Hamadjam Abboubakar
email: abboubakarhamadjam@yahoo.fr

Jean Claude Kamgang
email: jckamgang@gmail.com

Daniel Tieudjo
email: tieudjo@yahoo.com

Ensai

Backward Bifurcation and Control in Transmission Dynamics of Arboviral Diseases

Keywords: Compartmental model, Arboviral diseases, Vaccination, Vector control strategies, Stability, Backward bifurcation, Sensitivity analysis AMS Subject Classification (2010): 34D20, 34D23, 37N25, 92D30

In this paper, we derive and analyze a compartmental model for the control of arboviral diseases which takes into account an imperfect vaccine combined with individual protection and some vector control strategies already studied in the literature such as the use of adulticides, destruction of breeding site, and reduction of eggs and larvae through chemical interventions. After the formulation of the model, a qualitative study based on stability analysis and bifurcation theory reveals that the phenomenon of backward bifurcation may occur. The stable disease-free equilibrium of the model coexists with a stable endemic equilibrium when the reproduction number, R 0 , is less than unity. Using Lyapunov functions theory, we prove that the trivial equilibrium is globally asymptotically stable; when the disease-induced death is not considered, or/and, when the standard incidence is replaced by the mass action incidence, the backward bifurcation does not occur. Under a certain threshold condition, we establish the global asymptotic stability of the disease-free equilibrium of the full model.

Through sensitivity analysis, we determine the relative importance of model parameters for disease transmission. Numerical simulations show that the combination of several control mechanisms would significantly reduce the spread of the disease, if we maintain the level of each control high, and this, over a long period.

Introduction

Arboviral diseases are affections transmitted by hematophagous arthropods. There are currently 534 viruses registered in the International Catalog of Arboviruses and 25% of them have caused documented illness in human populations [START_REF] Chippaux | Généralités sur arbovirus et arboviroses overview of arbovirus and arbovirosis[END_REF][START_REF] Karabatsos | International Catalogue of Arboviruses, including certain other viruses of vertebrates[END_REF][START_REF] Gubler | Human arbovirus infections worldwide[END_REF]. Examples of those kinds of diseases are Dengue, Yellow fever, Saint Louis fever, Encephalitis, West Nile fever and Chikungunya.

A wide range of arboviral diseases are transmitted by mosquito bites and constitute a public health emergency of international concern. According to World Health Organisation (WHO), Dengue, caused by any of four closely-related virus serotypes (DEN-1-4) of the genus Flavivirus, causes 50-100 million infections worldwide every year, and the majority of patients worldwide are children aged 9 to 16 years [START_REF] Sanofi Pasteur | Dengue vaccine, a priority for global health[END_REF][START_REF]Dengue and severe dengue[END_REF][START_REF]Dengue and dengue haemorhagic fever[END_REF]. The dynamics of arboviral diseases like Dengue or Chikungunya are influenced by many factors such as human and mosquito behaviours. The virus itself (multiple serotypes of dengue virus [START_REF]Dengue and severe dengue[END_REF][START_REF]Dengue and dengue haemorhagic fever[END_REF], and multiple strains of chikungunya virus [START_REF] Moulay | Modélisation et analyse mathématique de systèmes dynamiques en épidémiologie. application au cas du chikungunya[END_REF][START_REF] Parola | Novel chikungunya virus variant in travelers returning from indian ocean islands[END_REF]), as well as the environment directly or indirectly affects all the present mechanisms of control [START_REF] Brasseur | Analyse des pratiques actuelles destinées á limiter la propagation d'aedes albopictus dans la zone sud est de la france et propositions d'amélioration[END_REF][START_REF] Carvalho | Mathematical modeling of dengue epidemic: Control methods and vaccination strategies[END_REF].

For all mentioned diseases, only yellow fever has a licensed vaccine. Nonetheless, considerable efforts are made to obtain the vaccines for other diseases. In the case of Dengue for example, the scientists of French laboratory SANOFI have conducted different tries in Latin America and Asia. Thus, a tetravalent vaccine could be quickly set up in the coming months.

The tries in Latin America have shown that vaccine efficacy was 64.7%. Serotype-specific vaccine efficacy was 50.3% for serotype 1, 42.3% for serotype 2, 74.0% for serotype 3, and 77.7% for serotype 4 [START_REF] Villar | Efficacy of a tetravalent dengue vaccine in children in latin america[END_REF]. The tries in Asia have shown that efficacy was 30.2%, and differed by serotype [START_REF] Sabchareon | Protective efficacy of the recombinant, live-attenuated, cyd tetravalent dengue vaccine in thai schoolchildren: a randomised, controlled phase 2b trial[END_REF]. In any case, it is clear that this vaccine will be imperfect.

Host-vector models for arboviral diseases transmission were proposed in [START_REF] Aldila | An optimal control problem arising from a dengue disease transmission model[END_REF][START_REF] Antonio | Optimal and sub-optimal control in dengue epidemics[END_REF][START_REF] Cannon | An epidemiology model suggested by yellow fever[END_REF][START_REF] Coutinho | Threshold conditions for a non-autonomous epidemic system describing the population dynamics of dengue[END_REF][START_REF] Cruz-Pacheco | Seasonality and outbreaks in west nile virus infection[END_REF][START_REF] Derouich | Dengue fever: mathematical modelling and computer simulation[END_REF][START_REF] Dumont | Vector control for the chikungunya disease[END_REF][START_REF] Esteva | Analysis of a dengue disease transmission model[END_REF][START_REF] Esteva | A model for dengue disease with variable human population[END_REF][START_REF] Feng | Competitive exclusion in a vector-host model for the dengue fever[END_REF][START_REF] Garba | Backward bifurcations in dengue transmission dynamics[END_REF][START_REF] Rodrigues | Vaccination models and optimal control strategies to dengue[END_REF][START_REF] Blayneha | Backward bifurcation and optimal control in transmission dynamics of west nile virus[END_REF][START_REF] Maidana | Dynamic of west nile virus transmission considering several coexisting avian populations[END_REF][START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF][START_REF] Moulay | Optimal control of chikungunya disease: larvae reduction,treatment and prevention[END_REF][START_REF] Poletti | Transmission potential of chikungunya virus and control measures: the case of italy[END_REF][START_REF] Abboubakar | Modeling the dynamics of arboviral diseases with vaccination perspective[END_REF] with the focus on the construction of the basic reproductive ratio and related stability analysis of the disease free and endemic equilibria. Some of these works in the literature focus on modelling the spread of arboviral diseases and its control using some mechanism of control like imperfect vaccines [START_REF] Garba | Backward bifurcations in dengue transmission dynamics[END_REF][START_REF] Rodrigues | Vaccination models and optimal control strategies to dengue[END_REF][START_REF] Abboubakar | Modeling the dynamics of arboviral diseases with vaccination perspective[END_REF] and other control tools like individual protection and vector control strategies [START_REF] Aldila | An optimal control problem arising from a dengue disease transmission model[END_REF][START_REF] Antonio | Optimal and sub-optimal control in dengue epidemics[END_REF][START_REF] Dumont | Vector control for the chikungunya disease[END_REF][START_REF] Blayneha | Backward bifurcation and optimal control in transmission dynamics of west nile virus[END_REF][START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF][START_REF] Moulay | Optimal control of chikungunya disease: larvae reduction,treatment and prevention[END_REF].

In [START_REF] Dumont | Vector control for the chikungunya disease[END_REF], Dumont and Chiroleu proposed a compartmental model to study the impact of vector control methods used to contain or stop the epidemic of Chikungunya of 2006 in Réunion island.

Moulay et al. [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF] studied an optimal control based on protection and vector control strategies to fight against Chikungunya. In [START_REF] Rodrigues | Vaccination models and optimal control strategies to dengue[END_REF], Rodrigues et al. simulate an hypothetical vaccine as an extra protection to the human population against epidemics of Dengue, using the optimal control theory. In those models [START_REF] Dumont | Vector control for the chikungunya disease[END_REF][START_REF] Rodrigues | Vaccination models and optimal control strategies to dengue[END_REF][START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF], (i) the population is constant, (ii) the disease-induced death in humans is not considered, (iii) the complete stage progression of development of vectors is not considered, (iv) none of the above mentioned models takes into account the combination of the mechanisms of control already studied in the literature, such as vaccination, individual protection and vector control strategies (destruction of breeding site, eggs and larvae reduction).

The aim of this work is to propose and study a arboviral disease control model which takes into account human immigration, disease-induced mortality in human communities, the complete stage structured model for vectors and a combination of human vaccination, individual protection and vector control strategies to fight against the spread of these kind of diseases.

We start with the formulation of a constant control model, which is an extension of the previous model developed in [START_REF] Abboubakar | Modeling the dynamics of arboviral diseases with vaccination perspective[END_REF]. We include the complete stage progression of development of vectors, the waning vaccine, and four others controls (individual protection, the use of adulticides, destruction of breeding site, and reduction of eggs and larvae through chemical interventions). We compute the net reproductive number N , as well as the basic reproduction number, R 0 , and investigate the existence and stability of equilibria. We prove that the trivial equilibrium is globally asymptotically stable whenever N < 1. When N > 1 and R 0 < 1, we prove that the system exhibit the backward bifurcation phenomenon. The implication of this occurrence is that the classical epidemiological requirement for effective eradication of the disease, R 0 < 1, is no longer sufficient, even though necessary. However, considering two situations: the model without vaccination and the model with mass incidence rates, we prove that the disease-induced death and the standard incidence functions, respectively, are the main causes of the occurrence of backward bifurcation. We find that the disease-free equilibrium is globally asymptotically stable under certain threshold condition. Through local and global sensitivity analysis, we determine the relative importance parameters of the model on the disease transmission. By using the pulse control technique (the control is not continuous in time order is effective only one day every T days [START_REF] Dumont | Vector control for the chikungunya disease[END_REF]) in numerical simulations, we evaluate the impact of different control combinations on the decrease of the spread of these diseases.

The paper is organized as follows. In Section 2 we present the transmission model and in Section 3 we carry out some analysis by determining important thresholds such as the net reproductive number N and the basic reproduction number R 0 , and different equilibria of the model. We then demonstrate the stability of equilibria and carry out bifurcation analysis. In section 4, both local and global sensitivity analysis are used to assess the important parameters in the spread of the diseases. Section 5 is devoted to numerical simulations. A conclusion rounds up the paper.

The formulation of the model

The model we propose here is an extension of the previous model studied in [START_REF] Abboubakar | Modeling the dynamics of arboviral diseases with vaccination perspective[END_REF], and is based on the modelling approach given in [START_REF] Dumont | Vector control for the chikungunya disease[END_REF][START_REF] Esteva | Analysis of a dengue disease transmission model[END_REF][START_REF] Esteva | A model for dengue disease with variable human population[END_REF][START_REF] Feng | Competitive exclusion in a vector-host model for the dengue fever[END_REF][START_REF] Garba | Backward bifurcations in dengue transmission dynamics[END_REF][START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF][START_REF] Moulay | Optimal control of chikungunya disease: larvae reduction,treatment and prevention[END_REF]. It is assumed that the human and vector populations are divided into compartments described by time-dependent state variables.

The compartments in which the populations are divided are the following ones:

-For humans, we consider susceptible (denoted by S h ), vaccinated (V h ), exposed (E h ), infectious (I h ) and resistant or immune (R h ); So that, N h = S h + V h + E h + I h + R h . Following Garba et al. [START_REF] Garba | Backward bifurcations in dengue transmission dynamics[END_REF] and Rodrigues et al. [START_REF] Rodrigues | Vaccination models and optimal control strategies to dengue[END_REF], we assume that the immunity, obtained by the vaccination process, is temporary. So, we denote by ω, the waning rate of vaccine. The recruitment in human population is at the constant rate Λ h , and newly recruited individuals enter the susceptible compartment S h . Are concerned by recruitment people that are totally naive from the disease. Each individual human compartment goes out from the dynamics at natural mortality rates µ h . The human susceptible population is decreased following infection, which can be acquired via effective contact with an exposed or infectious vector at a rate

λ h = aβ hv (η v E v + I v ) N h [23],
where a is the biting rate per susceptible vector, β hv is the transmission probability from an infected vector (E v or I v ) to a susceptible human (S h ). The expression of λ h is obtained as follows: the probability that a vector chooses a particular human or other source of blood to bite can be assumed as 1 N h . Thus, a human receives in average a N v N h bites per unit of times. Then, the infection rate per susceptible human is given by

aβ hv N v N h (η v E v + I v ) N v
. In expression of λ h , the modification parameter 0 < η v < 1 accounts for the assumed reduction in transmissibility of exposed mosquitoes relative to infectious mosquitoes [START_REF] Garba | Backward bifurcations in dengue transmission dynamics[END_REF][START_REF] Abboubakar | Modeling the dynamics of arboviral diseases with vaccination perspective[END_REF] (see the references therein for the specific sources). Latent humans (E h ) become infectious (I h ) at rate γ h . Infectious humans recover at a constant rate, σ or dies as consequence of infection, at a disease-induced death rate δ. After infection, immune humans retain their immunity for life.

-Following [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF], the stage structured model is used to describe the vector population dynamics, which consists of three main stages: embryonic (E), larvae (L) and pupae (P). Even if eggs (E) and immature stages (L and P) are all aquatic, it is important to dissociate them because, for the control point of view, drying the breeding sites does not kill eggs, but only larvae and pupae. Moreover, chemical interventions on the breeding sites has impact on the larvae population, but not on the eggs [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF]. The number of laid eggs is assumed proportional to the number of females. The system of stage structured model of aquatic phase development of vector is given by (see [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF] for details)

           Ė = µ b 1 - E Γ E (S v + E v + I v ) -(s + µ E )E L = sE 1 - L Γ L -(l + µ L )L Ṗ = lL -(θ + µ P )P
Unlike the authors of [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF], we take into account the pupal stage in the development of the vector. This is justified by the fact that they do not feed during this transitional stage of development, as they transform from larvae to adults [START_REF] Carvalho | Mathematical modeling of dengue epidemic: Control methods and vaccination strategies[END_REF][START_REF] Oliva | études biologiques et comportementales de deux espèces de moustiques (aedes albopictus et anopheles arabiensis) vectrices de maladies en vue du développement de la technique de linsecte stérile (tis) contre ces vecteurs à l'ile de la réunion[END_REF]. So, the control mechanisms cannot be applied to them.

