R 0 AND THE GLOBAL BEHAVIOR OF AN AGE-STRUCTURED SIS EPIDEMIC MODEL WITH PERIODICITY AND VERTICAL TRANSMISSION

. In this paper, we study an age-structured SIS epidemic model with periodicity and vertical transmission. We show that the spectral radius of the Fr´echet derivative of a nonlinear integral operator plays the role of a threshold value for the global behavior of the model, that is, if the value is less than unity, then the disease-free steady state of the model is globally asymptotically stable, while if the value is greater than unity, then the model has a unique globally asymptotically stable endemic (nontrivial) periodic solution. We also show that the value can coincide with the well-know epidemiological threshold value, the basic reproduction number R 0 .


1.
Introduction. Seasonal fluctuations in the incidence of infectious diseases is an important aspect of epidemics occurrence and an interesting subject in the field of mathematical epidemiology. In this context, systems of nonlinear differential equations with periodic coefficients are a natural mathematical tool for modeling purposes and several authors have adopted this approach to explain the periodic outbreak and the oscillations in the endemic presence of a disease in a population [2,3,12,13,15,16,18,19,21].
The basic reproduction number R 0 , which is defined as the expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population [8], is known as a good indicator of the future spread pattern of disease. That is, it can be expected that a disease dies out if R 0 < 1, while it remains endemic if R 0 > 1. The mathematical definition of R 0 as the spectral radius of a linear integral operator called the next generation operator was firstly given for autonomous cases [8], and has recently been extended to periodic cases [2,3,19] and to more general nonautonomous cases [12,18]. Since R 0 is defined for linearized systems around the disease-free steady states, it plays the role of a threshold for the local behavior of the original systems.
However, whether R 0 plays the same role of the threshold and determines the global behavior of these systems is generally an open question, and thus, we have to clarify the relation between R 0 and the global behavior of epidemic systems for each case. In this paper, we investigate such relation for an age-structured SIS epidemic model with periodicity and vertical transmission.
The global behavior of age-structured SIS epidemic models without periodicity was successfully investigated in [4,5,6,9], while the periodic case was investigated in [15], where it has been proved that an endemic (nontrivial) periodic solution is unique and even globally stable if it exists. However, no threshold-like condition for the existence of such a solution was given. The purpose of this paper is to show that the basic reproduction number R 0 , obtained for our periodic age-structured SIS epidemic model, plays the role of such a threshold and, applying the previous significant results in [15], we see that R 0 is a threshold for the global behavior of the model. To our knowledge, this is the first study of the relation between R 0 and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission (see [11] for the analysis of another age-structured epidemic model with vertical transmission). In particular, this is in contrast to the instability results obtained for age-structured SIR epidemic models (see, for instance, [1,7,17]).
Here we note that in this paper we use the SIS epidemic model as a case study.
The organization of this paper is as follows. In Section 2, we formulate the model and normalize it. In Section 3, we show the well-posedness of our problem. In Section 4, we show that if the spectral radius ρ (F) of a linear operator F is greater than unity, then the normalized system has an endemic periodic solution. Moreover, applying the results obtained in [15], we obtain uniqueness and global stability results for the periodic solution . In Section 5, we show that if the spectral radius ρ (F) is less than unity, then the disease-free of the normalized system is globally asymptotically stable. In Section 6, we investigate the relation between our threshold value ρ (F) and the basic reproduction number R 0 .
2. A basic model. Let p (a, t) be the age-density of the host population at time t (a ∈ [0, a † ] and t ≥ 0, where a † ∈ (0, +∞) is the maximum age for the population). Let µ (a) be the age-specific mortality rate and β (a) be the age-specific birth rate. Let us assume that the host population dynamics is described by the following von Foerster equation with initial and boundary conditions: where p 0 (a) is the given initial age distribution. µ and β are assumed to be nonnegative and measurable. In addition, β is assumed to be uniformly bounded above, and the (demographic) basic reproduction number of the host population is assumed to satisfy Thus, the demographic steady state R0 AND THE GLOBAL BEHAVIOR OF AN SIS MODEL Since we can scale the size of b 0 arbitrary, we set and obtain In what follows, we assume that the density of the host population has reached the steady state, that is, p (a, t) ≡ p * (a). As far as the epidemics is concerned, the host population is divided into the two epidemiological subclasses of susceptibles s (a, t) and infectives i (a, t). That is, we have For each t ≥ 0, let γ (a, t) be the age-specific recovery rate, λ (a, t) be the force of infection to susceptible individuals aged a and k (σ, a, t) be the transmission coefficient between susceptible individuals aged a and infective individuals aged σ. In order to model the vertical transmission process of the disease, we introduce a coefficient q ∈ (0, 1) which is the proportion of newborn offspring of infective parents who are themselves infective. Under this setting, the SIS epidemic model we consider in this paper is formulated as follows: where s 0 (a) and i 0 (a) are given initial distributions. γ and k are assumed to be nonnegative, measurable and uniformly bounded above by positive constants γ + < +∞ and k + < +∞, respectively. In addition, in order to model the seasonally fluctuating process of the disease, γ and k are assumed to be time-periodic with common period T > 0, that is, for all a, t and σ.

