On the input admittance of many violins a typical broad frequency peak, called "bridge hill" appears around 2.5 kHz. The physical parameters of a violin bridge have a significant influence on this feature, and then on the tonal colouration of the produced sound. The effect of the bridge characteristics (mass, stiffness and foot spacing) on the frequency response have been revealed by using bridge models through several studies.

Here the parameters of the bridge response are changed, by applying an active control method. Such a correction, very useful in noise reduction, enables to change separately the position and the shape of the bridge hill. It is first tested on a simplified bridge model, then on a real bridge by placing 2 actuators and an accelerometer at strategic positions.

Introduction

The sound production of most bowed string instruments is due to three essential components: the strings which are excited by the musician, the instrument body which radiates sound all around , thanks to the movement of the top and back plates, and the bridge, Fig. 1, which transmits the vibrations between the string and the body. Its response has a determining influence on the resulting sound. The instrument makers are aware of its role and attach importance to its adjustement. Several frequency response measurements of high quality violins revealed the existence of a broad peak between 2 and 3 kHz, followed by a steady drop in amplitude at higher frequencies [START_REF] Cremer | Physik der Geige (The physics of the violin)[END_REF]; an example is plotted in Fig. 2. This feature is called "bridge hill" because first, Cremer attributed its existence to the lowest resonance of the bridge [START_REF] Jansson | Admittance measurements of 25 high quality violins[END_REF].

Figure 2: Input admittance of a real violin bridge.

Then many studies have proposed bridge models in order to explain the cause of this hill and to make clear the effect of the bridge physical parameters on it. By analysing Reinicke's mass-spring model, Beldie showed that the hill center frequency depends on its stiffness [START_REF] Reinicke | Die Übertragungseigenschaften des Streichinstrumentensteges[END_REF][START_REF] Beldie | About the bridge hill mistery[END_REF]. Then Woodhouse carried out a simplified model, Fig. 3, and studied the influence of its mass and of its stiffness on the "bridge hill" shape [START_REF] Woodhouse | On the "bridge hill" of the violin[END_REF]. This paper proposes an other idealised bridge model in Fig. 4, simple enough to foresee its behaviour. First it is described and compared to the models introduced above. Then its physical parameters are changed in order to find out their effect on its frequency response. In the last part an additional couple of forces is exerted on the model to modify the bridge hill shape in real time.

F e jωt (l 0 cos(θ) + a) -sin(2θ) d 2 k 4 = Jjω V b l 0 + a ( 1
)
where

J(= m 1 a 2 + m2d 2 12
) is the moment of inertia of the bridge upper part and V b is the linear velocity of the point M . The bridge admittance Y b , defined as the quotient θ+θv F e jωt is obtained by differentiating the equation [START_REF] Cremer | Physik der Geige (The physics of the violin)[END_REF]. Assuming that θ is a very small angle, cos(2θ) ≈ cos(θ) ≈ 1 and sin(θ) << cos(θ); hence

Y b = (l 0 + a) 2 jω + kd 2 R 2 (jω) 2 J + kd 2 2 (1 + JRjω) (2)
Woodhouse [START_REF] Woodhouse | On the "bridge hill" of the violin[END_REF] led an equivalent calculation by using Reinicke's model with a torsion spring k , a punctual mass m and a rotating link length a . He gets the same results with k = kd 2 /2, m = J/(l 0 + a) 2 and a = l 0 + a. As its resonance frequency is

1 a k m , in our model it is defined by Ω b = d k 2J . Therefore k = 2J( Ω b d ) 2 .
The rotational admittance is expressed by using the violin model of Woodhouse [START_REF] Woodhouse | On the "bridge hill" of the violin[END_REF]. It incorporates the top and back plates of the body as two rectangular bending plates with hinged boundary conditions all around. It also includes the soundpost as a massless and rigid link between both plates. The velocity of the plates at the contact points is the same. R is plotted in Fig. 5. With such a model, it is composed by a set of eigen modes.

The constants l 0 , a, d and J that appear in the expression of Y b are estimated with a real violin bridge:

-l 0 = 0.01m -a = 0.03m -d = 0.03m -m tot = 0.5g and m 1 = m 2 = m tot /2 -J = m 1 a 2 + m2d 2
12 = 2.4375.10 -7 kg.m 2 On the measurement of bridge mobility carried out by Jansson and Bissinger [START_REF] Jansson | Violin frequency response -bridge mobility and bridge feet distance[END_REF][START_REF] Bissinger | The violin bridge as filter[END_REF], no isolated bridge resonances were observed. Here the chosen value of Ω b is 2π × 3kHz, so that it approximatively fits in with the lowest bridge in-plane resonance. Consequently the stiffness of each spring is k = 192457N.m -1 . On Fig. 6, Y b is plotted and the bridge hill is guessed around 2.8kHz. Thus the described model with two springs is in accordance with the earlier models using a torsion spring.

Figure 6: Bridge mobility calculated with the bridge model described here

Changing the bridge physical parameters

The previous part was dedicated to describe a bridge model and to plot its frequency response. Now it would be interesting to assess the influence of its physical parameters on it. It is therefore essential to display the bridge hill without the resonance peaks of the rotationnal admittance, in order to measure its characteristics accurately.

How to measure the bridge hill characteristics

Woodhouse explains why those peaks disappear if the violin top and back plates are replaced by infinite plates [START_REF] Woodhouse | On the "bridge hill" of the violin[END_REF]. Actually no wave propagating on such a body is reflected, because its boundaries are infinitely far away. Thus no resonance peak occurs, since no standing wave settles.

If the violin body model is modified, then the expression of R also changes. Actually it depends on the violin body admittance at the bridge feet and at the soundpost position [START_REF] Woodhouse | On the "bridge hill" of the violin[END_REF], and now those magnitudes are different since the violin plates are infinitely large. The new rotationnal admittance is called R ∞ . The bridge admittance is calculated again, Y b∞ while replacing R by R ∞ in (2):

Y b∞ = (l 0 + a) 2 jω + kd 2 R∞ 2 (jω) 2 J + kd 2 2 (1 + JR ∞ jω) (3)
It is plotted in dashed line on Fig. 6. The resonance peaks due to the rotationnal admittance have been removed. Such a smooth bridge response is called a skeleton curve by Woodhouse [START_REF] Woodhouse | On the "bridge hill" of the violin[END_REF]. Now the bridge appears clearly and looks like the frequency response of a second order filter. Its eigen frequency, its damping factor and its maximum gain can be easily measured and compared to those of other bridge responses.

Influence of some bridge physical parameters on the bridge hill

The next stage consists in discussing the effect of some physical bridge characteristics on the width, the height and the place of the hill. While a violin maker can not easily change the geometric characteristics of the bridge without modifying the playing comfort, he would rather control independently the bridge mass and the stiffness. The total mass can be reduced by removing matter from the "bridge heart", whereas the stiffness can decrease by digging the "bridge ears", cf Fig. 1. The result of these alterations is looked in the bridge model by varying the values of the mass and of the stiffness in the expression of Y b∞ : On the Fig. 7, the total mass m total is increased while keeping k fixed. In this case, the bridge hill center frequency, the damping factor and the maximum gain decreases.

On the Fig. 8, the stiffness k is increased while keeping m fixed. Hence the hill frequency gets higher whereas the damping factor and the maximum gain decreases. These results confirm those achieved by Woodhouse [START_REF] Woodhouse | On the "bridge hill" of the violin[END_REF] and Beldie [START_REF] Beldie | About the bridge hill mistery[END_REF]. They are also in accordance with the behaviour of a simple 2 nd order harmonic oscillator mass M , stiffness K, damping R which resonance frequency and damping factor are respectively K M , and KM R .

Correction of the violin sound parameters by active control

The bridge filters the excitation provided by the strings vibrations. Thus, while changing its frequency response, the instrument timbre gets modified.

Definition of the parameters to change in the violin response

Some harmonics of the played notes can be emphasized or dampened by changing the position of the bridge response bandwidth, its amplitude and its selectivity. Consequently controlling the resonance frequency, the damping factor and the maximum gain of the bridge admittance should enable to modify the violin's sound identity.

The expression of Y b∞ is very similar to the frequency response of a 2 nd order harmonic oscillator. Indeed, by using Eq.( 3), its eigen frequency, damping factor and maximum gain are respectively defined as:

f 0b = d k 2J = d k 2(m 1 a 2 + m2d 2 12 ) ( 4
)
Q b = 1 dR kJ 2 = ( 2
)
dR k(m 1 a 2 + m2d 2 12 )

(5)

H 0b = (l 0 + a) 2 J ( 1 jω + 2 kRd 2 ) (6)
Since the moment of inertia J and the total mass m total vary in the same way, the phenomena observed on the skeleton curves (figures [START_REF] Boutin | Contrôle actif sur instruments acoustiques[END_REF] and [9]) become clear: Q b and H 0b are all the more decreased when J or k are increased, Eq.( 5) and ( 6), and f 0b gets higher when k increases and lower when m increases, Eq.( 4).

Correction by active control

Changing the bridge mass and stiffness causes simultaneous modifications of the height, the width and the place of the hill. A PID correction applied to a 2 nd order harmonic oscillator enables to bring its resonance frequency, its Q-factor and its maximum gain under independant control.

Figure 9: Feedback correction applied on a real bridge

In such a looped system, plotted in Fig. 9, u is the oscillation velocity. The feedback force generated by the PID corrector is f corr = P u + I u jω + Djωu, so that the controlled oscillator gets new parameters f 0 , Q and H 0 . In this case the coefficients P , I and D can be expressed in function of their relative variations ∆ω 0 , ∆Q and ∆H 0 [START_REF] Boutin | Contrôle actif sur instruments acoustiques[END_REF].

