ON SOME REACTION-DIFFUSION EQUATIONS GENERATED BY NON-DOMICILIATED TRIATOMINAE, VECTORS OF CHAGAS DISEASE

. In this work, we study some reaction-diﬀusion equations set in two habitats which model the spatial dispersal of the triatomines, vectors of Chagas disease. We prove in particular that the dispersal operator generates an analytic semigroup in an adequate space and we prove the local existence of the solution for the corresponding Cauchy problem.

1. Introduction. Chagas disease or American trypanosomiasis is a life-threatening disease caused by the flagellated protozoan parasite Trypanosoma cruzi (T.cruzi). It is mainly transmitted by blood-sucking bugs belonging to the subfamily of triatominae. Via these vectors, the parasite can infect humans as well as a large number of domestic or wild mammalians. If bugs live in the nests of non domesticated mammals or birds, they are said to be sylvatic. If they lives in shelters neighboring human habitations, they are said to be domestic.
The different process involved in T.cruzi transmission by non-domiciliated bugs are very complex and their understanding goes through knowledge of vector ecology. In particular, demography and spatial dispersal are important processes during the re-infestation of a domestic area. In most cases, they are not captured by means of laboratory studies. Mathematical modeling stand a good tool to gain insights into these processes [21]. These approach used, for example, ordinary differential equations [15], partial differential equations [24] and integro-difference equations [20], [18]. Other studies based on scientific calculus were also used as cellular automaton [23], [5] and agent-based model [6], [9].
In this study we consider the infestation of a village by the domestic household species T. Dimidiata. The village adjoins a forest representing the habitat of the vectors. The latter move to the village for food [22]. Food consists of a blood meal on humans or the mammals they raise. The transmission of T. Cruzi from the vector to the host takes place mainly during this phase. After the kissing bug, individuals move again to seek habitat either at the village or return to the forest to trigger their fecundity.
In this work, we consider a triatomine population structured in time and space. Demography and spatial dispersal processes are captured by the reaction-diffusion equations in a two-dimensional space. In adequate functional spaces, the partial differential equations system is transformed into an abstract differential equation. Our first aim is to show that the operator generates an analytic semigroup. We prove then the existence of a local solution to the corresponding Cauchy problem.
To simplify the study, we assume that the parts of the border of Ω − ∪ Ω + defined by The common border Γ = ∂Ω − ∩ ∂Ω + , called interface ( Figure 1), plays an important role in the spatio-temporal dynamics of the population. Indeed, the triatomines in the buffer zone Ω + neighboring ∂Ω − ∩ ∂Ω + are attracted towards the village to feed or find refuge. Their movement defines then a skew Brownian motion [2].

Biological considerations.
Although the life cycle of triatomines consists of seven stages of development: an egg stage, five larval stages and an adult stage, the development of the egg in the fifth larval stage is considered to be a single stage which will be called the juvenile stage (Figure2). This assumption is realistic since the adults longevity is greater than the developmental time from the egg stage to the fifth larval stage [19]. Let us denote by J(t, x, y) and A(t, x, y) the respective densities of juveniles and adults classes at time t > 0 and at a point (x, y). Assuming a balanced sex-ratio τ = 1 : 1 (one female for one male). We focus our modeling on female bugs density in the domain Ω − ∪ Ω + . We assume that, inside this domain, the demographic parameters do not depend on the spatial position. During a time step dt between t and t + dt, juveniles having survived until t with a probability s j (t) will remain juveniles with a probability w j (t) or will transit to adult stage with a probability (1 − w j (t)). Adults who survived with a probability s a (t) will lay eggs with a rate f a (t). The entire life cycle is shown in (Figure 2).

