MATHEMATICAL ANALYSIS OF AN AGE-STRUCTURED SIR EPIDEMIC MODEL WITH VERTICAL TRANSMISSION

In this paper, we consider a mathematical model for the spread of a directly transmitted infectious disease in an age-structured population. We assume that infected population is recovered with permanent immunity or quarantined by an age-specific schedule, and the infective agent can be transmitted not only horizontally but also vertically from adult individuals to their newborns. For simplicity we assume that the demographic process of the host population is not affected by the spread of the disease, hence the host population is a demographic stable population. First we establish the mathematical well-posedness of the time evolution problem by using the semigroup approach. Next we prove that the basic reproduction ratio is given as the spectral radius of a positive operator, and an endemic steady state exists if and only if the basic reproduction ratio R0 is greater than unity, while the disease-free steady state is globally asymptotically stable if R0 < 1. We also show that the endemic steady states are forwardly bifurcated from the disease-free steady state when R0 crosses the unity. Finally we examine the conditions for the local stability of the endemic steady states.

1. Introduction.In this paper, we consider a mathematical model for the spread of a directly transmitted infectious disease in an age-structured population.We assume that infected population is recovered with permanent immunity or quarantined by a given age-specific schedule, and the infective agent can be transmitted not only by horizontally but also vertically from adult individuals to their newborns.On the other hand, for simplicity, we assume that the demographic process of the host population is not affected by the spread of the disease.Then the host population is assumed to be a demographic stable population, that is, its total size is growing exponentially but its age profile is not changing through time.
In a series of papers ( [10], [11], [12]), assuming that the proportionate mixing assumption (that is, the transmission kernel is given by the type of separation of variables) holds and the host population is in a demographic steady state, Cha, Iannelli and Milner studied the age-structured SIR model for vertically transmitted diseases.They calculated the basic reproduction ratio R 0 and conclude that if R 0 < 1, there is no endemic steady state and the disease-free steady state is locally stable, while if R 0 > 1 there exists at least one endemic steady state.They have also provided conditions for the existence of a unique endemic steady state.A local stability condition for the endemic steady state is also given, and they show an example of unstable endemic steady state.However, so far there is no result for this model with general transmission rate (non proportionate mixing case).
Hence our main purpose of this paper is to establish a most general approach to deal with the age-structured SIR epidemic model with vertical transmission and to extend the existing threshold and stability results to the case of general (nonseparable) transmission rate.Another point is to extend the theory so that we can deal with the case of non-stationary host population.In this paper, however, we only deal with the case that the host population already attains the stable age distribution.
First the basic epidemic system is formulated as a homogeneous dynamical system.Assuming that the host population attains the stable age distribution, we introduce the normalized system.Then we will describe the semigroup approach to the time evolution problem of the normalized epidemic system.Next we consider the disease invasion process to calculate the basic reproduction ratio R 0 , then we prove that the disease-free steady state is globally asymptotically stable if R 0 < 1. Subsequently we show that at least one endemic steady state exists if the basic reproduction ratio R 0 is greater than unity.By introducing a bifurcation parameter, we show that the endemic steady state is forwardly bifurcated from the disease-free steady state when the basic reproduction ratio crosses unity.Finally we consider the conditions for the local stability of the endemic steady states.
2. The basic system.First as a host population, we consider a closed one-sex age-structured population under the demographic stable growth.Let P (t, a) be the age-density at time t of the host population, µ(a) the age-specific natural death rate and f (a) the age-specific fertility rate.Then we assume that the host population dynamics is described by the McKendrick equation as follows: where P 0 (a) is given initial data and ω < ∞ is the upper bound of age.The system (2.1) is well known as the stable population model in demography.
It follows from the stable population theory (see [27], [29]) that the system (2.1a) and (2.1b) has a unique persistent age profile as Since ω is the maximum attainable age, that is, (ω) = 0, we assume that µ ∈ L 1 +,loc (0, ω) and ω 0 µ(σ)dσ = ∞.Moreover, for given initial data (2.1c) there exists a constant Q > 0 and a function η(t, a) such that 3) where lim t→∞ η(t, a) = 0 uniformly for a ∈ [0, ω].Then as time evolves, the age distribution converges to the persistent age profile: That is, ψ is the relatively stable age distribution and once it is attained, its profile is persistent.In fact, if P 0 (a) = Cψ(a) with a positive constant C, then P (t, a) = Ce r0t ψ(a) for t > 0.
In the following we assume that the stable age distribution is already attained, the age density of the host population is given by P (t, a) = N (t)ψ(a) where N (t) = ω 0 P (t, a)da is the total size of the population.
