The type-reproduction number of sexually transmitted infections through heterosexual and vertical transmission

Multiple sexually transmitted infections (STIs) have threatened human health for centuries. Most STIs spread not only through sexual (horizontal) transmission but also through mother-to-child (vertical) transmission. In a previous work (Ito et al. 2019), we studied a simple model including heterosexual and mother-to-child transmission and proposed a formulation of the basic reproduction number over generations. In the present study, we improved the model to take into account some factors neglected in the previous work: adult mortality from infection, infant mortality caused by mother-to-child transmission, infertility or stillbirth caused by infection, and recovery with treatment. We showed that the addition of these factors has no essential effect on the theoretical formulation. To study the characteristics of the epidemic threshold, we derived analytical formulas for three type-reproduction numbers for adult men, adult women and juveniles. Our result indicates that if an efficient vaccine exists for a prevalent STI, vaccination of females is more effective for containment of the STI than vaccination of males, because the type-reproduction number for adult men is larger than that for adult women when they are larger than one.


Model
We consider a compartment model with six compartments: susceptible juveniles S j (t), susceptible adult female S f (t), susceptible adult male S m (t), infected juveniles I j (t), infected adult female I f (t), and infected adult male I m (t). These variables represent the numbers of individuals belonging to the compartments at time t. Juveniles cannot have sexual intercourse, whereas adults can. Moreover, we assume that each adult has different sexual activity, as a congenital attribute. The model variables and parameters are summarized in Table 1.
Dynamics of juveniles. The dynamics of S j (t) and I j (t) are assumed to be Here, B is the number of births per unit of time, δ is the rate of infertility or stillbirth, and α is the rate of vertical transmission from mother to infant. The parameters λ and λ′ are the maturing rates for susceptible and infected juveniles, respectively. Juveniles become adults at the maturing rate. Because the pathogenicity of infection may reduce the growth of infected juveniles, we set λ′ ≤ λ. The parameter η j stands for the juvenile cure rate, where an infected juvenile becomes susceptible with the cure rate of η j . The pathogenic premature mortality rate μ ′ j is at least as large as the natural premature mortality rate μ j (thus, j j μ μ ≥ ′ ). Here, the juvenile's sex is not considered because there is no need to distinguish between sexes in the juvenile stage in this model.
∞ ∞ a a p a da ap a da ( ) ( ) 1 (2) The total number of sexual contacts per unit of time is assumed to be f(N f (t), N m (t)), where N f (t) = S f (t) + I f (t) and N m (t) = S m (t) + I m (t). Thus, the rates of having a sexual contact for a woman and a man are af N t N t N t af N t N t N t ( ( ), ( ))/ ( ), ( ( ), ( ))/ ( ), f m f f m m respectively. For example, if sexual contacts are modeled by mass action, f(N f (t), N m (t)) ∝ N f (t)N m (t). In this case, the rates of having a sexual contact for a woman and a man are proportional to aN m (t) and aN f (t), respectively. This is not very realistic because the number of sexual contacts a person has increases along with the population. Alternatively, if women dominate sexual contact, f(N f (t), N m (t)) ∝ N f (t). In this case, the rates of having a sexual contact for a woman and a man are proportional to a and aN f (t)/N m (t), respectively. This may be more suitable because men have less chance of contacting women when the number of women per men decreases. In any case, the sexual contacts are assumed to be well mixed. The average numbers of sexual contacts per unit of time for women and men are given as respectively, because we set 〈a〉 = 1. Moreover, it is convenient to define the averages weighted by sexual activity: These values correspond to the effective average over the distribution by degree of sexual activity, as defined by May and Anderson 21 , where c f and c m are not simply the mean but the mean plus the ratio of variance to the mean. High heterogeneity of sexual contacts means c f ≫ k f and c m ≫ k m .
