THE MALTHUSIAN PARAMETER AND R 0 FOR HETEROGENEOUS POPULATIONS IN PERIODIC ENVIRONMENTS

Since the classical stable population theory in demography by Sharpe and Lotka, the sign relation sign(λ0) = sign(R0 − 1) between the basic reproduction number R0 and the Malthusian parameter (the intrinsic rate of natural increase) λ0 has played a central role in population theory and its applications, because it connects individual’s average reproductivity described by life cycle parameters to growth character of the whole population. Since R0 is originally defined for linear population evolution process in a constant environment, it is an important extension if we could formulate the same kind of threshold principle for population growth in time-heterogeneous environments. Since the mid-1990s, several authors proposed some ideas to extend the definition of R0 so that it can be applied to population dynamics in periodic environments. In particular, the definition of R0 in a periodic environment by Bacaër and Guernaoui (J. Math. Biol. 53, 2006) is most important, because their definition of R0 in a periodic environment can be interpreted as the asymptotic per generation growth rate, so from the generational point of view, it can be seen as a direct extension of the most successful definition of R0 in a constant environment by Diekmann, Heesterbeek and Metz (J. Math. Biol. 28, 1990). In this paper, we propose a new approach to establish the sign relation between R0 and the Malthusian parameter λ0 for linear structured population dynamics in a periodic environment. Our arguments depend on the uniform primitivity of positive evolutionary system, which leads the weak ergodicity and the existence of exponential solution in periodic environments. For typical finite and infinite dimensional linear population models, we prove that a positive exponential solution exists and the sign relation holds between the Malthusian parameter, which is defined as the exponent of the exponential solution, and R0 given by the spectral radius of the next generation operator by Bacaër and Guernaoui’s definition.


(Communicated by Yang Kuang)
Abstract.Since the classical stable population theory in demography by Sharpe and Lotka, the sign relation sign(λ 0 ) = sign(R 0 − 1) between the basic reproduction number R 0 and the Malthusian parameter (the intrinsic rate of natural increase) λ 0 has played a central role in population theory and its applications, because it connects individual's average reproductivity described by life cycle parameters to growth character of the whole population.Since R 0 is originally defined for linear population evolution process in a constant environment, it is an important extension if we could formulate the same kind of threshold principle for population growth in time-heterogeneous environments.
Since the mid-1990s, several authors proposed some ideas to extend the definition of R 0 so that it can be applied to population dynamics in periodic environments.In particular, the definition of R 0 in a periodic environment by Bacaër and Guernaoui (J.Math.Biol.53,2006) is most important, because their definition of R 0 in a periodic environment can be interpreted as the asymptotic per generation growth rate, so from the generational point of view, it can be seen as a direct extension of the most successful definition of R 0 in a constant environment by Diekmann, Heesterbeek and Metz (J.Math.Biol.28,1990).
In this paper, we propose a new approach to establish the sign relation between R 0 and the Malthusian parameter λ 0 for linear structured population dynamics in a periodic environment.Our arguments depend on the uniform primitivity of positive evolutionary system, which leads the weak ergodicity and the existence of exponential solution in periodic environments.For typical finite and infinite dimensional linear population models, we prove that a positive exponential solution exists and the sign relation holds between the Malthusian parameter, which is defined as the exponent of the exponential solution, and R 0 given by the spectral radius of the next generation operator by Bacaër and Guernaoui's definition.

1.
Introduction.Since the brilliant study for the stable population theory by Alfred Lotka and his collaborators, the threshold condition for population growth formulated by the basic reproduction number R 0 and the Malthusian parameter (the intrinsic rate of natural increase), denoted by λ 0 here, have been recognized as a most important result in population dynamics and its applications.For example, the basic reproduction number for infectious diseases is essential to quantify the 314 HISASHI INABA disease prevention policy, and the Malthusian parameter is seen as a measure of fitness of mutant population in evolutionary biology.
In the stable population theory, the basic population system has a positive exponential solution with the growth rate λ 0 which dominates the asymptotic behavior of the population, and the following sign relation holds: sign(λ 0 ) = sign(R 0 − 1), (1) where the basic reproduction number R 0 is the asymptotic growth factor of the size of successive generations (the generational interpretation).Therefore R 0 can play a role of the threshold condition for population growth based on parameters capturing the average behavior of individuals, that is, it connects individual life cycle parameters to growth character of the whole population.Then we can summarize the essential feature of this threshold theory of population growth as follows: 1.There exists a positive dominant exponential solution, whose exponent gives the Malthusian parameter λ 0 of the population, 2. The basic reproduction number R 0 that allows the generational interpretation can be defined and the sign relation (1) holds between R 0 and λ 0 .
The above threshold theory was successfully established for heterogeneous populations in constant environments by Diekmann, Heesterbeek and Metz ( [20], it is called the DHM definition here for short).Then it would be an important extension if we could formulate the same kind of threshold theory for structured populations in time-heterogeneous environments.
During the last two decades, in the context of infectious disease epidemiology, several authors ([2]- [7], [27], [28], [49], [52]) proposed some ideas to extend the definition of R 0 so that it can be applied to population dynamics in periodic environments.In particular, the definition of R 0 in periodic environments by Bacaër and Guernaoui ( [2]) (which is called the BG definition for short in the following) is most important, because as is shown by Bacaër and Ait Dads ( [6], [7]) and Inaba ( [37]), their definition of periodic R 0 allows the generational interpretation, that is, it is interpreted as the asymptotic ratio of successive generation size, so it is a direct extension of the DHM definition of R 0 in a constant environment.
Although the threshold property of the BG definition of R 0 has been proved by several authors ( [46], [38], [49]), here we propose an another approach to establish the sign relation for R 0 and λ 0 in a periodic environment by focusing the existence of exponential solution.
