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The effect of men who have sex with men (MSM) on the spread of sexually transmitted infections
Theoretical Biology and Medical Modelling volume 18, Article number: 18 (2021)
Abstract
Sexually transmitted infections (STIs) have remained a worldwide public health threat. It is difficult to control the spread of STIs, not only because of heterogeneous sexual transmission between men and women but also because of the complicated effects of sexual transmission among men who have sex with men (MSM) and mothertochild transmission. Many studies point to the existence of a ‘bisexual bridge’, where STIs spread from the MSM network via bisexual connections. However, it is unclear how the MSM network affects heterosexual networks as well as mothertochild transmission. To analyse the effect of MSM on the spread of STIs, we divided the population into four subpopulations: (i) women, (ii) men who have sex with women only (MSW), (iii) men who have sex with both men and women (MSMW), (iv) men who have sex with men exclusively (MSME). We calculated the typereproduction numbers of these four subpopulations, and our analysis determined what preventive measures may be effective. Our analysis shows the impact of bisexual bridge on the spread of STIs does not outweigh their population size. Since MSM and mothertochild transmission rates do not have a strong synergistic effect when combined, complementary prevention measures are needed. The methodologies and findings we have provided here will contribute greatly to the future development of public health.
Introduction
The relationship between men who have sex with men (MSM) and sexually transmitted infections (STIs) has been extensively studied because the infection rates of various STIs in MSM are greater than those in women and men who have sex with women only (MSW) [1]. MSM have been widely studied in research on HIV; however, MSM were exposed to many STIs before the spread of HIV [2]. The term MSM has been used since 1990 to reflect the epidemiological belief that behaviour is responsible for the risk of STIs [3]. MSM are defined only by their behaviour, not their sexual identity. Many studies have estimated the proportion of MSM among the male population in various countries [4,5,6,7,8,9,10,11,12] (Table 1). According to these studies, the proportion of MSM among the male population is approximately 1% to 5% in many countries. MSM should be classified into two subpopulations: men who have sex with men exclusively (MSME) and men who have sex with men and women (MSMW). In the research studies shown in Table 1, MSME and MSMW are not distinguished; thus, the proportions of MSME and MSMW are unknown.
When we consider the persistence of STIs in human society, transmission routes are key factors [13, 14]. Hepatitis B virus (HBV) and human Tcell leukaemia virus type I (HTLV1) are endemic STIs that are spread by sexual (horizontal) transmission and mothertochild (vertical) transmission. What makes these viruses so persistent is that vertically infected infants can develop to reproductive age and spread the infections through sexual transmission [15,16,17,18]. We suspect that the presence of STIs, which have robust intragenerational routes of transmission, combined with mothertochild transmission, which allows transmission across generations, have contributed significantly to the persistence of these viruses. If we can mathematically estimate the spreading efficiency of each of these transmission routes, we can understand persistence strategies in the context of the ecological and evolutionary biology of STIs. This will also help us develop vaccination strategies for HBV, for which a safe vaccine exists. To address the question of whether an STI can persist, we need to take a longterm view, and it is necessary to consider not only transmission due to MSM but also mothertochild transmission.
There are many mathematical models of STIs focusing on human sexual networks [19, 20]. It is well known that human sexual networks are heterogeneous; most people have sexual contact with only a few partners, but a small number of sexually active people have sexual contact with hundreds of partners. This heterogeneity is believed to contribute to the persistence of STIs [21]. In a simple epidemic model of a network that does not consider sex [22], the basic reproduction number \({R}_{0}\), which measures the number of infections produced by an infected individual on average, is determined by the fluctuation in sexual activity \(a\):
Here, \(a\) is the degree of sexual activity, which is proportional to the number of sexual contacts, and thus, \(C\) is the variation of \(a\) [19]. If we consider heterosexual contacts between men and women (i.e., the network is bipartite), \({R}_{0}\) is proportional to the geometric mean of the value \(C\) for men and women [19]. If the contact frequency distribution is a power distribution and the network is scalefree [23, 24], then \({R}_{0}\) and \(C\) can approach infinity. Many studies have noted that sexual contact networks are scalefree [25,26,27,28].
