Yonago Acta medica 1996;39:83–98

Simulation of the Heterosexual HIV/AIDS Epidemic in Japan by a Fuzzy
Mathematical Model
Shinsuke Morio, Kenji Soda*, Shuji Hashimoto†, Kazuo Fukutomi‡,
Seiichi Ichikawa§, Mitsuhiro Kamakura# and Hideaki Nakayama
Department of Hygiene, Faculty of Medicine, Tottori University, Yonago 683, *Departmentof Public Health, School of Medicine, Yokohama City University, Yokohama 236, †Department of Epidemiology and Biostatistics, School of Health Sciences and Nursing, Universityof Tokyo, Bunkyo-ku 113, ‡The Institute of Public Health, Minato-ku 108, §Laboratory of
Public Health, Faculty of Hygienic Sciences, Kanagawa Prefectural College of Nursing andMedical Technology, Yokohama 241, #Department of Hygiene and Public Health, School of
Medicine, Keio University, Shinjuku-ku 160, Japan
A study with a mathematical model was conducted for the objectives of analyzingheterosexual human immunodeficiency virus (HIV) transmission in Japan andidentifying effective countermeasures against HIV/acquired immunodeficiency
syndrome (AIDS) epidemics in communities in which the transmission is mainly throughcommercial sex workers (prostitutes). A fuzzy mathematical model which examined 4groups (highly and moderately sexually active males, and highly and moderately sexuallyactive females) was developed to simulate the heterosexual HIV/AIDS epidemic in Japan.
The simulation generated bell-shaped epidemic curves for the numbers of male andfemale AIDS cases. At the early stage of the epidemic, the calculated incidence of maleand female AIDS cases was similar to the reported incidence in each sex, respectively,
in Japan. The results obtained with the scenario which assumed rapid spread of condom
usage among highly sexually active females disclosed that this countermeasure had thebiggest effect. The results obtained with either the scenario which assumed the spreadof condom usage among males who travel outside of the community, or which assumed
a decrease in the number of commercial sex workers from other communities, disclosedthat these countermeasures had little preventive effect. The results suggest thatcountermeasures designed to decrease heterosexual HIV transmission inside of thecommunity are the most effective, even if the inflow of HIV from outside communities
does not decrease.
Key Words: AIDS; epidemiology; fuzzy theory; mathematical model; sexually transmitted disease

Some Asian countries have been spared thepresent worldwide human immunodeficiency
virus/acquired immunodeficiency syndrome(HIV/AIDS) epidemic (WHO, 1995). In Japan,
the reported cumulative number of HIV-infected
persons and AIDS patients was 3,524 and 1,154,
respectively, at the end of 1995 (Soda et al.,1996). Many of these cases are in haemophili-acs, and the number of HIV/AIDS cases resulting from infection through heterosexual intercourse is not large. However, the number of
haemophiliac HIV cases does not show any
increase, while the number of heterosexual HIV/AIDS cases has become the 2nd largest groupand shows the most rapid rate of increase(Tables 1 and 2). Countermeasures against an

Abbreviations: AIDS, acquired immunodeficiency syndrome; CSW, commercial sex worker; HIV, humanimmunodeficiency virus; TFN, triangle fuzzy number.

83

S. Morio et al.

Table 1. Cumulative reported number of AIDS and HIV-infected persons by risk factors
Risk factors
Heterosexual (male)
Heterosexual (female)
Homosexual
Intravenous drug users
Materno-infant
Haemophiliacs
OthersUnknown
Total

Japanese

AIDS
foreigner

Total

JapaneseHIVforeigner

Total

135
23
148
2
5
582
12
82
987

35
20
31
3
2
0
4
72
167

170
43
179
5
7
582
16
152
1154

283
118
294
3
4
1806
29
72
2609

84
381
56
8
5
0
8
373
915

367
499
350
11
9
1806
37
445
3524

HIV/AIDS epidemic through transmission bycommercial sex workers (CSWs) or prostitutes
are thus an urgent issue in Japan.
Countermeasures against infectious diseases are well known among experts in the fieldof public health. However, the most effective
countermeasure against one infectious disease
is not necessarily identical to that for anotherinfectious disease. The effectiveness of countermeasures depends upon many factors which
relate not only to the disease itself but also tohuman behavior, and it is particularly difficult
to identify the most effective countermeasureagainst an HIV/AIDS epidemic, as it is related
to the complex human behavior of sexual
activity.
Simulation with a mathematical model is
one of the scientific methods of analyzing epidemics of infectious diseases and of evaluating
countermeasures against them. Many mathematical models have been developed and applied
to the HIV/AIDS epidemic since its initial
occurrence (Knox, 1986; Anderson, 1988, 1989;
De Gruttola et al., 1988; Dietz et al., 1988; Gupta
et al., 1989; Van Druten et al., 1990; Jager et al.,1992). Mathematical models which simulate
HIV transmission through heterosexual intercourse require analysis of multiple groups(multigroup transmission model). The models
should also assume an “open” model whichassumes the inflow of HIV from outside of thecommunity, and to examine the internal spreadof HIV among at least 4 groups (highly promiscuous and less promiscuous males, and highly
promiscuous and less promiscuous females).

However, the more complex the models become,the more data are necessary. This defect may
play a part in the infrequent usage of mathematical models of HIV/AIDS transmission
among public health decision makers.
In this study, we applied fuzzy theory to amathematical model of heterosexual HIV/AIDS
epidemics in order to compensate for this fact.
We used the reported numbers of HIV-infected
persons and AIDS patients in Japan to confirm
the effectiveness of countermeasures against
the transmission of HIV/AIDS.

