数理解析研究所講究録 1432 巻 2005 年 148-162

148

Hiromi SENO Naoki SATO
・

Mathematical Models for Epidemic Dynamics with Adult
Vaccination against Waning Immunity
免疫失活に対する成人ワクチン接種を伴う伝染病感染動態に関する数理モデル
Version date: December 20, 2004
Abstract. We analyze two simple mathematical models taking account of the waning immunity, specificallyformulated with measles in mind, to give some insight about the effect of vaccination in adult age classaccording to the elimination of measles within the whole population. First model is non-age-structured one,and the second model is with three age classes: infant, young, and adult. With mathematical and numericalanalyses making use of parameter values which could correspond to the demographic situation of a Japanesecommunity, we show the result emphasizing that the vaccination program should be planned with taking
account of the size of population within which the disease transmission is taking place. Especially in case ofJapan, although our numerical calculations imply that it would be hard to increase the infant vaccination rateenough to make the population approach the disease-free equilibrium, the increase of infant vaccination rate
above the present level would be more effective than the promotion of secondary vaccination for an adult age
class in order to reduce the infective population.
免疫が一時的であり, 失活を伴う場合の単純な伝染病感染動態に関する数理モデル 2 つ, 年齢構造のないものと,
3 つの年齢グループ (幼年, 少年, 成年) からなるもの, を考察する。 数理モデルには, 成入に対するワクチン接種の効果を導入する。 特に, 具体例として, 麻疹感染を取り上げ, 麻疹撲滅に対するワクチン接種の効果に関する試論を展開する。 数学的解析, および, 高知県の人ロデータおよび麻疹感染データを参照したパラメータ値による数値計算の結果は, 幼年期におけるワクチン接種に加えて, 成年期でのワクチン接種 行っても, その効果は期待できないということ, ただし, 成年期でのワクチン接種が有効であるような人ロサイズもありえることを示唆するものである。 特に, 大きな人口集団における伝染病感染に対しては, ワクチンの 2 次接種よりも, 幼年期でのワク$\acute{\text{を}}$チン接種率の向上が感染者数減少により効果的であると示唆された。

1. Introduction
Epidemic dynamics has been attracting not a few theoretical and mathematical researchers (for instance, see [1-5] and their references). In this paper, we analyze two simple mathematical models
taking account of the waning immunity, specificaliy formulated with measles in mind, to give someinsight about the effect of vaccination in adult age class according to the elimination of measles within
the whole population. First model is non-age-structured one, with which we see the interaction be
tween the immunity waning and the secondary vaccination. Second model is with three age classes:
infant, young, and adult. With the model, we discuss the efflciency of adult vaccination for the elimination of measles, with numerical analysis making use of parameter values which could correspondto the demographic situation of a Japanese community, the Kochi Prefecture, which is one of locat
communities that have experienced some serious outbreaks of measles in several years (see Fig. 1) [6].
In recent Japan, measles cases counts 11,000-22,000 a year in a certain average, and estimated at
286,000 especially for 2001, according to the report of infectious disease surveillance by the Infectious
Disease Surveillance Center (IDSC) of the Japanese Ministry of Health, Labour and Welfare $[7,8]$ .
More than 60% of the patients in the data are infants below two years old, and more than 95% are$\sim$

Hiromi SENO [corresponding author]: Department of Mathematical and Life Sciences, Graduate School of
Science, Hiroshima University, Higashi-hiroshima 739-8526, JAPAN.sci hiroshima-u.
jp Phone & Fax: $+81-(0)82\sim 4247394$
瀬野裕美広島大学大学院理学研究科数理分子生命理学専攻
739-8526 広島県東広島市鏡山
Naoki SATO: Department of Mathematics, Faculty of Science, Hiroshima University, JAPAN佐藤直樹広島大学理学部数学科$\mathrm{e}rightarrow \mathrm{m}\mathrm{a}\mathrm{i}\mathrm{i}:\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{o}\Phi \mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}$$\mathrm{a}\mathrm{c}.$$\mathrm{i}-3-\mathrm{i}$$\overline{\mathrm{T}}$Key words: waning immunity-

boosting effect-two-dose vaccination-

mathematical model-

adult measles

148
Hiromi SENO, Naoki SATO

京都大学数理解析研究所講究録$\frac{y)}{\mathrm{c}}$$. \frac{.\underline{\Phi}}{\mathrm{a}\iota \mathrm{w}}$$.\underline{\tilde{arrow\varpi \mathrm{c}\mathrm{c}}}$$\mathrm{c}\circ$$.\overline{\in \mathrm{O}\subset\Phi\supset}$’$\mathrm{B}\not\subset\Phi \mathrm{q}\mathrm{a}$monthFig. 1. Number of infant measles cases reported by pediatric sentinel clinics in the Kochi Prefecture, JaPan,from July 1979 (the first month) to February 2004 (the 296th month), given by the Kochi Prefectural Infectious
Disease Surveillance Center [6].

$2\mathrm{c}\Phi$$\overline{\alpha 0}$.

$\mathrm{o}$$.\overline{3\in \mathrm{Q}\subset\Phi}$$\mathrm{B}\approx 0\Phi\Phi \mathrm{a}$Fig. 2. Age distribution of measles cases in Japan. Drawn by the cumulative data from theto the 29th in 2002 [9].

$14\mathrm{t}\mathrm{h}$week in 1999

non-vaccinated. As indicated by Fig. 2, most infected age class is from one to five years old. Every
year local outbreak of measles is repeated in Japan [7-9].
Adult measles is defined for patients over 18 years old. Since several years, it has been reported
that teenagers vaccinated in the infant period are infected by measles $[9, 10]$ . As for the adult measles
in Japan, it has no decreasing tendency with about 3-4% of all cases, and the reported number ofmeasles cases decreases in age over 24 years old (see Fig. 3) $[8,9]$ .
Today, as seen from Fig. 4, the measles vaccination rate for infants is more than 80% in averageover the whole Japan, although there are some local communities with the rate around 50-60% [7-9].
It is said that 3-5% of vaccinated individuals fail to get the immunity $[11, 12]$ . On the other hand, thereport for the Sapporo City in Japan indicates that around 40% of the measles patients were nonvaccinated [8]. So only such failure of immunization is insatisfactory to explain the measles infection tovaccinated individuals. It is likely that the effective period of immunity becomes shorter than before.
In fact, some sero epidemiological estimates of antibody decay suggest that 25 years after vaccination,
measles antibody levels have waned to below protection levels [13-17]. A hypothesis is that, in past,
the imrnunity was reinforced by -encountering the pathogen [18-22], what is sometimes called the$\mathrm{r}$150
Hiromi SENO Naoki SATO

京都大学数理解析研究所講究録,$\alpha\frac{\underline{\Phi}}{\alpha}\frac{u}{\mathrm{c}}$$\mathrm{o}$$\dot{\mathrm{n}=\in \mathrm{c}\mathrm{n}’}$$\mathfrak{D}\epsilon 0\Phi\alpha\Phi$Fig. 3. Age distribution of adult measles cases in Japan. Drawn by the cumulative data from the 14th week
in 1999 to the 29th in 2002 [9].