With a rate θ, pupae become female Adults. Each individual vector compartment goes out from the dynamics at natural mortality rates µ v . The vector susceptible population is decreased following infection, which can be acquired via effective contact with an exposed or infectious human at a rate

λ v = aβ vh (η h E h + I h ) N h [23],
where β vh is the probability of transmission of infection from an infectious human (E h or I h ) to a susceptible vector (S v ). The modification parameter 0 ≤ η h < 1 accounts for the relative infectiousness of exposed humans in relation to infectious humans. Here too, it is assumed that susceptible mosquitoes can acquire infection from exposed humans [START_REF] Coutinho | Threshold conditions for a non-autonomous epidemic system describing the population dynamics of dengue[END_REF][START_REF] Garba | Backward bifurcations in dengue transmission dynamics[END_REF][START_REF] Abboubakar | Modeling the dynamics of arboviral diseases with vaccination perspective[END_REF]. Latent vectors (E v ) become infectious (I v ) at rate γ v . The vector population does not have an immune class, since it is assumed that their infectious period ends with their death [START_REF] Esteva | A model for dengue disease with variable human population[END_REF].

Then, we add new terms in the model to assess the different control tools studied:

(i) α 1 represents the efforts made to protect human from mosquitoes bites. It mainly consists to the use of mosquito nets or wearing appropriate clothes [START_REF] Moulay | Optimal control of chikungunya disease: larvae reduction,treatment and prevention[END_REF]. Thus we modify the infection term as follows:

λ c h = (1 -α 1 )λ h , and λ c v = (1 -α 1 )λ v , with 0 ≤ α 1 < 1; (1)
(ii) η 1 and η 2 are respectively, egg and larval mortality rates induced by chemical intervention, (iii) c m is the additional mortality rate due to adulticide, (iv) α 2 is the parameter associated with the efficacy of the mechanical control.

The above assumptions lead to the following non-linear system of ordinary differential equations

                                             Ṡh = Λ h + ωV h -(λ c h + ξ + µ h ) S h Vh = ξS h -[(1 -ǫ)λ c h + ω + µ h ] V h Ėh = λ c h [S h + (1 -ǫ)V h ] -(µ h + γ h )E h İh = γ h E h -(µ h + δ + σ)I h Ṙh = σI h -µ h R h Ṡv = θP -λ c v S v -(µ v + c m )S v Ėv = λ c v S v -(µ v + γ v + c m )E v İv = γ v E v -(µ v + c m )I v Ė = µ b 1 - E α 2 Γ E (S v + E v + I v ) -(s + µ E + η 1 )E L = sE 1 - L α 2 Γ L -(l + µ L + η 2 )L Ṗ = lL -(θ + µ P )P (2)
The description of state variables and parameters of model ( 2) are given in table 1 and table 2, respectively.

It is important to note that no intervention measure is performed to kill the pupae for two reasons: the first reason is the fact that at this stage, no food is absorbed by the insect, so it is impossible to make her ingested a toxic substance; the second reason is that the soluble products in the water deposits by contact are not selective for mosquito nymphs and act on all aquatic wildlife.

Hosts Adult vectors

Immature vectors

S h V h E h I h R h S v E v I v P L E
Figure 1: A compartmental model for the transmission dynamics of vector-borne diseases, which takes into account a waning vaccine and the complete phase of aquatic development of vector.

Well-posedness of the model

We now show that the system (2) is mathematically well defined and biologically feasible. Let us set

k 1 := ξ + µ h ; k 2 := ω + µ h ; k 3 := µ h + γ h ; k 4 := µ h + δ + σ; k 5 := s + µ E + η 1 ; k 6 := l + µ L + η 2 ; k 7 := θ + µ P ; k 8 := µ v + c m ; k 9 := µ v + γ v + c m ; K E := α 2 Γ E ; K L := α 2 Γ L ; π := 1 -ǫ, τ := a(1 -α 1 ). (3)
System (2) can be rewritten in the following way

dX dt = A(X)X + F (4) with X = (S h , V h , E h , I h , R h , S v , E v , I v , E, L, P ) T , A(X) = A 1 (X) 0 0 A 4 (X)
with

A 1 (X) =       -λ c h -k 1 ω 0 0 0 ξ -πλ c h -k 2 0 0 0 λ c h πλ c h -k 3 0 0 0 0 γ h -k 4 0 0 0 0 σ -µ h       and A 2 (X) =         -(λ c v + k 8 ) 0 0 0 0 θ λ c v -k 9 0 0 0 0 0 γ v -k 8 0 0 0 A 96 A 96 A 96 -A 97 0 0 0 0 0 A 109 -A 10 0 0 0 0 0 l -k 7        
where

A 96 = µ b 1 - E K E , A 97 = µ b N v K E + k 5 , A 109 = s 1 - L K L and A 10 = sE K L + k 6 ;
and F = (Λ h , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) T .

Note that A(X) is a Metzler matrix, i.e. a matrix such that off diagonal terms are non negative [START_REF] Berman | Nonnegative matrices in the mathematical sciences[END_REF][START_REF] Jacquez | Qualitative theory of compartmental systems[END_REF], for all X ∈ R 11 + . Thus, using the fact that F ≥ 0, system (4) is positively invariant in R 11 + , which means that any trajectory of the system starting from an initial state in the positive orthant R 11 + , remains forever in R 11 + . The right-hand side is Lipschitz continuous:

there exists a unique maximal solution.

By adding the first four equations of model system (2), it follows that

Ṅh (t) = Λ h -µ h N h -δI h ≤ Λ h -µ h N h
So that 0 ≤ N h (t) ≤ Λ h µ h + N h (0) - Λ h µ h e -µ h t Thus, at t -→ ∞, 0 ≤ N h (t) ≤ Λ h µ h .
By adding the equations in S v , E v and I v of system (2), it follows that

Ṅv (t) = θP -µ v N v So that 0 ≤ N v (t) = θP µ v + N v (0) - θP µ v e -µv t Thus, at t -→ ∞, 0 ≤ N v (t) ≤ θlK L µ v k 7 since P ≤ lK L k 7 .
Therefore, all feasible solutions of model system (2) enter the region:

D = (S h , V h , E h , I h , R h , S v , E v , I v , E, L, P ) ∈ R 11 + : N h ≤ Λ h µ h ; E ≤ K E ; L ≤ K L ; P ≤ lK L k 7 ; N v ≤ θlK L k 7 k 8 ,
which is a positively invariant set of system [START_REF] Karabatsos | International Catalogue of Arboviruses, including certain other viruses of vertebrates[END_REF].

Furthermore, the model ( 2) is well-posed epidemiologically and we will consider dynamic behaviour of model (2) on D.

Mathematical analysis

In this section, we focus on the qualitative analysis of our model. We determine the different equilibrium points which are governed by two epidemiological thresholds, the net reproductive number denoted by N and the basic reproduction number denoted by R 0 . Then, with respect on the values of these thresholds, we study the stability (local as well as global) of the different equilibrium points, and also we test the occurrence of the backward bifurcation phenomenon.

These stability analysis are also done for the different corresponding models (model without vaccination and model with mass-action incidences), to determine the causes of the occurrence of backward bifurcation phenomenon.

The disease-free equilibria and its stability

In the absence of disease in the both population (human and Adult vector), i.e.

λ c h = λ c v = 0 (or E h = I h = E v = I v = 0)
, we obtain two equilibria without disease: the trivial equilibrium (equilibrium without vector and disease) E 0 = (S 0 h , V 0 h , 0, 0, 0, 0, 0, 0, 0, 0, 0) and the disease-free equilibrium (equilibrium with vector and without disease) E 1 = (S 0 h , V 0 h , 0, 0, 0, N 0 v , 0, 0, E, L, P ) with

S 0 h = Λ h k 2 µ h (k 2 + ξ) , V 0 h = ξΛ h µ h (k 2 + ξ) , N 0 v = K E K L k 5 k 6 (N -1) µ b (K E s + k 6 K L ) , P = K E K L k 5 k 6 k 8 (N -1) µ b θ (K E s + k 6 K L ) , L = K E K L k 5 k 6 k 7 k 8 (N -1) µ b θl (K E s + k 6 K L ) , E = K E K L k 5 k 6 k 7 k 8 (N -1) s (µ b lK L θ + k 5 k 7 k 8 K E ) . ( 5
)
where N is the net reproductive number [START_REF] Moulay | The chikungunya disease: Modeling, vector and transmission global dynamics[END_REF][START_REF] Cushing | An Introduction to Structured Population Dynamics[END_REF][START_REF] Cushing | The net reproductive value and stability in matrix population models[END_REF] given by

N = µ b θls k 5 k 6 k 7 k 8 (6)

Local stability of disease-free equilibria

The local asymptotic stability result of equilibria E 0 and E 1 is given in the following.

Theorem 1. Define the basic reproductive number [START_REF] Diekmann | Mathematical Epidemiology of Infectious Diseases. Model building, analysis and interpretation[END_REF][START_REF] Van Den Driessche | Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF]

R 0 = a 2 (1 -α 1 ) 2 β hv β vh µ h k 5 k 6 (γ h + k 4 η h ) (γ v + k 8 η v ) (πξ + k 2 ) α 2 Γ E Γ L (N -1) k 3 k 4 k 8 k 9 µ b Λ h (ξ + k 2 ) (k 6 Γ L + sΓ E ) (7)
Then, (i) if N ≤ 1, the trivial equilibrium E 0 is locally asymptotically stable in D;

(ii) if N > 1, the trivial equilibrium is unstable and the disease-free equilibrium E 1 is locally asymptotically stable in D whenever R 0 < 1.

Proof. See appendix B.

The basic reproduction number of a disease is the average number of secondary cases that one infectious individual produces during his infectious period in a totally susceptible population. The epidemiological implication of Theorem 1 is that, in general, when the basic reproduction number, R 0 is less than unity, a small influx of infectious vectors into the community would not generate large outbreaks, and the disease dies out in time (since the DFE is locally asymptotically stable) [START_REF] Garba | Backward bifurcations in dengue transmission dynamics[END_REF][START_REF] Diekmann | Mathematical Epidemiology of Infectious Diseases. Model building, analysis and interpretation[END_REF][START_REF] Van Den Driessche | Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF][START_REF] Cruz-Pacheco | Modelling the dynamics of west nile virus[END_REF]. However, we will show in the subsection 3.2 that the disease may still persist even when R 0 < 1.

Global stability of the trivial equilibrium

The global stability of the trivial equilibrium is given by the following result:

Theorem 2. If N ≤ 1
, then E 0 is globally asymptotically stable on D.

Proof. To prove the global asymptotic stability of the trivial disease-free equilibrium E 0 , we use the direct Lyapunov method. To this aim, we set Y = X -T E with

X = (S h , V h , E h , I h , R h , S v , E v , I v , E, L, P )
T and rewrite (2) in the following manner

dY dt = B(Y )Y.
The global asymptotic stability of E 0 is achieved by considering the following Lyapunov function L(Y ) =< g, Y > where

g = 1, 1, 1, 1, 1, 1, 1, 1, k 8 µ b , k 5 k 8 µ b s , k 5 k 6 k 8 µ b sl .
See appendix C for the details.

Global stability of the disease-free equilibrium

We now turn to the global stability of the disease-free equilibrium E 1 . we prove that the diseasefree equilibrium E 1 is globally asymptotically stable under a certain threshold condition. To this aim, we use a result obtained by Kamgang and Sallet [START_REF] Kamgang | Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (dfe)[END_REF], which is an extension of some results given in [START_REF] Van Den Driessche | Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF]. Using the property of E 1 , it is possible to rewrite (2) in the following

manner ẊS = A 1 (X)(X S -X DF E ) + A 12 (X)X I ẊI = A 2 (X)X I (8)
where X S is the vector representing the state of different compartments of non transmitting individuals (S h , V h , R h , S v , E, L, P ) and the vector X I represents the state of compartments of different transmitting individuals (E h , I h , E v , I v ). Here, we have

X S = (S h , V h , R h , S v , E, L, P ) T , X I = (E h , I h , E v , I v ) T , X = (X S , X I ) and X DF E := E 1 = (S 0 h , V 0 h , 0, 0, 0, N 0 v , 0, 0, E, L, P ) T , A 1 (X) = A (1) 1 A (2) 1 A (3) 1 A (4) 1 , with A (1) 1 (X) =     -(λ c h + k 1 ) ω 0 0 ξ -(πλ c h + k 2 ) 0 0 0 0 -µ h 0 0 0 0 -(λ c v + k 8 )     , A (2)
1 (X) =     0 0 0 0 0 0 0 0 0 0 0 θ     , A (3)
1 (X) =     0 0 0 µ b 1 - E K E 0 0 0 0 0 0 0 0     , A (4)
1 (X) =       -k 5 + µ b S 0 v K E 0 0 s 1 - L K L -k 6 + sE * K L 0 0 l -k 7       , A 12 (X) =                  0 0 - τ β hv η v S 0 h N h - τ β hv S 0 h N h 0 0 0 0 0 - τ β hv η v πV 0 h N h - τ β hv πV 0 h N h 0 0 0 0 σ 0 0 0 0 0 - τ β vh η h S 0 v N h - τ β vh S 0 v N h 0 0 0 0 0 0 0 µ b 1 - E K E µ b 1 - E K E 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0                  , A 2 (X) =        -k 3 0 τ β hv η v (S h + πV h ) N h τ β hv (S h + πV h ) N h γ h -k 4 0 0 τ β vh η h S v N h τ β vh S v N h -k 9 0 0 0 γ v -k 8       
.

