Mostafa Adimy

Abdennasser Chekroun
email: chekroun@math.univ-lyon1.fr

Toshikazu Kuniya

Global asymptotic stability for a distributed delay differential-difference system of a Kermack-McKendrick SIR model

Keywords: SIR epidemic model, Delay differential-difference system, Lyapunov-Krasovskii functional, Local and global asymptotic stability AMS CLASSIFICATION 34K60, 37N25, 92D30

We investigate a system of distributed delay differential-difference equations describing an epidemic model of susceptible, infected, recovered and temporary protected population dynamics. A nonlocal term (distributed delay) appears in this model to describe the temporary protection period of the susceptible individuals. We investigate mathematical properties of the model. We obtain the global asymptotic stability of the two steady states: disease-free and endemic. We construct appropriate Lyapunov functionals where the basic reproduction number appears as a threshold for the global asymptotic behavior of the solution between disease extinction and persistence.

Introduction

Recently, Adimy et al. [START_REF] Adimy | Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase[END_REF] (see also [START_REF] Adimy | Traveling waves of a differential-difference diffusive Kermack-McKendrick epidemic model with age-structured protection phase[END_REF]) proposed and analyzed a new age-structured partial differential system describing the dynamics of an epidemic model of susceptible, infected and recovered individuals. More precisely, they considered an age-structured phase where individuals can be protected from disease for a limited period of time, for example by vaccination or drugs with temporary immunity. In the same way as in their previous works (see [START_REF] Adimy | Age-structured and delay differential-difference model of hematopoietic stem cell dynamics[END_REF][START_REF] Adimy | Global asymptotic stability for an age-structured model of hematopoietic stem cell dynamics[END_REF] and subsequent works), this modeling corresponds, in a sense to be defined according to the studied problem, to the fact that there are mainly two phases: an active phase and an inactive one. By integrating the agestructured equation using the method of characteristics, the system can be reduced to distributed delay differential-difference equations. Contrary to the model obtained here, the epidemic system studied in [START_REF] Adimy | Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase[END_REF] is with a discrete delay. Adimy et al. [START_REF] Adimy | Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase[END_REF], focused on the mathematical analysis of the behavior of the steady states (local and global asymptotic stability). Our objective in this paper is to further analyze this epidemic model by considering a distribution delay which is more realistic, but also more complicated, than a discrete delay.

The global asymptotic stability of an SIR epidemic model with distributed delay was first studied in [START_REF] Beretta | Global stability of an SIR epidemic model with time delays[END_REF]. The authors proved the global asymptotic stability of the disease-free steady state when the basic reproduction number R 0 < 1. They also gave some sufficient conditions for the global asymptotic stability of the endemic steadystate when R 0 > 1 (see also [START_REF] Beretta | Global asymptotic stability of an SIR epidemic model with distributed time delay[END_REF][START_REF] Takeuchi | Global asymptotic properties of a delay SIR epidemic model with finite incubation times[END_REF]). In [START_REF] Mccluskey | Complete global stability for an SIR epidemic model with delay -distributed or discrete[END_REF], McCluskey proposed a method based on a Lyapunov functional to prove the global asymptotic stability of the endemic steadystate when R 0 > 1. His technique has been used to study the global asymptotic stability of a variety of epidemic models such as an SIR model with general nonlinear incidence [START_REF] Mccluskey | Global stability of an SIR epidemic model with delay and general nonlinear incidence[END_REF], a multigroup SEIR model [START_REF] Li | Global stability of multi-group epidemic models with distributed delays[END_REF], and an SIRS model with general nonlinear incidence [START_REF] Enatsu | Lyapunov functional techniques for the global stability analysis of a delayed sirs epidemic model[END_REF]. The global dynamics and the existence of traveling waves in epidemic models with diffusion and distributed delay have also been studied (see for instance, [START_REF] Chekroun | An infection age-space structured SIR epidemic model with Neumann boundary condition[END_REF][START_REF] Bai | Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay[END_REF]). In most of these studies, the distributed delay was incorporated into the force of infection term.

We consider four compartments of individuals: susceptible S(t), infectious I(t), recovered R(t) and protected P (t). As in [START_REF] Adimy | Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase[END_REF], we can consider the density of protected individuals p(t, a), where a ∈ (0, τ ) is the time since an individual entered this compartment. The length of the protection phase is assumed to be distributed according to a probability density -a probability kernel -denoted by g(a), with τ 0 g(a)da = 1. This last property means that all protected individuals lose their protection after a maximum fixed time τ > 0. There is also a minimum period after which an individual is effectively protected. So, we assume that there exists τ 0 ∈ [0, τ ) such that g(a) = 0 for a ∈ [0, τ 0 ]. If we put u(t) := p(0, t), we obtain a Kermack-McKendrick type system composed of delay differential equation and renewal difference one, with distributed delay, t > 0

             S (t) = Λ -(γ S + h)S(t) -βS(t)I(t) + (1 -α) τ 0 e -γP a g(a)u(t -a)da, I (t) = -(γ I + µ)I(t) + βS(t)I(t), u(t) = hS(t) + α τ 0 e -γP a g(a)u(t -a)da, (1)
with initial conditions S(0) = S 0 , I(0) = I 0 and u(t) = φ(t) for -τ ≤ t ≤ 0.

(

) 2
The function u represents the new protected individuals. The total population of protected individuals is then given by P (t) = τ 0 e -γP a u(t -a)da. The recovered individuals that get permanent immunity, satisfy the equation

R (t) = -γ R R(t)+µI(t), R(0) = R 0 .
All the parameters of the model are constant and nonnegative. Λ is the birth rate in the compartment of the susceptible individuals, γ S , γ I , γ P and γ R are the death rates in each compartment. The parameter h describes the protection rate of susceptible individuals through for instance vaccination or drugs with temporary immunity. The constant β is the contact rate per infectious individual that results in infection. µ is the recovering rate with permanent immunity. α ∈ [0, 1) is the specific protection rate with temporary immunity. The integral τ 0 e -γP a g(a)u(t -a)da in [START_REF] Adimy | Global dynamics of a differential-difference system: a case of Kermack-McKendrick SIR model with age-structured protection phase[END_REF] gives the rate at which individuals (who have survived) leave the protection phase. A fraction α of these individuals returns to the protection phase (for example by updating their vaccine). These individuals re-enter the protection phase with a rate α τ 0 e -γP a g(a)u(t -a)da, then their age is reset to 0 (because the age here is the time since an individual entered this compartment). The other individuals lose their protection and become susceptible with a rate (1 -α) τ 0 e -γP a g(a)u(t -a)da. Through this paper we put f (a) = e -γP a g(a)

and ν(τ

) := τ 0 f (a)da ≤ 1. (3)
The quantity ν represents the proportion of individuals leaving the protection phase.

It also contains the survival rate in this phase. A part of these individuals returns to the protection phase with a rate α and the other part becomes susceptible again with a rate 1 -α. We remark that the equations of S , I and u are independent on R and P . Then, we can focus only on the reduced system without R and P components.

In the following sections, we perform a detailed mathematical analysis and obtain global stability properties of the two steady states: disease-free equilibrium and endemic equilibrium. More precisely, we are interested in the validity for our model, of the threshold theorem known in epidemiology and based on the basic reproduction number

R 0 = Λβ(1 -αν(τ )) (µ + γ I )(γ S + h -(αγ S + h)ν(τ ))
.

Summarizing the main results of this paper, we obtain the following threshold theorem.

Theorem 1.1.

-If R 0 < 1, then the disease-free equilibrium of System (1) is the only equilibrium and it is globally asymptotically stable.

-If R 0 > 1, then a unique endemic equilibrium of System (1) exists and it is globally asymptotically stable.

This result supports the claim that the basic reproduction number R 0 of our model serves as a threshold parameter that determines the outcome of the initial outbreak.

Preliminaries

Let us introduce C := C([-τ, 0], R), the space of continuous functions on [-τ, 0] and C + := C([-τ, 0], R + ), the set of nonnegative continuous functions on [-τ, 0]. Throughout this paper, we assume S 0 ≥ 0, I 0 ≥ 0, φ ∈ C + . The existence and uniqueness of nonnegative solutions of (1)-( 2) can be obtained as for neutral differential equations with delay (see [START_REF] Hale | Existence, uniqueness and continuous dependence for hereditary systems[END_REF]). Consider the auxiliary linear homogeneous difference equation

u(t) = D(u t ), t ≥ 0, (4)
where the history function u t ∈ C is defined for t ≥ 0 and u ∈ C([-τ, +∞), R), by

u t (θ) = u(t + θ) for θ ∈ [-τ, 0]. The linear operator D : C([-τ, 0], R) → R is given by D(ψ) = α τ 0 f (a)ψ(-a)da, ψ ∈ C. Remark that Condition (3) implies that D := sup ψ ≤1 |D(ψ)| < 1, (5)
with ψ = sup θ∈[-τ,0] |ψ(θ)|. Condition (5) says that the zero solution of the linear difference equation ( 4) is globally asymptotically stable [START_REF] Hale | Introduction to functional differential equations[END_REF]. From the last equation of (1), we can write

u(t) -D(u t ) = hS(t), t > 0. (6)
Using Eq. ( 6), Condition ( 5) and ( [START_REF] Cruz | Stability of functional differential equations of neutral type[END_REF], Lemma 3.5), we have the following result.

Lemma 2.1. Let (S(t), u(t)) be a sub-solution of System (1). Then, lim t→+∞ u(t) > 0 if and only if lim t→+∞ S(t) > 0.

We now consider the following fundamental auxiliary system, for t > 0,

             dS(t) dt = Λ -(γ S + h)S(t) + (1 -α) τ 0 f (a)u(t -a)da, u(t) = hS(t) + α τ 0 f (a)u(t -a)da, S(0) = S 0 , u(s) = φ(s), for -τ ≤ s ≤ 0. (7)
The system (7) has a unique steady state

(S 0 , u 0 ) = Λ(1 -αν(τ )) γ S + h -(αγ S + h)ν(τ ) , Λh γ S + h -(αγ S + h)ν(τ ) . (8)
We have the following result.