120$\hat{\grave{\vee 0^{\Phi}}}\{00$$\frac{\mathrm{q})}{1\mathrm{d},\llcorner}80$$. \frac{.\underline{\mathrm{o}\subset}}{16}60$$. \frac{\mathrm{c}}{8}40$$>\varpi 20$

0

year

Fig. 4. Measles vaccination rate for infant in Japan [9]. The rate in the data is calculated by the ratio ofvaccinated infant population to the PoPuiation of one year old.

Fig. 5. Estimated number of measles susceptibles in Japan [25]“boosting effect” , whereas the chance to re–encounter the pathogen recently becomes less and less [23,24]. Indeed, as shown in Fig. 5, the age distribution of Japanese susceptible population indicates asmall peak in teenagers and twenties [25]. Besides, it is said that the Japanese vaccine would be lesseffective against mutated strains of measles virus $[8,9]$ .
Promoting the vaccination program for infants is a possible public health strategy against themeasles outbreak. Another strategy is to carry out the secondary vaccination for older age class, which

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京都大学数理解析研究所講究録

can be regarded as playing role of an artificial “boosting effect”. Such two dose vaccination programis recently under consideration for a teenager class in Japan $[8,9]$ , while, in the other countries, forexample, the United States, Canada, and Australia, it has been promoted for the younger class aroundor less than 10 years old. Some mathematical models have been analyzed to discuss problems related
to such two-dose vaccination schedule [26-30]. In this paper, we consider the secondary vaccinationin the adult age class over 18 years old, focusing the adult measles transmission, to get some insights
about the efficiency of such two-dose vaccination program for Japanese situation.
2. Preliminary Model

2. 1 Modeling
We consider the epidemic dynamics given by the following SIRM model (see Fig. 6) :(1)$\frac{dS}{dt}=-\beta SI-\omega_{2}S+\rho_{1}(I)R+\rho_{0}(I)M+(1-\omega_{1})bN-\mu S$$\frac{dI}{dt}=\beta SI-\gamma(I\rangle$(2)$I-\mu I$(3)$\frac{dR}{dt}=\gamma(I)I-\rho_{1}(I)R-\mu R$$\frac{dM}{dt}=\omega_{1}bN+\omega_{2}S-\rho_{0}(I\rangle$$M-\mu M, (4)$where $S(t),$ $I(t),$ $R(t)$ and $M(t)$ are respectively the susceptible population, the infective population,
the recovered and infection-induced immune population, and the vaccine induced immune population
at time . Although not a few models have taken account of latent period [1-3, 31,32], with an additional
subpopulation frequently denoted by , exposed to the epidemic disease, we do not consider it in our
$t$

$E$

model.

are positive constants. Parameter is the birth rate, the natural, andParameters
introduces the mass-action type of disease transmission
Termdeath rate, the infection rate.
Kermack-McKendrick SIR model [33] (see also [1-3]).
well-known
susceptible,
likefrom infective to
can berate in the infant period, while
vaccination
theisParameter
period,elder
regarded as that in theto be a function of infective population density, and so are theis
The recovery rate
as well. Vaccine induced immune individual loses its immunity
andimmunity waning rates, while the infection-induced immune individual does withand comes back to susceptible with rate
the infection. Waning rate of the vaccine induced immunity is in general less than that of#-dependence,
rate
these
induced one, as indicated by some serological studies [13-15, 19, 22,34,35]. With
we introduce the boosting effect into our model. As the infective population density gets larger, therecovery of infective individual becomes harder because the infective individual gets more chance fordecreasing
the re-infection of antigen. So the recovery rate averaged over the population is in generalof antigen
theforchanceofin terms of infective population density. Similarly, the increasewaningrate
immunity
the
that
so
antigen,themakes the reinforcement of immune system against
density.
population
infectiveofaveraged over the population is decreasing in terms
at
We consider a stationary population, that is, a constant total population $N=S+I+R+M$,
any time . Hence, from (1-4), at any time
$b,$

$b$

$\beta,$$\mu,$$\mu$$\omega_{2}$$\omega_{1}$$\beta SI$$\beta$$\omega_{2}(0\leq\omega_{2}\rangle$$\omega_{1}(0\leq\omega_{1}\leq 1)$$\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{u}\mathrm{m}\mathrm{e}\Lambda$$\gamma(I)$$\rho_{0}(I)$$\rho_{1}(I)$$\rho \mathrm{o}(I)$$\rho_{1}(I)$$\mathrm{r}\mathrm{e}rightarrow \mathrm{i}\mathrm{n}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$

$t$

$t$

$\frac{dN}{dt}=\frac{dS}{dt}+\frac{dI}{dt}+\frac{dR}{dt}+\frac{dM}{dt}=(b-\mu)N=0$.

Therefore, we assume $\mu=b$ hereafter.

2. 2. Disease-free Equilibrium
From (1-4), we can easily find that the disease-free equilibrium
always exists, where$S^{*}= \frac{(1-\omega_{1})b+\rho_{0}(0)}{b+\rho_{0}(0)+\omega_{2}}N$.

$(S^{*}, I^{*}=0, R^{*}=0, M^{*}=N-S^{*})$

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京都大学数理解析研究所講究録

Fig. 6. SIRM model (1-4).

By means of the standard eigenvalue analysis, we can easily obtain the following condition for its
locally asymptotic stability:$N< \frac{\{b+\rho_{0}(0)+\omega_{2}\}\{b+\gamma(0)\}}{\{(1-\omega_{1})b+\rho_{0}(0)\}\beta}=N_{\mathrm{c}}^{*}$.

(5)

For the population of size
, the epidemic disease eventually becomes endemic. This result
simultaneously means that the basic reproductive number
is given by$N>N_{\mathrm{c}}^{*}$$R_{0}$

島$= \frac{\{(1-\omega_{1})b+\rho_{0}(0)\}\beta N}{\{b+\rho_{0}(0)+\omega_{2}\}\{b+\gamma(0)\}}$.

(6)

Disease free equilibrium is asymptotically stable if, while it is unstable if $R_{0}>1$ . The basic
reproductive number, that is,is monotonically increasing in terms of
becomes larger as
the wane of immunity gets faster. However,does not infinitely increase but has the upper bound, where$R_{\mathrm{O}}<1$$R_{0}$$\rho_{0}(0)$$R_{0}$$R_{0}$$N/N_{\mathrm{c}1}$$N_{\mathrm{c}1}= \frac{\gamma(0)+b}{\beta}$.

