J Hillairet
email: julien.hillairets@cea.fr

P Mollard

L Colas

J.-M Bernard

J.-M Delaplanche

F Durand

N Faure

P Garibaldi

G Lombard

C Bourdelle

C Desgranges

E Delmas

R Dumont

A Ekedahl

F Ferlay

M Goniche

C Guillemaut

G T Hoang

P Maget

R Volpe

W Helou

G Urbanczyk

Y Song

Q Yang

Z Chen

Y Wang

H Xu

S Yuan

Y Zhao

F Durodie

E Lerche

R Ragona

N Bertelli

M Ono

S Shiraiwa

V Bobkov

C Klepper

C Lau

E Martin

B Lu

R Maggiora

D Milanesio

K Vulliez

G Wallace

WEST Actively Cooled Load Resilient Ion Cyclotron Resonance Heating System Results

Keywords: ICRH, ICRF, WEST

Three identical new WEST Ion Cyclotron Resonance Heating (ICRH) antennas have been designed, assembled then commissioned on plasma from 2013 to 2019. The WEST ICRH system is both load-resilient and compatible with long-pulse operations. The three antennas have been successfully operated together on plasma in 2019 and 2020. The load resilience capability has been demonstrated and the antenna feedback controls for phase and matching have been developed. The breakdown detection systems have been validated and successfully protected the antennas. The use of ICRH in combination with Lower Hybrid has triggered the first high confinement mode transitions identified on WEST.

Introduction and antennas description

One of the objectives of the WEST machine being to study high heat loads and particle fluxes in a fully metallic tokamak environment [START_REF] Bucalossi | Operating a Full Tungsten Actively Cooled Tokamak: Overview of WEST First Phase of Operation[END_REF][START_REF] Bourdelle | WEST Physics Basis[END_REF], the WEST external heating systems must be compatible with long-pulse ELMy H-mode and steady-state plasma operation. No Ion Cyclotron Resonance Heating (ICRH) system before ITER has had to deal with these two challenges simultaneously, i.e. ELMs resilience and Continuous Wave (CW) RF operation. To meet these objectives, three new identical ICRH antennas have been designed through a European collaboration and with CAS/ASIPP [START_REF] Helou | Radio-Frequency Electrical Design of the WEST Long Pulse and Load-Resilient ICRH Launchers[END_REF][START_REF] Hillairet | Ion Cyclotron Resonance Heating Systems Upgrade toward High Power and CW Operations in WEST[END_REF][START_REF] Chen | Design and Optimization of the WEST ICRH Antenna Front Face Components Based on Thermal and Hydraulic Analysis[END_REF][START_REF] Vulliez | The Mechanical Structure of the WEST Ion Cyclotron Resonant Heating Launchers[END_REF], within the framework of the Associated Laboratory IRFM-ASIPP [START_REF] Yang | Manufacturing and Process Research of the West ICRH Antenna[END_REF]. The antennas have been manufactured at the Keye Company in Hefei under the supervision of ASIPP.

The three WEST ICRH antennas have been designed to satisfy the ICRH power requirements for WEST plasma scenarios, i.e. a total ICRH power up to 9 MW (high power scenarios) or 3 MW (high fluence scenarios) [START_REF] Bourdelle | WEST Physics Basis[END_REF]. RF frequency can be varied from 46 to 65 MHz to fit the main WEST ICRH scheme, which consists of hydrogenminority in deuterium plasmas.

Each antenna is made of four short straps (2 toroidal × 2 poloidal) located in separated boxes behind a Faraday Screen (figure 1). The tilt angle of the Faraday Screen bars is close to the direction of the total magnetic field lines facing the antennas (7°). Each toroidal side of an antenna is fed by a separate high power source. The phase between the left and right sides of each antenna is defined by the plasma control system, the default being dipole configuration. An FPGA-based PID controller modulates the right side generator source frequency (the left side being the reference) to maintain the requested phase with a response time of approximately 50 µs. Moreover, the frequencies at which antennas are operated are slightly shifted from the scenario request (around ±0.3 MHz) to reduce cross-talk between antennas. All antennas are movable radially for optimizing the power coupling and thermal loads handling.

To cope with the fast changes of the electron density facing the antenna during ELMs, the antennas rely on internals conjugate-T [START_REF] Bosia | High-Power Density Ion Cyclotron Antennas for Next Step Applications[END_REF]: poloidal pairs of straps are connected to an adjustable series capacitor and fed from a common T-junction. Matching is achieving by tuning these capacitors such that the two parallel branches have complex conjugate admittances. If the impedance seen at the T-junction is in the range of 3 to 6 Ω, this feeding circuit allows maintaining the Standing Wave Ratio (SWR) below reasonable values for the generator (below 2:1 for 3 MW per antenna on WEST) within a large increase in antenna resistive loadings, such as the ones expected during ELMs. This load-tolerant electrical circuit has been successfully tested in few seconds pulses in Tore Supra [START_REF] Vulliez | Validation of the Load-Resilient Ion Cyclotron Resonance Frequency Antenna Concept on Tore Supra Plasmas[END_REF] and JET [START_REF] Durodié | Physics and Engineering Results Obtained with the Ion Cyclotron Range of Frequencies ITER-like Antenna on JET[END_REF]. This choice was also motivated by the fact that no 3 dB Continuous Wave (CW) couplers were available (and still) on the market and required important R&D. The low impedance at the conjugate-T junction is then transformed to the feeding 30 Ω transmission lines by a fixed two-stage quarterwave transformer.

Thanks to the collaboration between LPP-RMA and CEA, the experience gained with the JET ILA [START_REF] Durodié | Physics and Engineering Results Obtained with the Ion Cyclotron Range of Frequencies ITER-like Antenna on JET[END_REF] has been used during the WEST antenna design. In particular, the reliability of the operation of the capacitors has been improved with reduced mechanical stresses and the possibility to perform remote-diagnostic of the capacitor internal vacuum. Lessons were also learned from Tore Supra prototypes, to improve vacuum tightness and to bring water in all the parts of the antenna [START_REF] Vulliez | Validation of the Load-Resilient Ion Cyclotron Resonance Frequency Antenna Concept on Tore Supra Plasmas[END_REF][START_REF] Vulliez | The Mechanical Structure of the WEST Ion Cyclotron Resonant Heating Launchers[END_REF]. The vacuum pumping of the impedance transformer sections has been improved and the pressure is monitored with two pressure gauges. Strap voltages are measured using D-dot electric field probes located between the capacitors and the straps (figure 2). Strap currents are deduced analytically from these voltages, assuming the strap impedance is mostly reactive. Forward and reflected powers are measured by bi-directional couplers located at the rear of the antenna and at the generator plant. The forward RF power is feedback-controlled to keep voltages and currents below their maximum safety thresholds, set to 28 kV and 915 A respectively, with a fast response time of the order of 10 µs to avoid possible voltage or current overshoots during plasma disruptions.

A unique feature of these antennas is to satisfy the long pulse requirement of WEST. Antennas are fully water-cooled using two distinct water circuits: i) a high temperature/high pressure (up to 200 °C during baking operation and 70 °C/30 bar bar during plasma operation) water circuit and ii) a low temperature/low pressure (25 °C/4 bar) specifically designed to cool the antenna matching capacitors [START_REF] Chen | Design and Optimization of the WEST ICRH Antenna Front Face Components Based on Thermal and Hydraulic Analysis[END_REF][START_REF] Vulliez | The Mechanical Structure of the WEST Ion Cyclotron Resonant Heating Launchers[END_REF]. However, some limitations remain in the thermal evolution of non-actively watercooled components that could not be replaced or actively cooled, such as in the generator plant, transmission lines or vacuum feed-throughs. This implies a tuning of the total power as a function of the pulse duration, i.e. to 9 MW /30 s, 6 MW/60 s and 3 MW/1000 s [START_REF] Hillairet | Ion Cyclotron Resonance Heating Systems Upgrade toward High Power and CW Operations in WEST[END_REF].

All antenna RF surfaces, such as the Faraday Screen, antenna box, straps, internal and external conductors, are silver-coated (10 µm Ni and around 50 µm Ag, i.e. more than five times the skin depth). Lateral limiters are located on both sides of each antenna and protrude by 2 cm from the edges of the Faraday Screen. These limiters are the ones used on Tore Supra and are made of CFC tiles coated with a double coating made of a first 80 µm molybdenum layer followed by an 80 µm tungsten (W) coating [START_REF] Firdaouss | Tungsten Coating by ATC Plasma Spraying on CFC for WEST Tokamak[END_REF][START_REF] Firdaouss | Overview of the Different Processes of Tungsten Coating Implemented into WEST Tokamak[END_REF]. While having high Z materials close to an ICRH antenna was known to increase the tungsten sputtering attributed to the acceleration of light impurity ions in the rectified RF sheaths [START_REF] Vl | Assessment of Compatibility of ICRF Antenna Operation with Full W Wall in ASDEX Upgrade[END_REF], this choice was motivated by the desire to keep a full W, reactor relevant machine [START_REF] Bucalossi | The WEST Project: Testing ITER Divertor High Heat Flux Component Technology in a Steady State Tokamak Environment[END_REF].