Methods
Mathematical model

Overview
In the model, we assume HIV/AIDS transmission through heterosexual intercourse in acommunity. This model consists of the following 4 groups: (a) highly sexually active males(Group 1); (b) highly sexually active females(Group 2); (c) moderately sexually active males(Group 3) and (d) moderately sexually active
females (Group 4). Individuals in Group 1 aremales who have regular sexual intercourse with
CSWs. They are assumed to have sexual intercourse with those in Groups 2 and 4, and with
CSWs outside of the community when they
travel or stay in the foreign country. Those inGroup 2 are female CSWs, who work at specialized Japanese-style bath rooms or work at othersex-related services. This group includes
84

Fuzzy mathematical model of HIV/AIDS epidemic

Table 2. Reported incidence of AIDS by sex
and year
Year

Male

Femaleintercourse, the prevalence rate of HIV among
these CSWs, and the probability of HIV
transmission per CSW. It is also assumed thatsome individuals of Group 1 are infected withHIV through heterosexual intercourse with
HIV-infected CSWs who come into thecommunity from outside communities, eventhough the individuals from Group 1 do not go
outside of the community. The probability of
the infection is a function of the number of
CSWs with whom they have sexual intercourse,
the rate of CSWs from outside communities
among individuals in Group 2, the prevalence
rate of HIV among these foreign CSWs, and theprobability of HIV transmission per CSW.
Mathematical formulas are shown in the Appendix (Jager and Ruitenberg, 1992).

Total

1985
6 ( 0)
0 ( 0)
6 ( 0)
1986
5 ( 0)
0 ( 0)
5 ( 0)
1987
9 ( 1)
5 ( 4)
14 ( 5)1988
16 ( 4)3 ( 3)
19 ( 7)1989
15 ( 3)1 ( 0)
16 ( 3)1990
28 ( 7)3 ( 1)
31 ( 8)1991
38 ( 9)0 ( 0)
38 ( 9)1992
49 ( 20)
2 ( 1)
51 ( 21)1993
73 ( 26)
14 ( 8)87 ( 34)1994
120 ( 39)
17 ( 7)137 ( 46)1995
140 ( 61)28 (19)168 ( 80)Total
499 (170)
73 (43)572 (213)
The number of heterosexial cases is indicated inparentheses. Haemophiliacs are excluded.

CSWs who always live in the community andthose who come into the community fromoutside communities where HIV/AIDS is
epidemic. Individuals in Group 3 are assumed
to have sexual contact with those in Groups 2
and 4, and those in Group 4 to have sexual
contact with those in Groups 1 and 3. Figure 1 isa diagrammatic representation of this multigrouptransmission model.

Spread of HIV infection and occurrence of
AIDS patients
We assume that newly infected individuals
in Group 1 begin to transmit HIV to susceptible
persons in Group 2 or 4 through heterosexualintercourse. The probability of the infection ofsusceptible persons is a function of the numberof sex partners, the prevalence rate of HIV in
the group of the sex partners, and the probability of HIV transmission per sex partner. In
the same way, it is assumed that newly infected

The 1st stage of HIV/AIDS epidemic
At the 1st stage, there is
no HIV-infected person nor
AIDS patient in the community. It is assumed that someof the males who travel outof the community are infectedwith HIV through heterosexual intercourse with HIVinfected CSWs there, andGroup 1that they return to the community. The probability of
HIV infection is a functionof the number of CSWs withGroup 3whom they have had sexual

Sexual Workers

OutsideInsideGroup 2

Group 4

Fig. 1. Multigroup transmission model. HIV transmission among 4 groups. Group 1: highly sexually active
males; Group 2: highly sexually active females; Group 3: moderately sexually active males; Group 4:
moderately sexually active females. The arrows in the figure indicate HIV transmission. The arrow from the“Sexual Workers” to the “Group 2” indicates not only HIV transmission but also the inflow of commercial
sex workers from outside to inside.

85

S. Morio et al.

Table 3A. Summary of data: baseline estimates and ranges
Data

Baselineestimate

Range

Probability of HIV transmission
From male to female(person-year)
0.05 (mean) 0.012 (SD)
From female to male(person-year)
0.01 (mean) 0.0024 (SD)Incubation period of HIV
Median (Weibul distribution)(year)
10
Shape parameter (Weibul distribution) (year)
2.29
Pathogenicity rate(%)
85
Rate of continuance of infectivity(%)
50
Case fatality rate(/year)0.15
Mortality rate of health individuals(/year)0.001
Pathogenicity rate, rate of persons who develop AIDS among HIV-infected persons.

individuals in Group 2 or 4 transmit HIV tosusceptible persons in Group 1 or 3. In the community, the number of HIV-infected persons
gradually increases with the repetition of theseheterosexual (male-to-female and female-tomale) acts of intercourse.
Many HIV-infected persons graduallydevelop AIDS, and cease to transmit HIV tosusceptible persons. However, it is assumed
that some of them do not develop AIDS andcontinue their infectious status. The incidenceof AIDS among newly HIV-infected persons is

References

14, 15, 18, 41
14, 15, 18. 41
5, 31, 33
5, 31, 33
7, 22, 33, 44, 46
Assumed7, 42, 45
21

a function of the rate of persons who develop
AIDS among the infected persons (hereafter,“pathogenicity rate”) and the probability density which indicates development of AIDS after
an incubation period.
HIV-infected persons and AIDS patients dieof many causes, most of them related to AIDS.
We assume that the mortality rate of HIVinfected persons is the same as the rate of susceptible persons, but that the rate in AIDS patients is much higher than that in susceptible
persons.