From the condition (5), as shown by Fig. 7 we can see three different cases: i) the disease free equilibrium can be asymptotically stable even without any vaccination; ) only the infant vaccination canbring the disease free equilibrium even without secondary vaccination; ) the secondary vaccinationis necessary for the disease elimination.
The first case corresponds to the sufficiently small size of total population such that.
In this case, the condition (5) is satisfied for any pair ofand . Besides, as mentioned for the$\mathrm{i}\mathrm{i}$$\mathrm{i}\mathrm{i}$$N\leq N_{\mathrm{c}1}$$\omega_{1}$$\omega_{2}$153
Hiromi SENO, Naoki SATO

京都大学数理解析研究所講究録

果-dependence of the local stability of disease free equilibrium. The boundary is a line depending
Fig. 7.
on the total population size $N$ . Boundary line moves upper as $N$ gets larger. As for the detail explanation,
see text.
$(\omega_{1} , \omega_{2})$, so that the, then $R_{0}<1$ for any value ofbasic reproductive number in the above, if
effect.
boostingofstrengthoftheindependently
stableasymptoticallyisdisease free equilibrium
that
such
population
of
total
size
an
intermediateisfor
The second case$\rho_{0}(0)$$N\leq N_{\mathrm{c}1}$$N_{\mathrm{c}1}<N<N_{\mathrm{c}2}= \frac{\rho_{0}(0)+b}{\rho_{0}(0)}N_{\mathrm{c}1}$.

(7)

In this case, with the infant vaccination satisfying the below condition, the disease free equilibrium
can be asymptotically stable even without secondary vaccination:$\omega_{1}\geq\omega_{1\mathrm{c}}^{1\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}.=\frac{\rho_{0}(0\rangle+b}{b}(1-\frac{N_{\mathrm{c}1}}{N})$(8).

In the same case, even without infant vaccination, the disease-free equilibrium can be asymptoticallystable if the secondary vaccination satisfies$\omega_{2}\geq\omega_{2\mathrm{c}}^{\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r}}=\{\rho_{0}(0)+b\}(\frac{N}{N_{\mathrm{c}1}}-1)$(9).

. In this case, even withThe last case is for sufficiently large population such thatis, the disease endemic. This is the caseperfect vaccination for infant age class, that is, withsame as that discussed by [36]. So the secondary vaccination is necessary for the asymptotic stabilityof disease free equilibrium. For the asymptotic stability of disease free equilibrium, the rate w2 of
secondary vaccination must satisfy the condition that
$N\geq N_{\mathrm{c}2}$$\omega_{1}=1$$\omega_{2}>\omega_{2\mathrm{c}}^{1\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}=\{\rho_{0}(0)+b\}(\frac{N}{N_{\mathrm{c}2}}-1)$

.

(10)

, the disease becomes endemic independently of any infant vaccination. The secondaryIf. If is large enough tovaccination could contribute to the elimination of disease only if
infant vaccination.
withoutevensatisfy (9), the disease free equiiibrium is asymptotically stable
population size $N$ , the
the
total
of
increasinginterms
are linearlyandSince both
higher rate of secondary vaccination is required for the asymptotic stability of disease free equilibrium,
as the total population size gets larger.
$\omega_{2}\leq\omega_{2\mathrm{c}}^{1\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}$$\omega_{2}>\omega_{2\mathrm{c}}^{1\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}$$\omega_{2\mathrm{c}}^{1\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{r}}$$\omega_{2\mathrm{c}}^{\mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r}}$$\omega_{2}$154
Hiromi SENO, Naoki SATO

京都大学数理解析研究所講究録

Fig. 8. Three age class model (11-22). As for the detail explanation, see text.

We remark that these results are independent of mathematically detail features of functions
and $\rho_{i}(I)(\mathrm{i}=0,1)$ except for values
and. Therefore, our results are valid for a family offunctions for.
andWhen the disease free equilibrium is unstable, the system approaches some endemic state. Depending of the detail features of -dependence of functions, the detail features ofandendemic state, such as its bifurcation structure, could appear mathematically interesting even for oursimple model. However, we do not analyze them here, and will present elsewhere, In this paper, we arefocusing the invasion of epidemic disease, so that the main subject is the local stability of disease freeequilibrium. If the disease free equilibrium is asymptotically stable, we translate the situation as thatan invading small infective population decreases toward zero and its invasion fails. Otherwise, theinvading infective population increases and occurs an outbreak of disease transmission.
$\gamma(I)$$\gamma(0)$$\gamma(I)$$\rho_{0}(0)$$\rho_{i}(I)(\mathrm{i}=0,1)$$I$$\gamma(I)$$\rho_{i}(I)(\mathrm{i}=0,1)$

3. Three Age Class Model
3.1

Modelin9

POPULATION DYNAMICS
Now, we construct a mathematical model with three age classes: infant age class (Group 1) from age
0 to , young age class (Group 2) from age
to , and adult age class (Group 3) from age
to(see Fig. 8).
Let us consider the susceptible, the infective, the recovered, and the
subpopulations in each age class $(i=1,2,3)$ :$a_{1}$$a_{1}$$a_{2}$$a_{2}$$a_{3}$$\mathrm{v}\mathrm{a}\mathrm{c}\mathrm{c}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d}/\mathrm{i}\mathrm{m}\mathrm{m}\mathrm{u}\mathrm{n}\mathrm{i}\mathrm{z}\mathrm{e}\mathrm{d}$155
Naoki SATO
京都大学数理解析研究所講究録–Hiromi$8\mathrm{E}\mathrm{N}\mathrm{O},$: the susceptible population in Group at time
: the infective population in Group at time ;: the recovered population in Group at time ;at time ;$M_{i}(t)$ : the vaccinated or immunized population in Group.
$t:G_{i}(t)=S_{i}(t)+I_{:}(t)+R_{i}(t)+M_{i}(t)$: the population size of each age class at time$tj$$\mathrm{i}$$S_{i}(t)$$I_{i}(t)$$t$$\mathrm{i}$$R_{i}(t)$$\mathrm{i}$