However the described bridge model placed on an infinitely large body is not really a 2 nd order harmonic oscillator. Indeed its parameters f 0 , Q and H 0 are not constant since the rotational admittance depends on the frequency Eq.( 4),( 5) and [START_REF] Jansson | Violin frequency response -bridge mobility and bridge feet distance[END_REF]. Despite that, let us focus on the effect of a PID correction on the bridge hill parameters. A couple of forces are applied on the bridge upper part, so that their resulting moment is P

Y b∞ = (l 0 + a) 2 (jω + k R ∞ ) (jω) 2 α + jωβ + γ -Iδ jω (7)
where

k = kd 2 2 , α = (J -D), β = (J -D)R ∞ k -P , γ = -I + k (1 -R ∞ P ), and δ = k R ∞ .
In Eq.( 7), the denominator remains a 2nd degree polynomial only if I = 0. In this case, the new parameters ω 0 , Q and H 0 depend on P and D:

ω 0 = k (1 -R ∞ P ) J -D (8)
Q = k (1 -R ∞ P )(J -D) (J -D)k R ∞ -P (9)
H 0 = (l 0 + a) 2 (J -D)k R ∞ -P (1 - jk R ω ) (10)
This equations system gives expressions of P and D in function of the input values ∆ω 0 and ∆Q:

P = 1 (1+∆Q)(1+∆ω0) 2 R∞k J(∆Q-∆ω0) + R ∞ (11)
D = J(1 - 1 + ∆Q (1 + ∆Q)(1 + ∆ω 0 ) 2 + R 2 ∞ k J(∆Q -∆ω 0 ) ) (12)
An other relation is given by the system, between ∆H 0 , and the coefficients P and D:

∆H 0 = 1 J D+ P k R∞ -1 (13)
Therefore ∆H 0 can be deduced from ∆Q and ∆ω 0 . As a conclusion, while I = 0 the feedback loop imposes the value of two parameters, Q and ω 0 for instance, without controlling the third variation ∆H 0 . The Fig. 11 and Fig. 12 display different bridge responses resulting of some P D corrections: First, on Fig. 10, the damping factor is varied while imposing ∆ω 0 = 0. The skeleton curves have been normalized. In this way, ∆H 0 can not appear and the Q variations are clearly perceived. All the same, the resonance frequency of the bridge hill changes although it should remain constant. Especially, ω 0 decreases while ∆Q < 0. Then, on Fig. 11, ω 0 is modified while keeping ∆Q = 0. Here the curves are not normalized. That is why some gain modifications appear. The Q-factor seems to remain constant. However the ω 0 variations do not achieve the exact desired correction. In both extreme cases ∆ω 0 > -18% and ∆ω 0 < 18%.

Actually as R ∞ is a complex value, ω 0 , Q and H 0 are also complex. Therefore their value are not the resonance frequency, damping factor and maximum gain that are actually measured on the figure, because their imaginary part is not equal to zero. That is why the hill is not exactly centered in 3kHz. Moreover ω 0 , Q and H 0 depend on R ∞ and then also depend on ω 0 . Consequently, as the P and D coefficients of the controller are deduced from those parameters, they are not constant values either. To carry out a simplified correction on a real bridge, R ∞ just needs to be replaced by a constant value in the expression of P and D.

How to apply the correction on a real bridge

This last paragraph proposes a method for controlling a real violin's bridge response. The P D correction described above is obtained by applying an additionnal couple of forces, which moment depends on the bridge's angular velocity. This force signal can be generated by using an accelerometer that measures θ b . Then the controller calculates the P and D coefficients in function of the ∆ω 0 and ∆Q variation desired by the user in real time.It deduces the value of the following force signal:

F corr = 1 d (P . θ b +D ..
θ b u r ) Two actuators have been wedged in the "bridge ears" to apply this correction to the mobile part of the bridge. The system is described in Fig. 12. Thus, the movement of the real bridge should be described by Eq. [START_REF] Bissinger | The violin bridge as filter[END_REF]. Consequently its frequency response must get modified as observed in Fig. 10 and Fig. 11, according to the correction imposed by the instrumentist.

Conclusion

Applying active control to a vibrating system modifies the parameters of its movement. This technique has been used here to change the frequency behaviour of a violin bridge. In particular the purpose was to control the shape of the "bridge hill", since its influence on the violin sound is significant. Thanks to a feedback correction, the resonance frequency and the damping factor of the bridge response have been brought under independent control. Those results have been obtained by controlling a very simplified bridge model. Such a correction has also been carried out on a real bridge and has permitted to hear timbre modifications in the produced violin's sound.

Figure 1 :

 1 Figure 1: Example of a violin bridge.

Figure 3 :

 3 Figure 3: The simplified bridge model proposed by Woodhouse.

Figure 4 :

 4 Figure 4: An other violin bridge model.

Figure 5 :

 5 Figure 5: Rotational admittance calculated with Woodhouse's violin model.

Figure 7 :

 7 Figure 7: Variation of the bridge's mass

Figure 8 :

 8 Figure 8: Variation of the bridge's stiffness

.

  θ b +Iθ b +D .. θ b , cf Fig.4. Because of this additionnal moment, the bridge mobility is modified:

Figure 10 :

 10 Figure 10: Correction of the damping factor with an "ideal" PD corrector

Figure 12 :

 12 Figure 12: Real bridge provided with 2 actuators and 1 accelerometer

Acknowledgments

A lot of thanks to Nicolas Pailheret that has actively participated to this study.


Sten Madec
email: csten.madec@univ-tours.fr

Erida Gjini
email: degjini@igc.gulbenkian.pt

Predicting N-strain coexistence from co-colonization interactions: epidemiology meets ecology and the replicator equation

Keywords: SIS model, slow-fast dynamics, weak selection, competition-cooperation, multispecies coexistence, broken symmetries, invasion fitness network, replicator equation

Multi-type spreading processes are ubiquitous in ecology, epidemiology and social systems, but remain hard to model mathematically and to understand on a fundamental level. Here, we describe and study a multi-type susceptibleinfected-susceptible (SIS) model that allows for up to two co-infections of a host. Fitness differences between N infectious agents are mediated through altered susceptibilities to secondary infections that depend on colonizer-cocolonizer interactions. By assuming small differences between such pairwise traits (and other infection parameters equal), we derive a model reduction framework using separation of timescales. This 'quasi-neutrality' in strain space yields a fast timescale where all types behave as neutral, and a slow timescale where non-neutral dynamics take place.

On the slow timescale, N equations govern strain frequencies and accurately approximate the dynamics of the full system with O(N 2 ) variables. We show that this model reduction coincides with a special case of the replicator equation, which, in our system, emerges in terms of the pairwise invasion fitnesses among strains. This framework allows to build the multi-type community dynamics bottom-up from only pairwise outcomes between constituent members. We find that mean fitness of the multi-strain system, changing with individual frequencies, acts equally upon each type, and is a key indicator of system resistance to invasion. Besides efficient computation and complexity reduction, these results open new perspectives into high-dimensional community ecology, detection of species interactions, and evolution of biodiversity, with applications to other multi-type biological contests. By uncovering the link between an epidemiological system and the replicator equation, we also show our co-infection model relates to Fisher's fundamental theorem and to conservative Lotka-Volterra systems.

Introduction

Many factors have been shown to be important for ecological biodiversity. These include inter-vs. intra-species interactions [START_REF] Tilman | The importance of the mechanisms of interspecific competition[END_REF], population size [START_REF] Taylor | Evolutionary game dynamics in finite populations[END_REF], number and functional links with resources [START_REF] Armstrong | Coexistence of species competing for shared resources[END_REF], and movement in space [START_REF] Nowak | Evolutionary games and spatial chaos[END_REF]. Diversity of an ecosystem is closely related to its stability, resilience to perturbations and productivity [START_REF] Hooper | Effects of biodiversity on ecosystem functioning: a consensus of current knowledge[END_REF]. Co-colonization processes appear in many diverse ecological communities, from plant and marine ecosystems to infectious diseases: two species encountering and coexisting together in a unit of space or resource. Interactions between resident and co-colonizer entities upon encounter may exhibit asymmetries, randomness, and special structures, and may range from facilitation to competition. Although special cases for low dimensionality seem to be tractable analytically, the entangled network that arises between N interacting entities in co-colonization, and its consequences for multi-type dynamics over short and long time scales remain elusive. Importantly, the question of how the net behavior of a collective of types arises from pairwise outcomes in co-colonization constitutes a nontrivial analytical challenge.

In infectious diseases characterized by polymorphic agents, such as influenza, dengue, malaria, and pneumococcus, the mechanisms allowing many different co-circulating pathogen types to be maintained have long been considered [START_REF] Gog | Dynamics and selection of many-strain pathogens[END_REF][START_REF] Gupta | Population structure of pathogens: the role of immune selection[END_REF]Cobey and Lipsitch, 2012a). Typically, strain-specific and cross-immunity have been studied using SIR frameworks, as a main structuring force of such polymorphic systems [START_REF] Kucharski | Capturing the dynamics of pathogens with many strains[END_REF]. In contrast, the influence of interactions through co-colonization (or co-infection, implying simultaneous carriage of two strains), which reaches for example 10-20% prevalence in asymptomatic carriage in pneumococcus [START_REF] Valente | Decrease in pneumococcal co-colonization following vaccination with the seven-valent pneumococcal conjugate vaccine[END_REF], has received less mathematical attention on the SIS modeling spectrum [START_REF] Adler | The dynamics of simultaneous infections with altered susceptibilities[END_REF][START_REF] Lipsitch | Vaccination against colonizing bacteria with multiple serotypes[END_REF][START_REF] Gjini | How direct competition shapes coexistence and vaccine effects in multi-strain pathogen systems[END_REF]. In pneumococcal bacteria, displaying more than 90 serotypes [START_REF] Park | Discovery of a new capsular serotype (6c) within serogroup 6 of Streptococcus pneumoniae[END_REF], carriage of one serotype has been shown to alter, mostly reduce, the acquisition rate of a second serotype [START_REF] Auranen | Between-strain competition in acquisition and clearance of pneumococcal carriage: Epidemiologic evidence from a longitudinal study of day-care children[END_REF][START_REF] Lipsitch | Estimating rates of carriage acquisition and clearance and competitive ability for pneumococcal serotypes in Kenya with a Markov transition model[END_REF]. Despite competition, much diversity at the serotype and genotype level persists over long time, even in the face of vaccines [START_REF] Weinberger | Serotype replacement in disease after pneumococcal vaccination[END_REF]. Understanding how cocolonization interactions may promote coexistence patterns within and between species, in pneumococcus [START_REF] Bogaert | Variability and diversity of nasopharyngeal microbiota in children: A metagenomic analysis[END_REF][START_REF] Dunne | Nasopharyngeal microbial interactions in the era of pneumococcal conjugate vaccination[END_REF][START_REF] Shrestha | Identifying the interaction between influenza and pneumococcal pneumonia using incidence data[END_REF]) or other pathogen systems [START_REF] Cohen | Modeling the effects of strain diversity and mechanisms of strain competition on the potential performance of new tuberculosis vaccines[END_REF][START_REF] Abdullah | Host-multi-pathogen warfare: pathogen interactions in co-infected plants[END_REF] remains a challenge.

