Athanasios Laliotis
email: laliotis@univ-paris13.fr

Santiago Villalba
email: svillalb@fing.edu.uy

Horacio Failache

Horacio Failache Simple

Simple fabrication of miniaturized glass cells for atomic vapor spectroscopy

Keywords: glass cell, miniaturized cell, atomic spectroscopy, photonics devices

teaching and research institutions in France or abroad, or from public or private research centers.

Simple fabrication of miniaturized glass cells for atomic vapor spectroscopy

Athanasios Laliotis 1 , Santiago Villalba 2 and Horacio Failache 2*

1 Introduction

Spectroscopic studies on diluted atomic vapors were essential for the development of atomic physics. Most of this research was done, over the last hundred years, on low density (highly diluted) and high purity alkali vapors in glass cells. Glass is today the preferred material for vapor cell fabrication as it is inert enough when exposed to alkali atoms (this was verified by testing very old cells). Moreover, glass has a very low permeability, with a probable exception for the lightest alkalies as H and Li, and a good transmission for most light field wavelengths used in spectroscopy research. Most macroscopic glass cells are fabricated using traditional glass blowing techniques. However, technological applications of atomic physics like atomic frequency references and atomic clocks [1], magnetometers [2][3] and gyroscopes [START_REF] Noor | IEEE International Symposium on Inertial Sensors and Systems (INERTIAL)[END_REF], which require miniaturization and mass production of spectroscopy cells need a different approach.

Several technological improvements were recently made towards fabricating miniaturized, mass produced cells [START_REF] Liew | [END_REF][6][7][8][9]. Anodic bonding [10] of glass-silicon pieces is the most used technique, as described in [9]. This technique allows the fabrication of miniaturized cells where the body of the cell is etched in a wafer of Si to which the cell windows are bonded. It allows mass production, as several identical cells can be obtained from the same wafer of Si, with a cell geometry that is easily defined by the etching pattern. Anodic bonding requires, however, to heat the entire cell to around 300 °C. This high temperature is a drawback because the thermal expansion coefficients of materials must be matched to avoid cracks and hinder the use of anti-relaxation polymer coatings in the internal surface of the cell because these polymers have low melting point temperatures. Moreover, another drawback of this technique is that alkali atoms, or their precursors, must be placed inside the cell before sealing it. At these elevated temperatures, alkali atoms may diffuse in the glass, reducing the number of available alkali atoms and eventually darkening the glass. Several methods have been developed to fill in the cells [7][9] [11][12][13][14][15]. Some involve the use of alkali chlorides, chromates or azides, that are placed inside the cell, together with other components, before sealing the cell. Pure alkali atoms are then produced in a chemical reaction that takes place at relatively high temperatures. These reactions have some by-products that in some cases must be pumped before sealing the cells, that must be therefore placed inside a vacuum chamber during the manufacturing procedure. Most applications require a buffer gas, or a mixture of them, together with alkali atoms. More recently a low-temperature bonding was tested as an alternative to anodic-bonding [16]. Cell production using these above elaborate techniques is nevertheless complex and relatively expensive.

In this work, we present a procedure to fabricate miniature spectroscopy glass cells based on the idea of cutting a tiny cell from a larger one, that was previously filled with the required molecular or atomic vapor. This strategy preserves therefore all the flexibility of the filling procedure of macroscopic cells. This miniaturization process is based on a cutting element that cuts We also show the volume of unconfined Rb vapor where spectroscopy measurement can be done (the porous glass is colored by Rb clusters, that migrated away from the cutting point when its temperature increased [17]). (e) A standard cell was divided to make two smaller cells.

while simultaneously seals the otherwise exposed internal volume of the cell. This technique has the advantage of being simple, fast, and inexpensive. The procedure was tested on glass spectroscopic cells filled with pure Rb vapor, on which we have verified that the purity of the alkali vapor was preserved during the cutting process.

Miniature cell fabrication

To extract small spectroscopic cells from a larger one, we have used a cutting element that is hot enough to melt glass. While this element moves it drags melted glass creating sealing walls that separate the larger cell in two smaller ones, as sketched in the inset of Fig. 1a,b. This cutting element is an electrically heated NiCr (Nichrome alloy) wire with a circular cross-section. This alloy is one of the most commonly used electric heating elements, because of its relatively high electrical resistivity, if compared to other metallic conductors, and for its stability at high temperatures. This alloy shows high resistance to oxidation which preserves the smoothness of the wire surface at high temperature. The shape and dimensions of the cross-section, as well as the thickness of the cell's glass wall, are relevant parameters that determine the amount of melted glass that is dragged in and become available to build the seal between the walls of the cell. The amount of this melted glass determines therefore the maximum extension of the sealing walls (or the maximum cell vacuum gap that could be sealed). The wire temperature is also an important parameter, as it determines the temperature and therefore fluidity of the melted glass. We have performed the cutting process in the open air, therefore, if the melted glass is too fluid, the sealing wall could collapse under the pressure difference between the external atmospheric pressure and the vacuum in the cell. On the other hand, a less fluid glass hinders the wire movement. All these parameters, together with the velocity at which the wire moves, should be chosen for a given particular cell geometry. We have started roughly optimizing these parameters testing the cutting procedure with several dummy cells, evacuated but not containing any vapor.

