A two-group age of infection epidemic model with periodic behavioral changes

In this paper we propose a two-group SIR age of infection epidemic model by incorporating periodical behavioral changes for both susceptible and infected individuals. Our model allows different incubation periods for the two groups. It is proved in this paper that the persistence and extinction of the disease are determined by a threshold condition given in term of the basic reproductive number \begin{document}$ R_0 $\end{document} . That is, the disease is uniformly persistent if \begin{document}$ R_0 >1 $\end{document} with the existence of a positive periodic solution, while the disease goes to extinction if \begin{document}$ R_0 with the global asymptotic stability of the disease free periodic solution. The model we have proposed is general and can be applied to a wide class of diseases.


1.
Introduction. Mathematical modeling has become increasingly sophisticated and precise in the description of population dynamics. Several aspects are incoporated in models depending on the nature of the problem being studied. Structuring such as age of infection, chronological age, space ... have emerged as important elements in understanding population dynamics. We refer to the monographs [1,7,8,16,23,28,34] for an extensive discussion and numerious examples. Here we are interested not only in the age structure of infection but also in the behavior of individuals in the face of a disease. When a disease is spread by close contact between two individuals, behavior between individuals becomes a significant factor in the spread of the disease. Diseases such as Tuberculosis, HIV/AIDS, SARS, H1N1 are concrete examples of these situations. In fact its well known that individuals are able to change their behavior to avoid the risk of infection when they are aware of the apparent signs or not of an infection [5,20]. Here we have opted to model the behavior change by dividing the population into two groups. The first is made up of knowledgeable individuals who behave in a manner that avoids contamination, and the second group consists of individuals who behave normally. Both groups are compartmentalized into Susceptible-Infected-Removed classes. The model we consider in this paper is the following [β 11 (a)i 1 (t, a) + β 12 (a)i 2 (t, a)]da, supplemented with the initial conditions S k (0) = S k0 ∈ R + and i(0, ·) = i k0 ∈ L 1 + (R + , R), for k = 1, 2.
The state variables S k (t) and i k (t, a), k = 1, 2 describe respectively the number of susceptible at time t and the density of infected individuals at time t with time since infection a. The functions t → λ k (t), k = 1, 2 are the time periodic rates of recruitment of individuals in the susceptible classes while the parameter µ is the natural death rate. The age dependent function a → d k (a), k = 1, 2 are the rates at which individuals are removed due to death, recovery or isolation. The time periodic functions t → k (t) and t → ν k (t) describe the rates at which individuals move from one group to another respectively for the susceptible class and the infected class. Finally the map a → β ij (a), 1 ≤ i, j ≤ 2 is the rate at which suscepetible individuals of group i are contaminated by infected individuals of group j with age of infection a.
There have been many non-autonomous periodic coefficients epidemic models that have flourished in the literature in recent years. However most are systems of ordinary differential equations. Periodic delay differential equations have also gained many attentions and several works have been done in this subject. We refer among other to the works of [3,11,13,24,37,32,36]. The definition of a threshold by the basic reproduction number to know if it will be extinguished or persistence of a disease in the case of non-autonomous models has receive a lot of attentions. We can mention the works of [33,37] respectively on the systems of differential equations and delay differential equations with periodic coefficients. Age of infection structured models with periodic coefficients are slightly less numerous in the literature. However, the work of [29] concerning the renewal theorem can be of great help in the study of these problems. The results in [2,9,18,31] have also contributed a lot to these issues. One of the most common mathematical questions in non-autonomous epidemiological models is the question of uniform persistence and the coexistence of the endemic state and the disease-free state. The works of [6,12,19,26,27,38] allowed for a better understanding of these issues. We refer for instance to [26,38] for further comments on this subject.