Theorem 2.2. The steady state (S 0 , u 0 ) of System ( 7) is globally asymptotically stable.

Proof. We put, for t > 0, Ŝ(t) = S(t) -S 0 and û(t) = u(t) -u 0 . Then, we get the linear differential-difference system

       Ŝ (t) = -(γ S + h) Ŝ(t) + (1 -α) τ 0 f (a)û(t -a)da, û(t) = h Ŝ(t) + α τ 0 f (a)û(t -a)da. (9)
Let us consider the following Lyapunov functional L 0 : R + × C + → R + defined by

L 0 (S 0 , φ) = S 2 0 2 + 1 ν(τ ) γ S (1 -(αν(τ )) 2 ) + h(1 -α(ν(τ )) 2 ) 2h 2 τ 0 f (a) t t-a φ 2 (θ)dθda. We set η 1 (s) = s 2 /2 and η 2 (s) = ((1/2) + τ ϑ)s 2 , with ϑ := γ S (1 -(αν(τ )) 2 ) + h(1 -α(ν(τ )) 2 ) 2h 2 . ( 10
)
Then, this functional satisfies the inequalities η 1 (S 0 ) ≤ L 0 (S 0 , φ) ≤ η 2 (||(S 0 , φ)||). We can notice that System (9) is input-to-state stable (see [START_REF] Adimy | Age-structured and delay differential-difference model of hematopoietic stem cell dynamics[END_REF][START_REF] Gu | Lyapunov-Krasovskii functional for uniform stability of coupled differentialfunctional equations[END_REF]). More precisely, there exist constants C > 0 and σ > 0 such that the solution ( Ŝ, û) of ( 9) satisfies |û(t)| ≤

C φ e -σt + sup 0≤s≤t | Ŝ(s)| . The above estimation is an immediate consequence of ( [START_REF] Hale | Introduction to functional differential equations[END_REF], Theorem 3.5, page 275). By differentiating the functional t → L 0 ( Ŝ(t), ût ) along the solution ( Ŝ, û) of the system (9), we obtain

d dt L 0 ( Ŝ, ût ) = Ŝ(t) Ŝ (t) + ϑ ν(τ ) τ 0 f (a)[û 2 (t) -û2 (t -a)]da, = -(γ S + h -ϑh 2 ) Ŝ2 (t) + Ŝ(t) τ 0 f (a)û(t -a)da[(1 -α) + 2ϑhα] - ϑ ν(τ ) τ 0 f (a)û 2 (t -a)da + ϑα 2 τ 0 f (a)û(t -a)da 2 .
Moreover, Jensen's integral inequality implies that

1 ν(τ ) τ 0 f (a)û(t -a)da 2 ≤ 1 ν(τ ) τ 0 f (a)û 2 (t -a)da.
Then, we get

d dt L 0 ( Ŝ, ût ) ≤ -(γ S + h -ϑh 2 ) Ŝ2 (t) + Ŝ(t) τ 0 f (a)û(t -a)da[(1 -α) + 2ϑhα] - ϑ (ν(τ )) 2 τ 0 f (a)û(t -a)da 2 + ϑα 2 τ 0 f (a)û(t -a)da 2 .
Consequently,

d dt L 0 ( Ŝ, ût ) ≤ -(γ S + h -ϑh 2 ) Ŝ2 (t) + [(1 -α) + 2ϑhα] τ 0 f (a)û(t -a)da Ŝ(t) - ϑ (ν(τ )) 2 [1 -α 2 (ν(τ )) 2 ] τ 0 f (a)û(t -a)da 2 .
We set

a = γ S + h -ϑh 2 , b = [(1 -α) + 2ϑhα], c = ϑ (ν(τ )) 2 [1 -α 2 (ν(τ )) 2 ]. (11)
Then, the derivative of t → L 0 ( Ŝ(t), ût ) along the solution is given by

d dt L 0 ( Ŝ(t), ût ) = -a Ŝ2 (t) + b Ŝ(t) τ 0 f (a)û(t -a)da -c τ 0 f (a)û(t -a)da 2 , ≤ -c τ 0 f (a)û(t -a)da - b 2c Ŝ(t) 2 + 4ac -b 2 4c 2 Ŝ2 (t) .
Remember that α ∈ [0, 1). Then by using (3) and [START_REF] Li | Global stability of multi-group epidemic models with distributed delays[END_REF], we can show that 4ac -b 2 > 0. This is equivalent to

[(1 -α)ν(τ ) + 2ϑhαν(τ )] 2 -4(γ S + h -ϑh 2 )ϑ[1 -α 2 (ν(τ )) 2 ] = 4h 2 ϑ 2 -4(γ S (1 -α 2 (ν(τ )) 2 ) + h(1 -α(ν(τ )) 2 ))ϑ + (1 -α) 2 (ν(τ )) 2 < 0.
Since c > 0, we obtain

d dt L 0 ( Ŝ(t), ût ) ≤ b 2 -4ac 4c Ŝ2 (t) = -Ŝ2 (t),
with := (4ac -b 2 )/4c > 0. Hence (0, 0) is a globally asymptotically stable steady state of (9) (see, [START_REF] Adimy | Age-structured and delay differential-difference model of hematopoietic stem cell dynamics[END_REF][START_REF] Gu | Lyapunov-Krasovskii functional for uniform stability of coupled differentialfunctional equations[END_REF]). This completes the proof of Theorem 2.2.

Back to System (1). We investigate a fundamental result about the boundedness of solutions of (1). Proposition 2.3. Suppose that τ 0 > 0. All the solutions of System (1) are bounded.

Proof. Let (S, I, u) be the solution of (1) associated to the initial condition (S 0 , I 0 , φ) ∈ R + × R + × C + . We remark that the system [START_REF] Takeuchi | Global asymptotic properties of a delay SIR epidemic model with finite incubation times[END_REF],

           dS b (t) dt = Λ -(γ S + h)S b (t) + (1 -α) τ 0 f (a)u b (t -a)da, u b (t) = hS b (t) + α τ 0 f (a)u b (t -a)da, S b (0) = S 0 , u b (s) = φ(s), for -τ ≤ s ≤ 0,
represents an upper bound of System (1). Moreover, it is shown in Theorem 2.2 that S b (t) converges to a finite constant as t tends to infinity. This implies that S b is bounded. We denote S := S b -S and u := u b -u. Clearly, we have, for t > 0,

             d S(t) dt ≥ -(γ S + h) S(t) + (1 -α) τ 0 f (a) u(t -a)da, u(t) = h S(t) + α τ 0 f (a) u(t -a)da, S(0) = 0, u(s) = 0, for -τ ≤ s ≤ 0.
Suppose by contradiction that there exist t 1 ∈ R + and 0 < τ 0 such that (i) S(t) ≥ 0 for t ∈ [0, t 1 ), S(t 1 ) = 0 and S(t) < 0 for t ∈ (t 1 , t 1 + ], or (ii) u(t) ≥ 0 for t ∈ [0, t 1 ), u(t 1 ) = 0 and u(t) < 0 for t ∈ (t 1 , t 1 + ].

We distinguish three cases:

(1) Assume that (i) holds. If u(t) ≥ 0 and u(t) ≡ 0 on [0, t 1 + ], then we have

S (t 1 ) ≥ (1 -α) τ 0 f (a) u(t 1 -a)da > 0.
This gives a contradiction. If u(t) = 0 on [0, t 1 + ], then there exists a t ∈ (t 1 , t 1 + ] such that S( t), S ( t) < 0 and we get, for

t 1 -τ < t -a ≤ t 1 + -τ 0 , S ( t) ≥ -(γ S + h) S( t) + (1 - α) τ τ0 f (a) u( t -a)da > 0.
This gives also a contradiction.

(2) Assume that (ii) holds and S(t) ≥ 0 for all t ∈ [0, t 1 + ]. By using the equation of u, we get u(t 1 + ) = h S(t 1 + )+α τ τ0 f (a) u(t 1 + -a)da ≥ 0. This contradicts the fact that u(t 1 + ) < 0.

(3) Assume that (i) and (ii) hold at the same time. We have

S (t 1 ) ≥ (1 - α) τ 0 f (a) u(t 1 -a)da ≥ 0. If S (t 1
) > 0, we get directly a contradiction and if S (t 1 ) = 0, then we can proceed as (i). There exists a ť ∈ (t 1 , t 1 + ] such that S( ť), S ( ť) < 0 and we get, for

t 1 -τ < ť -a ≤ t 1 + -τ 0 , S ( ť) ≥ -(γ S + h) S( ť) + (1 -α) τ τ0 f (a) u( t -a)da > 0.
This leads to a contradiction with S ( ť) < 0.

Finally, we conclude that S(t) ≤ S b (t) and u(t) ≤ u b (t).

As a consequence by this comparison, S is also bounded. Then, there exists S > 0 such that S(t) < S , t > 0. On another side, we have (see Theorem 3.5 -page 275 in [START_REF] Hale | Introduction to functional differential equations[END_REF]), |u(t)| ≤ C φ e -αt + h sup 0≤s≤t |S(s)| , t > 0, with α > 0, C > 0 and φ = sup θ∈[-τ,0] |φ(θ)|. Then,

|u(t)| ≤ C φ e -αt + S . (12)
This implies that u is bounded. Consequently, there exists u > 0 such that u(t) < u , for all t > 0.

Let focus now on the component I. By summing the equations of S and I, we get, for > 0, (S + I)

(t) < Λ -(γ S + h)S(t) -(γ I + µ)I(t) + (1 -α)ν(τ )u . Thus, we obtain (S + I) (t) < Λ + (1 -α)ν(τ )u -ω(S + I)(t), with ω := min{γ S + h, γ I + µ}.
As a consequence, I is also bounded.

