ñ8λ1^pα^qτ1. First we We compute the gradient ofv in terms ofw

Bx1v w ^p1y1^qBy1wy2^By2w

Bx²v 2w2y1^By¹w ^p12y2^qBy²w

Bx3v 3w3y1^By1w3y2^By2w and obtain a new equation satisfied byw

Btw max

, this equation writes

Btw max

Remark that Equation (5.32) is a retrograd equation with final data w^py, T^q 1. Now we present numerical simulations of the solution at timet0,for different choices of coefficients. The final time is T 20 so that the shape of the solution is observed to be stationnary at timet0.

First case: α ^{P p}α_min, α_max^q. We choose α_min 1 and α_max 4, and coefficients of the matrix M which allow to haveα^Ppα_min, α_max^q. The existence ofα is ensured by Proposition 5.1.4 forτ2^¡2τ1, and its value is betweenαmin andαmaxfor an accurate choice of β2 and β3.The numerical solution to Equation (5.32) is plotted in Figure 5.3 on the simplex Σ2 at time t0.In this case the simplex Σ2 is divided into two zones: one where the maximum of the bracket is reached foraαmax and the other where it is reached foraαmin.These zones are separated by a border which containsV1^pα^qand which is close to the line defined by

H:^ty: φ1^pαopt^qF x0, x^py1, y1,¹

3

p1y12y2^qqT

u

as we can see in Figure 5.4. This line also containsV1^pα^qsinceM^1pα^qF and, thanks to Theorem 5.1.5, φ1M^1pα^qV1λ¹₁^pα^q0.

Figure 5.3: Solution to Equation (5.32) at timet0 plotted on the simplex Σ2 for coefficientsτ10.1, τ2

4, β³2β21.For these coefficients, the maximalλ1

is obtained forα^Ppαmin, αmax^qp1,4^q.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.4: In green: the border between the zones where maxa is reached in αmin (on the left) or αmax

(on the right). In blue: the line H. In black: the set

tV1^pα^q, α^Ppαmin, αmax^qu.In red: V1^pα^q.

On both sides of the border, the solution appears to be almost linear as we can see by plotting the level sets or the gradient (see Figure 5.5).

Second case: α ^¡ α_max and λ1^pα_max^{q ¡} τ1. In this case α_opt α_max and we can see in Figure 5.6 that the border and the lineHare merged. But the border does not pass troughV1^pα_opt^qand we try to understand why. Since forr1 we haveM^pa^qG aF, the maximum in Equation (5.29) is given by the sign ofF x∇v.Now we can check in Figure 5.7 that the gradient in the zone where the maximum is reached foraαmaxis constant. A good candidate for this constant gradient isφ1^pαopt^q.Indeed we can easily check thatv^px, t^qφ1^pαopt^qx e^λpαoptqt is a particular solution becauseαoptαmax.So, assuming that the gradient ofvis indeedφ1^pαopt^q,the border corresponds to the set ofxwhereφ1^pαopt^qF xchanges its sign. We restrict to finding the intersection of the border with the set of eigenvectors^tV1^pα^q, α^¥0^u.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.5: Left: Level sets of the solution to Equation (5.32) at timet0 and the border. Right: The partial derivative^By1w^py, t0^q.

For this we write

λ1^pα^qφ1^pαopt^qV1^pα^q φ1^pαopt^qpG αF^qV1^pα^q

φ1^pα_opt^qpG α_optF^qV1^pα^{q p}αα_opt^qφ1^pα_opt^qF V1^pα^q

λ1^pαopt^qφ1^pαopt^qV1^pα^{q p}ααopt^qφ1^pαopt^qF V1^pα^q and we obtain

φ1^pαopt^qF V1^pα^q λ1^pα^qλ1^pαopt^q

αα_opt φ1^pαopt^qV1^pα^q. (5.33) This quantity is positive forα α˜ and negative for α^¡α,˜ where ˜α is the only value greater than α such that λ1^pα˜^qλ1^pαopt^q.That gives an explanation of the fact that V1^pαopt^qdoes not belong to the border and conjectures that the intersection of the set^tV1^pα^q, α^¥0^uwith the border isV1^pα˜^q.Moreover the set ^tV1^pα^q, α ^{P p}α_min, α_max^qu is expected to be in the zone of α_max and it is indeed the case on numerical simulations (see Figure 5.6).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.6: The same plots as in Figures 5.3 and 5.4 but forτ1 0.4.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.7:The same plots as in Figure 5.5 but for τ10.4.

Third case: α ^¡ αmax and λ1^pαmax^q  τ1. In this case there is no value ˜α^¡ α such that λ1^pα˜^q λ1^pαopt^q.This is in accordance with the numerical simulations becauseHis out of the simplex and there is no border in the interior of Σ2 (see Figure 5.8). The maximum in HJB is reached everywhere for aαmaxαoptand the solution is expected to be the particular solutionv^px, t^qφ1^pαopt^qx e^λ1pαoptqt, up to a multiplication by a constant.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Figure 5.8: Simulations for coefficients τ1 1, τ² 4 and β3 2β2 1. Left: the solution w^py,0^q on Σ². Middle: the levelsets. Right: the border in green is merged with the egde of the simplex, andHin blue is out of the simplex.

Aknowledgements.