There are few models that consider the following two points simultaneously. The first point is mothertochild (vertical) transmission. Mothertochild transmission (e.g., via the placenta, birth blood exposure, breast feeding) is an important transmission route of STIs [14, 29, 30]. Therefore, in our previous studies, we built models that simultaneously considered sexual (horizontal) and mothertochild (vertical) transmission under the scalefree property of sexual contact frequency [31, 32]. In particular, a realistic model presented in our latest work took into account adult and infant mortality from infection, infertility and stillbirth caused by infection, as well as recovery with treatment [32]. Although many STIs can cause serious harm to infants infected via mothertochild transmission, HBV and HTLV1 do not immediately adversely affect infected infants, and infected infants can grow and spread the infection [33, 34].
The second point is how the MSM network indirectly affects mothertochild transmission through heterosexual networks via bisexual connections, which are called ‘bisexual bridges’ [35]. Bisexual and homosexual contacts play an important role in the spread of STIs [36]. Fernando argued that the current Centers for Disease Control (CDC) risk subpopulation classification, in which MSMW and MSME are included in a single MSM subpopulation, makes it impossible to know the extent of STI (e.g., HIV) transmission from MSMW to heterosexual women [37]. Thus, to model MSM accurately, we must divide MSM into men who have sex with men exclusively (MSME) and men who have sex with men and women (MSMW) because the effects of these two behaviours on public health are very different.
In this study, we simultaneously considered (1) the network heterogeneity of human sexual contacts, (2) mothertochild (vertical) transmission and (3) MSM (i.e., MSMW and MSME) to formulate typereproduction numbers; the typereproduction number is defined as an extension of the basic reproduction number, \({R}_{0}\). The typereproduction number rather than \({R}_{0}\) is required when the population is classified into several subpopulations according to epidemiological characteristics [38, 39]. Here, the typereproduction number for type \(i\), \({T}_{i}\), is the average number of secondary cases of type \(i\) produced by a primary case of type \(i\). Since \({T}_{i}<1\Leftrightarrow {R}_{0}<1\) regardless of type \(i\), a \({T}_{i}\) value less than one indicates that STIs will be eliminated. The spread of epidemics is prevented if we effectively vaccinate at least a fraction \((11/{T}_{i})\) of the susceptible target type [38]. We considered four subpopulations, women, MSMW, MSME, and MSW, and derived a formula to identify which types should be targeted for public health interventions. When it is possible to concentrate vaccination on a subpopulation, the target of vaccination is not necessarily determined by the relative sizes of the typereproduction numbers, because the cost of vaccines depends on the size of the subpopulation. However, since public health authorities do not know who is in which subpopulation, all they can do is to promote the vaccine and educate people about safe sex. It is difficult to focus promotion solely on MSME or MSMW, because they cannot be identified by the others. Therefore, even if the size of the subpopulation is small, the cost may not be so small. Thus, the typereproduction number can be a good indicator of public health.
Material and methods
The outline of the model is illustrated in Fig. 1. To clarify the effect of MSM, we adapt a susceptibilityinfectionsusceptibility (SIS) model where the population is divided into four subpopulations: \(\{\mathrm{w},\overset{\sim }{\mathrm{m}},\overline{\mathrm{m} },\mathrm{m}\}\): women, MSMW, MSME, and MSW, respectively. It is assumed that a type \(\overset{\sim }{\mathrm{m}}\) person has contact not only with types \(\overset{\sim }{\mathrm{m}}\) and \(\overline{\mathrm{m} }\) but also with type \(\mathrm{w}\), while a type \(\overline{\mathrm{m} }\) person has sexual contact with only types \(\overset{\sim }{\mathrm{m}}\) and \(\overline{\mathrm{m} }\). Here, we ignore sexual contact between women; thus, a person of type \(\mathrm{w}\) can have sexual contact with persons of only types \(\mathrm{m}\) and \(\overset{\sim }{\mathrm{m}}\). Furthermore, the effect of generational change is also taken into consideration. Here, \(B\) is the number of births per unit time, \(\delta\) is the rate of infertility or stillbirths, and a newborn individual is infected at a probability of vertical transmission rate α times the female infection rate. The proportion of types of births is \({\gamma }_{\mathrm{w}}:{\gamma }_{\mathrm{m}}:{\gamma }_{\overset{\sim }{\mathrm{m}}}:{\gamma }_{\overline{\mathrm{m}} }\). Since it is not known when and how sexual orientation is determined, this model assumes that the types are fixed at birth for simplicity. The natural death rate is \({\mu }_{i}\), and the death rate for infected persons is \({\mu }_{i}^{^{\prime}}\) for type \(i\in \{\mathrm{w},\overset{\sim }{\mathrm{m}},\overline{\mathrm{m} },\mathrm{m}\}\). Thus, in the equilibrium state in the absence of disease (in the case of \({I}_{i}\left(t\right)=0\), as we will see later), the population in each type is