To the best of our knowledge the model (1)- (2) presented in this article has not yet been studied in the literature. However there are models that come close [14,22,24,35] but remain different. In [14] the authors studied an autonomous two group age of infection epidemic model with criss-cross transmission. In [24] the author dealt with a one group model structured in age of infection with periodic coefficients whereas in [22] the author dealt with an autonomous one group age of infection model with immigration of new individuals into the susceptible, latent and infectious classes. In [35] an SIS autonomous age dependent infection model on a heterogeneous network were discussed. Let us also mention the work of [11] where an age structured model is considered by allowing the birth and death functions to be density dependent and periodic in time. In this manuscript we consider a two groups model with periodic coefficients that incorporates the criss-cross transmission, the within contamination and the change of behaviors of the individuals towards the disease. We also allow the possibility of having different incubation periods which makes our model applicable to several kind of diseases.
The paper is organized as follow. In Section 2 we deal with the existence of non negative solutions of system (1)-(2). We give a Volterra integral formulation and prove some properties of the model. The boundedness (upper and lower bound) of the solutions and the dissipativity are studied in Section 3. In Section 4 we give a description of the disease free periodic solution and give its relationship with the solutions of system (1)- (2). Section 5 is devoted to some preparatory results. In fact in Section 5 we consider a perturbed linear age structured model and study its asymptotic properties by using the renewal theorem [29]. The results obtained from the perturbed linear equation are combined with comparison principles in Section 6 in order to prove the global stability of the disease free periodic solution for system (1)-(2) when the basic reproductive is strictly less that one. In Section 7 we prove the uniform persistence of the periodic semiflow when the basic reproductive number is strictly greater that one. We also prove some extinction results of the model without any condition on the basic reproductive number whenever the initial distributions are take in some sub domain. We end up the paper in Section 8 by proving a coexistence result. More precisely we prove that when the basic reproductive is strictly greater that one then there exists an endemic periodic solution in addition to the disease free periodic solution.
2. Existence of non negative solutions. The existence of non negative globally defined solutions of (4) in [0, +∞) is classical and can be done by using either Volterra integral formulation or integrated semigroup formulation [7,15,30,34]. So in this section we will only describe the the solutions by mean of Volterra integral equations. This will allow us to deal with the renewal equation as well as the basic reproductive number by using the results in [29].
The following hypothesis will be of concern.
We denote the maximum age of infection by a . The supports Supp(β ij ), and there exists 0 ≤ a k < +∞ for k = 1, 2 such that We also assume that a − a > 0 (3) with a = max(a 1 , a 2 ). Remark 1. Condition iii) in Assumption 2.1 states that we may have different incubation periods for each group. The incubation periods are taken into account by a 1 for group 1 and a 2 for group 2. The condition Supp(β 11 ) ⊂ Supp(β 21 ) Supp(β 22 ) ⊂ Supp(β 12 ) is made to encounter the possibility of criss-cross transmission that is when β kk ≡ 0 for k = 1, 2. We will rewrite system (1)-(2) into a more compact form by considering the following state variables Next we introduce the following notations for each t ∈ R and a ≥ 0 Because (1)-(2) is non autonomous we will consider the following system with initial In the sequel the following notations will be used . Since we are considering a population of individuals, the sum norm in R 2 is considered that is Furthermore if M = (m ij ) 1≤i,j≤2 is a 2 × 2 matrix we define its norm as We denote by 1 the vector with all elements equal to 1 that is Note that with the above notations we have the supremum norm of b and d are defined by b ∞ := sup where · ∞ denotes the usual supremum norm and Volterra integral formulation: The integration along the characteristics for nonautonomous linear age structured system in general cases have been done in Inaba [10]. Inspired by [10] we will give the Volterra integral formulation for the non linear coupled system (4). We briefly give some details for completeness and later references. The characteristic lines of system (4) are given by dt dl = 1 and da dl = 1.