The basic reproduction number and existence of steady states

The basic reproduction number, R 0 , is used to measure the potential for disease transmission. It is the average number of secondary infections produced by an infected individual in a population completely susceptible. By dividing the equation of I(t), in the system (1), by (µ + γ I )I we get I (t)/[(µ + γ I )I(t)] = -1 + βS(t)/(µ + γ I ). The fraction β/(µ + γ I ) can be interpreted as the number of contacts per infected individuals during their infectious period that lead to the transmission of the disease. If βS(t)/(µ + γ I ) > 1, the disease persists, otherwise, it disappears. Then, the basic reproduction number of the disease is defined by R 0 := βS 0 /(µ + γ I ), where S 0 is the disease-free steady state of the susceptible population.

Next, we establish the existence of the steady states of System (1). We will see that the condition of existence of an endemic equilibrium is equivalent to R 0 > 1. Let (S * , I * , u * ) be a steady state of (1). Then, we have u * = hS * /[1 -αν(τ )], and (βS * -µ -γ I )I * = 0. We distinguish two cases I * = 0 or S * = (µ + γ I )/β. Suppose that I * = 0. Then, since γ S + h -(αγ S + h)ν(τ ) > 0,

S * = Λ(1 -αν(τ )) γ S + h -(αγ S + h)ν(τ ) and u * = Λh γ S + h -(αγ S + h)ν(τ )
.

We obtain the disease-free steady state (S * , I * , u * ) := (S 0 , 0, u 0 ),

= Λ(1 -αν(τ )) γ S + h -(αγ S + h)ν(τ ) , 0 , Λh γ S + h -(αγ S + h)ν(τ ) .
It describes the disappearance of the epidemic. Then, we can write

R 0 := βS 0 µ + γ I = Λβ(1 -αν(τ )) (µ + γ I )(γ S + h -(αγ S + h)ν(τ ))
.

Suppose now that I * > 0. Then, S * = (µ + γ I )/β. We have also u * = h(µ + γ I )/[β(1αν(τ ))] > 0. and

I * = Λ -(γ S + h)S * + (1 -α)ν(τ )u * βS * = Λ µ + γ I - γ S + h -(αγ S + h)ν(τ ) β(1 -αν(τ )) .
So, the existence of an endemic steady state is equivalent to R 0 > 1. That is equivalent to

Λ µ + γ I > γ S + h -(αγ S + h)ν(τ ) β(1 -αν(τ )) . ( 13
)
We set (S * , I * , u * ) := (S, I, u),

= µ + γ I β , Λ µ + γ I - γ S + h -(αγ S + h)ν(τ ) β(1 -αν(τ )) , h(µ + γ I ) β(1 -αν(τ )) . (14)
We summarize the existence of the two steady states in the following result.

Theorem 3.1. Assume that (13) holds. Then, System (1) has two distinct steady states: a disease-free steady state (S 0 , 0, u 0 ), which is given by (3), and an endemic steady state (S, I, u), which is given by [START_REF] Hale | Existence, uniqueness and continuous dependence for hereditary systems[END_REF]. If (13) does not hold, then (S 0 , 0, u 0 ) is the only steady state.

Stability of the disease-free steady state

Local asymptotic stability of the disease-free steady state

We recall that a steady state of ( 1) is locally asymptotically stable if all roots of the associated characteristic equation have negative real parts, and unstable if a root with positive real part exists (see [START_REF] Adimy | Age-structured and delay differential-difference model of hematopoietic stem cell dynamics[END_REF] and [START_REF] Diekmann | On the characteristic equation λ = α 1 + (α 2 + α 3 λ)e -λ and its use in the context of a cell population model[END_REF] for more details).

The linearized system of (1) about the equilibrium (S 0 , 0, u 0 ) is

             S (t) = -(γ S + h)S(t) -βS 0 I(t) + (1 -α) τ 0 f (a)u(t -a)da, I (t) = -(γ I + µ)I(t) + βS 0 I(t), u(t) = hS(t) + α τ 0 f (a)u(t -a)da,
and the characteristic equation is given by (see [START_REF] Adimy | Age-structured and delay differential-difference model of hematopoietic stem cell dynamics[END_REF] for more details)

∆ 0 (λ) = λ + µ + γ I -βS 0 × λ + γ S + h -[α(λ + γ S + h) + h(1 -α)] τ 0 f (a)e -λa da = 0. ( 15
)
The instability of the disease-free steady state is a directly obtained from the above equation.

Proposition 4.1. Assume that R 0 > 1. Then, the steady state (S 0 , 0, u 0 ) is unstable.

Proof. Assume that R 0 > 1. From the characteristic equation ( 15), there exists a positive real root of [START_REF] Hale | Introduction to functional differential equations[END_REF]. In fact, we have the following eigenvalue:

λ = -µ-γ I +βS 0 = (µ + γ I )(R 0 -1) > 0.
Then, the steady state (S 0 , 0, u 0 ) is unstable. Now, we consider the case R 0 < 1. Then, λ = (µ + γ I )(R 0 -1) < 0, and the local asymptotic stability of (S 0 , 0, u 0 ) is determined by the sign of the real part of λ ∈ C satisfying (see [START_REF] Adimy | Age-structured and delay differential-difference model of hematopoietic stem cell dynamics[END_REF] and [START_REF] Diekmann | On the characteristic equation λ = α 1 + (α 2 + α 3 λ)e -λ and its use in the context of a cell population model[END_REF]),

D 0 (λ) = λ + γ S + h -[α(λ + γ S + h) + h(1 -α)] τ 0 f (a)e -λa da = 0. ( 16
)
We have the following theorem.

Theorem 4.2. Assume that R 0 < 1. Then, the steady state (S 0 , 0, u 0 ) of System (1) is locally asymptotically stable.

Proof. The equation ( 16) can be rewritten as Hence, by taking the modulus of ( 17), we have |λ + γ S + h| ≤ |α(λ + γ S ) + h|. Taking square of both sides, we get

λ + γ S + h = [α(λ + γ S + h) + h(1 -α)] τ 0 f (
(λ + γ S + h)(λ + γ S + h) ≤ [α(λ + γ S ) + h][α(λ + γ S ) + h].
Arranging this inequality, we obtain

(1 -α 2 )|λ + γ S | 2 + (1 -α)h(λ + λ + 2γ S ) ≤ 0.
However, since 0 ≤ α < 1 and λ + λ = 2µ ≥ 0, we have

(1 -α 2 )|λ + γ S | 2 + (1 -α)h(λ + λ + 2γ S ) > 0,
which is a contradiction. Hence, µ < 0. We conclude that (S 0 , 0, u 0 ) is locally asymptotically stable.

Global asymptotic stability of the disease-free steady state

Let > 0 and consider the set

Ω := (S, I, u) ∈ R + × R + × C + : 0 ≤ S ≤ S 0 + and 0 ≤ u(s) ≤ u 0 + , for all s ∈ [-τ, 0] .
Lemma 4.3. Suppose that τ 0 > 0. For any sufficiently small > 0, the subset

Ω of R + × R + × C + is a global attractor for System (1).
Proof. The solutions of (1) satisfy, for all t > 0,

       S (t) ≤ Λ -(γ S + h)S(t) + (1 -α) τ 0 f (a)u(t -a)da, u(t) = hS(t) + α τ 0 f (a)u(t -a)da.
By the comparison principle (see the proof of Proposition 2.3), we have S(t) ≤ S b (t) and u(t) ≤ u b (t) for all t > 0, where (S b , u b ) is the solution of the system (7). Theorem 2.2 shows that S b (t) → S 0 and u b (t) → u 0 as t → +∞. This convergence implies that Ω is a global attractor for the system (1) in R + × R + × C + . This completes the proof.

Thanks to Lemma 4.3, we can restrict the global stability analysis of the disease-free steady state of (1) to the set Ω . Theorem 4.4. Suppose that τ 0 > 0. Assume that R 0 < 1. Then, the disease-free steady state (S 0 , 0, u 0 ) of System (1) is globally asymptotically stable.

Proof. It suffices to consider the solutions in Ω for any sufficiently small > 0. We then have, for t > 0,

I (t) ≤ -(γ I + µ)I(t) + β(S 0 + )I(t) = -(γ I + µ) 1 - β(S 0 + ) µ + γ I I(t).
Since R 0 < 1, we can choose > 0 such that the right-hand side of the above inequality is negative. This implies that lim t→+∞ I(t) = 0. From the above result, we see that for any > 0, there exists a T > 0 such that I(t) ≤ for all t ≥ T . We then have, for t > T

       S (t) ≥ Λ -(γ S + h)S(t) -βS(t) + (1 -α) τ 0 f (a)u(t -a)da, u(t) = hS(t) + α τ 0 f (a)u(t -a)da.
Consequently, we have S(t) ≥ S (t) and u(t) ≥ u (t) for all t ≥ T , where (S , u ) is the solution of the following problem

           dS (t) dt = Λ -(γ S + h)S (t) -βS (t) + (1 -α) τ 0 f (a)u(t -a)da, u (t) = hS (t) + α τ 0 f (a)u(t -a)da, S (0) = S 0 , u (s) = φ(s), for -τ ≤ s ≤ 0. (18)
As in the proof of Theorem 2.2, we can show that S (t) → S 0 and u (t) → u 0 as t → +∞, where (S 0 , u 0 ) is the steady state of [START_REF] Diekmann | On the characteristic equation λ = α 1 + (α 2 + α 3 λ)e -λ and its use in the context of a cell population model[END_REF]. Then, there exists a T > T > 0 such that, for t ≥ T , S 0 -≤ S(t) ≤ S 0 + and u 0 -≤ u(t) ≤ u 0 + . Since > 0 is arbitrary, S 0 → S 0 and u 0 → u 0 as → 0, we have that lim t→+∞ S(t) = S 0 and lim t→+∞ u(t) = u 0 . Recalling from Theorem 4.2 that (S 0 , 0, u 0 ) is locally asymptotically stable. Then, it is globally asymptotically stable. This completes the proof.