The total number per unit time of sexual contacts between individuals of type \(i\) and individuals of type \(j\) is \({f}_{ij}\) for \(i\ne j\) and \(\frac{1}{2}{f}_{ii}\) when \(i=j\). Although not specified for simplicity, \({f}_{ij}\) is a function of \({N}_{i}\) and \({N}_{j}\). From the definition, \({f}_{ij}={f}_{ji}\). From the assumptions of the model, \({f}_{\mathrm{ww}}={f}_{\mathrm{mm}}={f}_{\mathrm{m}\overset{\sim }{\mathrm{m}}}={f}_{\overset{\sim }{\mathrm{m}}\mathrm{m}}={f}_{\mathrm{w}\overline{\mathrm{m}} }={f}_{\overline{\mathrm{m}}\mathrm{w} }={f}_{\mathrm{m}\overline{\mathrm{m}} }={f}_{\overline{\mathrm{m}}\mathrm{m} }=0.\)
We assume that each individual in type \(i\) has a sexual activity value of \({a}_{i}\), and the value of \({a}_{i}\) follows the distribution of \({p}_{i}({a}_{i})\), where the mean of sexual activity is set to one:
The number per unit time of sexual contacts of an individual is proportional to the individual’s sexual activity value \({a}_{i}\) (thus \({a}_{i}\) is dimensionless). It is thought that the amount of \({a}_{i}\) changes with age, but in the model, where age is ignored, it is assumed to be constant for each individual. Thus, this model assumes that the values of \({a}_{i}\) is fixed at birth. The fluctuation in sexual activity defined as Eq. (1) is given as follows:
It is assumed that sexual contact is well distributed, ignoring monogamy and marriage, where the rate at which individuals with \({a}_{i}\) in subpopulation \(i\) sexually contact someone in subpopulation \(j\) is \({a}_{i}{f}_{ij}/{N}_{i}\).
The infection dynamics are as follows: \({S}_{i}(t)\) and \({I}_{i}(t)\) represent the numbers of susceptible and infected individuals in subpopulation \(i\), respectively. Assuming a susceptibleinfectedsusceptible model (SIS model) without carriers of immunity, the number of individuals in subpopulation \(i\) is given as \({N}_{i}\left(t\right)={S}_{i}\left(t\right)+{I}_{i}(t)\). The numbers of susceptible and infected individuals in subpopulation \(i\), whose sexual activities comprise infinitesimal interval \([{a}_{i},{a}_{i}+d{a}_{i}]\), are denoted as \({S}_{i}(t,{a}_{i})d{a}_{i}\) and \({I}_{i}(t,{a}_{i})d{a}_{i}\), respectively. Thus, the numbers of susceptible and infected individuals in type \(i\in \{\mathrm{w},\overset{\sim }{\mathrm{m}},\overline{\mathrm{m} },\mathrm{m}\}\) are given in integral form as follows:
The dynamics of \({S}_{i}(t,{a}_{i})\) and \({I}_{i}(t,{a}_{i})\) for \(i\in \{\mathrm{w},\overset{\sim }{\mathrm{m}},\overline{\mathrm{m} },\mathrm{m}\}\) are as follows:
where the parameter \({\beta }_{j\to i}\) is the probability of transmission per sexual contact from a person in subpopulation \(j\) to a person subpopulation \(i\), the parameter \({\eta }_{i}\) is the cure rate, and \({\Theta }_{i}\left(t\right)/{N}_{i}\left(t\right)\) represents the probability that the sexual partners are infected:
Results
In the absence of infection (\({I}_{i}\left(t,{a}_{i}\right)=0\)), we obtain a stationary solutions of Eq. (7):
To calculate the typereproduction number, we consider the linear dynamics of the infected state, following the methods of the previous studies (for example [39, 40]). In other words, we linearize Eq. (8) near the diseasefree solution given by Eq. (10). Here, we use Eqs. (6) and (9) to derive the dynamics of \({I}_{i}(t)\) and \({\Theta }_{i}(t)\) for \(i\in \{\mathrm{w},\overset{\sim }{\mathrm{m}},\overline{\mathrm{m} },\mathrm{m}\}\):
Here, \({\tilde{f }}_{ij}={f}_{ij}\left({\tilde{N }}_{i},{\tilde{N }}_{j}\right)\). Thus, the situation with very few infected persons is represented by a linear differential equation system closed by eight variables. According to the traditional method of calculating the typereproduction number by Diekmann et al. [39], the Jacobi matrix \(J\) presented in Eq. (11) is divided into part \(T\), which is related to infection of the target type, and part \(Q=JT\), and the dominant eigenvalue of \({TQ}^{1}\) is calculated. Thus, we need to perform complicated algebraic operations on an eightdimensional matrix.