Table 3B: Summary of data: baseline estimates and ranges
Data
Population ofHighly sexually active male*
Highly sexually active female*Moderately sexually active male*Moderately sexually active female*Individuals of G1 who travel abroad*Individuals of G1
who have sexual intercourse abroad*
Prevalence rate of HIV among CSWs abroad
Rate of foreign CSWs
among CSWs in a country*
Prevalence rate of HIV among foreign CSWs
Frequency of sexual intercourse
Between G2 and G1 (person/month)*
Between G3 and G2 (person/year)*
Between G1 and G4 (person/year)*
Between G3 and G4 (person/year)**Triangular fuzzy number.
CSW, commercial sex worker.

Baselineestimate

Range

References

300,000
10,0004,200,000
4,090,000
20,000150,000 – 600,000
5,000 – 15,000
3,900,000–4,350,000
4,085,000–4,095,000
10,000 – 30,000

21, 2521, 2521, 2521, 25
39

(%)
(%)

10
20

5–15

25
4, 36, 50

(%)
(%)

5
5

3–10

25
28

20
6
0.01
0.001

10–25
3–12
0.005–0.02
0.0005–0.002

25, 2825, 28AssumedAssumed86

Fuzzy mathematical model of HIV/AIDS epidemic

Through these processes, HIV/AIDS becomes epidemic in the community. (Mathematical formulas are shown in the Appendix.)

Phillips et al., 1992; Selwyn et al., 1992; Stoneet al., 1992; The Italian Seroconversion Study,
1992; Chang et al., 1993; Kamakura et al.,
1993; Kihara et al., 1993; Japan Health andWelfare Statistics Association, 1994). Where
data were incomplete, we used a fuzzy number.
The interval of confidence of the fuzzy number
was estimated by a Delphi survey, experts’opinions and project team consensus. In thismodel, the unit time is assumed to be a year,
because we compare the calculated results ofthe model with the reported number of HIV/AIDS cases in the surveillance, and thesenumbers are combined into an annual total.
Furthermore, there are some other data which
are reported as annual statistics.

Assumed countermeasures
We assume 4 types of countermeasures, as
follows: (a) spread of the usage of condoms
among males who travel outside of the community; (b) spread of the usage of condoms
among CSWs in Group 2; (c) spread of theusage of condoms among males in Groups 1
and 3 and (d) decrease of the number of CSWs
from other communities. The speed of the
spread of condom usage and of the decrease of
the number of CSWs is assumed to be either
rapid or slow. Consequently, there are 8 scenarios, which incorporate rapid or slow changeof the 4 factors affecting the HIV transmission
route. The outcome of each of these countermeasures is evaluated by measuring the numberof AIDS cases generated in the model. The decrease of frequency of heterosexual intercourse
among persons at risk is a preventive strategy
against HIV/AIDS epidemic, and similar to the
spread of condom usage. However, it is known
that the latter is more effective than the former,
and we assumed the spread of condom usage inthis model (Mann, 1994). (Mathematical formulas are shown in the Appendix.)
The community assumed in this model
simulates the young population of a metropolitan area (Tokyo, Kanagawa, Saitama andChiba) in Japan. The reported numbers of HIVinfected persons and AIDS patients, anddemographical and social data in this area areincorporated in order to make simulation withthe model realistic and to enhance the evaluation of the effect of each countermeasure.

Size of population
The number of persons aged 20–39 years isabout 8,600,000 (4,500,000 males and 4,100,000
females) in the metropolitan areas mentioned
above (Japan Health and Welfare Statistics Association, 1994). We used these numbers. The
numbers of individuals in Groups 1, 2, 3 and 4
were not studied directly. One Delphi survey
estimated that about 20% of young males (25thand 75th percentile; 10% and 30%; respectively) have several sexual partners in a year, andthat the number of female CSWs is about 10,000(3,500 and 15,000, respectively) in the metropolitan area (Kamakura et al., 1993). We used thetriangular fuzzy number (TFN) for these parameters (Table 3B).
It is known that about 2,500,000 Japanesemales aged 20–39 years travel abroad in a year
(Ministry of Justice in Japan, 1991). We estimated the number of individuals in Group 1
who travel abroad by extrapolation of the following 2 rates against this value: the rate ofGroup 1 to the total of Groups 1 and 3, and therate of the number of males in the metropolitan
area to all Japanese males. This estimated TFNof the number of individuals in Group 1 who
travel abroad was (10,000, 20,000, 30,000).
We assumed that the rate of CSWs fromoutside communities entering Group 2 was notconstant, that this rate rapidly increased in theearly stages of the HIV/AIDS epidemic, and thengradually decreased with course of the epidemic(see Assumed countermeasures above).

Data

We collected and synthesized data from published and unpublished sources (Tables 3A and3B) (Lui et al., 1988; Bacchetti and Moss, 1989;
Jason et al., 1989; Anderson et al., 1991; Kuo etal., 1991; Ministry of Justice in Japan, 1991;
Padian et al., 1991; Weniger, et al., 1991; European Study Group on Heterosexual Transmission of HIV, 1992, 1994; Haverkos et al., 1992;
87