An infrared thermography system monitors the antenna front-faces with a spatial resolution better than 5 mm/pixel [START_REF] Courtois | Full Coverage Infrared Thermography Diagnostic for WEST Machine Protection[END_REF]. Antenna limiters are protected with a real-time reduction of the RF power if the luminance temperature (i.e. with unity emissivity) exceeds a threshold initially determined either from material test bed or experience. The real-time protection of the ICRH antenna front-face is more challenging in WEST than in Tore-Supra [START_REF] Colas | Ion Cyclotron Resonant Heating in Tore Supra[END_REF], in particular due to the low emissivity of their silver coating which reduces the surface temperature estimation accuracy [START_REF] Aumeunier | Infrared Thermography in Metallic Environments of WEST and ASDEX Upgrade[END_REF]. So far, the front-faces are monitored but without real-time feedback on RF power during plasma operations.

The WEST ICRH antenna design phase has been conducted from 2013 to 2014, followed by manufacturing in ASIPP from 2015 to 2017. Antenna elements were shipped to CEA, then silver coated and the antennas assembled in series from 2016 to 2019. The antennas have been commissioned on TITAN [START_REF] Litaudon | Physics and Technology in the Ion-Cyclotron Range of Frequency on Tore Supra and TITAN Test Facility: Implication for ITER[END_REF], then installed on WEST and operated progressively since 2018 [START_REF] Bernard | Design and Construction of the First ELM Resilient Long Pulse ICRH Antenna for WEST[END_REF][START_REF] Bernard | Commissioning of the First WEST Load-Resilient Long Pulse ICRF Launcher in the TITAN Testbed and on WEST Plasmas[END_REF][START_REF] Helou | Characterizations and First Plasma Operation of the WEST Load-Resilient Actively Cooled ICRF Launchers[END_REF]. In 2019, all three antennas were operated on WEST (figure 4). This paper reports the commissioning and the key results obtained on plasma with the full WEST ICRH system.

RF Modelling

The WEST ICRH RF modelling, described in more details in [START_REF] Helou | SIDON: A Simulator of Radio-Frequency Networks. Application to WEST ICRF Launchers[END_REF][START_REF] Hillairet | RF Network Analysis of the WEST ICRH Antenna with the Open-Source Python Scikit-RF Package[END_REF], is summarized in this section. The antenna RF model uses a combination of full-wave models of the antenna's elements and equivalent lumped elements circuit of the tuning capacitors deduced from full-wave simulations (Figure 2). The evaluation of the scattering parameters of the 4-ports front-end facing the plasma computed using bespoke codes such as TOPICA [START_REF] Milanesio | A Multi-Cavity Approach for Enhanced Efficiency in TOPICA RF Antenna Code[END_REF] or Petra-M [START_REF] Bertelli | 3D Full Wave Fast Wave Modeling with Realistic Antenna Geometry and SOL Plasma[END_REF][START_REF] Syun | Towards Integrated RF Actuator Modeling: Whole Device Scale RF Fullwave Simulation Including Hot Core and 3D SOL/Antenna Regions[END_REF], or using commercial FEM software such as COMSOL in RAPLICASOL [START_REF] Tierens | Recent Improvements to the ICRF Antenna Coupling Code "RAPLICASOL[END_REF]. The complete electrical circuit a WEST ICRH antenna is assembled using the open-source Python package scikit-rf ‡ and can be found ‡ http://scikit-rf.org and used online §. The overall model allows comparing voltages, currents, powers and phases to measurements as illustrated in Section 4 and can be used in the control room to assist ICRH operators in antenna tuning. Future work will focus on making "digital twins" of each antenna, enabling operators to interpret the antenna measured responses (post-pulse) and to guide operators in optimizing the antennas for the next plasma pulse.

Antennas Commissioning

Low power RF tests

During the antenna assembly, calibrations have been performed of components not accessible in later stages, such as voltage probes and vacuum RF cables. Once done, cold RF tests were performed to check the antennas RF responses [START_REF] Bernard | Commissioning of the First WEST Load-Resilient Long Pulse ICRF Launcher in the TITAN Testbed and on WEST Plasmas[END_REF][START_REF] Helou | Characterizations and First Plasma Operation of the WEST Load-Resilient Actively Cooled ICRF Launchers[END_REF]. A dedicated tank filled with salty water used as a dummy load has been placed in front of the antenna to increase the antenna loading. Even though the maximum coupling resistance offered by the water load is no more than 0.5 Ω, the first validations of the load resilience of the antennas were already demonstrated [START_REF] Helou | Characterizations and First Plasma Operation of the WEST Load-Resilient Actively Cooled ICRF Launchers[END_REF].

Leak tests and First High RF Power Tests

Before being installed in WEST, each of the three ICRH antennas has been commissioned following the same procedure described in [START_REF] Bernard | Commissioning of the First WEST Load-Resilient Long Pulse ICRF Launcher in the TITAN Testbed and on WEST Plasmas[END_REF] and recalled here. Each antenna has been installed in the TITAN vacuum chamber to realize vacuum and water leak tests (figure 5). Once the tank is vacuum pumped, antenna water circuits are then sequentially filled with Helium for leak detection. Leak-tests are repeated at room and hot temperature (< 200 °C) during two consecutive cycles of temperature increase/plateau/decrease. An example of temperature and pressure traces during standard cycles is illustrated in figure 6 for the third antenna (Q4).

High voltage RF tests are made in TITAN alternatively on both sides of an antenna to check the voltage stand-off of the antenna up to 27 kV/3 s. The antenna energized side is matched while the other side is detuned to reduce cross-coupling and voltage breakdowns in the non-energized side. Once both sides are validated, antennas are ready to be mounted on WEST and the measurement chain calibrated.

Fault Detection

The WEST ICRH system uses a combination of several interlocks to detect faults and RF breakdowns throughout the antennas [START_REF] Hillairet | Ion Cyclotron Resonance Heating Systems Upgrade toward High Power and CW Operations in WEST[END_REF][START_REF] Helou | Characterizations and First Plasma Operation of the WEST Load-Resilient Actively Cooled ICRF Launchers[END_REF]. In case of fault detection, the two generators feeding an antenna are cut-off within 10 to 50 µs then re-applied after 30 to 50 ms (depending on the interlock, which in turns can give an idea on the interlock that has triggered). In addition to the standard reflected power (P ref /P fwd ) and Sub-Harmonic Arc Detection (SHAD) systems [START_REF] Berger-By | Arc Security System Based on Harmonics Detection for the Tore Supra ICRH Transmitter[END_REF], a new optical arc detection system monitoring the low impedance regions around the T-junctions come in reinforcement. This system relies on 8 optical fibres per antenna associated with fast photodetectors (< 10 µs) looking at various regions of the low impedance T-junctions. Abnormal events can also be detected from the antenna voltage measurements. Hence, RF power is stopped within 10 µs (then reapplied after 30 ms) in case of i) voltages lower § https://github.com/jhillairet/WEST_IC_antenna than 10 kV when the forward power larger than 100 kW, meaning RF power does not reach properly the antenna front-face or ii) if a relative difference in voltages between straps is higher than 50% (unbalance). A turbomolecular pump connected to at the back of each antenna helps to improve the vacuum level inside both impedance transformers. To prevent sustaining undetected RF plasma discharges which may arise inside these antenna vacuum sections, the RF power is stopped if the pressure measured by vacuum gauges close to the pump is higher than 4.5 × 10 -3 Pa, then reapplied when the pressure is below 2.2 × 10 -4 Pa.