To meet this challenge, mathematical frameworks need to integrate diversity and inter-dependence in such multitype systems with ecological principles and simpler formulations. Unfortunately however, mechanistic insights for high-dimensionality are not always available. In pneumoccocus, previous epidemiological studies, when including co-colonization, have either adopted low system size for analytical results (N = 2) [START_REF] Lipsitch | Vaccination against colonizing bacteria with multiple serotypes[END_REF][START_REF] Gjini | How direct competition shapes coexistence and vaccine effects in multi-strain pathogen systems[END_REF], focusing on vaccination outcomes, or combined niche and neutral mechanisms exclusively through simulations for larger number of strains (large N) (Cobey and Lipsitch, 2012a;[START_REF] Bottomley | A mathematical model of serotype replacement in pneumococcal carriage following vaccination[END_REF][START_REF] Nurhonen | Pneumococcal transmission and disease in silico: a microsimulation model of the indirect effects of vaccination[END_REF]. In other systems, advances for N = 3 and special parameter combinations, have only recently been made [START_REF] Pinotti | Interplay between competitive and cooperative interactions in a three-player pathogen system[END_REF].

To advance our understanding and predictability of multi-type systems, there is a need to overcome both these limitations, increasing our analytical power over a larger and more realistic system size. For example, N ≈ 30 is a typical number of co-circulating pneumococcus serotypes in any given setting [START_REF] Lipsitch | Estimating rates of carriage acquisition and clearance and competitive ability for pneumococcal serotypes in Kenya with a Markov transition model[END_REF], but comprehensively, the quadratic explosion of number of serotype pairs with N makes it very hard to predict and mechanistically understand epidemiological variables describing co-colonization.

Limitations in modeling interacting disease systems persist even in more general theoretical contagion studies of SIS [START_REF] Hébert-Dufresne | Complex dynamics of synergistic coinfections on realistically clustered networks[END_REF][START_REF] Chen | Fundamental properties of cooperative contagion proc[END_REF] and SIR type [START_REF] Sanz | Dynamics of interacting diseases[END_REF][START_REF] Miller | Cocirculation of infectious diseases on networks[END_REF], which have recognized the rich behavior of such systems, especially when cooperative interactions between entities are at play. These studies have also considered very low system size (N = 2), and either only cooperative or only competitive interactions, highlighting more critical transitions in contact networks [START_REF] Hébert-Dufresne | Complex dynamics of synergistic coinfections on realistically clustered networks[END_REF][START_REF] Miller | Cocirculation of infectious diseases on networks[END_REF]. Joint analyses of cooperation and competition under the same framework, for an arbitrary number of interacting entities, that avoid combinatorial explosion of complexity, remain undeveloped.

In this study, we provide a fundamental advance on this challenge. We describe and analyze a very general system of multi-type interactions that arise in SIS epidemiological dynamics with co-infection (co-colonization). We show that by assuming small differences in altered susceptibilities to coinfection between N types, an explicit reduced system emerges via a timescale separation. This enables to express the total dynamics as a fast (neutral) plus a slow (non-neutral) component, the latter related to variation in co-colonization interactions. Multi-strain collective dynamics can be very complex, but we find that they are explicitly modulated by global parameters such as overall transmission intensity R 0 , and mean interaction coefficient between strains, k. We derive a closed analytic solution for strain frequencies over long time-scales in a changing mean fitness landscape. We further show that the emergent equation from this N-dimensional model reduction corresponds to a version of the classical replicator equation from game theory [START_REF] Taylor | Evolutionary stable strategies and game dynamics[END_REF][START_REF] Weibull | Evolutionary game theory[END_REF][START_REF] Hofbauer | Evolutionary game dynamics[END_REF], in our case, appearing in terms of pairwise invasion fitnesses between strains.

Although the model is treated in an epidemiological spirit, parallels and conceptual analogies with other systems can be easily drawn, where multiple types (species/strains/entities) in a homogeneous mixing scenario compete for free and singly-occupied niches via generic colonizer-cocolonizer interactions. In the paper, we will use the terms co-infection and co-colonization interchangeably, referring always to an avirulent scenario, and thus reinforcing the potential application of the framework both in epidemiology and ecology.

The modeling framework

N-strain SIS model with co-colonization

We consider a multi-type infectious agent, transmitted via direct contact, following susceptible-infected-susceptible (SIS) transmission dynamics, with the possibility of co-infection. The number of potentially co-circulating strains is N. With a set of ordinary differential equations, we describe the proportion of hosts in several compartments: susceptibles, S , hosts colonized by one type I i , and co-colonized hosts I i j , with two types of each combination, independent of the order of their acquisition. We have:

                         Ṡ = m(1 -S ) -S N j=1 F j , İi = F i S -mI i -I i j K i j F j , 1 ≤ i ≤ N İ i j = I i K i j F j -mI i j , 1 ≤ i, j ≤ N (1)
where

F i = β I i + N j=1 1 2 (I i j + I ji )
gives the force of infection of strain i. We assume that hosts colonized with a mixture of two subtypes i and j, I i j transmit either with equal probability. For more information and interpretation of parameters see Table S1 . Notice that S = 1 -(I i + I i j ), thus the dimension of the system is indeed N + N 2 . In practice, I i j and I ji cannot be distinguished so the dimension is effectively N + N(N -1)/2.

We study pathogen diversity only in relation to how strains interact with each other upon co-colonization (K i j ), Hosts move from susceptible to singly-colonized state, and from singly-colonized to co-colonized state. Clearance happens at equal rate for single and co-colonization. Co-colonization rate by strain j of singlycolonized hosts with i is altered by a factor K i j relative to uncolonized hosts. b. Complex epidemiological dynamics can be represented in two interrelated timescales. Assuming that pairwise interaction coefficients in co-colonization can be represented as: K i j = k + εα i j , the global compartmental dynamics can be decomposed in a fast and slow component. On the fast time-scale (o(1/ε)), strains follow neutral dynamics, driven by mean-field parameters, where total prevalence of susceptibles S , singly-infected hosts, I and dually-infected hosts, D, are conserved. On a slow time-scale, εt, complex non-neutral dynamics between strains takes place, depicted here by the constituent variations within the blue and green. These non-neutral dynamics are explicitly derived in this paper, and yield an explicit equation for strain frequency dynamics. assuming equivalence in transmission β and clearance rate γ. The coefficients K i j , denote factors of altered susceptibilities to secondary infection when a host is already colonized, and can be above or below 1, indicating facilitation or competition between colonizer and co-colonizer strain. In the above notation m = γ + r, encapsulating both clearance rate γ of colonization episodes and recruitment rate of susceptible hosts r, equal to the mortality rate from all compartments (r = d). For a model summary diagram see Fig. 1a-b. Since we model N closely-related strains, we can write each co-colonization coefficient as: K i j = k + εα i j , where 0 ≤ ε << 1. This will form the basis of our model reduction framework, and system decomposition into smaller sub-systems.

Results

Re-writing the system in aggregated form

In order to derive a reduced model for such a general N strain system, displaying complicated and high-dimensional dynamics (Fig. 1b), we use the following aggregation of variables:

J i = I i + N j=1 1 2 (I i j + I ji ) and T = i I i + i I ii + i j I i j . (2)
for the prevalence of strain i in the population, and the total prevalence of all strains, respectively. Total prevalence satisfies T = i J i and the forces of infection are: F i = βJ i . With these notations, the system (1) can be rewritten as:

Ṡ = m(1 -S ) -βS T, Ṫ = βS T -mT, İi = βJ i S -mI i -βI i N j=1 K i j J j , 1 ≤ i ≤ N (3) Ji = (βS -m)J i + β 2 N j=1 I i K i j J j -I j K ji J i , 1 ≤ i ≤ N İ i j = I i K i j βJ j -mI i j , 1 ≤ i, j ≤ N
This system of 2 + N + N 2 equations, now displays a convenient structure, that we exploit for our analysis: i) First we describe the block of 2 equations (S , T ) that do not depend on K i j . ii) Next, we study the block of 2N equations (I i , J i ), which is the most complicated. iii) Lastly, we deal with the block of the N 2 equations of I i j , which is simple once the dynamics of I i and J i are known (see Supplementary material 2 for details). Clearly, if the basic reproduction number R 0 = β m > 1, as typical in SIS models, then there is an endemic equilibrium, whereby (S , T ) [START_REF] Dietz | The estimation of the basic reproduction number for infectious diseases[END_REF]. By reducing the system to the invariant manifold (S , T ) = (S * , T * ) ≡ 1 R 0 , 1 -1 R 0 , we obtain a simpler system for the second block of equations. Once it is known that (I i , J j ) → (I * i , J * j ), then the N 2 equations for co-colonization compartments imply I i j → β m I * i K i j J * j . Thus, once the dynamics of the second set of equations are explicit, so are the dynamics of co-colonization variables, and ultimately of the entire system.