A direct computation shows that the eigenvalues of A 1 (X) have negative real parts. Thus the system ẊS = A 1 (X)(X S -X DF E ) is globally asymptotically stable at X DF E . Note also that A 2 (X) is a Metzler matrix.

We now consider the bounded set G:

G = (S h , V h , E h , I h , R h , S v , E v , I v , E, L, P ) ∈ R 11 + : S h ≤ N h , V h ≤ N h , E h ≤ N h , I h ≤ N h , R h ≤ N h , Nh = Λ h (µ h + δ) ≤ N h ≤ N 0 h = Λ h µ h ; E ≤ K E ; L ≤ K L ; P ≤ lK L k 7 ; N v ≤ θlK L k 7 k 8
Let us recall the following theorem [START_REF] Kamgang | Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (dfe)[END_REF] (See [START_REF] Kamgang | Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (dfe)[END_REF] for a proof in a more general setting).

Theorem 3. Let G ⊂ U = R 7 × R 4 . The system (2) is of class C 1 , defined on U. If
(1) G is positively invariant relative to (8).

(2) The system ẊS = A 1 (X)(X S -X DF E ) is Globally asymptotically stable at X DF E .

(3) For any x ∈ G, the matrix A 2 (x) is Metzler irreducible.

(4) There exists a matrix Ā2 , which is an upper bound of the set

M = {A 2 (x) ∈ M 4 (R) : x ∈ G} with the property that if A 2 ∈ M, for any x ∈ G, such that A 2 (x) = Ā2 , then x ∈ R 7 × {0}. (5) The stability modulus of Ā2 , α(A 2 ) = max λ∈sp(A 2 ) Re(λ) satisfied α(A 2 ) ≤ 0.
Then the DFE is GAS in G.

For our model system (2), conditions (1-3) of the theorem 3 are satisfied. An upper bound of the set of matrices M, which is the matrix Ā2 is given by

Ā2 =         -k 3 0 τ β hv η v (S 0 h + πV 0 h ) Nh τ β hv (S 0 h + πV 0 h ) Nh γ h -k 4 0 0 τ β vh η h S 0 v Nh τ β vh S 0 v Nh -k 9 0 0 0 γ v -k 8         , where Nh = Λ h (µ h + δ) .
To check condition (5) in theorem 3, we will use the useful lemma [START_REF] Kamgang | Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (dfe)[END_REF] in appendix A. To this aim, let

A = -k 3 0 γ h -k 4 , B =   τ β hv η v (S 0 h + πV 0 h ) Nh τ β hv (S 0 h + πV 0 h ) Nh 0 0   , C =   τ β vh η h S 0 v Nh τ β vh S 0 v Nh 0 0   , and D = -k 9 0 γ v -k 8 .
Clearly, A is a stable Metzler matrix. Then, after some computations, we obtain D-CA -1 B is a stable Metzler matrix if and only if

R c ≤ 1 (9)
where

R c = a 2 (1 -α 1 ) 2 β hv β vh k 5 k 6 (γ h + k 4 η h ) (γ v + k 8 η v ) K E K L (k 2 + πξ)(N -1) k 3 k 4 k 8 k 9 µ b (k 2 + ξ)(k 6 K L + K E s)Λ h (µ h + δ) 2 µ h . ( 10
)
We claim the following result

Theorem 4. If N > 1 and R c ≤ 1, then the disease-free equilibrium E 1 is globally asymptoti- cally stable in G. Remark 1. (i) From (10), we have R c = (µ h + δ) µ h R 0 > R 0 ,
showing that R c is not necessarily an optimal threshold parameter.

(ii) Note that inequality (9) is equivalent to

R 0 ≤ R G = µ h µ h + δ < 1, (11)
which means that for

R 0 ≤ R G , (12)
the disease-free equilibrium E 1 is the unique equilibrium (no co-existence with endemic equilibria). If

R G < R 0 ≤ 1, ( 13
)
then it is possible to have co-existence with endemic equilibria and thus, the occurrence of the backward bifurcation phenomenon.

(iii) Note that in the absence of disease-induced death, i.e. δ = 0, inequalities (9) and (11

) become R 0 ≤ 1. ( 14
)
This suggests that the disease-induced death may be a cause of the instability of the disease-free equilibrium E 1 when R 0 < 1, and thus, one cause of the occurrence of the backward bifurcation phenomenon.

Remark 2. The above results show that if, at any time, through appropriate interventions (eg the destruction of breeding sites, massive spraying, personal protection, vaccination, effective treatment,...), we are able to reduce N or R 0 and R c less than 1 for a sufficiently long period, then, the disease may disappear. The same remark is done in [START_REF] Dumont | Vector control for the chikungunya disease[END_REF].

The backward bifurcation phenomenon, in epidemiological systems, indicate the possibility of existence of at least two endemic equilibria when R 0 is less than unity. Thus, the classical requirement of R 0 < 1 is, although necessary, no longer sufficient for disease elimination [START_REF] Garba | Backward bifurcations in dengue transmission dynamics[END_REF][START_REF] Arino | Global results for an epidemic model with vaccination that exhibits backward bifurcation[END_REF][START_REF] Brauer | Backward bifurcations in simple vaccination models[END_REF][START_REF] Sharomi | Role of incidence function in vaccine-induced backward bifurcation in some hiv models[END_REF]. In some epidemiological models, it has been shown that the backward bifurcation phenomenon is caused by factors such as non-linear incidence (the infection force), diseaseinduced death and vaccine (perfect or not) [START_REF] Garba | Backward bifurcations in dengue transmission dynamics[END_REF][START_REF] Sharomi | Role of incidence function in vaccine-induced backward bifurcation in some hiv models[END_REF][START_REF] Buonomo | On the backward bifurcation of a vaccination model with nonlinear incidence[END_REF][START_REF] Buonomo | A note on the direction of the transcritical bifurcation in epidemic models[END_REF][START_REF] Dushoff | Backward bifurcations and catastrophe in simple models of fatal diseases[END_REF][START_REF] Safan | Vaccination based control of infections in sirs models with reinfection: special reference to pertussis[END_REF]. To confirm whether or not the backward bifurcation phenomenon occurs in this case, one could use the approach developed in [START_REF] Van Den Driessche | Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF][START_REF] Dushoff | Backward bifurcations and catastrophe in simple models of fatal diseases[END_REF][START_REF] Castillo-Chavez | Dynamical models of tuberculosis and their applications[END_REF], which is based on the general centre manifold theorem [START_REF] Guckenheimer | Dynamical Systems and Bifurcations of Vector Fields[END_REF]. We will explore this method in the next section.

Endemic equilibria and bifurcation analysis 3.2.1 Existence of endemic equilibria

We turn now to the existence of endemic equilibria. Let us introduce the following quantity R 1 = R 0,δ=0 . We prove the following result Theorem 5. We assume that N > 1, then (i) In the absence of disease-induced death in human population (δ = 0), model system (2) have 1. A unique endemic equilibrium whenever R 1 > 1.

No endemic equilibrium otherwise.

(ii) In presence of disease-induced death in human population (δ > 0), model system (2) could have 3. At least one endemic equilibrium whenever R 0 > 1.

4. Zero, one or more than one endemic equilibrium whenever R 0 < 1.

Proof. See appendix D.

Note that case 4 of Theorem 5 indicate the possibility of existence of at least one endemic equilibrium for R 0 < 1 and hence the potential occurrence of a backward bifurcation phenomenon.

Backward bifurcation analysis

In the following, we use the centre manifold theory [START_REF] Blayneha | Backward bifurcation and optimal control in transmission dynamics of west nile virus[END_REF][START_REF] Van Den Driessche | Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF][START_REF] Dushoff | Backward bifurcations and catastrophe in simple models of fatal diseases[END_REF][START_REF] Castillo-Chavez | Dynamical models of tuberculosis and their applications[END_REF] to explore the possibility of backward bifurcation in (2). To do so, a bifurcation parameter β * hv is chosen, by solving for β hv from R 0 = 1, giving

β * hv = k 3 k 4 k 8 k 9 µ b Λ h (ξ + k 2 ) (k 6 K L + sK E ) a 2 (1 -α 1 ) 2 β vh µ h k 5 k 6 (γ h + k 4 η h ) (γ v + k 8 η v ) (πξ + k 2 ) K E K L (N -1) . ( 15
)
Let J β * hv denotes the Jacobian of the system (2) evaluated at the DFE (E 1 ) and with β hv = β * hv .

Thus,

J β * hv = J 1 J 2 J 3 J 4 , (16)
where

J 1 =       -k 1 ω 0 0 0 ξ -k 2 0 0 0 0 0 -k 3 0 0 0 0 γ h -k 4 0 0 0 0 σ -µ h       , J 4 =         -k 8 0 0 0 0 θ 0 -k 9 0 0 0 0 0 γ v -k 8 0 0 0 K 1 K 1 K 1 -K 2 0 0 0 0 0 K 3 -K 4 0 0 0 0 0 l -k 7         . J 2 =             0 - τ β * hv η v S 0 h N 0 h - τ β * hv S 0 h N 0 h 0 0 0 0 - τ β * hv πη v V 0 h N 0 h - τ β * hv πV 0 h N 0 h 0 0 0 0 τ β * hv η v H 0 N 0 h τ β * hv H 0 N 0 h 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0             , J 3 =             0 0 - τ β vh η h S 0 v N 0 h - τ β vh S 0 v N 0 h 0 0 0 τ β vh η h S 0 v N 0 h τ β vh S 0 v N 0 h 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0             , with H 0 = S 0 h + πV 0 h , K 1 = µ b 1 - E * K E , K 2 = k 5 + µ b K E S 0 v . K 3 = s 1 - L * K L , and
K 4 = k 6 + sE * K L .
Note that the system (2), with β hv = β * hv , has a hyperbolic equilibrium point (i.e., the linearised system (2) has a simple eigenvalue with zero real part and all other eigenvalues have negative real part). Hence, the centre manifold theory [START_REF] Guckenheimer | Dynamical Systems and Bifurcations of Vector Fields[END_REF][START_REF] Carr | Applications of Centre Manifold Theory[END_REF] can be used to analyze the dynamics of the model (2) near β hv = β * hv . The technique in Castillo-Chavez and Song (2004) [START_REF] Castillo-Chavez | Dynamical models of tuberculosis and their applications[END_REF] entails finding the left and right eigenvectors of the linearised system above as follows.

The left eigenvector components of J β * hv , which correspond to the uninfected states are zero (see Lemma 3 in [START_REF] Van Den Driessche | Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF]). Thus a non-zero components correspond to the infected states. It follows that the matrix J β * hv has a left eigenvector given by v = (v 1 , v 2 , . . . , v 11 ), where

v 1 = v 2 = v 5 = v 6 = v 9 = v 10 = v 11 = 0; v 3 = k 8 N 0 h a(1 -α 1 )β * hv H 0 v 8 ; v 4 = a(1 -α 1 )β vh S 0 v (η v k 8 + γ v ) k 4 k 9 N 0 h v 8 , v 7 = (η v k 8 + γ v ) k 9 v 8 , v 8 > 0.
The system (2) has a right eigenvector given by w = (w 1 , w 2 , . . . , w 11 ) T , where

w 11 > 0, w 8 > 0, w 10 = k 7 l w 11 , w 9 = K 1 θ k 5 k 8 w 11 , w 7 = k 8 γ v w 8 , w 6 = θ k 8 w 11 - k 9 γ v w 8 , w 5 = γ h σk 8 k 9 N 0 h a(1 -α 1 )β vh µ h γ v S 0 v (η h k 4 + γ h ) w 8 , w 4 = µ h σ w 5 , w 3 = k 4 γ h w 4 , w 2 = - a(1 -α 1 )β * hv (η v k 8 + γ v ) γ v N 0 h (k 1 k 2 -ξω) (ξS 0 h + k 1 V 0 h )w 8 , w 1 = ω k 1 w 2 - a(1 -α 1 )β * hv S 0 h k 1 N 0 h (η v w 7 + w 8 ) .
Theorem 4.1 in Castillo-Chavez and Song [START_REF] Castillo-Chavez | Dynamical models of tuberculosis and their applications[END_REF] is then applied to establish the existence of backward bifurcation in [START_REF] Karabatsos | International Catalogue of Arboviruses, including certain other viruses of vertebrates[END_REF]. To apply such a theorem, it is convenient to let f k represent the right-hand side of the k th equation of the system (2) and let x k be the state variables whose derivative is given by the k th equation for k = 1, . . . , 11. The local bifurcation analysis near the bifurcation point (β hv = β * hv ) is then determined by the signs of two associated constants, denoted by A 1 and A 2 , defined by

A 1 = n k,i,j=1 v k w i w j ∂ 2 f k (0, 0) ∂x i ∂x j and A 2 = n k,i=1 v k w i ∂ 2 f k (0, 0) ∂x i ∂φ (17)
with φ = β hvβ * hv . It is important to note that in f k (0, 0), the first zero corresponds to the disease-free equilibrium, E 1 , for the system (2). Since

β hv = β * hv is the bifurcation parameter, it follows from φ = β hv -β * hv that φ = 0 when β hv = β * hv which is the second component in f k (0, 0).
It follows then, after some algebraic computations, that

A 1 = Γ 1 -Γ 2 with Γ 1 = τ β * hv (2V 0 h w 1 + πS 0 h w 2 ) (N 0 h ) 2 (η v w 7 + w 8 )v 3 + τ β vh S 0 v N 0 h (η h w 3 + w 4 ) 1 S 0 v + η h w 3 + 1 S 0 v w 4 w 6 v 7 , Γ 2 = 2 τ β vh S 0 v (N 0 h ) 2 5 i=1 w i (η h w 3 + w 4 )v 7 + τ β * hv (S 0 h + πV 0 h )(N 0 h + 1) (N 0 h ) 2 5 i=3 w i (η v w 7 + w 8 )v 3 ,
and

A 2 = a(S 0 h + πV 0 h ) N 0 h (η v w 7 + w 8 ) v 3 .
Note that the coefficient A 2 is always positive. Thus, using Theorem 4.1 in [START_REF] Castillo-Chavez | Dynamical models of tuberculosis and their applications[END_REF], the following result is established.