The RF conditioning after an air venting of the WEST ICRH antennas is performed as follows. Each antenna side is separately energized for 20 ms pulses until to reach a stable maximum voltage of 20 kV. Then, both sides are combined in dipole phasing and forward power then duration are progressively increased. Interestingly, the kind of interlock which triggers, indicated by the marker type and colour scale in figure 7, vary throughout the conditioning process, i.e. the progression of the RF conditioning from low voltages/durations (5 kV /20 ms) up to high voltages/durations (27 kV/5 s). While both reflected power and SHAD detectors trigger mostly for high voltage breakdowns (simultaneously or not) [START_REF] Inca | Progress in the Design of a Sub-Harmonic Arc Detector for ITER[END_REF], other kinds of breakdowns often undetected with the two later systems were detected with the new optical arc detectors. These breakdowns generally occur during the beginning of the conditioning phases, but also during plasma operation. Max pressure interlocks trigger in case of severe outgassing, which occurs naturally as the RF energy increases and as long as the antenna RF surfaces are not yet sufficiently conditioned. During plasma operation, all of these interlocks can be triggered, in combination or not. These preliminary results confirm the importance of combining operational safety systems as the location and the breakdown type vary depending on the RF circuit, voltages and material surface properties' evolution [START_REF] Inca | Arc Detection for the ICRF System on ITER[END_REF]. No significant damage has been observed on the antenna front-faces and internal elements since their installation.

Plasma Experiments

Key Achievements

The commissioning phase on WEST plasmas was relatively short, only two days for each antenna were required to reach 1 to 1.5 MW. After this commissioning phase, all three antennas have been operated simultaneously on WEST plasmas. The antennas are operated at a slightly different frequencies (around 150 kHz shift between antennas) to reduce cross-talks between antennas. A maximum coupled power of 5.8 MW with all three antennas combined has been obtained after around 50 plasma discharges (close to 2 MW per antenna). In combined operation with the Lower Hybrid Current Drive (LHCD) system, a total injected RF power of up to 8 MW has been achieved during few seconds [START_REF] Bucalossi | Operating a Full Tungsten Actively Cooled Tokamak: Overview of WEST First Phase of Operation[END_REF].

Antenna Load Resilience and Operation

The intrinsic load tolerance property of the WEST ICRH antennas, already demonstrated during commissioning and first plasma experiments [START_REF] Helou | Characterizations and First Plasma Operation of the WEST Load-Resilient Actively Cooled ICRF Launchers[END_REF], has been confirmed in the last experimental campaigns. An example of the load resilience of the WEST antennas is illustrated in figure 8 with WEST pulse # 55589. During this pulse, 5 deuterium pellets have been injected from the high field, leading to fast density increases, correlated with fast changes in the antenna coupling resistance (from 0.5 to 2 Ω). Note that the brief SWR increases at 4.6 s and 7.6 s are the consequence of an inward radial shift (around 2 cm) of the perturbed plasma. Figure 9 shows the SWR evolution with coupling resistance during the third pellet injection and is compared to the antenna RF model simulation. The red dashed line illustrates the generator SWR limit to couple 3 MW per antenna (1.5 MW per generator). The gray dashed line represents for reference the SWR that would have been reached if the antenna was not load-resilient (cf. [START_REF] Helou | Characterizations and First Plasma Operation of the WEST Load-Resilient Actively Cooled ICRF Launchers[END_REF] for additional details).

The operation of the antennas is relatively straightforward, as once the antenna tuning capacitors are properly set up for a given frequency, the antenna load tolerance accepts small capacitance deviations from the ideal match point. Excluding effects on plasma heating efficiency and impurities production, changes in plasma parameters such as minority concentration, fluctuations of the fast wave cut-off layer or minor changes in resonance layer location do therefore not affect the antenna operation significantly. This feature was particularly appreciated during the C4 experimental campaign, as it was still possible to couple RF power to various plasma scenarios despite that few obsolete motor controllers had broken down, prohibiting the movement of some capacitors on two distinct antennas. The motorization system has been fully refurbished for the C5 campaign.

All antennas are equipped with dedicated gas puffing valves, poloidally distributed 15 to 20 cm behind the antenna lateral limiters, to improve the coupling conditions. However, while the antenna coupling slightly increased at the modest levels of gas flux injected by these valves (1 × 10 21 el/s), the core plasma density was no more controlled due to the limited pumping capabilities of WEST at that time. Improvement of the WEST pumping capabilities for future experimental campaigns gives hope to use local gas puffing as an effective operational tool to improve the ICRH performance, like adopted in other devices such as JET and AUG [START_REF] Jacquet | Maximization of ICRF Power by SOL Density Tailoring with Local Gas Injection[END_REF][START_REF] Lerche | Optimization of ICRH for Core Impurity Control in JET-ILW[END_REF].

Automatic Matching

The use of real-time matching feedback control to optimize the WEST ICRH antenna response during plasma has been successfully demonstrated [START_REF] Helou | Characterizations and First Plasma Operation of the WEST Load-Resilient Actively Cooled ICRF Launchers[END_REF]. The algorithm follows the same principle as the JET ILA [START_REF] Durodié | Circuit Model of the ITER-like Antenna for JET and Simulation of Its Control Algorithms[END_REF] and is described below for convenience. The capacitors are actuated using a negative feedback loop (figure 10) which aims to minimize the difference Z T,SP -Z T , where Z T is the input impedance at the T-junction (figure 2) and Z T,SP the impedance target (SetPoint), which eventually complexvalued. The impedance Z T depends on both capacitors, ie. Z T = Z T (C top , C bot ). It is numerically deduced from power and phase measurements made at the bidirectional couplers located at the rear of the antenna, using a combination of RF measurements and full-wave simulations [START_REF] Helou | Characterizations and First Plasma Operation of the WEST Load-Resilient Actively Cooled ICRF Launchers[END_REF]. Close to the setpoint Z T,SP = Z T (C SP ) = Z T (C top,SP , C bot,SP ), Z T can be approximated to:

Z T (C) = Z T (C SP + δC) ≈ Z T,SP + g T δC ( 1
)
where g is the gradient vector of Z T at the setpoint. Separating real and imaginary parts and reordering leads to the desired incremental capacitance changes:

δC ≈ D -1 (Z T -Z T,SP ) (2)
Where the D is the matrix of partial derivatives around the setpoint and Z T = ( (Z T ), (Z T )). However, since this matrix cannot be evaluated in real-time, in practice it is approximated with the sign of its elements, evaluated numerically using the antenna RF model, to give the proper descent direction. Depending on the choice of these signs, the operator can choose one of the two matching solutions. The velocities V at which the capacitors are moved are proportional to the incremental change request δC. Figure 11 illustrates an example of the usage of this method on plasma. As the Radial Outer Gap (ROG) of the plasma decreases, the coupling resistance (R c ) increases and the capacitors move apart to improve the antenna matching (SWR decreases). Because it is based on a RF model linearized close to the match point, the algorithm converges only if the initial guess point is properly chosen, i.e. as long as the gradient estimate corresponds to a negative descent direction. Future work could investigate further improvements such as non-linear converging (faster velocities when far from the set point, slower when closer) [START_REF] Durodié | ITER-like Antenna for JET First Results of the Advanced Matching Control Algorithms[END_REF] or control methods allowing to increase the range of the starting point. As the capacitors of each antenna side are currently controlled in parallel, future work will consist of taking into account crosscoupling between left and right sides in a self-consistent way and/or minimize voltage unbalance rather than SWR. The possibility to automatically deactivate the real-time control during fast changes of the plasma is envisaged, if it becomes necessary, for example during ELMy phases, using signals from the WEST interferometry system.

Phase Control

On WEST ICRH antennas, the forward power is generated by two tetrodes [START_REF] Kuus | The ICRH System for TORE SUPRA[END_REF], one for each side of the antenna. The toroidal phase between the left and right sides is achieved by frequency-modulating the right side, the left side being the reference.

Without phase control, the toroidal phase can change in an uncontrolled manner due to any plasma variation (e.g. coupling resistance) or interactions with other IC (or LH) antennas. Figure 12 illustrates a (typical) case, in which despite a relatively constant ROG (±3 mm), the toroidal phase of an antenna (here Q2, but the behaviour is the same with other antennas) vary by ± 15°or more during the plasma discharge.

Initially, the input signal was taken from the toroidal phase between forward powers and the toroidal phase between strap voltages was derived from a linear combination of additional phase measurements. However, this method gave unstable results due to antenna-dependent phase roll-overs occurring in electronic cards. The system has been further improved in the C5 experimental campaign, to measure directly the toroidal phase between (top) strap voltages as the input signal to control. The phase is regulated using an FPGA-based PID whose typical response time to change request is approximatively 50 µs and has been found very robust on each antenna to plasma changes. As an example, figure 13 illustrates the performances of the toroidal phase control for the Q2 antenna on plasma in which the Radial Outer Gap (ROG) is oscillating by ±5 mm. The phase feedback control has been found very robust to plasma changes. Despite the changes in the coupling resistance, the accuracy of the toroidal phase is in the order of ±2°. The phase error can be increased to ±5°w hen multiple antennas are used at the same time or in case of bad absorption, as the phase measurements do not discriminate with frequency and are thus affected by antennas cross-talk and/or multiple passes in the plasma. The role of the antenna toroidal phase in impurity production is discussed in the next section.