→ 1 R 0 , 1 -1 R 0

Slow-fast decomposition of the system

We use a similar trick as in [START_REF] Gjini | A slow-fast dynamic decomposition links neutral and non-neutral coexistence in interacting multi-strain pathogens[END_REF]. We assume the N strains are closely-related, thus each cocolonization coefficient can be rewritten as:

K i j = k + εα i j
, where 0 ≤ ε << 1, and k a suitable reference. Replacing these in (3), re-arranging, and analyzing the cases of ε = 0 and ε > 0 but small, we obtain the slow-fast representation of the system (See Materials and Methods, and Supplementary Material 2 for details).

Neutral system

If ε = 0, then we obtain the Neutral model, where all strain co-colonization coefficients are equal to k and strains behave as equivalent. We find that on the fast time-scale o(1/ε), the key dynamics are given by these two quantities:

H i = I * T * 2(T * ) 2 -D * I * I i I * - J i T * , and
z i = I i I * + 2(T * ) 2 2(T * ) 2 -D * I * J i T * - I i I * .
which obey: Ḣi = -ξH i , żi = 0. The quantity H i measures the difference between the part occupied by strain i in single colonization versus the part of strain i in total carriage. Thus, H i → 0 means that, on the fast timescale, the proportion of strain i in single colonization ( I i I * ), tends to equalize the proportion of strain i in overall colonization ( J i T * ). This implies it also tends to be equal to the proportion occupied by strain i in co-colonization:

j i I i j /2 + I ii = D i /D * .
Because H i = 0 is the only equilibrium, we can infer that during fast dynamics z i tends to the limit:

z i = J i T * = I i I * = D i D * , (4)
whereby z i becomes the variable in the system that exactly describes exactly the frequency of strain i in the host population, with z i = 1.

The slow dynamics: strain frequencies Analyses of the system for ε > 0, reveal that on the slow timescale τ = εt, strain frequencies, z i , obey explicit dynamics (See Supplementary material 3). These dynamics are given by the N-dimensional system:

żi = Θz i              N j=1 j i µ(α ji -α i j ) + α ji z j + α ii z i -q(z)              , 1 ≤ i ≤ N, (5)
where the constants Θ, µ > 0 are explicit functions of the global steady state (T * , I * , D * ) of the neutral model:

Θ = βT * I * D * 2(T * ) 2 -I * D * ; µ = I * D * = 1 k(R 0 -1) ,
and the term q(z) is a quadratic term given by:

q(z) = 1≤k, j≤N α k j z k z j .
The above expression encapsulates how the ultimate competitive hierarchies between N co-colonizing strains, circulating in the same host population, are driven by the magnitude of asymmetries in within-vs. between-strain interactions (the α's), as well as by mean-field global quantities, such as R 0 and k, affecting Θ and µ. Thus, the strain selection occurring in the slow time scale, on the basis of small differences in co-colonization interactions, becomes entirely explicit.

From (5), we can notice that these equations resemble conservative Lotka-Volterra dynamics [START_REF] Lotka | Elements of physical biology[END_REF][START_REF] Volterra | Fluctuations in the abundance of a species considered mathematically[END_REF], but with an extra term q(z), which changes during time. This term q(z) represents the evolving impact of all the strains on their 'common environment', which in turn modifies their own fitness landscape. A more explicit way to interpret q(z) is in terms of relative change in 'effective' mean interaction coefficient between all extant types in the system, which if negative, indicates a global trend toward more competition in co-colonization, and if positive, indicates a global trend toward more facilitation. Formally one would write this mean trait dynamics as ke f f ective (t) = k + εq(t).

Link with pairwise invasion fitnesses λ j i and the replicator equation

Next, we uncover an equivalent very useful representation of the model reduction by using the notion of pairwise invasion fitness [START_REF] Metz | How should we define ?fitness? for general ecological scenarios[END_REF][START_REF] Geritz | Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree[END_REF][START_REF] Meszéna | Link between population dynamics and dynamics of darwinian evolution[END_REF]. Let λ j i be the exponential growth rate of strain i evaluated when introduced at the trivial endemic equilibrium of the strain j alone. If the fitness λ j i > 0, strain i will invade j, and viceversa. By considering the rate of growth of strain i in an endemic equilibrium set by j (dz i /dt (5) in the special case where all z k = 0 for k j), we find the exact formulation of pairwise invasion fitness, in our model, is given by:

λ j i = µ(α ji -α i j ) + α ji -α j j . (6)
We can thus recast the strain frequency equations (system (5)) using only these fitness notations (for details see Supplementary Material 3) :

d dτ z i = Θz i •         j i λ j i z j - 1≤k j≤N λ k j z j z k         , (7)
which can be made even more compact by denoting the pairwise invasion fitness matrix Λ = (λ j i ) i, j , and using vector notation:

d dτ z = Θz • Λz -z t Λz . (8)
Ultimately, it is this matrix Λ that defines all 'edges' of the rescaled interaction network between N strains.

For N = 2 and λ 2 1 λ 1 2 0, similar to the classical competitive Lotka-Volterra model, in our model, as previously shown [START_REF] Gjini | A slow-fast dynamic decomposition links neutral and non-neutral coexistence in interacting multi-strain pathogens[END_REF], there are only four possible outcomes between 2 strains (edge linking 1 and 2): i)

λ 2 1 > 0 , λ 1 2 > 0 : stable coexistence of 1 and 2; ii) λ 2 1 < 0 , λ 1 2 < 0: bistability of 1-only and 2-only; iii) λ 2 1 > 0 , λ 1 2 < 0 :

1-only competitive exclusion; iv) λ 2 1 < 0 , λ 1 2 > 0: 2-only competitive exclusion. Knowing all pairwise invasion fitnesses between each couple of strains, via expression (8) we can reconstitute the ultimate dynamics of the full system with N types and co-colonization. Seen more closely, this model reduction in strain frequency space, corresponds to an instance of the classical replicator equation for multiplayer games, long studied in evolutionary game theory [START_REF] Taylor | Evolutionary stable strategies and game dynamics[END_REF][START_REF] Weibull | Evolutionary game theory[END_REF][START_REF] Hofbauer | Evolutionary game dynamics[END_REF]. Remarkably, however, in contrast to assuming it heuristically a priori, here we have derived it from basic aggregation and timescale principles in an epidemiological context. Moreover, the pairwise traits in the emergent expression are indeed special traits, denoting pairwise invasion fitnesses between co-colonizing strains; invasion fitness being a central quantity in adaptive dynamics used to predict trait evolution [START_REF] Metz | How should we define ?fitness? for general ecological scenarios[END_REF][START_REF] Geritz | Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree[END_REF][START_REF] Meszéna | Link between population dynamics and dynamics of darwinian evolution[END_REF].

The entire dynamics of the N-strain 'game' can now be recapitulated based only on knowledge of the pairwise invasion network between each two 'players'. By having heterogeneous interactions across strain space, and receiving heterogeneous interactions from other strains in the network, each strain, competing for susceptibles and singly-colonized hosts, "perceives" its own unique environment, creates its own niche, which ultimately enables its persistence or extinction. However, the global fitness of each strain in such coupled multi-player competition, depends not only on its own individual fitness and frequency at any time, but also on the fitnesses and frequencies of all other strains. This feature can lead to a multitude of outcomes, as already recognized in replicator equation studies [START_REF] Nowak | Evolutionary dynamics of biological games[END_REF][START_REF] Cressman | The replicator equation and other game dynamics[END_REF].

For illustration, in Figure 2 and in Supplementary Movie S1, we provide an example of the modeling framework and the coexistence dynamics that arise among a number of strains (here N = 6) for an arbitrary co-colonization interaction matrix K. Another combination of parameters, leading to a limit cycle for N = 6, is illustrated in Supplementary Figure S1 and Supplementary Movie S2. These examples demonstrate that even weak asymmetries between apparently similar types, in altered susceptibilites to co-colonization, have the potential to generate rich and hierarchical collective behavior over long time.

Note that in Equations ( 5) and ( 8), the constant Θ is the rate that sets the tempo of multi-type "motion" on the slow timescale towards an equilibrium. The pre-factor Θ depends specifically on the absolute transmission rate of the pathogen β, which calibrates the slow dynamics, but also on mean traits R 0 and k, via the conserved aggregated quantities T * , I * , D * , where R 0 and k are nonlinearly coupled (see Supplementary Figure S2 ). This shows how global environmental variables (similar to the notions of effective population size in population genetics) affect the speed of non-neutral dynamics between types in the system. The quantity µ = I * /D * on the other hand represents the ratio between single colonization and co-colonization prevalence in the overarching neutral system, a critical factor that amplifies the importance of type asymmetry in the slow dynamics due to deviation from neutrality [START_REF] Gjini | The key to complexity in interacting systems with multiple strains[END_REF]. A summary of key model quantities in terms of mean field parameters R 0 and k is given in Table 1.

The quadratic term Q and evolution of mean fitness

In the equivalent representation of the slow dynamics equation, (8), the common quadratic term Q acting on each strain's frequency, relates the common changing environment to pairwise invasion fitnesses [START_REF] Metz | How should we define ?fitness? for general ecological scenarios[END_REF][START_REF] Geritz | Evolutionarily singular strategies and the adaptive growth and branching of the evolutionary tree[END_REF]:

Q(z) = k j (λ k j + λ j k )z j z k /2.
This sum over all extant pairs in the system reflects the mean 'pairwise invasibility' of the system as a whole, changing over time with strain frequencies z j (τ), z k (τ). Upon closer inspection of (8), information on the resilience of a group of strains may be derived from the sign of Q: if Q > 0 then each existing strain is less competitive within the group, but the overall community is more resistant to invasion by a new outsider strain, and viceversa, if Q < 0, then each existing strain grows more within the group, but the overall community would inevitably be also more permeable to invasion by invader strains. This feature of the model, embodies an instance of a classical trade-off between individual and group-selection in evolution [START_REF] Michod | Darwinian dynamics: evolutionary transitions in fitness and individuality[END_REF].