S. Morio et al.

Sexual activity
Since the beginning of the epidemic of HIV/
AIDS, many studies of sexual activity havebeen carried out in Western countries (DeBuono et al., 1990; ACSF investigators, 1992;
Johnson et al., 1992; Mann, 1992a; Ward et al.,
1993; Lemp et al., 1994). However, it is not
appropriate to use these data in this study,
because the sexual activity within one community is quite different from that in another community, and this difference may be especiallygreat when cultures differ.
Studies of the sexual activity of lay people
or CSWs in Japan are very few. One study disclosed that 50% of young Japanese males who
had stayed abroad more than 2 years had theexperience of having sexual intercourse withsome unspecified number of females during
their stay (Jitsukawa and Okamoto, 1994). The
before-mentioned Delphi survey estimated thatabout 25% (10% and 40%) of young Japanesemales who travel abroad have sexual intercourse with native females there (Kamakura etal., 1993). We assumed that about 10% of
individuals in Group 1 who travel abroad would
have sexual intercourse with a CSW there
without condom usage. This parameter was
also assumed as the TFN (5%, 10%, 15%).
Another study revealed that the mean numberof customers per foreign female CSW in amonth was 18.8 ± 23.8 (mean ± SD) (Kihara etal., 1993). We assumed that about one female
sexual worker, independently of her nationality,
has about 20 customers in a month. This parameter (person/month) was assumed as the TFN(10, 20, 25).

3(*1000)Incidence of AIDS

4

2

1

0

25

50

In addition to these parameters, there aresome sex parameters with which we must
estimate real values, i.e. the frequencies ofsexual intercourse between individuals inGroups 2 and 3, Groups 1 and 4 and Groups 3
and 4. However, we could not find any studies
giving estimates of these values. We assumed
these values as the TFN, and estimated theirmagnitudes based on experts’ opinions andproject team consensus. The assumed TFNs forthe frequency (person/year) of sexual intercourse between individuals in these Groups areshown in Table 3B.
We assumed that there was no use of condoms at the early stage of the HIV/AIDS epidemic, and that the use of condoms either
rapidly or slowly increased with the epidemic(see Assumed countermeasures on p. 87).

HIV transmission, pathogenicity and mortality
The probability of HIV transmission throughsexual intercourse is one of the parameterswhich are difficult to estimate. Earlier studies
estimated this value as the probability per
sexual act, while more recent studies have estimated it as the probability per sexual partner or
person-year. We used the results of more recent
studies, and assumed that this value was expressed as the probability per person-year. We
further assumed that this value could be expressed with the normal distribution: the mean
± SD of the probability of male-to-femaletransmission and female-to-male transmission
was 0.05 ± 0.012 and 0.01 ± 0.0024/person-year,
respectively (Padian et al., 1991; European
Study Group on Heterosexual Transmission ofHIV, 1992, 1994; Haverkos and Battjes,
1992).
We assumed the Weibull distribution, and set 10 years as the median of
the incubation period and 2.29 years as
the shape parameter (Lui et al., 1988;
Bacchetti and Moss, 1989; Kuo et al.,1991). According to these values, theestimated proportion of individuals
Male
who develop AIDS within 5 years is
Female0.15, and that within 10 years is 0.53.

Time (year)

Fig. 2. Basic model (Models A and B).

88

Fuzzy mathematical model of HIV/AIDS epidemic

Table 4. Reported and calculated incidencesof AIDS cases by heterosexual infection by
sex and year
Year (i)Reported AIDS Calculated AIDS
Male Female Male Female

1981 ( 0)1982 ( 1)1983 ( 2)1984 ( 3)1985 ( 4)1986 ( 5)1987 ( 6)1988 ( 7)1989 ( 8)1990 ( 9)1991 (10)1992 (11)1993 (12)
Total

0
0
0
0
0
0
1
4
3
7
8
16
31
70

0
0
0
0
0
0
4
2
1
1
0
0
9
17

0
0
0
0
0
0
1
2
4
9
15
25
38
94

and 2). By project team consensus, we assumed
that most of these AIDS patients were illegal
CSWs from foreign countries, and that theyeither returned voluntarily or were returned totheir home country because it became impossible for them to continue working for various
reasons. This assumption was incorporated into
the program of the mathematical model.
The program of this mathematical model iswritten in C language, and executed with aUNIX computer. The source program is available upon written request.

0
0
0
0
0
0
0
0
1
1
1
2
3
8

Results
Basic Model

The simulation of the mathematical model
without any countermeasure showed bellshaped epidemic curves of male and female
AIDS cases (Model A, Fig. 2). However, theamplitude of the epidemic curve of the male
AIDS cases was much bigger than that of thefemale AIDS cases. The 1st case of AIDS occurred 4 years after the 1st case of HIV infection, and the epidemic of AIDS continued forabout 40 years. (We assumed that the end of theepidemic was the point in which the incidenceof HIV-infected persons became almost constant.) At the early stage of the epidemic, thecalculated incidence of male and female AIDS
cases was similar to the reported incidence of
AIDS cases, respectively (Table 4). The cumulative numbers of HIV-infected persons and
AIDS patients by the end of the epidemic wereabout 283,900 (male: 275,619; female: 8,279)
and 39,200 (male: 38,241; female: 1,066), respectively. The AIDS epidemic curves by Groups
showed that the HIV/AIDS epidemic occurredin Groups 1 and 2 at the 1st stage, and that the
epidemic spread into Groups 3 and 4 at the nextstage. The AIDS epidemic in Groups 1 and 2
started in the 6th and 8th years, respectively,
and ended in the 36th and 33rd years,
respectively; the epidemic in Groups 3 and 4
started in the 11th and 25th years, respectively,
and ended in the 46th and 48th years,
respectively. The incidence rate of AIDS in

It is known that the “pathogenicity rate” of
AIDS is relatively high in comparison to otherviral infections. The Delphi survey estimatedthat the “pathogenicity rate” of AIDS was about
85%, so we assumed the rate of 85% in themodel. It is also known that the case fatality
rate of AIDS patients without proper treatment
is very high (Lui et al., 1988; Jason et al., 1989;
The Italian Seroconversion Study, 1989; Selwyn
et al., 1992; Chang et al., 1993). We assumed
that about 85% of the patients died of AIDSrelated diseases within 5 years after the onset of
AIDS, and that the annual case fatality rate isstable within 5 years. The annual mortality rateof members of the 4 groups was assumed to be
the same as that of Japanese individuals aged
20–39 years, i.e., 0.001/year (Japan Health andWelfare Statistics Association, 1994).