Subsequently let us divide the host population into three subpopulations; the susceptible class, the infective class and the recovered class, the age-density functions of each class are denoted by S(t, a), I(t, a) and R(t, a).Let β(a, σ) be the transmission rate between the susceptible individuals aged a and the infective individuals aged σ, γ(a) the rate of recovery at age a and θ(a) the rate of removal (by vaccination or by quarantine) at age a. Then the basic system (age-structured SIR epidemic model) with vertical transmission can be formulated as follows: where the force of infection λ(t, a) is given by and q is the proportion of newborns produced from infected individuals who are vertically infected.Since we assume that the demographic parameters are not affected by the epidemic, it is convenient to introduce the ratio age distributions for each epidemiological class as follows: Then the new system for the ratio age distributions is given as Moreover of course, it follows from the definition that s(t, a) + i(t, a) + r(t, a) = 1. (2.8) In the following, we mainly consider the normalized system (2.7) under the condition (2.8) and the following technical assumption: Here we remark that the basic system (2.5) can be seen as the homogeneous dynamical system in the sense of Hadeler, et al. [23], Webb [52] and Iannelli and Martcheva [28], since (2.5) is a semilinear evolution equation whose nonlinear term is homogeneous of degree one.Then it follows that the steady state solutions of the normalized system correspond to the persistent solutions of the original homogeneous system.Though in this paper we only consider the normalized system (2.7), we can prove that as long as the disease does not affect the demographic vital rates of the host population, the local stability of the steady state solutions of the normalized system implies that of the persistent solutions of the basic homogeneous system, even if the age distribution of the host population is not necessarily the stable age distribution.This point of issue will be treated in a separate paper ( [33]).
3. The semigroup formulation.From the normalized condition (2.8), instead of the full system (2.7) we can consider the IR system: where λ[a|i] is given by the integral operator defined by The state space of the IR system is where E + is the positive cone of E := L 1 (0, ω) × L 1 (0, ω).Though we can also consider the SI system, the advantage of the IR system is that it has a linear homogeneous boundary condition.Let us define operators A and F acting on E as follows: where φ = (φ 1 , φ 2 ) T ∈ E, T denotes the transpose of the vector and the domain of the differential operator A is defined by Then U j (t) is a strongly continuous semigroup on L 1 (0, ω) and it is easy to see that U j (t) can be represented as where B(t) is a solution of the following renewal equation Observe that B(t) is given by the limit of iterative sequence B n (n = 0, 1, 2, ...) such that Then it is easy to see that 0 ≤ B n ≤ 1 for all n.Then we know that That is, Ω is positively invariant with respect to the semiflow defined by e tA .
Lemma 3.2.Under the Assumption 2.1, the map F : Ω → E is Lipschitz continuous and there exists a small number α ∈ (0, 1) such that Proof of Lemma 3.2.Since the Lipschitz continuity is clear, let us prove (3.3).Let Let θ + := sup θ, γ + := sup γ and β + := sup β.Then we have λ where α is chosen such that (3.3) is satisfied.Then the mild solution of this Cauchy problem is given as the solution of the integral equation (the variation of constants formulaj( [43], Chapter 6)F Let S(t)u 0 , t > 0 be the semiflow defined by the solution of the above variation of constants formula.Then S(t)u 0 can be given as the limit of the iterative sequence u n such that It is observed that if u n ∈ Ω, then e tA u 0 , e (t−s)A [u n (s) + αF (u n (s))] ∈ Ω and u n+1 ∈ Ω since it is a linear convex combination of two elements in Ω.It follows from the Lipschitz continuity of F that u n converges to the mild solution S(t)u 0 ∈ Ω uniformly.
4. The disease invasion process.It is easy to see that the basic system (2.Here we remark that in our case the disease-free steady state is not a completely susceptible population but it is partially immunized or quarantined.If a very small number of infected individuals enter into the disease-free steady state, the initial phase of epidemic could be described by the linearized system at the disease-free steady state.Since the linearized equations for infective population does not include other subpopulations, we can only consider the single equation for infective population as where operators A 0 and F 0 acting on E 0 := L 1 (0, ω) as follows: where φ ∈ E 0 , T and the domain of A 0 is given by Here we adopt the following technical assumption: Assumption 4.1.The transmission coefficient β satisfies the following: The following holds uniformly for ζ ∈ R: 3. There exists a nonnegative function η(σ) such that η(σ) > 0 for a left neighborhood at σ = ω and β(a, σ) ≥ η(σ) for almost all (a, σ) ∈ R × R.