Next, we highlight the meaning of Q by considering a few special cases for the invasion fitness matrix between strains (see Supplementary Material 4), illustrated in Figure 3: i) Symmetric matrix. In fact, a symmetric λ j i structure between strains in mutual invasion leads to a general feature of the dynamics whereby Q always increases over time (Fig. 3a). This case of our N-dimensional system ( żi ), namely the replicator equation for doubly symmetric games, is formally equivalent to the continuous time model of natural selection at a single (diploid) locus with N alleles, known as Fisher's fundamental theorem of natural selection [START_REF] Fisher | The genetical theory of natural selection[END_REF][START_REF] Price | Fisher's 'fundamental theorem' made clear[END_REF][START_REF] Edwards | The fundamental theorem of natural selection[END_REF]. In this case, it can be shown that the population mean fitness increases over time, with the rate of change in mean fitness equal to the trait variance at any point (see S4 for full verification of this feature also in our version of this model). In our modeling context, where λ i j denote pairwise invasion fitness between any two strains, the increase in mean fitness during selective dynamics among N strains, implies that when pairwise invasion 'games' are symmetric, the system becomes more resistant to invasion by outsider strains over time.

ii) Invader-driven invasion. In this case, columns of Λ are equal, meaning it's differences in 'attack rates' (invasiveness) of types that are defining their hierarchical dynamics (Fig. 3b). Mean fitness Q again evolves over slow time, reflecting the selection occurring in the multi-type system, and again tends to increase toward positive values, suggesting coexistence is more likely, although in special cases competitive exclusion may occur.

iii) Resident-driven invasion. In this case, multi-strain dynamics are driven by variation in 'defense' or invasability (rows of Λ are equal), and the principle of competitive exclusion (with possible multi-stability) applies more often, whereby the weakest strains are excluded, and only the best 'defender' of its territory (equilibrium when alone) survives. Competitive exclusion obviously implies Q should tend to 0, verified in Fig. 3c. In exceptional Figure 3: Canonical pairwise invasion structures (Λ) between N types and collective dynamics evolution. We generated random λ j i matrices from 6 special cases and simulated multi-type dynamics (N = 10) under many realizations of the model (( 8)), starting from random initial conditions on the slow manifold, for each case. Q is the mean fitness term in the system (the common 'environment' for all types) changing differently depending on the pairwise invasion fitness matrix. In the third column, the thin blue lines indicate Q evolution for each realization, the thick blue line indicates Q evolution for the z i dynamics shown in the second column, and the thick red line depicts the mean over all 30 realizations. a) Symmetric matrix. This corresponds to the same dynamics captured by Fisher's fundamental theorem. b) Invader-driven fitnesses ('hierarchical attack'). Large potential for coexistence. c) Resident-driven fitnesses ('hierarchical defense'). Large potential for competitive exclusion. d) Anti-symmetric invasion fitnesses. Q is exactly zero over all time and there is large potential for complex multi-strain behavior. e) Almostantisymmetric invasion fitnesses. Maintenance of potential for complex dynamics (e.g. limit cycles) leading to periodicity (but positivity) in Q. f) Random mutual invasion. Rich model behavior is possible. On average coexistence is more likely, but increases as well as decreases in Q over a single realization are possible. cases coexistence may also be possible in this case (see S4 for details). iv) Antisymmetric matrix. This is the case when λ j i = -λ i j and the propensity for complex coexistence dynamics between strains is very high (Fig. 3d). Q is exactly zero in this case, corresponding to zero-sum-games in evolutionary game theory [START_REF] Hofbauer | Evolutionary game dynamics[END_REF]. Like in the classical prey-predator Lotka-Volterra system, there exists a unique center surrounded by a family of cycles [START_REF] Chawanya | Large-dimensional replicator equations with antisymmetric random interactions[END_REF]. This type of oscillatory coexistence is structurally unstable.

v) Almost-antisymmetric. In this case, a small perturbation of the pure anti-symmetric structure in mutual invasion fitnesses disrupts the center leading to a stable or an unstable node. This gives rise to positive and periodic Q, where limit-cycles, heteroclinic cycles or chaos are more likely for multi-strain coexistence.

vi) Random mutual invasion. In this case, which is the most general case, captured by our framework, the dynamics of Q(z) can be arbitrary, and increase or decrease over the same realization of multi-strain dynamics, thus encapsulating dynamic shifts in 'environment quality', and unpredictable emergent dynamics of mean fitness over time (Fig. 3e).

When comparing these 6 special cases, we find that over many stochastic realizations of matrices with such canonical structures and initial conditions, the symmetric invasion matrix results in fastest global mean increase in fitness, as opposed to the random case which leads to the slowest increase in Q (see Supplementary Figure S3), in support of the strong stabilizing effect of pairwise symmetry for collective community dynamics.

It is worth noting that special cases of Q in λ j i space are more straightforward to analyze and understand than special cases in K i j trait space, because for each λ j i representation (see Eq. ( 6)), there is an infinite number of possible K i j in this model, leading to the same pairwise invasion network between strains. In particular, when co-colonization coefficients K i j display a row-wise or column-wise structure (see [START_REF] Lipsitch | Estimating rates of carriage acquisition and clearance and competitive ability for pneumococcal serotypes in Kenya with a Markov transition model[END_REF] for such hypotheses in pneumococcus), invoking a strain-specific hierarchy in this trait, for N = 2 the principle of competitive exclusion applies, but for general number of strains N, such case collapses to the Q = 0 case (antisymmetric invasion matrix above) and complex dynamics are expected.

Going back to the original N + N(N -1)/2 epidemiological variables Now let us make the link to the original system with N strains, given by the SIS model with co-colonization interactions (1). Assume the K i j and the global epidemiological parameters are known. The aim here would be to use the model reduction for computational reasons or for analytical insights. Notice that the framing K i j = k + εα i j is mathematically non-unique, and can be applied with respect to any reference k, provided that the resulting ε is small. However, one convenient choice is to define k as the average of the original interaction matrix entries K i j :

k = i, j K i j N 2 , (9)
and to define deviation from neutrality, ε, as the root mean square distance of the K i j from the mean interaction coefficient k:

ε = i, j (K i j -k) 2 N 2 ,
thus representing the standard deviation of the K i j traits in the pool of N available strains. The direction of deviation from neutrality (bias) for the interaction between strain i and j is then obtained as:

α i j = K i j -k ε .
Thus, the matrix

A = α i j 1≤i, j≤N
is the normalized interaction matrix, with A 2 = 1≤i, j≤N α 2 i j = N. This matrix A, and the ratio µ, determine the pairwise invasion fitness matrix (Eq.( 6)), which contains nearly all the qualitative information about the non-neutral dynamics (Eq.( 8)). So far, we have shown that provided the deviation from neutrality, ε, is small, the behavior of our approximation (5) describes very well the long term dynamics of the original system (1). To recover the original variables, after solving for strain frequencies z i on the slow manifold, as the dynamics in Eqs.(1) are well approached by those in Eqs.( 5), we can use the relations:

S (t) = S * := m β = 1 R 0 , T (t) = T * := 1 -S * = 1 - 1 R 0 , I * (t) := mT * m + βkT * = T * 1 + R 0 kT * ,
to obtain the total prevalence of uncolonized hosts S , total prevalence of colonized hosts T , and prevalence of single colonization I. The prevalence of co-colonization follows simply from D * = T * -I * . Further, to recover strain-specific single colonization, and co-colonization prevalences in the system, within such a 'conservation law', reminiscent of other conservation laws in ecology [START_REF] Hubbell | The unified neutral theory of biodiversity and biogeography[END_REF], we have:

I i (t) := I * z i (τ), I i j (t) = D * z i (τ)z j (τ).
where the slow time scale is τ = εt, and the strain frequencies z i (τ) verify i z i (τ) = 1 and follow explicit dynamics (8). The recovery of the epidemiological variables from the replicator equation exposes two special features of the multi-strain dynamics: on one hand, the slow variables z i (1 ≤ i ≤ N), describing relative strain frequencies in the host population, tend to be necessarily equal in single and co-colonization (Figure 4a); on the other hand, the prevalence of co-colonization with strains i and j in this system (Figure 4b), is proportional to the product between single prevalences of i and j in the population (I i j ∼ I i I j ). Notice, that there is an explicit mean-field pre-factor, dependent on R 0 and k, determining this constant of proportionality (see Table 1). These two quasi-neutrality principles, are preserved on the slow timescale, independently of strain identities and for all time, and moreover, independently of the complexity of the dynamics.

Figure 4: Illustration of invariant principles for strain coexistence on the long timescale. a) Strain frequencies tend to be the same in single and co-colonization, for all strains and for all time when on the slow manifold (( 4)). b) Prevalence of co-colonization I i j tends to scale with the product of strain prevalences in single colonization (I i , I j ), for all strain pairs and for all time, when on the slow manifold (Table 1). This example is simulated using a random matrix K, with N = 10. Each trajectory corresponds to a given strain in the system (a), or a given strain pair (b).

Another example for N = 20 is shown in Supplementary figure S7.

Table 1: Key model quantities in terms of basic reproduction number R 0 and reference interaction coefficient in co-colonization k.