Additional assumptions
In Japan, the time trend of the reported
number of heterosexually HIV-infected persons
has not been the same as that of heterosexual
AIDS patients. The reported number of heterosexually HIV-infected foreign females increased from 1987, the beginning of surveillance forHIV-infected persons, to 1992, while the reported number of heterosexually infected foreign
female AIDS patients did not increase (Tables 1
89

S. Morio et al.

3(*1000)Incidence of AIDS

4

2

1

Male
Female02550Time (year)

Fig. 3. Model C1.

3(*1000)Incidence of AIDS

4

2

1
Male
Female02550Time (year)

Group 2 was the highest in the 4 Groups;
the rate at its peak was about 1,000/105 .
The rate at the peak in Groups 1, 3 and 4
was about 900, 60 and 0.08/105, respectively.
The ratio of the calculated number andthe reported number of HIV-infected
persons at the early stage of the epidemic
was 0.20 (658/3373). This ratio indicatesthe rate of the reported HIV-infected
persons in all HIV-infected persons (hereafter “capture rate”) in an HIV/AIDS
surveillance.
The result shows that HIV spreads
among highly sexually active males andfemales at the 1st stage of HIV/AIDS
epidemic, that many of the highly sexually active males and females are infectedwith HIV during the epidemic, and that alow, but constant, number of AIDS cases
occur among moderately sexually active
males and females for a long time after the
epidemic.

Fig. 4. Model C2.

Comparison between fuzzy and nonfuzzy models

8(*1000)Incidence of AIDS

10

6

4

2

Male
Female02550Time (year)

Fig. 5. Model D1.

20(*1000)Incidence of AIDS

25

15

10
Male

5

Female02550Time (year)

Fig. 6. Model D2.

90

In order to study the effects of the fuzzy
mathematical model, we compared theresults obtained with the fuzzy and nonfuzzy models. In the non-fuzzy model,
parameters which are the TFNs in the
fuzzy model were treated as ordinary
numbers. The baseline estimates of the
TFNs are transformed to the ordinary
numbers.
The simulation of this non-fuzzymodel also showed bell-shaped epidemic
curves of male and female AIDS cases(Model B). The cumulative numbers of
HIV-infected persons and AIDS patientsby the end of the epidemic were about
254,600 (male: 246,511; female: 8,078) and
35,600 (male: 34,560; female: 1,051),
respectively. The AIDS epidemic curves
by Groups in the non-fuzzy model were
almost the same as those in the fuzzymodel.

Fuzzy mathematical model of HIV/AIDS epidemic

8060(*1000)Incidence of AIDSThe result shows that the effect of the
fuzzy model is small in simulation, andthat the non-fuzzy model can be usefulwhen highly reliable baseline estimates areavailable. However, in this study, allbaseline estimates may not be reliable, andusing only a non-fuzzy model is doubtful.

40

20
Male
FemaleModels which includes assumed
countermeasures

0

25

50

Time (year)(*1000)Female25

50Time (year)(*1000)Incidence of AIDS

Incidence of AIDS

Male

Male
Female25

50Time (year)(*1000)Incidence of AIDS

The models under each of the 8 scenarios Fig. 7. Model E1.
were: rapid (Model C1) or slow (Model
C2) spread of condom usage among males
200
who travel out of the community; rapid(Model D1) or slow (Model D2) spread of150
condom usage among CSWs in Group 2;
rapid (Model E1) or slow (Model E2)100
spread of condom usage among males inGroup 1 and 3; and rapid (Model F1) or
50
slow (Model F2) decrease in the numberof CSWs from other communities.
0
In Models C1 and C2, the annual rateof the spread of condom usage among
males who travel outside of the commu- Fig. 8. Model E2.
nity was assumed to be 1.25/year and4
1.05/year, respectively, and the rate in
Models D1 and D2 was also assumed to3
be 1.25/year and 1.05/year, respectively.
In Models E1 and E2, respectively, the2annual rate of spread of the usage ofcondom was assumed to be 1.01/year and1
1.002/year. These 2 values (1.01/year and1.002/year) are values reflecting the0annual rates in Models D1 and D2 adjusted by the ratio of population size ofGroups 2 and 3 (10,000/4,200,000). In Fig. 9. Model F1.
Models F1 and F2, the annual rate ofdecrease of the number of CSWs from4other communities was assumed to be
0.75/year and 0.95/year, respectively.
3
All 8 models generated bell-shaped
2epidemic curves of male and female AIDS
cases. The amplitudes of the epidemic
1
curves of male AIDS cases were muchbigger than those of female AIDS cases
0(Figs. 3–10). Among the 8 models,Model D1, rapid spread of condom usage
among CSWs in Group 2, showed the Fig. 10. Model F2.
91

Male
Female25

Time (year)

50

S. Morio et al.

Table 5. Cumulative numbers of HIV-infected persons and AIDS patients by models
Model

Peak ofepidemic

Total

HIV
Male

FemaleTotal

A
29283,900
274,700
8,200
39,200B
27254,600
246,500
8,000
35,600C129
28,300274,700
8,200
39,200C2
29
283,900
274,700
8,200
39,200
D1
14
1,200
1,100
100
150
D2
15
2,100
2,000
100
290
E1
19
6,800
6,400
400
920
E2
25
21,00020,000
1,000
2,730
F1
29
283,900
274,700
8,200
39,200F2
29
283,900
274,700
8,200
39,200
Values shown in the table are the nearest rounded whole numbers.