$t$

$t$

$\mathrm{i}$$G_{i}(8)$includes the individuals vaccimeans the vaccinated infant population. In contrast,
Especiallyinfant age class. We assume
in
the
the
infectionfromrecovered
those
age
and
class
nated in the infantaneffective immunity even in
carry
age
class
can
in
the
infantimmunity
get
the
who
can
that thoseof the waning immunity.
because
age
class
adult
susceptible
intheclass,
becomes
age
and
young
the
Hence,
includes the individuals
ageclass.
adultin
thesecondaryvaccinationtheWe introduce
adult age class.
in
the
those
vaccinated
and
age
class
young
in
the
infectionfromtherecoveredsystem:followingbythegoverned
dynamics
populationthe
We consider$M_{2}$$M_{1}$$M_{3}$$\frac{dS_{1}}{dt}=bG_{3}-\lambda_{1}(I_{1}, I_{2}, I_{3})S_{1}-\omega_{1}S_{1}-\phi_{1}S_{1}-\mu_{1}S_{1}$(11)$\frac{dI_{1}}{dt}=\lambda_{1}(I_{1}, I_{2}, I_{3})S_{1}-\gamma_{1}I_{1}-\phi_{1}I_{1}-\mu_{1}I_{1}$(12)$\frac{dR_{1}}{dt}=\gamma_{1}I_{1}-\phi_{1}R_{1}-\mu_{1}R_{1}$(13)$\frac{dM_{1}}{dt}=\omega_{1}S_{1}-\phi_{1}M_{1}-\mu_{1}M_{1}$(14)$\frac{dS_{2}}{dt}=-\lambda_{2}(I_{1}, I_{2}, I_{3})S_{2}+\phi_{1}S_{1}-\phi_{2}S_{2}-\mu_{2}S_{2}$(15)$\frac{dI_{2}}{dt}=\lambda_{2}(I_{1}, I_{2}, I_{3})S_{2}-\gamma_{2}I_{2}+\phi_{1}I_{1}-\phi_{2}I_{2}-\mu_{2}I_{2}$(16)$\frac{dR_{2}}{dt}=\gamma_{2}I_{2}-\phi_{2}R_{2}-\mu_{2}R_{2}$(17)$\frac{dM_{2}}{dt}=\phi_{1}(M_{1}+R_{1})-\phi_{2}M_{2}-l^{\lambda_{2}M_{2}}$(18)$\frac{dS_{3}}{dt}=-\lambda_{3}(I_{1}, I_{2}, I_{3})S_{3}-\omega_{2}S_{3}+\phi_{2}(S_{2}+M_{2})-\phi_{3}S_{3}-\mu_{3}S_{3}$(19)$\frac{dI_{3}}{dt}=\lambda_{3}(I_{1}, I_{2}, I_{3})S_{3}-\gamma_{3}I_{3}+\phi_{2}I_{2}-\phi_{3}I_{3}-\mu_{3}I_{3}$(20)$\frac{dR_{3}}{dt}=\gamma_{3}I_{3}-\phi_{3}R_{3}-\mu_{3}R_{3}$(21)$\frac{dM_{3}}{dt}=\omega_{2}S_{3}+\phi_{2}R_{2}-\phi_{3}M_{3}-\mu_{3}M_{39}$(22)where the disease infection force functions$\lambda_{i}(\mathrm{i}=1,2,3)$

are given by$\lambda_{i}(I_{1}, I_{2}, I_{3})=\beta_{i1}I_{1}+\beta_{i2}I_{2}+\beta_{i3}I_{3}$.

are all positive constants. Meaning of each, andParametersparameter is given in Table 1. As indicated in (11), the net birth rate is proportional to the populationof Group 3. We assume that any reproductive individual belongs to Group 3.
size
$b,$

$\phi_{i},$$\beta_{ij},$$\gamma_{i},$$\mu_{i},$$\omega_{1}$$\omega_{2}(\mathrm{i}, j=1,2,3)$$G_{3}$

ASSUMPTION OF CONSTANT POPULATION SIZE
constant, that is,
We assume that the population size of each age class, $G_{i}(\mathrm{i}=1,2,3)$ , is temporally$N=G_{1}+G_{2}+G_{3}$assume a stationary age class distribution. Eventually, the total population size
according tois also assumed constant. By this assumption, the following equations must be satisfied(11-22):$\frac{dG_{1}}{dt}=bG_{3}-(\mu_{1}+\phi_{1})G_{1}=0$

(23)

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Hiromi SENO, Naoki SATO

京都大学数理解析研究所講究録

Table 1. Parameter list in the three age class model (11-22).
$b$birth ratenatural eath rate in roup
population transfer rate from Group to the subsequent age class
infection rate within the same GrouPin ection rate rom rouP to GrouP
recovery rate of the infective in GrouP
vaccination rate in the infant age classvaccination rate in the ult age claes$i$$\mu_{i}$$i$$\phi_{t}$$i$$\beta_{\mathrm{i}i}$$\beta_{ij}$$(i\neq j)$

$i$

$j$

$i$

$\gamma_{i}$$\omega_{1}$$\omega_{2}$$\frac{dG_{2}}{dt}=\phi_{1}G_{1}-(\mu_{2}+\phi_{2})G_{2}=0$

(24)

.

(25)

$\frac{dG_{3}}{dt}=\phi_{2}G_{2}-(\mu_{3}+\phi_{3})G_{3}=0$

BALANCE EQUATIONS
Next, we consider the population renewal in the infant age class. As indicated in (11), the newborn
populationenters into the infant age class during $[t, t+\Delta t)$ . This cohort of newborn population
decreases to
at time $t+a_{1}$ with the natural death of rate . On the other hand,
populationof ageis transfered from Group 1 to Group 2 during at. Thus, we obtain thefollowing equation:$bG_{3}\Delta t$$bG_{3}e^{-\mu_{1}a_{1}}\Delta t$$\phi_{1}G_{1}\Delta t$$\mu_{1}$$a_{1}$$bG_{3}e^{-\mu_{1}a_{1}}\Delta t=\phi_{1}G_{1}\Delta t$

(26)

.

In the same way, we can get the similar equations for each age class:$\phi_{1}G_{1}e^{-\mu 2(a_{2}-a_{1})}\Delta t=\phi_{2}G_{2}\Delta t$

;

(27)

$\phi_{2}G_{2}e^{-\mu s(a\mathrm{a}-a_{2})}\Delta t=\phi_{3}G_{3}\Delta t$

.

(28)

From these equations (26-28), we can obtain the expression foreters:$\phi_{1},$$\phi_{2)}$and$\phi_{3}$with the other param-$\phi_{1}=b\frac{G_{3}}{G_{1}}e^{-\mu_{1}a_{1}}$(29)$\phi_{2}=b\frac{G_{3}}{G_{2}}e^{-\{\mu_{1}a_{1}+\mu_{2}(a_{2}-a_{1})\}}$(30)$\phi_{3}=be^{-\{\mu_{1}a_{1}+\mu_{2}(a_{2}-a_{1})+\mu_{3}\langle a_{3}-a_{2}\rangle\}},$

.

(31)

With the assumption of a stationary age class distribution, those equations (23-25) and (29-31)
give the following relation between birth rate and death rates, and :$b$$\mu_{1},$$b= \frac{\mu_{3}e^{\mu_{1}a_{1}+\mu_{2}(a_{2}-a_{1})}}{1-e^{-\{\mu_{3}(a_{3}-a_{2})\}}}$$\mu_{2}$$\mu_{3}$.