Radiated Power and Impurity Production

When the isotopic ratio n H /(n H +n D ) is in the expected range for H minority heating scenarios (i.e. from 5 to 10%) and the toroidal phase properly tuned to dipole, the radiated power measured in 2019 and 2020 scales with the total power (Ohmic + total RF coupled power) and weak specific effect of LH or ICRH RF power has been found. This is illustrated in figure 14 for a large set of plasma scenarios, where each point represents a stable 200 ms time window during which the considered quantity has a relative change below 10%. High-performance plasma experiments conducted on WEST in 2019 have proven that the ICRH system is a key player in achieving edge density pedestal and H-mode transitions [START_REF] Goniche | Developing High Performance RF Heating Scenarios on the WEST Tokamak[END_REF][START_REF] Vermare | Formation of the Radial Electric Field Profile in WEST Tokamak[END_REF]. Several multiple transitions from L to H modes have been obtained and an H-mode phase of up to 4 s has been sustained with the combination of ICRH and LHCD [START_REF] Bucalossi | Operating a Full Tungsten Actively Cooled Tokamak: Overview of WEST First Phase of Operation[END_REF]. However, the sustainment of stable H-modes in WEST is presently impeded by the concomitant increase of core radiation level after a transition. It results in oscillatory edge dynamics also impacting ICRH antennas, decreasing the coupling and eventually the coupled power. Additionally, WEST scenarios are also currently hampered by the issue of burning through the tungsten radiation peak at 1.5-3 keV, due to a limited level of central electron heating including when applying ICRH [START_REF] Goniche | Developing High Performance RF Heating Scenarios on the WEST Tokamak[END_REF]. Future work will concentrate on optimizing the RF heating scheme to ICRH+LHCD plasma scenarios, especially to increase the central temperature as soon as possible in the discharge. A 3MW/CW ECRH system is also under preparation for WEST [START_REF] Bucalossi | Operating a Full Tungsten Actively Cooled Tokamak: Overview of WEST First Phase of Operation[END_REF]. This system will be used in particular for the central heating of the plasma during the very early phase of current ramp up.

The specific role of the ICRF waves in modifying the Scrape-Off has been evidenced using visible spectroscopy and infrared measurements [START_REF] Colas | First Application of Ion Cyclotron Resonant Frequency Waves on WEST Plasma Scenarios[END_REF][START_REF] Urbanczyk | Interaction of High-Power Waves with the Plasma Periphery of WEST & EAST Tokamaks[END_REF][START_REF] Urbanczyk | Perspective of Analogy between Heat Loads and Impurity Production in L-Mode Discharges with ICRH in WEST[END_REF]. 17 lines of sight are looking at both lateral limiters of the Q2 antenna (8/9 lines on the left/right limiters respectively). The W-I line (400.8 nm ) brightness is used as a proxy of the W gross erosion. When an antenna is powered, the brightness on its limiters is larger (for a constant total power) than when the antenna is not energized and the amplification factor depends on the antenna voltages. Figure 15 illustrates that a toroidal phase departing from the standard dipole phasing (180°) also led to larger brightness on the antenna side limiters and on the magnetically connected Antenna Protection Limiter (APL) but not elsewhere [START_REF] Urbanczyk | Interaction of High-Power Waves with the Plasma Periphery of WEST & EAST Tokamaks[END_REF]. Although dipole phasing is a local minimum, the W-I line brightness is larger than in ohmic regime (dashed lines). As the toroidal phase decreases from dipole (∆φ = 180°) to 80°, the brightness on the right limiter increased by a factor 1.5. Although the coupled power decreases by a factor of 2 as the antenna departs from its matching point, applying a factor √ P ICRF (homogeneous to the antenna electric fields which are a proxy to RF sheath amplitudes [START_REF] Colas | The Geometry of ICRF-Induced Wave-SOL Interaction. A Multi-Machine Experimental Review in View of ITER Operation[END_REF]) to compensate for lower RF power leads to a normalized brightness increases by around 1.8.

However, W-I line brightness is not related in a simple way to the W core contamination because of the large prompt redeposition of W and transport in the divertor and SOL. Future work will consist of a deeper characterization of the impurity sources (W, Cu or Ag) and the effect on plasma core using a combination of experiments measuring low ionisation stages (W-II to VIII) and edge transport simulations. At this stage, the operational domain explored so far does not allow us to differentiate the impact of the heating method on the core radiation level. New low-Z materials antenna lateral limiters are under design for both LHCD and ICRH antennas. It is expected to install them progressively, allowing direct comparisons between antennas equipped with W or low-Z antenna limiters on WEST plasma.

Conclusion and perspectives

Three identical new WEST ICRH antennas have been designed, assembled then commissioned on plasma from 2013 to 2019, in collaboration with European laboratories and ASIPP within the framework of the Associated Laboratory IRFM-ASIPP. These antennas are load-resilient and compatible with long-pulse operations, a unique combination that no ICRH system before ITER has had to deal with simultaneously. The three antennas have been successfully operated together on plasma in 2019 and 2020. The load resilience capability has been demonstrated and the antenna feedback controls (phase, matching) successfully used. All high confinement mode transitions identified on WEST have been obtained with the combination of LHCD and ICRH systems. Future work will consist of optimizing the combined RF heating scenarios in order to maximize the central temperature and minimize the radiated power.

Figure 1 .

 1 Figure 1. Picture of one WEST ICRH Antenna during its assembly and main dimensions.

Figure 2 .

 2 Figure 2. Block diagram of a WEST ICRH antenna.

Figure 3 .

 3 Figure 3. WEST Top and 3D cut view. ICRH antennas are named after the equatorial port (Q) number they are inserted in: Q1, Q2 and Q4.

Figure 4 .

 4 Figure 4. Picture of a WEST ICRH antenna inside the WEST machine, surrounded by its lateral W-coated limiters and steady-state heat flux shields @C.Roux

Figure 5 .

 5 Figure 5. WEST ICRH Antenna Q4 installed in the TITAN vacuum chamber. The high power RF input (visible at the right side in yellow) is not yet connected to one antenna side.

Figure 6 .

 6 Figure 6. WEST ICRH Antenna IC Q4 baking and leak tests. Top: chamber and antenna temperatures. Bottom: chamber and impedance transformer pressures.

Figure 7 .

 7 Figure 7. Example of vacuum conditioning process (from Q1 antenna in 2020), with mean strap voltage as a function of the RF pulse duration. RF conditioning starts at low forward power and duration (20 ms) and the power then the duration are increased until achieving stable pulses (no interruption).

Figure 8 .

 8 Figure 8. Antenna Load Resilience Demonstration using pellet injections.

Figure 9 .

 9 Figure 9. Standing Wave Ratio (SWR) versus coupling resistance (Rc) for the WEST ICRH antenna Q4 (WEST pulse #55589) and RF modelling (plain line).The behaviour of a non-resilient antenna is indicated for reference (dashed line).

Figure 10 .

 10 Figure 10. Real-Time Matching Algorithm Block Diagram.

Figure 11 .

 11 Figure 11. Example of real-time matching: As the Radial Outer Gap (ROG) of the plasma decreases, the coupling resistance (Rc) increases and the capacitors move to improve the antenna matching (SWR decreases toward 1).

Figure 12 .

 12 Figure 12. Example of phase evolution of an antenna during a plasma discharge without phase control activated.

Figure 13 .

 13 Figure 13. Example of toroidal phase control on an oscillating radial outer gap plasma.

Figure 14 .

 14 Figure 14. Radiated Power versus Total Power during the 2019 and 2020 WEST experimental campaigns. Each point represents a stable 200 ms plateau, independently of the plasma scenario. Plotted data have been filtered for plasma current Ip in [480, 520] kA and proper conditions of ratio and antenna toroidal phase. The colour indicates the relative fraction of the ICRH Power in the mix. The dashed line is the best fit linear regression giving P rad = 0.47Ptot + 0.23 with powers in MW.

Figure 15 .

 15 Figure 15. Averaged W-I brightness on the Q2 lateral limiters and the Antenna Protection Limiter (APL) during a toroidal phase scan of the Q2 antenna. Dashed lines correspond to the brightness level during the ohmic phase. The evolution of the average coupled power is also illustrated.