Dynamics of adults. The number of susceptible adult women whose sexual activity is in the infinitesimal interval [a, a + da] is denoted as S f (t, a)da. The same applies to S m (t, a), I f (t, a) and I m (t, a). Thus, we have The dynamics of S f (t, a), I f (t, a), S m (t, a) and I m (t, a) are expressed as The parameter γ in the first terms on the right hand side of the above equations represents the proportion of men at the time of coming of age. If γ = 0.5, the same number of male and female juveniles grow to adulthood.
The parameters μ f , f μ ′ , μ m , and μ ′ m are the death rates for susceptible adult women, infected adult women, susceptible adult men, and infected adult men, respectively. Here, we set μ ′ f ≥ μ f and μ ′ m ≥ μ m because of pathogenicity. The parameters η f and η m are the cure rates for women and men, respectively. Note that the death and cure rates are assumed not to be directly dependent on sexual activity a. If η j = η f = η m = 0, the infection is incurable, meaning that the model is of the susceptible-infected type. The probabilities of transmission per sexual contact are β m→f and β f→m for male-to-female and female-to-male transmission, respectively (0 ≤ β m→f , β f→m ≤ 1). Here, the variables Θ f (t) and Θ m (t) stand for the probabilities that the sexual partners of a man and a woman are infected, respectively. Because a sexual partner with sexual activity a is selected with the probability aN f (t, a)/N f (t) or aN m (t, a)/N m (t) and the probability that the sexual partner is infected is I f (t, a)/N f (t, a) or I m (t, a)/N m (t, a), we have There is a stable equilibrium state www.nature.com/scientificreports www.nature.com/scientificreports/ where the tildes over the variables indicate that the values are for the disease-free equilibrium. In this case, from Eq. (4), we have

Linearization.
To derive the basic reproduction number, we linearize Eqs. (1) and (7) near the disease-free case-that is, we consider only the f irst order of (1) and (7).
case without sexual transmission. Here, we consider a simple case where there is only mother-to-child transmission (β m→f = β f→m = 0). In this case, Eq. (12) is simplified to a three-dimensional system: In this case, we do not need to consider the dynamics of θ f (t) and θ m (t) because they do not affect I j (t), I f (t) or I m (t). We write the linearized system Eq. (13) , where matrix T corresponds to transmissions and matrix Q to transitions: Then, the spectral radius (dominant eigenvalue) of the next generation matrix −TQ −1 gives the reproduction number 41,42 . After some elementary algebra, we obtain the basic reproduction number for the case without sexual transmission.
eff f j Here, the amount α eff is the efficient vertical transmission rate. Equation (15) can be intuitively understood as follows (see Fig. 1a). The number of births per unit of time in the disease-free equilibrium is Because the average duration for infected adult women is 1/(μ ′ f + η f ), the average number of children born to an infected adult woman is The probability that a juvenile survives and becomes an adult woman is The average number of daughters infected vertically by an infected adult woman is given by the product of Eqs. (18) and (19), which leads to Eq. (15). The value of α eff represents the average number of infected adult daughters of an infected adult woman in a completely susceptible population. Thus, the epidemic threshold is given by and δ ≥ 0, α eff is always less than one. Therefore, STIs cannot survive if they spread only by mother-to-child transmission. In the same way (see Fig. 1a), we can show that the average number of sons infected vertically by an infected adult woman is As is shown in Fig. 1b, the average number of adult women infected through consecutive mother-to-child transmissions from an infected adult woman is As is shown in Fig. 1c, the average number of adults men infected through consecutive mother-to-child transmissions from an infected adult woman is eff eff type-reproduction numbers for the general case. Here, we calculate the type-reproduction numbers for three compartments. First, to derive the type-reproduction number for women infected horizontally, we focus on adult women infected through sexual transmission and regard vertical transmissions as transitions. We rewrite the linearized system Eq. (12) in the form  Here, for example, A 11 = λ′ + η j + μ ′ j and A k www.nature.com/scientificreports www.nature.com/scientificreports/ If R f > 1, the STI can spread over the population. Our previous study treating a simpler model gave a similar formula 35 , where we called Eq. (24) the basic reproduction number over generations. This metric is the average number of sexually infected adult women generated from a sexually infected woman in a completely susceptible population. As is shown in Fig. 2b, the first term represents the propagation through only two types of horizontal transmission (female-to-male and male-to-female); the second term represents the propagation through vertically infected women and these two types of horizontal transmission; and the third term represents the propagation through vertically infected men and male-to-female horizontal transmission. Note that the first term is dominant in Eq. (24) because c m ≫ k m and c f ≫ k f if α eff is not close to one.