First we briefly review the DHM definition of R 0 and the sign relation between the Malthusian parameter and R 0 in a constant environment.Second, we review basic results for the BG definition of R 0 when the state space is finite-dimensional, that is, the basic population evolution process is formulated by an ordinary differential equation system with periodic coefficients.Thirdly we formulate the definition of R 0 for an age-dependent population with periodic parameters (the multistate periodic Lotka-McKendrick population model) and show that a dominant positive exponential solution exists and the sign relation holds between its exponent and the BG definition of R 0 .Finally, as a more complex example, we introduce the BG definition of R 0 for the age-duration dependent SIR epidemic model with periodic coefficients.We again show that there exists a dominant positive exponential solution and the sign relation holds between its exponent and R 0 .Through those examples, our basic recipe is as follows: First we construct the solution evolutionary system and prove its uniform primitivity under appropriate conditions.The uniform primitivity of the periodic evolutionary system guarantees the existence of a dominant exponential solution, so we can define the Malthusian parameter by its exponent.Subsequently we introduce Bacaër and Guernaoui-type next generation operator to establish the sign relation between its spectral radius and the Malthusian parameter.In Appendix (section 7), we summarize the positive operator theory and show some new results for uniformly primitive periodic evolutionary system in addition to classical results by G. Birkhoff ([8]- [14]) and Inaba ([32]).
2. Preliminary: The Malthusian parameter and R 0 in a constant environment.First we review the definition of the basic reproduction number R 0 for heterogeneous population with finite i-state space in a constant environment.Although the definition of R 0 for heterogeneous population in a constant environment has been studied by demographers since 1970s ( [36]), it was first successfully established by Diekmann, Heesterbeek and Metz in 1990 ( [20], [21]) for epidemic models and it is still evolving ( [50], [35], [49], [22]).
Suppose that the state space of individuals, denoted by Ω = {1, 2, ..., N }, is a finite set and we neglect age structure.A subspace Ω b ⊂ Ω is called the birth state space1 if individuals can be born at state j ∈ Ω b .
Let p k (t), k ∈ Ω be the size of population in k-th state and p(t) = (p 1 (t), • • •, p N (t)) T be the density vector of multistate population.For the finite-dimensional autonomous case, the basic population evolution equation is given by where A = Q+M , M is a N ×N nonnegative, nonzero matrix called the reproduction matrix whose (j, k)-th entry m jk denotes the number of newborns produced at state j per unit time and per individual at state k, and Q is a N × N transition intensity matrix whose (j, k)-th entry q jk ≥ 0 (j = k) denotes the transition intensity from k-th state to j-th state.The diagonal element q kk (t) of Q is given by where µ k denotes the removal rate (by death or other causes) at k-th state.Then we assume that Q is a nonzero essentially nonnegative matrix, that is, all off-diagonal entries are nonnegative 2 .Then the survival rate matrix is given by L(t) = exp(Qt).If a square matrix A is irreducible and essentially nonnegative, it is called essentially positive.Then the following holds: Lemma 2.1 ([51], Section 8.2).A matrix A is essentially positive if and only if A + sI d3 is nonnegative, irreducible and primitive matrix for all sufficiently large s > 0.
Proof.Since "if" part is clear, we show "only if" part.If a matrix A is essentially positive, it is clear that A + sI d is nonnegative and irreducible for all sufficiently large s > 0. Then we can assume that A + (s − 1)I d is a nonnegative and irreducible n × n matrix for a large s > 0. From a well-known result ( [25], Chapter XIII, Lemma 1; [51], Lemma 2.2), we have Let s(Q) be the spectral bound of matrix Q, that is, s(Q) := max λ∈σ(Q) λ where σ(Q) is the set of eigenvalues of Q.Then it is natural to assume s(Q) < 0, because individuals die out as time evolves.Then the following holds (see [21], Lemma 6.12): Applying the variation-of-constants formula to (2), we have Multiplying M from the left hand side, we obtain a renewal equation formulation of system (2): where M p(t) gives a vector of newborns at time t and Ψ(τ ) := M L(τ ) is the net reproduction matrix.
Let v(t) := M p(t) be the birth rate vector of newborns.Then we obtain where g(t) = M L(t)p(0).Since g(t) is the density of newborns produced by the initial population p(0), we can define the successive generation of newborns by Then the solution of the renewal equation ( 3) is given by the generation expansion where v m (t) is the density of newborns produced as the m-th generation at time t.
Integrating both sides of the above iterative relation from 0 to ∞, we have Then if we define the time-aggregated density vector of m-th generation by The (i, j)-th entry of (−Q) −1 = ∞ 0 e Qτ dτ denotes the expected sojourn time that an individual born in state j spends in i-th state, so M (−Q) −1 maps a timeaggregated density vector of newborns of m-th generation to a time-aggregated density vector of the next generation of newborns.
If K is a nonnegative, irreducible and primitive matrix, it follows from the Perron-Frobenius Theorem that Since v m gives the total size of the m-th generation, the size of each generation asymptotically grows with geometric growth rate r(K), which is called the generational interpretation.Then K is called the next generation matrix ( [20], [21]), if every state is the birth state (Ω = Ω b ) 5 .
Recently, several authors established the following relation between r(K) and the spectral bound of A ( [22] Theorem A.1.; [49] From Proposition 2.4, we know that the stability of the trivial steady state (the steady state corresponding to population extinction) can be formulated by the spectral radius of the next generation matrix K. Based on the generational interpretation and the sign relation (4), the basic reproduction number R 0 for multistate (heterogeneous) populations is defined by R 0 = r(K).Although there are often surrogate threshold parameters which share the same kind of sign relation with R 0 ( [29]), R 0 = r(K) is distinguished from other indices by its generational interpretation.
Lemma 2.5.Under the assumption of Proposition 2.4, there exists a nonnegative exponential solution e λ0t φ 0 for the basic system (2), where φ 0 is a nonnegative eigenvector of A associated with eigenvalue λ 0 = s(A).
If we ensure the existence of a positive dominant6 exponential solution, we need additional conditions: Proposition 2.6 (Birkhoff [9]).If A is essentially positive, it has a unique (up to a constant factor) strictly positive eigenvector φ 0 associated with a real, simple eigenvalue λ 0 such that λ 0 > λ for any λ ∈ σ(A) \ {λ 0 }.Then the basic system (2) has a positive dominant exponential solution e λ0t φ 0 .
We remark that if e Qt > 0, that is, an individual born in any state can reach every state, Q is essentially positive, hence A is also essentially positive and there exists a dominant positive exponential solution.