Our initial tests on Rb-filled cells were done on porous glass cells [17] [18]. These home-made cells were created by sintering pyrex glass powder inside a glass tube to create a porous medium. The tube was then sealed after filling it with pure Rb vapor [17] (Fig. 1d). These porous glass cells are for example useful for the development of robust atomic frequency references [18]. We have used porous glass cells to test our cutting procedure because they have the advantage that the vacuum gap to be sealed is determined by the size of the pores (typically 50 µm) in our cells. Additionally, there is ample glass available to build these seals. We have used a hot wire with a diameter of 700 µm, and a temperature of approximately T w ∼ 1100 °C°C. Despite our initial tests on dummy cells, the cutting procedure on two porous cells failed, compromising the Rb purity. The analysis of these two cells using an optical microscope revealed a failed seal that shows bubbles. Two other porous cells were later successfully cut using a colder wire (T w ∼ 1050 °C) to increase the viscosity of the melted glass. After successfully cutting few porous cells, we proceed with a standard cell. We have specially made a cell using a pyrex-glass cylindrical tube, with an external diameter of 5 mm and a wall thickness of 1.0 mm, that was partially flattened to define an internal gap of approximately 1 mm. This cell was connected to a vacuum system composed of a turbo-molecular pump and an ion-pump and was evacuated down to a pressure of 10 -9 Torr. During this pumping process, the cell was baked at 200 °C for 24 hrs. A small dot of high purity Rb was then distilled into the cell before sealing and extracting it from the vacuum system. To extract the cell we followed the standard procedure that uses the flame of a torch to soften the glass tube to make it collapse under atmospheric pressure. The cell can be then sealed and separated from the vacuum system by melting the glass at the point where the tube collapsed.

In order to reduce the risk of failure of the seal during the cutting process of this cell, we have used a wire temperature slightly lower than the one used on porous cells to have a less fluid glass. The cutting process lasted approximately 2 minutes. Moreover, we have used in this case a curved wire, like the one shown in Fig. 1(b,c), to drag and feed the seal with more glass and reducing burrs. The result of this cutting process is shown in Fig. 1d. During the cutting procedure, it is important to reduce the number of Rb atoms in contact with hot or melted glass, because alkali atoms diffuse into the glass at high temperature reducing the amount of pure Rb in the cell and darkening the glass. To circumvent this issue, we have defined a cold point in the cell, away from the cutting place, where the Rb remains condensed. The ends of the cell were placed in good thermal contact with aluminum bodies of several cm 3 that fulfill as temperature reservoirs. Small cavities to accommodate the ends of the cell were machined in the metallic bodies, and thermal conductive paste was used to reduce the thermal resistance of the glass-metal contact. Using this procedure, the temperature of the glass at the cold point was kept below XX °C during the cutting procedure. As a primary indication of a successful cutting, in Fig. 2 we show an example of saturated absorption spectrum obtained with one of the cells shown in Fig. 1e. In particular, this spectra was measured in a miniaturized saturated absorption spectrometer that was only possible after miniaturizing the spectroscopy cell [19].

Spectroscopy analysis

All our cell cutting procedures were done in the open air. Earth's atmosphere is composed essentially of N 2 (78 %) and O 2 (21% ), therefore a possible leakage during the cut procedure would essentially introduce N 2 as contaminating gas. We assume that O 2 is eliminated in a fast redox reaction with the highly reactive rubidium atoms. Such N 2 impurities introduce inelastic collisions that in particular broaden the D1 and D2 transitions of Rb with a typical coefficient of 18 MHz/Torr [20]. This broadening could be measured using Doppler-free saturated absorption spectroscopy [START_REF] Demtroder | Laser Spectroscopy[END_REF] and used to determine the pressure of the potential N 2 contamination. Using a carefully aligned setup, with an extended cavity diode laser and a reference cell with pure Rb vapor, we have Simple fabrication of miniaturized glass cells for atomic vapor spectroscopy measured the dependence of the saturated absorption resonance width with the light intensity. To neglect the power broadening, we have extrapolated to zero light intensity a width of 15 MHz (full-width half-maximum). This width is much larger than the expected natural width of approximately 6 MHz for the Rb 5P levels, most probably due to frequency noise of the laser source. This low spectral resolution limits the pressure of N 2 impurities that can be measured by the saturated absorption technique to a fraction of Torr.

