Global stability in a competitive infection-age structured model

We study a competitive infection-age structured SI model between two diseases. The well-posedness of the system is handled by using integrated semigroups theory, while the existence and the stability of disease-free or endemic equilibria are ensured, depending on the basic reproduction number $R_0^x$ and $R_0^y$ of each strain. We then exhibit Lyapunov functionals to analyse the global stability and we prove that the disease-free equilibrium is globally asymptotically stable whenever $\max\{R_0^x, R_0^y\}\leq 1$. With respect to explicit basin of attraction, the competitive exclusion principle occurs in the case where $R_0^x\neq R_0^y$ and $\max\{R_0^x,R_0^y\}>1$, meaning that the strain with the largest $R_0$ persists and eliminates the other strain. In the limit case $R_0^x=R^0_y>1$, an infinite number of endemic equilibria exists and constitute a globally attractive set.


Quentin Richard 1
Abstract. We study a competitive infection-age structured SI model between two diseases. The wellposedness of the system is handled by using integrated semigroups theory, while the existence and the stability of disease-free or endemic equilibria are ensured, depending on the basic reproduction number R x 0 and R y 0 of each strain. We then exhibit Lyapunov functionals to analyse the global stability and we prove that the disease-free equilibrium is globally asymptotically stable whenever max{R x 0 , R y 0 } ≤ 1. With respect to explicit basin of attraction, the competitive exclusion principle occurs in the case where R x 0 = R y 0 and max{R x 0 , R y 0 } > 1, meaning that the strain with the largest R0 persists and eliminates the other strain. In the limit case R x 0 = R 0 y > 1, an infinite number of endemic equilibria exists and constitute a globally attractive set.
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Introduction
In [26], Kermack and McKendrick proposed the first ODE epidemic model. Since then, the literature on this topic is wide and such models are commonly used to predict the evolution of a disease and eventually prevent the apparition of epidemics. Incorporating another continuous variable such as the age since infection [29,30,33,44], the infection-load [38,39] or the time remaining before disease detection [27], the so-called structured epidemiological models are described by transport equations (we refer e.g. to [1,24] for an introduction of such models) and sometimes by transport-diffusion equations [3,4]. In the present paper, we consider the following infection-age structured SI model, that describes the competition between two diseases for a same susceptible Keywords and phrases. Lyapunov function, integrated semigroup, global stability, dynamical systems, structured population dynamics, competitive exclusion population: for every t ≥ 0 and a ≥ 0. More precisely, S(t), x(t, a) and y(t, a) respectively denote the density of susceptible individuals at time t and both infected populations of age a and at time t. The parameter Λ represents the recruitment flux into the susceptible class while µ S , µ x and µ y are the mortality rates of the three populations.
Finally β x and β y describe the transmission rates of both infected populations x and y. Let β x = sup(supp(β x )), β y = sup(supp(β y )) (supp(·) denoting the support of any function) be the maximal age of infectiousness of the corresponding disease.
In the sequel, we will make the following assumption.
Consequently to the latter assumption, the probabilities functions π x : a → e − a 0 µx(s)ds , π y : a → e − a 0 µy(s)ds describe the survival of the corresponding infected population. In the case β y ≡ 0, the system (1) becomes the following infection-age structured model with only one disease: (2) This latter model (2) has been investigated by Thieme and Castillo-Chavez [43,44] with the study of the uniform persistence and local exponential asymptotic stability of the endemic equilibrium. Related epidemic models with delay can be found e.g. in [34,35]. Thereafter, Magal, McCluskey and Webb [29] handled the global stability of the endemic equilibrium of (2), by proving the result below. First define the quantity describing the number of secondary infections produced by a single infected patient. This latter threshold is commonly used in the litterature (see e.g. [21] or more recently [37] for an introduction). First appareared in a demographic context with the work of Dublin and Lotka [13] (see more recently [25,Chapter 9] for more references), it is now frequently used in epidemiology (see e.g. [10,11]) to state if a disease will persist or disappear.
The same result holds when interchanging the indexes x and y. At this point we can note that in [29], the authors mentioned the global asymptotic stability of the disease-free equilibrium in the delicate case R 0 = 1. However, it seems that only the attractiveness is proved, by using Lyapunov functional. The same lack of proof seems to appear also e.g. in [33,38]. The reason for this is twofold. Firstly, in infinite dimensional systems, the stability property is not ensured even if the attractiveness property is (see the Lasalle invariance principle [36]). Secondly, the principle of linearisation used to get the local asymptotic stability fails when R 0 = 1: indeed, we obtain eigenvalues with real part equals to zero. However, we will show in Section 5.3 how to overcome the stability in that case, by using some Lyapunov functional.
