A basic general model of vector-borne diseases

. We propose a model that can translate the dynamics of vector-borne diseases, for this model we compute the basic reproduction number and show that if R 0 < ζ < 1 the DFE is globally asymptotically stable. For R 0 > 1 we prove the existence of a unique endemic equilibrium and if R 0 ≤ 1 the system can have one or two endemic equilibrium, we also show the existence of a backward bifurcation. By numerical simulations we illustrate with data on malaria all the results including existence, stability and bifurcation.


Introduction
The vector-borne diseases are responsible for more than 17% of infectious diseases, and causes over one million deaths each year [18]. Mathematical models help to better understand and propose solutions to reduce the negative impact of these disease in society, and several studies have already been made on diseases such as malaria, leishmaniasis, trypanosomiasis just to name a few (see [15,6,19,16,1,2,10,11]). We realise that the behavior of these diseases can be modeled by a generic model. Thus in this first work, we propose a model of one population that can translate the dynamics of several vector-borne diseases.
We take a different approach in modeling vectors, drawing on the work Ngwa, Ngonghala [8,4,5] which integrates the three phases of the gonotrophic cycle. We assume as in [13] that rest and laying occur in the same place ie we have a questing phase and another phase resting.
In addition to the consideration of the gonotriphic cycle, we integrate the management of the sporogonic cycle, that is to say that we consider the fact that after the first meal infecting the vector can transmit the disease only after a certain number of meals (This number depends on the species). In the population of host we introduce the two parameters u, v ∈ {0, 1} opposite to [12] where u, v ∈ ]0, 1[ thereby controlling the presence or absence of the compartments E of exposed and R of immune. This allows us to place ourselves in one of SIS, SIRS, SEIS or SEIRS dynamics, we study the existence and stability of equilibrium and the bifurcation.

Model description and mathematical specification
In our model, we consider two populations, namely a population of host which may be humans or animals and a vector population that can be specified according to the disease that one wishes to model.

Host population structure and dynamics
The host population is subdivided in four compartments: susceptible, infected, infectious and immune as shown in the graph above. The parameters u, v ∈ {0; 1} are used to choose the dynamics in the host population. Thus according to the values of u and v we can have the dynamics SIS, SIRS, SEIS or SEIRS.

Mosquito population structure and dynamics
In the vector population we adopt the questing-resting as described in [13], but to simplify the calculations we consider the questing-resting phase number equal to 1. The figure below illustrates the dynamics in Population of vectors.

Model equation
The diagrams (1) and (2) allow us to have the following system of equations: (1)

Well-posedness, dissipativity
In this section we demonstrate well-posedness of the model by demonstrating invariance of the set of non-negative states, as well as boundedness properties of the solution. We also calculate the equilibria of the system.

Positive invariance of the non-negative cone in state space
The system (1) can be rewritten in the matrix form as 2) is defined for values of the state variable x = (x S ; x I ) lying in the non-negative cone of R 10 + . Here x S = S h ; S q represents the naive component and x I = E h ; E 0 r ; E 1 q ; E 1 r ; I r ; I q ; I h ; R h represents the infected and infectious components of the state of the system.
The matrix A S (x), A SI (x) and A I (x) are define as For a given x ∈ R 11 + , the matrices A(x), A S (x) and A I (x) are Metzler matrices. The following proposition establishes that system (1) is epidemiologically well posed.

Boundedness and dissipativity of the trajectories
proposition 3.2. The set G defined by

Disease free equilibrium
We obtain the disease free equilibrium DFE after solve the system The disease free equilibrium of the system (1) is given by: are respectively the questing and the resting frequencies of mosquitoes.

Basic reproduction number R 0
Unlike the method proposed in [17] we will use the one given in [14] which is more appropriate for systems like what we describe The basic reproduction number is given by: Proof: The matrix of the infected A I (x ) can be written in the form We apply the algorithm given in the proposition to the matrix A I (x ) we have : is always a Metzler stable matrix. The condition of being a Metzler stable it is a 6 × 6 square matrix that can be decomposed in the following block matrix form : is a 2 × 2 square matrix given by: We make another iteration of the algorithm given by the proposition The last iteration of the algorithm, since N 22 (x ) is negative coefficient leads to the consideration of the matrix L(x ) and thus L(x ) of being Meztler stable on the unique condition

Endemic equilibrium
The system (1) admit two equilibriums, one named Disease Free Equilibrium (DFE) defined in the previous subsection and the other named Endemic Equilibrium (EE).
To determine the endemic equilibrium (EE) we must solve equation A(x) × (x * ) T = 0. The model (1) has: (a) if R 0 > 1, the system has a unique endemic equilibrium (b) if R 0 = 1 and R c < 1, the system has a unique endemic equilibrium, (c) if R c < R 0 < 1 and R 0 = R 1 0 or R 0 = R 2 0 , the system has a unique endemic equilibrium, The proof is given in the appendix A.