Acknowledgements

This work was carried out under the framework of the Sino-French Fusion Energy centeR (SIFFER, http://www.siffer.science). This research made use of scikit-


G Nakamura

S Plaszczynski

B Grammaticos

M Badoual

Modelling the effect of virulent variants with SIR

Keywords: 34A34, 37M05, 37N25, 39A30, 92D30 epidemic, vaccination, seasonality, recruitment, SIR model

We study the effect of an emerging virus mutation on the evolution of an epidemic, inspired by the appearance of the delta variant of SARS-CoV-2. We show that if the new variant is markedly more infective than the existing ones the epidemic can resurge immediately. The vaccination of the population plays a crucial role in the evolution of the epidemic. When the older (and more vulnerable) layers of the population are protected, the new infections concern mainly younger people, resulting into fewer hospitalisations and a reduced stress on the health system. We study also the effects of vacations, partially effective vaccines and vaccination strategies based on epidemic-awareness. An important finding concerns vaccination deniers: their atttude may lead to a prolonged wave of epidemic and an increased number of hospital admissions.

Introduction

Mathematical models can be particularly useful when applied to the study of epidemics. The domain is such that any experimentation or result replication attempt is out of question when the epidemic concerns the human population. A well conceived model can, through robust conclusions, guide policies aiming at the mitigation and/or confinement of the epidemic. Building an efficient model is always a tall order [START_REF] Landau | The Hard Lessons of Modeling the Coronavirus Pandemic[END_REF]. The first difficulty in the case of epidemic modelling is associated with long-term predictions. Epidemics having a major societal impact, make it necessary for the governments to introduce measures which have as a consequence the modification of the model parameters making their predictions, at times, questionable. The second difficulty stems from the fact that the said parameters are not always accurately known. Thus, when one decides to enrich a model by adding more effects this usually introduces more uncertainties which may influence in a non-insignificant way the results. Faced with this difficulty, the present authors' choice is to keep a model as simple as possible, albeit without sacrificing realism, and thus use it to infer only the most salient conclusions.

Our work is motivated by the ongoing Covid-19 pandemic. Given the global disruption caused by the emergence of the SARS-CoV-2 virus it is natural that major efforts would aim at the mitigation of the epidemic. The approach is two-prong: treating infected individuals and preventing the spread of the epidemic. The first lies in the domain of medicine and pharmacology and thus somehow less amenable to mathematical modelling. Prevention on the other hand is where modelling can be most useful. Since the onset of the epidemic the main strategy of the governments, and to tell the truth the only one initially available, was that of non-pharmaceutical interventions (NPI), consisting in a series of restrictions and recommendations. The various NPI strategies have been the object of a substantial literature. N. Perra [START_REF] Perra | Non-pharmaceutical interventions during the COVID-19 pandemic: A review[END_REF] published recently a review article containing a bibliography of close to 40 articles dealing with the matter. Several governments have set up panels monitoring the effect of NPI but the general conclusion is that those interventions are not always effective [START_REF] Brauner | Inferring the effectiveness of government interventions against COVID-19[END_REF]. This is due to the fact that epidemic-inducing diseases and human behaviour are intimately related and, in fact, in ways not clearly understood. The European Centre for Disease Prevention and Control, in their report [START_REF]and Control Guidelines for non-pharmaceutical interventions to reduce the impact of COVID-19 in the EU/EEA and the UK 24[END_REF] points out that 1 while the NPI did succeed in reducing the impact of COVID-19 in Europe, they also had a negative impact on the general well-being of people, the functioning of society, and the economy in general. Their recommendation is that the NPI be tailored to the local situation, avoiding "one size fits all" type measures. Our work [START_REF] Nakamura | Confinement strategies in a simple SIR model[END_REF] adopted precisely such a point of view. In that work we analysed the first wave of the epidemic in France, which was accompanied by a lockdown mandate, and investigated the possible lockdown-exit strategies. Given that the relaxation of the confinement measures could be accompanied by a new epidemic wave we investigated possible ways to moderate its amplitude. In a similar work Fokas and collaborators [START_REF] Fokas | Easing COVID19 lockdown measures while protecting the older restricts the deaths to the level of the full lockdown[END_REF] studied the affect of lockdown measures in Greece, investigating the possibility of age-specific exit strategies.

While NPI can indeed contain an epidemic or at least limit its spreading, the best prevention strategy is vaccination. And, of course, it is not limited to the Covid-19 epidemic. The Vaccine Impact Modelling Consortium [START_REF] Li | Estimating the health impact of vaccination against ten pathogens in 98 low-income and middleincome countries from 2000 to 2030: a modelling study[END_REF] investigated the impact of vaccination against a collection of pathogens concluding that millions of deaths can be avoided with timely vaccination. In the case of the Covid-19 pandemic the production, in record time, of efficient vaccines, moreover available in large numbers, was a real game-changer. The large-scale vaccination campaigns, first in Israel, then in the US and now in most European countries have established the efficiency of the vaccines. The data from Israel allowed Mahase [START_REF] Mahase | Covid-19: Israel sees new infections plummet following vaccinations[END_REF] to conclude that "Israel sees new infections plummet following vaccination". A report for the US Center for Disease Control [START_REF]COVID-19 Vaccine Breakthrough Infections Reported to CDC United States[END_REF] followed 100 million vaccinated parsons registering around 10 thousand breakthrough infections, confirming the exceptional vaccine efficiency. In [START_REF] Nakamura | Vaccination strategies for a seasonal epidemic: a simple SIR model[END_REF] we studied the effect of vaccinations on the number of hospital admissions, focusing on the possible delays on the launching of the vaccination campaign as well as the consequences of age-prioritisation.

What has motivated the present study is the appearance of a new virus mutation, the variant know today as "delta" [START_REF]SARS-CoV-2 Variant Classifications and Definitions[END_REF]. First detected in India, the variant is characterised by a substantially enhanced infectivity [START_REF] Campbell | Increased transmissibility and global spread of SARS-CoV-2 variants of concern as at June[END_REF],, even when compared to the variant known as "alpha", first detected in the United Kingdom [START_REF] Davies | Estimated transmissibility and impact of SARS-CoV-2 lineage B.1.1.7 in England[END_REF], which was largely responsible for the appearance of a third epidemic wave in a slew of countries. The appearance of the delta variant has raised several questions, among which the prominent one is "will there be a fourth wave", and if yes, "when will it arrive", "will it become the dominant variant" [START_REF] Ito | Predicted dominance of variant Delta of SARS-CoV-2 before Tokyo Olympic Games, Japan[END_REF],"who will it mainly affect" [START_REF] Sheikh | SARS-CoV-2 Delta VOC in Scotland: demographics, risk of hospital admission, and vaccine effectiveness[END_REF], "what will be its impact on the health system" and so on. We intend to address these questions in the framework of a model based on the SIR one used in previous works of ours. Moreover we shall investigate the effect of population circulation and concentration associated to the vacation period. Since the first assessments point towards a slightly diminished efficiency of the existing vaccines in dealing with the delta variant [START_REF] Planas | Reduced sensitivity of SARS-CoV-2 variant Delta to antibody neutralization[END_REF], we will include such an effect in our model. By going back to our two (young-old) population model, introduced in [START_REF] Nakamura | Vaccination strategies for a seasonal epidemic: a simple SIR model[END_REF], we shall try to identify the sub-population mainly concerned by the epidemic resurgence, the impact of the latter on hospitalisations and the possibility to mitigate the consequences by a targeted acceleration of the vaccination campaign. Finally we shall study the (spoiler:dire) consequences of the attitude of vaccination deniers of the evolution of the fourth wave.

The model

The continuous model

The model we shall use in the present study is the venerable SIR model. It was introduced a century ago by Kermack and Mc Kendrick [17] and, in its simple, original form, considers a population spilt into three parts: the 'susceptible' individuals who can catch the infection S, the 'infective' ones who can transmit the disease I, and the 'removed' R who either died from the disease or, having recovered, are immune to it. The SIR model is expressed as a nonlinear differential system

dS dt = -aSI dI dt = aSI -λI (1)
dR dt = λI,
where a is the infection rate and λ the removal rate of the infected individuals. The form of (1) ensures that the total population S + I + R is a conserved quantity. Assuming that S, I and R are fractions of the population, we can normalise this quantity to 1. The ratio a/λ defines what is called the basic reproduction number, usually referred to as R 0 . It corresponds to the number of infections in the susceptible population resulting from a single infection. The parameter λ fixes the time scale. In the case of the COVID-19 epidemic the value of λ -1 is 5 days, as we argued in [START_REF] Nakamura | Confinement strategies in a simple SIR model[END_REF]. Many modelling studies, aimed at the Covid-19 epidemic, favour the use of the SEIR model over that of SIR [START_REF] Li | Projecting the impact of SARS-CoV-2 variants on the COVID-19 epidemic and social restoration in the United States: a mathematical modelling study preprint medRxiv[END_REF]. However, as argued in [START_REF] Nakamura | Recruitment effects on the evolution of epidemics in a simple SIR model[END_REF] we prefer the use of the simpler model. Our main argument, based also on the work of Aleta and Merano [START_REF] Aleta | Evaluation of the potential incidence of COVID-19 and effectiveness of containment measures in Spain: a data-driven approach[END_REF], is that there exists a perfect correlation between the results of the two models. The main effect of the presence of the extra, "exposed", component in SEIR is just a tunable delay in the growth of the epidemic compared to the SIR results.