Symbol Interpretation

Features R 0 Basic reproduction number R 0 = β/m > 1 k
Reference interaction coefficient between types in co-colonization (e.g. mean of

K i j ) k        < 1 : competition, > 1 : cooperation S * Equilibrium prevalence of susceptibles S * = 1 R 0 I *
Equilibrium prevalence of singlycolonized hosts

I * = R 0 -1 R 0 [1+k(R 0 -1)] D *
Equilibrium prevalence of co-colonized hosts

D * = (R 0 -1) 2 k R 0 [1+k(R 0 -1)]
µ Ratio between single and co-colonization

µ = I * D * = 1 (R 0 -1)k
Θ Rate of slow dynamics for strain frequencies (z i (τ))

Θ = β 1 -1 R 0 µ 2(µ+1) 2 -µ I i j
Co-colonization prevalence with i and j

I i j = kR 0 [1 + k(R 0 -1)]I i I j
The fact that prevalence of co-colonization with strains i and j in this system, involves a product between single prevalences of i and j in the population (I i j ∼ I i I j ), even though the strains are interacting, is contrary to the independence closure assumption under a purely statistical perspective, adopted heuristically in epidemic models [START_REF] Kucharski | Capturing the dynamics of pathogens with many strains[END_REF]. As we show here, depending on epidemiological details, the feedbacks between interacting strains may mathematically lead to overall multiplicative effects between individual and dual strain prevalences in colonization, with an explicit pre-factor determined by R 0 and k.

To test the quality of the slow-fast approximation with respect to the original system, we verify that the error between the two is small. This is made precise via numerical simulations (Figure S4), where the neutral model is shown to be a good approximation of the original system in a fast time-scale, o(1/ε), and the slow-dynamics reduction a good approximation on the longer time-scale (εt). Finally, we show that this approximation is also advantageous in terms of efficient computation of dynamics for an arbitrary number of strains N (Figures S5-S6 ). While reinforcing the validity of our method, these two important features can aid parameter inference frameworks for epidemiological time-series in multi-strain systems, known to pose many challenges [START_REF] Shrestha | Statistical inference for multi-pathogen systems[END_REF][START_REF] Gjini | How direct competition shapes coexistence and vaccine effects in multi-strain pathogen systems[END_REF].

Overall this multi-strain model, the slow-fast decomposition, and the key features outlined above provide a crucial missing link towards the full characterization of conservative multi-type SIS dynamics with interactions in cocolonization. Our conceptual and analytical framework suggests that in understanding multi-type communities, a closer integration between different temporal scales on one hand, and demographic vs. selective processes on the other, may be possible. Ultimately, the key mechanistic link with the replicator equation, uncovered here, could have wide-ranging applications from microbial ecology to complex systems and public health.

Discussion

One of the fundamental questions in ecology and evolutionary biology is the generation and maintenance of biodiversity [START_REF] Gause | The struggle for existence[END_REF][START_REF] Macarthur | The theory of island biogeography[END_REF][START_REF] Hubbell | The unified neutral theory of biodiversity and biogeography[END_REF]. Most theoretical approaches consider multi-species interactions with classical Lotka-Volterra ODE systems [START_REF] Pascual | Ecological networks: linking structure to dynamics in food webs[END_REF][START_REF] Mougi | Diversity of interaction types and ecological community stability[END_REF] or with evolutionary game theory models [START_REF] Nowak | Evolutionary games and spatial chaos[END_REF][START_REF] Traulsen | Evolution of cooperation by multilevel selection[END_REF], where the interplay between cooperation and competition is key. Typically when more population structure is involved (e.g. in epidemiology [START_REF] Kucharski | Capturing the dynamics of pathogens with many strains[END_REF]) and in high-dimensional spaces of diversity, analysis is very challenging, calling for new theoretical tools and methods, beyond numerical simulation. So far, neither Lotka-Volterra models, nor the replicator equation have been explicitly adopted in mathematical epidemiology, typically based on multi-type compartmental descriptions of host population structure, with only relatively recent attention on evolutionary dynamics [START_REF] Day | Insights from price?s equation into evolutionary[END_REF] and typically focusing on virulence evolution [START_REF] Berngruber | Evolution of virulence in emerging epidemics[END_REF].

In this study, we have bridged further such analytical fronts, by considering a new system of multi-type interactions that arise in SIS transmission dynamics with co-infection and altered susceptibilities. We have derived an explicit link with the replicator equation [START_REF] Taylor | Evolutionary stable strategies and game dynamics[END_REF][START_REF] Weibull | Evolutionary game theory[END_REF][START_REF] Hofbauer | Evolutionary game dynamics[END_REF][START_REF] Cressman | The replicator equation and other game dynamics[END_REF] for strain frequencies, which in turn, is mathematically related to Lotka-Volterra multi-species models [START_REF] Lotka | Elements of physical biology[END_REF][START_REF] Volterra | Fluctuations in the abundance of a species considered mathematically[END_REF]. The continuous replicator equation on N types is topologically equivalent to the Generalized Lotka-Volterra equation in N -1 dimensions [START_REF] Bomze | Lotka-volterra equation and replicator dynamics: a two-dimensional classification[END_REF][START_REF] Bomze | Lotka-volterra equation and replicator dynamics: new issues in classification[END_REF][START_REF] Hofbauer | Evolutionary game dynamics[END_REF]. Thus, by uncovering the replicator equation at the heart of our co-colonization system, we offer new and promising avenues of Our expression for strain frequency evolution (( 8)) makes it also explicit that 'environmental deterioration' may be seen as a cost for the existing collective (since it reduces each strain's rate of growth), but it serves as a protective mechanism against invasion by an outsider strain, and viceversa: 'amelioration' may on one hand seem like it benefits all strains, but on the other it also benefits any outsiders, which eventually may invade more easily. Central to these insights is having made explicit in this particular model the dependence on environmental dynamics of the selective dynamics between types, both in invasion fitness trait space (λ j i ), and in cocolonization trait space (α i j ). This uncovers, in this system, the role of environmental feedback on eco-evolutionary processes, highlighted as a key challenge in evolutionary ecology [START_REF] Lion | Theoretical approaches in evolutionary ecology: environmental feedback as a unifying perspective[END_REF].

Invariant principles in N-type co-colonization

As known from frequency-dependent selection in evolutionary games, the final outcome among N players can be complex, represent a non-fitness-maximizing equilibrium and include oscillations and chaos [START_REF] Nowak | Evolutionary dynamics of biological games[END_REF][START_REF] Cressman | The replicator equation and other game dynamics[END_REF]). Yet, an important finding of our study concerns invariant principles emerging in non-equilibrium multi-type dynamics: the first one being about the dominance of types in single and co-colonization, which is expected to be equal, and the second one being about the co-colonization prevalences as a function of single colonization prevalences of strains (Figure 4, Supplementary Figure S6 ), and these could be used as a practical test for quasi-neutrality, when strain prevalence data are available. Our recapitulation of co-colonization dynamics from slow variables, sheds new analytical light on pathogen interactions and their manifestation at the epidemiological level [START_REF] Kucharski | Capturing the dynamics of pathogens with many strains[END_REF], providing the link between within-host co-occurrence and population-level prevalences between strains. While independence underpins a majority of methods for detecting pathogen interactions from crosssectional survey data (see e.g. [START_REF] Valente | Decrease in pneumococcal co-colonization following vaccination with the seven-valent pneumococcal conjugate vaccine[END_REF]), it is being recognized that even simple epidemiological models challenge the underlying assumption of statistical independence [START_REF] Hamelin | Co-infections by non-interacting pathogens are not independent and require new tests of interaction[END_REF]. These studies are showing that even if pathogens do not interact, other epidemiological feedbacks can induce positive correlation between their prevalences, which leads the proportion of co-infected hosts to be higher than multiplication would suggest.

Along similar logic, our results here expose very clearly, that even if pathogens interact (e.g. via altered susceptibilities to coinfection), multiplicative effects between their prevalences emerge epidemiologically in co-colonization, but with an explicit pre-factor dependent on overall transmission intensity and mean interaction coefficient. Such finding invites a revision of methods to identify interactions between pathogens in endemic systems from cross-sectional data, based on a deeper mathematical understanding of underlying feedbacks. However, as already cautioned in previous studies (Cobey and Lipsitch, 2012b), patterns of interaction are subtle to detect from data and may require very large sample sizes to assess statistically, even with more sophisticated mathematical expectations. neutral limit, delineated here, sampling noise is expected to become increasingly important. Utilisation of fast-variable elimination techniques is leading to growing awareness that noise-induced selection appears across several evolutionary systems in ecology, epidemiology and population genetics (reviewed in [START_REF] Constable | Exploiting fast-variables to understand population dynamics and evolution[END_REF]). For instance, in a similar SIS model to ours, for two strains [START_REF] Kogan | Two-strain competition in quasineutral stochastic disease dynamics[END_REF], it was found that such noise generated selection for one of the strains, depending on the type of initial conditions, even though this was not predicted by the deterministic theory. Exploration of noise-induced selection during the fast (neutral) time-scale of our multi-strain system, where all strains are expected to behave as equivalent, would be interesting for the future.

In summary, although motivated by infectious disease transmission with altered susceptibilities in co-colonization [START_REF] Gjini | How direct competition shapes coexistence and vaccine effects in multi-strain pathogen systems[END_REF][START_REF] Gjini | A slow-fast dynamic decomposition links neutral and non-neutral coexistence in interacting multi-strain pathogens[END_REF], the global contagion dynamics captured here provide compelling parallels with other systems and should have broader applications. The coinfection model could be applied to study more mechanistically opinion propagation dynamics, coexistence in microbial consortia, plant ecology, and other multitype systems where colonizer-cocolonizer interactions matter. Thanks to its powerful abstraction and generality, this model and closed analytical form for frequency dynamics, offers a new bridge between population dynamics in epidemiology, community ecology, and Darwinian evolution. Because on the slow manifold, we have H i = 0, we can infer that during fast dynamics z i tends to:

z i = J i T * = I i I * = D i D * ,
whereby z i describes exactly the frequency of strain i in overall colonization, with z i = 1.