Therefore in order to integrate system (4) along the characteristics we define for any c 1 ∈ R and c 2 ≥ 0 the map l → L(l; c 1 , c 2 ) ∈ L(R 2 ) as the unique solution of the ordinary differential equation (5) Then along the characteristic lines the i-equation of (4) satisfies the ordinary differential equation Hence Therefore if t − t 0 > a by setting c 2 = 0, l = a and c 1 = t − a > t 0 we obtain Therefore by setting By combining (4) and (6) we see that the maps t → B S (t, t 0 ; S 0 , i 0 ) and t → B I (t, t 0 ; S 0 , i 0 ) must satisfy respectively for each t ≥ 0 and We conclude that t ∈ R + → (B S (t, t 0 ; S 0 , i 0 ), B I (t, t 0 ); S 0 , i 0 ) is the unique continuous solution of the following Volterra integral system of equations with t ≥ 0. Therefore we define for each t 0 ≥ 0 and the unique mild solution of (4) that is to say that Hence {U(t, t 0 )} t≥t0 is an evolution family on R 2 + × L 1 + (R + , R) 2 . More precisely U(t, t 0 ) maps R 2 + × L 1 + (R + , R) 2 into itself for all t ≥ t 0 and satisfies U(t, t) = I, ∀t ≥ 0 and U(t, s)U(s, t 0 ) = U(t, t 0 ), ∀t ≥ s ≥ t 0 .
Furthermore if with t → (S(t), i(t, ·)) the mild solution of (4). Note that since it is clear from (6) and (9) that U is a τ -periodic evolution semiflow that this Moreover (7) and (15) also imply that and Assumption 2.1 ensures that Ψ has a compact support with respect to a. More precisely we have Ψ(t, a) = 0 L(R 2 ) , ∀a ≥ a , ∀t ≥ 0.  Therefore 1 T L(l; c 1 , c 2 ) ≤ e −µl 1 T , ∀l ≥ 0. To obtain (20) we note that since the off diagonal entries of the matrix l → −d(l + c 2 ) + k(l + c 1 ) are non negative and the diagonal entries are uniformly bounded for l + c 1 ∈ R and l + c 1 ≥ 0 there exists γ > 0 such that It is convenient for later references to show the dependence of G and Ψ with respect to the support of β 12 and β 21 . The next two lemmas give our precise statements. The first one is a direct consequence of (17), (18) and (19). Lemma 2.3. Let Assumption 2.1 be satisfied. Then Ψ is a periodic kernel with compact support. More precisely we have and with 1 T = (1, 1). Thus using (8) and (19) we get for each t ≥ 0, each t 0 ≥ 0 and the result follows easily.
3. Boundedness of the solutions. In this section we will prove that the mild solutions of (4) have upper and lower bounds. Define Before proving the boundedness properties we recall that R 2 is endowed with the sum norm. More precisely we have set Then we have Upper bound: Using the initial conditions of (4) giving classical solutions we obtain that the map t → (S(t), I(t)) satisfies Since the maps t → S(t) and t → I(t) are non negative it follows that t → S(t) and t → I(t) are derivable with respect to t. Furthermore we have for each t > t 0 Observe that for all t ∈ R and x ∈ R 2 + we have 1, e(t)x = 0 and 1, k(t)x = 0.
Hence taking the scalar product with 1 in the both sides of the equality of (25) we obtain and by summing the two equations of (27) we get providing that and we deduce that Using the fact that the set of initial conditions giving classical solutions is dense in the space of the state variables [15] and (29) we obtain the following lemma.
Lemma 3.1. Let Assumption 2.1 be satisfied. Let S 0 ∈ R 2 + and i 0 ∈ L 1 + (R + , R) 2 be given and define Then we have In order to state a uniform boundedness in bounded sets we recall that for each satisfying (9) is given by Then a direct consequence of Lemma 3.1 is the following Lower bound: In order to obtain a lower bound for the map t → S(t) we will also first consider the initial conditions giving classical solutions so that system (25) holds true. Next we observe that Hence by setting Thus setting one gets from the S-equation of (25) that Hence and we deduce that The density of the set of initial conditions giving classical solutions [15] allows us to conclude to the following lemma.
We end this section by the following lemma.
Then for each t ≥ t 0 the following estimate hold Proof. Define and observe that Then by using (6) we have Using Lemma 3.1 we obtain The result follows by using Gronwall's inequality, Lemma 2.4 and (37).