Stability of the endemic steady state

Local asymptotic stability of the endemic steady state

We now show the local asymptotic stability of the endemic steady state (S, I, u) for R 0 > 1. The linearized system of (1) around (S, I, u) is given by

             S (t) = -(γ S + h)S(t) -βIS(t) -βSI(t) + (1 -α) τ 0 f (a)u(t -a)da, I (t) = βIS(t), u(t) = hS(t) + α τ 0 f (a)u(t -a)da.
Substituting S(t) = Xe λt , I(t) = Y e λt and u(t) = Ze λt , and dividing them by e λt , we obtain

               λX = -(γ S + h)X -βIX -βSY + (1 -α) τ 0 f (a)e -λa da Z, λY = βIX, Z = hX + α τ 0 f (a)e -λa da Z.
From the third equation, we get

Z = hX 1 -α τ 0 f (a)e -λa da -1
and injecting it into the first equation, we obtain Thus, the characteristic equation is given by

           (λ + γ S + h + βI)X + βSY 1 -α τ 0 f (
[λ 2 + (k 1 + h)λ + k 2 ] 1 -α τ 0 f (a)e -λa da -(1 -α)hλ τ 0 f (a)e -λa da = 0,
where k 1 := γ S + βI and k 2 := β 2 S I.

Theorem 5.1. Assume that R 0 > 1. Then, the steady state (S, I, u) of System (1) is locally asymptotically stable.

Proof. The characteristic equation can be rewritten as

λ 2 + (k 1 + h)λ + k 2 = α(λ 2 + k 1 λ + k 2 ) + hλ τ 0 f (a)e -λa da. (19)
Suppose that λ = µ + iω (µ, ω ∈ R) and µ ≥ 0. Note that | τ 0 f (a)e -λa da| ≤ 1. By taking the modulus of ( 19), we have

λ 2 + (k 1 + h)λ + k 2 ≤ α(λ 2 + k 1 λ + k 2 ) + hλ .
Taking square of both sides, we get

λ 2 + k 1 λ + k 2 + hλ λ 2 + k 1 λ + k 2 + hλ ≤ α(λ 2 + k 1 λ + k 2 ) + hλ α(λ 2 + k 1 λ + k 2 ) + hλ .
Arranging this inequality, we obtain

(1 -α 2 )|λ 2 + k 1 λ + k 2 | 2 + (1 -α)h (|λ| 2 + k 2 )(λ + λ) + 2k 1 |λ| 2 ≤ 0.
However, since 0 ≤ α < 1 and λ + λ = 2µ ≥ 0, we have

(1 -α 2 )|λ 2 + k 1 λ + k 2 | 2 + (1 -α)h (|λ| 2 + k 2 )(λ + λ) + 2k 1 |λ| 2 > 0.
This gives a contradiction. Therefore, µ < 0 and the steady state (S, I, u) is locally asymptotically stable.

Global asymptotic stability of the endemic steady state

This section provides the global asymptotic stability of the unique endemic steady state (S, I, u) given by ( 14). First, we notice that the component I of System (1) is persistent for R 0 > 1, it remains above a positive level ˆ for all sufficiently large times. Indeed, we consider the following sub-system (without difference equation), for t > 0

     S (t) = Λ -(γ S + h)S(t) -βS(t)I(t), I (t) = -(γ I + µ)I(t) + βS(t)I(t), S(0) = S 0 , I(0) = I 0 .
It is a classical SIR system and we known that it is persistent (see for instance Theorem 3.7.1 and 3.7.2 of [START_REF] Li | An Introduction to Mathematical Modeling of Infectious Diseases[END_REF], for a detailed proof). In fact, we have I (t) = βS 0 I(t) S(t)/S 0 -1/R 0 . Formally, if S is sufficiently close to S 0 , then I (t) > 0. This implies that I(t) increases. By the comparison principle, we can conclude that the solution I of System (1) satisfies lim inf t →+∞ I(t) > ˆ > 0. Let (S, I, u) be the endemic steady state with S > 0, I > 0 and u > 0. We can write the system (1) for the variables S(t) := S(t) -S and ũ(t) := u(t) -u and by using the fact that βS = µ + γ I as

               S (t) = -(γ S + h) S(t) -β S(t)I(t) -βSI(t) + βS I + (1 -α) τ 0 f (a)ũ(t -a)da, I (t) = -(γ I + µ)I(t) + β S(t)I(t) + βSI = β S(t)I(t), ũ(t) = h S(t) + α τ 0 f (a)ũ(t -a)da. (20)
Theorem 5.2. Assume that R 0 > 1. Then, the steady state (S, I, u) of System (1) is globally asymptotically stable.

Proof. Let us consider the following Lyapunov functional L e : R

+ × R + × C([-τ, 0], R + ) → R + defined by L e (S 0 , I 0 , φ) = S 2 0 2 + S I 0 -I -I ln I0 I + 1 ν(τ ) γ S (1 -(αν(τ )) 2 ) + h(1 -α(ν(τ )) 2 ) 2h 2 τ 0 f (a) t t-a φ 2 (θ)dθda.
The function g(x) = x -x -x ln(x/x) for x > 0, satisfies g(x) ≥ 0 for all x > 0 and g(x) = 0 if and only if x = x. Then, L e (S 0 , I 0 , u 0 ) = 0 if and only if (S 0 , I 0 , u 0 , ) = (0, I, 0). Recall that the constants a, b and c are given in [START_REF] Enatsu | Lyapunov functional techniques for the global stability analysis of a delayed sirs epidemic model[END_REF]. The derivative of t → L e ( S(t), I(t), ũt ) along the solution trajectory is given by As the function S (t) is uniformly bounded and S(t) is uniformly continuous. Then, the Barbalat's Lemma [START_REF] Hou | New versions of Barbalat's lemma with applications[END_REF] applied to the function t → t 0 S2 (s)ds, shows that lim t→+∞ S(t) = 0. The result in Lemma 3.5 of [START_REF] Cruz | Stability of functional differential equations of neutral type[END_REF] implies that lim t→+∞ ũ(t) = 0. Then, the expression of the functional L e implies that lim This prove the global asymptotic stability of (S, I, u).

Disclosure statement

No potential conflict of interest was reported by the authors.

  a)e -λa da. (17) Suppose that λ = µ + iω (µ, ω ∈ R) and µ ≥ 0. Note that τ 0 f (a)e -λa da ≤ τ 0 f (a)e -µa e -iωa da = τ 0 f (a)e -µa da ≤ 1.

0 f

 0 a)e -λa da -(1 -α)hX τ (a)e -λa da = 0, -βIX + λY = 0.

0 fc τ 0 f 2 - 0 S2 (s)ds ≤ 1 L

 00201 d dtL e ( S(t), I(t), ũt ) = -a S2 (t) + b S(t)τ (a)ũ(t -a)da -(a)ũ(t -a)da βI(t) S2 (t), S(t), I(t), ũt ) ≤ b 2 -4ac 4c S2 (t) =: -S2 (t). (21)It is shown in the proof of theorem 2.2 that c > 0 and > 0. Then, the functional t → L e ( S(t), I(t), ũt ) is nonincreasing and we have L e ( S(t), I(t), ũt ) -→ t→+∞ inf s≥0 L e ( S(s), I(s), ũs ) =: L * ∈ R + . By integration, (21) implies that t 0 S2 (s)ds ≤ L e ( S(0), I(0), ũ0 ) -L e ( S(t), I(t), ũt ). (22) The both sides of the inequality (22) are nondecreasing functions. Then, the limits exist and satisfy lim t→+∞ t e ( S(0), I(0), ũ0 ) -L * .

  t→+∞ g(I(t)) = L * /S. Furthermore, the function S(t) is bounded and differentiable, then the fluctuations Lemma implies that there exists a sequence t k → +∞ such that lim k→+∞ S (t k ) = 0. Then, the first equation of (20), implies that lim k→+∞ I(t k ) = I. The continuity of the function g gives lim k→+∞ g(I(t k )) = g(I) = 0. Then, L * = 0. From the properties of the function g, we conclude that lim t→+∞ I(t) = I.

FundingA.

  Chekroun thanks grant, PRFU: C00L03UN220120180004, from DGRSDT of Algeria. T. Kuniya is supported by JSPS Grant-in-Aid for Early-Career Scientists [grant number 19K14594] and the Japan Agency for Medical Research and Development [grant number JP20fk0108535].


domain: scco shs

text:
Olivier Dodier

Henry Otgaar

Steven Jay Lynn

A Critical Analysis of Myths About Dissociative Identity Disorder Une Analyse Critique des Mythes Relatifs au Trouble Dissociatif de l'Identité

Keywords: Amnesia, Dissociative identity disorder, Dissociation, Trauma, Dissociative Amnesia Résumé

 identified what they purported to be myths about dissociative identity disorder (DID) and dissociative amnesia.

When responding to these beliefs, they supported the Trauma Model of dissociation and argued for a causal etiological link between trauma and dissociative conditions. In contrast, they challenged the Sociocognitive Model (SCM), which they claimed rejects the existence of DID and associated disorders (e.g., dissociative amnesia) and considers symptoms to be the byproduct of fantasy, suggestion, and the iatrogenic effect of psychotherapies. In this article, we critically evaluate the authors' arguments and propose a more balanced, accurate, and comprehensive view of the sociocognitive model. We demonstrate that this model neither rejects the existence of DID, nor a link between trauma and dissociation potentially mediated by a variety of cognitive-affective-behavioral variables. We argue, contrary to Piedfort-Marin et al., that the tendency to confabulate and other cognitive and socio-cultural variables may also contribute to the development of DID. We contend that a multifactorial integrative etiological perspective can move the field beyond a limited focus on controversies that divide the TM and SCM models of dissociation.