In the following, if a dominant positive exponential solution exists, we call its exponent λ 0 the Malthusian parameter (the intrinsic rate of natural increase) of the heterogeneous population in a sense that every state-specific population asymptotically grows with a common exponent λ 0 .
3. Finite-dimensional periodic evolutionary system.Now we assume that individuals' reproduction and survival parameters are changing with time.If we consider the case that density-dependent effects can be neglected, p(t) satisfies a linear differential equation system as follows 7 .
Define the survival rate matrix L(t) as the solution of a matrix differential equation where I d denotes the N ×N identity matrix.In other words, L(t) is the fundamental matrix of the N -dimensional nonhomogeneous ODE system dx(t)/dt = Q(t)x(t), so it is invertible.Let ij (t) be the (i, j)-th entry of L(t).Then ij (t) for j ∈ Ω b is the probability that a newborn produced at state j and time zero will survive in state i and time t.
Define a 2-parameter system (the transition matrix) L(t, s) := L(t)L(s) −1 , t ≥ s.Then it forms an evolutionary system8 generated by Q(t), that is, the following properties hold: In the following, we call {L(t, s) : t ≥ s} the survival evolutionary system.The transition matrix is nonnegative; L(t, s) ≥ 0, since Q(t) is essentially nonnegative.The exponential growth bound of the evolutionary system L(t, s) is defined by where |A| denotes the norm of a matrix A.
In this paper, we adopt the following assumption: Assumption 3.1.We assume that Q(t) and M (t) are uniformly bounded and continuous on R, so For the survival evolutionary system, we assume that ω(L) < 0.
The above assumption ω(L) < 0 is biologically reasonable, because it implies that a closed population goes to extinction if there is no reproduction.For example, if we assume that inf j,t µ j (t) =: µ > 0, then we can observe that |L(t, τ )| ≤ e −µ(t−τ ) , t ≥ τ , so it follows that ω(L) ≤ −µ.
Next let U (t, s), t ≥ s be the evolutionary system associated with the generator A(t).By using a fundamental matrix Φ(t) of (5), we have U (t, s) = Φ(t)Φ(s) −1 .Then the solution of ( 5) is given by p(t) = U (t, s)p(s) and {U (t, s) : t ≥ s} is an evolutionary system acting on R n such that Applying the variation-of-constants formula, we have Then L(t, s) ≥ 0 implies U (t, s) ≥ 0, so {U (t, s) : t ≥ s} is a positive evolutionary system.Moreover, we note that U (t, s) is strongly positive if L(t, s) is strongly positive 9 .Under some additional conditions, a positive evolutionary system becomes weakly ergodic, that is, every positive solutions are asymptotically proportional to each other (see Appendix).A useful sufficient condition to guarantee the weak ergodicity is uniform primitivity (see section 7.3).Here we show a simple condition to guarantee the uniform primitivity of the evolutionary system U (t, s) associated with the ODE system (5) ( [53]): Proposition 3.2.Suppose that there exists an essentially positive matrix B such that B ≤ A(t) for all t ∈ R10 .Then the positive evolutionary system U (t, s), t > s is uniformly primitive.
Proof.From our assumption here and the assumption 3.1, there exists an essentially positive matrix C such that A(t) ≤ C for all t ∈ R. From the variation-of-constants formula, we can observe that Then we obtain 0 < e (t−s)B ≤ U (t, s) ≤ e (t−s)C .Therefore if p(0) > 0, then p(t) > 0 for all t > 0 and it is easy to see that the projective diameter of U (s + 1, s) is uniformly bounded, so U (t, s) forms a uniformly primitive multiplicative process in the sense of G. Birkhoff.
In the following, let us consider the periodic case.Define the state space E := C θ (R; R n ), which is a Banach space of all θ-periodic continuous functions from R to R n equipped with maximum norm.Let E + be its positive cone.Suppose that Q(t) and M (t) are θ-periodic (matrix-valued) functions.Then it follows that If there exist a number λ ∈ C and φ ∈ E \ {0} such that e λt φ(t) satisfies ( 5) for all t ∈ R, we call it the exponential solution with exponent λ.If the basic system (5) has a positive exponential solution and any solution is asymptotically proportional to the exponential solution, we call the exponent of the exponential solution the Malthusian parameter (or the intrinsic rate of natural increase).
For the weakly ergodic evolutionary system, if there exists a positive exponential solution, any positive solution is asymptotically proportional to the exponential solution by the weak ergodicity, so the positive exponential solution necessarily dominates the asymptotic behavior of the evolutionary system, and the exponent of the positive exponential solution is the Malthusian parameter.
From Proposition 3.2 and general results Proposition 7.19 and Proposition 7.24, we have Corollary 3.3.Under the assumption of Proposition 3.2, the basic system (5) with periodic coefficients has a positive exponential solution and every positive solutions are asymptotically proportional to the exponential solution.
From the dominant property of the exponential solution and the above sign relation, we know that r(U (s + θ, s)) is "the most obvious choice as threshold quantity for periodic system" ( [21], p. 149), so several authors developed this type of approach to R 0 ( [27], [28], [4]).However, the monodromy operator is not the next generation operator, so r(U (s + θ, s)) does not have the generational interpretation and we can not see r(U (s + θ, s)) as an extension of the DHM definition of R 0 in a constant environment.
Instead of the monodromy matrix approach, Bacaër and his collaborators ([2]- [7]) have developed a definition of R 0 in periodic environments based on the theory of periodic renewal equation theory ( [46], [38]).For our ODE model ( 5), the next generation operator K θ in the Bacaër-Guernaoui definition is defined as follows: The basic reproduction number in a periodic environment by the BG definition is the spectral radius of the positive operator K θ .
As is shown in separate papers ( [6], [7], [37]), the basic reproduction number given by r(K θ ) satisfies the generational interpretation, that is, it gives the asymptotic ratio of successive generation size.Thus it is reasonable to define R 0 by r(K θ ) from the generational point of view.In the following, we show that the sign relation (1) holds between the Malthusian parameter λ 0 , whose existence is guaranteed by Corollary 3.3, and R 0 given by the BG definition 11 .