We have considered an alternative technique based on the measurement of the lifetime of an excited state of Rb by measuring, in the time domain, the transient of the spontaneously emitted fluorescence. Using this method the pressure of N 2 impurities can be estimated with good accuracy without being limited by the linewidth or frequency noise of the laser sources. We have in particular measured the lifetime of 85 Rb(5D 5/2 ) levels following the method described in [START_REF] Failache | [END_REF]. These Rb levels were excited using the ladder-level excitation 780-776 nm technique [23] described in the inset of Fig. 3. We have measured the transient decay of the blue fluorescence intensity at 420 nm after suddenly shutting down the 776 nm field using a fiber-coupled Mach-Zender intensity modulator (model PHOTLINE NIR-MX800-LN-10), and measuring the blue fluorescence transient using a solid state photon counting detector (model Perkin Elmer SPCM-AQRH-13). The 85 Rb(5D 5/2 ) level lifetime was then measured by adding approximately 10 6 measurements of the fluorescence transient and fitting the resulting curve with a theoretical model. Fig. 3 shows the transients of the blue fluorescence in three different cells: a reference commercial cell that has pure Rb vapor (Toptica), one of the cells shown in Fig. 1e, and one of the porous cell that was contaminated during the cutting procedure. We also show a fit of the theoretical model from which the τ 5D lifetime value was determined. The uncertainty on the value of this lifetime was estimated to be ± 5 ns [START_REF] Failache | [END_REF]. The mean lifetime of the 5D level τ 5D can be reduced due to quenching collisions with N 2 molecules. This reduction can be estimated with the equation τ -1 =τ -1 5D +σ q 8

πµkT p [START_REF] Demtroder | Laser Spectroscopy[END_REF], where σ q is the cross-section of quenching collisions for the 5D levels, µ is the reduced mass of N 2 -Rb two-body system, k is the Boltzmann constant, T the absolute temperature, and p the pressure. Unfortunately, σ q has not been previously measured for the 5D levels of Rb. We can, however, make a rough estimate of the magnitude of this cross-section from the measurements of Supronowicz et al. [24], who report the quenching collisions cross-section of the 7D and 8D Rb level by N 2 collisions to be σ q = 30 ± 30 Å2 . We expect a similar cross-section for the 5D energy level considering that this value is also close in magnitude to those of intermediate nP states in Rb [25] and Cs [26], nS (5 < n < 9) [27] and D states in Na [28]. The 85 Rb(5D 5/2 ) has a natural lifetime τ 5D = 238 ns [29], that due to quenching collisions is then reduced as δτ ∼ -τ 2 σ q 8 πµkT δp ∼ 360 ns/Torr of N 2 at 100 °C. Therefore, with this procedure, we can estimate the pressure of N 2 impurities with an accuracy of ∼ 14 mTorr.

Alternative techniques to measure the presence of N 2 impurities could be considered. One of them is the Electromagnetically Induced Transparency (EIT) spectroscopy of Rydberg atoms [30]. In the particular case of Rb, the cross-section of dephasing collisions of Rydberg states with N 2 molecules lie in the range of 1000 Å2 [31]. This implies a broadening coefficient of approximately 25 MHz/Torr at 100 o C. The width of EIT resonances for Rydberg states of Rb could be as low as a fraction of MHz [32]. Assuming such narrow resonances, this technique could also be used to detect a pressure of N 2 impurities of few tens of mTorr. An other possible technique is to detect the shift and broadening of the Rb 5S 1/2 → 5D 5/2 two-photon transition, due to collisions with impurities and particularly N 2 [33]. In this case an accuracy on the N 2 pressure of several mTorr to few tens of mTorr could be achieved, although requiring a precision on the measure of the width and/or shift of this transition of few hundred kHz. However, these two techniques are be affected by the frequency noise of the laser sources.