(32)

Since the total population size $N$ is constant, from (23-25) and (29-31), we can obtain the following
ations, too:$\mathrm{e}\mathrm{q}\mathrm{u}$$\frac{G_{1}}{N}=\frac{b\mu_{2}(1-e^{-\mu_{1}a_{1}})}{b\mu_{2}(1-e^{-\mu_{1}a_{1}})+b\mu_{1}e^{-\mu_{1}a_{1}}(1-e^{-\mu_{2}(a_{2}-a\iota)})+\mu_{1}\mu_{2}}$(33)$\frac{G_{2}}{N}=\frac{b\mu_{1}e^{-\mu_{1}a\mathrm{z}}(1-e^{-\mu 2(a_{2}-a_{\mathrm{I}}\rangle})}{b\mu_{2}(1-e^{-\mu_{1}a_{1}})+b\mu_{1}e^{-\mu_{1}a_{1}}(1-e^{-\mu_{2}\langle a_{2}-a_{1})})+\mu_{1}\mu_{2}}$(34)$\frac{G_{3}}{N}=\frac{\mu_{1}\mu_{2}}{b\mu_{2}(1-e^{-\mu \mathrm{x}a_{1}})+b\mu_{1}e^{-\mu_{1}a_{1}}(1-e^{-\mu 2(a_{2}-a_{1})})+\mu_{1}\mu_{2}}$

.

(35)

Consequently, for given natural death rates, the birth rate is uniquely determinedby (32), and then, from (33-35), ratios $G_{i}/N(\mathrm{i}=1,2,3)$ are determined, and vice versa. Further,from (29-31), population transfer rates
are determined, too.
$\mu_{i}(\mathrm{i}=1,2,3)$$\phi_{i}(\mathrm{i}=1,2,3)$

$b$

157
Hiromi SENO, Naoki SATO

京都大学数理解析研究所講究録

Ihble 2. Parameter values for numerical calculation*Data of the Kochi Prefecture in Japan, 2000.
**Group 3 consists of every individual over 18 years old.

3.2. Parameters for Numerical Calculation
For our three age class model, we analyze it mainly with numerical calculations, making use of measles
data for the Kochi Prefecture in Japan [6]. Parameter values used for numerical calculations are given
in Table 2.

AGE CLASSIn Japan, the infant measles infection occurs mostly from one to five years old [9], so that we put
Japan, we put$a_{1}=5$ . Since the adult mealses infection is defined for individual over 18 years old inyears old.
$a_{2}=18$ , and $a_{3}=\infty$ in our model. Therefore, Group 3 consists of every individual over 18
BIRTH RATE, DEATH RATE, AND POPULATION TRANSFER RATE. In contrast,
In our model, the population renewal is assumed proportional to the population size
numbers to
newborn
ratio
ofgiven ‘birth rate’ in the demographic data is generally defined by the:andthe total population $N$ . Hence, we get the following relation between$G_{3}$$\tilde{b}$$\tilde{b}$$b= \tilde{b}\cdot\frac{N}{G_{3}}$$b$.

From a demographic data for in Japan [37], we roughly choose the parameter value of as in Table2.
and #2 from (33-35),
With thle estimated value of , we can numerically estimate values of,
put $\phi_{3}=0$ because
we
ratetransfer
population
then can get the value of from (32). As for theandvalues ofobtaincanwe(29-31),
From
years
old.
Group 3 consists of every individual over 18$b$$\tilde{b}$$b$$\mu_{1}$$\phi_{3}$$\mu_{3}$$\phi_{1}$$\phi_{2}$.

RECOVERY RATE
individual,
We assume that the mean duration of infection till recovery is common for any infective
independently of age. From the general knowledge about the measles infection, we put the mean
duration of infection 9 days. Therefore, from $\gamma_{i}=\gamma=1/9(\mathrm{i}=1,2,3)$ per day, we can get the valueof$\gamma$per year in Table 2.

158
Hiromi SENO, Naoki SATO

京都大学数理解析研究所講究録

Table 3. Infection rate for numerical calculation.
VaIue year
Infection rate
1.3
2.5$\beta 13=\beta 23=\beta 33=\beta_{32}=\beta_{3l}$3.3$\beta_{11}$$\mathrm{x}$$\beta_{12}=\beta_{21}=\beta_{22}$$\mathrm{x}$$\mathrm{x}$$10^{-}$$10^{-}$$10^{-}$INFANT VACCINATION RATEIn the surveillance data, the vaccination rate is defined by the ratio of vaccinated infant population
to the population of one year old. On the other hand, the measles vaccination for Japanese infant isscheduled at present for those from twelve to ninety months after birth. In this reason, the vaccination
rate in the surveillance data can be beyond 100% (see Fig. 4).
In our numerical calculation, we give the population of one year old, , by$\tilde{\omega}$$\overline{\omega}$$J$$J= \int_{1}^{2}bG_{3}e^{-\mu\iota^{t}}dt=\frac{bG_{3}(e^{-\mu_{1}}-e^{-2\mu 1}\rangle}{\mu_{1}}$

.

(36)

. If the, we assume that
Then, at an equilibrium state with the equilibrium value
vaccination rate in data is 80%, we use the correspondence such that $M_{1}^{*}/J=0.8$ .
from (11-14) as follows:
At the equlibrium state for our model, we can get the expression of$\tilde{\omega}=M_{1}^{*}/J$$M_{1}^{*}$$M_{1}^{*}$$M_{1}^{*}= \frac{\omega_{1}\{bG_{3}-(\gamma_{1}+\phi_{1}+\mu_{1})I_{1}^{*}\}}{(\phi_{1}+\mu_{1})(\omega_{1}+\phi_{1}+\mu_{1})}$

(37)

.

Hence, for instance, according to the surveillance data for the Kochi Prefecture in 2002 and 2003 [6],except for years when the measles infection outbreaks, if we put $I_{1}^{*}=16$ , then, with $\tilde{\omega}=M_{1}^{*}/J=0.8$ ,.
we can get the infant vaccination rate
4.483
According to the disease-free equilibrium with parameter values in Table 2, making use of (37)with $I_{1}^{*}=0$ , we can numerically calculate the correspondence between and :2.856$\omega_{1}=$$\mathrm{x}10^{-2}$$\tilde{\omega}$$\omega_{1}$$\mathrm{x}10^{-2}$$\omega_{1}=$

for
for
4.604
for
3.411
3.993
, and
5.246
for
.
With parameter values in Table 2, becomes beyond 100% at relatively small value of . Besides,
the value of significantly depends on the infective population size
becomes beyond 100% withzy more than about 132. This indicates that, even for relatively small infective population size , thevaccination rate corresponding to that in the demographic data is beyond 100%.
for$\tilde{\omega}=0.6,$$\tilde{\omega}=0.9$$\mathrm{x}10^{-2}$$\omega_{1}=$$\mathrm{x}10^{-2}$$\omega_{1}=$$\tilde{\omega}=0.7,$$\omega_{1}=$$\mathrm{x}10^{-2}$$\tilde{\omega}=0.8,$$\mathrm{x}10^{-2}$$\omega_{1}=$$\tilde{\omega}=1.0$$\tilde{\omega}$$\omega_{1}$$\tilde{\omega}$$I_{1}^{*}:$$\tilde{\omega}$$I_{1}^{*}$$\tilde{\omega}$