biggest effect as a countermeasure. The cumulative numbers of HIV-infected persons and AIDS
patients by the end of the epidemic were about
1,200 (male:1,096; female:113) and 150(male:134; female:19), respectively. The ratiosof these cumulative numbers of HIV-infected
persons and AIDS patients in Model D1 toModel A (the basic model) were both 0.004.
Model D2, slow spread of condom usage
among CSWs in Group 2, showed the 2nd biggest effect, and Model E1, rapid spread ofcondom usage among males in Groups 1 and 3,
showed the 3rd biggest effect, followed byModels E2, C1, C2, F1 and F2 in order of themagnitude of the effect (Table 5).
The results of these 8 models showed thatprevention of the inflow of HIV from outsidecommunities is not an effective countermeasure
once the spread of HIV among highly sexuallyactive groups has taken place, and that the decrease of HIV transmission through heterosexual intercourse within the community is themost effective countermeasure, even if the
inflow of HIV from outside communities does
not decrease.

AIDS
Male

Female38,20034,60038,20038,200
130
260
870
2,600
38,20038,200

1,000
1,000
1,000
1,000
20
30
50
130
1,000
1,000

through heterosexual intercourse. Even if the
inflow of HIV from outside communities continues through male travelers and foreign female
CSWs, the most effective countermeasures in
an isolated community is decreasing HIV transmission through heterosexual intercourse within the community. This means that the promotion of the use of condoms among CSWs andtheir customers, detection of HIV-infected
CSWs, and promotion of having steady sex
partners are highly effective preventive methods.
These preventative measures have already been
carried out in many countries (Chin and Mann,
1990; Dumasia, 1990; Weniger et al., 1991;
Mann, 1992b, 1992c; Thomas et al., 1993). However, there are also many other countermeasuresagainst the HIV/AIDS epidemic, some of whichhave been carried out without objective evaluation. It is difficult to show the quantitative
effect of a single countermeasure, in comparison to other countermeasures, even if theincidence of new HIV-infected persons or AIDS
patients suddenly decreases in association withthe expansion of HIV/AIDS countermeasures.
Thus, the effects of some relatively poorly
effective countermeasures may be overestimated. For example, in Japan, the high risk of HIV
infection through sexual intercourse with
CSWs in foreign countries has been widely announced, and leaflets/pamphlets to this effectare distributed to travelers abroad. However,there is no governmental program to promote
the usage of condom among CSWs and their

Discussion
This quantitative study disclosed that decreaseof HIV transmission is the most effective countermeasure against HIV/AIDS epidemic in an isolated community where HIV is mainly spread
92

Fuzzy mathematical model of HIV/AIDS epidemic

customers. Countermeasures targeting travelers abroad are among the preventive countermeasures, but should not be pursued without
countermeasures designed to decrease the HIVtransmission through heterosexual intercourse
within the community.
Many mathematical models have beendeveloped since the inception of the HIV/AIDS
epidemic (Knox, 1986; Anderson, 1988, 1989;
De Gruttola and Mayer, 1988; Dietz andHadler, 1988; Gupta et al., 1989; Van Druten etal., 1990; Jager and Ruitenberg, 1992), but inonly a few, the effectiveness of different countermeasures has been evaluated (Anderson, 1989;
Van Druten et al., 1990; Jager and Ruitenberg,
1992). In many models, the effectiveness of
only a single countermeasure is monitored,
rather than comparison of various countermeasures. We are aware of no model in which
the effectiveness of efforts to prevent the inflow
of HIV from outside is compared with that ofefforts to prevent the transmission of HIV within the community. Presently, foreign CSWs area major infectious source of the HIV/AIDS epidemic, not only in Japan but also many othercountries as well. This “open” model, which
includes the inflow of HIV through foreign
CSWs, is quite useful in decision-making related
to countering the HIV/AIDS epidemic.
However, the comparison of effectivenessusing a mathematical model is not easy, because the solutions yielded are largely affectedby the values of parameters entered into themodel. The more realistic the algorithm becomes, the more uncertain the parameters are.
In many models, instead of a complex algorithm which includes uncertain parameters, a
simple algorithm which does not include theseparameters is used. In this study, we selected acomplex algorithm which includes uncertain
parameters, because simulation with more
realistic models is known to yield more useful
information for decision-making in countering
the HIV/AIDS epidemic. The results of thisstudy are almost certainly affected by the valuesof the parameters; when the parameteric values
are changed, the relative effectiveness of various countermeasures can be changed; at least in
simulation using the model. In this study, inorder to compensate for this defect, we used thefuzzy theory to generate the parameteric values.
The application of fuzzy mathematicalmodels in engineering and industry has beenfavorably evaluated (Jain, 1976; Prade, 1979;
Kaufmann and Gupta, 1991). However, it isdifficult to evaluate their application in publichealth, because of the impossibility of testing
the results yielded by the fuzzy models inexperiments. In general, the mathematicalmodels used in public health have uncertain
effect and include uncertain values which aredependent on human behavior. The fuzzymodel is one of the newest methods used to
calculate these uncertain values in mathematical models. In this study, we incorporated
the results of the Delphi survey, which areindicated in an actual-values range and can beinterpreted as TFNs, into the fuzzy model. We
used the fuzzy model to make maximum use of
the results of the Delphi survey. The combination of the fuzzy model with the Delphi
survey results may be an effective model for use
in the public health field.
The basic model generated a bell-shaped
epidemic curve of AIDS cases. Epidemic
curves with this shape have been observed inepidemics of some other infectious diseases,
and mathematical models of epidemiology ofthese diseases have yielded theoretical epidemic curves which were similar to the observed epidemic curves (Kermack and Mckendrick,
1927; Morio et al., 1985; Bailey, 1986; Cvjetanovic
et al., 1986). The HIV/AIDS epidemic is still inprocess, and thus the entire epidemic curve of
HIV/AIDS cannot be known. However, someother mathematical models have also yielded abell-shaped theoretical epidemic curve of AIDS
cases (Anderson, 1988; Koopman et al., 1991).
The epidemic curves of AIDS cases in thesestudies and ours indicate that the incidence of
AIDS patients exponentially increases at theearly stage of the epidemic, begins to decrease
after many susceptible persons are infected,
then decreases to a low point and becomes almost constant.
The incidence at the end stage of the epidemic curve is almost null. It may not be acceptable that the incidence of a sexually trans93