Mots clés : Amnésie, Trouble dissociative de l'identité, Dissociation, Trauma, Amnésie dissociative

Introduction

Dissociative Identity Disorder (DID) is as controversial in the scientific literature as it is fascinating and compelling to the public in popular culture. The intriguing question of the coexistence of several distinct identities resident in one individual has been dramatized in many successful movies, such as Fight Club or, more recently, Split. DID is characterized in the DSM-5 (American Psychiatric Association, 2013, p. 291) "by the presence of two or more distinct personality states or an experience of possession and recurrent episodes of amnesia" (see [START_REF] Piedfort-Marin | Le trouble dissociatif de l'identité : les mythes à l'épreuve des recherches scientifiques[END_REF], for a more exhaustive presentation of this disorder). As [START_REF] Piedfort-Marin | Le trouble dissociatif de l'identité : les mythes à l'épreuve des recherches scientifiques[END_REF] pointed out, DID is also the subject of many misconceptions about the occurrence of its symptoms and its etiology. In their literature review, Piedfort-Marin and colleagues (2021) addressed five myths they believe are often encountered when discussing DID: (i) DID is schizophrenia; (ii) DID does not exist; (iii) DID develops in response to the influence of the media and/or therapists in suggestible people who are fantasy prone; (iv) The etiology of DID is iatrogenic and not posttraumatic; and (v) Amnesia for the terrifying traumatic stories that DID people report in therapy is not likely, and such stories have been invented.

To discuss and share the scientific view regarding these myths, Piedfort-Marin et al.

(2021) compared two models, namely the trauma model (TM) of dissociation and the sociocognitive model (SCM) to explain the development of DID. The TM asserts a direct causal link between trauma and the development of DID (e.g., [START_REF] Brand | Trauma-related dissociation is no fantasy: addressing the errors of omissions and errors commission in Merckelbach and Patihis (2018)[END_REF]Dalenberg et al., 2013), whereas the SCM is an open model that considers the potential influence and interaction among social and cognitive variables (e.g., fantasy proneness; cognitive failure; suggestibility; peer, therapist or media influence; e.g., [START_REF] Giesbrecht | Cognitive processes in dissociation: an analysis of core theoretical assumptions[END_REF][START_REF] Lynn | The trauma model of dissociation: inconvenient truths and stubborn fictions. Comment on Dalenberg et al[END_REF] as antecedents of dissociative experiences and symptoms. [START_REF] Piedfort-Marin | Le trouble dissociatif de l'identité : les mythes à l'épreuve des recherches scientifiques[END_REF] mainly rejected the tenets of the SCM and were more in favor of the TM. Specifically, they concluded the following: DID is distinct from schizophrenia; it develops independent of influence of therapists, media, or more generally sociocognitive variables; the origin of DID is traumatic in nature; and people with DID do not exhibit proneness to fantasy or confabulation.

Rather, the horrifying events of childhood sexual abuse reported by people with DID in therapy are real and known to law enforcement. In this critical commentary, we challenge a number of interpretations of the literature that Piedfort-Marin et al. ( 2021) advanced and argue for a more balanced, accurate, and comprehensive interrogation of potential antecedents of DID beyond trauma. While we agree that DID is not schizophrenia and recovered memories of terrifying childhood events may be accurate, we view other claims as more questionable. We will argue that some contentions are fallacious in that they inaccurately attribute claims to proponents of the SCM that they have not advanced (e.g., "dissociative amnesia cannot exist"). Furthermore, we will show that the authors failed to consider recent contributions to the literature that would have provided a more accurate view of the complexity that DID represents. Finally, we will posit that accounts advanced by the SCM are more nuanced than what the authors portray.

To structure our commentary, we focus on aspects of Piedfort-Martin et al.'s review (2021) that we judge to be the most problematic. We will not elaborate on possible differences between DID and schizophrenia, as there is scant disagreement that DID should be confused with schizophrenia. However, we will focus on the following issues: 1) Does DID exist, 2) Does DID have a traumatic origin, 3) Does dissociative amnesia exist, and 4) Are the horrific stories of people with DID unfounded?

Does Dissociative Identity Disorder Exist?

Piedfort-Marin et al. ( 2021) criticized three issues that they attributed to the SCM: DID is caused by media influence, it has an iatrogenic origin (i.e., an influence of therapy and therapists), and people with DID are fantasy prone. We do not find their arguments convincing, either because they are not supported by compelling evidence, or because they disregard certain aspects of the literature supportive of the SCM. We will therefore follow the organization of the authors' three subsections and provide a critical evaluation of their contentions.

Media and DID

According to [START_REF] Piedfort-Marin | Le trouble dissociatif de l'identité : les mythes à l'épreuve des recherches scientifiques[END_REF], proponents of the SCM identified the release of a novel (and its movie adaptation), Sybil, as the source of the outbreak of DID diagnoses and that DID would only be a fad. We agree with Piedfort-Marin et al. ( 2021) that, contrary to [START_REF] Paris | The rise and fall of dissociative identity disorder[END_REF] it does not appear to be the case that DID only manifested in the 1980-1990 period before disappearing. A quick search on Medline using the keywords "Dissociative Identity Disorder" and "Multiple Personality Disorder" shows that, as illustrated in Figure 1, a rise in publications on these topics occurred in the 1980s, and the publication hits remain quite high and relatively stable. However, the fact that papers are published on the topic of DID should not be conflated with valid and reliable evidence for the TM. Considering the popularity (even among scientists) of a disorder as evidence of its existence would be exemplify an argumentum ad populum. Critically, the authors omitted the fact that the case described in Sybil is based on fabricated and/or misleading information, as exposed by the investigative journalist Deborah [START_REF] Nathan | Sybil exposed: The extraordinary story behind the famous multiple personality case[END_REF]. According to Shirley Mason, the woman renamed Sybil in the book, stated that her therapist suggested symptoms of DID, and that she enacted the role of a person with "multiple personalities" largely to please her therapist [START_REF] Paris | The rise and fall of dissociative identity disorder[END_REF]. Another significant omission by the authors was that there was no credible or corroborated evidence of childhood abuse in the case, a problem that plagues much of the research on DID. In fact, Mason's childhood appeared to be quite ordinary [START_REF] Paris | The rise and fall of dissociative identity disorder[END_REF].

In support of the claim that the media does not affect the genesis of DID, [START_REF] Piedfort-Marin | Le trouble dissociatif de l'identité : les mythes à l'épreuve des recherches scientifiques[END_REF] referred to Dell's study (2006) showing that of all the symptoms of DID, 83-95% of people diagnosed with DID exhibited 15 symptoms (out of 23) that were "unknown to the media, to the general public, but also to the majority of health care professionals" (p. 379) because they would be subjective and unobservable. Such a finding, according to [START_REF] Dell | A new model of dissociative identity disorder[END_REF] and [START_REF] Piedfort-Marin | Le trouble dissociatif de l'identité : les mythes à l'épreuve des recherches scientifiques[END_REF] would contradict the SCM. However, the authors did not disclose several critical limitations of this study. First, the assertion that the 15 symptoms in question are unknown to the public, the media and health practitioners is not supported by any evidence in Dell's article (2006), or any evidence we are aware of. The fact that the symptoms are experiential, subjective, and not evident to onlookers, for example, does not signify that they are not known to exist, particularly given that publicized accounts of DID are widely disseminated in the media.

Second, the DSM-5 (American Psychiatric Association, 2013) lists in the differential diagnosis section no less than 10 other disorders or types of disorders that could be confused with DID (e.g., post-traumatic stress disorder, bipolar disorder, depressive disorder, psychotic disorder). Thus, many of the 15 symptoms frequently evident in DID patients, which are claimed to be unknown, are present in one or more disorders and not specific to DID. Among the eight remaining symptoms frequently found in DID there exist (i) those known to the public, the media, and health professionals, and (ii) symptoms that make it possible to rule out other disorders from the differential diagnosis; for example, the presence of two (or more) distinct personality states. In short, this study alone has no apparent bearing on the SCM.

The authors observed that, despite the release of several movies featuring DID, "there does not seem to have been an increase in treatment demands for DID" (Piedfort-Marin et al., 2021, p. 379). Furthermore, the authors concluded that they find it "difficult to draw the conclusion that DID is always the result of media influence" (p. 379). The SCM does not assert the extreme case that "DID is "always the result of media influence." To the contrary, according to the SCM, media exposure is only one of multiple pathways by which DID might develop [START_REF] Lilienfeld | Dissociative identity disorder and the sociocognitive model: recalling the lessons of the past[END_REF][START_REF] Lynn | The trauma model of dissociation: inconvenient truths and stubborn fictions. Comment on Dalenberg et al[END_REF]. Moreover, the SCM does not assert a monotonic relation between media exposure and the genesis of DID (although such a relation is possible) or demand for treatment of DID.

Is DID Iatrogenic?

According to [START_REF] Piedfort-Marin | Le trouble dissociatif de l'identité : les mythes à l'épreuve des recherches scientifiques[END_REF], what makes the proponents of the SCM reason that DID could be the result of therapist influence is that "the clinical presentation of DID becomes more explicit after the beginning of treatment" and that presentation would be a reflection of the therapist who would "take advantage, most of the time unconsciously, of the patient's cognitive failures, high level of imagination and suggestibility" (p. 379). Proponents of the SCM have expressed concerns regarding the use of suggestive techniques in psychotherapy in diagnosing and treating DID [START_REF] Lilienfeld | Psychological treatments that cause harm[END_REF][START_REF] Spanos | Multiple identities and false memories: a sociocognitive perspective[END_REF]. These methods include journaling, hypnosis, referring to different personality states with different names, mapping purported personality systems, attempts to recover memories associated with different personalities, and so forth [START_REF] Lilienfeld | Psychological treatments that cause harm[END_REF]. These techniques arguably would reify presentations of identity confusion and the expression of puzzling and disturbing symptoms in terms of a narrative of "multiple selves" consistent with DID. Narratives that emerge in therapy may thus be a function of suggestive methods, but also can arise from widely available cultural beliefs about DID and idiosyncratic ways individuals represent their subjective experience of self-fragmentation. Even adherents of the TM concede that some cases of DID can be iatrogenic [START_REF] Dalenberg | Evaluation of the evidence for the trauma and fantasy models of dissociation[END_REF], although the rate of occurrence of such cases is unknown.