From the above assumption and the well known compactness criteria in L 1 , we obtain the following ( [53], Chapter X, p. 275): Then it is easy to see that A 0 + F 0 is a generator of an eventually compact semigroup T 0 (t) = exp((A 0 + F 0 )t), since A 0 is the generator of the population semigroup and F 0 is a compact perturbation ( [42], p. 87; [29]; [51], Prop.4.14).Since the spectral mapping theorem holds for the eventually compact semigroup, we know that where ω 0 (A) denotes the growth bound of the semigroup exp(tA) and σ(A) denotes the spectrum of A. Then if γ > ω 0 (A), there exists a number M (γ) ≥ 1 such that exp(tA) ≤ M (γ) exp(γt) for t ≥ 0. In particular, if ω 0 (A) < 0, the equilibrium i = 0 of (4.1) is asymptotically stable.From the principle of linearized stability ( [13]; [51], Prop.4.19), the stability of the equilibrium i = 0 in (4.1) implies the local asymptotic stability of the disease-free steady state of (2.7).
For u ∈ D(A 0 ) and v ∈ E 0 , let us consider the resolvent equation: Then we have where we use the notation as < f, g >:= ω 0 f (a)g(a)da.By the variation of constants formula, we can obtain the expression where w(a) := Θ(a)λ[a|v] and Multiplying qπ to the both sides of (4.5) and integrating from zero to ω, we have ) where we use v(0) = q < π, v > and Then (4.6) can be written as follows: where Again multiplying Θ(a)β(a, σ)ψ(σ) to the both sides of (4.5) and integrating from zero to ω with respect to σ, we obtain where Then (4.8) can be written as follows: where and a 22 (z) is a linear operator from L 1 (0, ω) into itself defined by Let us define a linear operator T (z) from C × L 1 (0, ω) into itself as Then under our condition, T (z), z ∈ C is an analytic family of compact operators with respect to z.By using T (z), we can formulate (4.7) and (4.9) as a simultaneous equation as follows: Thus the solution (v(0), w) is uniquely determined, that is, the resolvent (z Proof of Lemma 4.3.Since T (z) is an analytic family of compact operators and I − T (z) is invertible for z with sufficiently large real part, then (I − T (z)) −1 is meromorphic with respect to z (see [45]).Then we have For z such that I − T (z) is invertible, A 0 + F 0 has a compact resolvent, so we have Σ = P σ (A 0 + F 0 ) (see Theorem 6.29 of [34]).If I − T (z) is invertible, we obtain v(0) and w uniquely, hence we can calculate v from (4.5).That is, z ∈ ρ(A 0 + F 0 ), where ρ(A 0 + F 0 ) denotes the resolvent set of A 0 + F 0 .On the other hand, if (I − T (z)) is not invertible for some z, z is a pole of the meromorphic family . This completes our proof.
Note that the spectrum of the compact operator is a countable set with no accumulation point different from zero ( [34], Theorem 6.26), so (4.11) implies that Σ is composed of complex numbers z such that T (z) has an eigenvalue one.Now we can define T (0) as the next generation operator for the invasion at the partially immune population (s * , i * , r * ) = (Θ(a), 0, 1 − Θ(a)), since T (0) maps the density of primary cases (v(0), w) to the density of secondary cases.Hence the per-generation growth factor of the infectious population density, called as the basic reproduction ratio, denoted by R 0 , is given by the spectral radius, denoted by r(T (0)), of the next generation operator T (0) (see [14], [15]) 1 .
If the positive operator T (0) has the Perron-Frobenius properties, r(T (0)) = R 0 is the dominant eigenvalue of the next generation operator T (0) (see [38]).Roughly speaking, if r(T (z)) is decreasing for real z, we could expect that there exists a unique real eigenvalue z 0 of A 0 + F 0 such that r(T (z 0 )) = 1, z 0 > 0 if r(T (0)) > 1 and z 0 < 0 if r(T (0)) < 1.That is, the disease can invade into the host population if R 0 > 1, while it cannot if R 0 < 1.This is the expected role of the basic reproduction ratio.
In the following, to check the above intuition, let us summarize some ideas from positive operator theory.For more detail, the reader may refer to [31], [24], [38] and [44].
Let E be a real or complex Banach space and let E * be its dual (the space of all linear functionals on E).We write the value of f ∈ F * at ψ ∈ E as < f, ψ >.A nonempty closed subset E + is called a cone if the following holds: (1) + is the subset of E * consisting of all positive linear functionals on E, that is, Let B(E) be the set of bounded linear operators from , there exists a positive integer p = p(ψ, f ) such that < f, T n ψ >> 0 for all n ≥ p.The spectral radius of T ∈ B(E) is denoted as r(T ).σ(T ) denotes the spectrum of T and P σ (T ) denotes the point spectrum of T .
From results by Sawashima [44] and Marek [38], we can state the following: From the above result, we can expect that for compact and nonsupporting operators in the ordered Banach space, the Perron-Frobenius properties hold just the same as the case of positive irreducible matrices.Lemma 4.5.For z ∈ R, T (z) is compact and nonsupporting.
