An SEIR epidemic model with constant latency time and infectious period.

We present a two delays SEIR epidemic model with a saturation incidence rate. One delay is the time taken by the infected individuals to become infectious (i.e. capable to infect a susceptible individual), the second delay is the time taken by an infectious individual to be removed from the infection. By iterative schemes and the comparison principle, we provide global attractivity results for both the equilibria, i.e. the disease-free equilibrium E0 and the positive equilibrium E+, which exists iff the basic reproduction number R0 is larger than one. If R0 > 1 we also provide a permanence result for the model solutions. Finally we prove that the two delays are harmless in the sense that, by the analysis of the characteristic equations, which result to be polynomial trascendental equations with polynomial coefficients dependent upon both delays, we confirm all the standard properties of an epidemic model: E0 is locally asymptotically stable for R and unstable for R0 > 1 , while if R0 > 1 then E+ is always asymptotically stable.

(Communicated by Yang Kuang) Abstract. We present a two delays SEIR epidemic model with a saturation incidence rate. One delay is the time taken by the infected individuals to become infectious (i.e. capable to infect a susceptible individual), the second delay is the time taken by an infectious individual to be removed from the infection. By iterative schemes and the comparison principle, we provide global attractivity results for both the equilibria, i.e. the disease-free equilibrium E 0 and the positive equilibrium E + , which exists iff the basic reproduction number R 0 is larger than one. If R 0 > 1 we also provide a permanence result for the model solutions. Finally we prove that the two delays are harmless in the sense that, by the analysis of the characteristic equations, which result to be polynomial trascendental equations with polynomial coefficients dependent upon both delays, we confirm all the standard properties of an epidemic model: E 0 is locally asymptotically stable for R 0 < 1 and unstable for R 0 > 1, while if R 0 > 1 then E + is always asymptotically stable.
1. Introduction. In recent years, attempts have been made by many authors to analyse the global stability properties of delay epidemiological models with a general nonlinear infection rate. For example, Takeuchi and coworkers analysed SIR, SIS, SEIR and SEI epidemic delay models [4,5,6] by constructing Lyapunov functionals and thus generalizing to the delay case the general approach by Lyapunov functions proposed by Korobeinikov for non-delayed epidemic models with a very general infection rate ( [7] and the references therein). Even very interesting contribution to the topic has been recently presented by Xu and Ma [12] and by Xu and Du [11], who analysed, by using iterative schemes and comparison principles, global attractivity properties of the equilibria respectively of an SEIRS delay model and of an SIR delay epidemic model, where the non-linear infection rate is the one introduced by Capasso and Serio [2] with a saturated incidence rate with respect to the infectious individuals. In [11] there is one constant delay which represents the constant infectious period after which the infected individuals are removed, whereas in [12] the delay represents the constant latency time which is the time taken by an infected individual to become infectious. In [11,12] the authors have paid attention also to the analysis of the characteristic equation. While in [12] they show that the delay is harmless in inducing stability switches that modify the standard stability behaviour of the equilibria, for the delay SIR model in [11] they confirm the local stability property only for the disease-free equilibrium (i.e. if the basic reproduction number R 0 < 1 then it is asymptotically stable and if R 0 > 1 it becomes unstable), whereas for the endemic equilibrium (i.e. R 0 > 1) they refer to a result by Beretta and Kuang [1] but leave the problem open.
In our paper, we modify the SIR delay epidemic model presented in [11] by introducing the class of the exposed individuals and by considering two (constant) delays. The first delay τ 1 is the constant latency time and represents the time taken by a susceptible individual that, infected at time t becomes infectious I, i.e. capable to infect other susceptibles, but only at a time t + τ 1 : the infected individuals who are not yet infectious are called exposed E and they stand in the exposed class for the time τ 1 . The second delay τ 2 is the constant infectious period and represents the time necessary to remove the infectious individuals I from the cycle of the infection. Therefore, an individual infected at time t will be removed at time t + τ 1 + τ 2 and it stands in the infectious class I for a time τ 2 . We assume that the removed individuals R cannot return to the susceptible class S. A possible interpretation is that τ 2 is the time taken by an infectious individual before presenting the symptoms of the infection, with the assumption that the infectious individuals I are removed from the infection as soon as the symptoms appear. In this case the infection is transmitted by the asymptomatic infectious I. This could be the case, for example, of SARS where all individuals with symptoms are removed ( [9] and the references therein).