When a new variant of the virus appears, with characteristics different form the existing one and starts infecting people the equations should be modified accordingly. We introduce a new infective population J and obtain the system

dS dt = -aSI -bSJ dI dt = aSI -λI (2)
dJ dt = bSJ -λJ dR dt = λ(I + J),
where b is the infection rate of the new variant. Since we assume that the new variant appears only at some point during the evolution of the epidemic, the quantity J is initially zero and starts growing when a small, seed, quantity is introduced at some time step. For simplicity we assume the same value of λ for the two variants, accounting for any possible difference through the value of the infection rate.

After a year and a half into the epidemic a feature is emerging: it displays seasonal variations. Already at the appearance of the epidemic, laboratory studies [START_REF] Merow | Seasonality and uncertainty in global COVID-19 growth rates[END_REF], [START_REF] Carleton | Evidence for ultraviolet radiation decreasing COVID-19 growth rates: global estimates and seasonal implications[END_REF] of the effect of ultraviolet radiation on the virus had hinted at a possible seasonality. More studies are now supporting this argument based on an analysis of existing data [START_REF] Sajadi | Temperature, humidity, and latitude analysis to predict potential spread and seasonality for COVID-19[END_REF]. Watanabe [START_REF] Watanabe | Early detection of seasonality and second-wave prediction in the COVID-19 pandemic[END_REF] compared the evolution of the epidemic over the first few months of the year in the north and south hemispheres obtaining substantial differences, providing thus a clear statistical evidence in favour of seasonality. Introducing seasonality in the SIR model is straightforward: it suffices to make the parameter a time dependent and assume that it varies periodically around some mean value with a period corresponding to one year [25]. A simple way to do this is by introducing the sinusoidal expression a(t) = α + β cos(2πt/T ) where T stands for the duration of a year. The phase of the cosine chosen corresponds to the maximum of a(t) occurring at the beginning of the year, representing the situation in the northern hemisphere. A phase shift of T /2 would be necessary in the case of the southern hemisphere. All our simulations will deal with a northhemisphere epidemic which exhibits seasonality, waning during winter, waxing during summer and reappearing again during the autumn period.

Starting from the simple SIR model we shall now modify it (minimally) so as to take into account the various effects we wish to study, starting with vaccination. One important question that can be raised is that of immunity. Several studies have concluded that a Covid-19 infection generates a substantial immunological memory [START_REF] Dan | Immunological memory to SARS-CoV-2 assessed for up to 8 months after infection[END_REF] and thus individuals who have recovered can mount an effective response upon re-exposure [START_REF] Gaebler | Evolution of antibody immunity to SARS-CoV-2[END_REF].

Concerning the immunity conferred by vaccination we believe that the extensive statistics of the CDC settle the question. Thus we construct a model sans-reinfection. Assuming a constant-rate vaccination we have

dS dt = -a(t)SI -v dI dt = a(t)SI -λI (3)
dR dt = λI + v.
where v is the (constant) vaccination rate. This is not an essential constraint. We can easily made the rate timedependent and also dependent on the other components of the model as explained in [START_REF] Nakamura | Vaccination strategies for a seasonal epidemic: a simple SIR model[END_REF] where we simulated the effect of vaccination awareness. The tacit assumption in ( 3) is that the vaccine efficiency is perfect. However, in view of the first results concerning the delta variant, this may be particularly optimistic. Relinquishing this assumption does not present any problem. We assume that the vaccine has an efficiency f , smaller than 1 [START_REF] Abu-Raddad | Effectiveness of the BNT162b2 Covid-19 Vaccine against the B.1.1.7 and B.1.351 Variants[END_REF], and this results into a number of vaccinated individuals who can be infected and introduce the population U of persons who remain susceptible after having been vaccinated. (We can also assume that the fact that they are vaccinated attenuates somewhat the probability that they be infected, resulting in an infection rate c smaller than the initial one a. However in the absence of hard evidence on this point we shall not allow for such a freedom in our simulations, taking c = a). The SIR system for partially efficient vaccination is

dS dt = -a(t)SI -v dU dt = vf -c(t)U I dI dt = a(t)(S + U )I -λI (4)
dR dt = λI + v(1 -f ).
The SIR model, simple and realistic though it is, has a serious drawback: it lacks spatial dependence. The main assumption in (1) is that everybody is in contact with everybody else, as if they were concentrated on a single point. This is not what is observed in the evolution of the epidemics, which have usually a rich spatial structure [START_REF] Shi | An extended Weight Kernel Density Estimation model forecasts COVID-19 onset risk and identifies spatiotemporal variations of lockdown effects in China[END_REF]. Travelling plays an essential role in the spreading of the epidemics. Several approaches [START_REF] Zhao | Impact of population recruitment on the HIV epidemics and the effectiveness of HIV prevention interventions[END_REF], [START_REF] Saito | Extension and verification of the SEIR model on the 2009 influenza A (H1N1) pandemic in Japan[END_REF] have been contemplated, in the pre-Covid era, in order to palliate this SIR shortcoming. In [START_REF] Nakamura | Recruitment effects on the evolution of epidemics in a simple SIR model[END_REF] we introduced our own approach to population recruitment, which consists in assuming that there is an influx of susceptible individuals. Introducing the recruitment rate µ(t) we obtain the system

dS dt = -a(t)SI + µ(t)(1 -S) dI dt = a(t)SI -λI -µ(t)I (5)
dR dt = λI -µ(t)R.
Given the form of (4) it is clear that if we start with S + I + R = 1 the recruitment effect does not modify this constant of motion. The simplest case, explored already in [START_REF] Nakamura | Recruitment effects on the evolution of epidemics in a simple SIR model[END_REF] (and which will be considered in this paper), is the vacation-associated recruitment. In this case µ(t) is a step function, µ = 0 for t < 7 months, µ = µ 0 for 7 < t < 9 months, and µ = 0 for t > 9 months (for European countries).

Discretising the SIR model

In order to proceed to simulations it is necessary to obtain a discrete form of the SIR equations. Instead of using a black-box integration routine we prefer to construct our own discretisation based on the method we have presented in [START_REF] Grammaticos | Revisiting the Human and Nature Dynamics model[END_REF]. It follows the ideas of Mickens [START_REF] Mickens | Exact solutions to a finite-difference model of a nonlinear reaction-advection equation: Implications for numerical analysis[END_REF], [START_REF] Grammaticos | Discretising the Painlevé equations a la Hirota-Mickens[END_REF] and is based on the fact that the quantities appearing in the SIR model are by definition positive. (In the case of SIR, Hethcote [START_REF] Hethcote | The Mathematics of Infectious Diseases[END_REF] has shown that the positivity of the solution is guaranteed provided one starts from positive initial conditions). The prescription proposed in [START_REF] Grammaticos | Revisiting the Human and Nature Dynamics model[END_REF] can be cast in a most simple form: 'if all quantities are positive, no minus sign should appear anywhere'. We start from (3), introduce a forward difference of the time derivative, with time step δ, a staggering in the nonlinear terms. The only remaining minus sign comes from the vaccination term. The same difficulty was encountered in [START_REF] Grammaticos | Revisiting the Human and Nature Dynamics model[END_REF] where we proposed a simple solution. It suffices to replace the -v term by -vS/S and introduce the staggering -vS n+1 /S n . We obtain thus the discrete analogue of (3)

S n+1 -S n δ = -a n I n S n+1 -v S n+1 S n I n+1 -I n δ = a n I n S n+1 -λI n+1 . (6)
R n+1 -R n δ = λI n+1 + v S n+1 , S n .
and, solving for the points at (n + 1), we have

S n+1 = S n 1 + a n δI n + vδ/S n I n+1 = I n (1 + a n δS n+1 ) 1 + λδ (7)
R n+1 = R n + λδI n+1 + vδ S n+1 S n .
Given the structure of (7) it is clear that S n + I n + R n is a conserved quantity and, since we shall work with fractions of the population, we normalise it to 1.

For completeness sake we give also the discrete equations corresponding to the population recruitment, as modelled by system [START_REF] Nakamura | Confinement strategies in a simple SIR model[END_REF]. We start from the solution at point (n + 1) obtained, say, from [START_REF] Li | Estimating the health impact of vaccination against ten pathogens in 98 low-income and middleincome countries from 2000 to 2030: a modelling study[END_REF] and introduced the recruitment function M = δµ. We find Sn =

S n + M n 1 + M n Ĩn = I n 1 + M n (8) Rn = R n 1 + M n .
It is clear that, thanks to the presence of the denominators, we have Sn + Ĩn + Rn = S n +I n +R n = 1. As already mentioned the estival habits in most European countries are such that a non-negligible part of the population travels to the preferential destinations of vacationers. Our recruitment function will be a simple constant M , non-zero over two summer months.