The authors want to thank St´ephane Gaubert for precious help and discussions.

Chapitre 6

Un sch´ ema d’ordre ´ elev´ e pour des mod` eles avec coalescence

Ce travail a ´et´e r´ealis´e lors du Cemracs’09 sur la mod´elisation math´ematique en m´edecine, en collaboration avec L´eon Matar Tine dans le projet PRION.

L’objectif est d’´ecrire un sch´ema num´erique d’ordre ´elev´e et pr´eservant la masse pour des mod`eles d’agr´egation-fragmentation avec coagulation. Nous ´ecrivons tout d’abord les termes int´egraux de coagulation et de fragmentation sous forme conserva-tive afin de pouvoir utiliser un sch´ema de transport qui pr´eserve la masse. Nous choi-sissons alors une discr´etisation WENO d’ordre 5 pour ses propri´et´es anti-dissipatives et nous pr´esentons des simulations num´eriques obtenues `a partir de donn´ees exp´eri-mentales. Ceci a fait l’objet d’une publication dans ESAIM-Proceedings sous le titre High-order WENO scheme for polymerization-type equations [60].

6.1 Introduction

The central mechanism of amyloid diseases is the polymerization of proteins : PrP in Prion diseases, APP in Alzheimer, Htt in Huntington. The abnormal form of these proteins is pathogenic and has the ability to polymerize into fibrils. In order to well understand this process, investigation of the size repartition of polymers is a crucial point. To this end, we discuss in this paper the mathematical modeling of these polymerization processes and we propose numerical methods to investigate the mathematical features of the models.

Mathematical models are already widely used to study the polymerization mechanism of Prion diseases [20, 38, 43, 63, 64, 80, 91, 92, 83, 117], Alzheimer [32, 87, 108] or Huntington [16]. Such models are also used for other biological polymerization processes [4, 13] and even for cell division [10, 37, 110] or in neurosciences [107].

Another field where we find aggregation-fragmentation equations is the physics of aggregates (aerosol and raindrop formation, smoke, sprays...). Among these models (see [77] for a review), one can mention the Smoluchowsky coagulation equation [45, 54, 55, 78, 101] with fragmentation [44, 46, 47, 75, 76, 74] and the Lifshitz-Slyosov system [21, 29, 30, 53, 103, 104, 105, 106]. In [28, 67] a Smoluchowsky coagulation term is added to the Lifshitz-Slyosov equation.

In this paper we are interested in a model including polymerization, coagulation and fragmentation phenomena. We consider a medium where there are monomers (normal proteins for instance) character-ized by the concentrationV^pt^qat timet and polymers (aggregates of abnormal proteins) of sizexwith

115

the concentrationu^pt, x^q.The dynamics of the density functionu^pt, x^qis driven by the system

The monomers are attached by polymers of sizexwith the polymerization ratekon^px^q.Depolymerization occurs when monomers detach from polymers with a ratekoff^px^q. Hence the transport term writes

T^pV, x^qV kon^px^qkoff^px^q. (6.2) The two functionskonandkoffare piecewise derivable but can be discontinuous. They are positive except thatkoncan vanish at zero. In this case, or more generally whenT^pV^pt^q,0^q¤0,the boundary condition on u^pt,0^q is not necessary since the characteristic curves outgo from the domain. The choice of the boundary conditionu^pt,0^q0 is justified later.

The coalescence of two polymers and the fragmentation of a polymer into two smaller ones are taken into account by the operatorQ.More precisely, denoting byA_x an aggregate of sizexwe have

Ax Ay

Thus the coagulation-fragmentation operator isQQ_cQ_fwith Q_cpu^qpx^q1 This rate is a nonnegative function as the fragmentation symmetric ratekf^px, y^qkf^py, x^qwith which a polymer of sizex y produces two fragments of sizexandy.

There is a difference between V^pt^q and u^pt, x 0^q. In biochemical polymerization processes, small polymers are very unstable and thus do not exist. When they appear by detachment from a longer polymer, they are immediately degraded into monomers. Thus, the quantity of small polymers vanishes while the quantity of monomers is very high. To reflect this in the mathematical model, a quantity V^pt^qof monomers is introduced, which is different from the quantity of small polymersu^pt, x0^q. The evolution of the first one is given by an ODE while the second one is forced to be equal to zero through the boundary conditionu^pt,0^q0.A consequence of this distinction is that starting fromu0^px^q0 and V0 ^¡0 there is no evolution : the concentration of monomers is constant in time, V^pt^q V0, and the concentration of polymers remains null,u^pt, x^q0.This is a very intuitive and natural behaviour which is important to preserve for biological applications.

In the modeling, the distinction between V and u^px 0^q induces a separation of the polymerization-depolymerization process from the coagulation-fragmentation. Indeed the aggregation of a monomer to a polymer can be seen as a coagulation but the resulting polymer has same sizexthan the initial one, since a monomer is very small compared to the typical size of a polymer. So a transport term is more accurate to model this phenomenon than an integral term (see [38]).

There is also the fact that when a small polymer is degraded into monomers, it increases the quantity of monomers. In a discrete model, this term appears in the equation onV (seen0in [92]). In the continuous model (6.1) this term can be neglected since the quantity of monomers produced by degradation of small polymers is very small compared to the total quantity of monomers.