Simulation studies reveal that healthy people are readily able to mimic symptoms of DID in laboratory contexts with minimal cues to do so (see Boysen & van Bergen, 2014;[START_REF] Spanos | Multiple personality: a social psychological perspective[END_REF][START_REF] Stafford | Cultural scripts, memories of childhood abuse, and multiple identities: a study of role-played enactments[END_REF], implying that sociocultural narratives regarding DID are widely accessible to the general public. However, the ability to simulate DID should not be conflated with the idea that DID itself is typically simulated or faked, which is not the case, other than in probably rare forensic contexts. The SCM does make such a claim.

Piedfort-Marin and colleagues (2021) rejected the idea that individuals with DID exhibit high levels of suggestibility stating that dissociation would explain only 1% of the variance in suggestibility. Piedfort-Martin et al.'s concern regarding suggestibility is based on Dalenberg et al.'s meta-analysis [START_REF] Dalenberg | Evaluation of the evidence for the trauma and fantasy models of dissociation[END_REF] whose study inclusion criteria have been criticized [START_REF] Lynn | The trauma model of dissociation: inconvenient truths and stubborn fictions. Comment on Dalenberg et al[END_REF][START_REF] Patihis | Psychometric comparison of Dissociative Experiences Scales II and C: a weak trauma-dissociation link[END_REF]. [START_REF] Dalenberg | Evaluation of the evidence for the trauma and fantasy models of dissociation[END_REF] used studies that measured dissociation mostly with the Dissociative Experience Scale (DES, [START_REF] Bernstein | Development, reliability, and validity of a dissociation scale[END_REF]. However, Dalenberg's analysis excluded studies conducted with a modified version of the DES, the DES-C [START_REF] Wright | Measuring dissociation: comparison of alternative forms of the Dissociative Experiences Scale[END_REF], which was developed to counteract the floor effect and skewness of the distribution encountered in nonclinical populations. scale (CEQ, [START_REF] Merckelbach | The Creative Experiences Questionnaire (CEQ): a brief self-report measure of fantasy proneness[END_REF]. The DID sample scores were lower than several control groups (e.g., actors, fantasy gamers, and patients diagnosed with borderline personality disorder). Nevertheless, this study has notable limitations that warrant caution in interpreting the results. First, crucial information is missing (e.g., the sample size of the control groups, the characteristics of these samples, the control participants' possible traumatic history, their score on dissociation scales). Second, clinical patients may be motivated to suppress endorsement of items that tap fantasy proneness in order to not invalidate their DID diagnosis, as merely the product of fantasy. Accordingly, there is reason to suspect strong biases, particularly among patients with DID in psychotherapy, to underreport scores on measures of fantasy-proneness and for correlations between measures of fantasy proneness and dissociation to be attenuated. However, this plausible contention remains to be confirmed empirically. Third, although scores on the CEQ were low in the DID condition, the researchers reported no statistical comparison of mean scores across comparison conditions. Fourth, Piedfort-Martin et al. ( 2021) conflated mean scores with correlations, which are the appropriate analysis to shed light on the association between fantasy proneness and dissociation, as mean scores are independent of correlations.

Importantly, in a recent meta-analysis of 132 articles with 24,007 participants, Merckelbach et al. (in press) found that the link between fantasy proneness and dissociative experiences and the link between fantasy proneness and an eight-item version of the DES that tapped more serious dissociative pathology was appreciably higher (respectively, r = .

Final remarks

The suggestion that proponents of the SCM consider that DID does not exist (p. 380) is not founded. Decades ago, [START_REF] Lilienfeld | Dissociative identity disorder and the sociocognitive model: recalling the lessons of the past[END_REF] dismissed this contention as a "pseudoissue" (p. 509) and conceded what is obvious: some people express symptoms of DID. Rather than quarrel about the existence of DID, the SCM addresses the etiology of the disorder and the role of trauma, sociocognitive variables presumed to be linked with dissociative experiences, and the mechanisms that contribute to the diagnosis. Current discussions still focus on these three points of controversy (Lynn et al., 2019a).

Piedfort-Marin and his colleagues (2021) claimed that SCM proponents use the DES as a measure of dissociation "while asserting that dissociation is a universal phenomenon without a traumatic origin, whereas the DES is built on the assumption of a traumatic origin of severe dissociation" (p. 380). It is unclear what the relevance is of the scale being "built on the assumption of a traumatic origin…" in terms of how it is related to the validity and psychometric properties of the scale. What is relevant to the SCM is that weak-to-moderate correlations between reports of trauma and dissociation, which we consider in the next section, imply that other variables or mechanisms must be elucidated to more fully account for dissociative experiences and disorders.

Does DID Have a Traumatic Origin?

Piedfort-Marin et al. ( 2021) state that SCM researchers rejected the idea of an etiological association between trauma and dissociation because DID patients reporting childhood traumatic experiences would suffer the effects of "the malleability of memory which would be vulnerable to suggestion" (p. 380). We contend that this is possible and add the imperative that memory reports-whether of traumatic or nontraumatic events [START_REF] Bernsten | The reconstructive nature of involuntary autobiographical memories[END_REF]-should be corroborated by objective evidence where possible, given the widely accepted finding that memory is reconstructive (e.g., Barlett, 1932), prone to inaccuracies, and sensitive to the influence of external sources, including the media [START_REF] Bernsten | The reconstructive nature of involuntary autobiographical memories[END_REF][START_REF] Nahleen | Memory consistency for sexual assault events[END_REF].

As [START_REF] Chalavi | Abnormal hippocampal morphology in dissociative identity disorder and posttraumatic stress disorder correlates with childhood trauma and dissociative symptoms[END_REF]. They stated that "this study concludes that DID is closely related to PTSD and that both disorders are traumatic in origin" (p. 381).

Nevertheless, we argue that this conclusion is not supported by the quality of the evidence. Given the impossibility-for obvious ethical reasons-of inducing sufficient levels of stress in the laboratory to induce trauma, many extant biomarker findings are correlational and not longitudinal, and therefore do not indicate the direction of the association. Indeed, the associations between hippocampal volume and trauma may indicate a risk factor for the experience of distress or trauma due to pre-existing hippocampal abnormality [START_REF] Childress | Hippocampal volumes in patients with chronic combat-related posttraumatic stress disorder: a systematic review[END_REF]. In addition, biomarkers studies generally do not control for the confounding effects of comorbid conditions, include comparison groups matched for general psychopathology, and distinguish traumatic events from generalized stressors and daily life stress or hassles (Lynn et al., 2019a). Nor do they typically compare dissociative conditions associated with reports of a history of trauma versus no history of trauma to isolate the role of trauma, if any.

To our knowledge, there exists no reliable, replicable neurobiological signature specific to trauma, much less a biomarker that reflects the diversity of traumatic experiences, just as there is no set of personality characteristics specific to or indicative of trauma or sexual abuse.

Accordingly, we contend that the following conclusion is not warranted: "(…) the most recent and extensive (neuroimaging) studies confirm that DID is a trauma-induced disorder and that it is at the extreme end of a continuum of trauma-induced disorders" (Piedfort-Marin et al.,

2021, p. 381).

These considerations aside, we do not reject the potential etiological role of trauma in dissociative disorders such as DID, but we assume that sociocognitive and other variables need to be considered in a complete account of dissociative conditions. (2019) and others discuss, dissociative symptoms can occur in disorders that also are associated with sleep dysfunctions, cognitive-affective-behavioral dysregulation (e.g., borderline personality disorder, schizophrenia spectrum disorders), and alexithymia/poor meta-cognition, indicating shared vulnerabilities among different manifestations of psychopathology that are comorbid with dissociative conditions. Whether trauma is causally connected with these variables remains an open question, but even if it were found to be the case, it would still not support the proposition that different personality states are separated by amnesic barriers and associated with independent systems of control, which is a highly questionable assumption that is not supported by the available evidence, as we will note below in our discussion of interidentity amnesia.

Does Dissociative Amnesia Exist?

This section in Piedfort-Marin et al.'s article ( 2021) is possibly the most problematic.

Here, the authors again respond to arguments or claims that do not accurately reflect those advanced by proponents of the SCM. First, they argued that for some SCM proponents, dissociative amnesia "cannot exist" (p. 381). This statement is not consistent with arguments voiced by critics of dissociative amnesia. For example, [START_REF] Otgaar | The return of the repressed: the persistent and problematic claims of long-forgotten trauma[END_REF] stated that "the idea of repressed memories runs counter to well-established principles of human memory" (p.

1074), not that the phenomenon "cannot exist." Also, recently, Mangiulli and colleagues (in press) stated explicitly that "we do not want to imply that dissociative amnesia is a nonexisting diagnostic entity. Rather, our findings highlight that case study data surrounding the nature and etiology of dissociative amnesia are unconvincing, lacking of convergence and cohesion across clinicians and academics." What these authors argue is that the mechanisms described in the DSM-5 as "dissociative amnesia" do not correspond to current knowledge of how memory works, either in an ordinary way or in intense stress circumstances. In a recent

French review of the critical literature on dissociative amnesia, Dodier (in press) stated in an introductory statement: "(...) it is necessary to note that the purpose of this article is not to assert that dissociative amnesia does not exist. Rather, the purpose is to emphasize both a lack of evidence for a link between trauma and forgetting but also alternative, more parsimonious empirical evidence, thus creating an obstacle to the claim that dissociative amnesia exists." To say that evidence is lacking for the existence of a phenomenon or process is not the same as saying that the phenomenon or process does not exist. This epistemological caution is important in that, contrary to what Piedfort-Marin and his colleagues (2021) claim, skeptical scholars of dissociative amnesia remain quite open to the existence of dissociative amnesia.