Figure 1 :

 1 Figure 1: Model summary diagram. a. Co-colonization model structure.Hosts move from susceptible to singly-colonized state, and from singly-colonized to co-colonized state. Clearance happens at equal rate for single and co-colonization. Co-colonization rate by strain j of singlycolonized hosts with i is altered by a factor K i j relative to uncolonized hosts. b. Complex epidemiological dynamics can be represented in two interrelated timescales. Assuming that pairwise interaction coefficients in co-colonization can be represented as: K i j = k + εα i j , the global compartmental dynamics can be decomposed in a fast and slow component. On the fast time-scale (o(1/ε)), strains follow neutral dynamics, driven by mean-field parameters, where total prevalence of susceptibles S , singly-infected hosts, I and dually-infected hosts, D, are conserved. On a slow time-scale, εt, complex non-neutral dynamics between strains takes place, depicted here by the constituent variations within the blue and green. These non-neutral dynamics are explicitly derived in this paper, and yield an explicit equation for strain frequency dynamics.

Figure 2 :

 2 Figure 2: Example dynamics of our model for N = 6. a) The matrix of interaction coefficients in co-colonization, generated randomly, with mean k = 1, and standard deviation ε = 0.1. b) The corresponding pairwise invasion fitness matrix for assumed R 0 = 2. c) The network where each edge displays the outcome of pairwise competition between any couple of strains, where the direction of grey edges denotes the winner in competitive exclusion. d) Slow epidemiological dynamics resulting from these interactions. A dynamic display of the trajectory is shown in Supplementary Movie S1.

theoretical cross-fertilization betwen ecology, epidemiology and evolution.

We investigated how co-colonization interaction coefficients (K i j ), be they cooperative or competitive on average (k above/below 1) and with arbitrary among-strain variation (α i j ), drive coexistence in a system with N similar types.

We obtained a timescale separation for the dynamics, which links explicitly variation in interaction traits among cocolonizing strains (K i j = k + εα i j ), with the slow timescale (εt) under which their coupled selective dynamics unfold.

Model reduction for small deviations from neutrality allowed us to express strain frequencies on the slow manifold via only N equations (instead of N + N(N -1)/2 equations in the full system). Strain frequencies can be easily re-mapped to the more complex epidemiological variables reflecting population structure.

From pairwise invasion to collective coexistence

We have demonstrated that the dynamics of the full system can be expressed entirely in terms of pairwise invasion fitnesses of each of the strains (Eq. ( 8)). This is a novel and important finding that links mathematically pairwise outcomes to emergent community dynamics. Our results suggest that a bottom-up approach can be applied to understand and exactly predict community structure. In a recent experimental study, investigating assembly rules in microbial communities [START_REF] Friedman | Community structure follows simple assembly rules in microbial microcosms[END_REF], survival in three-species competitions was predicted by the pairwise outcomes with an accuracy of 90%. Yet, a similar level of accuracy in competitions between sets of seven or all eight species was harder to obtain, and required additional information regarding the outcomes of the three-species competitions. Although their ecological dynamics was based on the generalized Lotka-Volterra framework, and ours here on a multi-type SIS dynamics, the key to higher-dimensional dynamics may be in exploiting the full structure of the nonlinear coupling between pairwise invasion fitnesses, made explicit here.

Environmental feedback from higher-order interactions

Higher-order interactions are expected to emerge whenever the presence of an additional species changes the interaction between two existing species, and can impact on the maintenance of diversity [START_REF] Billick | Higher order interactions in ecological communities: what are they and how can they be detected?[END_REF].

In our co-colonization system, modeling explicitly type interactions with two types of resources: susceptibles S and singly-colonized hosts I, a certain type of higher-order interactions arise naturally because of the indirect effects that altered susceptibilities between any pair i and j in co-colonization have on suppressing or augmenting the available resources I i and I j for the rest of the community, and thus when summed, contribute to mean fitness among everybody in the system. In this entangled network, the multiple types modulate their common environment through the changing term Q in (8), which can mean 'deterioration' of the environment if Q > 0 or 'amelioration' of the environment if Q < 0. This does not necessarily mean strains become more cooperative or competitive in epidemiological cocolonization, as the dynamics of q (in Eq. ( 5)) can be different from the dynamics of Q.

Extensions and outlook

In our system, microbial strains can infect a host simultaneously, and here we concentrated only on the case of up to 2 strains co-colonizing a host. Extension to higher multiplicity of infection (MOI) could be studied in the future, making use of previous models addressing arbitrary MOI and altered susceptibilities in co-infection [START_REF] Adler | The dynamics of simultaneous infections with altered susceptibilities[END_REF]. The transmission and clearance rates of all strains here were assumed equal, except for the interactions in co-colonization given by an N × N matrix. In an ongoing work we show that this assumption can be relaxed leading to the same replicator equation with invasion fitnesses. The new invasion fitness expression emerging from these additional asymmetries in other traits is then a combination of all the deviations from neutrality (unpublished) .

Past theoretical work has considered vulnerability to co-infection modeling it as a single mean-field parameter [START_REF] Alizon | Co-infection and super-infection models in evolutionary epidemiology[END_REF]. Other studies have studied how this trait at the host-pathogen interface impacts disease persistence [START_REF] Gaivão | Transmission fitness in co-colonization and the persistence of bacterial pathogens[END_REF], coexistence and vaccination effects [START_REF] Lipsitch | Vaccination against colonizing bacteria with multiple serotypes[END_REF][START_REF] Gjini | How direct competition shapes coexistence and vaccine effects in multi-strain pathogen systems[END_REF], and contributes to diversity in other traits, e.g. virulence [START_REF] Alizon | Multiple infections and the evolution of virulence[END_REF] and antibiotic resistance [START_REF] Davies | Within-host dynamics shape antibiotic resistance in commensal bacteria[END_REF]. With the here-proposed analytical framework, exploration of these other biological aspects of systems with co-colonization could be made deeper and more insightful. We expect our results to stimulate further progress for the investigation of coexistence and evolution in multi-strain epidemiological systems such as the one of pneuomococcal bacteria (Cobey and Lipsitch, 2012a), where explicit mathematical results linking neutral and niche mechanisms, for high-dimensional coexistence, are still needed.

Our main goal here was to present the timescale decomposition for this system and highlight the fundamental features of the reduced model. While there is a lot of complex dynamics for different N and different matrix structures that we have not explored here, including multistability, limit cycles and chaos, our results thusfar clearly delineate promising avenues for new theoretical work. The wider and more complete ecological picture of co-colonization, as well as the gradients in diversity-stability regimes in coexistence is the focus of another study [START_REF] Gjini | The key to complexity in interacting systems with multiple strains[END_REF]. A natural next step is comparison with the classical Lotka-Volterra model [START_REF] Lotka | Elements of physical biology[END_REF][START_REF] Volterra | Fluctuations in the abundance of a species considered mathematically[END_REF], widely studied in ecology and adopted in applications [START_REF] Mougi | Diversity of interaction types and ecological community stability[END_REF][START_REF] Friedman | Community structure follows simple assembly rules in microbial microcosms[END_REF], with increasing impact in the microbiome era. A great advantage however, of our framework, uncovering the replicator equation [START_REF] Cressman | The replicator equation and other game dynamics[END_REF] at the core of the dynamics, is because thanks to our mechanistic link, many mathematical results for general and special cases derived over decades for the replicator equation (and its links with LV models) [START_REF] Hofbauer | Evolutionary game dynamics[END_REF][START_REF] Sandholm | Population games and evolutionary dynamics[END_REF], would carry over automatically in this setting, and yield important insights for epidemiology.

The slow-fast framework allows to reduce the complexity of multi-type co-colonization systems, and understand transient and long-term coexistence on conserved manifolds near neutrality [START_REF] Hubbell | The unified neutral theory of biodiversity and biogeography[END_REF]. Inevitably, in the quasi-APPENDIX 1: More details on Materials and Methods Here, we briefly outline the materials and methods used. A more detailed treatment of the theoretical analysis and additional figures are presented in the online SI Appendix.

Slow-fast representation of the dynamics

Since we model N closely-related strains, we can write each co-colonization coefficient as: K i j = k + εα i j , where 0 ≤ ε << 1. Replacing these in (3), and re-arranging, we obtain:

where 1 ≤ i ≤ N, and I = I j .


Henri Boutin
email: boutin@lam.jussieu.fr

IJLRDA Charles Besnainou
email: besnainou@ccr.jussieu.fr

PHYSICAL PARAMETERS OF AN OSCILLATOR CHANGED BY ACTIVE CONTROL APPLICATION TO A XYLOPHONE BAR

By applying active control to an oscillator, its modal behaviour changes. This paper makes a comparison between a second order damped harmonic oscillator and a xylophone bar's mode. Then it proposes a method for acting on an eigen resonance of a xylophone bar. The purpose is to get sound modifications, by bringing under quantitative and independent control its pitch and its duration. Thus it extends our previous work [1], by using a digital feedback controller.

INTRODUCTION

The sound emitted by a xylophone bar strongly depends on the localization and the shape of its eigen resonances. Thus changing the relative position and amplitude between the resonances should alter the bar's sound identity. This paper proposes a method for controlling in real time the resonance frequency and the damping factor of a xylophone bar mode. The variation of these parameters is intended to be independent and under quantitative control.

Active control of sound

Active control of sound is sound field modification. Various technological applications have been reported since the beginning of the last century, particularly around noise cancellation. The earliest patent, expressing how to compensate a sound signal by another one in antiphase, was submitted in 1933, by the German inventor P. Lueg [START_REF] Lueg | Process of silencing sound oscillations[END_REF]. The device is described by D. Guicking [START_REF] Guicking | On the invention of active noise control by paul lueg[END_REF] and presented in Fig1.

Figure 1: Illustration page from P. Lueg's U.S. patent [START_REF] Lueg | Process of silencing sound oscillations[END_REF] In 1995, an active control method was applied to a xylophone bar in order to cancel the damping on a chosen vibration mode [START_REF] Besnainou | Modal stimulation: a sound synthesis new approach[END_REF]. At the time we were using an analog feedback corrector which prevented us from achieving totally independent variations of the resonance frequency and the damping factor. The method described in this paper also uses an active control technique but with a digital feedback controller in order to get more accuracy on the mode parameters to modify.

