The Prion Equation: Existence of Periodic Solutions

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Prion diseases are believed to be due to self-replication of a pathogenic protein through a polymerization process not yet very well understood (see [83] for more details). To investigate the replication process of this protein, a mathematical PDE model was introduced by [63]. We recall this model here under a form slightly different from the original one (see [20, 38] for the motivations to consider this form)

$

'

'

&

'

'

%

dVptq

dt λVptq

δ ˆ 8

0

τpxqupt, xqdx

,

B

Btupt, xq Vptq B

Bx τpxqupt, xq

µpxqupt, xq Fupt, xq.

(4.66)

In this equation,upt, xq represents the quantity of polymers of pathogenic proteins of size xat time t, andVptq the quantity of normal proteins (also called monomers). The polymers lengthen by attaching monomers with the rateτpxq,die with the rateµpxqand split into smaller polymers with respect to the fragmentation operatorF.The quantity of monomers is driven by an ODE with a death parameterδand production rateλ.This ODE is quadratically coupled to the growth-fragmentation equation because of the polymerization mechanism which is assumed to follow the mass action law.

This system admits a trivial steady state, also called disease-free equilibrium since it corresponds to a situation where no pathogenic polymers are present: V λδ and u0. The stability of this steady state has been investigated under general assumptions on the coefficients in [19, 20, 125, 131]. The results indicate that the disease-free equilibrium is stable when it is the only steady state, and becomes unstable when a nontrivial steady state appears (also called endemic equilibrium). It is proved in [17] that several such nontrivial steady states can exist. But the stability (even linear) of these steady states is a difficult and still open problem for general coefficients. The only existing results concern the “constant case”

(τ constant, β linear and κ constant) initially considered by [63], since then the model reduces to a closed system of ODEs. In this case, the problem has been entirely solved by [43, 63, 117]: the endemic equilibrium is unique and it is globally stable when it exists.

A new more general model has been introduced in [64] and takes into account the incidence of thetotal massof polymersPptq:´

xupt, xqdxon the polymerization process. More precisely, they consider that the presence of many polymers reduces the attaching process of monomers to polymers by multiplying the polymerization rate by 1 ωP1

ptq withω a positive parameter. Then they prove similar results about the existence and stability of steady states, still in the case of constant parameters.

Here we look at a generalization of the influence of polymers on the polymerization rate by considering the system

where p ¥ 0 and f : R Ñ R is a differentiable function. In this framework, the model of [64]

corresponds to p1 and fpPq 1 1ωP,together with τ constant, β linear and κconstant. Using the reduction method to ODEs, we prove that such a system can exhibit periodic solutions. For this we consider the following system, whereMpMprUs,

which is a particular case of System (4.67) with coefficients satisfying the assumptions of Theorem 4.2.1 and up to a dilation off. We prove that, under some conditions on the incidence functionf, there exist values ofpfor which System (4.68) admits periodic solutions. The method is to considerpas a varying bifurcation parameter and prove that Hopf bifurcation occurs whenpincreases.

Theorem 4.5.1. Define on and consider a positive differentiable function f satisfying

D!x0¡0 s.t. fpx0qgpx0q and moreover 0 f1px0q g1px0q. (4.69) Assume thatµ¤pk µ1qδ,then there existsp¡0for which Equation (4.68)admits a periodic solution.

More precisely, there existV, W andQperiodic such that Vptq, QptqUpWptq;xq

is solution to Equation (4.68).

Proof. First step: reduced dynamic in E

We look at the dynamic of System (4.68) on the invariant eigenmanifoldE.For any initial conditionu0

inE,there existQ0 andW0 such thatu0 writes u0pxq 1

M1

Q0UpW0;xq.

Consider the solution to System (4.68) corresponding to this initial data and defineW as the solution to

Then we know from Theorem 4.2.1 that the solution satisfies upt, xq Q0

M1

UpWptq;xqe´0tpWpsqµqds and this allows to compute

ˆ 8

and System (4.68) reduces to

$

Now we prove that System (4.71) admits a unique nontrivial steady state which undergoes a supercritical Hopf bifurcation whenpincreases from 0.