TOSHIKAZU KUNIYA AND MIMMO IANNELLI
From (5), we have s (a, t) = p * (a) − i (a, t). Hence, substituting this into (6), we can obtain the following single equation for i (a, t): Using u (a, t) := i (a, t) /p * (a) and equation (4), the system is normalized as where and the solution must satisfy 0 ≤ u(a, t) ≤ 1 .
Note that, from the definition, κ is also T -periodic with respect to time t. The global behavior of system (7) is the main problem we consider in the following sections.
3. Abstract formulation. Let E := L 1 (0, a † ). System (7) can be formulated as the abstract Cauchy problem d dt in E, where A is a linear operator on E defined as and {F (t, ·)} t∈R+ is a family of nonlinear operators on E where It is easy to see that the convex feasible region is positively invariant under the strongly continuous semigroup e tA t∈R+ defined by where b is the solution of integral equation That is, e tA (C) ⊂ C. We assume that the domain of {F (t, ·)} t∈R+ is limited to C ⊂ E. As in [4], we can prove the following Lemma: The proof is omitted here (see the proof of Proposition 3.1 of [4]). Using α in (ii) of Lemma 3.1, we rewrite problem (9) as d dt The mild solution of (16) is given by the solution of the integral equation Consider the scheme for the standard iterative procedure. From Lemma 3.1, we see that u n+1 ∈ C if u n ∈ C. Hence, according to the argument in [4], we can prove the following proposition:

4.
Existence of an endemic periodic solution. In this section, we investigate the existence of an endemic periodic solution u * to system (7). Such a solution must satisfy

TOSHIKAZU KUNIYA AND MIMMO IANNELLI
Integrating the first equation of (19) along the characteristic lines, we have Substituting (20) into the second and third equations of (19), we have and respectively. Thus, if we find nontrivial, positive, T-periodic λ * and u * (0, ·), satisfying (21) and (22), by (20) we also obtain a nontrivial positive T-periodic u * , because so are λ * and u * (0, ·). Such u * can be regarded as the desired endemic periodic solution of system (7) in a weak sense, namely in (7) the differential operator Thus, in what follows, we look for nontrivial positive periodic λ * and u * (0, ·) satisfying (21) and (22).
The sets X T and Y T are actually Banach spaces when respectively endowed with the norms Let us define a nonlinear positive operator Φ : where and Then, from (21) and (22), we see that a nontrivial positive fixed point (ϕ * 1 , ϕ * 2 ) ∈ (X T,+ \ {0}) × Ỹ T,+ \ {0} of the operator Φ corresponds to the desired λ * and u * (0, ·). Therefore, in what follows, we look for such a fixed point (ϕ * 1 , ϕ * 2 ) of Φ. First, we investigate the mathematical properties of the operator Φ. We have the following lemma: (ii) Φ is uniformly bounded on X T,+ ×Ỹ T,+ ; (iii) Φ is monotone nondecreasing on X T,+ ×Ỹ T,+ .
As in the previous studies [4,5,9,10], we can expect that the spectral radius of the Fréchet derivative DΦ [0] of Φ at zero plays the role of a threshold for the existence of the desired fixed point (ϕ * 1 , ϕ * 2 ) of Φ. Thus, we consider the positive linear operator and and show that the spectral radius ρ (F) of F plays the role of such a threshold. In order to establish the main theorem of this section, we need some additional assumptions on the parameters. First we assume Assumption 1. k (σ, a, t) = 0, β (σ) = 0 and γ (σ, t) = 0 for all σ ∈ (−∞, 0) ∪ (a † , +∞).