Theorem 6. The model (2) exhibits a backward bifurcation at R 0 = 1 whenever A 1 > 0. If the reversed inequality holds, then the bifurcation at R 0 = 1 is forward.

The associated bifurcation diagrams are depicted in Figures 2 and3. Parameter values used in figure 2 correspond to those in Table 3 Parameter values used in figure 3 correspond to those in Table 3, except Λ h = 10, β vh = 0.8, 2) of the number of infected humans, E h , and the number of infected vectors, E v , for parameter values given in the bifurcation diagram in Figure 2 with β hv = 0.0105. So R 0 = 0.29 < 1, for two different set of initial conditions. The first set of initial conditions (corresponding to the solid trajectory) is

, except Λ h = 10, ǫ = 1, β vh = 0.8, η h = 1, η v = 1, σ = 0.01428, δ = 1, α 1 = 0.001, α 2 = 1, c m = 0.0001, Γ E =
η h = η v = 0 = δ = c m = α 1 = 0, α 2 = 1, Γ E =
S h = 700, V h = 10, E h = 220, I h = 100, R h = 60, S v = 3000, E v = 400, I v = 120, E = 10000, L = 5000 and P = 3000.
The second set of initial conditions (corresponding to the dotted trajectory) is

S h = 489100, V h = 10, E h = 220, I h = 100, R h = 60, S v = 3000, E v = 400, I v = 120, E =
10000, L = 5000 and P = 3000. The solution for initial condition 1 approaches the locally asymptotically stable endemic equilibrium point, while the solution for initial condition 2 approaches the locally asymptotically stable DFE.

The different causes of the backward bifurcation

From theorem 5, item (i), it follows that the disease-induced death in human (δ) may be a cause of the occurrence of the backward bifurcation phenomenon. In the following, we prove that the backward bifurcation phenomenon is caused by the disease-induced death in human and/or the standard incidence functions (λ c h and λ c v ). To this aim, we will consider, in this section, two variants of the model ( 2): the corresponding model without vaccination and the corresponding model with mass action incidence.

Analysis of the model without vaccination

The model without vaccination is given by

                                         Ṡh = Λ h -(λ c h + µ h ) S h Ėh = λ c h S h -(µ h + γ h )E h İh = γ h E h -(µ h + δ + σ)I h Ṙh = σI h -µ h R h Ṡv = θP -λ c v S v -(µ v + c m )S v Ėv = λ c v S v -(µ v + γ v + c m )E v İv = γ v E v -(µ v + c m )I v Ė = µ b 1 - E α 2 Γ E (S v + E v + I v ) -(s + µ E + η 1 )E L = sE 1 - L α 2 Γ L -(l + µ L + η 2 )L Ṗ = lL -(θ + µ P )P (18)
where λ c h and λ c v are given at (1). Model system [START_REF] Derouich | Dengue fever: mathematical modelling and computer simulation[END_REF] is defined in the positively-invariant set

D 1 = (S h , E h , I h , R h , S v , E v , I v , E, L, P ) ∈ R 10 : N h ≤ Λ h /µ h ; E ≤ K E ; L ≤ K L ; P ≤ lK L k 7 ; N v ≤ θlK L k 7 k 8 .
Without lost of generality, we assume that N > 1. The corresponding disease-free equilibria of model ( 18) are given by E nv 0 = (N 0 h , 0, 0, 0, 0, 0, 0, 0, 0, 0) which correspond to the trivial equilibrium and E nv 1 = (N 0 h , 0, 0, 0, N 0 v , 0, 0, E, L, P ) which correspond to the disease-free equilibrium, with N 0 h = Λ h µ h and N 0 v , E, L and P are the same, given by ( 5). The associated next generation matrices, F 1 and V 1 , are, respectively, given by

F 1 =       0 0 τ β hv η v τ β hv 0 0 0 0 τ β vh η v N 0 v N 0 h τ β vh N 0 v N 0 h 0 0 0 0 0 0       and V 1 =     k 3 0 0 0 -γ h k 4 0 0 0 0 k 9 0 0 0 -γ v k 8     .
It follows that the basic reproduction number for the model without vaccination, denoted by

R nv = ρ(F 1 V -1 1 )
, is given by

R nv = a 2 (1 -α 1 ) 2 β hv β vh (γ h + k 4 η h )(γ v + k 8 η v )N 0 v k 3 k 4 k 8 k 9 N 0 h . (19)
Using Theorem 2 of [START_REF] Van Den Driessche | Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF], we establish the following result:

Theorem 7. Assumed that N > 1.
For basic arboviral model without vaccination, given by (18), the corresponding disease-free equilibrium is locally asymptotically stable (LAS) if

R nv < 1, and unstable if R nv > 1.
Existence of endemic equilibria. Here, the existence of endemic equilibria of the model (18) will be explored. Let us set the following coefficients

R c 1 = {2k 8 (k 3 k 4 -δγ h ) + (η h k 4 + γ h )aµ h (1 -α 1 )β vh } k 3 k 4 k 8 , d 2 = -k 9 µ b Λ h (sK E + k 6 K L ) (k 3 k 4 -δγ h ) ((η h k 4 + γ h )aµ h (1 -α 1 )β vh + (k 3 k 4 -δγ h )k 8 ) < 0, d 1 = k 2 3 k 2 4 k 8 k 9 (sK E + k 6 K L )µ b Λ h µ h (R 2 nv -R 2 c 1 ), d 0 = k 2 3 k 2 4 k 8 k 9 (sK E + k 6 K L )µ b Λ h µ 2 h R 2 nv -1 . (20)
We claim the following:

Theorem 8. The arboviral diseases model without vaccination [START_REF] Derouich | Dengue fever: mathematical modelling and computer simulation[END_REF] has:

(i) A unique endemic equilibrium if d 0 > 0 ⇔ R nv > 1;
(ii) A unique endemic equilibrium if d 1 > 0, and

d 0 = 0 or d 2 1 -4d 2 d 0 = 0; (iii) Two endemic equilibria if d 0 < 0 (R nv < 1), d 1 > 0 (R nv > R c 1 ) and d 2 1 -4d 2 d 0 > 0;
(iv) No endemic equilibrium otherwise.

Proof. Solving the equations in the model [START_REF] Derouich | Dengue fever: mathematical modelling and computer simulation[END_REF] in terms of λ c, * h and λ c, * v , gives

S * h = Λ h µ h + λ c, * h , E * h = λ c, * h S * h k 3 , I * h = γ h λ c, * h S * h k 3 k 4 , R * h = σγ h λ c, * h S * h µ h k 3 k 4 , (21)
and

S * v = θP (λ c, * v + k 8 ) , E * v = θP λ c, * v k 9 (λ c, * v + k 8 ) , I * v = γ v θP λ c, * v k 8 k 9 (λ c, * v + k 8 ) , E = µ b θK E P (k 5 k 8 K E + µ b θP ) , L = µ b θsK E K L P k 6 K L (k 5 k 8 K E + µ b θP ) + sµ b θK E P , (22)
Substituting ( 21) and ( 22) into the expression of λ c, * h and λ c, * v and simplifying, lead the nonzero equilibria of the model without vaccination satisfy the quadratic equation

d 2 (λ c, * h ) 2 + d 1 λ c, * h + d 0 = 0 (23)
where d i , i = 0, 1, 2, are given by [START_REF] Esteva | Analysis of a dengue disease transmission model[END_REF].

Clearly, d 2 < 0 and d 0 > 0 (resp.

d 0 < 0) if R nv > 1 (resp. R nv < 1)
. Thus Theorem 8 is established.

It is clear that cases (ii) and (iii) of theorem 8 indicate the possibility of backward bifurcation (where the locally-asymptotically stable DFE co-exists with a locally asymptotically stable endemic equilibrium when R nv < 1) in the model without vaccination [START_REF] Derouich | Dengue fever: mathematical modelling and computer simulation[END_REF]. This is illustrated by numerical simulation of the model with the following set of parameter values (it should be noted that these parameters are chosen for illustrative purpose only, and may not necessarily be realistic epidemiologically):

Λ h = 5, β hv = 0.03, η h = η v = 1, δ = 1, σ = 0.01, c m = 0.1, β vh = 0.4, α 1 = 0.
7 and α 2 = 0.5. All other parameters are as in Table 3. With

this set of parameters, R c 1 = 0.0216 < 1, R nv = 0.2725 < 1 (so that R c 1 < R nv < 1)
. It follows: The associated bifurcation diagram is depicted in figure 5. This clearly shows the coexistence of two locally-asymptotically stable equilibria when R nv < 1, confirming that the model without vaccination [START_REF] Derouich | Dengue fever: mathematical modelling and computer simulation[END_REF] undergoes the phenomenon of backward bifurcation too. Thus, the following result is established. Two endemic equilibrium points coexist for values of R 0 in the range (0.2286, 1) (corresponding to the range (0.0211, 0.4040) of β hv ). The notation EE and DFE stand for endemic equilibrium and disease-free equilibrium, respectively. Solid lines represent stable equilibria and dash lines stand for unstable equilibria.

d 2 = -0.0263 < 0, d 1 = 4.8763 × 10 -4 and d 0 = -3.5031 × 10 -7 , so that d 2 1 -4d 2 d 0 = 2.0093 × 10 -7 > 0. The resulting two endemic equilibria E nv = (S * h , E * h , I * h , R * h , S * v , E * v , I * v , E
Non-existence of endemic equilibria for R nv < 1 and δ = 0. In this case, we have the following result.

Lemma 2. The model [START_REF] Derouich | Dengue fever: mathematical modelling and computer simulation[END_REF] without disease-induced death (δ = 0) has no endemic equilibrium when R nv,δ=0 ≤ 1, and has a unique endemic equilibrium otherwise.

Proof. Considering the model [START_REF] Derouich | Dengue fever: mathematical modelling and computer simulation[END_REF] without disease-induced death in human, and applying the same procedure, we obtain that the non-zero equilibria of the model without vaccination satisfy the linear equation

p 1 λ c, * h + p 0 = 0, where p 1 = k 9 k 10 K 12 aµ b Λ h µ h (1 -α 1 )β vh + k 3 (µ h + σ)k 8 k 9 K 12 µ b Λ h and p 0 = -µ h k 3 k 4 k 8 k 9 K 12 µ b Λ h R 2 nv,δ=0 -1 .
Clearly, p 1 > 0 and p 0 ≥ 0 whenever R nv,δ=0 ≤ 1, so that λ c, * h = -p 0 p 1 ≤ 0. Therefore, the model ( 18) without disease-induced death in human, has no endemic equilibrium whenever

R nv,δ=0 ≤ 1.
The above result suggests the impossibility of backward bifurcation in the model [START_REF] Derouich | Dengue fever: mathematical modelling and computer simulation[END_REF] without disease-induced death, since no endemic equilibrium exists when R nv,δ=0 < 1 (and backward bifurcation requires the presence of at least two endemic equilibria when R nv,δ=0 < 1) [START_REF] Garba | Backward bifurcations in dengue transmission dynamics[END_REF][START_REF] Sharomi | Role of incidence function in vaccine-induced backward bifurcation in some hiv models[END_REF].

To completely rule out backward bifurcation in model ( 18), we use the direct Lyapunov method to prove the global stability of the DFE.

Global stability of the DFE of (18) for δ = 0.

Define the positively-invariant and attracting region

D 2 = (S h , E h , I h , R h , S v , E v , I v , E, L, P ) ∈ D 1 : S h ≤ N 0 h ; S v ≤ N 0 v . We claim the following result. Theorem 9. The DFE, E nv 1 , of model (18) without disease-induced death (δ = 0), is globally asymptotically stable (GAS) in D 2 if R nv,δ=0 < 1.
Proof. See appendix E.