Global asymptotic stability of the Disease Free Equilibrium (DFE)
In this section we analyze the stability of the system equilibria given in Proposition .
We have the following results for the global asymptotic stability of the disease free equilibrium: Theorem 5.1 Let ζ = µ µ+d , andG = {x ∈ G : x = 0} a positively invariant space. When R 0 ≤ ζ , then the DFE for system (1) is GAS in the sub-domain {x ∈G : x I = 0}.
Proof: Our proof is based on Theorem 4.3 of Kamgang & Sallet [14] , which establishes global asymptotic stability for epidemiological systems that can be expressed in the matrix form (2). We need only establish for the system (1) that the five conditions (h1-h5) required in Theorem 4.3 of Kamgang & Sallet [14] are satisfied when R 0 ≤ ζ .
(h1) The system (1) is defined on a positively invariant set R 10 + of the non-negative orthant. The system is dissipative onG .
The DFE, satisfying the hypotheses H 2 .
(h3) The matrix A I (x) given by (4) is Metzler. The graph shown in the figure below, whose nodes represent the various infected disease states is strongly connected, which shows that the matrix A I is irreductible. In this case, the two properties required for condition After tree iterations, we have The last iteration gives α(Ā I ) < 0 ⇐⇒ R 0 < µ µ + d Since the five conditions for Theorem 4.3 of Kamgang & Sallet [14] are satisfied, the DFE is GAS when R 0 < µ µ + d .
Corollary 5.1 If the disease-induced death rate is 0 (d = 0) then, when R 0 ≤ 1, then the DFE for system (1) is GAS in the sub-domain {x ∈G : x I = 0}.

Bifurcation analysis
To explore the possibility of bifurcation in our system at critical points, we use the centre manifold theory [3]. A bifurcation parameter m is chosen, by solving R 0 = 1, we have is the Jacobian matrix of of system (1) evaluated at the DFE and for m = m The eigenvalues of this matrix are the eigenvalues of the sub-matrix The caracteristic polynom of the matrix J 2 m is given by: When R 0 = 1, 0 is a eigenvalue of the matrix J m The components of the left eigenvector of J(x , m ) are given by A non-zero components correspond to the infected states.
The proof of the previous theorem is based on Theorem 4.1 in [9],

Local sensibility analysis
We have an explicit expression for R 0 , we can evaluate the sensitivity index of different parameters intervening in this expression. In addition to the various parameters involved in the model, we also evaluate the sensitivity index for f q and f r which are calculated parameters derived from the parameters β and χ.
The formula below proposed in [7] gives us the expression of the index of sensitivity of a parameter q to R 0 .

Numerical Simulation
We performed simulations for the particular case of malaria and for SEIRS i.e u = 1 and v = 1 dynamics in humans. All the results of stability, existence of endemic equilibrium and bifurcation established above are illustrated by graphs.

Conclusion
At the end of this work, we studied a generic model of vector-borne diseases incorporating questing-resting dynamics in vectors. We determined the basic reproduction rate and showed that the DFE is GAS when R 0 < ζ . We have also shown that there exist two endemic equilibria when R 0 < 1, for R 0 > 1 the system admits a unique endemic equilibrium and that for R 0 = 1 the system admits a backward bifurcation. We also have at the end of a sensitivity analysis show that the rate of transition from the questing state to the resting f q state and the transition from the resting state to the questing f r state are the parameters More sensitive, which reflects the importance of considering this dynamic.
If R 0 = 1, then the equation (10) has a unique solution − A 1 A 2 which is positive if A 1 < 0 ie R c < 1.
If R 0 = 1 (A 0 = 0), then the equation has a unique solution which is α * = − A 1 A 2 , or A 2 > 0 so this solution is positive if If R 0 < 1 (A 0 > 0), the discriminant of (10) is given by: ∆ is a second degree equation in R 0 and his discriminant ∆ r is define by: So ∆ = 0 admits two distinct solutions R 1 0 and R 2 0 If R 0 = R 1 0 or R 0 = R 2 0 alors ∆ = 0 and equation (10) has a unique solution α * = − If R 0 ∈ [0; R 1 0 [∪]R 1 0 ; +∞[ then ∆ > 0 Hence the equation has two solutions, so the sum and the product are given by s = − A 1 A 2 and p = A 0 A 2 > 0. These two solutions are positive if s > 0 ie