We use the network diagram method according to the work by Lewis et al. [41], which is an intuitive and easytounderstand method. Infected persons of each type are further divided into those by mothertochild (vertical) transmission and those by sexual (horizontal) transmission. Thus, there are eight infection states, and we consider their network (see Fig. 2). In Fig. 2, red arrows indicate that infected persons in the state at the end of the arrow are infected from infected persons in the state at the beginning of the arrow. Blue arrows indicate that infected persons in the state at the ends of the arrow are born from an infected mother in the state at the beginning of the arrow. The quantities near the arrows give the number of new infections born per unit time divided by the duration of the original infected state. \({R}_{ij}^{\mathrm{v}}\) and \({R}_{ij}^{\mathrm{h}}\) represent the average number of the original persons in subpopulation \(j\) infected through sexual transmissions from a typical vertically and horizontally infected person in subpopulation \(i\), respectively, as follows [32]:
where \({C}_{i}\) is the fluctuation of sexual activity \({a}_{i}\) in subpopulation \(i\) (see Eq. (4)), and \({b}_{i}\) represents direct vertical transmission as follows:
The network diagram in Fig. 2 is represented by the following transition matrix:
The dominant eigenvalue of the matrix \(A\) gives the basic reproduction number, but it cannot be given in an analytical form because the characteristic equation becomes a quartic equation. On the other hand, the typereproduction number can be written in a relatively simple formula. For example, if sexually infected females are the focus, the target matrix \(B\) has only nonzero entries \({B}_{2k}={A}_{2k}\) for \(1\le \mathrm{k}\le 8\). In this case, the typereproduction number is given by the dominant eigenvalue of the matrix \(B(IA+B)\) [39]. Since the rank of the matrix \(B(IA+B)\) is one, the calculation of eigenvalues is easy. The typereproduction number can be calculated using numericalanalysis software (the Mathematica source code is included in the Additional file 1). By performing some troublesome formula transformations, we obtain the typereproduction numbers of persons in the four subpopulations who are infected sexually as follows if the denominators are positive:
Here, \({A}_{i}\) represents the average number of persons in each subpopulation (\(i\in\left\{\mathrm{w},\mathrm{m},\overset{\sim }{\mathrm{m}},\overline{\mathrm{m}}\right\}\)) who are infected through consecutive vertical transmissions from a sexually infected adult woman:
Note that in Eq. (15), \({R}_{i\to j}^{v}\) are always multiplied by \({A}_{i}\), such as \({A}_{i}{R}_{i\to j}^{v}\). Therefore, no vertical transmission (\(\alpha =0\)) always makes \({A}_{i}{R}_{i\to j}^{v}=0\), even if \({R}_{i\to j}^{v}\) is positive. If the denominators in Eq. (15) are not positive, they are not well defined, which means that infectious diseases are not extinct although the infection rate for the type of interest can approach zero. In particular, if the STI increases in only MSM,
the infection cannot be controlled without measures for MSM. On the other hand, if Eq. (16) does not hold, it is difficult to eradicate infections by targeting only MSM.