Second step: Hopf bifurcation for the reduced system

First we look for a positive steady state of System (4.71). Such a steady state is unique and given by W8 positive. Finally there exists a unique positive steady state and we prove that a Hopf bifurcation occurs at this point. The linear stability of the steady state is given by the eigenvalues of the Jacobian matrix

J aceq

where the8 indices are suppressed for the sake of clarity. The trace of this matrix is TδµkQfpQqµ

k pµV Qf1pQq

which is negative forp p1and positive forp¡p1 with p1: δ µ{k µkQfpQq

µV Qf1pQq ¡0.

The determinant is

D µ

kV Q δf1pQqµkf2pQq

. It is independent ofpand negative sincef1px0q g1px0qand

g1px0q

δµk 2

pλµk 1x0q2

µk δ f2px0q. The sum of the three 22 principal minors is

M δpV Qf1pQq µδ

k µkpµ 1

kqQfpQqµ

kV Qf1pQq.

To use the Routh-Hurwitz criterion, let defineψppq:M TD and look at its sign. Forp0 we have ψp0qµ

k

δ2 δQfpQq µδ

k δpk µ1qµ

µkQfpQq

V Q µk1pk µ1qf2pQqf1pQq

µkQfpQq µ k

, and it is negative sinceµ¤pk µ1qδandf1pQq g1pQqµδkf2pQq¤µk1pk µ1qf2pQq.Forpp1, it is positive because ψpp1qD ¡0. Now we investigate the variations ofψ between 0 andp1. The first derivative ofT, D andM are given by

T1ppqµV Qf1pQq, M1ppqδV Qf1pQq, D1ppq0, and the second derivatives are all null

T2ppqM2ppqD2ppq0.

So we have

ψ2ppq2M1ppqT1ppq 0

andψis concave. Thus there exists a uniquep0Pp0, p1qsuch thatψpp0q0.Now we can use the Routh-Hurwitz criterion (see [68] for instance). For 0¤p p0we haveT 0, D 0 andM T  D,so the steady state is linearly stable with one real negative eigenvalue and two complex conjugate eigenvalues with a negative real part. Forp0 p p1 we haveT  0, D 0 andM T ¡D,so the steady state is linearly unstable with one real negative eigenvalue and two complex conjugate eigenvalues with a positive real part. The two conjugate eigenvalues cross the imaginary axis whenpp0so there is a Hopf bifurcation at this point. To prove that a periodic solution appears with this bifurcation, it remains to check that the complex eigenvalues cross the imaginary axis with a positive speed (see [56] for instance). Denote by aib the two conjugate eigenvalues andc   0 the real one. We have to prove that the derivative a1pp0q¡0.For this we expressψppqin terms ofappq, bppqandcppq, and we use the concavity ofψ. We have for anyp

T 2a c, Dcpa2 b2q, M a2 b2 2ac, so

ψppq2apa2 b2q 4a2c 2ac2. Then, using thatapp0q0 by definition ofp0,we obtain

ψ1pp0q2pb2 c2qa1.

Butψ1pp0q¡0 becauseψis concave and increasing on a neighbourhood ofp0,so necessarilya1pp0q¡0.

This proves the existence of a periodic solutionpV, W, Qqto System (4.71) for a parameterp¥p0 close top0.Then the functionsVptqandQptqUpWptq, xqare periodic and solution to System (4.68).

The question to know if such a periodic solution is stable is difficult, even for the reduced dynam-ics (4.71). Nevertheless we give in Figure 4.3 evidences that it should be the case. This simulation is made with parameters and a functionf satisfying Assumption (4.69), for a value of parameterp¡p0.It seems to indicate that the periodic solution persists forpaway fromp0.

0 0.5 1 1.5 2 2.5 3 3.5

0.5 1 1.5 2 1 1.5 2 2.5 3 3.5

Dans le document The DART-Europe E-theses Portal (Page 102-106)