Recently, some papers considered structured epidemiological models with two groups of infections, or two paths of infection (see e.g. [6,8,30]) by adding some interaction between the two groups. The global stability of the equilibria and the persistence of the diseases are investigated, leading the to the existence of a R 0 threshold. A very similar model to (1) was analysed by Martcheva and Li [33], where they considered a SIR model with n ≥ 2 different groups of infectious individuals, to see how the emergence of other diseases can affect the dynamics of the susceptible population. The analyse leads to the existence of n thresholds, one for each disease. Then, using persistence results and proving existence of a global attractor as in [29], they enlighten a competitive exclusion principle, meaning that the disease with the biggest R 0 value will asymptotically survive, while the other strains will disappear. This fundamental result in ecology was first postulated by Gause [17]. We refer e.g. to [2,9,12] for similar structured models where this principle occurs.
We first define the following thresholds The system (1) always admits the disease free equilibrium When R x 0 > 1 (resp. R y 0 > 1), we also have an endemic equilibrium given by for every a ≥ 0. Finally, when R x 0 = R y 0 > 1, we have an infinite number of equilibria, given by for every a ≥ 0 and where we can note that E * 1 = E 1 and E * 2 = E 2 . In order to analyse the asymptotic behaviour of the solutions, we let X + = R + × L 1 + (0, ∞) × L 1 + (0, ∞) and we define the sets containing initial infected populations that are in age to contaminate susceptible individuals, with the corresponding disease, now or in the future. The convergence results, obtained in the present paper, that depend on the thresholds R x 0 , R y 0 and on the initial condition, are summed up in the following table: We notice that for each k ∈ {x, y}, when taking an initial condition in ∂S k the solutions behave as in the case (2), that is to say either the initial condition is taken in ∂S k and the solution goes to E 0 , or it is taken in S k and the fate of the solution depends on the threshold R k 0 . Furthermore, we prove that for each value R x 0 and R y 0 , the equilibria E 0 , E 1 and E 2 are globally asymptotically stable in the corresponding basin of attraction, according to Figure 1. The competitive exclusion principle is then verified and we also handle e.g. the global asymptotic stability of E 0 in X + when max{R x 0 , R y 0 } = 1. However, the stability of the set of equilibria {E * α , α ∈ [1, 2]} is left open.
We first use the integrated semigroup theory, following [29], to get an appropriate framework in order to prove that (1) is well-posed. It also allows us to linearise the system around each equilibrium, obtaining linear C 0 -semigroups, then we use spectral theory to get the local stability of the equilibria (see e.g. [15,46,47] for more results on this topic). In [29], the authors combine uniform persistence results due to Hale and Waltman [20], with results obtained in [32], to get the existence of a global attractor. While the same approach was used in [33], we follow [38] and we take advantages of an explicit formulation of the semiflow that enables us to exhibit the compactness of the orbits.
The method then used to perform the global analysis is based on the existence of a Lyapunov function (see e.g. [23] for a survey of such functions in various ecological ODE and reaction-diffusion models). We therefore use the following key non-negative function: that was first used by Goh [18] and Hsu [22]. For the present model, we shall also use the following Volterra-type Lyapunov, incorporating the age-structure: for any function φ > 0 a.e. with x * the equilibrium and φ some appropriate function. It was introduced in [29], and was later used e.g. in [12,30,33,38] for structured models. Note that similar functionals are used for delayed equations (see e.g. [40] and the references therein). The latter attractiveness combined with the stability then yield the global asymptotic stability of the corresponding equilibrium.
Note that the technique used in the present paper, contrarily to [33], allows us to study the case where the maximal reproduction number is not unique, that is when R x 0 = R y 0 . As written in Figure 1, the set of equilibria {E * α , α ∈ [1, 2]} is proved to be globally attractive in S x ∩ S y . Finally, following [16] and [40], we handle the stability of the disease-free equilibrium E 0 in the case max{R x 0 , R y 0 } = 1, by making use of the Lyapunov functionals.
This article is structured as follows: in Section 2 we give the preliminaries results concerning existence, uniqueness and boundedness of the solutions. In Section 3 we handle the stability of each equilibrium. Section 4 then deals with the existence of a compact attractor for the dynamical system and the identification of the basins of attraction. In Section 5 we investigate the global analysis of (1). We start by defining suitable Lyapunov functionals and proving their well-posedness. It allows to prove on one hand the global attractiveness of each equilibrium, by using a Lasalle invariance principle theorem, and on the other hand the stability of the disease free equilibrium when the principle of linearisation fails. Finally, we conclude about the global stability of each equilibrium. We end the paper with some numerical simulations in Section 6 to illustrate the above results.