In the inset of Fig. 3 and Table .1, we show some measurements of τ 5D on several cells. For the reference cell, we obtain a τ 5D value of 235 ± 5 ns, close to the theoretical value. We have measured τ 5D on the cells shown in Fig. 1e before and after the cutting procedure and we do not observe a significant difference. We have also measured approximately the same value for τ 5D in the unconfined Rb vapor region (see Fig. 1d) in a cut porous cell, a reference porous cell from the same batch, and in another cell from a different batch. We attribute the small difference observed on these measurements of τ 5D with the theoretical value to some imperfection in our filling procedure of spectroscopy cells. We can conclude that the pressure of N 2 impurities potentially introduced during a cutting procedure can be considered lower than the sensitivity limit of our measurement method, i.e. few tens of mTorr. In the inset of Fig. 3, we also show the value of τ 5D measured in a porous cell that is clearly detected as contaminated after a failed cutting procedure. This value of τ 5D corresponds to a N 2 contamination pressure of about 595 mTorr. When we have used this cell in a saturated absorption spectroscopy setup, where we have measured an absorption resonance full-width-half-maximum of 17 MHz (at the limit of zero light intensity), only slightly larger than the value obtained for a pure Rb cell. This result is consistent with our previous considerations about the low sensitivity of the saturated absorption technique.

Discussion and conclusion

The present work is a proof of principle of a new method for miniaturized cell fabrication. In Fig. 4 we sketch a geometry that could be potentially used to produce miniaturized spectroscopy pre-filled cells. The minimum size of the cells that could be cut is limited by the maximum stress created by the glass temperature difference between the hot cutting point and the glass near the metallic Rb, that should be kept at a much lower temperature. After our experiments, in the case of pyrex glass this minimum size is well below 5 mm. This procedure has the advantages that is simple, fast and inexpensive. Moreover, most miniaturized atomic clocks and magnetometers require the use of pumping system and atoms reservoir Fig. 4 A possible configuration to fabricate small spectroscopy cells from a larger one. Black dashed lines show the cutting places. A metallic structure could be glued to the cell to reduce the fragility of the system and to define cold points where alkali atoms remain condensed during the cutting procedure.

an alkali vapor and a mixture of buffer gases with specific partial pressures. In principle, the technique presented here would allow to fill a large "mother" cell with an alkali and buffer gas mixture that would be replicated in the tiny "daughter" cell. The technique presented here should be further improved, reducing the risk of failure of the seal, by performing the cutting procedure in a chamber with an atmosphere of controlled composition and pressure. The cutting element could be also improved using a more complex device. For example, instead of using a single cutting element that works as a knife, as in this work, a two or multiple elements cutting device, acting like scissors, could be used to reduce burrs and have better control of the shape of the miniaturized cell.

Fig. 1 (

 1 Fig. 1 (Color online) (a),(b) Diagram describing the proposed method of cutting and sealing miniature spectroscopy cells. In (a) are represented two stages of the cutting process, showing a longitudinal section of a portion of a cylindrical sealed cell, and in (b) the corresponding transversal section. (c) A photograph of the hot wire. (d) Photograph of a porous cell and the tiny cell extracted from it.We also show the volume of unconfined Rb vapor where spectroscopy measurement can be done (the porous glass is colored by Rb clusters, that migrated away from the cutting point when its temperature increased[17]). (e) A standard cell was divided to make two smaller cells.

Fig. 2

 2 Fig.2Probe beam transmission through one of the miniaturized cells shown in Fig.1ein a saturated absorption spectroscopy setup[19]. The full-width-half-maximum of the saturated absorption resonances is ∼16 MHz (essentially determined by frequency noise in the laser field). This spectrum was measured in the D1 line of Rb. The frequency detuning in this figure is measured from the cross-over of the 85 Rb 5S 1/2 (F=3) → 5P 1/2 transition.

Fig. 3 (

 3 Fig. 3 (Color online) Transient decay of the blue fluorescence intensity, recorded after turning off the 776 nm laser. We show the transient for a reference cell (filled with pure Rb, in black), the transient for a miniaturized cell (blue), and the corresponding model fit (red). For comparison, we also show the transient for a slightly contaminated cell (magenta). Insets: Rb energy levels involved in the experiment. 85 Rb(5D 5/2 ) level mean lifetime τ 5D measured by fitting the theoretical model. The corresponding N 2 pressure is shown in the right vertical axis. •: reference cell, : a cell before cutting and after cutting, : porous cell after cutting, : porous cell from the same batch, : cell from another batch, : slightly contaminated cell.

Table 1

 1 Measured values of τ 5D on different Rb cells. These cells were produced in 3 different batches which are arbitrarily numbered.

	Cell	τ 5D (ns) Batch
	Standard cell (reference)	235 ± 5	-
	Standard cell (before cutting) 217 ± 5	1
	Standard cell (after cutting)	223 ± 5	2
	Porous cell (after cutting)	222 ± 5	2
	Porous cell (batch reference)	217 ± 5	2
	Porous cell (batch reference)	220 ± 5	3
	Porous cell (failed cutting)	126 ± 5	2

Acknowledgments. We gratefully acknowledge fruitful discussions with Arturo Lezama and financial support from CSIC (Universidad de la República).