According to the above remarks, our model is an SEIR model with two delays and we assume for it the same nonlinear rate of infection as in [11,12]. With these assumptions it turns out to be a generalization of the delay SIR model in [11] and [13] with constant infectious period. Moreover in [13] the infection rate is a bilinear function in S and I. Our model becomes coincident with the SIR model in [11] if we assume τ 1 = 0, thus implying that the exposed class E(t) is identically vanishing. If we further assume a bilinear infection rate in S and I we also obtain the model in [13]. An interesting aspect of the model is that the characteristic equation at the endemic equilibrium can be reduced to a second order trascendental polynomial equation with two delays, where the polynomial coefficients are real functions of both delays. By its analysis, we prove that, whenever it exists, the endemic equilibrium is locally asymptotically stable, that is, the delays τ 1 and τ 2 are harmless in inducing stability switches. Though we prove that the endemic equilibrium is globally attractive only if the sup of the incidence rate is less than the susceptibles death rate constant (as in [11]), our feeling is that we should be able to prove that, whenever it exists, the endemic equilibrium is globally asymptotically stable, for example by using the Lyapunov functional approach (see [8,Section 2.5]) but with different Lyapunov functions with respect to those considered in [4,5,6] since their method seems not working in our model. However, this is left as a future work.
The structure of the paper is the following: in Section 2 we introduce the model equations with their main properties and we also present all the results that we are going to prove in the paper. In Section 3 we analyse the characteristic equation at the disease-free equilibrium as well as at the endemic one. Section 4 (together with Appendix A) is devoted to a permanence result for the solutions of the model. In Section 5 we prove attractivity results for both equilibria. Conclusions are driven in Section 6. 2. The model equations. Herefollowing we introduce the necessary notation and the model equations.
We denote by S the susceptible individuals; by E the exposed individuals, who have been infected and take a time τ 1 to become infectious, i.e. capable to infect the susceptible individuals S; by I the infectious individuals, capable to infect the susceptibles and who take a time τ 2 to be removed from the infection; finally, R denotes the removed individuals, for which we assume that they cannot return to the susceptible class because they have been "quarantined" and/or they acquire permanent immunity. It is assumed τ := (τ 1 , τ 2 ) ∈ R 2 + . As far as the parameters of the model are concerned, Λ and µ 1 are the constant recruitment and death rate, respectively, of susceptibles S; µ 2 is the constant death rate for both exposed E and infectious I; µ 3 is the constant death rate for removed R; it is assumed Λ ∈ R + as well as µ i ∈ R + , i = 1, 2, 3. Of course, we could assume different death rate constants for exposed E and infectious I individuals, but this would not change the results, while enabling us to slightly simplify the notation.
While denoting the rate of infection f (S, I), we assume the structure (see [2,11,12]) f (S, I) := g(I)S (1) with a saturated incidence rate g(I) := βI 1 + αI (2) with respect to the number I of the infectious individuals. With β, α ∈ R + , βI is a measure of the force of infection and 1 1+αI accounts for the inhibition effect on the rate of infection when I becomes large.
By assuming and by taking into account that the rate of infection at time t is g(I(t))S(t) and that the exposed individuals that become infectious I at time t are those infected at the previous time t − τ 1 , multiplied for the fraction e −µ 2 τ 1 of the exposed survived in the time interval [t − τ 1 , t], we get the evolution equation for the exposed E(t). By similar arguments we can write the evolution equations also for I(t) and R(t), while the one for S(t) is standard. Thus, the model equations are
2.1. Positivity. We see that the positivity of the above initial conditions for S and I in [−(τ 1 + τ 2 ), 0] imply positivity for all solutions (S(t), E(t), I(t), R(t)), t > 0, of system (3) or (4), simply considering recurrence arguments applied to the integral forms for E(t), I(t) and R(t) in (4). We further note that S(t) can never vanish since at each time t > 0 where S(t) vanishes it is dS(t) dt = Λ > 0. We can also prove the following.
is globally attractive and invariant for the solutions of (3).
for t ≥ 0 with initial condition N (0) > 0. Thus, we obtain for all t ≥ 0, which proves the Lemma.