Integrated semigroup formulation
In this section, we handle the well-posedness of (1). To this end, we follow [29] and we use integrated semigroups theory (see e.g. [31] and the references therein for more details), whose approach was introduced by Thieme [42]. First we consider the spaceX = R × L 1 (0, ∞) then we define the linear operatorsÂ x : D(Â k ) ⊂X →X andÂ y : D(Â k ) ⊂X →X bŷ If λ ∈ C is such that ℜ(λ) > −µ 0 , then λ ∈ ρ(Â x ) ∩ ρ(Â y ) (the resolvent sets ofÂ x andÂ y respectively), and we have the following explicit formula for the resolvent ofÂ k (with k ∈ {x, y}): We can notice that (1) is equivalent to we can then rewrite (5) as an ordinary differential equation coupled with two non-densely defined Cauchy problem: Consider the sets and define the linear operator A : We then see that (the closure of D(A)), so that D(A) is not dense in X. Now, define the non-linear function F : and its positive cone We can thus rewrite (1) as the following abstract Cauchy problem: where u(t) := (S(t), x(t, ·), y(t, ·)) and u 0 = (S 0 , x 0 , y 0 ).

Local existence and positivity
Using the above semigroup formulation, we can state the classical following result: Proposition 2.1. Suppose that Assumption 1.1 holds. Then there exists a unique continuous semiflow {U (t)} t≥0 on X 0+ such that for every z ∈ X 0+ there exist t max ≤ ∞ and a continuous map U ∈ C([0, t max ), X 0+ ) which is an integrated solution of (6), i.e. such that and Proof. On one hand, the explicit expression (4) of the resolvent ofÂ k for each k ∈ {x, y} ensures us that for some c > 0 and for every n ≥ 1, so that A is a Hille-Yosida operator with (−µ 0 , ∞) ⊂ ρ(A). On the other hand, we can check that the non-linear function F is Lipschitz continuous. Using [28,Proposition 3.2] or [5, Proposition 4.3.3, p. 56] we get the local existence. Now, from (4) we deduce that A is resolvent positive, that is to say (λI − A) −1 X + ⊂ X + , ∀λ ∈ ρ(A). Moreover, the expression of the non-linearity F implies that for every r > 0, there exists c ≥ 0 such that where B(0, r) denotes the ball of X, centred in 0 ∈ X and with radius r. Finally, using [28, Proposition 3.6], we get the non-negativity of the solution.

Boundedness and global existence
Let the Banach space X := R × L 1 (0, ∞) × L 1 (0, ∞) endowed with the usual norm and denote by X + its positive cone. We are ready to give the main result of this section: Theorem 2.2. Suppose that Assumption 1.1 holds. Then for every z = (S 0 , x 0 , y 0 ) ∈ X + , there exists a unique mild solution (S, x, y) ∈ C(R + , X + ), that induces a continuous semiflow via: rewrites using Duhamel formulation as follows: with Φ S t (z) > 0 for every t > 0 and every z ∈ X + . Finally, Φ x t and Φ y t are given by: where χ denotes the characteristic function. Moreover, there exists a constant k (independent of z), such that Proof. Let z := (S 0 , x 0 , y 0 ) ∈ X + and (S, x, y) ∈ C([0, t max ), X + ) be the solution of (1). Suppose by contradiction that t max < ∞. It would imply by [ From (1) we see that by using a Gronwall argument. Now, an integration of (1) leads to ). Thus we have x(t, a)da and then lim sup Similarly, we get lim sup Consequently, we get a contradiction with (6) and then t max = ∞. Finally, from (12)-(13)- (14), we deduce that the solutions are asymptotically uniformly bounded, since the bound do not depend on the initial condition.