sa(t)fa(t) wj(t)sj(t) Adults sa(t) Juveniles
Except in the neighborhood of the interface, the biological population spatial dispersal in the domain Ω − ∪ Ω + is modeled by a diffusion process for juveniles and adults with respective constants d j > 0 and d a > 0. These coefficients are nonnegative as, whatever their developmental stage, bugs must move for their blood meal [16].
If during the demographic process we assumed identical demographic parameters in Ω − ∪ Ω + , it is clear that the diffusion constants depend on the nature of each part of the domain. We will therefore denote, d j+ and d a+ the diffusion coefficients constants in Ω + and d j− and d a− the diffusion coefficients constants in Ω − . The demographic and diffusion parameters of T. Dimidiata population are summarized in Table 1. 2.3. Reaction-diffusion system. By denoting J − and A − ( respectively J + and A + ) the densities of the triatomines in Ω − , (respectively in Ω + ), the biological system is modeled by the following system of reaction-diffusion equations: on Ω + on Ω + and boundary conditions:  [3], we will also consider the skew brownian motion conditions and the continuity of the densities. (Int.C) where p > 1/2 is the probability to cross the interface from Ω + to Ω − . We put: These two matrices are operator matrices.
The following operator L (which acts with respect to the spatial variables (x, y)) is then defined by: where q ∈]1, +∞[. The following classic operational vector notation will be adopted So the previous problem is written in the form of an abstract reaction-diffusion system: which is equivalent to 3. The spectral equation. The study of problem (1) is based on the spectral equation for complex λ in a sector to specify and on the good behavior of: , in order to show that L generates an analytic semigroup. So, after the resolution of the spectral equation, we have to estimate: where: then (3) gives: with the transmission conditions We explain the previous spectral system: This system can be divided into two subsystems; one of which is governed by and the other by the couple (w − , w + ) (of adults) checking a similar system. The difference between them lies in the diffusion coefficients, which are all different (here, we have assumed them to be constant). Therefore, it will be sufficient to analyze one system. 4. The operational formulation. Let's introduce, in Banach space E = L q (0, 1), operator Λ defined by: For ω ∈ [0, π] we define the sector: It is known that this operator is a closed linear operator with a dense domain and verifies: and there exists a ball B(0, δ), δ > 0, such that ρ(Λ) ⊃ B(0, δ) and the above estimate is true in S π−η ∪ B(0, δ). Here ρ(Λ) denotes the resolvent set of Λ. The following usual operational notation of vector-valued functions: v ± (x)(y) := v ± (x, y) , leads the previous system (3) for the couple (v − , v + ) , in space E, to be formulated by: (S1) Boundary conditions (EC 1v ) are implicit and expressed by the action of Λ.
To estimate the resolvent operator, we must estimate: Note that: 5. Some technical results. We will use the following results.
We then write: K ∈ Sect(ω). The following angle is called the spectral angle of K. Statement2. implies necessarily that K is closed.
Here we will assume that λ is such that where ε 0 is a small fixed number. Consider the two following operators: which have the same domain then for λ such that |arg(λ)| < π − ε 0 , and by an explicit calculus we obtain: We have two possible cases: 1. if |arg(λ)| < π/2 then for all µ > 0 : where α ∈]ε 0 , π/2]; therefore: we deduce that there exists a constant C independent of λ such that is independent of λ, see Proposition 2.1.1 in [13]. In a similar way, we obtain We also deduce that the two following operators: , are well defined and generate analytic semigroups on E, see [13] p.81 and also [1].
Lemma 6.1. Let −∞ < a < b < +∞. Then: bounded, analytic for t > 0 and strongly continuous for t ≥ 0 satisfies moreover There exists M > 0 such that for any f ∈ L q (a, b; E) and any λ ∈ S ω We must show that: ) and all boundary and transmission conditions are verified. For x ∈] − d, 0[, we have: , can be treated similarly.
We have, for x ∈ ]−d, 0[ : , from which we get , so, the first term belongs to an interpolation space and the second is very regular, therefore x → (I)(x) ∈ L q (−d, 0; E); the same is true for the two terms (II)(x) and (III)(x).
For the fourth term we have: due to the Dore-Venni theorem.
and λ is such |arg(λ)| < π − ε 0 . Now, we use the lemma 6.1: We have thanks to lemma 6.1 and (8); there exists two constants M 1 > 0 and M 2 > 0 (independent of λ) such that: hence the existence of a constant C > 0 (independent of λ) such that: A similar estimate is obtained for the other terms. Summing up, we obtain By the same methods, we obtain:

Then in the space
we conclude by the theorem: 7. Return to the evolution equation. We go back to our abstract evolution equation: Since L generates an analytic semigroup in E = [L q (Ω)] 2 (which is exponentially decreasing), then necessarily the solution (if it exists) is written as: with where we have used the operational notation: with, for example, for t > 0 : Here, the representation (10) is well defined, if for all This is checked if, for example, the rates ω j , s j , f a and s a are bounded. Therefore

7.2.
Application of the fixed point theorem. We will apply the fixed point theorem to the equation: Note that we can also consider this equation as a non autonomous linear equation since we know that L + B(t) for every t ≥ 0 generates an analytic semigroup (B(t) is bounded). It can be written as It is well known that the equation: whith g ∈ L q (0, T ; E) has the Lq-maximal regularity. So for all g ∈ L q (0, T ; E), there exists a unique solution Φ ∈ W 1,q 0 (0, T ; E) ∩ L q (0, T ; D(L)), and there exists a constant C 0 > 0 such that On the other hand we know that (D (L) ; E) 1/q,q for the interpolation space, see [12] with continuous injection and therefore there exists a constant C 1 > 0 such that These constants do not depend on T here as L is invertible, the semigroup is exponentially decreasing, from which we obtain the maximal Lq-regularity on (0, +∞). Now let us consider the problem: which has a unique solution given by: Let us introduce the following closed ball of L q (0, T ; E) of center Φ * and radius r ∈ ]0, 1[: ,q 0 (0, T ; E) ∩ L q (0, T ; D(L)) and W − Φ * 1 ≤ r} ; We are going to apply the fixed point theorem for contraction mappings.
1. For every W ∈ B r , the problem has a unique solution V ∈ W 1,q (0, T ; E) ∩ L q (0, T ; D(L)). Let us define the following application: 2. Let us prove thatΨ is a strict contraction on B r . We have where obviously G(., 0) = 0.

Recall that
We know that W − Φ * ∈ W 1,q 0 (0, T ; E) ∩ L q (0, T ; D(L)) and by definition of B r , we have so for all t ∈ [0, T ]: We have 4. Note that all these functions in the sup are positive and are not vector-valued only coefficients. Then in the space E = [L q (Ω)] 2 , we have from wich we obtain that the function: Ψ : B * r −→ W 1,q (0, T ; E) ∩ L q (0, T ; D(L)) W −→ Ψ(W ) = V.