4. The disease-free periodic solution. In this section we will give a complete description of the disease free periodic solution as well as its relationship with the mild solutions of (4). The disease-free periodic solution is given by In order to better describe the disease-free periodic solution we consider for each t 0 ∈ R the ordinary differential equation The foregoing ordinary differential equation (39) generates an evolution family {E(t, t 0 )} t≥t0 on R 2 . In particular for each t 0 ∈ R and each x 0 ∈ R 2 we have Furthermore since the off-diagonal entries of t → e(t) are non negative we deduce that {E(t, t 0 )} t≥t0 is a non singular linear evolution family with we obtain from (39) and the fact that 1, Let Assumption 2.1 be satisfied. Then the disease-free periodic solution is given by and for each c 2 ≥ 0 we have Moreover we have with α 2 defined in (34).
Proof. Formulas (41) and (42) will be proved simultaneously since (41) is a particular case of (43) with c 2 = 0. Let c 2 ≥ 0 be given. Then By using the variation of constants formula we have By (40) and the uniform boundedness of t → S(t) it is easy to see that the integrals The equality (42) follows by letting t 0 → −∞ in (45). In order to prove (43) we start with some observations. Let α 2 be defined in (34). Let t 0 ∈ R be given. Define for all t ≥ t 0 Then the maps t → x 1 (t) and t → x 2 (t) satisfy the following ordinary differential equationsẋ (46) Let us now proceed to the proof of (43). The right hand side of (43) follows by simple computations. To prove the left hand side of (43) note that and by the variation of constants formula we have for all t ≥ 0 Next observe that Finally we have for each t ≥ 0 and the result follows by the periodicity of t → S(t).
The next lemmas will play an important role in the asymptotic analysis of (4).
Then for any > 0 there exists t 1 := t 1 ( , S 0 ) ≥ t 0 such that Proof. Using the variation of constants formula the S-equation of (4) solves as for all t ≥ t 0 . Hence providing that for some 0 ≥ 0 then by using (48) we obtain that for each t ≥ t 0 The next lemma shows that the number of susceptible of each group becomes positive even if we start with zero susceptible in both groups. Indeed this due to the recruitment rate t → Λ(t) which is assumed to satisfy Λ(t) 0 R 2 for all t ∈ R.
Then there exists t 1 ≥ t 0 such that Proof. First note that Lemma 3.1 ensures that Then there exists a constant c > 0 large enough such that Hence using comparison principle for ordinary differential equations and a variation of constants formula we obtain The result follows.
Lemma 4.4. Let Assumption 2.1 be satisfied. Let (S 0 , i 0 ) ∈ R 2 × L 1 + (R + , R 2 ) be given and define Assume that there exists c > 0 such that then there exists t 1 := t 1 (S 0 , c) ≥ t 0 such that Hence using the S-equation of (4) one has and by using comparison principle for ordinary differential equations and Variation of constants formula we obtain Thus noting that from (45) we have Next we note that The inequality (52) implies that the result follows.
A consequence of Lemma 4.4 is the following Corollary 2. Let Assumption 2.1 be satisfied. Let (S 0 , i 0 ) ∈ R 2 × L 1 + (R + , R 2 ) be given and define Assume that there exists c 0 > 0 such that for each Proof. It is easy to see that and we apply Lemma 4.4 with c = b ∞ 2c 0 .

5.
The perturbed linear problem. In this section we will study a perturbed linear problem that will allow us to use comparison principles in order to prove the asymptotic properties of (4). From now on we fix then from Lemma 4.1 we have We consider for each ≥ −η the following linear equation Remark 2. Note that for = 0, system (58) coincide the linearised i-equation of (4) around the disease free periodic solution. The foregoing system (58) will play an important role in the asymptotic analysis.