Analysis of the model with mass action incidence

Consider the model (2) with mass action incidence. Thus, the associated forces of infection, λ c h and λ c v , respectively, reduce to

λ mh = C h (η v E v + I v ) and λ mv = C v (η h E h + I h ), (24)
where, C h = a(1α 1 )β hv and C v = a(1α 1 )β vh . The resulting model (mass action model), obtained by using ( 24) in (2), has the same disease-free equilibria given by [START_REF]Dengue and severe dengue[END_REF]. Without lost of generality, we consider that N > 1. The associated next generation matrices, F m and V m are given by

F m =     0 0 C h η v H 0 C h H 0 0 0 0 0 C v η v S 0 v C v S 0 v 0 0 0 0 0 0     , V m =     k 3 0 0 0 -γ h k 4 0 0 0 0 k 9 0 0 0 -γ v k 8     ,
where H 0 = S 0 h + πV 0 h . It follows that the associated reproduction number for the mass action model, denoted by R 0,m = ρ(F m V -1 m ), is given by

R 0,m = R m hv R m vh , (25)
where

R m hv = C h Λ h (γ h + k 4 η h ) (πξ + k 2 ) µ h k 3 k 4 (ξ + k 2 ) and R m vh = C v (γ v + k 8 η v ) θP k 2 8 k 9 .
Using Theorem 2 of [START_REF] Van Den Driessche | Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF], the following result is established:

Theorem 10. Assume that N > 1.
For the arboviral disease model with mass action incidence, given by (2) with (24), the DFE,

E 1 , is LAS if R 0,m < 1, and unstable if R 0,m > 1.
Existence of endemic equilibria. Solving the equations in the model ( 2) with ( 24) in terms of λ * mh and λ * mv , gives

S * mh = Λ h (πλ c, * mh + k 2 ) λ c, * mh (k 2 + π(k 1 + λ c, * mh )) + k 1 k 2 -ωξ , V * mh = ξS * mh k 2 + πλ c, * mh , E * mh = λ c, * mh S * mh k 3 , I * mh = γ h λ c, * h S * mh k 3 k 4 , R * mh = σγ h λ c, * mh S * mh µ h k 3 k 4 , (26)
and

S * mv = θP (λ c, * mv + k 8 ) , E * mv = θP λ c, * mv k 9 (λ c, * mv + k 8 ) , I * mv = γ v θP λ c, * mv k 8 k 9 (λ c, * mv + k 8 ) . (27)
Substituting ( 26) and ( 27) in the expression of λ * mh and λ * mv and simplifying, shows that the nonzero equilibria of the model ( 2) with ( 24) satisfy the quadratic equation

e 2 (λ c, * mh ) 2 + e 1 λ c, * mh + e 0 = 0, (28)
where e i , i = 0, 1, 2, are given by

e 2 = k 8 k 9 π [(γ h + k 4 η h ) C v Λ h + k 3 k 4 k 8 ] e 1 = k 3 k 4 k 2 8 k 9 κπ (πξ + k 2 ) R 2 cm -R 2 0,m , e 0 = k 3 k 4 k 2 8 k 9 κ 1 -R 2 0,m , with κ = k 1 k 2 -ξω > 0 and R cm = [(γ h + k 4 η h ) (πξ + k 2 )Λ h C v + (k 1 π + k 2 )k 3 k 4 k 8 ] (πξ + k 2 ) k 3 k 4 k 8 κπ .
e 2 is always positive and e 0 is positive (resp. negative) whenever R 0,m is less (resp. greater) than unity. Thus, the mass action model admits only one endemic equilibrium whenever R 0,m > 1.

Now, we consider the case R 0,m < 1. The occurrence of backward bifurcation phenomenon depend of the sign of coefficient e 1 . The coefficient e 1 is always positive if and only if R 0,m < R cm . It follows that the disease-free equilibrium is the unique equilibrium when N > 1 and

R cm < 1. Now if R cm < R 0,m < 1
, then in addition to the DFE E 1 , there exists two endemic equilibria whenever ∆ m = e 2 1 -4e 2 e 0 > 0. However,

R cm < R 0,m < 1 ⇒ R cm < 1 ⇔ β vh < - k 3 k 4 k 8 (ξωπ + k 1 π 2 ξ + k 2 (πξ + k 2 )) a(1 -α 1 ) (γ h + k 4 η h ) (πξ + k 2 )(πξ + k 2 )Λ h < 0.
Since all parameter of model are nonnegative, we conclude that the condition R cm < R 0,m < 1 does not hold. And thus, the model with mass-action incidence does not admit endemic equilibria for R 0,m < 1. We claim the following:

Lemma 3. The arboviral diseases model (2) with mass-action incidences (24) has:

(i) a unique endemic equilibrium if R 0,m > 1;

(ii) no endemic equilibrium otherwise.

Global stability of the DFE for the model with mass action incidence.

Since the DFE of the model with mass action incidence is the unique equilibrium whenever the corresponding basic reproduction number R 0,m is less than unity, it remains to show that the DFE is GAS. To this aim, we use the direct Lyapunov method. Let us define the following positive constants:

p 1 = 1 k 3 , p 2 = C h H 0 (η v k 8 + γ v ) k 8 k 9 C v S 0 v k 3 k 4 , p 3 = p 1 C h H 0 (η v k 8 + γ v ) k 8 k 9 , p 4 = C h H 0 k 3 k 8 .
Consider the Lyapunov function

L = p 1 E h + p 2 I h + p 3 E v + p 4 I v .
The derivative of L is given by

L = p 1 Ėh + p 2 İh + p 3 Ėv + p 4 İv = (p 1 C h η v H + p 4 γ v -p 3 k 9 )E v + (p 1 C h H -p 4 k 8 )I v + (p 3 C v η h S v + p 2 γ h -p 1 k 3 )E h + (p 3 C v S v -p 2 k 4 )I h
Replacing p i , i = 1, . . . 4, by their respective term, and using the fact that H = (S h + πV h ) ≤

H 0 = (S 0 h + πV 0 h ) and S v ≤ N 0 v in D 3 = (S h , V h , E h , I h , R h , S v , E v , I v , E, L, P ) ∈ D : N h ≤ Λ h µ h , S v ≤ N 0 v = θP, E ≤ K E , L ≤ K L , P ≤ lK L k 7 k 8 , we obtain L ≤ R 2 0,m -1 E h . We have L ≤ 0 if R 0,m ≤ 1, with L = 0 if R 0,m = 1 or E h = 0. Whenever E h = 0, we also have I h = 0, E v = 0 and I v = 0. Substituting E h = I h = E v = I v = 0 in the first, fourth
and fifth equation of Eq. ( 2) with mass action incidence [START_REF] Rodrigues | Vaccination models and optimal control strategies to dengue[END_REF] gives

S h (t) → S 0 h , V h (t) → V 0 h , R h (t) → 0, and S v (t) → S 0 v = N 0 v as t → ∞. Thus [S h (t), V h (t), E h (t), I h (t), R h (t), S v (t), E v (t), I v (t), E(t), L(t), P (t)] → (S 0 h , V 0 h , 0, 0, 0, N 0 v , 0, 0, E, L, P ) as t → ∞.
It follows from the LaSalle's invariance principle [START_REF] Hale | Ordinary Differential Equations[END_REF][START_REF] Lasalle | Stability theory for ordinary differential equations[END_REF][START_REF] Lasalle | The stability of dynamical systems[END_REF], that every solution of (2) with mass action incidence, with initial conditions in D 3 converges to the DFE, as t → ∞. Hence, the DFE, E 1 , of the model with mass action incidence, is GAS in

D 3 if R 0,m ≤ 1.
Thus, we claim the following result.

Theorem 11. The DFE, E 1 , of the model (2) with mass action incidence, is globally asymptotically stable (GAS) in

D 3 if R 0,m < 1.
Thus, the substitution of standard incidence with mass action incidence in the arboviral model (2) removes the backward bifurcation phenomenon of the model. It should be mentioned that a similar situation was reported by Garba et al. in [START_REF] Garba | Backward bifurcations in dengue transmission dynamics[END_REF] and by Sharomi et al. in [START_REF] Sharomi | Role of incidence function in vaccine-induced backward bifurcation in some hiv models[END_REF].

We summarize the previous analysis of subsection 3.3 as follows:

Lemma 4. The main causes of occurrence of backward bifurcation phenomenon in models [START_REF] Karabatsos | International Catalogue of Arboviruses, including certain other viruses of vertebrates[END_REF] and ( 18) are the disease-induced death and the standard incidence rates.

Sensitivity analysis

As shown in the previous sections, model [START_REF] Karabatsos | International Catalogue of Arboviruses, including certain other viruses of vertebrates[END_REF] may admit single or multiple steady states according to the value of the basic reproduction number R 0 . In turn, R 0 depends on the parameters of the model. The various uncertainties encountered in data collection and the estimated values lead us to evaluate the robustness of the model predictions with the parameter values and, in particular, to estimate the effect on R 0 of varying single parameter. To this aim, we use sensitivity analysis and calculate the sensitivity indices of R 0 to the parameters in the model using both local and global methods.

Local sensitivity analysis

The local sensitivity analysis, based on the normalised sensitivity index of R 0 (see [START_REF] Chitnis | Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model[END_REF]), is given by

S Ψ = Ψ R 0 ∂R 0 ∂Ψ
where Ψ denotes the generic parameter of (2).

This index indicates how sensitive R 0 is to changes of parameter Ψ. Clearly, a positive (resp.

negative) index indicates that an increase in the parameter value results in an increase (resp.

decrease) in the R 0 value [START_REF] Chitnis | Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model[END_REF].

For instance, the computation of the sensitivity index of R 0 with respect to a is given by

S a = a R 0 ∂R 0 ∂a = 1 > 0.
This shows that R 0 is an increasing function of a and the parameter a has an influence on the spread of disease.

We tabulate the indices of the remaining parameters in Table 2 using parameter values on Table 3. The results are displayed in Table 4 and Figure 7a. The parameters are arranged from most sensitive to least. The model system ( 2) is most sensitive to a, the average number of mosquitoes bites, followed by µ v , ǫ, s, Λ h , β hv , β vh , Γ E , Γ L and α 2 . It is important to note that increasing (decreasing) a by 10% increases (decreases) R 0 by 10%. However, increasing (decreasing) the parameters µ v by 10% decreases (increases) R 0 by 9.190%. The same reasoning can be done for other parameters.

Uncertainty and global sensitivity analysis

Local sensitivity analysis assesses the effects of individual parameters at particular points in parameter space without taking into account of the combined variability resulting from considering all input parameters simultaneously. Here, we perform a global sensitivity analysis to examine the model's response to parameter variation within a wider range in the parameter space.

Following the approach by Marino et al. [START_REF] Marino | A methodology for performing global uncertainty and sensitivity analysis in systems biology[END_REF] and Wu et al. [START_REF] Wu | Sensitivity analysis of infectious disease models: methods, advances and their application[END_REF], partial rank correlation coefficients (PRCC) between the basic reproduction number R 0 and each parameter are derived from 5,000 runs of the Latin hypercube sampling (LHS) method [START_REF] Stein | Large sample properties of simulations using latin hypercube sampling[END_REF]. The parameters are assumed to be random variables with uniform distributions with their mean value listed in Table 3.

With these 5,000 runs of LHS, the derived distribution of R 0 is given in Figure 6. This sampling shows that the mean of R 0 is 2.0961 and the standard deviation is 2.7910. This implies that for the mean of parameter values given in Table 3, we may be confident that the model predicts an endemic state. The probability that R 0 > 1 (the disease-free equilibrium is unstable and there is exactly one endemic equilibrium point) is 55.60%. We also evaluate the probabilities that condition in Theorem 4 and inequalities [START_REF] Villar | Efficacy of a tetravalent dengue vaccine in children in latin america[END_REF], ( 12) and ( 13) are satisfied. Let us set P[X] the probability of X, and the sets of parameter values for which (N > 1) is true by Φ,

P [¬Φ] = P [N ≤ 1] = 0.0096, (29a)
P [Φ and (R 0 ≤ 1)] = 0.4440, (29b)

P [Φ and (R 0 ≤ R G )] = 0.0220, (29c)
P [Φ and (R G < R 0 ≤ 1)] = 0.4220. ( 29d
)
Therefore, the probability that the trivial disease-free equilibrium is locally asymptotically stable is 0.0096 (from (29a)), the probability that the disease free equilibrium point is locally asymptotically stable is 0.440 (from (29b)), the probability that the disease free equilibrium point is locally asymptotically stable and (i) there are no endemic equilibrium points is 0.0220 ((29c)); and (ii) there are two endemic equilibrium points is 0.4220 (from (29d)). This implies that for the ranges of parameter values given in Table 3, the disease-free equilibrium point is likely to be locally asymptotically stable but, the probability of co-existence of a locally asymptotically stable endemic equilibrium point (occurrence of backward bifurcation phenomenon) is very significant.

We now use sensitivity analysis to analyze the influence of each parameter on the basic reproductive number. From the previously sampled parameter values, we compute the PRCC between R 0 and each parameter of model ( 2). The parameters with large PRCC values (> 0.5 or < -0.5) statistically have the most influence [START_REF] Wu | Sensitivity analysis of infectious disease models: methods, advances and their application[END_REF]. The results, displayed in Table 5 and We note that the order of the most important parameters for R 0 from the local sensitivity analysis does not match that of the global sensitivity analysis, showing that the local results are not robust, and depend of the parameters values. So, for this kind of situation, we must to consider the results of the global sensitivity analysis. kill the eggs and larvae, and mechanical control to reduce the number of breeding sites at least near habited areas) [START_REF] Dumont | Vector control for the chikungunya disease[END_REF]. Here, we investigate and compare numerical results, with the different scenario. We use the following initial state variables S h (0) = 700, V h (0) = 10, E h (0) = 220,

I h (0) = 100, R h (0) = 60, S v (0) = 3000, E v (0) = 400, I v (0) = 120, E(0) = 10000, L = 5000, P = 3000.

Strategy A: Vaccination combined with individual protection only

In this strategy, we consider the model ( 2) without vector control. we set α 2 = 1 and c m = η 1 = η 2 = 0 and vary the parameter related to individual protection, namely α 1 , between 0 and 0.8.

The values of other parameters are given in Table 3. Figure 8 shows that the increase of the individual protection level, permit to reduce the total number of infected humans, and the total number of infected vectors, but has no impact on the populations of eggs and larvae. However, from this figure, it is clear that, this reduction is significant if the level of protection must turn around 80% at least, and this, over a long period. Thus, continuous education campaigns of local populations, on how to protect themselves individually, are important in the fight against the spread of arboviral diseases. 3.

Strategy B: Vaccination combined with adulticide

Nowadays, Deltamethrin is the most used insecticide for impregnation of bednets, because it is a highly effective compound on mosquitoes, and this, at of very low doses [START_REF] Darriet | Insecticides larvicides et adulticides alternatifs pour les opérations de démoustication en france, synthèse bibliographique[END_REF]. However, when sprayed in an open environment, Deltamethrin seems to be effective only during a couple of hours [START_REF] Dumont | Vector control for the chikungunya disease[END_REF][START_REF] Bosc | Premier Bilan sur les Impacts des Traitements Anti-moustiques, dans le cadre de la[END_REF][START_REF] Sofia | Optimal control and numerical optimization applied to epidemiological models[END_REF]. Also, its use over a long period and continuously, leads to strong resistance of the wild populations of Aedes aegypti, for example [START_REF] Darriet | Insecticides larvicides et adulticides alternatifs pour les opérations de démoustication en france, synthèse bibliographique[END_REF]. The mortality of the mosquitoes after spraying varied between 20% and 80%. To be more realistic, we will consider the technique called "pulse control" (the control is not continuous in time order is effective only one day every T days) [START_REF] Dumont | Vector control for the chikungunya disease[END_REF]. To this aim, we consider that spraying is done once a week, and this, for 100 days. We set α 1 = η 1 = η 2 = 0 and α 2 = 1.