First observe that e λt φ is an exponential solution of ( 5) if and only if the following eigenvalue problem has a solution (λ, φ) ∈ C × E12 : According to the standard split of the population generator ( [21], [40], [50]), we define where φ ∈ E.
Consider the resolvent equation as For any s < t, by using the variation of constants formula, we have Observe that there exists a number M ≥ 1 such that Then if we let s → −∞ with λ > ω(L), we have Therefore the resolvent operator exists for λ > ω(L) and it is positive for real λ > ω(L) 13 : For λ > ω(L), let K(λ) be an operator14 defined by Using the operator K(λ), we can rewrite the eigenvalue problem (6) as follows: On the other hand, we can also introduce a similar operator K(λ) as follows: Then we know that if ψ = K(λ)ψ, λ > ω(L), then φ := (λ − Ã1 ) −1 ψ gives the eigenfunction of ( 6) associated with eigenvalue λ.Then it holds that P σ (K(λ)) = P σ ( K(λ)) and r(K(λ)) = r( K(λ)) , although we omit the proof.
In order to reduce K(λ) to a compact operator, using the well-known calculation ( [2], [4]), we define an integral operator J(λ) on Then Ψ and Θ have a period θ.Observe that Then a matrix Θλ (t, σ) is well defined for λ > ω(L) on (t, σ) ∈ [0, θ] × [0, θ] and it is continuous except for t = σ.From the Ascoli-Arzelá Theorem, it follows that J(λ) is a compact operator on Z.
Let P : Z → E be a periodization operator defined by As is shown by [4] and [6], we can observe that for φ ∈ E where Proposition 3.5.Suppose that the basic system (5) with periodic coefficients has a positive exponential solution with exponent λ 0 .If J(λ) or J(λ) is compact and strongly positive for all λ > ω(L), then the sign relation holds: where R 0 = r( K(0)) = r(K(0)).
Proof.It is sufficient to prove for the case that J(λ) is compact and strongly positive for all λ > ω(L).From the well-known Krein-Rutman theorem and its stronger version based on the strong positivity ( [17], Theorem 19.2, 19.3), r( J(λ)) is an eigenvalue of J(λ) and r( J(λ)) is strictly monotone decreasing for λ, because J(λ 1 ) J(λ 2 )  [32], [34], [35]).Here we introduce the definition of R 0 for the multistate Lotka-McKendrick model with time periodic parameters 16 and establish the sign relation.
In general, the age-dependent population dynamics in a heterogeneous environment is described as follows ( [31], [32], [35]): where p(t, a) = (p 1 (t, a), ..., p N (t, a)) T is the population density vector composed of the age density function p j (t, τ ) at each state j ∈ Ω, a is the chronological age and β is the length of reproductive period, which is assumed to be finite, Q(t, a) is an essentially nonnegative matrix whose (j, k) (j = k) element q jk (a) is the force of transition from k-th state to j-th state at age a at time t, and the diagonal element is given by is the removal rate (caused by death, emigration, etc.) and M (t, a) is a nonnegative matrix whose jk-th element m jk (a) is the age-dependent fertility rate of children with state j who are produced by individuals at state k.We assume that M (t, a) ≡ 0 for a > β.
First we define the survival rate matrix L(x; t, a), x ≥ 0 as the solution of the matrix ODE system: Integrating the McKendrick equation along the cohort line ( [32]), the solution p(t, a) of ( 7) with initial data p(s, a), s ≤ t is given as p(t, a) = (U (t, s)p(s, •))(a), where two-parameter family of linear positive operators U (t, s), s ≤ t from the state space where B(ξ; φ, s), ξ > 0 denotes the number of newborns per unit time at time s + ξ with initial data φ at time s, it is the solution of the renewal equation: Although the expression ( 8) is not necessarily differentiable with respect to time and age, it has a directional derivative along the cohort line, so we consider (8) as a generalized solution of the Lotka-McKendrick system (7) with initial data φ ∈ X + ( [30]).
Then U (t, s), s ≤ t, forms an evolutionary system (time-inhomogeneous multiplicative process), so it satisfies the law of evolution: Moreover, here we assume that M (t, a) and Q(t, a) are θ-periodic with respect to time t.Then it follows that L(x; t + θ, a) = L(x; t, a), U (t + θ, s + θ) = U (t, s).
For the periodic evolutionary system U (t, s), if there exists a θ-periodic X +valued function φ such that e λt φ(t) = U (t, s)e λs φ(s) for all t > s, it is called the exponential solution.As is shown by Lemma 7.23, if U is a strictly positive evolutionary system, there exists an exponential solution with exponent λ if and only if the monodromy operator U (s + θ, s) has an eigenvector associated with a positive eigenvalue e θλ .
In the following, we assume the following assumption: Although we omit the proof here, the above assumption is sufficient to show the the uniform primitivity of U by using the same kind of argument as [32]: 32]).Under the assumption 4.1, U is a uniformly primitive evolutionary system in X + .
From the above proposition and Proposition 7.24, we conclude that Corollary 4.3.Under the assumption 4.1, if U is a periodic evolutionary system on X + , then it has a dominant positive exponential solution.
From the above corollary, the periodic evolutionary system U has the Malthusian parameter λ 0 .Then we establish the sign relation between the Malthusian parameter λ 0 and R 0 calculated as the spectral radius of the the next generation operator for the periodic Lotka-McKendrick system (7): where f is a θ-periodic locally integrable function 18 .
First note that if the exponential solution e λt φ(t, a) satisfies the basic system (7) in the sense of generalized solution, (λ, φ) ∈ R × F is a solution of the eigenvalue problem: where D is a directional derivative operator defined by and F is a Banach lattice composed of θ-periodic, locally integrable function R × [0, β]: , where its norm is given by In order to formulate a next generation operator, we formally rewrite (9) as an abstract eigenvalue problem in an extended state space 19 to use the standard split of the population generator as in the previous section.Let Y be a Banach lattice composed of θ-periodic, R n -valued locally integrable function on R: where Y -norm is given by Let W := Y × F be an extended state space for the eigenvalue problem (9).Then F is identified with the subspace W 2 := {0} × F ⊂ W and we write φ ∈ F as (0, φ) when we see φ ∈ F as an element of W 2 .Moreover, Y is identified with the subspace Then the eigenvalue problem ( 9) is formulated as follows: Then we have Therefore we can define an operator K(λ) : the eigenvalue problem (9) can be written as In fact, if there exists a number λ such that K(λ)(f, 0) = (f, 0), (λ, φ) is a solution of the eigenvalue problem (9) where (λ − A 1 ) −1 (f, 0) = (0, φ).
Based on the above formal discussion, again we define a linear integral operator where we adopt a convention that Ψ = 0 for a ∈ R \ [0, β].Then we obtain Lemma 4.4.The periodic evolutionary system U has a positive exponential solution with exponent λ 0 if and only if K(λ 0 ) has a positive eigenvector associated with eigenvalue unity.
By using the same kind of argument as Norton's lemma ( [32], Lemma 4.1), we here prepare the following technical lemma: Lemma 4.5.Under the assumption 4.1, for any δ > 0 such that 0 < δ < (γ 2 − γ 1 )/2, it follows that is defined iteratively as follows: Proof.From our assumption, (10) holds for n = 1.Assume that (10) holds for n = k.If we assume that t − a ∈ S k , we have t − a ≥ γ 1 and Let I = I 1 ∩ I 2 where If t − a ∈ S k+1 , it follows that I 2 ⊂ I 1 , hence the length of the interval I, denoted by |I|, is greater than Now we define I = I 1 ∩ I 2 where

Then again we have
Therefore we conclude that (10) holds for n = k + 1.By mathematical induction, (10) holds for every positive integer.Lemma 4.6.Under the assumption 4.1, the operator J(λ) is nonsupporting 20 for λ ∈ R.

Proof. Observe that for f
where f = P f Z .From Lemma 4.5, we have If we take a large n such that |S n | > θ and N n > 0, for each t ∈ [0, θ] there exists an integer k such that

Then it follows that
Therefore we obtain for t ∈ [0, θ] Let F ∈ Z * + 21 be a positive functional.From the above estimate, it follows that 20 For the definition of nonsupporting operators, the reader may refer to section 7. 21 Z * is the dual cone of the positive cone Z + .
which implies that the operator J(λ) is nonsupporting.
From the well-known compactness criterion in L 1 and the assumption 4.7, it is easy to see that J(λ) is a compact operator on Z. Proposition 4.8.Under the assumption 4.1 and 4.7, there exists a positive exponential solution with exponent λ 0 and the sign relation holds: Proof.From Lemma 4.4, K(λ 0 ) and so J(λ 0 ) has a positive eigenvector associated with eigenvalue unity.Since J(λ 0 ) is nonsupporting and compact, its spectral radius r(J(λ)) is a positive dominant eigenvalue of J(λ) associated with a (essentially unique) positive eigenvector and it is a strictly decreasing function of λ (Corollary 7.6, Proposition 7.7).Therefore we have r(J(λ 0 )) = r(K(λ 0 )) = 1 and we obtain the sign relation (11).
Since K θ = K(0), we conclude that the BG definition R 0 = r(K θ ) is the reasonable extension of the definition of R 0 in a constant environment in a sense that R 0 allows the generational interpretation ( [7], [37]), the Malthusian parameter λ 0 is well-defined and the sign relation holds between R 0 and λ 0 .
5. Infinite-dimensional periodic evolutionary system II: The age-duration dependent SIR epidemic model.Finally, as a more complex example, let us consider an age-duration structured epidemic model with periodic coefficients.
Let us introduce I(t, τ ; a) as the density of infected population at time t and infection-age τ whose chronological age at infection is a.Let S(t, a) be the age density of susceptibles at time t and age a and R(t, τ ; a) the density of recovered individuals at time t, duration (the time elapsed since recovery) τ and age at recovery a.Then the age-duration-dependent homogeneous SIR epidemic system is formulated as follows: where B is the number of susceptible newborns per unit time, µ(a) the force of mortality, γ(τ ; a) the rate of recovery at infection-age τ and the age at infection a and κ is the force of infection 22 .We assume that the force of infection is given by where m(t, a, σ) can be interpreted as the probability that a susceptible individual with age a encounters with an infected individuals with age σ at time t, and the function f (τ ) is the probability of successful transmission of infective agents from infective individuals with infection-age τ .In the expression ( 13), we omit the scale factor 1/N (N is the total host population size), because we can assume that the host population is in a demographic steady state and so N is constant.
For simplicity, we assume that demographic parameters of the host population is time-independent, but the transmission coefficient m(t, a, σ) is θ-periodic with respect to time variable.Although this assumption is mathematically restrictive, it would be reasonable for common childhood infectious diseases with seasonally changing parameters.
Thus we can assume that the host population density, described by P (t, a) := S(t, a)+ where J(t, τ ; a) denotes the density of infected population at time t and infection-age τ whose age at infection is a in the initial invasion phase.First we check the uniform primitivity ergodicity of the linear evolutionary process given by (14).Integrating ( 14) along the characteristic line, we can obtain the expression as Inserting (15) into the boundary condition of ( 14), we obtain an abstract renewal equation for B(t, a) := J(t, 0; a) as where Here we assume that the infectious period is finite, so there exists a number β > 0 (the length of the period of infectiousness) such that f (τ ) = 0 for τ > β.Then Ψ(t, x) is a linear positive operator on L 1 (R + ) and Ψ = 0 for x > β.
Let R(t, z) be the resolvent kernel (operator on L 1 ) corresponding to Ψ(t, t − z) defined by the solution of the resolvent equation: Using the resolvent kernel, we can obtain the solution of ( 16) as follows: For simplicity, we here adopt the following assumption, although we could relax this positivity condition as assumption 4.1 in the previous section: Assumption 5.1.m, γ, f and µ are bounded nonnegative measurable functions for t ∈ R and a, σ, τ ∈ R + .In particular, suppose that f := inf f > 0 and β := inf β > 0.
Let us define the population evolution operator U (t, s), t > s on where Then the following holds: Proposition 5.3.Under the assumption 5.1, U is a uniformly primitive periodic multiplicative process.
Proof.Since U (t, s), t > s is a solution evolution operator for (14), it forms an nonnegative evolutionary system.It follows from Ψ(t+θ, x) = Ψ(t, x) and ) is a periodic multiplicative process.In order to see the uniform primitivity of U (t, s), it is sufficient to show that U (t, 0) has a finite projective diameter 23 for some t > 2β.In fact, if ∆(U (t, 0)) ≤ α, then ∆(U (t + nθ, nθ)) ≤ α for all integer n by its periodicity, so U is uniformly primitive for positive time.From ( 17) and ( 18), we have where J 0 (τ ; a) = J(0, τ ; a) ∈ X and Let us define Then for J 0 ∈ X, we have For a fixed t > 2β, λ is a positive functional and e is a quasi-interior point in X + .Therefore we conclude that ∆(U (t, 0)) ≤ 2 log α.Then U (t, s) is a uniformly primitive multiplicative process.
From Proposition 5.3, we know that there exists a dominant positive exponential solution and so the Malthusian parameter λ 0 is defined as its exponent.Inserting an exponential solution e λt ψ(t, τ ; a) into ( 14), we obtain the following eigenvalue problem: where where is a periodization operator defined by and its norm is given by Then it is easy to see that a function ψ given by satisfies the eigenvalue problem (22) if φ ∈ Y satisfies the boundary condition: The boundary value ψ(t, 0; a) = φ(t, a) denotes the density of newly produced infecteds at age a and time t.
Define a linear positive operator on L 1 (R + ) as where u ∈ L 1 (R + ).Then ( 23) can be written as follows: where K(λ) is a linear operator on Y defined by If φ = P φ Z ∈ Y for φ ∈ Z, it follows from the periodicity of φ and Ψ λ that (24) can be reduced to an equation on the space Z: where Therefore , we have K(λ)P = P J(λ) and r(K(λ)) = r(J(λ)).
Now we define a bounded measurable function Π λ (t, a, y, z) for t, y ∈ R and a, z ∈ R + as where we adopt a convention that Π λ (t, a, y, z) = 0 if a < 0, z < 0 and t < y.Then we have so we can see J(λ) as an integral operator from L 1 -space Z into itself.
Lemma 5.4.Suppose that the following holds uniformly for (y, z): Then the operator J(λ), λ ∈ R is a nonsupporting compact operator on Z.
Proof.Using the function Π λ , we can rewrite (23) as where Therefore it follows from the above assumption and the well-known Frechét-Kolmogorov compactness criterion in L 1 that J(λ) is compact.Next we can observe that Then it is easy to see that J(λ), λ ∈ R is a nonsupporting operator on Z.
By using the same kind of argument as Proposition 4.8, we can conclude as follows: Proposition 5.5.Under the assumption 5.1, there exists a positive dominant exponential solution with exponent λ 0 such that r(J(λ 0 )) = r(K(λ 0 )) = 1 and sign(λ 0 ) = sign(r(J(0)) − 1) = sign(r(K(0)) − 1), From the above result, we know that it is reasonable to define the basic reproduction number for the periodic epidemic system (12) by the spectral radius of the integral operator K(0) and we have the sign relation between λ 0 and R 0 = r(K(0)).

Conclusion.
As is seen in the main text and the appendix, the asymptotic behavior of a non-autonomous linear positive system can be characterized by the positive exponential solution if it exists and the evolutionary system is weakly ergodic.Uniform primitivity is a useful sufficient condition for weak ergodicity of positive evolutionary system.If the evolutionary system is periodic and uniform primitive, we can show that there exists a dominant positive exponential solution, so the basic system has the Malthusian parameter λ 0 .Moreover, as is shown for typical cases of linear population dynamics in heterogeneous environments, we can construct the next generation operator to show the sign relation between the Malthusian parameter λ 0 and R 0 given by the spectral radius of the next generation operator.As is shown by Bacaër and Ait Dads ( [7]) and Inaba ([37]), the generational interpretation holds for this BG definition of R 0 in periodic environments, that is, R 0 becomes the asymptotic growth ratio of successive generation size.Therefore, we can say that the classical threshold theory for R 0 and the Malthusian parameter in constant environments can be extended to the case of periodic environments without loosing its essential features.

Appendix.
7.1.Theory of linear positive operators.Let E be a real or complex Banach space and let E * be its dual space.Then E * is a space of all linear functionals on E. In the following, we write the value of f ∈ E * at ψ ∈ E as f, ψ .
A closed subset C ⊂ E is called the cone (or positive cone) if the following conditions hold: Proposition 7.1 (Krein-Rutman Theorem; [39], [45]).Suppose that C is total, a positive linear operator T : C → C is compact and r(T ) > 0. Then r(T ) is an eigenvalue of T corresponding to a positive eigenvector ψ ∈ C + .Proposition 7.2.Suppose that C is a solid cone and T : C → C is a compact linear strongly positive operator.Then it follows that (1) r(T ) > 0, r(T ) is a simple eigenvalue associated with an eigenvector in C o and there is no other eigenvalue with a positive eigenvector.(2) |λ| < r(T ) for all eigenvalues λ = r(T ).Definition 7.3 ([44], [41]).A positive operator T ∈ B(E) is called semi-nonsupporting if for any ψ ∈ C + and f ∈ C * \ {0}, there exists an integer p = p(ψ, f ) such that f, T p ψ > 0. A positive operator T ∈ B(E) is called nonsupporting if for any ψ ∈ C + and f ∈ C * \ {0}, there exists an integer p = p(ψ, f ) such that f, T n ψ > 0 for all n ≥ p.A positive operator T ∈ B(E) is called strictly nonsupporting if for any ψ ∈ C + , there exists a positive integer p = p(ψ) such that T n ψ is a quasi-interior point of C for all n ≥ p.
The idea of semi-nonsupporting is an infinite-dimensional extension of indecomposability of nonnegative matrices.Krasnoselskij called it irreducible.The idea of nonsupporting is an infinite-dimensional extension of primitivity of nonnegative matrices.Proposition 7.4 ([44], [41]).Suppose that the cone C is total, T ∈ B(E) is seminonsupporting with respect to C and r(T ) is a pole of resolvent R(λ, T ) = (λ−T ) −1 .Then the following holds: (1) r(T ) ∈ P σ (T ) \ {0} and r(T ) is a simple pole of the resolvent R(λ, T ); (2) The eigenspace corresponding to r(T ) is one-dimensional and its eigenvector ψ ∈ C is a quasi-interior point.Any eigenvector in C is proportional to ψ; (3) The adjoint eigenspace corresponding to r(T ) is one-dimensional and its eigenfunctional f ∈ C * \ {0} is strictly positive.
where Γ 0 is a positively oriented circle with center at r(T ) such that no points of the spectrum σ(T ) except r(T ) lie on and inside the circle Γ 0 .
From the above statement, we know that lim n→∞ r(T ) −n T n converges to a projection operator on one-dimensional eigenspace spanned by the positive eigenvector associated with the dominant positive eigenvalue r(T ).
Combining Krein-Rutman Theorem and Sawashima's theorem, we can obtain a useful statement: Corollary 7.6.Suppose that the cone C is total, r(T ) > 0, T is power compact 25  and nonsupporting with respect to C. Then all statements of Proposition 7.4 and 7.5 hold.
Proof.Suppose that T n is compact.Then the spectrum σ(T n ) is a countable set with no accumulation point different from zero.From the Spectral Mapping Theorem, we have σ(T n ) = {σ(T )} n , in particular r(T n ) = r(T ) n > 0 and 25 T ∈ B(E) is power compact if there is a positive integer n such that T n is compact.r(T ) = r(T n ) 1/n ∈ σ(T ).Since T is power compact, its nonzero eigenvalue r(T ) is a pole of the resolvent R(λ, T ) ( [23], p. 579).Therefore we can apply Proposition 7.4 to T and arrive at the conclusion.Proposition 7.7 ([41]).Let E be a Banach lattice.Suppose that S, T ∈ B(E) are positive operators.Then the following holds: (1) If S ≤ T , then r(S) ≤ r(T ).
7.2.The projective metric and uniform primitivity.We here summarize some basic concepts and results of the projective metric and uniform primitivity of positive operators, as long as they are needed to discuss the weak ergodicity of evolutionary system in a Banach lattice 26 .For omitted proofs, the reader may consult Birkhoff ([12], [13], [14]), Inaba ([32]) and its references.
Let E be a Banach lattice 27 with a total positive cone C. For (x, y) ∈ E × C + , we define sup(x/y) := inf{λ : x ≤ λy} and inf(x/y) := sup{µ : µy ≤ x}, where we adopt conventions such that inf ∅ = ∞ and sup ∅ = −∞.In C + , the Hilbert projective pseudometric is defined as follows: Then it is easy to see that d(x, y) has the following properties: By the metric d, {C + , d} becomes a pseudometric space.The connected component in {C + , d} is an equivalent class composed of elements such that d(x, y) < ∞.The ray is an equivalent class composed of elements such that d(x, y) = 0. Two elements x, y in C + are called comparable if there exist µ > 0 and α ≥ 1 such that µy ≤ x ≤ αµy.Then two elements x, y of C + belong to the same component if and only if they are comparable.
If two elements in C + are comparable, images of those elements by a strictly positive linear operator A are also comparable, and it follows that Then a strictly positive linear operator is a contraction mapping with respect to the projective metric d.The projective diameter of a strictly positive operator A is defined by ∆(A) := sup{d(Ax, Ay) : (x, y) ∈ C + × C + }. 26 The main results of Birkhoff's theory hold in Archimedian semi-ordered real linear space. 27For a real liner space X with partial order ≤, X is called lattice if for any two elements x, y ∈ X, there are the least upper bound by the partial order ≤, denoted by sup(x, y) =: x ∨ y and the the greatest lower bound, denoted by inf(x, y) =: x ∧ y.Moreover, the partial order ≤ satisfies the following condition (1) x ≤ y, then x + z ≤ y + z, (2) x ≤ y and λ ≥ 0, then λx ≤ λy, then X is called vector lattice.The absolute value of an element of a vector lattice x ∈ X is defined by |x| = x ∨ (−x).A real Banach space X is called Banach lattice if it is a vector lattice and its norm • satisfies x ≤ y provided that |x| ≤ |y|.
For a linear strictly positive operator A, it is called uniformly positive if ∆(A) < ∞.If a power of A becomes uniformly positive, A is called uniformly primitive28 .Lemma 7.9.Let A and B be strictly positive linear operators.Then it follows that ∆(AB) ≤ min{∆(A), ∆(B)}.
Proof.From the definition, we have for any x ∈ C.
Corollary 7.11.If A is a uniformly primitive operator such that (25) holds for a quasi-interior point e, then A strictly nonsupporting.
Proof.Suppose that for some integer n, A n satisfies ( 25) with a quasi-interior point e.For any v * ∈ C * \ {0} and x ∈ C + , we have v * , A n x ≥ λ(x) v * , e > 0, which shows that A n x is a quasi-interior point.
Corollary 7.12.If a strictly positive linear operator A is uniformly primitive, there exists an integer n such that the range of A m (C + ) (m ≥ n) is included in a connected component K and K is invariant with respect to A, that is, A(K) ⊂ K.
For a strictly positive linear operator A and elements x, y such that 0 < d(x, y) < ∞, the projective norm A p or the Birkhoff contraction ratio k(A) is defined by and the oscillation ratio N (A) is defined by where osc(x/y) := sup(x/y) − inf(x/y).
Proposition 7.14 ([8]- [14]).Suppose that A is a strictly positive linear operator in E. Then it follows that From the above theorem, a uniformly positive linear operator A is a strictly contractive mapping with respect to the projective metric.That is, its contraction ratio is less than tanh(∆(A)/4): Proof.The original proof for the case of uniformly positive operator is given in [8].Suppose that for some integer k, A k is uniformly positive.Then ∆ Therefore {A k+j x} j=1,2,,.. is a Cauchy sequence in a complete connected component which shows that φ is a positive eigenvector of A.
Corollary 7.17.Let φ is a positive eigenvector of a uniformly primitive operator A associated with positive eigenvalue γ.Then there exist a strictly positive linear functional v * , a positive constant M (x) and 0 where v * is a strictly positive eigenfunctional of the dual operator A * associated with the positive eigenvalue γ.
Proof.Suppose that A k is uniformly positive.Observe that sup(A n x/A n φ), n = 1, 2, .. is monotone decreasing and inf(A n x/A n φ) is monotone increasing and positive for x ∈ C + because A n (C + ) is a connected component for n ≥ k, and it holds that 0 ≤ osc(A n x/A n φ) = sup(A n x/A n φ) − inf(A n x/A n φ) ≤ (e d(A n x,A n φ) − 1) sup(A k x/A k φ) → 0, (n → ∞).
Then we can define a strictly positive functional v * as v * , x := lim n→∞ inf(A n x/A n φ) = lim n→∞ sup(A n x/A n φ).
From the definition, we have inf(A n x/A n φ)A n φ ≤ A n x ≤ sup(A n x/A n φ)A n φ.
inf(A n x/A n φ)A n φ ≤ v * , x A n φ ≤ sup(A n x/A n φ)A n φ Proposition 7.18.If f (t) is consistent with U (t, s) and v * (t) is consistent with U * (s, t), it follows that for any t, s ∈ J v * (t), f (t) = v * (s), f (s) = const.
Let U (t, s), t ≥ s be a strictly positive evolutionary system for positive time on (E, C), it is called weakly ergodic if for any ψ, φ ∈ C + , lim t→∞ d(U (t, s)ψ, U (t, s)φ) = 0. Proposition 7.19 ([13], [32]).Let U be a strictly positive evolutionary system for positive time on (E, C).If U is uniformly primitive for positive time, it is weakly ergodic.
The functional v * (s) is consistent with the dual process U * (s, t), s ≤ t and, up to a positive constant factor, it is uniquely determined.
A strictly positive linear functional v * (s) defined in Proposition 7.20 is called importance functional.Proposition 7.21 ([13]).Let U (t, s), t ≥ s be a weakly ergodic time-inhomogeneous multiplicative process for positive time on a Banach lattice (E, C).Then the importance functional is a one and essentially only one positive linear functional consistent with the dual process U * (s, t), t ≥ s.
Proof.Suppose that a uniformly primitive periodic evolutionary system U has an exponential solution e λt φ(t).Then it follows from ( 28) that there exists a positive functional v * (s) such that for any x ∈ C + satisfying d(φ(s), x) < ∞, From (29), we know that for any function f (t) = U (t, s)x consistent with a uniformly primitive periodic multiplicative process, we obtain lim t→∞ f (t) f (t) − φ(t) φ(t) = 0, which shows that the normalized distribution f (t)/ f (t) converges to a periodic distribution φ(t)/ φ(t) .
7.5.Reproductive value in a periodic environment.Bacaër and Abdurahman ( [5]) proposed to define w * (s) as the reproductive value in the periodic environment.In fact, if the population vector p(t) is evolved as p(t) = U (t, s)p(s) and the total reproductive value is defined by V (t) := w * (t), p(t) , it follows from (25) that V (t) = w * (t), p(t) = e λt v * (t), p(t) = e λt v * (0), p(0) = e λt V (0), which shows that the exponential growth of the "total reproductive value", so it is seen as an extension of Fisher's theorem for the reproductive value in a constant environment ( [31]).On the other hand, the importance functional v * (s) is seen as the demographic potential defined by Ediev ([24]) in the context of Lotka's stable population model.Then Birkhoff's result (27) implies Ediev's observation that the total demographic potential v * (t), f (t) is constant in the time-inhomogeneous Lotka model.
If we consider a constant environment, v * (s) becomes a (essentially) unique persistent (exponential) solution of the dual evolution system (semigroup) and w * (s) is a time-independent positive eigenvector of the generator of the dual semigroup.

where u 0
and v 0 are the right and left positive eigenvectors of K associated with the dominant eigenvalue r(K).Therefore we have lim m→∞ m |φ m | = lim m→∞ m v m = r(K), where |φ m | = N j=1 |φ m j | is the norm of N -vector φ m := (φ m 1 , .., φ m N ) T and v m = ∞ 0 |v m (t)|dt.

( 1 )
C + C ⊂ C, (2) λ ≥ 0 ⇒ λC ⊂ C, (3) C ∩ (−C) = {0} and (4) C = {0}.With respect to the cone C, we write x ≤ y if y − x ∈ C, and x < y if y − x ∈ C + := C \ {0}.If the set {ψ − φ : ψ, φ ∈ C} is dense in E, the cone C is called total.If E = C − C, C is called a reproducing cone.If a cone C has nonempty interior C • , C is called a solid cone.Any solid cone is reproducing.We write x y if y − x ∈ C • Let B(E) be a set of bounded linear operators from E into itself.Let r(T ) be the spectral radius of T ∈ B(E) and let P σ (T ) be the point spectrum of T .The dual cone C * is a subset of E * composed of all positive linear functionals
2, Lemma 4; [51], Theorem 8.2).A square matrix A is essentially nonnegative if and only if e At ≥ 0 for all t ≥ 0, and A is essentially positive if and only if e At > 0 for all t > 0.

4 .
Infinite-dimensional periodic evolutionary system I: The Lotka-McKendrick age-depndent population model.If we take into account the age structure, the basic linear population system is formulated as the multistate stable population model (Lotka-McKendrick model), which has been developed in mathematical demography 15 ([31], On the other hand, it follows from d(ABx, ABy) ≤ d(Bx, By) that ∆(AB) ≤ ∆(B).This completes our proof.Lemma 7.10 ([32]).A strictly positive linear operator A is unifromly positive if and only if there exist e ∈ C + , α ≥ 1 and a strictly positive functional λ(x) such that λ(x)e ≤ Ax ≤ αλ(x)e, Proposition 7.15 ([8]).A connected component of a positive cone C in E is a complete pseudo metric space with respect to the projective metric.Proposition 7.16 (The projective contraction mapping principle).If a strictly positive bounded linear operator A on a Banach lattice E is uniformly primitive, A has a unique fixed point (positive eigenvector) φ ∈ C + with respect to d, and A n x (n = 1, 2, ..) converges to φ for any x ∈ C + .