Bruce Masonova

Solozafy Bemena

André Totohasina
email: andre.totohasina@gmail.com

Daniel Rajaonasy Feno
email: fenodaniel2@yahoo.fr

A model of the propagation endemic of emerging infectious diseases SEPI

Keywords: model Susceptible, Exposed, Precontaged, Infected

In the field of epidemiology, there are two classes of dynamic models. There is the class of temporary models specialized for short-lived epidemics and the class of endemic models reserved for epidemics that persist for a long time. In the light of the new temporary dynamic model Susceptible, Exposed, Precontaged, Infected (SEPI) (1 er class) that we have proposed and in order to make our model more efficient, we will develop the endemic model SEPI (2 th class) and to propose the different studies relating to this model. This endemic SEPI model is specialized in epidemics that persist for a long time and in cases where the infection spreads directly: first between precontagious individuals (asymptomatic) and susceptible individuals, second between infectious individuals (symptomatic) and susceptible individuals.

Introduction

In 1760, Daniel Bernoulli proposed the first epidemic model. In 1906, W. H. Hamer presented the first dynamic compartmental model. Since then, researchers have continued to improve the various existing models in order to arrive at a good model. A good model is one that is close enough to reality to provide valid and useful conclusions (J.M.M. ONDO (2012)).

In addition, it often happens that the epidemic persists and lasts for several periods in a population. This is called an endemic. This calls into question the effectiveness of the various models because they are developed in the case of temporary epidemic phenomena. To remedy the weaknesses of the epidemiological models, M. Fan (2001) incorporated the parameters immigration, birth and mortality by the disease in the Susceptible, Exposed, Infected (SEI) model proposed by (L.Q. Gao (1995)) in order to arrive at an endemic model. Then, other researchers such as (G. Li and al. (2006)) followed their method to apply to other models. But nowadays, this model presents weaknesses, because today, there are infectious diseases that spread with a very rapid speed and are transmitted even between individuals who do not yet present symptoms INSPQ (2021), Y.WELKER (2020), [START_REF] Alizon | modélisation de l'épidémie de COVID-19 : modèle SEAIR[END_REF]. This is what (OMS (2019)) and (Z. Hu and al. (2020)) called a new strain of coronavirus or COVID-19. This is inconsistent with the SEI model hypothesis.

In the light of the new dynamic process of infection of infectious diseases caused by global warming and in order to make our SEPI model into a good model, this paper proposes an endemic SEPI model adapted to this new behaviour of infectious diseases caused by global warming. In what follows, our work is divided into ten sections. Section 2 remember the definition of assumptions. Section 3 defines the assumptions of the SEPI endemic model. Section 4 presents a schematic of the SEPI endemic model. Section 5 develops the differential equations of the SEPI endemic model. Section 6 presents the simulation study of the SEPI model. Section 7 studies the equilibrium point of the SEPI endemic model. Section 8 determines the basic reproduction number. Section 9 proposes the study of the stability of the equilibrium point. Section 10 studies the local stability of the equilibrium point. Section 11 provides a conclusion.

Reminder of the definition of assumptions

According to the work of [START_REF] Masonova | A modelling of emerging and reemerging infectious diseases[END_REF] :

Definition .1. An individual who has been infected with the disease pathogen and can also transmit the disease, but has no symptoms, is referred to as a precontagious or precontaminated individual. Definition .2. Precontaged represents a compartment where the disease requires a period of precontagion. Pre-contagious individuals are capable of transmitting the disease into the population, but they do not yet show symptoms of the disease. They are therefore assigned to this compartment with the rate k called precontagiousness rate. In the following, the letter P will be used to refer to individuals who are infected and contagious, but do not yet represent symptoms of the disease. Definition .5. The period of contagiousness is a distinct phase of time when the sick individual (person with symptoms of the disease and whose health is impaired) transmits a disease to the other individual.

Definition

In the figure (1), we illustrate the context in which this event takes place : This schematic presentation shows the different phases (disease states).

FIG. 1: Representation of the contagion process 3 Assumptions of the endemic SEPI model

In this type of model, we consider that the disease persists and continues to spread during a time interval [t, t + ∆t] (equivalent to a month or a quarter).This leads us to take into account, over time, the birth rate, the natural mortality of the population and the loss of infectivity of the disease in the population. Furthermore, we consider that emigration is balanced with immigration of inhabitants.