Second, focusing on adult men infected through sexual transmission, the linearized system Eq. (12) in the form  x = (T + Q)x is given by Finally, focusing juveniles infected through mother-to-child transmission, the linearized system Eq. (12) in the form  x = (T + Q)x is given by Calculating the dominant eigenvalue of −TQ −1 , we obtain the type-reproduction number for adult men: ), R j diverges to infinity. The divergence of R j means that the spread of the STI cannot be stopped even if we eliminate mother-to-child transmission completely. It is obvious that R j > 1 if and only if R f > 1.
The type-reproduction numbers in Eqs. (24), (26) and (28) are independent of the number of births per unit of time B. Thus, regardless of the details of the birth process, these formulas are always valid for the equilibrium. Without mother-to-child transmission (α = 0 or α eff = 0), we obtain Graphical analysis. In Fig. 3, we show the relationship between the infection rate (vertical and horizontal) and the three type of reproduction numbers. Here, we set k f , k m = 0.8, c f , c m = 20, μ f , μ ′ f , μ m , m μ ′ , μ j , μ ′ j = 1/50, η f , η m , η j =0, δ = 0, and λ = λ′ = 1/15, and then α eff = α. When the vertical transmission rate is low or β m→f ≪β f→m , there is almost no difference between R f and R m . Otherwise, when STI is not widespread, the type-reproduction  Fig. 1b,c. Note that these processes contain vertical transmission starting from a vertically infected adult woman. The middle two processes represent horizontal transmission from a horizontally (blue) or vertically (brown) infected woman. The bottom two processes represent horizontal transmission from a horizontally (purple) or vertically (red) infected man. The average production numbers are given by the product of the transmissibility (β f→m or β m→f ), contact rate (c f , k f , c m or k m ), and duration (1/(μ ′ f + η f ) or 1/( m μ ′ + η m )). (b) There are three paths starting from a horizontally infected adult woman and ending with a horizontally infected adult woman: two horizontal transmissions (HH), one vertical and two horizontal transmissions (VHH), and one vertical and one horizontal transmission (VH). Because the HH case is made up of the processes shown in blue and purple in (a), the reproduction rate R HH for the HH case is given by the product of their average production numbers. We can calculate the reproduction rates R VHH and R VH for the VHH and VH cases in the same way. Thus, the typereproduction number for adult women is given by summing these rates: R HH + R VHH + R VH ; thus, Eq. (24) is obtained. (c) The paths starting with a horizontally infected adult man and ending with a horizontally infected adult man are more complicated. The HH case (purple and blue) and the VHH case (purple, yellow, and brown) are similar to (b) (see the top two parts). The average production numbers of these cases are R HH and R VHH . In addition to these two cases, secondary infected women may vertically infect their sons, who may then infect other women (i.e., we can insert the green and red parts in the way; see the middle two parts). The average production numbers of these cases are R HH R VH and R VHH R VH . This procedure can be repeated (see the bottom two parts). Thus, the type-reproduction number for adult men is derived by a geometric series with geometric ratio R VH , and Eq. (26) is obtained. (d) The paths starting with an infected juvenile and ending with an infected juvenile are also complicated. The case through one woman is simple, and it is obvious that its reproduction rate is α eff . There are two cases through two women: One includes men infected horizontally (brown and purple), and the other includes those infected vertically (red). Inserting the blue and purple part in the way, we obtain two cases through three women. This procedure can be repeated. Considering two geometric series with geometric ratio R HH , we obtain the type-reproduction number for juveniles-Eq. www.nature.com/scientificreports www.nature.com/scientificreports/ number of females exceeds that of males (i.e., 1 > R f > R m ); when the STI has already spread, the type-reproduction number of males is larger than that of females (i.e., R m > R f > 1). This result means that if an efficient vaccine for the STI exists, vaccination of females is more effective for containment of the STI than vaccination of males because herd immunity would be possible at vaccination rates above 1 − 1/R f for females or 1 − 1/R m for males.