Equilibria and their stability
As mentioned in the introduction, we have the following result concerning the existence of equilibria: We start by reminding the following classical definition Definition 3.2. Let S ⊂ X + be a subset of X + and E be an equilibrium of (1). Then we say that E is • locally asymptotically stable (L.A.S.) in S if E is stable and locally attractive in S; • globally attractive in S if for every z ∈ S, (15) is satisfied; • globally asymptotically stable (G.A.S.) in S if E * is stable and globally attractive in S.
In the following, for notational simplicity, we will not specify the subset S if the latter is the whole positive cone, i.e. when S = X + . We now handle the stability of the equilibria formerly defined. ( Let E := (S, x, y) be an equilibrium of (1), then the linearised system of (1) around E is: where DF E : X → X denotes the differential of F around E and is defined by: We then have on one hand: On the other hand, from its above expression, we see that DF E (X) is finite dimensional, so that DF E is a compact bounded operator. From [14, Theorem 1.2] we get is finite and composed (at most) of isolated eigenvalues with finite algebraic multiplicity, where σ(.) denotes the spectrum. Consequently, it remains to study the punctual spectrum of (A + DF E ) 0 . Using [46,Proposition 4.19,p. 206], we know that if s(A + DF E ) < 0 then E is L.A.S., while if s(A + DF E ) > 0 then E is unstable. We consider exponential solutions, i.e. of the form u(t) = e λt v, with 0 = v := (S, x, y) ∈ D(A) and λ ∈ C. We obtain the following system: We then get x(a) = x(0)π x (a), y(a) = y(0)π y (a) for every a ≥ 0, and with (S, x(0), y(0)) = (0, 0, 0).
so we have y(0) = 0. Likewise we deduce that x(0) = 0, but it then follows that S = 0, which is absurd. Consequently E 0 is L.A.S. (2) Let E := E 1 . From (16)-(17), we get: Suppose that R y 0 > R x 0 > 1, then when y(0) = 0, we obtain We see that the function is strictly decreasing, with f (0) = S * 1 r y = rx ry > 1, and we deduce that which is absurd, so y(0) = 0. We deduce that Considering real and imaginary parts of λ, we get: then identifying the real part, we obtain: It follows that Similar arguments as for the latter point allow us to prove the result for E 2 .
hence E * α is not locally attractive, and therefore not L.A.S.

Preliminaries
In the sequel, we will denote by O z = {Φ t (z), t ≥ 0} the orbit starting from z ∈ X + and the ω-limit set of z. We follow [38,Section 3], to prove the existence of a compact attractor.
Proof. Define the ball B r := {z ∈ X , z X ≤ r} for any r > 0. From (7)- (9), we see that for every r > 0 and Moreover we can prove that for any t ≥ 0, (Φ S t , Φ x,2 t , Φ y,2 t ) maps bounded sets of X + into relatively compact sets in X . Indeed, let M ⊂ X + be a bounded subset of X , i.e. there exists r > 0 such that z X ≤ r for any z ∈ M . First, we see that Φ S t (M ) is relatively compact since it is finite dimensional. Moreover, from Theorem 2.2, we deduce that for every t ≥ 0, there exists a constant c(r) > 0 such that for any (t, z) ∈ R + × M . Finally, the Frchet-Kolmogorov theorem ensures that the sets Φ x,2 t (M ) and Φ y,2 t (M ) are relatively compact. From [46, Proposition 3.1.3 p. 100], we deduce that for every z ∈ X + , the orbit O x is relatively compact.
The latter compactness result of the orbits then leads to the existence of a compact attractor in the following sense (see e.g. [ (1) ω(z) is non-empty, compact and connected; We remind the following classical result and we give its proof for completeness.
and u be the solution of the PDE: for every a ∈ (0, c) and every t ≥ 0. Suppose that u 0 ∈ L 1 + (0, c) \ {0} and that The same holds when replacing x by y.
Proof. Define the linear operator A : is compact, then A has a compact resolvent, and consequently the spectrum of A is composed at most of isolated eigenvalues with finite algebraic multiplicity. This follows from the fact that the canonical injection i : (D(A), · D(A) → (L 1 (0, c), · L 1 (0,c) ) is compact by the Rellich-Kondrachov Theorem. Any eigenvalue of A has to satisfy: where u ∈ D(A). We hence get the following characteristic equation: We see that We deduce that (λ − A) −1 is positivity improving. Using [7, p. 165], we deduce that {T A (t)} t≥0 is irreducible, i.e. for any φ ∈ L 1 + (0, ∞) \ {0} and any ψ ∈ L ∞ + (0, ∞) \ {0}, there exists t > 0 such that T A (t)φ, ψ > 0, where ·, · denotes the duality pairing between L 1 and L ∞ . Since the semigroup is positive, we know that Moreover, since the spectrum of A is punctual, then Consequently {T A (t)} t≥0 is both irreducible and has a spectral gap (i.e. ω 0 > ω ess ). On one hand we know that s(A) is a simple pole of the resolvent of A, with geometric multiplicity equal to one (see e.g. [7, p. 224