Hence by using the Volterra integral equation formulation as in Section 2 one can easily obtain that system (58) generates a periodic linear evolution family (59) where t → B (t, t 0 ; ϕ) is the unique continuous function on [0, +∞) satisfying the following Volterra integral equation where G and Ψ are defined respectively in (7) and (8). Let C τ (R, R 2 + ) denotes the space of continuous τ -periodic functions on R. Define for all λ ∈ R the linear operator L λ : with Ψ (t, a) = diag(S(t) + 1)Ψ(t, a), ∀t ∈ R, a ≥ 0, ∀ ≥ −η (62) and Ψ defined in (7). It is clear from Lemma 2.3 that the family of linear operators defined in (61) are well defined for all λ ∈ R and for each t ≥ 0 we have In order to use the results in [29] we will verify some conditions in addition to (63). The family of kernels { Ψ (t, a) : t ∈ R, a ≥ 0} is compact (then power compact) (See [2,29]) for each ε ≥ −η and satisfies Ψ (t, a)R 2 + ⊂ R 2 + , ∀t ∈ R, ∀a ≥ 0. Next we prove that for each fixed t 0 ≥ 0 the family of periodic kernels { Ψ (t+t 0 , a) : t ∈ R, a ≥ 0} is w-positive (See Proposition 1) for any ε ≥ −η. Before proceeding we prove the following lemma.
Proof. Let k 1 ≥ 1 be a fixed an integer large enough such that and n ≥ 1 an integer large enough such that and 2nτ > a + 4τ.
Since min(β 12 , β 21 ) is not zero a.e. [26,Corollary B.6] implies that there exists b > 0 depending only on β 12 , β 21 andφ such that Since for allφ(t 1 ) > 0 for some t 1 ∈ [0, τ ] and is continuous we have Proof of ii) : Let Let c 0 > 0 be large enough such that Let x = 0 R 2 be given. Using (62) and Lemma 2.3 we have Since we have used the sum norm it is clear that and we deduce that Hence , ∀t ∈ R, ∀t 0 ≥ 0, ∀a ≥ 0 and since · w(t) is a norm we obtain , ∀t ∈ R, ∀t 0 ≥ 0, ∀a ≥ 0.
In order to make use of [29,Remark 2.4] we note that using (62) and Lemma 2.3 it is straightforward that uniformly for t ∈ R, t 0 ≥ 0, a ≥ 0. Before stating the main results of this section let us introduce the metric d : As a consequence of (62) , (76), Proposition 1 and [29, Theorem 2.2, Theorem 2.3] we have the following result.
Proposition 2. Let Assumption 2.1 be satisfied. Then for each ∈ (−η, η) there exists a unique pair λ ∈ R andφ ∈ C τ (R, R 2 + ) such that the following hold where d is the metric defined in (77). Proof. Note that by (60) and Remark 3 there exists some constant C > 0 such that

Remark 3. Note that conditions i) and v) in Proposition 2 imply that there exists a constant
Therefore we obtain In order to deal with the uniform persistence we will use the following sets Proof. Recall that for each ≥ −η, each ϕ ∈ ∂ M 0 and each t 0 ≥ 0 we have is the unique continuous function on [0, +∞) satisfying the following Volterra integral equation Observe that if ϕ ∈ ∂ M 0 then by Lemma 2.4 we have Now using the fact that the matrix diag(S(t) + 1) is invertible for all t ∈ R and ≥ −η we get and the uniqueness of the solutions provides that Hence the result follows easily by combining (84) and (85).
The following proposition will be essential in the proof of our uniform persistence result.
Proposition 4. Let Assumption 2.1 be satisfied. For each ∈ (−η, η), each ϕ ∈ M 0 and each t 0 ≥ 0 there exists t 1 > t 0 such that and Proof. Let ∈ (−η, η), ϕ ∈ M 0 and t 0 ≥ 0 be given. Since it is clear that Also Lemma 2.2 ensures that and we deduce that Thus by property i) in Proposition 1 there exists t 1 > 0 large enough such that Condition (87) is a consequence of Lemma 5.2 and (91). Assume that λ < 0. Let us now prove (88). To do so we will make use of (91) and Proposition 2. In fact (90) ensures that the parameters α obtained by applying Proposition 2 to (60) is positive (α > 0). Furthermore we have lim t→+∞ d e λ t B (t, t 0 ; ϕ), α φ (t) = 0 where d is defined in (77). Let δ 0 > 0 be given. Then there exists t 2 > t 1 large enough such that Therefore using the definition of the metric d in (77) there exists t ∈ [t 2 , +∞) → θ(t) such that and Now using the periodicity ofφ combined with (91) and (93) we obtain that min t≥t2 φ (t) > 0.