INFECTION RATE
If we apply those infection rates used in [38], $\beta_{11}=2.4\mathrm{x}10^{-4},$ $\beta_{12}=\beta_{21}=\beta_{22}=2.6\cross 10^{-4}$ ,and $\beta_{13}=\beta_{23}=\beta_{33}=\beta_{32}=\beta_{31}=1.3\mathrm{x}10^{-4}$ , our numerical calculation with $\omega_{2}=0$ and4.483
which is derived in case of
and $\tilde{\omega}=M_{1}^{*}/J=0.8$ with (37), indicates$(I_{1},
I_{2},
I_{3})\approx(16,100,200)$
a fluctuation aroundin the stationary state. According to the case of the, and see Fig. 1), the corresponding values averaged over recent several years result
Kochi Prefecturein $(I_{1}+I_{2}, I_{3})\approx(16,1)$ , which seems much smaller than the above numerical result. So we turned
infection rates so that those values of (, I3) at the stationary state have order corresponding to
the data for the Kochi Prefecture, and as a result we choose those values in Table 3.
$\omega_{1}=$$\mathrm{x}10^{-2}$$\langle$$\mathit{1}_{1}^{*}=16$$[6]$$I_{1},$ $I_{2}$33. Disease-Free Equilibrium
As for the local stability analysis for the disease free equilibrium, we carry it out with numericallycalculating its eigenvalues with those parameter values in Tables 2 and 3. Our numerical calculations
show that the eigenvalues for the disease-free equilibrium are all real for any
and .
$\omega_{\mathrm{A}}1$$\omega_{2}$158
Hiromi SENO, Naoki SATO

京都大学数理解析研究所講究録

CRITICAL VA CCINATION RATE

. This is the case when the adult vaccination is
At first, Jet us consider a specific case withperfectly carried out so that no susceptible individual exists in Group 3 at any time . So $S_{3}=0$ at
the disease free equilibrium, too. In this case, the numerically obtained maximal eigenvalue is 22.526
with $\omega_{1}=0.0$ , and -29.907 with $\omega_{1}=1.0$ . It is numerically shown that the maximal eigenvalue ismonotonically decreasing in term of . As a result, we find that, even with the perfect vaccinationforsuch that the disease free equilibriumin the adult age class, there exists a criticai value. We can numericall obtain, and unstable withis asymptotically stable with
0.11367 in this case.
by (37) at the disease free
, we can numerically calculate the value ofWith
in Table 2.
population
size
175.08%with theequilibrium, and obtainis not carried out,
vaccination
adult
when
the
Next, we consider another specific case withway
for the previoussameasJapan.
In
thein
present
situationat all. This case corresponds to the$\omega_{1}=0.0$ , and -29.888 withwitheigenvalue,
22.541case, we can numerically obtain the maximal, the difference is very small: $10^{-2}-$. Compared to the maximal eigenvalue with
$10^{-1}\%$ in order. Indeed, the-dependence of maximal eigenvalue is rather similar with that for thein this case is numericallyis negligible. The critical value
,
that
the
differencesocase of
175.20%,correspondingly.11379,andasobtained. This is the case whenOn the other hand, we can consider another specific case with
the infant vaccination is perfectly carried out. In this case, no susceptible individuai exists in Group
1 at any time , and eventually no susceptible also in Group 2 at any time because of the effective
immunity, We can explicitly obtain the analytical expression of every eigenvalue for the disease freeequilibrium, and get the general condition for its asymptotic stability as follows:$\omega_{2}arrow\infty$$t$$\omega_{1}$$\mathrm{s}$$\omega_{1}$$\omega_{1}\mathrm{c}$$\omega_{1}<\omega_{1\mathrm{C}}$$\omega_{1}>\omega_{1\mathrm{C}}$$\omega_{1\mathrm{C}}=$$\omega_{1\mathrm{C}}=0.11367$$M_{1}^{*}$$J$$\tilde{\omega}=M_{1}^{*}/J=$$\omega_{2}=0$$\omega_{2}arrow\infty$$\omega_{1}=1.\mathrm{O}$$\omega_{1}$$\omega_{1}\mathrm{c}$$\omega_{2}arrow\infty$$\tilde{\omega}=$$\omega_{1\mathrm{C}}=0$$\omega_{1}arrow\infty$

$t$

$t$

$\omega_{2}>\frac{\phi_{3}+\mu_{3}}{\phi_{3}+\mu_{3}+\gamma_{3}}\cdot\{\beta_{33}G_{3}-(\phi_{3}+\mu_{3}+\gamma_{3})\}$

(38)

,

where we used the equations (23-25). This condition means the existence of case when the adultvaccination could be essential for the elim ination of disease. However, only in case when the right sideof adult age class is sufliciently great beyondof (38) is positive, that is, when the population size
a specific size, this condition could be meaningful. Indeed, in our case with parameter values given
in Tables 2 and 3, the right side of (38) has a negative value, so that the perfect vaccination in theinfant age class leads to the disease-free state, independently of the adult vaccination. Making use of(35), we can numerically estimate the population size with which the condition (38) is meaningful:.
$G_{3}>$ 1.236
that corresponds to the condition that the totai population size $N>$ 1.453
the
reach
never
could
millionThis result implies that a huge community with population over ten
disease-free state only vith the infant vaccination, whereas the secondary vaccination program might
be effective to eliminate the disease or decrease the infective population.
Consequently, our numerical calculations indicate that, almost independently of the value of ,is beyond athe population approaches the disease-free equilibrium if the infant vaccination rate
value.
thecritical
is
below
if
state
critical value, whereas it approaches the endemic$G_{3}$$\mathrm{x}10^{7}$$\mathrm{x}10^{7}$$\omega_{2}$$\omega_{1}$$\omega \mathrm{l}$EFFECT OF TOTAL POPULATION SIZE
depends on the total population size $N=G_{1}+G_{2}+G_{3}$ .
for the vaccination rate
Critical value
We numerically investigated the maximal eigenvalue for the disease-free equilibrium with parameterhas anvalues of Tables 2 and 3 except for population sizes, and found that the critical value
almost linear dependence on the total population size.
to the result appears negligible.
Also in these numerical investigations, the contribution ofwithpopulation is 5.277
total
of
size
,
criticaltheIndeed, even in the specific case of
and
3.094
with
,
5.276
of
while,
in
case
,
with
and 3.099.
with
In Table 4, we show numerically obtained critical infant vaccination rate in terms of the total
population size. We conclude that it would be hard for the population over eight hundred thousands toreach the disease-free equilibrium with the present ievel of infant vaccination rate, while the population$\omega_{1}$$\omega_{1\mathrm{C}}$$\omega_{1\mathrm{C}}$$\omega_{2}$$\mathrm{x}10^{5}$$\omega_{2}arrow\infty$$\omega_{1}=0$$\omega_{1}=1$$\mathrm{x}10^{6}$$\omega_{1}=1$$\omega_{2}=0$$\mathrm{x}10^{5}$$\omega_{1}=0$$\mathrm{x}10^{6}$

160
Hiromi SENO, Naoki SATO

京都大学数理解析研究所講究録

Table 4. Critical infant vaccination rate versus the total population size in case of
Total population size$<5.276$

6.000

6.010
6.152
6.302
6.624
7.000
8.000
9.000
10.000100.000$(\mathrm{x}10^{\mathrm{d}})$$\omega_{1}\mathrm{c}$0.00000
0.02818
0.02856
0.034110.039930.05246
0.06709
0.10600
0. 14492
O.IS3843.73842

Corresponding$\tilde{\omega}$$\mathrm{w}_{2}=0$

.