In order to develop the endemic model of SEPI, we make the following assumptions :  A1 : The size of the population is equal to N, assumed fixed.  A2 : The time variable t is of discrete type, such that t T or T is the total duration.  A3 : The period of time ∆t = dt represents hours or days or weeks.  A4 : At each time t, the population N is subdivided into four compartments : S(t) : set of susceptible individuals, E(t) : set of exposed individuals, P(t) : set of precontaged individuals, I(t) : set of infected individuals with S(0) = S 0 > 0, P (0) = P 0 > 0 and/or I(0) = I 0 > 0.  A5 : We admit that each susceptible individual in a period ∆t is exposed, precontaged then infected.  A6 : The transmission of the infection is done through a direct contact between : firstly, susceptible S and one or more precontaged P with a factor β p proportionality (also called rate of precontagion or rate of transmission or rate of transmission from the susceptible to the infected), secondly, susceptible S and one or more infected I with a factor β i of proportionality (also called rate of infection) and it is admitted that a factor β is the rate of total transmission or of exposure such as β = β p + β i .  A7 : An infected individual remains and remains contagious until the end of his life in the rate λ.  A8 : Compartment D is used to store individuals who have died as a result of the disease at a rate λ in the time interval [t, t + ∆t]. Here, with respect to the aggressiveness of the microorganism, it is difficult to determine the cause of death of each individual in the Exposed and Precontaged compartments if it is unnatural or not by the disease. Therefore, we consider here the deaths in these two compartments are already counted and included in the rate of λ.  A9 : During the time interval [t; t+∆t], we admit that the population studied increases (birth of children) with the birth rate δ ; it also undergoes natural death with the mortality rate µ and it suffers in addition the loss of infectiousness of the disease only for the sick individuals (infected) with the mortality rate λ.  A10 : We consider that the disease studied does not transmit vertically (vertical transmission is from mother to child or genetically).  A11 : We consider that a constant average number of contacts cannot apply to all diseases :

we can generalise a little by putting the proportionality coefficients β p and β i which depend on N.

Scheme of the endemic model of SEPI

We admit that an infected individual and a precontaminated individual meet on average β(S/N) individuals susceptible to be exposed per unit of time, with β = β p + β i . We note : β > 0 : the rate of exposure (or of transmission from the susceptible to the exposed), k > 0 : the rate of precontagiousness (or of transmission from the exposed to the precontaged), ν > 0 : the rate of contagiousness (or of transmission from the precontaged to the infected), λ > 0 : the rate of infected to die, µ > 0 : the rate of natural mortality, δ > 0 : the rate of birth. The endemic model can be schematised as in the figure (2) below : There is a unique solution for the model (1), under the initial conditions : S(0)=S 0 , E(0)=E 0 , P(0)=P 0 , I(0) =I 0 in particular, in the region Ω = which is positively invariant for the system ( a set G is said to be positively invariant if x 0 G, the trajectory passing through x 0 is contained in G after x 0 : if x is the solution of the system (with F of class ) verifying x(0) = x 0 , then t ≥ 0, x(t) G ).

Simulation of the SEPI endemic model

The different curves, obtained with Scilab, already give us an idea of the evolution of the epidemic. For the simulation, we consider here to have an individual precontaged at time t = 0 with N=1000, δ = 0.3, β p = 0.2, β i = 0.1, k=0.4, ν = 0.2, µ = 0.2 et λ=0.3. We consider a period t depends on the unit of transmission rates, and it is equivalent to day or week or month. By doing the simulation, we obtain the following curves :

FIG. 3: Curves of S(t) coloured in red and E(t) in blue

Interpretation :

From figure (3), we have noted that even with low precontagion and infection rates, the epidemic is spreading with a phenomenal and very rapid speed. Even with a birth rate ( δ = 0.3) that we consider to be the average population rate, any susceptible population is already exposed after only 2 e time period t

FIG. 4: Curve of E(t) during a phase of the epidemic

Interpretation :

According to figure (4), after the phenomenal evolution of the epidemic, the curve of the Exposure decreases, and stablilises and becomes endemic after the 7 e period of time t of the epidemic.

FIG.5: Curves of I(t) coloured in red and P(t) in blue

Interpretation :

From Figure ( 5), it appears that even after the 7 e period of time t, the curves for the Precontaged and the Infected are still increasing and are signs of a pandemic.

FIG. 6: Curves of I(t) coloured in red and D(t) in blue

Interpretation :

According to figure (6), parallel to the situation of the infected, the curve of D(t) also increases and we record the death of 450 individuals related to this disease in 7 e time period t only.