Figure 3 illustrates some typical results, and we make additional assumptions as follows:

The total numbers per unit time of sexual contacts are given as
$$\begin{array}{c}{f}_{\mathrm{wm}}={f}_{\mathrm{mw}}={k}_{\mathrm{hetero}}{N}_{\mathrm{m}}, {f}_{\mathrm{w}\overset{\sim }{\mathrm{m}}}={f}_{\overset{\sim }{\mathrm{m}}\mathrm{w}}={k}_{\mathrm{hetero}}{N}_{\overset{\sim }{\mathrm{m}}},\\ {f}_{\overset{\sim }{\mathrm{m}}\overset{\sim }{\mathrm{m}}}={k}_{\mathrm{homo}}\frac{{N}_{\overset{\sim }{\mathrm{m}}}^{2}}{{N}_{\overset{\sim }{\mathrm{m}}}+{N}_{\overline{\mathrm{m}}} }, {f}_{\overline{\mathrm{m} }\overline{\mathrm{m}} }={k}_{\mathrm{homo}}\frac{{N}_{\overline{\mathrm{m}} }^{2}}{{N}_{\overset{\sim }{\mathrm{m}}}+{N}_{\overline{\mathrm{m}}} },\\ {f}_{\overset{\sim }{\mathrm{m}}\overline{\mathrm{m}} }={f}_{\overline{\mathrm{m}}\overset{\sim }{\mathrm{m}} }={k}_{\mathrm{homo}}\frac{{N}_{\overset{\sim }{\mathrm{m}}}{N}_{\overline{\mathrm{m}}}}{{N }_{\overset{\sim }{\mathrm{m}}}+{N}_{\overline{\mathrm{m}}} }.\end{array}$$(17) 
As a result, for a man of types \(\mathrm{m}\) and \(\overset{\sim }{\mathrm{m}}\) and women, the average number of heterosexual contacts per unit time is \({k}_{\mathrm{hetero}}\) and \({k}_{\mathrm{hetero}}{N}_{\mathrm{m}}/{N}_{\mathrm{w}}\), respectively. They are approximately equal to each other because the sex ratio is approximately equal. For a man of types \(\overset{\sim }{\mathrm{m}}\) and \(\overline{\mathrm{m} }\), the average number of homosexual contacts per unit time is \({k}_{\mathrm{homo}}\).

Life history parameters (\({\gamma }_{i},{\eta }_{i},{\mu }_{i}^{^{\prime}}\)) and the fluctuation in sexual activity are set to \({C}_{i}=3\) for all categories \(i\in \left\{\mathrm{w},\mathrm{m},\overset{\sim }{\mathrm{m}},\overline{\mathrm{m}}\right\}\). The observational basis for the value of C will be explained later. Increasing \({C}_{i}\) reduces the effect of mothertochild transmission (\({b}_{\mathrm{w}}\)), but the results do not change qualitatively.

Consider only three values of infection rates: female to male (\({\beta }_{\mathrm{w}\to \mathrm{m}}={\beta }_{\mathrm{w}\to \overset{\sim }{\mathrm{m}}}={\beta }_{\mathrm{w}\to \overline{\mathrm{m}} }\)), male to female (\({\beta }_{\mathrm{m}\to \mathrm{w}}={\beta }_{\overset{\sim }{\mathrm{m}}\to \mathrm{w}}={\beta }_{\overline{\mathrm{m}}\to \mathrm{w} }\)), and male to male (\({\beta }_{\overset{\sim }{\mathrm{m}}\to \overset{\sim }{\mathrm{m}}}={\beta }_{\overline{\mathrm{m}}\to \overset{\sim }{\mathrm{m}} }={\beta }_{\overset{\sim }{\mathrm{m}}\to \overline{\mathrm{m}} }={\beta }_{\overline{\mathrm{m} }\to \overline{\mathrm{m}} }\)).