Basins of attraction
We now give some results about the attractive sets, depending on the initial condition as well as the thresholds R x 0 and R y 0 . Proposition 4.4. Suppose that Assumptions 1.1 holds, then: (1) the sets ∂S x and ∂S y are positively invariant, i.e. Φ t (∂S x ) ⊂ ∂S x and Φ t (∂S y ) ⊂ ∂S y , ∀t ≥ 0. Moreover, for every z := (x 0 , y 0 , z 0 ) ∈ ∂S x (respectively z ∈ ∂S y ), then for every t ≥ 0; (2) the equilibrium E 0 is globally exponentially stable for Φ t restricted to ∂S x ∩ ∂S y ; (3) there exists c > 0 such that for every z ∈ X + we have: (4) for every z ∈ S x (resp. z ∈ S y ), there exists τ ≥ 0 such that for every t ≥ τ . Moreover, the sets S x and S y are asymptotically positively invariant, i.e. for every (1) Let z ∈ ∂S x . We remind that the component in x of the semiflow rewrites as Φ where Φ x,1 t (z) and Φ x,2 t (z) are respectively defined in (7) and (8). We see that Then a Gronwall argument states that F (t) = 0 for every t ≥ 0 and we deduce from (8) that Φ x,2 t (z)(a) = 0 for every t ≥ 0 and every a ≥ 0. Consequently we get for every t ≥ 0, thus ∂S x is positively invariant. Moreover, we can deduce that for every t ≥ 0 by using (7) and Assumption 1.1. Similar arguments would prove on one hand that ∂S y is positively invariant, and on the other hand that (18) holds for every z := (x 0 , y 0 , z 0 ) ∈ ∂S y and every t ≥ 0 by using (10) and Assumption 1.1. (2) Let z := (x 0 , y 0 , z 0 ) ∈ ∂S x ∩ ∂S y . Using the first point, we have for every t ≥ 0. Using (18), we deduce which proves the second point.
(3) Let z ∈ X + and let (S, x, y) ∈ C(R + , X + ) the solution of (1). By Theorem 2.2, we know that there exists k > 0 (independent of z) such that lim sup Injecting the latter equation into (1) implies that for every ε > 0, there exists t 0 > 0 such that x 0 (a)da > 0.
By Assumption 1.1, we may find c ∈ (β x , ∞) such that β x (a) > 0 a.e. a ∈ [β x , c). Let t 0 = c − b 2 , then using (7), we see that Since β x > 0 a.e. on [β x , c] and Φ S t (z) > 0 for every t > 0 due to Theorem 2.2, then we get Φ x t0 (z)(0) > 0 by using (8). By continuity arguments, there exists t 1 > t 0 such that Let ∆ t = t 1 − t 0 > 0 and let 0 < ε << c − β x , then we see that Similarly we can prove that Let 0 < ε < c and t = t * + c + ε, then we get Hence we deduce that Φ x t (z)(0) > 0, ∀t ∈ [t * , t * + 2c] then repeating this argument we obtain Finally, we obtain for every t > t * , so that S x is asymptotically positively invariant. The same arguments would prove the result for y. We deduce that (S, y) satisfies (2). If R y 0 > 1 and z ∈ S y , then from Proposition 1.2 we obtain whence ω(z) ⊂ S y . If R y 0 ≤ 1, we deduce from Proposition 1.2 that (6) The latter arguments would prove the case z ∈ ∂S y . (7) Let z ∈ X + : (a) Suppose that R x 0 ≤ 1. A simple upper bound on (1) leads to since Φ y t (z)(a) ≥ 0 a.e. a ≥ 0. From Proposition 1.2, we obtain The same argument proves the result when R y 0 ≤ 1.
(8) Let z ∈ S x ∩ S y : (a) Suppose that max{R x 0 , R y 0 } > 1. Without loss of generality, we can suppose that R y 0 > 1. By continuity arguments, there exists β y < c < ∞ such that Λ c 0 β y (a)π y (a)da µ S > 1 and there exists ε > 0 small enough such that Let M ε := {z ∈ S x ∩ S y , z − E 0 X ≤ ε}. We first prove that for every z ∈ M ε , there exists t(z) such that Φ t (z) − E 0 X > ε (20) holds. By contradiction, suppose that there exists z := (S 0 , x 0 , y 0 ) ∈ S x ∩ S y such that We know by Proposition 4.4 (4) that there exists τ ≥ 0 such that Φ t (z) ∈ S x ∩ S y for every t ≥ τ . Thus, a Gronwall argument leads to , ∀t ≥ 0. Now, we denote for convenience y(t, a) = Φ y t (z)(a), and we deduce from (1) that y satisfies the following system: ∂t + ∂y(t, a) ∂a = −µ y (a)y(t, a),  (20) is proved. Therefore we obtain Now, consider z ∈ S x ∩ S y , and suppose by contradiction that there exists w ∈ ω(z) ∩ ∂S x ∩ ∂S y . The invariance of ω(z) (due to Lemma 4.2) then gives ω(w) = ω(z) and so A consequence of Lemma 4.2 and Proposition 4.4 (2), is that d(ω(z), E 0 ) = 0 and so {E 0 } ⊂ ω(z) which contradicts (22), whence ω(z) ⊂ S x ∪ S y for any z ∈ S x ∩ S y .
(b) Suppose that R x 0 > max{1, R y 0 }. First suppose that R y 0 ≤ 1, then using Proposition 4.4 (8.a) and (7.b), we deduce that ω(z) ⊂ (S x ∪ S y ) ∩ ∂S y = S x ∩ ∂S y ⊂ S x . Now suppose that R x 0 > R y 0 > 1. We see that r x > r y , so we can consider ε > 0 small enough such that r x (1/r y − ε) > 1. We then define the set M ε := {(S 0 , x 0 , y 0 ) ∈ S x ∩ S y , x 0 L 1 (R+) ≤ ε} and we aim to prove that for every z ∈ M ε , there exists t(z) such that By contradiction, suppose that there exists z := (S 0 , x 0 , y 0 ) such that Denoting (Φ S t (z), Φ x t (z), Φ y t (z)) = (S, x, y) for notational simplicity, we deduce from (1) that S satisfies the following inequalities where r y is defined in Section 1. Consequently, there existst ≥ 0 such that for every t ≥t, we have