Proof of Lemma 4.5.Under the Assumption 4.1, it is easy to see from the well known compactness criterium in L 1 that the operator T (z) is a compact operator for all z.Next for z ∈ R and φ := (x, f where φ := (x, f ) T and ã21 (z Then it is clear that A(z, φ) > 0 for φ ∈ Z + \ {0} and we have T (z)φ ≥ A(z, φ)e where e := (1, 1) T is a quasi-interior point in Z + .Moreover for any integer n, we have T (z) n+1 φ ≥ A(z, φ)A n (z, e)e.Then we obtain < F, T (z , that is, we know that T (z) is a nonsupporting operator.
By using the above results, we can relate the Malthusian parameter of the infected population to the next generation operator and its spectral radius:

and it is the dominant characteristic root as
Proof of Proposition 4.6.Since T (z), z ∈ R is nonsupporting, it follows from Proposition 4.5 that r(T (z)), z ∈ R is strictly decreasing.For z ∈ R, let F z be a positive eigenfunctional corresponding to the eigenvalue r(T (z)) of positive operator T (z).
Let e be a quasi-interior point.Then it follows from (4.12) that Since F z is strictly positive, we obtain r(T (z)) ≥ A(z, e).Then it follows from the Assumption 4.1 that lim z→−∞ r(T (z)) = +∞.On the other hand, it is clear that lim z→∞ r(T (z)) = 0. Then r(T (z)) is strictly decreasing from +∞ to zero when z moves from −∞ to +∞.Then the first half of the proposition is the direct consequence of this monotonicity of r(T (z)).Next we show the dominant property of z 0 .For any z ∈ Σ, there is an eigenfunction ψ z such that T (z)ψ z = ψ z .In the following, we write Let F z be the positive eigenfunctional corresponding to T ( z), we obtain that Hence we have r(T ( z)) ≥ 1 and z ≤ z 0 because r(T (z)) is strictly decreasing for z ∈ R and r(T , taking duality pairing with the eigenfunctional F z 0 corresponding to the eigenvalue r(T (z 0 )) = 1 on both sides yields which is a contradiction.Then we can write that |ψ z | = cψ z0 , where ψ z0 is the eigenfunction corresponding to the eigenvalue r(T (z 0 )) = 1.Hence, without loss of generality, we can assume that c = 1 and write (x z , f z ) = (x z 0 e iα , f z 0 exp(iβ(a)) for some real number α and real function β(a).If we substitute this relation into Then we have where Therefore we have From [24, Lemma 6.12], we obtain that a) , so θ = β(a), which implies that z = 0. Then there is no element z ∈ Σ such that z = z 0 and z = z 0 , hence z 0 is the dominant root in Σ.
From the above result, we can state the threshold criterion as follows: Proof of Proposition 4.7.From Proposition 4.6 and Lemma 4.3, if R 0 > 1, the linearized generator A 0 + F 0 has a positive eigenvalue z 0 , then the disease-free steady state is unstable.Next observe that the infective population in the original nonlinear system satisfies the Cauchy problem as where F 1 (t) is a time dependent operator such that where we see r(t, a) as a given function.By using the variation of constants formula and the iteration method , i(t) can be seen the limit of the sequence {i n } n=1,2,.. with the initial data i 0 defined by Let us consider the another sequence {w n } n=1,2,.. with the same initial value such that Then we have i 1 ≤ w 1 since F 1 (t)i 0 ≤ F 0 i 0 , hence iteratively we obtain that i n ≤ w n .Since w = lim n→∞ w n is the solution of the linearized system (4.1), then we have i(t) ≤ w(t) = e (A0+F0)t i 0 .If R 0 < 1, it follows from (4.13) that ω(A 0 +F 0 ) < 0, where Then we know that the semigroup e A1t is a nilpotent semigroup, that is, e A1t = 0 for t > ω.From the variation of constants formula, we have Since F 2 (t) is uniformly bounded and lim t→∞ F 2 (t) = 0, we know that lim t→∞ η(t) = 0 (see [30]).Then the disease-free steady state of (3.2) is globally asymptotically stable if R 0 < 1.
As an important special case, we here briefly consider the proportionate mixing assumption (in the following, we call it as the PMA), that is, the transmission rate β can be written as β(a, σ) = β 1 (a)β 2 (σ).In this case we can calculate the threshold condition explicitly.