Simulation results

In this section we are going to present results obtained from simulations based on equation [START_REF] Brauner | Inferring the effectiveness of government interventions against COVID-19[END_REF]. In all these simulations we assume that the epidemic is seasonal, the amplitude of its infectivity being modulated by a factor 1 + 0.25 cos(2πt/T ) and we fix λ -1 = 5 days. Moreover, unless otherwise explicitly stated, we assume that the vaccine efficiency is total. The first set of results concern a single population without any age stratification.

Results for a single component population

For all the figures presented in this section, the simulation starts with I(0) = 1. 10 -7 , R(0) = 0 and S(0) = 1-I(0) for the first variant. The second variant is introduced at time t 0 = 5 months and J(t 0 ) = 1. 10 -6 . When present, vaccination starts at t = 2 months. The reproduction number varies with time, because of seasonality: a Figure 1 shows what would be the expected evolution of a moderately virulent epidemic under a low-rate vaccination campaign. Under the influence of seasonality the number of infections abates during the summer months only to increase again with the arrival of autumn. Still the effect of vaccinations (provided the rhythm is not disrupted) is such that the second, winter, wave is of lower intensity than the first. Next we assume that a slightly more infective variant of the virus makes its appearance at the end of May (at which point 25 % of the total population have acquired immunity through vaccination of after having recovered from having been infected) and study its effect on the second wave under the assumption that the vaccination rate does not change. The results of the simulation are shown in Figure 2. We remark that the number of infections has increased substantially and that almost all are due to the new, more infective, variant which has essentially replaced the previous one. However, while the new variant becomes dominant, the temporal evolution of the epidemic does not change. The second wave arrives only in autumn.

In order to investigate the possible behaviour of an epidemic resurgence due to the delta SARS-CoV-2 variant some further assumptions are in order. First we assume that the new variant is substantially more infective than the already circulating ones. Second we tailor our conditions to those prevailing in France (and, in fact, in most European countries) where the vaccination rate is such that by the time the delta variant did appear (at the end of May), roughly 50 % of the population had already acquired immunity. Figure 3 shows the results of such a simulation. The situation here is totally different from that depicted in Figure 2. The new epidemic wave appears immediately, during the summer months. However, also due to the ongoing vaccinations, the amplitude of the wave is not excessively large and the wave disappears before the arrival of winter, when the quasi-totality of the population is vaccinated.

At this point it is interesting to study the effect of disrupting factors, ones that could alter the details of the new-variant-induced wave. The first such effect is that of vacations. In previous publications of ours we have introduced the notion of recruitment, whereupon, due to the population circulation and concentration in the locations preferred by vacationers, the number of susceptible individuals temporarily increases. The details of the recruitment mechanism were presented in [START_REF] Nakamura | Recruitment effects on the evolution of epidemics in a simple SIR model[END_REF] and we shall not go into them here. Suffices it to say that we use equation ( 5), or rather its discrete equivalent [START_REF] Mahase | Covid-19: Israel sees new infections plummet following vaccinations[END_REF], with a recruitment during the months of July and August. The results are shown in Figure 4. We remark that the effect of vacations is far from devastating. The amplitude of the wave does increase but its temporal extension is not affected. Just as in the non-recruitment case the wave disappears by autumn.

The second disrupting factor we have studied is the effect of partially efficient vaccinations. The matter is not yet settled but the consensus is that the current vaccines are slightly less efficient when it comes to the delta variant. We have thus investigated this effect by assuming a 90 % efficiency. Our results are presented in Figure 5. Just as in the case of the vacation effect, the slightly inefficient vaccines do not alter the epidemic dynamics. The wave is amplified but does not extend beyond autumn. Despite their diminished efficiency, the vaccines manage to curb the epidemic. And, though we are not presenting a figure for this, the same conclusion holds when one combines the effects of vacations and slightly less efficient vaccine. The amplitude of the wave increases but its temporal extension does not.

Next we investigate the behaviour of the new-variant-induced epidemic wave on a population where we distinguish two age groups.

Results for a two-component population

Up to now we have assumed that the population was constituted by a single group with the same characteristics as far as the infection is concerned. However it is clear that the various age groups do not respond in the same way to the infection risk and the requirement of hospitalisation, to say nothing of the probability of demise. In [START_REF] Nakamura | Vaccination strategies for a seasonal epidemic: a simple SIR model[END_REF] we presented an SIR-based model where the population was split into two subgroups. As explained there the infection rate a is not constant anymore. Instead we have a matrix a ij , which corresponds to the rate with which the subgroup j is infecting the subgroup i. We have argued in [START_REF] Nakamura | Vaccination strategies for a seasonal epidemic: a simple SIR model[END_REF] that , unless one wishes to ask a very specific question, it is better to stick to the simplest possible generalisation of the single-component population, namely a two-component one. One group consists of the younger population up to roughly 60 years of age, the remaining comprising the older individuals. We decided that the "young" correspond to 70 % of the total, while the "old" make up the remaining 30 % (these proportions are reasonable for most European countries). The young-old split allows one to answer in a more realistic way questions about the number of hospital admissions and/or deaths. The assumption is, that given the age split he have opted for, the probability for hospitalisation is 5 times smaller for the 'young' group as compared to that of the 'old'. As explained in [START_REF] Nakamura | Vaccination strategies for a seasonal epidemic: a simple SIR model[END_REF], the number of hospitalisations in a sub-group is proportional to the total number of "removed" that have not been vaccinated and the total number of hospitalisations is the sum of the hospitalisations in the two sub-groups. The model we used is:

dS o dt = -a oo S o I o -a oy S o I y -v o dS y dt = -a yo S y I o -a yy S y I y -v y dI o dt = a oo S o I o + a yo S y I o -λI o (9)
dI y dt = a oy S o I y + a yy S y I y -λI y dR o dt = λI o + v o dR y dt = λI y + v y ,
where a and v may vary with time. Again, the total population is conserved and we can introduce the normalisation

S o + I o + R o + S y + I y + R y = 1.
The seasonality applies to each element of the matrix: for the first variant, a ij (t) = a ij (1.0 + 0.25 cos(2πt/T )), with T = 12 months and we define R ij (t) = a ij (t)/λ. We have performed simulations using the same parameters for the initial reproduction matrix as in [START_REF] Nakamura | Vaccination strategies for a seasonal epidemic: a simple SIR model[END_REF]: R oo = 1.5, R yo = 1.0, R oy = 1.0, R yy = 1.2, where R oy represents the number of infections among the "old" from one infective "young" person. We multiply the reproduction numbers for the new variant by a global factor of 3, in order to account for its increased infectivity. For all the simulations I o (0) = 3.0 10 -6 and I y (0) = 7.0 10 -6 . The second variant always appears at t 0 = 5 months, J o (t 0 ) = 1.0 10 -6 and J y (t 0 ) = 1.0 10 -6 .

The vaccination rate was the same for the two groups but the vaccinations of the young subgroup started one month later than those of the old subgroup. As a result at the moment of the appearance of the new variant two-thirds of the old group were vaccinated while for the young group the proportion was not quite one fourth. The results of the simulation are shown in Figure 6. We remark that the epidemic extends now up to the onset of winter but it affects almost exclusively the 'young' population. Due to this fact the number of hospitalisations associated with this second wave are just 10 % of those of the first one. Moreover most are related to 'young' persons admissions, the old ones accounting for less than 20 % of the total. This is obviously due to the fact that the younger subgroup is not well protected by vaccination.

In order to investigate the effects of vaccination in the younger subpopulation we performed two different simulations. In the first we assumed that the vaccination campaign aimed at that group was intensified and a 30 % increase was introduced in the vaccination rate. The effect of the targeted vaccination we observe is to limit both the amplitude and the temporal extension of the second epidemic wave, see Figure 7. Moreover an analysis of the results predicts a substantial reduction of hospitalisations which are now a mere few percent of those of the first wave.