Since Ω is a limit set for system (3), in the sequel we assume initial conditions satisfying (3) it is easy to see that the Disease-Free Equilibrium (DFE) is which exists for all values of the parameters. As far as the interior (positive) equilibrium E + is concerned, first we need to define the basic reproduction number R 0 according to the definition in [4], extended to a delayed epidemic model. The basic reproductive number is the mean number of secondary cases that a typical infected case will cause in a population with no immunity to the disease in the absence of interventions to control the infection: where 1/µ 2 is the mean infection period and e −µ 2 τ 1 − e −µ 2 (τ 1+τ 2 ) is the probability of an infected people to be in the infectious period. Thus 1 is the mean infectious period. P i is the initial maximum infection rate: where S 0 is the initial susceptible population i.e., according to the first equation in (3), is the equilibrium value of susceptibles in the absence of infection: S 0 = Λ/µ 1 . Therefore Accordingly, the interior (positive) equilibrium is which exists iff the basic reproduction number, which according to the previous definition is The delay domain of existence of the positive equilibrium E + requires R 0 > 1. If we denote it by Ω + we have: and, of course, R 0 = 1 when τ 1 = h(τ 2 ) for (τ 1 , τ 2 ) ∈ R 2 + . Furthermore, notice that e. E + exists if the time taken to become infectious τ 1 is sufficiently small and the asymptomatic infectivity period τ 2 is sufficiently large, Figure 1. The above results can be summarized in the following Lemma.
Concerning the equilibria of the model we give below the main results, which will be proved in the forthcoming sections. In particular, Theorems 2.4 and 2.6 about global attractivity will be proved in Section 5; Theorems 2, 2.5 and 2.8 about stability will be proved in Section 3; finally Theorem 2.7 on permanence of solutions will be proved in Section 4. About this latter we recall the following. First of all, the following statement is a particular case of the forthcoming Theorem 2.4 since βΛ ≤ µ 1 µ 2 implies R 0 < 1. Corollary 1. If βΛ ≤ µ 1 µ 2 only the DFE E 0 is feasible and it is globally attractive, i.e. for all initial conditions the solutions of (3) satisfy Secondly, by assuming βΛ > µ 1 µ 2 we have the following results.
Theorem 2.8. Whenever it exists, the equilibrium E + is locally asymptotically stable.
3. The reduced system and the characteristic equation. In this section we want to prove Theorems 2.5 and 2.8 by the help of the characteristic equation at E 0 and E + , respectively. In particular, we assume the global attractivity results of Theorems 2.4 and 2.6, which will be proved in Section 5, to hold true.
Herefollowing we denote by p := (Λ, α, β, µ 1 , µ 2 , µ 3 , τ 1 , τ 2 ) the vector of all real parameters of model (3). They belong either to Γ := p ∈ R 8 + : when we are dealing with the characteristic equation at E 0 or to when we are dealing with the characteristic equation at E + , respectively. Since in (3) the evolution equations for S(t) and I(t) do not contain the variables E(t) and R(t), in order to compute the characteristic equation at any equilibrium E = (S * , E * , I * , R * ) it is sufficient to consider the characteristic equation of the reduced system 938 EDOARDO BERETTA AND DIMITRI BREDA at E = (S * , I * ). In fact, it is easy to check that the characteristic roots, i.e. the solutions of the characteristic equation, for the complete system (3) are either the negative ones λ = −µ i , i = 2, 3 (due to the second and fourth equations), or given by the solutions of the characteristic equation of the reduced system (7), which reads with p ∈ Γ or p ∈ Γ + according to which equilibrium is considered. Thus, at E + the characteristic equation is given by (8) whose associated characteristic function (9) has, according to (5), coefficients and p ∈ Γ + .