Under the above assumption, the relative sizes of \({R}_{i\to j}^{\mathrm{v}}\) and \({A}_{i}\) are as shown in Table 2. In the phase plane in Fig. 3, the region above the bold curve indicates the persistence region of STIs (i.e., \({T}_{\mathrm{w}},{T}_{\mathrm{m}},{T}_{\overset{\sim }{\mathrm{m}}},{T}_{\overline{\mathrm{m}} }>1\)), the four colours indicate which typereproduction numbers are the smallest, and the colourless region indicates that all \({T}_{i}\) are not well defined. The dashed curves indicate that \({T}_{\overset{\sim }{\mathrm{m}}}\) is not well defined above them (in Fig. 3a and c, the dashed curves coincide with the solid ones). If there are relatively few sexual contacts between men, it is more effective to focus infection control on women (w) (\({R}_{\overset{\sim }{\mathrm{m}}\to \overline{\mathrm{m}} }^{\mathrm{h}}={R}_{\overset{\sim }{\mathrm{m}}\to \overset{\sim }{\mathrm{m}}}^{\mathrm{h}}={R}_{\overline{\mathrm{m} }\to \overline{\mathrm{m}} }^{\mathrm{h}}={R}_{\overline{\mathrm{m}}\to \overset{\sim }{\mathrm{m}} }^{\mathrm{h}}=0.25\) in Fig. 3a and c) than on men. In this case, measures focused only on MSMW will not be able to suppress the STIs, because \({T}_{\overset{\sim }{\mathrm{m}}}\) is not well defined. On the other hand, if there are relatively more sexual contacts between men, there are some parameter areas where focusing infection control on MSMW (\(\overset{\sim }{\mathrm{m}}\)) is most effective (\({R}_{\overset{\sim }{\mathrm{m}}\to \overline{\mathrm{m}} }^{\mathrm{h}}={R}_{\overset{\sim }{\mathrm{m}}\to \overset{\sim }{\mathrm{m}}}^{\mathrm{h}}={R}_{\overline{\mathrm{m} }\to \overline{\mathrm{m}} }^{\mathrm{h}}={R}_{\overline{\mathrm{m}}\to \overset{\sim }{\mathrm{m}} }^{\mathrm{h}}=0.49\) in Fig. 3b and d). However, such a parameter range is narrow, and \({T}_{\overset{\sim }{\mathrm{m}}}\) is not well define in most of the blue regions where \({T}_{\mathrm{w}}\) is minimized. Moreover, Fig. 3e and f show that it would be most efficient to focus prevention measures on women (w) if there is little transmission among men and on MSME (\(\overline{\mathrm{m} }\)) otherwise. If there are many cases of both transmission types, infection cannot be suppressed by taking measures for only one subpopulation. This suggests that it is important to simultaneously prevent both homosexual and heterosexual transmission to suppress STIs. Moreover, in Fig. 4, we compare the influence of MSMW and MSME. Figure 4df show that when homosexuals contribute more to infectious diseases than heterosexuals, higher proportions of MSME require more measures against MSME than MSMW. On the other hand, Fig. 4ac show that if the contributions to homosexual and heterosexual infections are equal, measures for women are important in any case.
Discussion
In our previous paper [31], we proposed a simple STI model with heterosexual and vertical transmission and studied their mutual effect on the spread of STIs. In this model, people were divided into two types by sex. Moreover, we extended this model to include a juvenile type and showed that it is not necessary to include the juvenile type in the model because its effect can be mathematically reduced to postnatal effects [32]. In this paper, we formulated a typereproduction number for the STI model that simultaneously considered (1) the network heterogeneity of human sexual contacts, (2) mothertochild (vertical) transmission and (3) MSMW and MSME. These three factors greatly influence the spread of STIs, and we expect that the current approach will contribute to a comprehensive understanding of STI infection dynamics. It should be emphasized that the result given by Eq. (15) does not depend on the details of the model, such as the addition of the childhood stage.
In the current model, we assume a wellmixed population without consideration of the specific network structure, and we assume that each individual has a different level of sexual activity. This type of approximation is good for epidemic models with complex networks (e.g., [31, 42]). Although our model does not take into account the details of personal relationships (e.g., marital status, distinguishing between primary and casual sexual partners, repeated sexual contacts, and parent–child relationships), the result provides a good reference theory for complicated situations. Our model can reveal various trends in the population by changing the parameters. Several types of sexual contact (oral, anal and genital) can be considered differences in the level of sexual activity. Moreover, the transmission rates (\({\beta }_{i\to \mathrm{j}}\)) are dependent on sexual culture, which changes over time. In this study, we neglected women who have sex with women (WSW). Although WSW can potentially transmit STIs from current and prior male and female partners [42], it is unlikely that the WSW sexual network is a large reservoir of STIs, in contrast with the MSM network, because the prevalence of STIs in women who have sex with women exclusively (WSWE) is not higher than the prevalence in heterosexual women for many STIs [43, 44]. We considered only MSM, which has been confirmed to contribute significantly to the spread of STIs.