S. Morio et al.

mitted disease (HIV/AIDS in this case) becomes null. This is a result of using a static
population, if we use a dynamic population theincidence at the end stage of the epidemic curve
must be a constant value other than zero. However, the aim of this study is to perform a quantitative comparison among 8 scenarios, so there
is no reason to use a dynamic population and to
make the model more complex. We used a
simple static population.
Although the bell-shaped epidemic curve
was shown in both males and females, in thisstudy, the basic model disclosed a marked
difference in incidence for male and female
AIDS patients. There are 2 explanations forthis difference by sex, which has not been described in other studies of HIV/AIDS using
mathematical models. First, in the basic model,
the population size of highly sexually active
males is assumed to be much bigger than that ofhighly sexually active females (300,000:
10,000), and this assumption, of course, leads toa substantial difference between the 2 groups inincidence. Second, it is assumed that HIVinfected highly sexually active females, who
are from outside of the community and develop
AIDS, return to their home countries without
being counted as AIDS patients; this assumption also decreases the incidence of female
AIDS patients.
The epidemic curves by Groups generated
in the basic model showed that the HIV/AIDS
epidemic begins in Groups 1 and 2, and thenspreads into Groups 3 and 4 at the next stage.
These epidemic curves indicate that the HIV/AIDS epidemic in the community will be small
if the epidemic in Groups 1 and 2 can be prevented. These epidemic curves are also consistent with the before-mentioned results obtainedwith the assumed countermeasures, in that theresults for D models (assumption of the rapid/
slow spread of the usage of condoms among
CSWs), and E models (assumption of the rapid/
slow spread of the usage of condoms among
males) disclosed relatively high effectiveness.
The simulation with the basic model yielded
not only the estimated incidence (and prevalence) of AIDS cases but also the prevalence ofHIV-infected persons. The reported number of

AIDS cases is one of the indices of the magnitude of the HIV/AIDS epidemic. In many
countries, the surveillance of AIDS cases iscarried out in order to estimate HIV/AIDS
epidemic. However, AIDS occurs after a longincubation period, and the incidence (or prevalence) of AIDS is an index of the HIV epidemic
about 10 years before (the median of the incubation period). Also, AIDS patients are seldom
capable of transmitting HIV to susceptible
persons because they are sick. The prevalence(or incidence) of HIV-infected persons, who do
not have symptoms but are capable of transmitting HIV to susceptible persons, is a more
appropriate index of the magnitude of the HIV/AIDS epidemic. In some countries, including
Japan, even though the surveillance of HIVinfected persons is also carried out, the usage ofthese data is limited because there is no information about the capture rate (as mentioned
before, the rate of the reported HIV-infected
persons among all HIV-infected persons). Even
when an increase in HIV prevalence is observed, it cannot be determined whether the prevalence of HIV has actually increased or whether
the capture rate has merely become higher.
In this study, at the early stage of the HIV/AIDS epidemic, the calculated incidence of
AIDS in the model is similar to the reported
incidence of AIDS, and the calculated prevalence of HIV may be similar to the actual
prevalence of HIV. The comparison between
the calculated prevention of HIV and the actual
prevalence of HIV is also desired, but very
difficult, because it is almost impossible to get
the actual prevalence of HIV. In this study, thecapture rate is used to estimate the reliability of
the calculated prevalence of HIV. The capture
rate calculated from the prevalence of the reported and of the calculated HIV was 0.20.
This value is similar to the capture rate which
was calculated by the “back-calculation method”
with data regarding AIDS in Japan (Hashimoto
et al., 1993), and makes the data of HIV
surveillance more useful.
This mathematical model simulates an epidemic of HIV/AIDS transmitted solely throughheterosexual intercourse, but not through homosexual intercourse or intravenous drugs. Be94

Fuzzy mathematical model of HIV/AIDS epidemic589.
5 Bacchetti P, Moss AR. Incubation period of
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cause a mathematical model which includes not
only heterosexual intercourse but also
homosexual intercourse and intravenous drug
usage is very complex, we cannot get reliabledata for parameters in a model. This simulation
based exclusively on heterosexual transmission
may limit the applications of this study. However, among the many HIV transmission routes,
heterosexual HIV transmission is the one forwhich the effectiveness of preventive methods
is most difficult to evaluate, for a number ofreasons: the population at risk is large, mobility
makes follow-up difficult, and it is difficult toacquire exact information regarding sexual
activity. In many countries, however, theepidemic seems to involve primarily heterosexual transmission. The algorithm used in thisstudy can be applied to data acquired in many
countries, and the fuzzy model can be used evenwhen adequate data are not available. We thinkthat this fuzzy model is one of the tools in identifying effective countermeasures against the
HIV/AIDS epidemic with potential applications
in countries where heterosexual transmission of
HIV/AIDS is a major issue.