6. Global stability of the disease free periodic solution. In order to study the global stability of the disease free periodic solution we will introduce the basic reproductive number R 0 as in [2,24]. Define the basic reproductive number by Theorem 6.1. Let Assumption 2.1 be satisfied. If R 0 < 1 then the disease free periodic solution is globally asymptotically stable.
Proof. We will prove the theorem by applying Proposition 2 to (60) and using some comparison principles. Note that from condition iii) in Proposition 2 we have Thus thanks to iv) in Proposition 2 we can fix ∈ (0, η) small enough such that 0 < λ and ρ(L λ ) = 1. Let S 0 ∈ R 2 + and i 0 ∈ L 1 + (R + , R) 2 be given and define Using Lemma 4.2 one knows that there exists t 1 > t 0 such that Therefore we have Then using the comparison principal in [17] we have 0 ≤ i(t, ·) ≤ W (t, t 1 )(i(t 1 , ·)), ∀t ≥ t 1 .
Then according to Proposition 3 we obtain To conclude to the global asymptotic stability we will prove that To do so note that for all t ≥ t 1 and by using the variation of constants formula we have Observe that and by using (97) and (99) we also have We can conclude to lim t→+∞ S(t) − S(t) = 0 by using (102) and (103) together with Proposition 3.
7. Uniform persistence. In order to apply the theory of persistence for steady states we introduce the sets with M 0 defined in (80) and with ∂ M 0 defined in (81). Define the continuous map ξ : with i 01 , i 02 the components of i 0 and observe that we have Moreover we also have Observe that the family of maps {U(t, 0)} t≥0 is a periodic semiflow in the sense defined in [38]. More precisely it satisfies the following properties i) U(0, 0) = I, ii) U(t + τ, 0) = U(t + τ, τ )U(τ, 0) = U(t, 0)U(τ, 0), for all t ≥ 0, The main result of this section is the following.
Theorem 7.1. Let Assumption 2.1 be satisfied. Assume that R 0 > 1. Then U is ξ-uniformly persistent with respect to the decomposition (M 0 , ∂M 0 ) in the following sense : There exists δ 0 > 0 such that for each The proof of Theorem 7.1 will be decomposed into several intermediate results. We will now split the nonlinear operator U(t, t 0 ) defined by (11), (6) and (9) into two operators. Firstly we consider the non linear operator defined for all t ≥ t 0 and where we have set is the unique continuous solution of the Volterra integral system of equations (9). Secondly we consider the linear operator defined for each t ≥ t 0 , each Observe that with the above notations we have for each The following lemma can be proved by using similar arguments in [34] together with Corollary 1, the continuity of the coefficient of (4) as well as the uniform continuity of the contact rates. The proof may be very long but require any difficulty if we follow the same ideas in [34].
Lemma 7.2. Let Assumption 2.1 be satisfied. Then for all t > 0 the operator Proposition 5 (Extinction). Let Assumption 2.1 be satisfied. If S 0 i 0 ∈ ∂M 0 then for each t 0 ≥ 0 the mild solution Moreover the disease free periodic solution is globally asymptotically stable in ∂M 0 . More precisely we have Proof. Let t 0 ≥ 0 be given. Define Let 0 > 0 be large enough such that S(t 0 ) ≤ S(t 0 ) + 1 2 0 1. Then by Lemma 4.2 we have Therefore the map t → i(t, ·) satisfies Thus by using the comparison principle in [17] we have Since i 0 ∈ ∂ M 0 Lemma 5.2 provides that Because i 0 ∈ ∂ M 0 , Lemma 5.2 and (114) imply that By using the same arguments as in the proof of Theorem 6.1 we also have lim t→+∞ S(t) − S(t) = 0.