Simulation results on figure 9 show that a mortality rate induced by the use of larvicide, c m , greater than 60% has a significant impact on the decrease of the total number of infected humans and vectors, and on the decrease of eggs and larvae. 3.

Strategy C: Vaccination combined with larvicide

Since the efficacy and the duration of a larvicide (Bti=Bacillus thuringiensis var. israelensis) strongly depend on several factors like water quality, exposure, and even the type of breeding sites, we thus consider, to be more realistic, that the duration can vary between a couple of days and two weeks [START_REF] Dumont | Vector control for the chikungunya disease[END_REF][START_REF] Licciardi | [END_REF]. We consider that the larvicide spraying happens once every 15 days, and this, on a period of 100 days. We set α 1 = c m = 0 and α 2 = 1.

The figure 10 shows that the use of larvicide has no significant impact on the decrease of total number of infected humans and vectors, as well as on the number of eggs and larvae. This can be justified by the fact that the use of conventional larvicides requires certain constraints on their use: they can not be used continuously, their duration of action decreases with time.

In addition, eggs of certain populations of vectors such as Aedes albopictus, come into prolonged hibernation when conditions in the breading sites are not conducive to their good growth (this is justified by the control rate value η 1 = 0.001). Also, the pupae do not consume anything, until reaching the mature stage.

Strategy D: Vaccination combined with mechanical control

The effectiveness of this type of control depends largely on awareness campaigns of local people in the sense that, to reduce the proliferation of vectors, people should destroy and systematically, 3.

potential breeding sites. Thus, we consider that this type of control can be achieved by local 558 populations, and this, daily. We set

α 1 = c m = 0 = η 1 = η 2 .

559

The figure 11 shows that this type of control is appropriate in the fight against the prolifer-560 ation of vectors. This can only be possible by the multiplication of local populations awareness 561 campaigns. 3. Figure 12 shows that the use of the combination of these 568 controls decreases significantly the total number of infected humans, infected vectors as well as 569 the number of eggs and larvae, when its associated rates, namely α 1 and c m , are greater than

α 2 =1 α 2 =1/2 α 2 =1/3 α 2 =1/5
Figure 11: Simulations results showing how the total number of infected vectors, eggs and larvae populations decrease with the mechanical control associated parameter α 2 . All others parameters values are in Table 3. Like for strategy E, the simulations results on figure 13 show that the combined use of these 574 three types of controls has a positive impact in the vector control. We have begun by calculate the net reproductive number N and the basic reproduction 581 number, R 0 , and investigate the existence and stability of equilibria. The stability analysis 582 revealed that for N ≤ 1, the trivial equilibrium is globally asymptotically stable. When

583

N > 1 and R 0 < 1, the disease-free equilibrium is locally asymptotically stable. Under certain 584 threshold condition, the disease-free equilibrium is also globally asymptotically stable. We We have proved that the model admits at least one endemic equilibrium, and only one endemic equilibrium point in the model without disease-induced death, and in the model with mass action incidences, whenever the basic reproduction number is great than unity.

Using parameters value of Chikungunya and Dengue fever, we have calculated the sensitivity indices of the basic reproduction number, R 0 , to the parameters in the model using both local and global methods. Local sensitivity analysis showed that the model system is most sensitive to a, the average number of mosquitoes bites, followed by µ v , the natural mortality rate of vectors. Considering that all input parameters vary simultaneously, we have used the Latin Hypercube Sampling (LHS) to estimate statistically the mean value of the basic reproduction number. The result showed that the model is in an endemic state, since the mean of R 0 is greater than unity. We also estimated the probability that the model predicts the occurrence of backward bifurcation phenomenon and of the likely stability of the disease-free equilibrium point.

Then, using global sensitivity analysis, we have computed the Partial Rank Correlation

Coefficients between R 0 and each parameter of the model. Unlike the local sensitivity analysis, the global analysis showed that the parameters α 1 , the human protection rate, has the highest influence on R 0 . The other parameters with an important effect are α 2 , the efficacy of the mechanical control, β hv , the probability of transmission of infection from an infected vector to a susceptible human, β vh , the probability of transmission of infection from an infected human to a susceptible vector, and θ, the maturation rate from pupae to adult vectors. This showed that the order of the most important parameters for R 0 from the local sensitivity analysis does not match those from the global sensitivity analysis. So, the local sensitivity results are not robust.

To assess the impact of combination of different controls, we have conducted several simulations, using the called "pulse control" technique. According to the numerical results, we conclude that the use of an imperfect vaccine with low efficacy combined with high individual Thus, pending the development of a high efficacy vaccine and long-acting, individual protection and the various vector control methods are effective ways to combat the spread of arboviral diseases, for developing countries. In addition, the realization of the combination of these controls may be too expensive, because it means that, for constant controls, we must keep them at high levels, and this, for a long time. Thus it is important to know what happens when, instead of the constant controls, we use time dependent controls, in optimal control theory.

This represents a perspective of this work.

Acknowledgement

The first author (H. A) thanks the Direction of UIT of Ngaoundere for their financial assistance in the context of research missions in 2015.

A Useful result.

We use the following result, which is the characterization of Metzler matrices, to compute the threshold R c , obtained at Eq. ( 10). B Proof of Theorem 1.

The Jacobian matrix of f at the Trivial equilibrium is given by

Df (E 0 ) = Df 1 Df 2 Df 3 Df 4 . ( 30
)
where

Df 1 =         -k 1 ω 0 0 0 0 ξ -k 2 0 0 0 0 0 0 -k 3 0 0 0 0 0 γ h -k 4 0 0 0 0 0 σ -µ h 0 0 0 0 0 0 -k 8         , Df 3 =      
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 µ b 0 0 0 0 0 0 0 0 0 0 0 0

      , Df 2 =               - τ β hv η v S 0 h N 0 h - τ β hv S 0 h N 0 h 0 0 0 - τ β hv πη v V 0 h N 0 h - τ β hv πV 0 h N 0 h 0 0 0 τ β hv η v H 0 N 0 h τ β hv H 0 N 0 h 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 θ               , Df 4 =       -k 9 0 0 0 0 γ v -k 8 0 0 0 µ b µ b -k 5 0 0 0 0 s -k 6 0 0 0 0 l -k 7      
, and H 0 = S 0 h + πV 0 h .

The characteristic polynomial of Df (E 0 ) is given by:

P (λ) = -(λ + k 3 ) (λ + k 4 ) (λ + k 8 ) (λ + k 9 ) (λ + µ h ) φ 1 (λ)φ 2 (λ)
where

φ 1 (λ) = λ 2 + (k 2 + k 1 )λ + µ h (k 2 + ξ) and φ 2 (λ) = λ 4 + A 1 λ 3 + A 2 λ 2 + A 3 λ + A 4 . we have set A 1 = k 5 + k 6 + k 7 + k 8 , A 2 = k 8 (k 5 + k 6 + k 7 ) + k 7 (k 5 + k 6 ) + k 5 k 6 , A 3 = k 5 k 6 k 7 + k 8 (k 5 k 6 + k 7 (k 5 + k 6 )), A 4 = k 5 k 6 k 7 k 8 (1 -N ).
The roots of P (λ) are

λ 1 = -µ h , λ 1 = -k 1 , λ 2 = -k 3 , λ 3 = -k 4 , λ 4 = -k 8 , λ 4 = -k 9 ,
and the others roots are the roots of φ 1 (λ) and φ 2 (λ). The real part of roots of φ 1 (λ) are negative.

Since N < 1, it is clear that all coefficients of φ 2 (λ) are always positive. Now we just have to verify that the Routh-Hurwitz criterion holds for polynomial φ 2 (λ). To this aim, setting

H 1 = A 1 , H 2 = A 1 1 A 3 A 2 , H 3 = A 1 1 0 A 3 A 2 A 1 0 A 4 A 3 , H 4 = A 1 1 0 0 A 3 A 2 A 1 1 0 A 4 A 3 A 2 0 0 0 A 4 = A 4 H 3 .
The Routh-Hurwitz criterion of stability of the trivial equilibrium E 0 is given by

       H 1 > 0 H 2 > 0 H 3 > 0 H 4 > 0 ⇔        H 1 > 0 H 2 > 0 H 3 > 0 A 4 > 0 (31)
We have

H 1 = A 1 = k 5 + k 6 + k 7 + k 8 > 0, H 2 = A 1 A 2 -A 3 = (k 7 + k 6 + k 5 ) k 2 8 + k 2 7 + (2k 6 + 2k 5 ) k 7 + k 2 6 + 2k 5 k 6 + k 2 5 k 8 + (k 6 + k 5 ) k 2 7 + k 2 6 + 2k 5 k 6 + k 2 5 k 7 + k 5 k 2 6 + k 2 5 k 6 H 3 = A 1 A 2 A 3 -A 2 1 A 4 -A 2 3 = (k 6 + k 5 ) k 2 7 + (k 6 + k 5 ) k 7 + k 5 k 6 k 3 8 + µ b lsθ + (k 6 + k 5 ) k 3 7 + 2(k 6 + k 5 ) 2 k 2 7 + k 3 6 + 4k 5 k 2 6 + 4k 2 5 k 6 + k 3 5 k 7 + k 5 k 3 6 + 2k 2 5 k 2 6 + k 3 5 k 6 k 2 8 + (2k 7 + 2k 6 + 2k 5 ) µ b lsθ + k 2 6 + 2k 5 k 6 + k 2 5 k 3 7 + k 3 6 + 4k 5 k 2 6 + 4k 2 5 k 6 + k 3 5 k 2 7 + 2k 5 k 3 6 + 4k 2 5 k 2 6 + 2k 3 5 k 6 k 7 + k 2 5 k 3 6 + k 3 5 k 2 6 k 8 + k 2 7 + (2k 6 + 2k 5 ) k 7 + k 2 6 + 2k 5 k 6 + k 2 5 µ b lsθ + k 5 k 2 6 + k 2 5 k 6 k 3 7 + k 5 k 3 6 + 2k 2 5 k 2 6 + k 3 5 k 6 k 2 7 + k 2 5 k 3 6 + k 3 5 k 2 6 k 7
We always have H 1 > 0, H 2 > 0, H 3 > 0 and H 4 > 0 if N < 1. Thus, the trivial equilibrium E 0 is locally asymptotically stable whenever N < 1.

We assume the net reproductive number N > 1. Following the procedure and the notation in [START_REF] Van Den Driessche | Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF], we may obtain the basic reproduction number R 0 as the dominant eigenvalue of the next-generation matrix [START_REF] Diekmann | Mathematical Epidemiology of Infectious Diseases. Model building, analysis and interpretation[END_REF][START_REF] Van Den Driessche | Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF]. Observe that model (2) has four infected populations, namely

E h , I h , E v , I v .
It follows that the matrices F and V defined in [START_REF] Van Den Driessche | Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF], which take into account the new infection terms and remaining transfer terms, respectively, are given by

F =         0 0 τ β hv η v H 0 N 0 h τ β hv H 0 N 0 h 0 0 0 0 τ β vh η v S 0 v N 0 h τ β vh S 0 v N 0 h 0 0 0 0 0 0         , and V =     k 3 0 0 0 -γ h k 4 0 0 0 0 k 9 0 0 0 -γ v k 8     .
The dominant eigenvalue of the next-generation matrix F V -1 is given by [START_REF] Moulay | Modélisation et analyse mathématique de systèmes dynamiques en épidémiologie. application au cas du chikungunya[END_REF]. The local stability of the disease-free equilibrium E 1 is a direct consequence of Theorem 2 in [START_REF] Van Den Driessche | Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF]. This ends the proof.

C Proof of Theorem 2.

Setting Y = X -T E with X = (S h , V h , E h , I h , R h , S v , E v , I v , E, L, P ) T , H 0 = (S 0 h + πV 0 h ), A 99 = k 5 + µ b S v + E v + I v K E ,
where

B(Y ) = A(Y ) B(Y ) C(Y ) D(Y ) , with A(Y ) =       -(λ c h + k 1 ) ω 0 0 0 0 ξ -(πλ c h + k 2 ) 0 0 0 0 λ c h πλ c h -k 3 0 0 0 0 0 γ h -k 4 0 0 0 0 0 σ -µ h 0       , B(Y ) =             - τ β hv η v S 0 h N h - τ β hv S 0 h N h 0 0 0 - τ β hv η v πV 0 h N h - τ β hv πV 0 h N h 0 0 0 τ β hv η v H 0 N h τ β hv H 0 N h 0 0 0 0 0 0 0 0 0 0 0 0 0             , C(Y ) =         0 0 0 0 0 -(λ c v + k 8 ) 0 0 0 0 0 λ c v 0 0 0 0 0 0 0 0 0 0 0 µ b 0 0 0 0 0 0 0 0 0 0 0 0         , D(Y ) =         0 0 0 0 θ -k 9 0 0 0 0 γ v -k 8 0 0 0 µ b µ b -A 99 0 0 0 0 s -A 10 0 0 0 0 l -k 7         .
It is clear that Y = (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) is the only equilibrium. Then it suffices to consider the following Lyapunov function L(Y ) =< g, Y > were g = 1, 1, 1, 1, 1, 1, 1, 1,

k 8 µ b , k 5 k 8 µ b s , k 5 k 6 k 8 µ b sl . Straightforward computations lead that L(Y ) =< g, Ẏ > def = < g, B(Y )Y > = -µ h Y 1 -µ h Y 2 -µ h Y 3 -(µ h + δ)Y 4 -µ h Y 5 - k 8 K E (Y 6 + Y 7 + Y 8 ) - k 5 k 8 µ b K L Y 9 Y 10 + θ 1 - 1 N Y 11 We have L(Y ) < 0 if N ≤ 1 and L(Y ) = 0 if Y i = 0, i = 1, 2, . . . , 11 (i.e S h = S 0 h , V h = V 0 h and E h = I h = R h = S v = E v = I v = E = L = P = 0)
. Moreover, the maximal invariant set contained in L| L(Y ) = 0 is (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0). Thus, from Lyapunov theory, we deduce that (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0) and thus, E 0 , is GAS if and only if N ≤ 1.