Next, setting
and (note that each of the series is well-defined by Assumption 1), we make the following assumption: Assumption 2. The following equations hold uniformly for x ∈ [0, T ] and s ∈ [0, a † ]: These assumptions are required for proving the compactness of operator F, in view of the use of the Krein-Rutman theorem (see [14]). Thus, we proceed to prove the following:

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These are linear operators defined on X T or on Y T , precisely Then, from (28)-(30), we have and, to complete the proof, it suffices to show the compactness of each F i,j (i, j = 1, 2). First we consider F 1,1 . From Assumption 1, we have From (35)-(36) and Assumption 1, we have Then, regarding F 1,1 as an operator on L 1 ([0, a † ] × [0, T ]), from Assumption 2 and the well-known compactness criteria in L 1 (see, for instance, [20], p.275), we see that F 1,1 is compact. Of course F 1,1 is compact also regarded as an operator in X T . Similarly, we have AND THE GLOBAL BEHAVIOR OF AN SIS MODEL   939 and Thus, as in the above case of F 1,1 , we see that F 1,2 , F 2,1 and F 2,2 are compact.
The previous result can be implemented with the results obtained in [15]. In fact, once existence of an endemic periodic solution u * of system (7) is proved, we can resort to [15] to get uniqueness and global asymptotic stability. To this aim we consider the following assumption (ii) There exist a positive constant > 0 and nonnegative measurable functions κ 1 (a) and κ 2 (a) such that and κ 1 (a) = 0 on (0, a † ) and κ 2 (a) = 0 on (0, a 0 ).
Then, from Theorem 5.5 in [15], we have the following proposition.

5.
Global stability of the disease-free steady state. The global stability of the disease-free steady state u ≡ 0 of system (7), for ρ (F) < 1, can also be proved by using Theorem 5.6 in [15]. The theorem is applied to system (7) as the following lemma: Lemma 5.1. If system (7) has no endemic (nontrivial) periodic solution u * in Ω T , then the disease-free steady state u ≡ 0 of the system is globally asymptotically stable.
6. The basic reproduction number R 0 . Finally we investigate the relation between our threshold value ρ (F) and the basic reproduction number R 0 [8] which is a well-known epidemiological threshold value.
According to its epidemiological definition, R 0 is the average number of secondary cases produced by a typical infected individual, introduced into a completely susceptible population, during its entire period of infectiousness. From the mathematical viewpoint R 0 is the spectral radius of an integral operator called the next generation operator and, recently, its definition has been extended to the case of time periodic environments [2,3,12,18,19].
Linearizing system (6) around the disease-free steady state (p * (a) , 0), we have whereĩ denotes the perturbation from the disease-free steady state i ≡ 0. Integrating the first equation of (43) along the characteristic lines, we havẽ Substituting (44) into the second equation of (43), and using Assumption 1, we havẽ Similarly, substituting (44) into the third equation of (43), we havẽ Let us define the linear operator A (t, τ ) from L 1 (0, +∞) × R into itself as where Then, following the arguments in [2,3,12,18,19], we see that the basic reproduction number R 0 is obtained as the spectral radius of the next generation operator where V T denotes the space of T -periodic vector-valued functions ϕ = (ϕ 1 , ϕ 2 ) such that ϕ 1 (t) ∈ L 1 (0, +∞) and ϕ 2 (t) ∈ R for each t. Concerning the relation between this R 0 = ρ (K) and the threshold value ρ (F), we have the following proposition.
7. Discussion. We have formulated an age-structured SIS epidemic model (6) with periodicity and vertical transmission. The system was normalized to system (7), and the existence of an endemic (nontrivial) periodic solution u * of (7) was investigated. We have shown that the spectral radius ρ (F) of the Fréchet derivative F of a nonlinear operator Φ at 0 plays the role of a threshold for the existence of such u * , that is, if ρ (F) > 1, then u * is obtained as a nontrivial fixed point of Φ. The uniqueness and global stability results obtained in [15] were directly applied to our case, and thus, we have shown that ρ (F) is a threshold for the global behavior of system (7). Furthermore, we have shown that if ρ (F) < 1, then the diseasefree steady state u ≡ 0 of system (7) is globally asymptotically stable. The relation between ρ (F) and the basic reproduction number R 0 was also investigated. We have shown that the two threshold values coincide. Consequently, this study is regarded as the first one showing that R 0 plays the role of a threshold for the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission.