数理解析研究所講究録
第 1557 巻 2007 年 28-44

28

Homogeneous Epidemic Systems in the Stable
Population*
Hisashi Inaba
Department of Mathematical Sciences, University of Tokyo,
3-8-1 Komaba Meguro-ku, Tokyo 153-8914 Japan
E-mail: inaba@ms.u-tokyo.ac.jp
April 2, 2007

Abstract
In this paper, we develop anew approach to deal with asymptotic behavior of the age-structured homogeneous epidemic systems. For homogeneous systems, there is no attracting nontrivial equilibrium, instead wehave to examine exlstence and stability of persistent solutions. Assumingthat the host population dynamics is described by the stable population
model, we rewrite the basic system into asystem of ratio age distribution, which is the age profile divided by the stable age profile. If the hostpopulation has the stable age profile, the ratio age distribution system
is reduced to anormalized system. Then we prove alinearized stability
principle that the local stability [instability] of steady states of the normalized system implies the orbital stability [instability] of corresponding
persistent solutlons of the original homogeneous system.

Keywords: homogeneous epidemic system, stable population, age profile,linearized stability principle.
Mathematical Subject Classification (2000): $92D30,92D25$ .

1

Introduction: Homogeneous epidemic systems

First let us introduce a very general formulation for age-structured epidemic
systems. Suppose that the host population is a closed age-structured population divided into subpopulations (compartments), each of which corresponds
to an epidemic i-state, for example, susceptible, exposed, infective, removed,vaccinated state and so on. The state of such a population is described bymeans of age density functions $U_{j}(t, a))$ where denotes time, stands for age
$n$

$t$

$a$

’This paper is a revised version of the first half of Inaba [13]. We have removed theassumption used in [13] that the host population has a positive Malthusian parameter.

29
and indicates the j-th epidemic state. The dynamics of this model is describedby the system of equations$j$$\frac{\partial U_{j}(t,a)}{\partial t}+\frac{\partial U_{j}(t,a)}{\partial a}=-\mu_{j}(a)U_{j}(t, a)+\sum_{k=1}^{n}g_{jk}(a, U(t))U_{k}(t, a)$$U_{j}(t, 0)= \sum_{k=1}^{n}\int_{0}^{w}m_{jk}(a)U_{k}(t, a)da$$U_{j}(0, a)=U_{0j}(a)$

,

(1.1)

,

,

where denotes the upper bound of age,is the initial data. The function
denotes the natural death rate of the j-th state population and $m_{jk}(a)$ is thebirth rate of j-th newborns by k-th individuals. The function $g_{ij}(a, U),$ $i\neq j$denotes the density-dependent force of transition from j-th state to i-th stateandis defined as$\omega$$U_{0j}$$\mu_{j}$$g_{jj}$$g_{jj}(a, U):=- \sum_{i\neq j}g_{ij}(a, U)$.

We as sume that
represents a density. Moreoversinceeach age-specific fertility function
is assumed to be a bounded nonnegative
function with a compact support, where $0<a_{1}<a_{2}<\omega$ andiscalled the reproductive age period.
Let us define the survival function
at j-th state by$U_{j}(t, \cdot)\in L_{+}^{1}(0,\omega)$$U_{j}$$m_{ij}$$[a_{1}, a_{2}]$$[a_{1}, a_{2}]$$\ell_{j}(a)$$\ell_{j}(a)$$:= \exp(-\int_{0}^{a}\mu_{j}(\sigma)d\sigma)$

In order to guarantee $l_{j}(\omega)=0$ , we assume that
nonnegative function satisfying$\int_{0}^{w}\mu_{j}(\sigma)d\sigma=+\infty$.

$\mu_{j}$(1.2)

is a locally integrable(1. 3).

$\cdot$Moreover we assume that the incident rate of death is essentially bounded:$\sup\mu_{j}(a)\ell_{j}(a)<+\infty$

.

(1.4)

$a\in[0,\omega]$

Then it is reasonable to restrict the state space of age density functions suchthat $\mu_{j}(a)U_{j}(t, a)$ becomes integrable with respect to age . For this purpose,
we as sume that the state space of age-density function
is given by$a$$U_{j}$$X_{j}$$:= \{\phi\in L_{+}^{1}(0,\omega):\frac{\phi(\cdot)}{\ell_{j}(\cdot)}\in L_{+}^{1}(0,\omega)\}$

,

(1.5)

where its norm is given by. If there is no differential(natural) mortality among epidemic states, then each state space of age-density
functions is given by the same set as $\{U_{j}\in L_{+}^{1}(0,w):U_{j}(\cdot)/\ell(\cdot)\in L_{+}^{1}(0,\omega)\}$ .
$\Vert\phi\Vert_{X_{j}}=\Vert\phi(\cdot)/p_{j(\cdot)\Vert_{L^{1}}}$

30
Then the age-structured epidemic system (1.1) can be formulated as a semilinear dynamical system as follows:$\frac{dU(t)}{dt}=AU(t)+F(U)$ ,$U(0)=U_{0}\in E_{+}$

(1.6)

,

where $E=\Pi_{j=1}^{n}X_{j}$ is a Banach space of state vectors and $E+is$ its positive
cone, $U(t)=(U_{1}(t, \cdot),$is the state vector at time , andthe operator $A:D(A)\subset Earrow E$ is the population operator, given by$U_{2}(t, \cdot),$$\cdots,$$U_{n}(t, \cdot))^{T1}$$t$$(AU)(a)=(-dU_{1}/da-\mu_{1}(a)U_{1}, \cdots, -dU_{n}/da-\mu_{n}(a)U_{n})^{T}$.

The domain $D(A)$ is given as follows:$D(A)= \{U\in E:\frac{U_{j}}{p_{j}}\in AC[0,\omega],$$U( O)=\int_{0}^{w}M(a)U(a)da\}$ ,where $AC[0,w]$ denotes the set of absolutely continuous functions, $M(a)$ $:=$
fertility matrix, and the nonlinear term $F(U)$ is given
is a$n\cross n$$as(m_{ij}(a))_{1\leq i,j\leq n}$$F(U)(a)=G(a, U)U$,(1.7)

where $G(a, U)=(g_{ij}(a, U))_{1\leq i,j\leq n}$ is a $n\cross n$ transition matrix.
The transition rate from the susceptible class to the exposed or infected classis called the force of infection. For many classical epidemic models, the foroe of
infection is the only density-dependent transition rate (which depends on the
to bedensity of infectious population) and other transition rates are
given age-dependent functions or constants.
To ensure the existence and uniqueness of asolution, we as sume local Lipschitz continuity, that is, there exists an increasing function $k(r)$ such that
$V\in\{U\in E+:\Vert U||\leq r\}$ . Under the$\Vert F(U)-F(V)\Vert\leq k(r)\Vert U-V||$ for all
above as sumption, it can be shown that the epidemic system (1.1) has auniqueglobal mild solution $S(t)u_{0}$ such that $S(t)(E_{+})\subset E+and$ if $u_{0}\in E+\cap D(A)$ ,$S(t)u_{0}$ becomes aclassical solution. In general the evolution system (1.1) canbe seen as anonlinear extension of ademographic multistate stable population
model [11], which has been extensively studied by many authors ([17], [19], [9],[4]).
In this paper we focus on the special case in which the basic system (1.1)
defines ahomogeneous dynamical system. Adynamical system can be called homogeneous if its solution operator is homogeneous, that is, $S(t)(\alpha U)=\alpha S(t)(U)$for all $U\in E+and$ for all $\alpha>0$ . For the semilinear Cauchy problem (1.6), if $F$
is ahomogeneous of degree one nonlinear continuous operator:$a\epsilon sumed$$U,$$F(\alpha U)=\alpha F(U)$$1T$

denotes the transpose of the vector.

,$\forall U\in E+,$$\forall\alpha>0$

,

(1.8)

31
then (1.6) becomes a homogeneous dynamical system. In the following we callthe age-structured multistate model (1.1) the homogeneous epidemic system if
it describes an epidemic compartment model and $F$ satisfies the homogeneity
condition $(1.7)-(1.8)$ with $G(a, \alpha U)=G(a, U)$ .
The assumption of homogeneity has been widely used in the epidemic modeling to reflect the saturation of contact number in large scale populations ([3],[6], [18]). In general, the force of infection
can be seen as a product of threefactors, the transmission probability of infectious agents per one contact,
the number of contacts per capita and per unit time $(C(N))$ and the fractionof the population that is infectious $(I/N)$ . That is,$(\lambda)$$(\beta)$(1.9)$\lambda\approx\beta\cross C(N)\cross\frac{I}{N}$

where denotes the size of the infectious population and $N$ is the size of thetotal population. Though it is natural to suppose that the number of contacts$C(N)$ is an increasing function of the population size $N$ , for many cases it is also
believed that $C(N)$ levels off at high population densities ([1], [18], [14]). Thenit would be reasonable to adopt the simplified assumption that $C$ is constant if
the host population size is so large that $C(N)$ is essentially equal to its limiting
value ( $C(N)$ is assumed to be an increasing function bounded above). In sucha case, becomes homogeneous of degree zero with respect to the scale of
population densities, which leads to the homogeneous epidemic system.
Ifis an equilibrium solution of the system (1.6) with homogeneity condition (1.8), then so isfor any $c>0$ . Then there is no possibility ofan attracting nontrivial equilibrium, which is very different from the classicalepidemic models whose force of infection is a linear functional of the infected
population density. So our interest focuses on the existence and stability ofpersistent solutions defined as follows: The solution of (1.6) with (1.8) is calleda persistent solution if it has the form as
is a constant andwhere$U^{*}\in E$ . Then it follows from the homogeneous nonlinearity that a (biologicallymeaningful) persistent solution exists if and only if the nonlinear eigenvalue
problem$U^{*}\in E+$$AU^{*}+F(U^{*})=r^{*}U^{*}$ ,(1.10)’$I$$\lambda$$U^{*}$$cU^{*}$$e^{r^{*}}{}^{t}U^{*}$$r^{*}$

has a solution.
The homogeneous dynamical system has been studied by several authors who
have already developed some approaches to show stability results of persistentsolutions in the homogeneous (epidemic) dynamical systems ([5], [3], [6], [7], [8],[20], [21], [10]). However, in fact the above nonlinear eigenvalue problem is very
difficult to solve except for some special cases. For the homogeneous epidemic
system, the most important special case such that the eigenvalue problem (1.10)
could be solved is the case that the host population dynamics is independent
ffom the epidemic and described by the stable population model.
In order to consider the solvable case, we adopt the following additional
as sumption:

32
Assumption 1.1 There is no differential mortality, $\mu(a)=\mu_{j}(a)(1\leq j\leq n)$
is the common age-specific natural death rate. Moreover there is no differential$fertili\sim ty$ , so there exists a state-independent age-specific fertility
function $f(a)$such that $m_{ij}(a)=k_{i}f(a)$ , where
denotes the proportion of newbo $ms$ whose.
epidemic state is and $\sum_{i=1}^{n}k_{i}=1,$$k_{i}$$k_{i}\geq 0$$i$Note that under the Assumption 1.1 we can escape from the regulation
problem, that is, the effects of the epidemic on the demographic vital rates. Yet
demography influences the disease dynamics. For example, the host population
structure and the Malthusian parameter could affect the basic reproduction
ratio ([2], [12]).
If the Assumption 1.1 is satisfied, it is easy to see that the age density
function of the host population, denoted by $P(t, a)$ $:= \sum_{i=1}^{n}U_{i}(t, a)$ satisfies
the stable population model:$\frac{\partial P(t,a)}{\partial t}+\frac{\partial P(t,a)}{\partial a}=-\mu(a)P(t, a)$,$P(t, 0)= \int_{0}^{w}f(a)P(t, a)da$(1.11),$P(0, a)=P_{0}(a)$ ,where $P_{0}(a)= \sum_{j=1}^{n}U_{j}(0,a)$ is a given initial data. Moreover, by adding term
to term in the eigenvalue problem (1.10), we can obtain the following linear
eigenvalue problem:$(- \frac{\partial}{\partial a}-\mu(a))P^{*}(a)=r^{*}P^{*}(a)$,(1.12)$P^{*}( O)=\int_{0}^{w}f(a)P^{*}(a)da$,. From (1.12), we know that $P^{*}(a)=P^{*}(0)e^{-ra}\ell(a)$ ,where $P^{*}(a)$is the survival function. Inserting the expreswheremustsion ofinto the boundary condition, we know that the eigenvalue
be a root of the Euler-Lotka characteristic equation:$:= \sum_{i=1}^{n}U_{i}^{*}(a)$$\ell(a)$$:= \exp(-\int_{0}^{a}\mu(\sigma)d\sigma)$$r^{*}$$P^{*}$$\int_{0}^{w}e^{-za}f(a)\ell(a)da=1$,$z\in C$.

(1.13)

Then it is well known that (1.13) has a unique dominant real root , which iscalled the intrinsic rate of natural increase.
Therefore the host population system (1.11) has a unique persistent age
profile (normalized age distribution) given by$r_{0}$$\psi(a):=\frac{e^{-r_{0}a}\ell(a)}{\int_{0}^{\omega}e^{-r_{0}\sigma}l(\sigma)d\sigma}=\psi(0)e^{-r_{0}a}p(a)$,(1.14)and is called the stable age profile. Therefore under the Assumption 1.1, persistent solutions of the homogeneous epidemic system (1.1) have a unique growth$\psi$

33
rate . Then the remaining problem is to look for the persistent distributions
and to examine their stability.
Intuitively speaking, if the host population age distribution approaches astable age distribution independently from the epidemic, we could expect thatthe long-run behavior (stability and instability of the persistent solutions) of thebasic system would be determined by the limiting system whose host population
has already attained the stable age distribution. In order to show this intuition
to hold, first we have to consider the dynamics of age profile more precisely. In
the following we call the homogeneous epidemic system under the Assumption
1.1 the homogeneous epidemic system in the stable population.
$r_{0}$$U^{*}$

2

Age profile dynamics of the stable population

For the stable population model, its initial data are called trinial if $f(a+$$t)P_{0}(a)=0$ for almost all $a\in[0,\omega]$ and all $t\geq 0$ , which means that the
support ofstarts beyond the maximum reproductive age ([9], p.22) (that is,
the initial individuals are too old to be fertile), otherwise they are called nontrivial. It follows from the stable population theory ([11], [9]) that if a given
initial data
is non-trivial, the age distribution converges to the stable age$P_{0}$$P_{0}(a)$profile:$\lim_{tarrow\infty}\frac{P(t,a)}{\int_{0}^{\omega}P(t,a)da}=\psi(a)$

.

(2.1)

That is,is a relatively stable age distribution. Once the stable age profile
is attained, it is persistent and the population grows exponentially with theMalthusian parameter .
In order to show the stability principle for the epidemic system in the next
section, here we introduce some properties of the ratio age distnbution definedby:$\psi$$\psi$$r_{0}$$w$$w(t, a)= \frac{P(t,a)}{N(t)\psi(a)}$

(2.2)

,

where $N(t)= \int_{0}^{\omega}P(t, a)da$ is the total size of the population at time and$P(t, a)/N(t)$ is the age profile (normalized age distribution) of the host population at time , so is the ratio of the age profile at time to the stable age
profile. The ratio distribution is well defined as long as the initial dataisnontrivial.
Inserting $P=N\psi w$ into (1.11), we can derive that satisfies the following
system:
$t$

$t$

$w$

$t$

$P_{0}$$w$$\frac{\partial w(t,a)}{\partial t}+\frac{\partial w(t,a)}{\partial a}=(r_{0}-h(w(t, \cdot))w(t, a)$$w(t,0)= \int_{0}^{\omega}\pi(\sigma)w(t, \sigma)d\sigma$$w(0, a)=w_{0}(a)\in\Sigma$

,

,

,
(2.3)

34
where $\pi(a)$ $:=e^{-r_{O^{Q}}}f(a)\ell(a),$$\Sigma$$\Sigma$is the state space of$;=\{\phi\in L_{+}^{1}(0,\omega)$$w$

given by

: $\int_{0}^{w}\psi(a)\phi(a)da=1\}$ ,and $h:\Sigmaarrow R$ is a functional defined by$h(\phi)$$:= \int_{0}^{w}(f(a)-\mu(a))\psi(a)\phi(a)da$.

It is easy to see that $h(1)=r_{0}$ and $h(w(t, \cdot))$ gives the growth rate of the hostpopulation at time , that is, $N’(t)/N(t)=h(w(t, \cdot))$ .
Once the solution of (2.3) is given, the host population $P$ is recovered as
$t$

$w$

$P(t, a)=N(0)e^{\int_{0}^{t}h(w(\sigma,\cdot))d\sigma}\psi(a)w(t, a)$

.

(24)

It follows that $P(t, a)$ becomes the persistent solution $N(0)e^{r_{0}t}\psi(a)$ if and only
if $w(t, a)=w^{*}\equiv 1$ , which is the unique nontrivial equilibrium solution of (2.3).
From the well-known strong ergodicity theorem for the stable population, it
exponentiallyis easy to see that the ratio distribution converges to(see [9] Chapter II). However, more precisely, let us show
norm asinis decreasing exponentially, which result is needed inthat
the next section.
:
Let us consider the system (2.3) as a Cauchy problem in$w^{*}\equiv 1$$w$$tarrow\infty$$L^{1}$$\Vert w-w^{*}\Vert/\Vert w_{0}-w^{*}\Vert$$L^{1}(0, \omega)$$\frac{dw(t)}{dt}=A_{0}w(t)+G(w(t)),$$w(0)=w_{0}\in\Sigma$

,

(2.5)

is a population operator defined by $(A_{0}\phi)(a)=-d\phi(a)/da$ with thewheredomain given by$A_{0}$$D(A_{0})=\{\phi\in AC[0,\omega]$

and the nonlinear term$G$

: $\phi(0)=\int_{0}^{\omega}\pi(a)\phi(a)da\}$ .

is given by$G(\phi)$$:=(r_{0}-h(\phi))\phi$.

Since is a continuously Fr\’echet differentiable nonlinear perturbation, for) and a uniquethere exists a maximal interval of existenceeach$tarroww(t;w_{0})$
that
)
to
such
from
continuous function$G$$[0, T_{w_{0}}$$w_{0}\in L^{1}$$[0, T_{w_{0}}$$L^{1}$$w(t;w_{0})=e^{tA_{0}}w_{0}+ \int_{0}^{t}e^{A_{0}(t-\sigma)}G(w(\sigma;w_{0}))d\sigma,$$t\in[0,T_{wo}$

),

(2.6)

and $T_{w_{0}}=\infty$ or $\lim suptarrow T_{w_{0}}||w(t;w_{0})||=\infty$ and if $w_{0}\in D(A_{0})$ , then$w(t;w_{0})\in D(A_{0})$ and $w(t;w_{0})$ is continuously differentiable and satisfies (2.6)) ([16]; [19], Prop. 4.16). Since (2.3) is induced from the stable popon
ifulation model (1.11) by the transformation (2.4), we know thatis a nontrivial data, and the following holds:$[0, T_{w_{0}}$$T_{w_{0}}=\infty$$w_{0}$

35
Lemma 2.1 Let
be the set of nontrivial initial data. Thenis positively invariant with respect to the solution semiflow of the system (2.3).
$\Sigma\cap\Sigma_{0}$$\Sigma_{0}$$1$:

Next consider the linearized equation of (2.5) at the equilibrium point
,

(2.7)

$L^{1}arrow L^{1}$

is a linear operator with$\frac{d\zeta(t)}{dt}=(A_{0}+G’[w^{*}])((t)$$:=w(t, a)-w^{*}$ and $G’[w^{*}]$ :where
one-dimensional range defined as$\zeta(t, a)$$w^{*}\equiv$,$G’[w^{*}]\zeta:=-h(\zeta)w^{*}$

Then $A_{0}+G’[w^{*}]$ generates a strongly continuous semigroup $T(t)=\exp((A_{0}+$$G’[w^{*}])t)$ , because $A_{0}+G’[w^{*}]$ is a bounded linear perturbation of the population semigroup generator.
From the biological meaning, the state space of the perturbation is$A_{0}$$\zeta$$\Sigma^{*}$$:=\{\phi\in L_{+}^{1}(0,\omega)$: $\int_{0}^{\omega}\psi(a)\phi(a)da=0\}$ .

Then it is easy to see thatis positively invariant with respect to the semiflow$T(t)$ . So we write the restriction of $T(t)$ to the subspace.
as$\Sigma^{*}$$\Sigma^{*}$

Lemma 2.2 For the linearized semigroup
andsuch that$T_{\Sigma}\cdot(t)$$T_{\Sigma}\cdot(t)$, there exist numbers$\epsilon>\dot{0}$$M(\epsilon)\geq 1$$\Vert T_{\Sigma}\cdot(t)\Vert\leq M(\epsilon)e^{-\epsilon t}$

.

(2.8)

Proof: Since the semigroup generated by the population operator with a compact perturbation is eventually compact ([11], [19] Prop. 4.14), it follows that$\omega_{0}(A_{0}+G’[w^{*}])=\sup\{\Re z :z\in P_{\sigma}(A_{0}+G’[w^{*}])\}^{2}$. Hence the restriction$T_{\Sigma}\cdot(t)$is also an eventually compact semigroup. Then if there exists a negative domi, then there exist
nant eigenvalue for $A_{0}+G’[w^{*}]$ on the subspaceandsuch that. So let us consider the eigenvalue$A_{0}+G’[w^{*}]$
equation foron$M(\epsilon)\geq 1$$\Sigma^{*}$$\epsilon>0$$\Vert T_{\Sigma^{*}(t)}\Vert\leq M(\epsilon)e^{-\epsilon t}$$\Sigma^{*}:$$\phi\in D(A_{0})\cap\Sigma^{*}$.

$\phi(0)=\int_{0}^{w}\pi(a)\phi(a)da$,$(\lambda-(A_{0}+G’[w^{*}]))\phi=0$,

Then we have$\lambda\phi+\phi’+h(\phi)=0$,

which is solved formally as follows:$\phi(a)=\phi(0)e^{-\lambda a}-h(\phi)\int_{0}^{a}e^{-\lambda(a-s)}ds$

denotes the point spectrum of the operator(see [15])bound of the semigroup$2P_{\sigma}(A)$$e^{tA}$$A$.

and $w_{0}(A)$ denotes the spectral

36
Inserting the above formal solution into the boundary condition, we have$\phi(0)=\phi(0)\int_{0}^{\omega}\pi(a)e^{-\lambda a}da-h(\phi)\int_{0}^{\omega}\pi(a)\int_{0}^{a}e^{-\lambda(a-s)}dsda$

On the other hand, since$\phi\in\Sigma^{*}$.

, we obtain a condition$\phi(0)\int_{0}^{w}\psi(a)e^{-\lambda a}da-h(\phi)\int_{0}^{\omega}\psi(a)\int_{0}^{a}e^{-\lambda(a-s)}dsda=0$.

Then we have a simultaneous equation system for the unknown numbers
andas$\phi(0)$$h(\phi)$$(\begin{array}{ll}\Psi_{1l}(\lambda) \Psi_{12}(\lambda)\Psi_{2l}(\lambda) \Psi_{22}(\lambda)\end{array})(_{h(\phi)}^{\phi(0)})=0$

,

where
,
$\Psi_{12}(\lambda)\Psi_{22}(\lambda))$$\Psi_{21}(\lambda)\Psi_{11}(\lambda)$$:=(^{\int_{0}^{w}\pi(a)e^{-\lambda a}da-1} \int_{0}^{w}\psi(a)e^{-\lambda a}da$. Then$- \int_{0}\psi(a)\int_{0^{a}}e^{-\lambda(a-\epsilon)}dsda-\int^{w}\pi(a)\int_{0}^{a}e^{-\lambda(a-\epsilon)}dsda$

is an eigenvalue of our interest if and only$:=\{\lambda\in C : \det(I-\Psi(\lambda))=0\}$ . It is easy to see that:if
. By using the standard argument for the Euler-Lotka characteristic
haveequation, we know that all eigenvalues associated with eigenvectors innegative real part.

Let$\Psi(\lambda)=(\Psi_{ij}(\lambda))_{1\leq i,j\leq 2}$$\lambda$$\Lambda=\{\lambda\in C\backslash \{0\}$$\lambda\in\Lambda$$\Psi_{11}(\lambda)=0\}$$\Sigma^{*}$$\square$Proposition 2.3 Let $w(t)$ be the solution of (2.5). Then there emsts a nurnber$\eta>0$ such that if $w_{0}-w^{*}\Vert<\eta$ , it follows that
Il$\Vert w(t, \cdot)-w^{*}||\leq M(\epsilon)e^{-\epsilon t}\Vert w_{0}-w^{*}\Vert$for sonumbers
initial data .
$rne$$M(\epsilon)\geq 1$

and$\epsilon>0$, where$M(\epsilon)$

,

(2.9)

is independent from the$w_{0}$

Proof: Inserting$w^{*}+\zeta(t),$$\zeta\in\Sigma^{*}$

into (2.5), we have$\frac{d\zeta(t)}{dt}=(A_{0}+G’[w^{*}])\zeta(t)+R(\zeta(t))$(2.10),

where$R(\zeta(t))$ $:=G(w^{*}+\zeta(t))-G’[w^{*}]\zeta(t)\in\Sigma^{*}$.

IFlrom the IFMr\’echet differentiability, there exists a continuous increasing function$k(r),$ $r\geq 0$ with $k(O)=0$ such that$\Vert G(w^{*}+\zeta)-G(w^{*})-G’[w^{*}]\zeta\Vert=||R(\zeta)\Vert\leq k(r)\Vert\zeta\Vert$

for allsuch thatto (2.10), we have$\zeta\in L^{1}$$\Vert\zeta\Vert\leq r$$\zeta(t)=\tau_{\Sigma}$.

,. Applying the variation of constants formula$(t) \zeta(0)+\int_{0}^{t}\tau_{\Sigma}.(t-s)R(\zeta(s))ds$,(2.11)

37
withFrom Lemma 2.2, if we can choose $\gamma<0$ such thatsome $M(\gamma)\geq 1$ . Choose $r>0$ such that $k(r)\leq-\gamma/2M(\gamma)$ and let $\eta\leq r/M(\gamma)$ .
Suppose thatand let $0<t_{1}\leq\infty$ be the largest extended real number
such thatfor $0\leq t<t_{1}$ . Then for $0\leq t<t_{1}$ , we have$\Vert T_{\Sigma^{*}}(t)\Vert\leq M(\gamma)e^{\gamma t}$$\Vert\zeta(0)\Vert<\eta$$\Vert((t)\Vert\leq r$$\Vert\zeta(t)\Vert\leq M(\gamma)e^{\gamma t}\Vert\zeta(0)\Vert+M(\gamma)\int_{0}^{t}e^{\gamma(t-s)}\Vert R(\zeta(s))\Vert ds$,$\leq M(\gamma)e^{\gamma t}\Vert\zeta(0)\Vert+M(\gamma)e^{\gamma t}\int_{0}^{t}e^{-\gamma s}k(r)\Vert((s)\Vert ds$.

Therefore we obtain.

$e^{-\gamma t} \Vert\zeta(t)\Vert\leq M(\gamma)\Vert\zeta(0)\Vert+\frac{-\gamma}{2}\int_{0}^{t}e^{-\gamma s}\Vert\zeta(s)\Vert d_{8}$From the Gronwall inequality, we have$e^{-\gamma t}\Vert\zeta(t)\Vert\leq M(\gamma)\Vert\zeta(0)\Vert e^{-\iota_{t}}2$

Thus we arrive at the estimate asthat
and (2.9) holds $for-\epsilon=\gamma/2$ .

.

$\Vert((t)\Vert\leq M(\gamma)e^{2}2t\Vert\zeta(0)\Vert\leq r$$t_{1}=\infty$3, so we know$\square$Linearized stability for the homogeneous epidemic system

For the homogeneous epidemic system, once the stable age distribution is attained, we can rewrite the basic system into the normalized system in which
the nonlinear term becomes bilinear (the mass action type). In the following,
we prove that the local stability [instability] of steady states of the normalized
system implies the stability [instability] of persistent solutions of the original
homogeneous system.
Let us introduce the ratio $dist7\dot{\tau}butions$ for each of the epidemiological classes
as follows:$u_{j}(t, a)$(3.1)$:= \frac{U_{j}(t,a)}{N(t)\psi(a)}$In the following we assume that $N(t)>0$ for all $t\geq 0$ , which holds if the initial
age profile is non-trivial.
Note that under the Assumption 1.1, we have$w(t, a)= \frac{P(t,a)}{N(t)\psi(a)}=\sum_{j=1}^{n}u_{j}(t, a)=<e,u(t, a)>$,where we define $e:=(1,1, \cdot\cdot, 1)^{T}$ so that for $u\in Y:=(L^{1}(0,w))^{n}$ , we have$<e,$ $u(a)>:= \sum_{i=1}^{n}u_{i}(a)$

.

(3.2)

38
Now in order to make a concrete normalization argument possible, we introduce a special homogeneity condition:
Assumption 3.1 The force of infection is given by an expression of the form$\int_{0}^{\omega}\beta(a, \sigma)\frac{U_{j}(t,\sigma)}{N(t)}d\sigma$

,

(3.3)

where is the transmission probability function (which is assumed to be essentially bounded and nonnegative) andis an infectious class. Other transitionin the matrix $G(a;U)$ are given bounded nonnegative functions independent of the population density .
$\beta$$U_{j}$$entr\dot{v}es$$U$

Inserting $U_{j}(t, a)=N(t)\psi(a)u_{j}(t, a)$ into (1.1), the homogeneous epidemic
system in the stable population is reduced to a Cauchy problem on the Banach
space $Y+=(L_{+}^{1}(0,w)^{n}$ :$\frac{du(t)}{dt}=\tilde{A}u(t)+\tilde{F}(u(t))+B(u(t))$

where the operators$\tilde{A},\tilde{F}$$:=\{\phi\in Y$$u(O)=u_{0}$

.

(3.4)

and $B$ acting on $Y$ are defined as follows:$\tilde{A}\phi$$D(\tilde{A})$,:$:=-d\phi/da$ ,$\phi_{i}\in AC[0,\omega],$ $\phi(0)=\int_{0}^{w}\tilde{M}(a)\phi(a)da,$ $\}$$:=(k_{ij}\pi(a))_{1\leq i,j\leq n}$ andwhere
infection term in is given by the formula as$\tilde{F}(\phi)(a)=\tilde{G}(a, \phi)\phi$$\tilde{M}(a)$,where the force of$\tilde{G}$$\lambda[a|\phi_{j}]:=\int_{0}^{w}\beta(a, \sigma)\psi(\sigma)\phi_{j}(\sigma)d\sigma\phi_{k}$

.

(3.5)

are the same as the corresponding
Other elements of the matrix
is a generator of the multistate population
elements of . The operator$\tilde{G}(a;U)$$\tilde{A}$$G$

semigroup$e^{t\overline{A}}$([11]). The nonlinear term $B$ is given by$B(\phi)=z(\phi)\phi$

(36)

,

where $z:Y+arrow R+is$ a functional given by$z(\phi)$$:=r_{0}-h(<e, \phi>)$.

Conversely if the system (3.4) is given, by using its solution
the total size $N(t)$ from the equation:$\frac{1}{N(t)}\frac{dN(t)}{dt}=r_{0}-z(u(t))$

,

(3.7)
$u$

$N( O)=\int_{0}^{w}P(O, a)da$

we can recover,39
Once we can determine $N(t)$ from the above equation,by (3.1) and (3.2).
Let
be a closed convex set given by

and $P$ can be recovered$U_{j}$$K_{1}$$K_{1}$$:=\{\phi\in Y+:<e, \phi(a)>=1\}$.

Moreover, from the definition (3.2), biologically meaningful solutions $u(t)$ of(3.4) should take a value in the closed convex set:
$K_{2}$$:= \{\emptyset\in Y+:\int_{0}^{w}\psi(a)<e,$

Then it follows from$\int_{0}^{\omega}\psi(a)da=1$

Lemma 3.2 The sets
andthe solution $se7niflow$ of (S.4).
$K_{1}$

that$K_{2}\cap\Sigma_{0}$$\phi(a)>da=1\}$ .

$K_{1}\subset K_{2}$and $z(\phi)=0$ if

are positively$inva’\dot{\tau}ant$$\phi\in K_{1}$.

with respect to

Proof: In the system (2.5), if $w(O, a)=w^{*}$ , we have $w(t, a)=w^{*}$ for all $t>0$ ,which means that the positive orbit is included in
if $u(O)\in K_{1}$ . Next
suppose that
. Then it follows that
$K_{1}$$u$$u(0)\in K_{2}\cap\Sigma_{0}$$\frac{d<e,u(t)>}{dt}=A_{0}<e,$$u(t)>+z(u)<e,$ $u(t)>$ .

(3.8)

The equation (3.8) is equivalent to (2.5) with $w=<e,$ $u>$ , hence it follows $hom$Lemma 2.2 that $<e,$ $u(t)>stays$ in the state space, which implies that$\Sigma_{0}\cap\Sigma$$u(t)\in K_{2}\cap\Sigma_{0}$.

$\square$Now we adopt a technical as sumption, which would be satisfied for manycases:

Assumption 3.3 The operator : $Yarrow Y$ is locally Lipschitz continuous andcontinuously Fr\’echet differentiable.
$\tilde{F}$Lemma 3.4 Let $u(t)$ be a solution of (3.4). Then there enists $B_{0}>0$ andsuch that$\epsilon_{1}>0$$\Vert B(u(t))\Vert\leq B_{0}e^{-\epsilon_{1}t}\Vert u(t)||\Vert w_{0}-w^{*}\Vert$

,

(3.9)

wheoe $w_{0}(a):=<e,$ $u(O, a)>$ .

Proof: It is sufficient to show that there exists $B_{0}>0$ and$|z(u(t))|\leq B_{0}e^{-\epsilon_{1}t}\Vert w_{0}-w^{*}\Vert$such that. It follows from $r_{0}= \int_{0}^{w}(f(a)-\mu(a))\psi(a)da$ that$\epsilon_{1}>0$$|z(u(t))| \leq\int_{0}^{w}|f(a)-\mu(a)|\psi(a)|w^{*}-w(t, a)|da$.

IFhrom the estimate (2.9), there exist numbers $M>0$ and $\epsilon>0$ such that
. Hence if we choose as $B_{0}=M \sup_{a\in[0,w]}|f(a)-$$\mu(a)|\psi(a)$ , then (3.9) follows immediately.
$\Vert w(t, \cdot)-w^{*}\Vert\leq Me^{-\epsilon t}\Vert w_{0}-w^{*}\Vert$$B_{0}$$\square$40
From Lemma 3.4 we can expect that the asymptotic behavior of (3.4) is:determined by the system restricted to$K_{1}$$\frac{dv(t)}{dt}=\tilde{A}v(t)+\tilde{F}(v(t))$,$v\in K_{1}$.

(3.10)In the following, we call (3.10) the normalized system. From Lemma 3.4, thenormalized system can be seen as a limiting equation with respect to (3.4). The
normalized system is no other than the ratio age distribution system in case thatthe host population attains the stable age distribution, that is, we can prove
the following:
Lemma 3.5 Let $u(t)$ be the solution of (3.4). Then $u(t)\in K_{1}$ if and only if$P(t, a)$ is the persistent solution of (1.11).

Proof: If $u(t)\in K_{1}$ , we have $<e,$ $u(t, a)>=1$ which implies that$\sum_{j=1}^{n}u_{j}(t, a)=\frac{1}{N(t)\psi(a)}\sum_{j=1}^{n}U_{j}(t, a)=\frac{P(t,a)}{N(t)\psi(a)}$.

Therefore $P(t, a)=N(t)\psi(a)$ . Then $P(t, a)$ is a separation of variable typesolution of (1.11), we know that $N(t)=\Vert P_{0}||e^{r_{0}t}$ and becomes the persistentsolution. Conversely if we assume that $P(t, a)=Ce^{r_{0}t}\psi(a)(C$ is a positive
constant), then $N(t)=Ce^{r_{0}t}$ , hence $\sum_{j=1}^{n}u_{j}(t, a)=1$ .
$P$$\square$is positively invariant with respect to the solution semiflow of (3.4), it follows thatand (3.4) is reduced to (3.10) on

Since$K_{1}$$K_{1}$

Lemma 3.6 Let $V(t)$ be a solution semiflow such that $V(t)v_{0}=v(t;v_{0})$ where$v(t;v_{0})$ denotes the solutionof the norrnalized system (3.10) with the initial data$v(O)=v_{0}\in Y$ . Then $V(t)(K_{1})\subset K_{1}$ .

is an equilibnum point of (S.4) if and only if
Lemma 3.7
point of the norm alized system (S.10), that is,equilibrium
is
and
an

$u^{*}\in K_{1}$$u^{*}\in K_{2}$$u^{*}$$\tilde{A}u^{*}+\tilde{F}(u^{*})=0$(3.11).

be an equilibrium point of (3.4). Define a perturbation
by $\zeta=u-u^{*},$ $u\in K_{2}$ . Inserting $u=u^{*}+(into(3.4)$ , we have an equation forthe perturbation:Let$\zeta(t, a)$$u^{*}\in K_{1}$$\frac{d\zeta(t)}{dt}=\tilde{A}\zeta(t)+\tilde{F}’[u^{*}]\zeta+R(u^{*}, \zeta(t))$

where
by

$\tilde{F}’[u^{*}]$

is a FXr\’echet derivative of$\tilde{F}$at$u^{*}$(3.12),and the residual term $R$ is given$R(u^{*}, \zeta(t)):=\tilde{F}(u^{*}+((t))-\tilde{F}(u^{*})-\tilde{F}’[u^{*}]((t)+B(u^{*}+\zeta(t))$.

(3.13)

41
From Lemma 3.4, we obtain an estimate:$\Vert B(u^{*}+\zeta(t))\Vert\leq B_{0}e^{-\epsilon_{1}}{}^{t}(\Vert u^{*}\Vert+\Vert\zeta(t)\Vert)\Vert w_{0}-w^{*}\Vert$(3.14).

From the definition of Fr\’echet differentiability, there exists a continuous
increasing function $k(r),$ $r\geq 0$ such that $k(O)=0$ and$\Vert\tilde{F}(v^{*}+\zeta)-\tilde{F}(u^{*})-\tilde{F}’[u^{*}]\zeta\Vert\leq k(r)\Vert\zeta\Vert$

for allLet$\zeta\in Y$

such that$\Vert\zeta\Vert\leq r$(3.15),.

linearbe a strongly continuous semigroup of$A+F’[u^{*}]$. Thenoperators in the Banach space with infinitesimal generator
such that, there exists a constantiffollowing:. Now we can prove the$T^{*}(t),$$t\geq 0$$bou_{\sim}nded$$Y$$\Vert T^{*}(t)||\leq$$M(\omega^{*})\geq 1$$w^{*}>\omega_{0}(\tilde{A}+\tilde{F}’[u^{*}])$$M(\omega^{*})e^{w^{*}t}$

Proposition 3.8 Let $u^{*}\in K_{1}$ be an equilibrium solution of (3.4). If$\tilde{F}’[u^{*}])<0$ , there exist $\eta>0,$ $M\geq 1$ and $\gamma<0$ such that
if $u_{0}\in K_{2}$ and
1 $u_{0}-u^{*}||<\eta$ , then for all $t>0$ , it follows that$\omega 0(\tilde{A}+$$\Vert u(t;u_{0})-u^{*}\Vert\leq Me^{\gamma t}\Vert u_{0}-u^{*}||$

Proof: From (3.12), the perturbation$\zeta(t)$.

(3.16)$:=u-u^{*}$ is given

as the continuous

solution of the variation of constants formula:$\zeta(t)=T^{*}(t)((0)+\int_{0}^{t}T^{*}(t-s)R(u^{*}, \zeta(s))ds$(3.17).

where isLetbe a number such thatchosen in Lemma 3.4. Choose $r>0$ such that $k(s)\leq-\omega^{*}/4M(w^{*})$ for $s\in[0,r]$ .
Let choose $\eta<r$ so small such that
$0> \omega^{*}>\max\{w_{0}(\tilde{A}+\tilde{F}‘[u^{*}]), -\epsilon_{1}\}$$w^{*}$,$0< \eta\leq\min\{\frac{r}{M(w^{*})(1+\frac{B_{0}||u^{*}||}{w^{*}+\epsilon_{1}})},$ $- \frac{w^{*}}{4M(w^{*})B_{0}}\}$

Suppose that
number such that$\Vert\zeta(0)\Vert<\eta$$\epsilon_{1}$and let $0<t_{1}\leq\infty$ be the largest extended real
for $0\leq t<t_{1}$ . Note that$\Vert\zeta(t)\Vert\leq r$$\Vert w_{0}-w^{*}\Vert=\int_{0}^{w}|\sum_{i=1}^{n}(u_{i}(0, a)-u_{i}^{*}(a))|da\leq\Vert u(0)-u^{*}\Vert=\Vert\zeta(0)||<\eta$.