As far as the type-reproduction numbers are concerned, the effective increase in the cure rate η m is equivalent to the effective decrease in β m→f . Similarly, the increase in mortality μ ′ m due to the STI is equivalent to the decrease in β m→f . On the other hand, the increase in cure rate η f is equivalent to the decrease in both β f→m and α eff , and the increase in mortality f μ ′ is equivalent to the decrease in both β f→m and α eff . Equations (24), (26) and (28) do not explicitly include the mortality rate of juveniles (μ j ,μ ′ j ), but the type-reproduction numbers depend on μ j and μ ′ j through the efficient vertical transmission rate α eff , which decreases when j μ ′ increases. Keeping c f /k f (c m /k m ) constant, increasing k f (k m ) corresponds to increasing β f→m (β m→f ).

Discussion
In this article, we have presented a compartment model of STIs, considering both mother-to-child transmission and sexual transmission. The proposed model is of major importance because it takes into account the heterogeneity of sexual contacts, adult mortality from infection, infant mortality caused by mother-to-child transmission, infertility or stillbirth caused by infection, and recovery with treatment. Our model derives analytical formulas for the type-reproduction numbers R f , R m , and R j . Because these metrics give the same epidemic threshold, it is convenient to use the simplest formula, Eq. (24). Equation (24) coincides qualitatively with the basic production number over generations that we proposed in previous work 35 .
The model proposed here allows us to understand how the various effects of mother-to-child transmission on juveniles will change the epidemic threshold. It should be emphasized that all effects of STIs on the juvenile period before reproductive age are expressed mathematically in α eff . For example, infant mortality and various disturbances to growth reduce α eff . Because α eff is not in the dominant term in Eq. (24), α eff does not strongly affect the epidemic threshold. In other words, the spreading efficiency of STIs will not be strongly altered even if various problems in the juvenile stage (i.e., mortality or stillbirth) are solved. In sum, we suggest that the heterogeneity of sexual contacts highly contributes to the spread of STIs, and mother-to-child transmission may work as an auxiliary infection route, contributing to the survival of STIs. This fact was derived from the mathematical analysis and is consistent with our previous work, which did not consider mortality from STIs 35 .
The analytical formulas for type-reproduction numbers can provide us with important insight into strategies to prevent the spreading of STIs. If there was an efficient vaccine, herd immunity would be possible at vaccination rates above 1 − 1/R f for women or 1 − 1/R m for men. R m > R f , as was shown above, because of mother-to-child transmission; therefore, our result means that if the number of vaccines and the amount of funds are limited, it is more efficient to concentrate the vaccine in women only. For example, in many countries, publicly funded HPV immunization programs target young adolescent girls, who are at the border between the juvenile period and adulthood 43 . According to our results, this intensive vaccination investment in young girls makes sense mathematically. To apply our results quantitatively to actual STIs, we need to estimate the model parameters using clinical epidemiological and demographic data on sexual behavior. However, it is difficult to estimate the infection rate (i.e., β m→f , β f→m , and α) and mortality rate (i.e., j μ ′ , μ ′ f , and μ ′ m ) of specific individual STIs because people frequently have more than one STI at the same time 44 . In addition, the data on the distribution of human sexual activity are insufficient 45 .