(%)

0.000

59.30460.00070.00080.000100.000
121.014167.296203.293232.092
465.589

Fig. 9. Temporal variation toward the endemic state. Numerically calculated with parameter values given in. Initial conditions are given by $S_{1}(0)=G_{1}$ 76, $I_{1}(0)=76,$
Tables 2 and 3 with
) $=G_{2}$ 224,
I2(0) $=224,$ $S_{3}(0)=G_{3}$ , and I3(0) $=R_{i}(0)=M_{i}(0)=0(\mathrm{i}=1,2,3)$ , which correspond to the data fbr the
Kochi Prefecture [6]. (a); (b). (a-1) andshow the temPoral variationof infective populations in logarithmic value, (a-2) and (b-2) show that of the corresponding ratio $M_{1}/J$ .
$\omega_{2}=0$$\omega_{1}=3.0\mathrm{x}10^{-2}$$S_{2}\langle 0$–$\omega_{1}=5.0\mathrm{x}10^{-2}$–$(\mathrm{t}\succ 1)$

below seven hundreds thousands is likely to reach the disease free equilibrium. Furthermore, small
population below five hundreds thousands would eventually reach the disease free equilibrium without
any vaccination.

ENDEMIC STATE

As indicated by our numerical calculations in the previous section, with the present level of infant
vaccination rate, the population over seven hundred thousands would be on the way toward endemic
state. In our numerical calculations, such endemic state was always an equilibrium approached throughdumping oscillation with rather large amplitude, as shown in Fig. 9. Such nature of dumping oscillation
may be involved in typical repetitive outbreaks of measles as seen also for the case of the Kochi
Prefecture in Japan (Fig. 1). Characteristics of the dumping oscillation is sensitively affected by thevalue ofin our numerical calculations. We found such a tendency that the frequency of oscillation
becomes smaller asgets larger.
$\omega_{1}$$\omega_{1}$1Bl
Hiromi SENO, Naoki SATO

京都大学数理解析研究所講究録

4. Conclusion

Our results for both non-age-structured and three age class models clearly indicate that the vaccination
program should be planned with taking account of the size of population within which the diseasetransmission is taking place. It is expected that the vaccination program would be effective to reduce
drastically the incidence within relatively small population, whereas it could not eliminate the repeatedoutbreak of infection within relatively large population as in some Japanese local communities [7-9].
Especially with the present level of infant vaccination rate in Japan, it is likely that the measles
elimination would be far away from its realization. Besides, in case of Japan, the secondary vaccination
program for an adult age class would be little effective, compared to the promotion program of infant
vaccination. Although our numerical calculations imply that it would be hard to increase the infant
vaccination rate enough to make the population approach the measles-free equilibrium, the increaseof infant vaccination rate above the present level would be at least more effective than the secondary
vaccination for an adult age class in order to reduce the infective population.
For a relatively large population, the promotion of both infant vaccination and secondary vaccination for an adult age class would be the proper strategy for the public health. In such case, theselection of age class targeted by the secondary vaccination would be one of the most important factors
which determine the result of vaccination program. Many mathematical modeling considerations havestudied this subject [26-30]. However, as Guris [24] mentioned for the limited population of Palau,
we remark here that, for a relatively small population, such two-dose vaccination program would not
be necessarily more appropriate than the promotion of single dose vaccination program. Moreover,
for a sufficiently large population, the two-dose vaccination program would be less effective than the
promotion program of primary vaccination.
Although our models have mathematically simple structure with mass-action type of disease transmission, we conjecture that, except for the detail bifurcation structure of solution, the essential featureof solution would be similar to that for some SEIR model or even for the more sophisticated or complicated model with a range of corresponding parameter values.
Acknowledgernents. The authors greatly thank Prof. Masayuki Kakehashi for his valuable comments andencouragement to complete this work.
${\rm Re}$1.

ferences$\mathrm{R}.\mathrm{M}$. Anderson and$\mathrm{R}.\mathrm{M}$. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University

Press, Oxford, 1991.
2. F. Brauer and C. Castillo\leftrightarrow Ch\’avez, Mathematical Models in Population Biology and Epidemiology, Texts in
Applied Mathematics 40, Springer, New York, 2001.
3 0. Diekmann and J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Buitding,Analysis and Interpretation, Wiley Series in Mathematical and Computational Biology, John Wiley&Son,

Chichester, 2000.
. Murray, Introduction to Mathernatical Biology, Interdisciplinary Applied Mathematics 17, Springer,
New York, 2002.
. Murray, Mathematical Biology: Spatial Models and Biomedical Applications, 3rd Edition, Interdisci5.
plinary Applied Mathematics 18, Springer, New York, 2002.
6. 高知県健康福祉部健康政策課, 高知県における麻疹流行と取組報告, 高知, 2003 年 3 月.
7. Infectious Disease Surveillance Center (IDSC), Measles, Japan, 1999-2001, Infectious Agents Surveillance
Report $(IASR)22(11):273- 274(2001)$ .
8 Infectious Disease Surveillance Center (IDSC), Measles, JaPan, 2001-2003, Infectious Agents Surveillance$25(3)$ : 60-61 (2004).
Repo 尻月.
年
9. 国立感染症研究所感染症情報センタ一, 麻疹の現状と今後の麻疹対策について, 東京, 2003 1035-36, Geneva,
outbreaks,
measles
responsetoPP.
andpreparedness
epidemicforguidelines
10. WHO, WHO
Switzerland, 1999.
. Hoffman, et al., A measles outbreak at a college with prematriculation. Markowitz,. Hersh,11.
Public Heafth, 81: 360364 (1991).
immunization requirement, Am.
Hernandez,
et al., Loss of maternally acquired measles antibodies in wellAlvarez,
H.
.
Vaisberg,
12. A.
$736\sim 738$ (1990).
nourished infants and response to measles vaccination: Peru, Am. J. Publtc Health, 80:
titers, early
vaccination
long-termofcomparison
antibody
Bottiger,
MeaslesM.
and
13. B. Christenson
Vaccene,12:
virus,
measlesontheeffects
boosterandtoand naturally acquired-immunity