However, in the absence of convincing evidence, we contend that continued caution is warranted.

Second, the authors attributed to the proponents of the SCM the argument that dissociative amnesia may be ordinary forgetting, and that by forgetting, SCM proponents are suggesting that people can remember their trauma if they are helped to do so. Piedfort-Marin et al. ( 2021) therefore expressed surprise that the "danger of suggestion does not seem to be a problem here" (p. 381). We see two major concerns with this last claim: (i) the danger of suggestion is problematic for dissociative amnesia skeptics (see, e.g., [START_REF] Dodier | Recovered memories of child abuse outside of therapy[END_REF], and (ii) helping people to remember better can be done completely independently of any suggestion, as illustrated by the large body of research on non-suggestive forensic interviewing methods such as the cognitive interview (see [START_REF] Dodier | Recovered memories of child abuse outside of therapy[END_REF], for real-world highly stressful experiences; see [START_REF] Memon | The cognitive interview: a meta-analytic review and study space analysis of the past 25 years[END_REF], for a meta-analysis). It is also entirely conceivable that individuals will recover ordinarily forgotten (accurate) traumatic memories on their own and spontaneously in the presence of relevant retrieval cues in everyday life (e.

g., hearing the perpetrator's name, returning to the scene of the abuse). Interestingly, people who spontaneously recovered memories in this manner were found to be less susceptible to false memories than people who recovered their memories in therapy [START_REF] Geraerts | Cognitive mechanisms underlying recoveredmemory experiences of childhood sexual abuse[END_REF], and their memories were more likely to be corroborated by external evidence than memories recovered in therapy [START_REF] Geraerts | The reality of recovered memories: corroborating continuous and discontinuous memories of childhood sexual abuse[END_REF].

In the following discussion, Piedfort-Marin et al. ( 2021) relied on two types of studies to support the validity of dissociative amnesia. First, they presented data from retrospective studies (i.e., [START_REF] Briere | Self-reported amnesia for abuse in adults molested as children[END_REF]Hermann & Schatzow, 1987) showing that victims of childhood abuse reported "rates of amnesia (...) in 59% of the cases (out of 450 subjects), and 28% of the women (out of 53) in their sample, respectively" (p. 381). The authors also described the work of [START_REF] Elliott | Posttraumatic stress associated with delayed recall of sexual abuse: A general population study[END_REF] finding "20% total amnesia and 42% partial amnesia" (p. 381). Finally, they reported the findings of [START_REF] Wilsnack | Selfreports of forgetting and remembering childhood sexual abuse in a nationally representative sample of US women[END_REF] that 25% of a non-clinical population reported amnesia, of which only 1.8% recovered their memories in therapy. None of these studies, however, support the validity of dissociative amnesia. The question asked in these studies is not that of amnesia, but that of the period without memories of abuse between the event and the study. Researchers have argued that reporting periods without memories of the events can be explained in terms of alternative processes or phenomena related to ordinary memory [START_REF] Dodier | Recovered memories of child abuse outside of therapy[END_REF][START_REF] Loftus | Repression: a mistaken impression?[END_REF][START_REF] Mcnally | A new solution to the recovered memory debate[END_REF]. To address this issue, [START_REF] Dodier | Recovered memories of child abuse outside of therapy[END_REF] reported very large inconsistencies in the percentages of self-report of periods without abuse memories (rank = 6%-77%). They then hypothesized that, in addition to methodological weaknesses of studies, false positives (i.e., mistakenly reporting periods of time without memories) could explain these large variations in proportions. Using follow-up questions (i.e., asking participants what they meant exactly when they reported recovering memories of childhood abuse) to limit false positives, (i) 23% of people reporting memories of childhood abuse had experienced a period without memory of the events before recovering them at a specific time in their lives, and (ii) 31% of people who initially reported recovering previously forgotten memories had in fact always had memories, but had reinterpreted them over time. This second result aligns with hypotheses of alternate developmental phenomena or processes proposed earlier in the literature (e.g., [START_REF] Mcnally | A new solution to the recovered memory debate[END_REF]. In addition, almost 10% of recovered memories of childhood abuse that individuals were previously unaware of were recovered in therapy, and many of the contexts of recovery were consistent with the SCM (e.g., media exposure, personal documentation on repressed memory of child abuse, discussion with relatives about childhood abuse and repressed memories).

Additionally, [START_REF] Piedfort-Marin | Le trouble dissociatif de l'identité : les mythes à l'épreuve des recherches scientifiques[END_REF] relied on Williams's prospective study (1994), which reported that 38% of women who had reported abuse 17 years earlier did not spontaneously recall it during an interview. Nevertheless, this research does not constitute unambiguous evidence for dissociative amnesia. [START_REF] Loftus | Forgetting sexual trauma: what does it mean when 38% forget?[END_REF], for example, explained that not spontaneously recalling events by no means provides dispositive evidence that they have truly been forgotten; that is, people may simply have chosen not to talk about the abuse.

In addition, Piedfort-Marin et al. ( 2021) stated that this study found that "the closer the relationship to the perpetrator, the greater the possibility of forgetting the abuse in adulthood" (p. 382). Although the authors do not provide an explanation for this result, which [START_REF] Williams | Recall of childhood trauma: a prospective study of women's memories of child sexual abuse[END_REF] experiencing traumatic events can lead to memory problems (e.g., involuntary and persistent memories, encoding failure). However, such problems do not support the existence of amnesia for entire autobiographical experiences. Moreover, concerning the inter-identity amnesia often reported in the DID literature (e.g., [START_REF] Eich | Memory, amnesia, and dissocative identity disorder[END_REF], researchers have documented that--despite subjective reports of forgetting information (i.e., amnesia)-when objective indicators are employed with individuals with DID, information is, in fact, implicitly transferred from one "identity" to another [START_REF] Huntjens | Inter-identity autobiographical amnesia in patients with dissociative identity disorder[END_REF]. Moreover, performance on inter-identity episodic memory transfer tasks is similar among DID patients, simulators, and healthy control participants [START_REF] Huntjens | Interidentity amnesia for neutral, episodic information in dissociative identity disorder[END_REF].

Finally, Mangiulli and colleagues (in press) reviewed case studies on dissociative amnesia in the period of 2000-2020. They critically examined 128 case studies and assessed whether the information reported in these case studies was aligned with DSM-5 criteria for dissociative amnesia. The authors found, however, that the evidence to support dissociative amnesia was very weak and plagued with unexplained heterogeneity regarding the origin of the memory loss. None of the case studies completely satisfied all DSM-5 criteria, and most case studies failed to consider alternative explanations for the claimed memory loss.

Collectively, at present, evidence to support dissociative amnesia, as a phenomenon distinct from ordinary forgetting and ruling out memory loss in response to neurological injuries, is severely lacking. Hence, we encourage clinicians and researchers to consider alternative and more mundane memory-based explanations more critically for claimed autobiographical memory loss.

Are the Terrifying Stories of People with DID Unfounded?

The authors presented documented cases of paedophile networks in support of their claim that the terrifying and traumatic events that DID patients report can be real [START_REF] Piedfort-Marin | Le trouble dissociatif de l'identité : les mythes à l'épreuve des recherches scientifiques[END_REF]-because, in fact, such events do occur. When people claim to recall abuse that was formerly forgotten, one always needs to examine the validity of these claims, which may confirm the accuracy of recovered memories. The proponents of SCM do not deny the reality of sexual abuse, and they do not deny that recovered memories (even in therapy) can be accurate. Rather, we assert the imperative to explore a variety of ordinary and noncontroversial explanatory mechanisms before resorting to dissociative amnesia to account for memory loss (e.g., delayed understanding of the abuse, reluctance to report events, encoding failure, brain injury).

However, we cannot endorse the use of anecdotal evidence to contradict the false memory hypothesis. If we must rely on anecdotal cases, it is also possible to propose some that illustrate cases of false memories. For example, the Benoit Yang Ting case in France, widely reported in the media, concerns a clinical psychologist who induced false intrauterine memories in patients of attempted abortions by their mothers. Such memories are highly unlikely and the criminal investigation, as well as the court decision recognized these memories as false and induced during suggestive therapy and the therapist was given a one year suspended prison sentence for abusing vulnerable people. Does this case and the court finding reduce all memories of childhood abuse to false memories? Surely not. In the same way, the fact that paedophile networks have been found to be active does not imply that all memories of childhood abuse, which arise in therapy, are veridical. Only a careful, case-bycase examination of the events in question and potential factors that may be detrimental to memory can yield information that can corroborate, no corroborate, or determine the impossibility of corroborating memories in and out of the therapy context.

Conclusion

We argue that that the literature review presented by [START_REF] Piedfort-Marin | Le trouble dissociatif de l'identité : les mythes à l'épreuve des recherches scientifiques[END_REF] represents an imbalanced and inaccurate accounting of the literature on DID and misrepresents the SCM. Current proponents of the SCM do not contend that DID or dissociative amnesia do not exist; that trauma plays no role in the etiology of the disorder (although that can be the case), and that the memories of terrifying events reported by DID patients are necessarily false and implanted by unscrupulous or incompetent therapists.