Then for $0\leq t<t_{1}$ , we have,$\Vert\zeta(t)||\leq M(w^{*})e^{wt}||\zeta(0)\Vert+M(\omega^{*})\int_{0}^{t}e^{\omega^{*}(t-s)}\Vert R(u^{*}, \zeta(s))\Vert ds$$\leq M(w^{*})e^{\omega t}\Vert\zeta(0)\Vert+M(\omega^{*})e^{w^{*}t}\int_{0}^{t}e^{-\omega s}[k(s)+B_{0}e^{-\epsilon_{1^{S}}}||\zeta(0)\Vert]\Vert\zeta(s)||ds$$+M( \omega^{*})e^{wt}\int_{0}^{t}e^{-w\epsilon}B_{0}e^{-\epsilon_{18}}\Vert u^{*}\Vert||\zeta(0)||ds$.

42
Therefore we obtain.

$e^{-\omega^{*}t} \Vert\zeta(t)\Vert\leq M(\omega^{*})\Vert\zeta(0)\Vert(1+*\frac{B_{0}||u^{*}\Vert}{\omega+\epsilon_{1}})+\frac{-w^{*}}{2}\int_{0}^{t}e^{-\omega^{*}s}\Vert\zeta(s)\Vert ds$$Rom$ the Gronwall inequality, we have$e^{-wt} \Vert\zeta(t)||\leq M(\omega^{*})\Vert\zeta(0)\Vert(1+*\frac{B_{0}||u^{*}\Vert}{\omega+\epsilon_{1}})e^{-*t}$.

Then we arrive at the estimate as$\Vert\zeta(t)\Vert\leq M(\omega^{*})e^{\dot{\tau}^{t}}(1+*\frac{B_{0}||u^{*}\Vert}{w+\epsilon_{1}})\Vert\zeta(0)\Vert\leq r$

Then we know that $t_{1}=\infty$ and (3.16) holds for$\gamma=\omega^{*}/2$.

.

$\square$is the linearized generator of the norHere we should note thatmalized system (3.10) but it is not the linearized generator of the system (3.4),
since $B(u(t))$ includes a linear term. Nevertheless Proposition 3.8 tells us thatthe local stability of equilibrium point of the normalized system (3.10) implies
that of the ratio age distribution system (3.4).
is
In many cases, we can show that the linearized semigroupifalllocally
asymptoticallysteadystableisstateeventually compact, so thehave negative real part.
eigenvalues of the linearized generator
Here we clear the meaning of stability of the persistent solution implied in
Proposition 3.8. Observe that the initial data of the homogeneous system (1.1)
can be decomposed as follows:$\tilde{A}+\tilde{F}’[u^{*}]$$e^{(\tilde{A}+\tilde{F}’[u])t}$$\tilde{A}+\tilde{F}’[u^{*}]$$U_{j}(0,a)=N(0)\psi(a)(u_{j}^{*}(a)+r_{j}(a))$,$1\leq j\leq n$,

where the function $r(a)=(r_{1}(a), \cdots,r_{n}(a))$ satisfies the condition$\int_{0}^{w}\psi(a)<e,$$r(a)>da=0$ ,. Corresponding toand the condition $||u_{0}-u^{*}\Vert<\eta$ can be written as
persistentassolution
the equilibrium point , we can consider the$\Vert r||<\eta$$u^{*}$$U^{*}(t, a)=N_{\infty}e^{r_{0}t}\psi(a)u^{*}(a)$where$N_{\infty}$$:= \lim_{tarrow\infty}e^{-r_{0}t}N(t)$,. Then we can show that following:

Proposition 3.9 There exists a number $\eta>0$ such that if$\lim_{tarrow\ovalbox{\tt\small REJECT}}e^{-r_{0}t}\Vert U(t, \cdot)-U^{*}(t, \cdot)\Vert=0$.

$\Vert r||<\eta$, then(3.18)

43
Proof: Observe that$\int_{0}^{w}|U_{j}(t, a)-U_{j}^{*}(t, a)|da\leq\int_{0}^{w}|\frac{U_{j}(t,a)}{N(t)\psi(a)}-\frac{N_{\infty}e^{r_{0}t}}{N(t)}u_{j}^{*}(a)|N(t)\psi(a)da$$\leq N(t)\sup_{a\in[0,w]}|\psi(a)|[\int_{0}^{w}|\frac{U_{j}(t,a)}{N(t)\psi(a)}-u_{j}^{*}(a)|da+\int_{0}^{\omega}|u_{j}^{*}(a)|da|1-\frac{N_{\infty}e^{r_{0}t}}{N(t)}|]$

From (3.16) and $N(t)e^{-r_{0}t}arrow N_{\infty}>0$ , we obtain (3.18).

.

$\square$

For the system (3.4), if we choose the initial data such that $u(O)\in K_{1}$ ,then $u(t)\in K_{1}$ for all $t>0$ since
is positively invariant. In such case, the$u(t)\inK_{1}$
orbit
is described by the normalized system (3.10) since $B(u(t))=0$ .
Therefore the instability ofas the equilibrium point of (3.10) implies theinstability ofas the equilibrium point of (3.4):$K_{1}$$u^{*}$$u^{*}$

Proposition 3.10 Let
be an equilibrium solution of (S.4) and (S.10). If
is unstable with respect to the normalized system (3.10), it is also unstable forthe system (3.4).
$u^{*}$$u^{*}$

Rom the above results, we know that to examine asymptotic behavior of
the persistent solutions of the homogeneous epidemic system in the stable population, it is sufficient to study equilibrium points of the normalized system byusing the principle of linearized stability in the classical sense.
Acknowledgements: I would like to thank Dr. Katumi Kamioka for his
valuable comments to my paper [13].

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