First let us introduce another formulation of the next generation operator.Observe that (z where U (z) := F 0 (z − A 0 ) −1 .Note that (z − A 0 ) −1 exists for all z with nonnegative real part.Then for z with z ≥ 0, U (z) is formally calculated as Then it is reasonable to define U (0) = F 0 (−A 0 ) −1 as the next generation operator for non newborn population, since if z = 0, the first term of (4.15) describes the secondary cases infected by the infective newborns, who are produced from the primary cases, and the second term corresponds to the horizontally transmitted secondary cases.In fact, though the value of r(U (0)) is different from R 0 , we can prove that: Proof of Proposition 4.8.It follows from (4.14) that for z with z ≥ 0, z ∈ Σ if and only if 1 ∈ P σ (U (z)).Suppose that R > 1.Since U (z), z ∈ R + is compact and nonsupporting, r(U (z)) is strictly decreasing with respect to z ≥ 0. Then there exists z 0 > 0 such that r(U (z 0 )) = 1 ∈ P σ (U (z 0 )).That is, z 0 ∈ Σ and it follows from Proposition 4.6 that R 0 > 1. Conversely if R < 1, r(U (z)) = 1 has no nonnegative root, and by using the same argument as the proof of Proposition 4,6, we can show that complex number z such that U (z) has an eigenvalue one has negative real part.Then we have R 0 < 1. Finally if R = 1, we have r(U (0)) = 1 ∈ P σ (U (0)), which implies that Σ includes the dominant root zero and R 0 = 1.
If we assume the PMA, the range of U (0) becomes one-dimensional, it is easy to see that the only eigenvector is Θ(a)β 1 (a) and its eigenvalue is given by Dependence of R on the Malthusian parameter r 0 is an interesting question.Note that in our expression (4.16), π and ψ depend on r 0 .McLean [40] observed that R 0 is no greater in developing countries with high r 0 than in developed regions with low r 0 .By simple calculation, we can observe that where a 0 := ω 0 aψ(a)da is the average age of the host stable population.Let R 1 be the reproduction number by horizontal transmission: Then we have where If we assume that β 2 is constant and k is an increasing function (it is possible if θ = 0, β 1 and γ are constant respectively), then That is, we conclude that R 1 is decreasing with respect to the Malthusian parameter r 0 , which supports McLean's observation.However, for more general cases and the vertical transmission part of R, the effect of the Malthusian parameter on the reproduction numbers would be much more complex.

5.
Existence and bifurcation of endemic steady states.We have so far shown that there is no endemic steady state if R 0 < 1.In this section, we consider the existence of endemic steady states and their bifurcation from the disease-free steady state at R 0 = 1.Let (s * , i * , r * ) be the density vector at the endemic steady state, then it must satisfy the following system: By formal integration, we obtain the following expression: Applying π to the both sides of (5.3) and integrating from 0 to ω, we obtain where < π, i * >:= ω 0 π(a)i * (a)da.Then we know that i * (0) = q < π, i * > is given by the functional G as (5.4) Let us define a positive operator H : (5.5) for λ ∈ L 1 (0, ω).Then from (5.5) we know that the force of infection at the endemic steady state λ * is given by positive solutions of the fixed point equation: where we have used relations Then we know that Ω is invariant with respect to H, in fact more strongly we can state H(L 1 + ) ⊂ Ω.By using the same argument as the proof of Krasnoselski's fixed point theorem ( [35], p. 135, Theorem 4.11), we can conclude that H has at least one non-zero fixed point in the positive cone of L 1 + if r(W 0 ) > 1 (see also [31], Proposition 4.6).On the other hand, we can see that W 0 = L −1 U (0)L, where L is an operator defined by (Lλ)(a) := Θ(a)λ(a) for λ ∈ L 1 .Then we obtain r(W 0 ) = r(U (0)) = R. From Proposition 4.8, r(W 0 ) = R > 1 if and only if R 0 > 1, hence we conclude that there exists at least one endemic steady state if R 0 > 1.
Next suppose that R 0 ≤ 1, that is, r(U (0)) = r(W 0 ) ≤ 1.If there exists a positive fixed point λ * of H, we have λ * = H(λ * ) ≤ W 0 λ * .Let F 0 be the adjoint eigenvector of W 0 corresponding to r(W 0 ).Taking the duality pairing, we find that + \ {0} and F 0 is a strictly positive eigenfunctional.Then we have r(W 0 ) > 1, which contradicts our assumption.Therefore there is no endemic steady state if R 0 ≤ 1.
If we can adopt the proportionate mixing assumption, that is, the transmission rate can be factorized as β(a, σ) = β 1 (a)β 2 (σ), the force of infection at the endemic steady state λ * is given as λ * (a) = cβ 1 (a) with a positive number c. Then the fixed point equation (5.6) is reduced to the following characteristic equation for unknown number c: (5.8) Since Ĥ(0) = R and Ĥ(∞) = 0, we can again confirm that there exists at least one endemic steady state if R > 1 (equivalently if R 0 > 1).If Ĥ becomes a monotone function under some additional conditions, we can prove the uniqueness of the endemic steady state.For example, if we assume that there exists an age A ∈ (0, ω) such that β 2 (a) = 0 for a > A and β 1 (a) = 0 for a < A, then the second term of Ĥ in (5.8) becomes zero and Ĥ(c) is monotone.