4,$\mathrm{J}.\mathrm{D}$$\mathrm{J}.\mathrm{D}$$(IA\mathit{8}R)$$\mathrm{L}.\mathrm{E}$$\mathrm{B}.\mathrm{S}$$\mathrm{R}.\mathrm{E}$$J$$\mathrm{J}.\mathrm{O}$–vaccination titers

129-133 1994).
$\langle$

162
Hiromi SENO, Naoki SATO

京都大学数理解析研究所講究録. Markowitz, P. Albrecht, et al., Measles antibody –revaluation of protective titers, .
Infect. Dis., 162: 1036-1042 (1990).
. Preblid, P.E.M. Fine, et al., Duration of live measles vaccine-induced immunity,
15.
. Markowitz,. J., 9: 101-110 (1990).
Pediatr Infect .
. Edmunds, et al., Modeling the impact of subclinical measles transmission. Nokes,
16. J. Mossong,in vaccinated populations with waning immunity, A $m$ . J. Epiderniol., 1SO: 1238-1249 (1999).
. O’Callaghan, and S. Ratnam, Modelling antibody response to measles vaccine and
17. J. Mossong,subsequent waning of immunity in a low exposure population, Vaccine, 19: 523-529 (2001).
18. J. Bennett, H. Whittle, B. Samb, et al., Seroconversions in unvaccinated infants: Further evidence for
subcIinical measles from vaccine trials in Niakhar, Senegal, Int . Epidemiol., 28: 147-151 (1999).
19.
. Helfand,. Kim, . Gary, et al., Nonclassic measles infections in an immune population exposed
to measles during a college bus trip, J. Med. Virol., 56: 337-341 (1998).
20. S. Huiss, B. Damien, F. Schneider, et al., Characteristics of asymptomatic secondary immune response tomeasles virus in late convalescent donors, Clin . $Exp$ . Immunol., 109: 416-420 (1997).
. Pedersen,. Mordhorst, G. Glikmann, et al., Subclinical measles infection in vaccinated seropositive
21.
individuals in arctic Greenland, Vaccine, 7: 345-348 (1989).
. Muller, Modelling meaeles -emergence as a result of waning of immunity in vacci22. J. Mossong andnated populations, Vaccine, 21: 4597-4603 (2003).
. Grenfell, Antibody dynamics in childhood disease: Waning and boosting of immunity
23. K. Glass andand the impact of vaccination, J. theor, Biol., 221: 121-131 (2003).
24. D. Guris, J. Mccready,. Watson, et al., Measles vaccine effectiveness and duration of. $J.,$ $15:1082- 1086$
immunity in the absence of boosting from exposure to measles virus, Pediatr. Infect.
$(1996)$ .
25. 多屋馨子, 新井智, 松永泰子, 岡部信彦, 2000 年度麻疹血清疫学調査ならびにワクチン接種率調査\sim 感染症流行予測調査よりゝ, Infectious Agents Surveilfance Report (IA $SR$ ) $22(11)$ :275-277(2001).
. Anderson and. May, Age-related changes in the rate of disease transmission: Implication for26.
the design of vaccination programmes, J. Hygiene, 94: 365-436 (1985).
27.
. Dreesman, An array based simulation approach for predicting the impact of different measles vaccination strategies in Lower Saxony, $APL$ Quote Quad, 31: 5-12 (2001).
. Gay, L. Pelletier, and P. Duclos, Modelling the incidence of measles in Canada: An assessment of the28.
options for vaccination policy, Vaccine, 16: 794-801 (1998).
. Becker, Assessment of two-dose vaccination schedules: Availability for vaccination
29. V. Rouderfer andand catch-up strategies, Math. Biosci., 129: 41-66 (L995).
30. D. Schenzle, An age-structured modeI of Pre- and post-vaccination measles transmission, IMAJ Math.
App . Med. Biol., 1: 169-191 (1984).
. Duncan,31.
. Duncan, and S. Scott, The dynamics of measles epidemics, Theor, Pop. Biol., 52:
155-163 (1997).
32. P.E.M. Fine, Herd immunity: History, theory, practice, Epidemiol Rev., 15: 265-304 (1993).
. Kermack and AG. McKendrick, A contribution to the mathematical theory of epidemics, Phdos.
33.
. Soc. Lond. $A115$ : 700-721 (1927).
34.
. Mueller, Measles elimination: Old and new challenges?, Vaccine, 19: 2258-2261 (2001).
. Mueller, F. Fack, B. Damien, et al., Immunogenic measles antigens expressed in plants: Role as an
35.
edible vaccine for adults, Vaccine, 21: 816-819 (2003).
36, W. Katzmann and K. Dietz, Evaluation of age-specific vaccination strategies,, Theor. Pop. Biol., 25:
125-137 (1984).
37. 総務省統計局, 年報, 平成 15 年 10 月 1 日現在推計人口, 東京, 2004 年 3 月.
. Babad,38.
. Gay, et al., Predicting the impact of measies vaccination in England and. Nokes,
Wales: Model validation and analysis of policy options, Epidemiol. Infect., 114: $319\sim 344$ $(1994)$ .

14.

$\mathrm{R}.\mathrm{T}$. Chen,$J$$\mathrm{L}.\mathrm{E}$$\mathrm{L}.\mathrm{E}$$\mathrm{S}.\mathrm{R}$$D\mathrm{i}s$$\mathrm{W}.\mathrm{J}$$\mathrm{D}.\mathrm{J}$$\mathrm{C}.\mathrm{J}$$J$$\mathrm{R}.\mathrm{F}$$\mathrm{I}.\mathrm{R}$$\mathrm{H}.\mathrm{E}$$\mathrm{D}.\mathrm{K}$$\mathrm{C}.\mathrm{H}$$\mathrm{C}.\mathrm{P}$$\mathrm{r}\mathrm{e}$$\mathrm{B}.\mathrm{T}$$\mathrm{v}\mathrm{a}\mathrm{c}\mathrm{c}\mathrm{i}\mathrm{n}\mathrm{e}\frac{-}{}\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{e}\mathrm{d}$$\mathrm{J}.\mathrm{C}$$D\mathrm{i}s$$\mathrm{R}.\mathrm{M}$$\mathrm{R}.\mathrm{M}$$\mathrm{J}.\mathrm{M}$$\mathrm{N}.\mathrm{J}$$\mathrm{N}.\mathrm{G}$$l$$\mathrm{C}.\mathrm{J}$$\mathrm{S}.\mathrm{R}$$\mathrm{W}.\mathrm{O}$$R$$\mathrm{C}.\mathrm{P}$$\mathrm{C}.\mathrm{P}$$\mathrm{H}.\mathrm{R}$$\mathrm{D}.\mathrm{J}$$\mathrm{N}.\mathrm{J}$