If we assume R 0 = 1, i.e. µ 2 = βΛ , then the previous equation becomes Since at R 0 = 1 the DFE E 0 is globally attractive by virtue of Theorem 2.4, then all the characteristic roots λ have α ≤ 0.
. This inequality implies thus proving the local asymptotic stability.
These properties together with Rouché's Theorem imply that the number of zeros of G(λ; p) in C + (i.e. the right-hand side of C) can change only if a root appears on or crosses the imaginary axis. Furthermore, for any (λ, p) ∈ D ×Γ + the function (9) satisfies the Implicit Function Theorem extended to complex-valued functions (e.g. [10, Theorem A.3, p.152]), This ensures the continuous dependence of the roots λ of (8) upon p.
Proof of Theorem 2.8. E + exists if and only if τ = (τ 1 , τ 2 ) ∈ Ω + . As previously observed, the associated characteristic roots λ depend continuously on all parameters p ∈ Γ + and their multiplicities in C + can change only if at least one root appears on or crosses the imaginary axis.
By Theorem 2.6 we know that the global attractivity of E + holds if β α < µ 1 . Then all roots λ satisfy (λ) ≤ 0 for all τ ∈ Ω + if β α < µ 1 . Our aim here is to prove that for all p ∈ Γ + the characteristic roots λ cannot reach the imaginary axis. This implies that if β α < µ 1 then all charactreristic roots λ satisfy (λ) < 0 for all delays τ ∈ Ω + , and that all the characteristic roots remain with (λ) < 0 for all p ∈ Γ + .
Proof. From (A.2) and the positivity of the solutions we have By (2), the third equation in (4) and Lemma 4.2 we obtain lim sup From (17) and (18) By the first equation in (3) and this latter inequality, for sufficiently large times we get and, by the comparison principle, which completes the proof.
Since the solutions of (3) are ultimately bounded in Ω S , without loss of generality we assume that the initial conditions belong to Ω S . Now we are ready to prove Theorem 2.7.
Proof of Theorem 2.7. Since (3) is ultimately bounded in Ω S , it is sufficient to prove that there exist positive constants ν I , ν E and ν R such that The key point is to prove the first inequality, from which the other two easily follow. We start by proving that lim inf t→+∞ I(t) > 0. According to Lemma 4.3 we therefore give continuous and positive initial conditions S(θ) = ϕ 1 (θ) ≥ ν S and I(θ) = ϕ 2 (θ) ≥ ε I > 0 for θ ∈ Ω 0 := [−(τ 1 + τ 2 ), 0], where we have set

EDOARDO BERETTA AND DIMITRI BREDA
Denoted by Σ n := [nτ 1 , (n + 1)τ 1 ] , n ∈ N 0 , we consider the positive real axis of times R +0 as covered by the union of these infinitely many intervals Σ n in such a way that By the third equation in (4) we can write and we see that t ∈ Σ n implies t + u ∈ Ω n := [(n − 1) With k depending upon the value of τ 2 in relation to that of τ 1 (see Appendix A), we have From (20) we have that if t ∈ Σ n then from which, by defining we obtain Denoting by I n := min t∈Σn I(t) and by ε n a constant lower bound for I(t) over Σ n such that I n ≥ ε n , n ∈ N 0 , we can define the sequence {ε n } n∈N0 of lower bounds for I(t) over Σ n according to (19), (21), (22) and (23): for n ∈ N 0 such that n ≥ k. Of course, t → +∞ iff n → +∞. Now let us notice that the following inequalities are equivalent: where by (22). This shows that ε c > 0 if R 0 > 1 and β αµ 1 < R 0 . According to (24) and (25), for the sequence {ε n } n∈N0 we have the following results: (i) if ε I ≤ ε c then {ε n } n∈N0 is non-decreasing, i.e. ε n ≥ ε 0 > 0 for all n ∈ N 0 ; (ii) if ε I > ε c then {ε n } n∈N0 is strictly decreasing, i.e. ε n > ε n+1 for all n ∈ N 0 . Now we prove that in case (ii) the strictly decreasing sequence {ε n } n∈N0 must satisfy lim n→+∞ ε n ≥ ε c . If not (i.e. if lim n→+∞ ε n < ε c ), there exists n * ∈ N 0 such that , which implies ε n * < g(ε n * ) β Φ(R 0 ). Since {ε n } is strictly decreasing and ε n * +1 = g(ε n * ) β Φ(R 0 ), we get to the contradiction ε n * < ε n * +1 . Therefore, we can conclude that in either case (i) and (ii) and, by (26), which, again by Lemma 4.3, let the last differential equation in (3) imply that, for a sufficiently large time T > 0, for all t > T . Then, the comparison principle implies Finally, by applying Lemma 4.1 to the second equation in (4), i.e.