Nevertheless, we are able to form some preliminary qualitative conclusions. Many STIs have serious fetal consequences, such as TORCH infections 46,47 , which include toxoplasmosis, other diseases (syphilis, varicella-zoster, parvovirus), rubella, cytomegalovirus, and herpes infections. Most TORCH infections cause mild maternal morbidity and have serious fetal consequences, and the treatment of maternal infections often has no impact on fetal outcomes; thus, the recovery rate η j for juveniles is nearly zero 47 . For incurable STIs such as HIV, HSV, HPV, HTLV-1, and antibiotic-resistant STIs, the proposed model is a susceptible-infected model (η j = η f = η m = 0). For recovery with treatment, the recovery rates depend on the medical systems and treatment strategy. For example, it is possible that infected women are less aware of their infections than are infected men with the same STIs, η f < η m because the treatment of infected women and is not promoted to the same extent as is the treatment of infected men 48,49 . In addition, some STIs may inhibit children's growth even when the infections are not fatal. For example, HSV can bring on central nervous system disorder 30 , and cytomegalovirus can lead to long-term neurological sequelae including unilateral and bilateral sensorineural hearing loss, mental retardation, cerebral palsy, and impaired vision from chorioretinitis [50][51][52] . In these cases, the maturing rate λ' for infected juveniles is lower than the maturing rate λ for susceptible juveniles.
The model proposed here did not consider some important factors that are related to the diffusion of STIs. First, we neglected some routes of transmission, such as needle sharing [53][54][55] and blood transfusion 56 , because these routes were rare in ancient times and there are many efforts to reduce them now 57,58 . Moreover, only a few STIs are known to be transmitted by mosquito bites 59 . Here, we focused on understanding the diffusion contribution of sexual and mother-to-child transmission. Second, our model assumed that sexual contacts are well mixed, and the effects of sex workers 55 , marital status and age structure were not taken into account. The presence of sex workers will increase the heterogeneity of sexual contact. Marital status will affect the infection dynamics of STIs because marriage yields sustained sexual activity with a specific partner 60 . The age structure also will influence the diffusion of STIs 61 . Sexual transmission tends to occur among people of the same generation, whereas mother-to-child transmission propagates infections across generations; thus, our model may slightly underestimate the contribution of mother-to-child transmission. Third, homosexual contact was not considered here, although homosexual transmission is important, especially for the spread not of only HIV but also of Scientific RepoRtS | (2019) 9:17408 | https://doi.org/10.1038/s41598-019-53841-8 www.nature.com/scientificreports www.nature.com/scientificreports/ HBV, syphilis etc. 62 . In this article, our aim was to understand the complex infection dynamics, simultaneously considering both unequal sexual transmission rates between males and females and mother-to-child transmission. If we also considered homosexual and bisexual networks, the infection dynamics would become extremely complicated. Thus, here, we have not discussed the spread of STIs through homosexual networks, such as men who have sex with men. Fourth, we have assumed that sexual activity is not inherited and does not depend on whether an individual is infected or not. In addition, we did not consider the correlation between sexual activity and fecundity. Constructing a model that takes into account the inheritance of sexual activity may reproduce the heterogeneous distribution of this variable. These limitations should be addressed in future research.
In conclusion, the comprehensive model proposed in this article can clarify the complex transmission of STIs. This model is prospective: It is meant to predict the spread of various STIs. We derived analytical formulas for three type-reproduction numbers, R f , R m , and R j , and elucidated the relationships among them. However, the quantitative evaluation of these metrics for actual STIs remains a topic for future research. The quantitative application of our model has the potential to clarify which kinds of countermeasures will be effective in combating STIs.

Data availability
The authors declare that all data supporting the findings of this study are available within the article or from the corresponding author upon reasonable request. f m m f for constant vertical transmission rate (α = 0.5), and (d-i) they are plotted as a function of α for constant sexual transmission rates. The left panels (a,d,g) represent the same sexual transmission rate (β f→m = β m→f ), the middle panels (b,e,h) represent the case in which the sexual transmission rate from males to females is greater than that in the opposite direction (β m→f = 10β f→m ), and the right panels (c,f,i) represent the case in which the sexual transmission rate from females to males is greater than that in the opposite direction (β f→m = 10β m→f ). For the case that β f→m β m→f is large (d-f), R j is not plotted because it diverges. The other parameters are set as k f , k m = 0.8, c f , c m = 20, μ f , μ ′ f , μ m , μ ′ m , μ j , j μ ′ = 1/50, η f , η m , η j = 0, δ = 0, λ = λ′ = 1/15, then α eff = α.