Of course, Ĥ is a monotone function if q = 0 (no vertical transmission).However, such additional assumptions to guarantee the monotonicity of Ĥ are usually very restrictive, and so far we have no biologically reasonable one.Though here we do not examine such additional conditions to guarantee the uniqueness of endemic steady state, the readers who are interested in the uniqueness problem may consult Cha, et al. [10], [11].
An important basic observation is that the endemic steady states are given by forward bifurcation from the disease-free steady state.In fact, this is intuitively clear for the PMA case, since Ĥ (0) < 0.Here we give a proof for the general transmission case by using a bifurcation scenario as follows: Assumption 5.2.The transmission rate β is given by β 0 (a, σ) where is a bifurcation parameter and β 0 is a given standard schedule such that R = r(∂H(0)) = 1.
Proposition 5.3.Under the Assumption 5.2, the endemic steady states are forwardly bifurcated from the disease-free steady state at R = 1.
Proof of Proposition 5.3.Under the Assumption 5.2, the fixed point equation (5.6) is written as λ = H(λ).Define a mapping F : R + ×L 1 → L 1 as F (λ, ) := H(λ)−λ and assume that F (λ, ) is analytic with respect to (λ, ).Now we are interested in the structure of solution set F −1 (0) := {(λ, ) ∈ L 1 (0, ω) × R + : F (λ, ) = 0}.From the Implicit Function Theorem, we can expect a bifurcation from the trivial branch (0, ) only for those values such that the linear mapping is not boundedly invertible, where D 1 denotes the Fréchet derivative for the first element and I is the identity operator.It follows from our assumption that ∂H[0] has a positive eigenvalue r(∂H[0]) = 1 and for ∈ (0, 1) L( ) is invertible, then a bifurcation from the trivial branch can occur when crosses unity.Let σ( ) be the simple real strictly dominant eigenvalue of L( ), φ( ) the eigenvector of L( ) and φ * ( ) the eigenvector of L * ( ) (the adjoint operator of L( )) associated with σ( ) such that < φ( ), φ * ( ) >= 1, where < φ, φ * > is the value of φ * at φ. Since φ(1) is the Frobenius eigenvector of the nonsupporting operator ∂H[0] corresponding to the eigenvalue one, there exist a projection to the one-dimensional eigenspace spanned by φ(1).Then we can apply the standard argument of Lyapunov-Schmidt method to conclude that the bifurcation at (0, 1) is subcritical if τ 1 < 0, and it is supercritical if τ 1 > 0, where the parameter τ 1 is given by where D 2 1 denotes the second derivative with respect to the first element (see [46], Chapter VII, p. 57).For our case, it is not difficult to see that Therefore we conclude that the bifurcation at R = 1 is supercritical.
Finally note that we can define the next generation operator at the endemic steady state.From the variation of constants formula, it follows from (5.1b) that (5.10) Applying qπ to the both sides of (5.10) and integrating from zero to ω, we obtain an expression: Again applying s * (a)β(a, σ)ψ(σ) to the both sides of (5.10) and integrating from 0 to ω with respect to σ and multiplying s * , we obtain the following expression: (5.12) Now let us define a positive linear operator T * from R × L 1 (0, ω) into itself as Then from (5.11)-(5.12), the newly infected population (i * (0), s * (a)λ * (a)) can be formally seen as the positive eigenvector of the operator T * corresponding to the eigenvalue one: The equation (5.13) implies that at the endemic steady state the infected population simply reproduce itself.Therefore we can call T * the next generation operator at the endemic steady state.This fact will be used to show the stability of the endemic steady state in the next section.
6. Stability of the endemic steady states.In this section, we consider the stability of endemic steady states.For this purpose let us assume that (s, i) = (s * , i * ) + (ζ, η), that is, (ζ, η) is a small perturbation from the endemic steady state (s * , i * ).For convenience, we here use SI system instead of IR system, but they can always transform into each other.Then the linearized system of (2.7) at the endemic steady state is given as where λ * (a then the linearized system (6.1) at the endemic steady state can be written as where the linear operator F e is given by Since the linearized stability principle holds for the age-structured population system (2.7) ( [51]), the endemic steady state is locally asymptotically stable if the trivial equilibrium u = 0 of the linearized system (6.1e) is locally asymptotically stable, while the endemic steady state is unstable if u = 0 is unstable in (6.1e).