Framework

The first part is dedicated to the theoretical review. First is described the behaviour of a second order harmonic oscillator when a PID (Proportional, Integral, Derivative) correction is applied. In the second part the transfer function of this system is compared to the frequency response of a xylophone bar around one of its eigen resonance. Then the validity of such a physical model will be discussed. Lastly a method is proposed for controlling in real time the resonance frequency and the damping factor of an eigen mode on the real xylophone bar.

PID CORRECTION ON A SECOND ORDER HARMONIC OSCILLATOR

The frequency response of a xylophone bar around one eigen resonance looks like the response of a second order damped harmonic oscillator. This assumption is verified below (cf part 3). This part describes the effect of a PID correction on the resonance frequency, the damping factor and the static gain of such an oscillator.

Expression of the oscillator's response parameters

Let us consider a {mass M , stiffness K, damping R} mechanical system as our theoretical model: cf Fig2

M du dt + Ru + K u(t)dt = Fext ( 1
)
where u is the velocity oscillation. The expression of the resonance pulsation is:

ω0 = K M (2)
Assuming Fext and u are the respective system input and output, equation [START_REF] Besnainou | Modal stimulation: a sound synthesis new approach[END_REF] gives the transfer function in the Laplace space, plot-

H(s) = U (s) Fext(s) = 1 M s + R + K s ( 3
)
where s is the Laplace variable. The static gain H0 is defined as the maximum value of |H(jω)|

H0 = |H(jω0)| = 1 R (4)
The damping factor Q indicates how selective the frequency response is;

Q = ω 0 ω 2 -ω 1 , where |H(jω1)| dB = |H(jω2)| dB = H0| dB -3dB. Therefore, Q = √ KM R (5)
Figure 3: Visualization of the oscillator's parameters: f0, Q and H0

PID correction on the oscillator

A PID controller provides a force Fcorr, which value is a sum of three terms proportional to its input u, its integral u, and its derivative u:

Fcorr = P u + I u(t)dt + D du dt (6)
Then the relation ( 1) is modified: [START_REF] Cremer | Physik der Geige[END_REF] describes the motion of an other oscillator, whose parameters have been changed:

(M -D) du dt + (R -P )u + (K -I) u(t)dt = Fext (7) (
ω 0 = K -I M -D , Q = (K -I)(M -D) R -P , H 0 = 1 R -P (8)
While the variations of ω0, Q and H0 are defined by: ∆ω0 =

ω 0 ω 0 -1, ∆Q = Q Q -1 and ∆H0 = H 0
H 0 -1, the coefficients P , I and D can be expressed by:

P = ∆H0 H0(1 + ∆H0) (9)
I = Qω0 H0 (1 - (1 + ∆Q)(1 + ∆ω0) 1 + ∆H0 ) (10) D = Q H0ω0 (1 - (1 + ∆Q) (1 + ∆ω0)(1 + ∆H0) (11

PID CORRECTION ON A REAL XYLOPHONE BAR RESONANCE

The purpose is to apply a PID correction to a real xylophone bar in order to modify one of its eigen modes. It would impose independent and quantitative control of its pitch, duration or sound intensity. On a real instrument some eigen modes may be coupled. In this case, modifying the parameters of a mode may imply nonlinear variations on the next ones. However in a xylophone bar we assume that some eigen frequencies are isolated, far from the other. Thus a correction on one of them does not change the bar's frequency response much, in its vicinity. Here we chose to control the first eigen mode of the bar.

System description

It has been proved above that a PID correction is an effective method to control the movement of an oscillator. In the real case the correction force is applied to the xylophone bar thanks to an actuator. As this force should depend on the system's velocity, eq [START_REF] Besnainou | Comment changer la voix des instruments de musique[END_REF], a sensor is also required to catch the signal and to feed it to the controller. Thus the PID controller is integrated in a feedback loop: cf Fig5. As a conclusion both measured system resonances are very similar to second order harmonic oscillator's responses. However the phase difference between estimation and measurement is larger and almost constant on the frequency band [480Hz, 540Hz]. This ought to be due to the fact that the sensor and the actuator are not colocalized.

Results of a PID correction

Three PID corrections are carried out on the first measured bar resonance (493Hz), cf Fig7. The PID controller is the theoretical one, which has been described in part 2. While ∆Q or ∆H0 ∈ [-25%, 150%], these variations remain independent. However as soon as ∆ω0 < -0.5% or > 2% this resonance pulsation modification comes with a bending of the mode's shape, it means an unwanted change of Q and H0.

PID CORRECTION IN REAL TIME

Active control on an instrument becomes relevant as soon as its application is carried out in real time. This way, it enables the musician or the instrument maker to immediately modify the sound.

System description

In real time the feedback correction signal cannot be calculated instantaneously by the controller. That's why a variable delay is added into the loop so that the control signal is exactly 2πdephased from the theoretical one. Moreover the PID controller is coded in C language and integrated into a digital signal processor (ARM microcontroller), in order to limit the introduced delay. Its input AD converter adds a very low offset in the signal that is fed to the PID controller. Then this offset is integrated so that the output DSP's level get saturated. That's why a digital high-pass filter H(z) = G(1+z -1 ) 1+τ z -1 is inserted in the controller just behind the integrator (Fig8). Its cut-off frequency is as low as possible (60Hz here). Thus, since the PID transfer function has been modified, the relationship between P , I, D and ∆ω0, ∆Q, ∆H0 (eq(9),( 10) and (11)) must be adapted too.

How to correct quantitatively, independently and in real time the parameters of the resonance

Let us assume that a second order harmonic oscillator is a valid model of the measured first mode, whose response is called H. The closed loop transfer function is a rational fraction of the Laplace variable s:

Htot(s) = 1 1 H(s) -C(s) (12)
where C is the response of the PID controller containing the previous described integrator, cf Fig8. The method is intended to set up a one-to-one mapping table between each (P , I, D) and each (ω0c, Qc, H0c), where ω0c, Qc, H0c are the parameters of the target's transfer function. First in the expression of |Htot|, ω is replaced by ω0c + ∆ω. It is then developed in power series around 0 (ω ≈ ω0c):

Htot(s) = k∈N a k ∆ω k , when ∆ω ≈ 0 (13)
The maximum value of |Htot(s)| is intended to reach H0c as soon as ω = ω0c. So, when ω ≈ ω0:

∂|Htot| ∂ω |ω=ω 0c = 0 ⇔ k∈N * ka k ∆ω k-1 = 0 (14) ⇒ a1 = 0 (15)
The (ai)i∈N coefficients depend on the parameters of H (H0,Q and ω0) and of C (P , I, D and G, τ ). Therefore (15) gives an expression of I in function of the other variables. This way, an algorithm tests a large range of P and D values, deduces I, and then compares the obtained response with the target. The best couple (P , D), which fits to the minimum relative quadratic error, is held in the table. The table is then completed by choosing another target mode.

Results

The described method is applied to the measured frequency response of the real xylophone bar. Thus the method enables to have H0 and Q under quantitative control without modifying ω0.

CONCLUSION

Acting on an acoustic instrument with an active control method is a way of diverting this technique from the traditional noise cancellation. Indeed by enhancing or damping an eigen mode, or by modifying its shape, we exploit larger possibilities than those usually used to soften vibrations. This new application of active control, in the music composition field, comes up to its expectation of new sounds, and reaches more and more fusion between the musician and its instrument [START_REF] Besnainou | Comment changer la voix des instruments de musique[END_REF].

The study described here is included in a larger work intended to build a generic method to control various sets of acoustic instruments by an active control technique. Finding a robust physical model of its resonances and an adapted corrector is a required stage for having them under digital and real time control. The very simple model described in part 2 is quite close to a real xylophone bar's mode. It shows that a PID correction is a possible theoretical way to modify an isolated xylophone resonance (part 3). By adapting it to real time requirements, some resonance modifications should be achieved as described in part 4. Some sounds come with these results. Now we still have the resonance frequency variations to bring under control by using the same digital corrector.

Once this method will be completed, it will be needed to considerate the effect of the coupling between modes in order to control the whole frequency response. Other works are in progress to apply the same technique to the violin bridge, by using another simple model described by Reinicke and Cremer [START_REF] Cremer | Physik der Geige[END_REF], and by checking the results published by J. Woodhouse [START_REF] Woodhouse | On the bridge hill[END_REF].

Figure 2 :

 2 Figure 2: System Mass, Stiffness, Damping: second order harmonic oscillator

Figure 4 :

 4 Figure 4: (a)-PID correction of f0: -2% ≤ ∆ω0 ≤ 2% (b)-PID correction of Q:-25% ≤ ∆Q ≤ 150% (c)-PID correction of H0-25% ≤ ∆H0 ≤ 150%

Figure 5 :Figure 6 :

 56 Figure 5: Description of the looped system {oscillator + con-troller}

Figure 7 :

 7 Figure 7: PID correction, on the real transfer function, of: (a)-the of the resonance frequency: -2% ≤ ∆ω0 ≤ 2% (b)-the damping factor: -25% ≤ ∆Q ≤ 150% (c)-the static gain: -25% ≤ ∆H0 ≤ 150%

Figure 8 :

 8 Figure 8: Description of the real looped system {oscillator + real time controller}

  Various corrections are simulated by attempting to keep ω0 constant (ie f0c = 493Hz). The resulting responses plotted in Fig 9 are obtained by choosing random (P ,D) values. Each of them have a different static gain (cf Fig9a) and a different damping factor (cf Fig9b), but the same resonance frequency.

Figure 9 :

 9 Figure 9: (a)-Various corrections of Q and H0 with ω0 constant (b)-Same corrections after normalization (H0 = 1)