Study of the equilibrium point of the SEPI endemic model

According to the work of Lyapunov in (J.M.M.ONDO ( 2012)), we define the equilibrium point as follows :

Définition .1. Consider U; a non empty open of R n containing 0, and I a non empty interval of R, not bounded on the right. Let the two equations below (2) and (3) :

where the functions f : U R n for the system (2) and f : I × U R n for the system (3) sare assumed to be continuous. A point "a" is an equilibrium point or equilibrium state or singular point of the system (2) (resp. ( 3)), if f(a) = 0 ( resp. if, for all t I, f(t, a) = 0 ).

From the definition (1) of the equilibrium point above, we obtain the following proposition : Proposition 1. Let N>0. Then, the system (1), with the condition S(0) = S 0 , E(0) = E 0 , P (0) = P 0, I(0) = I 0 and Ω = , admits a unique solution (S,E,P,I) defined on [0, + [.

Proof :

We calculate the equilibrium points in the absence of infection and/or precontagion. The equilibrium point of the model (1) satisfies :

In the absence of the infection I=0 and the pre-contagion P=0, we obtain the following proposition : Proposition 2. Let N>0, in the absence of the infection I=0 and the pre-contagion P=0, then the system (1) admits the equilibrium point : = Proof : By replacing I=0 and P=0 in the first, second and third equations in the system (4), we obtain the first equilibrium point : = ( ) :

=

In the presence of the precontagion P 0 and in the absence of the infection I = 0, we obtain the following proposition :

Proposition 3. Let N>0, in the absence of the infection I=0 and in the presence of the precontagion P 0, then :

 the system (1) admits the following equilibrium point  moreover, for all t>0 we have δβ p > νµ.

Proof :

By replacing I=0 in the third equation of the system (4), implies : = .

By replacing in the first equation of ( 4) with P 0, we obtain : = .

Replacing in the first equation of ( 4), we obtain By replacing in , we then have the equilibrium point as follows :

(6)

Note that > : (7) From ( 7), we deduce that : (8)

And (9)

According to ( 7),( 8),( 9), we then have δ > νµ.

In the presence of the precontagion P 0 and the infection I 0, we obtain the following proposition :

Proposition 4. Let N>0, in the presence of the precontagion P 0 and the infection I 0, then :

 the system (1) admits the following endemic equilibrium point  ;

 moreover, for all t>0 we have δ(λβ p + β i ν) > µνλ.

Proof : If P 0 and I 0, the system (4) becomes :

The third and fourth equations of the system (10), imply : = and By replacing and in the second equation of the system (10), we obtain :

By replacing and in the first equation of the system (10), we obtain :

(12)
Replacing in and we then have the endemic equilibrium point  as follows :

 (13) Note that : (14)
So, we have :

δ( ) > µνλ. ( 15
)
From ( 15), we deduce that : > 0, > 0 and

The basic reproduction number R 0 of the SEPI endemic model

To find the of the SEPI endemic model, we apply the study condition in the work (L. Chahrazed (2002)) on :

 If < 1, the equilibrium point E 0 is locally asymptotically stable ;  If > 1, the equilibrium point E 0 is unstable.
First, we calculate the Jacobian matrix of the linearized system (1) at the equilibrium point E 0 ignoring the death compartment and we obtain :

(16)

The characteristic equation of the system (1) in the neighbourhood of the equilibrium point E 0 is given by : It comes :

det(J -θI) = = 0 (17)
Then, the first eigenvalue is θ = -µ and it remains to study the equation :

We admit that the equation ( 18) is the characteristic equation of the sub-matrix :

We have the trace ( ) = -[k + ν + λ] < 0, so :

If det(J 1 ) > 0, we get :

According to the Varga and Poincaré-Lyapunov theorem of linearization in (G. Sallet (2010)), R 0 is defined by the expression below :

9 Study of the stability of the equilibrium point E 0

Local stability

L. Chahrazed (2002) defines the local stability of E 0 as follows :

Definition .2. We say that E 0 is locally asymptotically stable if and only if the trace of the Jacobian matrix in the neighbourhood of E 0 is strictly negative and the determinant is strictly positive.

Indeed, according to the equations ( 19), ( 20) and ( 15) above :

So we see that the conditions in ( 22) are met. Then the equilibrium point E 0 of the system (1) is unique. And it remains locally asymptotically stable.

Global stability

According to Lyapunov's method in the works [START_REF] Richard | Etude de la stabilité d'un système dynamique stationnaire par la méthode de lyapounov[END_REF]), [START_REF] Moulay | Stabilité des équations différentielles ordinaires[END_REF]) and [START_REF] Richard | Stabilité. MR Smart[END_REF]), we obtain the definitions below .

We consider that U always designates a non-empty open of (n ) containing 0 and I a nonempty interval of R, not bounded on the right.