which completes the proof of the permanence of system (3).

5.
Global attractivity results. In this section we present the proofs of Theorem 2.4 on the global attractivity of the DFE E 0 when R 0 ≤ 1 and of Theorem 2.6 on the global attractivity of the positive equilibrium E + when β α < µ 1 . The proofs are performed with the help of Lemmas 4.1 and 4.2 of Section 4 and by using comparison arguments which are close to those alredy used in the recent paper [11] by Xu and Du on a SIR model with one delay, i.e. constant infectious period. We extend their results to the SEIR model with two delays (3). In particular, in the proof of Theorem 2.6 we introduce the explicit dependence of the sequences of upper and lower bounds upon the basic reproduction number R 0 , then proving the decreasing strict monotonicity of the upper bounds and the increasing one for the lower bounds. Hence, herefollowing we recall the key points of the proofs leaving most of the details to the appropriate references in the above mentioned paper.
Proof of Theorem 2.4. By the structure of the model equations in (4) we see that if for all initial conditions we prove that then it is easy to prove (see [11,Theorem 4 We have to prove that We proceed by constructing sequences S n n∈N0 , I n n∈N0 and R n n∈N0 of upper bounds S ≤ S n , I ≤ I n , R ≤ R n which are strictly decreasing, and sequences {S n } n∈N0 , {I n } n∈N0 , {R n } n∈N0 of lower bounds S n ≤ S, I n ≤ I, R n ≤ R Since n → +∞ implies that t → +∞, then: lim t→+∞ (S(t), E(t), I(t), R(t)) = E + .
In order to construct the above sequences (for the details see [11,Theorem 4 From the third equation in (4)  and we obtain Then, from the first equation in (4), we obtain Again from the third equation in (4) Again, from the first equation in (4), we obtain Of course, from the last equation in (3) we get for n ≥ 1.
Then it is sufficient to prove that S 2 − S 1 < 0 to prove that S n+1 − S n < 0 for all n ≥ 1. To check this, we take (38) when n = 1 to obtain By the hypotheses 1 − β αµ 1 > 0 and R 0 > 1 we get Thus S n n∈N0 is strictly decreasing and lower bounded. Therefore there exists lim n→+∞ S n and we can compute it by (38) obtaining lim n→+∞ S n = β + αµ 1 R 0 R 0 (β + αµ 1 ) (Notice instead that if β αµ 1 > 1, then S n ↑). From (36) it follows that lim n→+∞ I n = 1 α From (37) Now, from (37) and thanks to (40) and (42) we can easily prove Finally, from the second equation in (4)  Therefore completing the proof together with (40), (42) and (43) 6. Conclusions. As already pointed out in the Introduction, though the model equations (3) are delay differential equations with delay dependent parameters, the delays just influence the existence delay domain Ω + of the positive equilibrium E + , but are harmless to induce stability switches for example from asymptotic stability to instability within Ω + (Theorem 2.8).
We think that the limitation in Theorem 2.6 to the global attractivity of E + is only a technical result of the approach followed and that, perhaps, a different approach to the global asymptotic stability of E + by Lyapunov functionals should be possible.
As far as the DFE equilibrium E 0 is concerned, we see that all the classical results of the epidemic models hold true. If the basic reproduction number satisfies R 0 ≤ 1 then E 0 is globally attractive (Theorem 2.4) and also locally asymptotically stable if R 0 < 1 (Corollary 2), whereas if R 0 > 1 then E 0 becomes unstable (Theorem 2.5).
The epidemic model could also be improved for two aspects: