The Prion Equation: Existence of Periodic Solutions

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)

$

'

'

&

'

'

%

dV^pt^q

dt λV^pt^q

δ ˆ ⁸

0

τ^px^qu^pt, x^qdx

,

B

Btu^pt, x^q V^pt^{q B}

Bx τ^px^qu^pt, x^q

µ^px^qu^pt, x^q Fu^pt, x^q.

(4.66)

In this equation,u^pt, x^q represents the quantity of polymers of pathogenic proteins of size xat time t, andV^pt^q the quantity of normal proteins (also called monomers). The polymers lengthen by attaching monomers with the rateτ^px^q,die with the rateµ^px^qand 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 ^λ_δ 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 polymersP^pt^q:´

xu^pt, x^qdxon 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 ωP}¹

pt^q 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 f^pP^q ₁ ¹_ωP,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, whereMpMp^rUs,

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. f^px0^qg^px0^q and moreover 0 f^1px0^q g^1px0^q. (4.69) Assume thatµ^¤pk µ^1qδ,then there existsp^¡0for which Equation (4.68)admits a periodic solution.

More precisely, there existV, W andQperiodic such that V^pt^q, Q^pt^qU^pW^pt^q;x^q

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 u0^px^q 1

M1

Q0U^pW0;x^q.

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 u^pt, x^q Q0

M1

U^pW^pt^q;x^qe^{´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 Tδµ^kQf^pQ^qµ

k pµV Qf^1pQ^q

which is negative forp p1and positive forp^¡p1 with p1: δ µ^{k µ^kQf^pQ^q

µV Qf^1pQ^{q ¡}0.

The determinant is

D µ

kV Q δf^1pQ^qµ^kf^2pQ^q

. It is independent ofpand negative sincef^1px0^q g^1px0^qand

g^1px0^q

δµ^{k 2}

pλµ^{k 1}x0^q2

µ^k δ f^2px0^q. The sum of the three 22 principal minors is

M δpV Qf^1pQ^q µδ

k µ^kpµ 1

k^qQf^pQ^qµ

kV Qf^1pQ^q.

To use the Routh-Hurwitz criterion, let defineψ^pp^q:M TD and look at its sign. Forp0 we have ψ^p0^qµ

k

δ² δQf^pQ^q µδ

k δ^pk µ^1qµ

µ^kQf^pQ^q

V Q µ^k1pk µ^1qf^2pQ^qf^1pQ^q

µ^kQf^pQ^q µ k

, and it is negative sinceµ^¤pk µ^1qδandf^1pQ^q g^1pQ^qµ_δ^kf^2pQ^q¤µ^k1pk µ^1qf^2pQ^q.Forpp1, it is positive because ψ^pp1^qD ^¡0. Now we investigate the variations ofψ between 0 andp1. The first derivative ofT, D andM are given by

T^1pp^qµV Qf^1pQ^q, M^1pp^qδV Qf^1pQ^q, D^1pp^q0, and the second derivatives are all null

T^2pp^qM^2pp^qD^2pp^q0.

So we have

ψ^2pp^q2M^1pp^qT^1pp^q 0

andψis concave. Thus there exists a uniquep0^Pp0, p1^qsuch thatψ^pp0^q0.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 a^1pp0^q¡0.For this we expressψ^pp^qin terms ofa^pp^q, b^pp^qandc^pp^q, and we use the concavity ofψ. We have for anyp

T 2a c, Dc^pa² b^2q, M a² b² 2ac, so

ψ^pp^q2a^pa² b^2q 4a²c 2ac². Then, using thata^pp0^q0 by definition ofp0,we obtain

ψ^1pp0^q2^pb² c^2qa¹.

Butψ^1pp0^q¡0 becauseψis concave and increasing on a neighbourhood ofp0,so necessarilya^1pp0^q¡0.

This proves the existence of a periodic solution^pV, W, Q^qto System (4.71) for a parameterp^¥p0 close top0.Then the functionsV^pt^qandQ^pt^qU^pW^pt^q, x^qare 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