Instead, the SCM proposes that, as is the case in many disorders, sociocognitive and cultural factors play a role in DID. Nor do SCM proponents claim that all DID symptoms are invariably media or therapist induced or even necessary or exclusive etiological components.

  l'expression symptomatique est le résultat d'invention, de suggestion et d'effet iatrogène des psychothérapies. Dans ce commentaire, nous répondons aux arguments des auteurs en proposant une vision plus nuancée, plus exacte et plus complète du modèle sociocognitif. Nous démontrons ainsi que ce modèle ne rejette ni l'existence du TDI, ni celle d'un lien causal entre trauma et dissociation, et proposons qu'un tel lien, s'il existe, serait plutôt indirect et médié par une variété de dérégulations cognitives, affectives et comportementales. Nous soutenons, contrairement à Piedfort-Marin et al., que la tendance à l'affabulation et d'autres facteurs cognitifs et socioculturels peuvent aussi contribuer au développement du TDI. Nous plaidons ainsi pour une approche étiologique multifactorielle intégrative et invitons à sortir de l'opposition classique entre les modèles traumatique et sociocognitif de la dissociation.

Figure 1

 1 Figure 1

  [START_REF] Patihis | Psychometric comparison of Dissociative Experiences Scales II and C: a weak trauma-dissociation link[END_REF] found no empirical justification for excluding DES-C studies. In fact, the DES and the DES-C performed comparably across indices of psychopathology, highlighting the problem of comorbidity of psychological symptoms in discerning the specificity of trauma to dissociation. Additionally, both scales secured evidence for only weak correlations between trauma and dissociation, with correlations ranging between r = .122 and r = .269 across the two studies, whereas the correlation with the DES and both fantasy proneness and cognitive failures ranged between r = .50 and r = .60 across these measures (Study 1). In fact, the correlations between sexual abuse and harassment and dissociation were not statistically significant (Study 2), consistent with other studies cited by[START_REF] Dalenberg | Evaluation of the evidence for the trauma and fantasy models of dissociation[END_REF], which found little or no association of trauma with dissociative experiences. One could argue that the[START_REF] Patihis | Psychometric comparison of Dissociative Experiences Scales II and C: a weak trauma-dissociation link[END_REF] findings were secured in a non-clinical sample and therefore not relevant to severe dissociation. However, there was no evidence for the findings to be associated with different levels of trauma and, more generally, scant studies have directly compared the correlations across patient and healthy individuals or traumatized and non-traumatized individuals in the context of the same study. Still, we acknowledge the possibility that future research will reveal that trauma plays a more causative role in some populations of individuals and across different types of trauma.

  52 and r = .40). So, contrary to what Piedfort-Marin et al. (2021) contend, considerable evidence exists for a sizable and meaningful relation between fantasy proneness and dissociation. Unfortunately, adherents of the TM, like Piedfort-Marin et al. (2021), often overlook or minimize findings not entirely supportive of their perspective.

  [START_REF] Piedfort-Marin | Le trouble dissociatif de l'identité : les mythes à l'épreuve des recherches scientifiques[END_REF] pointed out, the proponents of the SCM do not reject a potential etiological role of trauma in dissociation (p. 378). Nevertheless, we maintain that the evidence for trauma playing a necessary, sufficient, specific, or direct role in the genesis of dissociation is not as compelling as the TM model and Piedfort-Marin et al. (2021), in specific, maintain. In fact, the evidentiary basis for the TM is neither compelling nor convincing: It rests largely on modest (on average) and inconsistent correlational findings based on uncorroborated reports of trauma, rather than on replicated positive findings based on longitudinal studies of confirmed traumatic events[START_REF] Giesbrecht | Cognitive processes in dissociation: an analysis of core theoretical assumptions[END_REF][START_REF] Patihis | Psychometric comparison of Dissociative Experiences Scales II and C: a weak trauma-dissociation link[END_REF].[START_REF] Piedfort-Marin | Le trouble dissociatif de l'identité : les mythes à l'épreuve des recherches scientifiques[END_REF] presented neuroimaging studies to support the idea of a causal relation between trauma and dissociation. For example, they described a study showing an association between childhood trauma and reduced hippocampal volume in patients diagnosed with post-traumatic stress disorder (PTSD) and/or DID

  Lynn et al. (2019), for example, suggested direct as well as indirect influences on dissociation via sleep disturbances (see also[START_REF] Van Der Kloet | Fragmented sleep, fragmented mind: the role of sleep in dissociative symptoms[END_REF] and emotion dysregulation. Additionally, as Lynn et al.

  Lynn et al. (2019) recommended moving beyond the traditional TM/SCM debate and proposed additional factors that may contribute to the development of dissociative disorders. Supplementing a growing body of research over the past 20 years, which has highlighted the relation between sleep disturbance and dissociation (e.g.,[START_REF] Van Der Kloet | Fragmented sleep, fragmented mind: the role of sleep in dissociative symptoms[END_REF][START_REF] Watson | Dissociations of the night: individual differences in sleep-related experiences and their relation to dissociation and schizotypy[END_REF][START_REF] Watson | Explicating the psychopathological correlates of anomalous sleep experiences[END_REF],Lynn et al. (2019) advanced the hyperassociativity hypothesis.This hypothesis was based on Lynn's observations of six DID patients who shifted rapidly and dramatically in their clinical presentation across cognitive-affective-behavioral states in response to internal and external triggers(Lynn et al., 2019b).Lynn et al. (2019a) defined hyperassociativity as the fluid association of concepts, memories, or emotions that are weakly associated semantically and emotionally. To onlookers, such extreme shifts could be perceived as manifestations of different identities based on traditional notions of DID(Lynn et al., 2019a).Lynn et al. (2019a, b], however, did not view such shifts to signify different "personalities" or identities with independent executive systems, memories, or behavioral patterns separated by amnesic barriers, as argued by the TM. Rather, Lynn and his colleagues (2019a, b) posited that cognitive, affective, and behavioral instability and hyperassociativity evidenced in DID reflects weakly monitored, poorly regulated and processed, dysfunctional response sets and self-representations that patients and therapists can come to construe as separate indwelling identities in DID.Still, this formulation does not exclude an indirect (e.g., via sleep, self-control deficits, emotional dysregulation) or even a potentially direct role for trauma (i.e., in cases of depersonalization/derealization in peritraumatic dissociation), among a variety of mechanisms of dissociative experiences and symptoms. Nor does this perspective dismiss the personal and social sequelae and costs of trauma and its aftermath. Nevertheless, we contend that scant evidence exists to support[START_REF] Piedfort-Marin | Le trouble dissociatif de l'identité : les mythes à l'épreuve des recherches scientifiques[END_REF] implication that trauma is a necessary, sole, or generally sufficient cause of dissociation.

  reported, non-pathological social factors might provide an explanation. Indeed,[START_REF] Leach | The relationship between children's age and disclosures of sexual abuse during forensic interviews[END_REF] observed that when abuse was intrafamilial, younger children were less likely to disclose it during a police interview. Accordingly, the authors explained, children, especially

if they are young, may not perceive the abuse as such, fear reprisal for disclosure, or feel loyalty to the abuser. Unfortunately, Piedfort-Marin et al. (

2021

) conflate cases of not spontaneously recalling events, or not reporting them for a period of time, with proof of dissociative amnesia, rather than consider more parsimonious and less controversial explanations that are available in the literature. In any case, further research is needed to illuminate the mechanisms underlying the phenomena of recovered memories. Of course,

Does DID Reflect Fantasy Proneness?

Piedfort-Marin and colleagues (2021) suggested that, contrary to what the SCM implies, "there is no strong evidence that DID is the product of a fantasy proneness (...)" (p. 380). The authors rejected the argument of positive correlations observed between DES scores and different measures of fantasy proneness, because "as the DES includes many items referring to absorption (the ability to do an action while in a non-pathological daydreaming state), it is not surprising that correlations are found between the DES and different measures of daydreaming tendency and imagination" (p. 380). In addition to the fact that fantasy proneness is not only characterized by a proneness to daydreaming and imagination, but also by difficulties in distinguishing between reality and fantasy, the proportion of variance explained by fantasy proneness in dissociation remains large (i.e., 32%, [START_REF] Pekala | Dissociation as a function of child abuse and fantasy proneness in a substance abuse population. Imagination[END_REF], even when the DES absorption items are excluded, which is more than the approximately 10% of variance that represents the average correlation (i.e., effect size, r = .32) between dissociation and trauma reported by [START_REF] Dalenberg | Evaluation of the evidence for the trauma and fantasy models of dissociation[END_REF]see Lynn et al., 2014). Although the authors accurately reported the findings of van der Boom et al. ( 2010) that fantasy proneness did not play a mediating role in the association between trauma and dissociation, van der Boom et al. ( 2010) themselves emphasized that their results did not support a direct causal relation between trauma and dissociation either, given findings of only a modest correlation.

Piedfort-Marin et al. ( 2021) criticized the fact that most studies investigating the associations between fantasy proneness and dissociation were not conducted with clinical samples but with general public samples (e.g., students) and claimed that these studies do not allow for satisfactory conclusions. To bolster this claim, the authors cited Nijenhuis and Reinders's study (2012) that reported low mean scores in DID patients on a fantasy proneness Dwelling on increasingly stale controversies and points of disagreement across perspectives will not advance the field forward in meaningful directions and will only further polarize the discussion on the origins of dissociation. Clearly, the generally unimpressive link between trauma and dissociation, when it is apparent, and limits in earlier sociocognitive accounts in providing comprehensive modeling of DID (see Lynn et al., 2019a), underscores the necessity of expanding our consideration of moderators and mediators of DID symptoms and other dissociative conditions. We suggest that an "open" multifactorial approach, which integrates the strengths of competing models (see Lynn et al., 2019a) and considers more recent data on the roles of sleep disturbance; generalized stress (apart from trauma); deficits of selfawareness, self-regulation, and representation; and hyperassociativity, provides the best opportunity to capture the complexity of dissociative disorders and to delineate multiple pathways to understanding and treating dissociative conditions.