Remark 4. Note that in Proposition 5 no condition is made on the basic reproductive number R 0 . More precisely the disease always dies out when the initial conditions are taken into ∂M 0 . In fact this not surprising because if we start with initial conditions in ∂M 0 the infected individuals in both groups are not capable to infect susceptible individuals even in the future. Therefore no new infected individual will appear in the future and the disease dies out as t goes to +∞.
Then we have Hence the map t → (B S (t, t 1 ; x 0 , ϕ), B I (t, t 1 ; x 0 , ϕ)) satisfies and for all t ≥ 0. Moreover by Lemma 3.2 one has where the constants α k , k = 1, 2 are defined respectively in (32) and (34) so that with Then using (9), (120) and Lemma 2.2 it is obvious that for all t ≥ 0 and we deduce that for all t ≥ 0. Using similar argument as in the proof of Proposition 4 we know that there exists s ∈ [0, a ) such that and since min(β 12 (a), β 21 (a)) is not zero for almost every a ≥ 0 we infer from [26,Corollary B.6] that there exists t 2 > t 1 large enough such that To end the proof we will argue by contradiction. Assume that there exists some t 3 > t 0 such that Thus by Proposition 5 we have which contradict (121).
Therefore since i(t 1 , ·) ∈ M 0 one can make use of Proposition 4 and (128) to obtain In what follows, we consider the map Θ : Thus by using the evolution semiflow properties of U and its τ -periodicity we obtain for each n ≥ 2 Hence for any n ≥ 0 and p ≥ 0 we have so that {Θ n } n≥0 is a discrete time semigroup. Let δ > 0 be given and consider the set As a consequence of (28) and (29) we have the following Lemma 7.5. Let Assumption 2.1 be satisfied. The set B δ is forward invariant with respect to the discrete time semiflow Θ that is to say that Moreover Θ is point dissipative which means that for each Lemma 7.2 will allow us to obtain that the discrete time semigroup Θ is asymptotically smooth.
Lemma 7.6. Let Assumption 2.1 be satisfied. Then the semigroup Θ is asymptotically smooth in R 2 Proof. First observe that for all Since and U 1 (nτ, 0) is compact for all n ≥ 1 the result follows by applying Lemma 2.3.2 in [6].
Since Θ is point dissipative there exists n 0 ≥ 0 large enough such that Θ n S 0 i 0 ∈ B δ , ∀n ≥ n 0 .
Proof of Theorem 7.1 . Since Θ has a global attractor and ∂M 0 is ξ-ejective with respect to the decomposition (M 0 , ∂M 0 ), by using [19,Proposition 3.2] we obtain that Θ is ξ-uniformly persistent that is there exists δ 0 > 0 such that for each 8. Coexistence: Existence of a positive periodic solution. In this section we will show that system (4) admits a non trivial positive periodic solution that is different from the disease free periodic solution. This will be performed by applying [19,Theorem 4.5.] to the discrete time semiflow Θ defined in (129).
In what follow we will always assume that R 0 > 1. We know from Lemma 7.5 and Lemma 7.6 that Θ is point dissipative and asymptotically smooth. Moreover Θ has a compact global attractor A by Lemma 7.7 and is ξ-ejective with respect to the decomposition (M 0 , ∂M 0 ) by Lemma 7.8. Then its ξ-uniformly persistent with respect to the decomposition (M 0 , ∂M 0 ) [19, Proposition 3.2]. Therefore we only have to show that Θ is condensing. Before stating the exact meaning of condensing we recall that the Kuratowski measure of non compactness κ is defined by κ(B) = inf{r > 0 : B has a finite open cover of diameter ≤ r}, for all bounded subset B ⊂ R 2 + × L 1 + (R + , R) 2 . Therefore we say that Θ is κcondensing if κ(Θ(B)) < κ(B), for all bounded subset B ⊂ R 2 + × L 1 + (R + , R) 2 . We refer to [4,21,25] for several properties of κ.
The following lemma shows that Θ is κ-condensing.