Définition .3. Let f : I ×U

be a continuous application and a Cauchy-Lipschitz function, we associate the system : = f(x, t) (*), t 0 > 0, t ≥ t 0 , we have x et x(t, t 0 , x 0 ) denotes a solution of the system such that x(t 0 ) = x 0 . An equilibrium point such that for all t, f( , t) = 0 is (globally) attractive if the function φ(t, t 0 , x 0 ) tends to when t tends to + . Définition .4. We say that is an asymptotically stable equilibrium point, if it is a stable equilibrium point and if the domain of attraction of x 0 is a neighbourhood of x 0 . Définition .5. Let be a non-empty compact of U, we consider the system (*). We say that is a globally asymptotically stable equilibrium point for the system (*) if :

1. is stable on the system (*) 2. for all t 0 I, and x 0 U, x(t, t 0 , x 0 ) is defined for all t ≥ t 0 and Lemme 1. The number of Suspects S in the model (1) admits that : Proof : According to the model (1), we have the equation , avec S(0) = S 0 .

And it is obvious that :

Suppose that :

With the initial condition Z(0) = Z 0 = S 0 .

And we have Z(t), S(t) [0, + ], Z(0) = Z 0 = S 0 > 0 et t [0,+ [. Solving the equation ( 24) which is a first order linear ordinary differential equation in time t > 0 with Z(0) =Z 0 , we obtain :

By determining the limit of Z(t) when t goes to infinity, we obtain : Therefore Theorem .1. If then the equilibrium point E 0 of the system (1) is globally asymptotically stable.

Proof :

Using (23) for η > 0 then there exists T 1 > 0 such that , for t > T 1 . According to the system (1), we obtain :

Then for t > T 1 , we pose :

We obtain the matrix D defined in the following way :

We have the trace(D)

And for the determinant, we have :

If and η <<< 0, then det(D) > 0 implies . It comes : .

So

, in comparison with the system (1) ; , in comparison with the system (1) ; , in comparison with the system (1)

As

, for t > T 1 .

And if

, then and , for all θ > 0 and σ > 0, there exists T 2 > 0 such that : P (t) < θ and I(t) < σ, for t > T 2 .

Let , = max( , ), for t > , we obtain :

It comes :

Thus

We consider :

Solving this first-order linear differential equation, we obtain :

We pose It comes :

Therefore

As S(t),V (t) ([0, + ]) and S( ) = V ( ), it comes : S(t) ≥ V (t), for t >

This means that

From ( 26) and ( 27) if we choose , θ and σ very small and t> > , then we get :

. Passing to the limit :

Hence the equilibrium point E 0 of the system (1) is globally asymptotically stable.

Local stability of the equilibrium

By studying the local stability of , we obtain the following theorem :

Theorem .2. If R 0 > 1, then the endemic equilibrium of the system (1) is locally asymptotically stable. Proof :

According to the proposition in the works of L. Chahrazed (2002) and [START_REF] Chabour | Global stability of an SEIR epidemic model with constant immigration[END_REF], we characterize the local stability of , as follows :

Proposition 5. The epidemic is locally asymptotically stable if and only if all the eigenvalues of the Jacobian matrix J have a negative real part.

Indeed, J is defined by :

The eigenvalues can be determined by solving the equation det(J -θI) :

the characteristic function is written with the coefficients defined below as follows :

. The coefficients are : Using the Routh-Hurwitz criterion in the work of (J.M.M.ONDO ( 2012)), we have : AB -C > 0.

We calculate AB -C, we then obtain : Therefore is locally asymptotically stable.

We were first interested in the formulation of an endemic model describing the dynamic behaviour of the transmission of mutant microorganisms on the population. The study proposed in this model, as well as the different numerical simulations, allow us to know the moment when control efforts must be applied, as well as their intensity in the face of emerging infectious diseases. The SEPI endemic model described above is a deterministic model.

. 3 .

 3 The period of precontagiousness is a period of time when an infected individual does not yet show symptomatic signs of the disease, but can transmit the disease to another individual. Definition .4. The infected compartment represents those who are not only already infected and have shown symptoms of the disease, but are also capable of transmitting the disease back into the population.

FIG. 2 :

 2 FIG. 2: Scheme of the endemic model of SEPI 5 Differential equation representation of the SEPI endemic model According to the hypothesis (A7) in section (3) above, we consider that during the time interval dt, the Susceptible comparison has increased in number δN of new born. But at the same time, it loses the number of individuals exposed by the disease and the number µS of individuals who

A

  = µ + ν + λ + k + B = (ν + λ + k) + k(ν + λ) + νλ ; C = kνλ + ; D = kνλ + .We have A > 0 et B, C, D > 0.