In order to see the linearized stability by calculating the resolvent spectrum, let us consider the resolvent equation for the linearized operator: Let v := (v 1 , v 2 ).From (6.2), we obtain the following ODE system: ) ) By formal integration, we can derive the following expressions: ) Inserting (6.4a) into (6.4b) and using u 1 (0) = −u 2 (0), we have Multiplying qπ to the both sides of (6.5) and integrating from zero to ω, we obtain where By changing the order of integrals, the equation (6.6) can be written as follows: Again multiplying s * (a)β(a, σ)ψ(σ) to the both sides of (6.5) and integrating with respect to σ from zero to ω, we obtain that where Therefore the equation (6.8) can be written as follows: Let us define a linear operator B(z) as Then under our condition, B(z), z ∈ C is an analytic family of compact operators with respect to z.By using B(z), we can formulate (6.7) and (6.9) as a simultaneous equation as follows: Proof of Lemma 6.1.Since B(z) is an analytic family of compact operators and I − B(z) is invertible for z with sufficiently large real part, then (I − B(z)) −1 is meromorphic with respect to z (see [45]).Then we have This completes our proof.
Lemma 6.2.The linearized semigroup e (A+F e )t is eventually compact and where s(A + F e ) is the spectral bound of the generator A + F e .
Proof of Lemma 6.2.The linearized generator A + F e can be decomposed as A + F 1 + F 2 where It is easy to see that the operator A + F 1 is the generator of a multistate stable population with a finite age interval (see [29]), then ω 1 (A + F 1 ) = −∞ where ω 1 (A + F 1 ) is the essential growth bound.Since under the Assumption 4.1, F 2 becomes a compact perturbation, so we have ω [51], Prop.4.14).Then it follows from Proposition 4.13 in [51] that we obtain (6.13).
From the above result and the principle of linearized stability ( [51], Proposition 4.17, 4.18), we conclude that Proposition 6.3.If z < 0 for any characteristic root z ∈ Σ * , the endemic steady state is locally asymptotically stable, while there exists a characteristic root with positive real part, the endemic steady state is unstable.
By using the positive operator argument again, we can prove the following sufficient condition for the stability of the endemic steady state: Proposition 6.4.Suppose that λ * is so small that the following inequality holds:  Since s * (0)Λ * (ω) < 1, we obtain which implies that B(z) is also compact and nonsupporting if T (z) is compact and nonsupporting and κ > 0. Then r(B(z)) is strictly decreasing for z ∈ R.Moreover, we can observe that B(0) ≤ T * , where T * is the next generation operator at the endemic steady state.Since T * is also compact and nonsupporting, its spectral radius is the Frobenius eigenvalue corresponding to the unique positive eigenvector.If R 0 > 1, T * has a positive fixed point (i * (0), λ * s * ), that is, r(T * ) = 1.Hence from the Proposition 4.4 we obtain that r(B(0)) < r(T * ) = 1.By using the same argument as the proof of Proposition 4.6, we know that the dominant characteristic root in Σ * is given as the unique real root of r(B(z)) = 1 and it is less than zero if r(B(0)) < 1.Then it follows from Proposition 6.3 that the endemic steady state is locally asymptotically stable.
Finally we again consider the proportionate mixing case to show the local stability of forwardly bifurcated endemic steady states.If we adopt the PMA, the range of operator B(z) becomes two-dimensional.In fact, if we set β(a, σ) = β 1 (a)β 2 (σ), we have where g is a functional given by g(u 2 ) := where Π c is given by Then the operator equation (6.11) can be reduced to a two-dimensional simultaneous equation as and χ 3 is defined by The (i, j)-th elements of Φ(z, c) are given as follows: where s * (a) is given as a function of the parameter c: .17)Let us define the perturbation matrix P (z, c) as P (z, c) = (p ij (z, c)) 1≤i,j≤2 := Φ(z, 0) − Φ(z, c).Moreover we introduce functions h(a, c) and k(a, σ, c) defined by Then it is easy to see that Then we know that the roots of equation f 1 (z) = 0 give the eigenvalues of linearized system at the disease-free steady state.Let us assume that f 1 (0) = 0, that is, R 0 = 1.Then we have f (0, 0) = 0 and we can see that Again from the Implicit Function Theorem, we obtain z (0) = − f c (0, 0) f z (0, 0) < 0, (6.21) which means the dominant eigenvalue goes to the left half plane when c increases from the origin.By using the well known technique based on the Rouché theorem, we can confirm that z(c) is the dominant root of the perturbed equation f 1 + f 2 = 0 as long as c is small enough ( [27], Chapter IV; [32]).Since the endemic steady state is forwardly bifurcating from the disease-free steady state at R 0 = 1, c increases from zero as R 0 crosses unity.Therefore we can conclude that Proposition 6.5.Under the proportionate mixing assumption, the endemic steady state forwardly bifurcated from the disease-free steady state is locally asymptotically stable if R 0 > 1 and |R 0 − 1| is small enough.
