Model formulation

We extend the classical SIR model23 incorporating vertical and horizontal transmission between male and female hosts (see Fig. 1 and Methods for details). We solve this model allowing pathogens to exhibit different virulence in men and women.

Figure 1: Diagram showing our epidemiological model of vertical and horizontal transmission in males and females. We model the continuous change in time of the fraction of susceptible (subscript s), infected (subscript i) and recovered (subscript r) individuals in males (m) and females (f). Males and females born, B m and B f , are infected with probability αf i . Susceptible males and females, m s and f s , become infected with probability given by the force of infection in males H m =β(δ m )γ mm m i +β(δ f )γ fm f i and females H f =β(δ f )γ ff f i +β(δ m )γ mf m i . In a population at equilibrium every individual that dies is replaced with a new-born. Full size image

There is a large literature on epidemiological models exploring the evolution of virulence in pathogens that are horizontally transmitted only24,25. There is a limited literature on epidemiological models exploring the evolution of virulence in pathogens that are vertically transmitted only26, showing that natural selection disfavours pathogens that are vertically transmitted only (when vertical transmission is uniparental) if they exhibit any degree of virulence26,27. Pathogens can only be maintained if they exhibit some special feature that compensates for the fitness loss caused by the death of their host26,28. Epidemiological models of pathogens that are only horizontally or vertically transmitted do not consider pathogens that may exhibit different virulence in men and women (but see ref. 29 for sex-specific virulence as a result of different resistance of male and female hosts).

There is also a large literature on behavioural models exploring the evolution of male killing by pathogens transmitted exclusively through cytoplasmic inheritance (male-killers)30,31, that is, equivalent to male limited virulence in pathogens that are vertically transmitted only. Natural selection does not disfavour male-killers because their virulence is limited to the sex that does not transmit them and thus does not translate in any fitness loss32. However, natural selection favours male-killers only when the death of a male translates into greater fitness of females closely related to the male-killer (fitness compensation)28.

Here we formulate an epidemiological model exploring the evolution of virulence in pathogens that are horizontally and vertically transmitted (mixed transmission). When there is mixed transmission, pathogens that are vertically transmitted can be maintained because they are also horizontally transmitted and not because they exhibit any form of fitness compensation26,27,33,34. Because of their complex nature, epidemiological models of mixed transmission are rare27,33,34,35,36, and they do not consider pathogens that may have different virulence in men and women. Thus, to the extent of our knowledge, ours is the first epidemiological model to consider pathogens with a sex-specific virulence. Most importantly, this model allows us to investigate the evolution of sex-specific virulence in pathogens causing infectious diseases—where horizontal transmission plays a role—when the only difference between the sexes is the presence of vertical transmission in women.

In our model, infection can take place either through vertical transmission at a rate α or horizontal transmission at a rate β per contact (Fig. 1). Horizontal transmission can either be from males or females, that is β m and β f , where subscripts m and f denote males and females. Contacts can be established between any of the sexes, taking place at rates γ mm , γ mf , γ fm , and γ ff with the first subscript indicating origin and the second destination. We assume that the number of contacts per unit of time is independent of the number of individuals in the population. An infected individual either recovers at a rate σ, dies from causes unrelated to the infection at a rate μ (natural mortality), or dies from causes related to the infection at a rate ν (virulence) (Fig. 1). Therefore, infections are lost from the population at a rate δ equal to the sum of these rates (δ=σ+μ+ν; Fig. 1). Another way of interpreting δ is as the reciprocal of the average duration of the infection: the higher the δ, the shorter the average duration of the infection will be. The rate of loss of infection δ is the pathogen’s strategy to exploit its host (exploitation strategy). Crucially, we allow the pathogen’s exploitation strategy to be different in men (δ m ) and women (δ f ) (Fig. 1).

Trade-off between virulence and recovery and transmission

The higher the virulence ν, (and therefore the shorter the duration of the infection), the higher the horizontal transmission rate β37,38,39,40, that is β′(δ)>0 where prime denotes the derivative (see Supplementary Methods, Supplementary Table 1 for the notation used). We make the assumption—standard in evolutionary epidemiology models24,25,41—that there is a saturating trade-off between the horizontal transmission rate of a pathogen β from males and females, that is β m and β f , and its virulence ν in males and females, that is ν m and ν f , mediated by the sex-specific strategies δ m and δ f (refs 24, 25, 41), that is β m (δ m ), β f (δ f ) and ν m (δ m ), ν f (δ f ). Following classic vertical transmission models33,34 we make the assumption that the vertical transmission of a pathogen, α, is independent of its virulence, ν. Because we are interested in researching whether adding vertical transmission favours the pathogen to have different strategies in men and women (that is, whether evolution favours values of δ m and δ f that are not equal), we make the conservative assumption that the transmission-virulence trade-off is the same in pathogens residing in hosts of either sex, that is β m =β f =β and ν m =ν f =ν. Because transmission and virulence depend on the sex-specific exploitation strategies, this allows for the transmission and virulence of pathogens in women (β(δ f ) and ν(δ f )) to be different from those residing in men (β(δ m ) and ν(δ m )).

ESS for a non-sex-specific strategy

We derive the fitness of a mutant pathogen and the exploitation strategy that once established in the population cannot be beaten by any alternative exploitation strategy (ESS)32,42. We do this by finding the exploitation for which the selection gradient is zero (see Methods section at the end of the paper and Supplementary Methods for details of the derivation). We show that the non-sex-specific (δ m =δ f =δ) ESS, , satisfies:

This result renders itself to a simple interpretation: can be interpreted as the fraction of time a pathogen spends in females in a population at equilibrium, can be interpreted as the average number of females that an infected son infects through horizontal transmission at equilibrium, and are the male and female birth rates at equilibrium (Fig. 2). The expression can be interpreted as the effective rate of vertical transmission (Fig. 2). In the absence of vertical transmission, α=0, we recover the classic result for the evolution of virulence in pathogens with horizontal transmission only24,25.

Figure 2: Interpretation of parameters. Interpretation of parameter and expression . is the average number of females horizontally infected by a son and the male infections he will give rise to and results from adding up all possible transmission events to females. is the effective rate of vertical transmission. It results from calculating the probability of vertical transmission to sons and daughters and horizontal transmission from son to females with whom he establishes contact. Full size image

ESSs for sex-specific strategies

Solving for a different exploitation strategy in males and females, δ m and δ f , we determine the ESS for a male-specific strategy, :

and the ESS for a female-specific strategy, :

Notice that the ESS male-specific strategy is the same as the ESS non-sex-specific strategy when virulence is not expressed in females, that is, when , and the ESS female-specific strategy is the same as the ESS non-sex-specific strategy when virulence is only expressed in females, . In the absence of vertical transmission, α=0, there is no difference in virulence between the male- and female-specific strategies. See Table 1 for a summary of results.

Table 1 Summary of results on the ESS host exploitation condition. Full size table

Using a graphical method43 to analyse result (1) we confirm existing results indicating that a pathogen that is horizontally and vertically transmitted evolves to be less virulent than a pathogen that is horizontally transmitted only (Fig. 3a)34,44. Using the same graphical method to analyse results (2) and (3) we make two novel predictions: first, a pathogen that is horizontally and vertically transmitted is under selective pressure to evolve sex-specific virulence with lower virulence in females than males (Fig. 3b). The intuitive reason is that the existence of an additional route of transmission makes the life of hosts that transmit vertically—females but not males—more valuable to the pathogen. Second, the greater the vertical transmission the greater the difference in virulence between males and females a pathogen is selected for (Fig. 4). Therefore, given two populations that differ in their rates of vertical transmission, sex-specific virulence is more likely to be observed in the population with the greater rate of vertical transmission. Although this result is generic, the magnitude of the difference can depend on the model parameters (Supplementary Methods).

Figure 3: Interpreting the ESS virulence levels with a graphical method. We use a graphical method similar to the one developed by van Baalen and Sabelis43 to determine which ESS virulence will be greater. (a) Comparison between the ESS virulence with and without vertical transmission in the absence of sex-specific strategies. Without vertical transmission (α=0) the ESS exploitation strategy ( ) is the point where the tangent of the horizontal transmission function (β′(δ)) crosses the abscissa in δ=0. With vertical transmission (α>0), the ESS exploitation strategy ( ) is the point where the tangent of the horizontal transmission function (β′(δ)) crosses the abscissa in . Notice that the tangent of the horizontal transmission function at the ESS is higher or equal with vertical transmission, that is . This implies that the ESS virulence is greater or equal without vertical transmission, that is . (b) Comparison between the ESS virulence when the virulence can differ in males and females. With male-specific virulence ( ) the ESS exploitation strategy ( ) is the point where the tangent of the horizontal transmission function crosses the abscissa in δ m =0. With female-specific virulence ( ) the ESS exploitation strategy ( ) is the point where the tangent of the horizontal transmission function crosses the abscissa in . Notice that the tangent of the horizontal transmission function at the ESS is higher or equal with female-specific virulence, that is . This implies that the ESS virulence is greater or equal in males, . Full size image

Figure 4: Evolutionarily stable virulence levels as a function of the rate of vertical transmission. The probability of dying of the disease over a lifetime ( ) at the ESS for male specific, female specific and non-sex-specific exploitation strategies as a function of the rate of vertical transmission, α Parameters=1/80 σ f =σ m =0; ϕ=0.5, ; ; and . For these parameters the probability of dying of the disease for males, under male specific virulence will go to an ESS value of 6%. Full size image

Evolutionary predictions are often hard to validate. However, in this case there is a unique natural experiment that allows us to test our prediction. HTLV-1 is a retrovirus that is horizontally transmitted via sexual intercourse and vertically transmitted via breast-feeding13. Infection with HTLV-1 can progress to ATL that is lethal. The Tax protein—encoded by the viral gene tax—orchestrates the oncogenic potential of the virus45,46, and therefore expression of the Tax protein is positively correlated to the virulence of the pathogen. The presence of anti-Tax antibodies is also an independent risk factor for sexual transmission47 but not for transmission through breast-feeding48. Therefore, the synthesis of Tax by HTLV-1 generates a trade-off between horizontal transmission and virulence in accordance with the assumptions of our model.

HTLV-1 is highly prevalent in two foci located in Southern Japan and the Caribbean13. These foci differ in the relative importance of each transmission route with breast-feeding being more important in Japan and sexual intercourse being more important in the Caribbean. This claim is sustained by three observations: (i) in Japan the spatial distribution of HTLV-1 has a patchy structure characteristic of vertical transmission49,50,51 while in the Caribbean the distribution is more uniform52,53; (ii) HTLV-1 transmission through breast-feeding (vertical transmission) is not sex-biased but transmission through sexual contact (horizontal transmission) is female-biased. In Japan prevalence of HTLV-1 is unbiased early in life and female biased later in life (from age 40-50) which is consistent with vertical transmission being more important there50,51,54 while in the Caribbean prevalence of HTLV-1 is female biased all through adult life (from age 20)53,55; (iii) Japanese women include breast-milk in their children’s diet for longer time and in greater proportion than Caribbean women56,57,58 thus increasing the rate of vertical transmission per child59.

That vertical transmission rate is larger in Japan than in the Caribbean leads us to predict that HTLV-1 should be less virulent in women relative to men in Japan than in the Caribbean (Fig. 5). This prediction is borne out by epidemiological data: Japanese men infected with HTLV-1 are between 2 and 3.5 times more likely to develop ATL than Japanese women are13,14,45. In contrast, Caribbean men infected with HTLV-1 are as likely to develop ATL as Caribbean women13,52,60 (Fig. 6). While it would be possible that this difference is caused by men being worse at fighting infectious diseases than women are in Japan but not in Jamaica it is unlikely. The male-to-female mortality ratio because of infectious diseases is 1.04 in Japan and 1.11 in Jamaica (GBD 2013 (ref. 61)). Notice that we would need the mortality ratio of those already infected in each country (which is not available) to provide a definitive answer but these figures suggest that the health of men relative to women in these countries do not differ significantly. There is also some indication that the virus acts differently between these two locations: significantly fewer Japanese carriers show an anti-Tax antibody response than Caribbean carriers do, which implies that the virus acts differently in the two regions and produces less Tax protein, and thus less leukaemia, in Japan45.

Figure 5: Predicted sex-specific virulence in two populations with different rates of vertical transmission. Predictions for the evolution of sex-specific virulence in two populations where the relative weight of vertical transmission is lower (a) (for example, the Caribbean where transmission through breast-feeding is low) and higher (b) (for example, Japan where transmission through breast-feeding is higher). Figures show how the difference in virulence between men and women is predicted to be greater in Japan than in the Caribbean. Full size image

Figure 6: Sex-differences in the virulence of HTLV-1 in Japan and the Carribbean. Data on the lifetime risk of progression to ATL among male HTLV-1 carriers relative to female carriers. The lifetime risk of progression to ATL is approximately the same in men and women in the Caribbean60 but between 2.0 and 3.5 times more likely in men than women in Japan14. Full size image

We suggest that the geographical differences regarding progression to ATL that have puzzled scientists in the last two decades13,14,45 are, at least in part, caused by a sex-specific adaptation of HTLV-1 virulence. In Japan, where the importance of breast-feeding transmission relative to sexual transmission is greater, natural selection on HTLV-1 favours slower progression to ATL in women than men thus preserving women as a viral route of transmission. However, in the Caribbean where the importance of breast-feeding relative to sexual transmission is lower, natural selection on HTLV-1 does not favour any difference in progression to ATL between women and men.