In the second simulation we introduce the vaccination-awareness effect first proposed in [START_REF] Nakamura | Vaccination strategies for a seasonal epidemic: a simple SIR model[END_REF]. We consider that the vaccination of the older subgroup pursues with constant rate, while for the younger subgroup we assume that the vaccination rate is directly related to the number of infected individuals, increasing or decreasing with the number of infective. The effect of this awareness is slightly counterintuitive. Clearly during the summer months, when due to the seasonality the epidemic ebbs and the number of infective diminishes, there are relative few among the younger population who think about getting vaccinated. But, since during winter the epidemic results in a large number of infections in the younger population, motivating thus accrued vaccinations, the percentage of younger individuals who are vaccinated when the virulent variant appears is now close to 40 %. Temporal evolution of the infective fraction of the population, with seasonality, and two variants, in the case of two age-groups in the population and with a constant rate of vaccination for the older group and a variable rate of vaccination for the younger one (proportional to the number of infective). Again, the continuous line represents the younger subgroup while the dashed line corersponds to the older population subgroup.Vaccination starts at t = 2 months for the older subgroup and at t = 3 months for the young erone. The rate of vaccination is v/λ = 1.1 10 -2 for the old ersubgroup. For the younger one, the rate of vaccination is proportional to the density of infective (with a factor of proportionality equal to 2.0 10 3 ). The efficiency of the vaccination is 100% (f = 0).

The results of the simulation are shown in Figure 8. Clearly the behaviour of the epidemic is somewhat intriguing. The wave, instead of reaching its apex during the summer months, is delayed towards autumn. Again it concerns almost exclusively the younger population. As a consequence the number of hospitalisations do not soar, as one could fear looking at the number of infections, and do not exceed a 25 % of the ones registered during the first wave.

Discussion

This paper had a double motivation. On the one hand, the appearance of the variant delta of SARS-CoV-2 is a source of real concern. Since the first assessments revealed an increased infectivity it was natural to wonder how the predictions of the various models of the Covid-19 epidemic stand up with this new challenge. On the other hand, since the various media propagate quasi-apocalyptic visions, with predictions often off the top of the head of imaginative reporters, it was imperative to reach some robust conclusions based on reliable models.

The approach we adopted here (and in fact in all our previous studies on Covid -19), was to keep our model as simple as possible. The advantage of such an approach is that in this way one can master the inputs of the model resulting into more robust results. We have thus opted for an as simple as possible version of the SIR model, following Holling's recommendations of a "somewhat more complex model" [START_REF] Holling | A Journey of Discovery[END_REF]. This means that our models tend to be simple, and we enrich them just when necessary but still shying away from making them too complex.

Starting from a simple SIR model we introduced a seasonally varying virus infectivity in agreement with what has been (and is being) observed in the evolution of the epidemic. The vaccination campaign, which started for most countries at the beginning of this year was also incorporated in our model, with the possibility to account for an incomplete protection from the emerging variant delta. We have also provided for what we have dubbed the recruitment effect, corresponding to local increase of the susceptible population due to travelling. Finally an extension of the basic, single population, model to one involving more than one age groups was introduced along the lines of the approach of [START_REF] Nakamura | Vaccination strategies for a seasonal epidemic: a simple SIR model[END_REF].

Our man conclusion is that the new variant will probably lead to the appearance of a wave of infections during the summer months. This is what the media refer to as the "fourth wave", the presages of which can be already seen in the evolution of cases in the UK and Portugal. However, and contrary to what is disseminated, the delta-induced wave is not expected to be of distressing consequences (at least for the majority of European countries). The successful vaccination campaign has offered a serious protection to the older, and more vulnerable, population. The delta-related wave is expected to be associated to the younger part of the population. and while, depending on the details of the scenario simulated, the absolute numbers of infected individuals can be appreciable (in particular if one considers the cumulative effects of vacation-related recruitment and slightly inefficient vaccines), the stress on the health system is expected to be manageable.

The key factor in all this are the vaccinations. It is, for instance, essential that the vaccination rate be maintained during the summer months. Policies aiming at encouraging and facilitating the vaccination of the younger layers of the population will have a most beneficial effect on the mitigation of the epidemic. Which brings us to the last point we wish to touch upon in this work. We have been worried from the outset about the assumption concerning the constant character of the vaccination rate. It is now clear that a certain group of the population are vaccination-deniers [START_REF] Hotez | COVID vaccines: time to confront anti-vax aggression[END_REF] and would not get vaccinated, despite been confronted with the devastating consequences of Covid-19. We have thus performed a final simulation where we assumed that 25 % of the population will not get vaccinated. The results are shown in Figure 9. Admittedly 25 % is rather high since it lumps together hard-core deniers and hesitant [START_REF] Schwarzinger | COVID-19 vaccine hesitancy in a representative working-age population in France: a survey experiment based on vaccine characteristics[END_REF]. Be that as it may the predictions of the simulations give pause for thought. If the vaccinations stop when just 75 % of the population are protected the fourth wave may develop into a huge one extending all the way to the end of spring. And, while spread over a larger time interval, the total number of hospitalisations can be comparable to that of the previous wave. Thus the one recommendation one can formulate based on the results of our simulations is that vaccinations should continue unabated and that no effort should be spared when it comes to convincing those who still hesitate.

Figure 1

 1 Figure1Temporal evolution of the infective fraction of the population under the action of the first variant, with seasonality and vaccination. The reproduction number is a 0 /λ = 1.4. For vaccination, v/λ = 5 10 -4 and its efficiency is 100% (f = 0).

Figure 2

 2 Figure 2 Temporal evolution of the infective fraction of the population, with seasonality and vaccination, for the two variants. For the first variant (black line), a 0 /λ = 1.4. The delta variant (red line) is introduced at t = 5 months and is characterized by an infectivity b 0 /λ = 1.8. The rate of vaccination is: v/λ = 5 10 -4 and

Figure 3

 3 Figure3Temporal evolution of the infective fraction of the population, with seasonality and vaccination, for the two variants. For the first variant (black line), a 0 /λ = 1.4. The delta variant (red line) is introduced at t = 5 months and is characterized by an infectivity b 0 /λ = 4.0. Vaccination starts at t = 2 months with a rate of v/λ = 1.8 10 -2 and an efficiency of 100% (f = 0).

Figure 4

 4 Figure 4 Temporal evolution of the infective fraction of the population, with seasonality, vaccination and vacations, for the two variants. For the first variant (black line), a 0 /λ = 1.4. The delta variant (red line) is introduced at t = 5 months and is characterized by an infectivity b 0 /λ = 4.0. Vaccination starts at t = 2 months with a rate of v/λ = 1.8 10 -2 and an efficiency of 100% (f = 0). For the vacations µ 0 /λ = 5.0 10 -4 .

Figure 5

 5 Figure5Temporal evolution of the infective fraction of the population, with seasonality and a partially inefficient vaccination against the second variant. For the first variant (black line), a 0 /λ = 1.4. The delta variant (red line) is introduced at t = 5 months and is characterized by an infectivity b 0 /λ = 4.0. Vaccination starts at t = 2 months with a rate of v/λ = 1.8 10 -2 and an efficiency of 90% (f = 0.1). For the vacations µ 0 /λ = 5.0 10 -4 .

Figure 6

 6 Figure6Temporal evolution of the infective fraction of the population, with seasonality and vaccination, for the two variants, in the case of two age-groups in the population. The continuous line represents the younger subgroup while the dashed line corersponds to the older population subgroup. Vaccination starts at t = 2 months for the older subgroup and at t = 3 months for the younger one with a rate of v/λ = 1.1 10 -2 for both subgroups. The efficiency of the vaccination is 100% (f = 0).

Figure 7

 7 Figure 7 Temporal evolution of the infective fraction of the population, with seasonality and vaccination, for the two variants, in the case of two age-groups in the population. The continuous line represents the younger subgroup while the dashed line corersponds to the older population subgroup.Vaccination starts at t = 2 months for the older subgroup and at t = 3 months for the younger one. The rate of vaccination is: v/λ = 1.1 10 -2 for

Figure 8

 8 Figure 8 Temporal evolution of the infective fraction of the population, with seasonality, and two variants, in the case of two age-groups in the population and with a constant rate of vaccination for the older group and a variable rate of vaccination for the younger one (proportional to the number of infective). Again, the continuous line represents the younger subgroup while the dashed line corersponds to the older population subgroup.Vaccination starts at t = 2 months for the older subgroup and at t = 3 months for the young erone. The rate of vaccination is v/λ = 1.1 10 -2 for the old ersubgroup. For the younger one, the rate of vaccination is proportional to the density of infective (with a factor of proportionality equal to 2.0 10 3 ). The efficiency of the vaccination is 100% (f = 0).

Figure 9

 9 Figure9Temporal evolution of the infective fraction of the population, with seasonality, for the two variants, in the case of only one age-group but with a fraction of vaccination-deniers. Vaccination starts at t = 2 months with a rate of v/λ = 1.8 10 -2 and an efficiency of 100% (f = 0). The fraction of vaccination-deniers is 25%.