The reader may refer to [12] for another type of proof about the local stability of the endemic steady state under the proportionate mixing assumption.7. Discussion.In this paper, we consider a mathematical model for the spread of a directly transmitted infectious disease in an age-structured population, for which infected population is recovered with permanent immunity or quarantined by an agespecific schedule and the disease can be transmitted not only horizontally but also vertically from adult individuals to their children.The basic system is formulated as a homogeneous dynamical system.
We have described the semigroup approach to the time evolution problem of the normalized epidemic system.Next we have calculated the basic reproduction ratio and proved that the disease-free steady state is globally asymptotically stable if R 0 < 1, and at least one endemic steady state exists if the basic reproduction ratio R 0 is grater than unity.Moreover, we have shown that the endemic steady state is forwardly bifurcating from the disease-free steady state at R 0 = 1.Finally we have shown sufficient conditions which guarantee the local stability of the endemic steady state.Roughly speaking, the endemic steady state is locally asymptotically stable if it corresponds to a very small force of infection, which is a natural result expected from the forward bifurcation of endemic steady states (the principle of exchange of stability).
Cha, Iannelli and Milner [12], Thieme [47] and Andreasen [4] have shown that for the SIR age-structured epidemic system the stability change of endemic steady states can occur, and in particular Andreasen [4] gave a numerical example of sustained oscillations in the prevalence.However it is still an interesting open problem whether the stability change to lead a bifurcation of periodic solutions could occur under biologically reasonable conditions.Moreover, biologically appropriate assumptions for the unique existence of an endemic steady state is also not yet known, even if we adopt the proportionate mixing assumption.For the case of general (non-separable) transmission rate, only restricted conditions to guarantee the uniqueness are known even for the case of purely horizontal transmission ( [31]).On the other hand, we do not have any realistic example of multiple endemic steady states for age-structured models in case that the endemic steady state bifurcates forwardly when R 0 crosses unity, though they may exist for ODE models (see [36]).Pathological cases of non-unique endemic steady states are reported in [41].
Another important future challenge is to study the case that the host demographic parameters are affected by the spread of diseases.Though for this problem we have already some results for models with constant parameters, there are few results for age-dependent cases (see [39], [40], [2], [6], [8], [37]).

Proposition 4 . 4 .
Let E be a Banach lattice and let T ∈ B(E) be compact and nonsupporting.Then the following holds:(1) r(T ) ∈ P σ (T ) \ {0} and r(T ) is a simple pole of the resolvent, that is, r(T ) is an algebraically simple eigenvalue of T .(2) The eigenspace corresponding to r(T ) is one-dimensional and the corresponding eigenvector ψ ∈ E + is a quasi-interior point.The relation T φ = µφ with φ ∈ E + implies that φ = cψ for some constant c. (3) The eigenspace of T * corresponding to r(T ) is also one-dimensional subspace of E * spanned by a strictly positive functional f ∈ E * + .(4) Let S, T ∈ B(E) be compact and nonsupporting.Then S ≤ T , S = T and r(T ) = 0 implies r(S) < r(T ).
{z ∈ C : (I − B(z)) is not invertible} = {z ∈ C : z is pole of (I − T (z)) −1 }.For z such that I − B(z) is invertible, A + F e has a compact resolvent, so we have Σ * = P σ (A + F e ) (see Theorem 6.29 of[34]).If I − B(z) is invertible, the resolvent equation (6.2) is solvable, then we have z ∈ ρ(A + F e ).On the other hand, if (I − B(z)) is not invertible for some z, z is a pole of the meromorphic family (I − B(z)) −1 .Moreover if z is a pole of (I − B(z)) −1 , it is also a pole of the resolvent (I − (A + F e )) −1 , hence it is an eigenvalue of A + F e (see Proposition 4.8 of[51]), that is, z ∈ P σ (A + F e ).In other words, z ∈ ρ(A + F e ) if and only if (I − B(z)) is invertible.Since σ(A + F e ) = P σ (A + F e ), we have {z ∈ C : (I − B(z)) is not invertible} = C \ ρ(A + F e ) = P σ (A + F e ).
The operator A generates a C 0 semigroup e tA and the state space Ω is positively invariant with respect to the semiflow defined by e tA .
Proof of Proposition 5.1.First we can observe that the Fréchet derivative W 0 := ∂H[0] of the operator H at the origin is given by .11) Then the resolvent equation (6.2) is solvable if and only if I − B(z) is invertible.Now we conclude that Lemma 6.1.Let Σ * be the spectrum of A + F e .Then it follows that