D Proof of Theorem 5.

In order to determine the existence of endemic equilibria, i.e., equilibria with all positive components, say

E * * = (S * h , V * h , E * h , I * h , R * h , S * v , E * v , I * v , E, L, P ) ,
we have to look for the solution of the algebraic system of equations obtained by equating the right sides of system (2) to zero. In this way we consider two case:

(i) Special case: Absence of disease-induced death in human (δ = 0).

Note that in the absence of disease-induced death in human population, we have

N * h = N 0 h = Λ h /µ h . Let λ c, * h = a(1 -α 1 )β hv (η v E * v + I * v ) N * h , λ c, * v = a(1 -α 1 )β vh (η h E * h + I * h ) N * h (33)
be the forces of infection of humans and vectors at steady state, respectively. Solving the equations in (2) at steady state gives

S * h = Λ h (πλ c, * h + k 2 ) µ h (k 2 + ξ) + λ c, * h (πλ c, * h + πk 1 + k 2 ) , V * h = ξS * h (πλ c, * h + k 2 ) , E * h = λ c, * h (S * h + πV * h ) k 3 , I * h = γ h λ c, * h (S * h + πV * h ) k 3 k 4 , R * h = σγ h λ c, * h (S * h + πV * h ) µ h k 3 k 4 , (34)
and

S * v = θP (λ c, * v + k 8 ) , E * v = θP λ c, * v k 9 (λ c, * v + k 8 ) , I * v = γ v θP λ c, * v k 8 k 9 (λ c, * v + k 8 ) , E = µ b θK E P (k 5 k 8 K E + µ b θP ) , L = µ b θsK E K L P k 6 K L (k 5 k 8 K E + µ b θP ) + sµ b θK E P , (35)
where P is solution of the following equation

f (P ) = -k 7 P [µ b θ(sK E + k 6 K L )P + k 5 k 6 k 8 K E K L (N -1)] = 0 (36)
A direct resolution of the above equation give P = 0 or P

= k 5 k 6 k 8 K E K L (N -1) µ b θ(sK E + k 6 K L ) .
Note that P = 0 corresponds to the trivial equilibrium E 0 . Now we consider P > 0 i.e.

N > 1. Replacing (34) and ( 35) in [START_REF] Berman | Nonnegative matrices in the mathematical sciences[END_REF] give

λ c, * h = a(1 -α 1 )β hv µ h Λ h η v θP λ * v k 9 (λ * v + k 8 ) + γ v θP λ * v k 8 k 9 (λ * v + k 8 ) (37)
λ c, * v = a(1 -α 1 )β vh µ h Λ h η h λ * h (S * h + πV * h ) k 3 + γ h λ * h (S * h + πV * h ) k 3 k 4 (38)
Substuting [START_REF] Van Den Driessche | Reproduction numbers and the sub-threshold endemic equilibria for compartmental models of disease transmission[END_REF] in [START_REF] Diekmann | Mathematical Epidemiology of Infectious Diseases. Model building, analysis and interpretation[END_REF] give

(k 6 K L + sK E ) λ * h a 2 (λ * h ) 2 + a 1 λ * h + a 0 = 0 ( 39
)
where a 2 , a 1 and a 0 are given by

R b = (πξ + k 2 ) π(ξ + k 2 ) (k 1 π + k 2 ) µ h + a(1 -α 1 )β vh (γ h + k 4 η h )(πξ + k 2 ) k 3 k 4 k 8 , a 2 = (a(1 -α 1 )β vh µ h (γ h + k 4 η h ) + k 3 k 4 k 8 ) k 9 µ b Λ h π, a 1 = k 3 k 4 k 8 k 9 µ b Λ h (ξ + k 2 ) µ h π (πξ + k 2 ) (R b -R 1 ), a 0 = µ h k 3 k 4 k 8 k 9 µ b Λ h (ξ + k 2 ) (1 -R 1 ) . (40)
The trivial solution λ * h = 0 of (39) corresponds to the disease-free equilibrium E 1 . Now, we just look the equilibria when λ * h > 0. Note that coefficient a 2 is always positive and a 0 is negative (resp. positive) if and only if R 1 > 1 (resp. R 1 < 1). Thus model system (2), in absence of disease-induced death in human population (δ = 0), admits only one endemic equilibrium whenever R 1 > 1. Since the sign of coefficient a 1 depend of the value of parameter, we investigate the possibility of occurrence of backward bifurcation phenomenon when R 1 < 1.

Furthermore, consider the inequality

R 1 < R b . (41)
Since a 2 is always positive and a 0 is always positive whenever R 1 < 1, then, the occurrence of backward bifurcation phenomenon depend of the sign of coefficient a 1 . The coefficient a 1 is always positive if and only if condition (41) holds (i.e R 1 < R b ). It follows that the disease-free equilibrium is the unique equilibrium when N > 1 and R 1 < 1.

Now if R b < R 1 < 1, then in addition to the DFE E 1 , there exists two endemic equilibria whenever ∆ = a 2 1 -4a 2 a 0 > 0. However,

R b < R 1 < 1 ⇒ R b < 1 ⇔ β vh < - [π 2 ξ 2 + (µ h π 2 + (2ω + µ h ) π) ξ + (ω + µ h ) 2 ] k 3 k 4 k 8 a(1 -α 1 )µ h (πξ + k 2 ) 2 (γ h + k 4 η h ) < 0.
Since all parameter of model ( 2)

are nonnegative, we conclude that the condition R b < R 1 < 1 does not hold. And thus, the backward bifurcation never occurs in the absence of disease-induced death in human.

(ii) Presence of disease induced death in human (δ = 0).

In this case, we have

N * h = Λ h -δI * h µ h
. Applying the same procedure as case (i), we obtain that λ * h at steady state is solution of the following equation

f (λ * h ) = λ * h c 4 (λ * h ) 4 + c 3 (λ * h ) 3 + c 2 (λ * h ) 2 + c 1 λ * h + c 0 = 0, (42)
Λ h µ h (1 -α 1 )β vh + 2k 1 k 3 k 4 k 8 k 9 K 12 µ b δΛ h γ h π -2k 1 k 2 3 k 2 4 k 8 k 9 K 12 µ b Λ h π -2k 2 k 8 k 9 K 12 µ b δ 2 Λ h γ 2 h + 4k 2 k 3 k 4 k 8 k 9 K 12 µ b δΛ h γ h -2k 2 k 2 3 k 2 4 k 8 k 9 K 12 µ b Λ h ), c 2 = k 3 k 4 k 5 k 6 k 10 k 11 a 2 µ 2 h (1 -α 1 ) 2 β hv nπ 2 β vh ξK E K L + k 1 k 3 k 4 k 5 k 6 k 10 k 11 a 2 µ 2 h (1 -α 1 ) 2 β hv nπ 2 β vh K E K L + 2k 2 k 3 k 4 k 5 k 6 k 10 k 11 a 2 µ 2 h (1 -α 1 ) 2 β hv nπβ vh K E K L + k 9 k 10 K 12 aµ b δΛ h µ h γ h π 2 (1 -α 1 )β vh ξ 2 -k 8 k 9 K 12 µ b δ 2 Λ h γ 2 h π 2 ξ 2 -k 1 k 3 k 4 k 9 k 10 K 12 aµ b Λ h µ h π 2 (1 -α 1 )β vh ξ + k 3 k 4 k 9 k 10 K 12 aµ b Λ h µ h ωπ(1 -α 1 )β vh ξ + 2k 2 k 9 k 10 K 12 aµ b δΛ h µ h γ h π(1 -α 1 )β vh ξ -k 2 k 3 k 4 k 9 k 10 K 12 aµ b Λ h µ h π(1 -α 1 )β vh ξ + 2k 1 k 3 k 4 k 8 k 9 K 12 µ b δΛ h γ h π 2 ξ -2k 3 k 4 k 8 k 9 K 12 µ b δΛ h γ h ωπξ + 2k
c 0 = k 2 3 k 2 4 k 8 k 9 K 12 µ b Λ h µ 2 h (k 2 + ξ) 2 R 2 0 -1 , with k 10 = γ h + η h k 4 , k 11 = γ v + η v k 8 , K 12 = (sK E + k 6 K L )
and n = N -1. Notes that c 4 is always negative and c 0 is positive (resp. negative) if R 0 is greater (resp. less) that the unity.

It follows, depending of the sign of coefficients c 3 , c 2 and c 1 , that the model system (2) admits at least one endemic equilibrium whenever R 0 > 1 and the phenomenon of backward (resp. forward) bifurcation can occurs when R 0 < 1 (resp. R 0 > 1). This ends the proof.

E Proof of Theorem 9

Consider the Lyapunov function G = q 1 E h + q 2 I h + q 3 E v + q 4 I v , where

q 1 = 1 k 3 ; q 3 = ζ 1 S 0 h k 3 k 8 (γ v + k 8 η v ) k 9 , q 2 = ζ 1 S 0 h k 3 k 8 (γ v + k 8 η v )ζ 2 S 0 v k 4 k 9 , q 4 = ζ 1 S 0 h k 3 k 8 .
and we have set ζ 1 = τ µ h β hv Λ h and ζ 2 = τ µ h β vh Λ h . The derivative of G is given by Ġ = q 1 Ėh + q 2 İh + q 3 Ėv + q 4 İv = q 1 (λ c h S h -

k 3 E h ) + q 2 (γ h E h -k 4 I h ) + q 3 (λ c v S v -k 9 E v ) + q 4 (γ v E v -k 8 I v ) = q 1 ζ 1 S h (η v E v + I v ) -q 3 k 9 E v + q 4 γ v E v -q 4 k 8 I v + q 3 ζ 2 S v (η h E h + I h ) -q 1 k 3 E h + q 2 γ h E h -q 2 k 4 I h = (q 1 ζ 1 S h η v + q 4 γ v -q 3 k 9 )E v + (q 1 ζ 1 S h -q 4 k 8 )I v + (q 3 ζ 2 S v η h + q 2 γ h -q 1 k 3 )E h + (q 3 ζ 2 S v -q 2 k 4 )I h ≤ (q 1 ζ 1 S 0 h η v + q 4 γ v -q 3 k 9 )E v + (q 1 ζ 1 S 0 h -q 4 k 8 )I v + (q 3 ζ 2 S 0 v η h + q 2 γ h -q 1 k 3 )E h + (q 3 ζ 2 S 0 v -q 2 k 4 )I h , since S h ≤ S 0 h , S v ≤ S 0 v
Replacing q i , i = 1, . . . , 4, by their value gives after straightforward simplifications Ġ ≤ R 2 nv,δ=0 -1 E h

We have Ġ ≤ 0 if R nv,δ=0 ≤ 1, with Ġ = 0 if R nv,δ=0 = 1 or E h = 0. Whenever E h = 0, we also have I h = 0, E v = 0 and I v = 0. Substituting E h = I h = E v = I v = 0 in the first, fourth and fifth equation of Eq. ( 18) with δ = 0 gives S h (t) → S 0 h = N 0 h , R h (t) → 0, and S v (t) → S 0 v = N 0 v as t → ∞. Thus

[S h (t), E h (t), I h (t), R h (t), S v (t), E v (t), I v (t), E(t), L(t), P (t)] → (N 0 h , 0, 0, 0, N 0 v , 0, 0, E, L, P ) as t → ∞.

It follows from the LaSalle's invariance principle [START_REF] Hale | Ordinary Differential Equations[END_REF][START_REF] Lasalle | Stability theory for ordinary differential equations[END_REF][START_REF] Lasalle | The stability of dynamical systems[END_REF] that every solution of (18) (when R nv,δ=0 ≤ 1), with initial conditions in D 2 converges to E nv 1 , as t → ∞. Hence, the DFE, E nv 1 , of model ( 18) without disease-induced death, is GAS in D 2 if R nv,δ=0 ≤ 1.

  10 5 , Γ L = 50000. In this case the conditions required by Theorem 6, are satisfied: A 1 = 0.0114 > 0 and A 2 = 1.1393 > 0.

Figure 2 :Figure 3 :

 23 Figure 2: The backward bifurcation curves for model system (2) in the (R 0 , E * h ), and (R 0 , E * v ) planes. The parameter β hv is varied in the range [0, 0.2810] to allow R 0 to vary in the range [0, 1.5]. Two endemic equilibrium points coexist for values of R 0 in the range (0.2894, 1)(corresponding to the range (0.0105, 0.1249) of β hv ). The notation EE and DFE stand for endemic equilibrium and disease-free equilibrium, respectively. Solid lines represent stable equilibria and dash lines stand for unstable equilibria.