Acknowledgments: We thank Dr. Shigesato Takahashi(Dept. of Population Dynamics, Inst. of Population
Problems, Ministry of Health and Welfare), Dr.
Yukihiro Matsuyama (Research Inst., Sumitomo
Insurance Co.) and Dr. Naoyuki Okamoto (Dept. of
Epidemiology, Clinical Research Inst., Kanagawa
Prefectural Cancer Center) for their valuable suggestions.

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Appendix: Mathematical Formulas
The 1st stage of HIV/AIDS epidemic
At time t, the number of susceptibles, infectives and AIDS patients in Group i (i = 1, 2, 3, 4) is xi(t), yi(t), zi(t),respectively. The number of individuals in Group i is ni(t), which includes the number of susceptibles,
infectives and AIDS patients. The number of infected males in Group 1 who return to the community from theoutside at time t (t = 0) is described by,
y1 (0) = T1F × w(0),where, T1F is the transmission parameter from groups of females CSWs to Group 1 (subscripts “1” and “F”
means Group 1 and females, respectively), and w(t) is the number of individuals in Group 1 who go theoutside of the community. The transmission parameter (T1F) is a complex combination of other underlying
parameters,
T1F = β1F × k1F × f 1F,
where,

f 1F = proportion of individuals in Group 1 who have sexual intercourse with females outside of the group, theFuzzy number,k 1F = mean number of females with whom there is sexual contact at the unit time by an individual in Group 1,
β1F = probability of infection from an infected females to a susceptible in Group 1 per partnership, the normal
distribution function.
The number of infected foreign females in Group 2 who enter the community from the outside at time t (t = 0)
is described by,y2 (0) = ρ(0) × v(0),where, ρ(t) is the prevalence rate of HIV among CSWs outside the community, and v(t) is the number of
CSWs who enter the community from the outside.

The HIV/AIDS epidemic at time t in Group 1
From time t to t + ∆t, the change of the number of susceptibles to HIV in Group 1 is described by,
x1 (t + ∆t) – x1 (t) = – T1F × x1 (t) × (y 2(t)/n 2(t)) – T1F × x 1(t) × (y4(t)/n4(t)) – T1F × w(t),where, (y2(t)/n2(t)) and (y4(t)/n4(t)) is the proportion of HIV infectives in Groups 2 and 4, respectively.
The change of the numbers of infectives and AIDS patients in Group 1 are described by,t

y1 (t + ∆t) – y1(t) = T1F × x1 (t) × (y2(t)/n2(t)) + T1F × x1(t) × (y4(t)/y4(t)) +T1F × w(t) –t

z1 (t + ∆t) – z1 (t) =00

γ y1(u)du,γ y1(u)du – z 1(t),twhere, γ is the probability of developing AIDS in infected individuals (the Weibul distribution), γ y1 (u)du is0
the number of AIDS patients who was infected with HIV at the time 0 and manifested AIDS by the time t, andδ is the case fatality rate of AIDS patients.

97

S. Morio et al.

The HIV/AIDS epidemic at time t in Groups 2, 3 and 4
In Group 2, x 2(t), y2(t) and z2 (t) are described by,
x 2(t + ∆t) – x2(t) = – T2M × x2(t) × (y1 (t)/n1 (t)) – T2M × x2 (t) × (y 3(t)/n 3(t)),
ty 2(t + ∆t) – y2(t) = T2M × x2 (t) × (y1(t)/n1(t)) + T2M × x2 (t) × (y 3(t)/n3(t)) –tz 2(t + ∆t) – z2(t) =00γ y2 (u)du,γ y2(u)du – δz2 (t),where, T2M is the transmission parameter from groups of males to Group 2 (subscripts “2” and “M” means
Group 2 and males, respectively).
In Groups 3 and 4, x3(t), y3 (t), z3(t), x4 (t), y4(t) and z4 (t) are described by the formulas similar to them inGroup 2.

Assumed countermeasures
At time t, under the condition of widespead usage of condoms, x1 (t) and y1(t) are described by,
x 1(t + ∆t) – x1(t) = – ε1(t) × T1F × x1(t) × (y2(t)/n2(t)) – ε1(t) × T1F × x1(t) × (y4(t)/n4 (t)) – ζ(t) × T1F × w(t),y 1(t + ∆t) – y1(t) = ε1(t) × T1F × x1 (t) × (y2(t)/n2(t)) + ε 1(t) × T1F × x1(t) × (y4(t)/n4 (t)) + ζ(t) × T1F × w(t)t–0

γ y1(u)du,

where, ε1(t) is the rate of condom usage in Group 1 when in the community, and ζ(t) is the rate of condom
usage in Group 1 when outside the comunity.
In Group 2, x2(t) and y2 (t) are described by,
x 2(t + ∆t) – x2(t) = – ε2(t) × T2M × x2 (t) × (y1 (t)/n 1(t)) – ε2(t) × T2M × x2 (t) × (y3(t)/n3(t)),
y 2(t + ∆t) – y2(t) = ε2(t) × T2M × x2 (t) × (y1(t)/n1(t)) + ε2 (t) × T2M × x2(t) × (y3(t)/n3 (t))
t–0γ 0y2(u)du,

where, ε2 (t) is the rate of condom usage in Group 2 when in the community. In Groups 3 and 4, x3 (t), y3(t),x4(t) and y 4(t) are described by the formulas similar to them in Group 2.

(Received April 22, Accepted May 20, 1996)

98