Consider the following models
By definition of ε and by continuity arguments, there exists β x < c < ∞ such that From (1), we deduce that x satisfies: (t, a), for every a ∈ [0, c] and every t ≥t. We then have x(t, a) ≥x(t, a) wherex is the solution of the latter system, with an equality instead of the inequality, for every t ≥ τ and a.e. a ∈ [0, c]. We see that the  (24), hence (23) holds. We deduce that Let z ∈ S x ∩ S y and suppose that there exists w ∈ ω(z) ∩ ∂S x , then by Lemma 4.2 and Proposition 4.4 (5.a), whence {E 2 } ⊂ ω(z) which contradicts (26). Consequently we have ω(z) ⊂ S x for any z ∈ S x ∩ S y .
(c) The latter argument proves that whenever R y 0 > max{1, R x 0 }, then ω(z) ⊂ S y for any z ∈ S x ∩ S y .
Remark 4.5. We can note that to prove the item 4 of the latter proposition, we may not need to assume the item 2 of Assumption 1.1: namely the existence of β x and β y . Indeed, we can make use of irreducible operators to prove the statement, as in [30,Lemma 5.1]. However we still make the assumption, since the sketch of proof would be tedious and not add much to the result.

Global analysis
In this section, we aim to prove that the equilibria defined in Section 1, satisfy a global stability property. To this end, we use Lyapunov functionals.

Lyapunov functionals
We define da for any z = (S, x, y) ∈ X , where Ψ x ∈ L ∞ + (0, ∞) and Ψ y ∈ L ∞ + (0, ∞) are defined by for every a ≥ 0, and we remind that the other parameters are defined in Section 1. We first start with a well-posedness result:

Proof.
(1) By Theorem 2.2, we know that the semiflow Φ t is positive, and that Φ S t > 0 for every t > 0, so it proves the first point.
(2) Suppose that R x 0 > max{1, R y 0 } and let z ∈ S x . Either z ∈ ∂S y , so from Proposition 4.4 (6.a), we deduce that ω(z) ⊂ S x ∩ ∂S y , or z ∈ S y and we deduce from Proposition 4.4 (8.b) that ω(z) ⊂ S x . Moreover, Proposition 4.4 (3) ensures us that is well-defined for every t ≥ 0 and every v ∈ ω(z). We now prove that there exists a positive constant c(z) > 0, such that for every a ≥ 0, t ≥ 0 and v ∈ ω(z). Following [38, Proposition 2], we note that the definition of the function g (in (3)), implies that the following inequality holds: Let t ≥ 0 and v ∈ ω(z), then we deduce that the middle term of (27) is given by Thus, to prove (27), it suffices to prove that there exists a constant c(z), such that for every t ≥ 0 and every v ∈ ω(z). From Proposition 4.4 (4), we know that there exists τ ≥ 0 such that We deduce that Since ω(z) is compact (by Lemma 4.2), then a continuity argument ensures us with the existence of a constant c(z) (independent of v) such that Suppose that (t, a) ∈ (R + ) 2 such that t > a. From (8), (29) and Proposition 4.4 (3), we know that there exist two constants δ > 0 and c(z) > 0 such that for every v ∈ ω(z). By definition of x * 1 (see Section 1), we see that for every v ∈ ω(z), and consequently which proves (28) for any v ∈ ω(z) and every (t, a) ∈ (R + ) 2 such that t > a. Now, suppose that a ≥ t.
Since ω(z) ⊂ S x is invariant under the semiflow, then using [41, p. 26], we deduce that for any v ∈ ω(z), there exists a full orbit ξ −→ u v (ξ), for every ξ ∈ R, passing through v, i.e. satisfying: It then suffices to consider s ∈ R, such that t + s > a. Since u v (−s) ∈ ω(z), we deduce from (30) that which proves (28) for any v ∈ ω(z) and every (t, a) ∈ (R + ) 2 such that a ≥ t. We have then proved that (28) (and consequently (27)) holds for every (t, a, v) ∈ R + × R + × ω(z). Finally, the integrability on R + of the functions (3) Suppose that R y 0 > max{1, R x 0 } and let z ∈ S y . Either z ∈ ∂S x , so we see that ω(z) ⊂ ∂S x ∩ S y by using Proposition 4.4 (5.a), or z ∈ S x and we deduce from Proposition 4.4 (8.c) that ω(z) ⊂ S y . Using Proposition 4.4 (3), we see that is well-defined for every t ≥ 0 and every v ∈ ω(z). Similar computations as for proving (27) imply that there exists a positive constant c(z) > 0 such that for every a ≥ 0, t ≥ 0 and v ∈ ω(z). Finally we prove as above that the function (t, v) −→ L y (Φ t (v)) is well-defined on R + × ω(z) for every z ∈ S y . (4) Suppose now that R x 0 = R y 0 > 1. From Proposition 4.4 (8.a), we know that ω(z) ⊂ S x ∪S y . Consequently, either ω(z) ⊂ S x and we use the first point, to prove that the function (t, v) −→ L x (Φ t (v)) is well-defined on R + × ω(z), or ω(z) ⊂ S y and we use the second point, to prove that the function (t, v) −→ L y (Φ t (v)) is well-defined on R + × ω(z).
We remind the following definition: Definition 5.2. Let S ⊂ X . A function L : X → R is called a Lyapunov function if there hold that: • L is continuous on S (the closure of S in X ); • the function R + ∋ t −→ L(Φ t (z)) is non-increasing for every z ∈ S.
We now show that L 0 , L x and L y are Lyapunov functionals.
then L x is a Lyapunov function on ω(z) for every z ∈ S x ∩ S y such that ω(z) ⊂ S x . Moreover, L y is a Lyapunov function on ω(z) for every z ∈ S x ∩ S y such that ω(z) ⊂ S y . Proof.
(2) Suppose that R x 0 > max{1, R y 0 } and let z ∈ S x . Then L x is well-defined on ω(z) from Proposition 5.1 (1), and is clearly continuous. Let v ∈ ω(z), then Now, we compute each term. The fact that Now, we compute the second term: We remark that Thus, after an integration by parts we obtain: da.
since Ψ x (∞) = 0. Using (31) and the fact that x * since S * 1 = 1/r x . After an integration by parts, we see that the third term reads as by using (32). Now, adding (34) and (35), we see that: We remark that and we deduce that Now, adding (36) and (37), and recalling that S * 1 = 1/r x , we obtain: for any t ≥ 0 since g is a non-negative function and the fact that Consequently L x is a Lyapunov function on ω(z) for every z ∈ S x when R x 0 > max{1, R y 0 }. (3) Suppose that R y 0 > max{1, R x 0 } and let z ∈ S y . Then L y is well-defined on ω(z) from Proposition 5.1 (2), and is clearly continuous. Let v ∈ ω(z). After similar computations as above, a differentiation of L y w.r.t. t along (1) gives: for any t ≥ 0. We deduce that L y is a Lyapunov function on ω(z) for every z ∈ S y whenever R y 0 > max{1, R x 0 }. (4) Now, suppose that R x 0 = R y 0 > 1 and let z ∈ S x ∩ S y . We know by Proposition 4.4 (8.a) that ω(z) ⊂ S x ∪ S y . If ω(z) ⊂ S x , then using Proposition 5.1 (3), we know that the function L x is well-defined on ω(z) and is continuous. Let v ∈ ω(z). From (38) we see that for any t ≥ 0 since R x 0 = R y 0 ⇐⇒ r x = r y . Thus L x is a Lyapunov function on ω(z) for every z ∈ S x ∩ S y such that ω(z) ⊂ S x . Similarly, if ω(z) ⊂ S y , we know by Proposition 5.1 (3) that L y is well-defined on ω(z) and is continuous. Let v ∈ ω(z). From (39) we deduce that for any t ≥ 0. Thus L y is a Lyapunov function on ω(z) for every z ∈ S x ∩ S y such that ω(z) ⊂ S y .

Attractiveness
Using the Lyapunov functionals defined above, we can compute the basin of attraction of each equilibrium, by means of the Lasalle invariance principle (see e.g. [36,Corollary 2.3]).
Remark 5.5. We can note that the first point could also be proved by using the Lyapunov functional L 0 .

Lyapunov stability
In this section, we handle the stability of E 0 in the cases where the principle of linearisation (Proposition 3.3) fails.
(2) Suppose that R x 0 > 1. Let z := (S 0 , x 0 , y 0 ) ∈ B(E 0 , ν) ∩ ∂S y . The former arguments and the fact that the function t −→ L 0 (Φ t (z)) is non-increasing imply that E 0 is stable in ∂S y whenever R x 0 > 1. (3) It follows from the last point and interchanging the index x and y.
While the stability of E 0 in the critical cases are handled in the latter proposition, the question of the stability of the set {E * α , α ∈ [1, 2]} when R x 0 = R y 0 > 1 is open. The use of Lyapunov functional in the latter proof will raise some problems due to the fact that L x and L y are not defined in X + .

Global asymptotic stability
We are ready to give the main result of the paper: Theorem 5.9. The following hold: (1) E 0 is G.A.S. in ∂S x ∩ ∂S y . Moreover, it is also G.A.S. in (a) X + if max{R x 0 , R y 0 } ≤ 1; (b) ∂S y if R x 0 ≤ 1; (c) ∂S x if R y 0 ≤ 1.
(a) Suppose that max{R x 0 , R y 0 } ≤ 1. From Proposition 5.4 (1), we know that E 0 is globally attractive in X + . Using Proposition 3.3 (1) and Proposition 5.6, we deduce that E 0 is Lyapunov stable, whence the global asymptotic stability in X + . (b) Suppose that R x 0 ≤ 1. It follows from Proposition 4.4 (6.b) that E 0 is globally attractive in ∂S y , and from Proposition 5.6 that E 0 is stable in ∂S y . (c) When R y 0 ≤ 1, the result follows from Proposition 4.4 (5.b) and Proposition 5.6.

Numerical simulations and final remarks
We start this section by some illustrations of the main results. We plot the total quantity of individuals, i.e. the L 1 -norm for x and y, in function of time. We also consider two different initial conditions (in line and dotted line) in S x ∩ S y . In Figures 2 and 3, the competitive exclusion principle applies: the disease with the biggest R 0 value persists while the other one go extinct. In Figure 4, the two solutions (corresponding to both initial conditions), converge to two different equilibria belonging to the set {E * α , α ∈ [1, 2]}. We can note that the results obtained in the paper could be extended to the general case (N ≥ 3): ∞ 0 β xn (a)x n (t, a)da, ∂x n ∂t (t, a) + ∂x n ∂a (t, a) = −µ xn (a)x n (t, a), x n (t, 0) = S(t) ∞ 0 β xn (a)x n (t, a)da, (S(0), x 1 (0, ·), · · · , x N (0, ·)) = (S 0 , x 0 1 , · · · , x 0 N ) ∈ R + × (L 1 + (0, ∞)) N for every n ∈ 1, N . As we noticed with (1), considering an initial condition in ∂S xn for some n ∈ 1, N amounts to study the N − 1 dimensional case. Therefore, even if the number of cases increase exponentially, only the set S x1 × · · · × S xN is important for the initial conditions. In that situation, the competition exclusive principle applies whenever there exists i ∈ 1, N such that R xi 0 > R x,j 0 for every j ∈ 1, N \ {i}, that is: the disease x i persists while all the other go extinct. When the maximum if not unique, we can prove the existence of an infinite number of equilibria, that constitute a global attractive set, whose stability is an open problem.