Figure 4 :

 4 Figure 4: Solutions of model (2) of the number of infected humans, E h , and the number of infected vectors, E v , for parameter values given in the bifurcation diagram in Figure2with β hv = 0.0105. So R 0 = 0.29 < 1, for two different set of initial conditions. The first set of initial conditions (corresponding to the solid trajectory) isS h = 700, V h = 10, E h = 220, I h = 100, R h = 60, S v = 3000, E v = 400, I v =120, E = 10000, L = 5000 and P = 3000. The second set of initial conditions (corresponding to the dotted trajectory) isS h = 489100, V h = 10, E h = 220, I h = 100, R h = 60, S v = 3000, E v = 400, I v =120, E = 10000, L = 5000 and P = 3000. The solution for initial condition 1 approaches the locally asymptotically stable endemic equilibrium point, while the solution for initial condition 2 approaches the locally asymptotically stable DFE.

Lemma 1 .Figure 5 :

 15 Figure 5: The backward bifurcation curves for model system[START_REF] Derouich | Dengue fever: mathematical modelling and computer simulation[END_REF] in the (R nv , E * h ) and (R nv , E * v ) planes. The parameter β hv varied in the range [0, 0.9090] to allow R 0 to vary in the range [0,1.5]. Two endemic equilibrium points coexist for values of R 0 in the range (0.2286, 1) (corresponding to the range (0.0211, 0.4040) of β hv ). The notation EE and DFE stand for endemic equilibrium and disease-free equilibrium, respectively. Solid lines represent stable equilibria and dash lines stand for unstable equilibria.

Figure 7 (FrequenciesFigure 6 :

 76 Figure 7 (b), show that the parameter α 1 , the human protection rate, has the highest influence on R 0 . This suggests that individual protection may potentially be the most effective strategy to reduce R 0 . The other parameters with an important effect are α 2 , β hv , β vh and θ.

(a) Local sensitivity indices for R 0 -

 0 Partial rank correlation coefficients for R 0

Figure 7 :

 7 Figure 7: Local (a) and global (b) sensitivity indices for R 0 against model parameters show that the order of the most important parameters for R 0 from the local sensitivity analysis does not match those from the global sensitivity analysis.

Figure 8 :

 8 Figure 8: Simulations results showing how the total number of infected humans and the total number of infected vectors decrease when the individual protection increase. All others parameters values are in Table3.

Figure 9 :

 9 Figure 9: Simulations results showing how the total number of infected vectors, eggs and larvae populations decrease when the aldulticide control parameter c m increase. All others parameters values are in Table3.

Figure 10 :

 10 Figure 10: Simulations results showing how the total number of infected humans, the total number of infected vectors, and the eggs and larvae populations decrease with the larvicide control associated parameters η 1 and η 2 . All others parameters values are in Table3.

562 5 . 5

 55 Strategy E: Combining vaccination, individual protection and 563 adulticide 564 In this strategy, we consider the model (2) without larvicide and mechanical control. we set 565 α 2 = 1 and η 1 = η 2 = 0 and vary the parameter related to individual protection and the 566 use of adulticide, namely α 1 and c m , respectively, between 0 and 0.8. The values of other 567 parameters are given in Table

5. 6

 6 Strategy F: Combining vaccination, individual protection and 572 mechanical control 573

576

  In this paper, we have derived and analyzed a deterministic model for the transmission of 577 arboviral diseases with non linear form of infection and complete stage structured model for 578 vectors, which takes into account a vaccination with waning immunity, individual protection 579 and vector control strategies.

  580

585Figure 12 :

 12 Figure 12: Simulations results showing the advantage that we have to combine vaccination, individual protection and adulticide.

Figure 13 :

 13 Figure 13: Simulations results showing the advantage that we have to combine vaccination, individual protection and mechanical control.

  protection and good vector control strategy (reduction of breeding sites by local populations action, chemical action using adulticides and larvicides), can effectively reduce the transmission of the pathogen and the proliferation of vector populations. However, due to lack of resources to implement these control mechanisms, developing countries should focus on the education of the local populations. Because, unlike diseases such as malaria whose breeding sites of Anopheles mosquitoes are known, those of Aedes (aegypti and albopictus) and Culex, for example, are smaller (old tires, flower pots, vases and other hollow...) and unknown for many local populations, which favour the development of vectors.

where c 4 = -π 2 k 9 K

 49 12 µ b Λ h (k 3 k 4δγ h ) (k 10 aµ h (1α 1 )β vh + k 8 (k 3 k 4δγ h )) , c 3 = π(k 3 k 4 k 5 k 6 k 10 k 11 a 2 µ 2 h (1α 1 ) 2 β hv nπβ vh K E K L + 2k 9 k 10 K 12 aµ b δΛ h µ h γ h π(1α 1 )β vh ξ k 3 k 4 k 9 k 10 K 12 aµ b Λ h µ h π(1α 1 )β vh ξ -2k 8 k 9 K 12 µ b δ 2 Λ h γ 2 h πξ + 2k 3 k 4 k 8 k 9 K 12 µ b δΛ h γ h πξ k 1 k 3 k 4 k 9 k 10 K 12 aµ b Λ h µ h π(1α 1 )β vh + 2k 2 k 9 k 10 K 12 aµ b δΛ h µ h γ h (1α 1 )β vh -2k 2 k 3 k 4 k 9 k 10 K 12 aµ b

Table 1 :

 1 The state variables of model[START_REF] Karabatsos | International Catalogue of Arboviruses, including certain other viruses of vertebrates[END_REF].

	Humans	Vectors
	S h : Susceptible	E: Eggs
	V h : Vaccines	L: Larvae
	E h : Infected in latent stage P : Pupae
	I h : Infectious	S v : Susceptible
	R h : Resistant (immune)	E v Infected in latent stage
		I v Infectious

Table 2 :

 2 Description and baseline values/range of parameters of model (2).

	Parameter Description	Baseline	Sources
			value/range
	Λ h	Recruitment rate of humans	2.5 day -1	[23]
	µ h	Natural mortality rate	1 (67×365) day -1	[23]
		in humans
	ξ	Vaccine coverage	Variable
	ω	Vaccine waning rate	Variable
	ǫ	The vaccine efficacy	Variable
	a	Average number of bites	1 day -1	[13, 23]
	β hv	Probability of transmission of	0.1, 0.75 day -1 [13, 23]
		infection from an infected vector
		to a susceptible human
	γ h δ	Progression rate from E h to I h Disease-induced death rate	1 15 , 1 3 10 -3 day -1 day -1	[19, 32] [23]
	σ	Recovery rate for humans	0.1428 day -1	[13, 23]
	η h ,η v	Modifications parameter	[0, 1)	[23]
	µ v γ v β vh	Natural mortality rate of vectors Progression rate from E v to I v Probability of transmission of	1 30 , 1 14 1 21 , 1 2 day -1 day -1 0.1, 0.75 day -1 [13, 23] [13, 23] [19, 32]
		infection from an infected human
		to a susceptible vector
	θ	Maturation rate from pupae	0.08 day -1	[19, 27, 28]
		to adult
	µ b	Number of eggs at each deposit	6 day -1	[19, 27, 28]
	Γ E	Carrying capacity for eggs	10 3 , 10 6	[13, 27]
	Γ L	Carrying capacity for larvae	5 × 10 2 , 5 × 10 5 [13, 27]
	µ E	Eggs death rate	0.2 or 0.4	[28]
	µ L	Larvae death rate	0.2 or 0.4	[28]
	µ P	Pupae death rate	0.4	Assumed
	s	Transfer rate from eggs to larvae	0.7 day -1	[28]
	l	Transfer rate from larvae to pupae 0.5 day -1	[27]
	η 1 , η 2	Eggs and larvae mortality rates	0.001,0.3	[28]
		induced by chemical intervention
	α 1	Human protection rate	[0, 1)	Assumed
	α 2	Efficacy of the mechanical control (0, 1]	[19]
	c m	Adulticide killing rate	[0,0.8]	[19]

  , L, P ), are:

	E nv 1 = (281, 70, 5, 1207, 5739, 182, 44, 22180, 10201, 9977) which is locally stable and E nv 2 = (6333, 67, 4, 1147, 5936, 37, 2, 22180, 10201, 9977) which is unstable.

Table 3 :

 3 Parameter values using to compute the sensitivity indices of R 0 .

	Parameter value Parameter value Parameter value
	c m	0.01	s	0.7	β vh	0.75
	µ b	6	η 2	0.3	Γ E	10000
	µ P	0.4	µ E	0.2	Γ L	5000
	θ	0.08	ǫ	0.61	α 2	0.5
	l a	0.5 1	Λ h β hv	2.5 0.75	µ h η v	1 67 * 365 0.35
	µ v γ h ξ	1 30 1 14 0.5	µ L η h ω	0.4 0.35 0.05	σ γ v η 1	0.1428 1 21 0.001
	δ	0.001	α 1	0.2

Table 4 :

 4 Sensitivity indices of R 0 to parameters of model (2), evaluated at the baseline parameter values given in Table3.

	Parameter	Index	Parameter	Index	Parameter	Index
	a	+1	σ	-0.2911	ξ	-0.0566
	µ v	-0.9190	c m	-0.2757	ω	+0.0565
	ǫ	-0.6223	α 1	-0.25	µ E	-0.0171
	s	+0.5172	η h	+0.2067	δ	-0.0020
	Λ h	-0.5	γ h	-0.2064	η 1	-0.0000858
	β hv , β vh , Γ E , Γ L , α 2	+0.5	η v	+0.1207
	µ h	+0.4996	γ v	+0.1174
	µ P	-0.4810	µ L	-0.1026
	θ	+0.4810	µ b	+0.0772
	l	+0.4489	η 2	-0.0770

Table 5 :

 5 Partial Rank Correlation Coefficients between R 0 and each parameters of model (2).

	Parameter Correlation Parameter Correlation Parameter Correlation
		Coefficients		Coefficients		Coefficients
	α 1	-0.6125	l	0.3767	γ v	0.0378
	α 2	0.5960	ǫ	-0.3348	µ L	-0.0357
	β hv	0.5817	s	0.2945	c m	-0.0271
	β vh	0.5815	σ	-0.1808	η h	0.0178
	θ	0.5078	µ P	-0.1594	η 1	-0.0161
	a	0.4810	µ h	0.1306	µ E	-0.0113
	µ v	-0.3911	γ h	-0.0605	ξ	-0.0109
	Γ L	0.4195	η v	0.0578	δ	-0.0077
	Γ E	0.3888	µ b	0.0439	η 2	0.0037
	Λ h	-0.3876	ω	0.0410
	5 Numerical simulations

In the previous model

[START_REF] Abboubakar | Modeling the dynamics of arboviral diseases with vaccination perspective[END_REF]

, we have shown that the use of a vaccine with efficacy of about 60%, should be accompanied by other control measures such as means of individual protection (spanning wearing clothes during hours of vector activity, use of repellents,. . .), vector control (combining the use of adulticide to kill adult vectors, chemical control with use of larvicide to

  and A 10 = k 6 + s

	manner	dY dt	E K L = B(Y )Y	. we can rewrite (2) in the following

  2 3 k 2 4 k 8 k 9 K 12 µ b Λ h ωπξ -2k 2 k 8 k 9 K 12 µ b δ 2 Λ h γ 2 h πξ + 2k 2 k 3 k 4 k 8 k 9 K 12 µ b δΛ h γ h πξ -2k 1 k 2 k 3 k 4 k 9 k 10 K 12 aµ b Λ h µ h (1α 1 )πβ vh + k 2 2 k 9 k 10 K 12 aµ b δΛ h µ h γ h (1α 1 )β vhk 2 2 k 3 k 4 k 9 k 10 K 12 aµ b Λ h µ h (1α 1 )β vh k 2 1 k 2 3 k 2 4 k 8 k 9 K 12 µ b Λ h π 2 + 4k 1 k 2 k 3 k 4 k 8 k 9 K 12 µ b δΛ h γ h π -4k 1 k 2 k 2 3 k 2 4 k 8 k 9 K 12 µ b Λ h π k 2 2 k 8 k 9 K 12 µ b δ 2 Λ h γ 2 h + 2k 2 2 k 3 k 4 k 8 k 9 K 12 µ b δΛ h γ hk 2 2 k 2 3 k 2 4 k 8 k 9 K 12 µ b Λ h , c 1 = ((k 1 k 3 k 4 k 5 k 6 k 10 k 11 a 2 µ 2 h (1α 1 )β hv nπ 2 + k 3 k 4 k 5 k 6 k 10 k 11 a 2 µ 2 h (1α 1 ) 2 β hv n(k 2ω)π)β vh ξ + (2k 1 k 2 k 3 k 4 k 5 k 6 k 10 k 11 a 2 µ 2 h (1α 1 )β hv nπ + k 2 2 k 3 k 4 k 5 k 6 k 10 k 11 a 2 µ 2 h (1α 1 )β hv n)(1α 1 )β vh )K E K L + (k 3 k 4 k 9 k 10 K 12 aµ b Λ h µ h ωπ(1α 1 )β vh -2k 3 k 4 k 8 k 9 K 12 µ b δΛ h γ h ωπ)ξ 2 + ((k 2 k 3 k 4 k 9 k 10 K 12 aµ b Λ h µ h ωk 1 k 2 k 3 k 4 k 9 k 10 K 12 aµ b Λ h µ h π)(1α 1 )β vh + (2k 1 k 2 3 k 2 4 k 8 k 9 K 12 µ b Λ h ω + 2k 1 k 2 k 3 k 4 k 8 k 9 K 12 µ b δΛ h γ h )π + (2k 2 k 2 3 k 2 4 k 8 k 9 K 12 µ b Λ h -2k 2 k 3 k 4 k 8 k 9 K 12 µ b δΛ h γ h )ω)ξ k 1 k 2 2 k 3 k 4 k 9 k 10 K 12 aµ b Λ h µ h (1α 1 )β vh -2k 2 1 k 2 k 2 3 k 2 4 k 8 k 9 K 12 µ b Λ h π + 2k 1 k 2 2 k 3 k 4 k 8 k 9 K 12 µ b δΛ h γ h -2k 1 k 2 2 k 2 3 k 2 4 k 8 k 9 K 12 µ b Λ h ,