This model assumed that MSMW and MSME are innate, and the proportion of the sum of MSMW and MSME was set to 4% according to the previous studies shown in Table 1. Again, it is difficult to distinguish between MSMW and MSME, and the population ratio of MSMW and MSME cannot be determined definitively. This problem makes the estimation of the costeffectiveness of prevention measures for each subpopulation difficult. The fluctuation of sexual activity is set to \({C}_{i}=3\) for all subpopulations according to the data of previous studies (see Table 3) [28, 45,46,47,48,49,50,51,52]. Although the values of \({C}_{i}\) may be slightly larger, our result does not change qualitatively when \({C}_{i}\) increases. There was almost no difference in \({C}_{\mathrm{i}}\) between men and women in Finland and Russia. In the UK, there was a large difference between men and women, with \({C}_{\mathrm{m}}=68\) for men and \({C}_{\mathrm{w}}=15\) for women. The C in Japan is large because the surveys cover all ages; thus, the variance in the total number of sexual partners is large. However, C in Japan may not be that large in reality since sexual activity in the model is assumed to be innate.
If the activity of homosexual individuals is similar to that of heterosexual individuals, the contribution of MSM to STI transmission is less than we suspected (as seen in Fig. 4). Under the realistic MSM population estimated from the previous studies shown in Table 1, prevention measures focusing on MSM are not efficient as long as there is not an explosive spread among the MSME population (Eq. (16) does not hold). It should be noted that in this model, MSMW is nearly twice as active as the other groups because it includes both homosexuality and heterosexuality. This seems to be an overestimation, but nevertheless, the impact of MSMW is not necessarily significant in this model. In the case that MSM is not important, it is more efficient to concentrate measures on women than men. Interestingly, a study published by Kahn et al. [53] in 1997 estimated that the annual number of HIV infections in the United States was approximately 40,000 and that infections transmitted by bisexual persons accounted for only 200–600 of those. Therefore, they concluded that transmission via bisexual contact was a relatively minor component of all HIV transmissions in the United States, and it seems that their findings are consistent with our model results [53].
Conclusion
In the current study, we constructed an STD model in which the population is divided into four subpopulations, women, MSMW, MSME, and MSW, and derived the analytical formula for typereproduction numbers. As research on actual sexual contacts, including homosexuality, progresses in the future, this formula will be useful for developing preventive strategies. What we can say now is that it is important to simultaneously prevent both homosexual and heterosexual transmission to suppress STIs because MSM and mothertochild transmission rates do not have a strong synergistic effect. Furthermore, our study is the first to quantify the effects of bisexual bridges on the spread of STIs. Understanding the potential role of MSMW and MSME in STI transmission from MSM to women is epidemiologically important. Our model shows the impact of bisexual bridge on the spread of STIs does not outweigh their population size.
Availability of data and materials
The authors declare that all data supporting the findings of this study are available within the article.
Abbreviations
 STIs:

Sexually transmitted infections
 MSW:

Men who have sex with women only
 MSM:

Men who have sex with men
 MSME:

Men who have sex with men exclusively
 MSMW:

Men who have sex with men and women
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This work was supported by the JSPS KAKENHI (Grant nos. 17H04731, 19KK0262 and 21H01575 to H.I.; 17H04659 and 19H05737 to T.Y.; 18K03453 and 21K03387 to S.M.), the Joint Usage/Research Center on Tropical Disease, Institute of Tropical Medicine, Nagasaki University (2020Ippan01) and the Japan Science and Technology Agency Crest to S.M.
This funding source had no role in the design of this study and or in its execution, analyses, interpretation of the data, or decision to submit results.
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H.I. and S.M. conceived the study and wrote the manuscript. S.M. constructed the mathematical model. H.I., T.Y. and S.M. assisted in the interpretation of the results. H.I. and S.M. generated the figures. H.I., T.Y., and S.M. revised the references and data. The author(s) read and approved the final manuscript.
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Additional file 1.
Mathematica code example for calculating the typereproduction number.
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Ito, H., Yamamoto, T. & Morita, S. The effect of men who have sex with men (MSM) on the spread of sexually transmitted infections. Theor Biol Med Model 18, 18 (2021). https://doi.org/10.1186/s12976021001489
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Keywords
 Sexually transmitted diseases
 Typereproduction number
 Complex networks
 MSM contact
 Bisexual bridge