Role of trade-off between sexual and vertical routes for evolution of pathogen transmission

Abstract

Many pathogens that are predominantly sexually transmitted can also be transmitted vertically. On the other hand, nonbeneficial pathogens that are predominantly vertically transmitted appear to be rare to absent. Many infections also exist that are only transmitted sexually. Using an empirically suggested trade-off between the horizontal and vertical transmission modes, we develop and analyze a model to study evolutionary dynamics of sterilizing, sexually transmitted infections which can also be transmitted vertically. We assume several flexible forms of the trade-off and ask under which conditions evolution in nonbeneficial pathogens favors vertical transmission, sexual transmission, or a mixture of the two. The evolutionary analysis of our model reveals a rich spectrum of evolutionary outcomes. In particular, evolution of pure sexual, pure vertical, and mixed transmission is possible, in addition to a frequent occurrence of evolutionary suicide. These outcomes can also arise via evolutionary branching and be combined in several evolutionary bistability regimes. We show that the shape of the trade-off between the two transmission modes significantly affects pathogen evolution. In particular, while vertical transmission dominates for concave and sigmoid trade-offs, sexual transmission is most commonly observed under convex trade-offs. Our analysis can shed more light on when an infection adopts a particular evolutionary behavior, and which region of the parameter space is realistic, so something about the evolutionary process itself.

Appendices

Appendix: Appendix A: Evolutionary analysis of model (1)

Here, we calculate the term for mutual invasibility M to see what particular factors directly influence the evolutionary outcome. Using Eqs. 6 and 7, we obtain

$$\begin{array}{@{}rcl@{}} \hat{i}^{\prime}(\beta^{*}) &=& \left.\left(1-\frac{b(1-\xi(\beta))(1-\sigma)}{\beta-\alpha(\beta)-b\sigma}\right)'\right|_{\beta=\beta^{*}}\\ &\overset{(6)}{=}&\frac{b(1-\sigma)\left((\beta^{*}-\alpha(\beta^{*})-b\sigma)\frac{\alpha^{\prime}(\beta^{*})-(1-\hat{i}(\beta^{*}))}{b(1-\sigma)}+(1-\alpha^{\prime}(\beta^{*}))(1-\xi(\beta^{*}))\right)}{(\beta^{*}-\alpha(\beta^{*})-b\sigma)^{2}}\\ &\overset{(7)}{=}& \frac{1}{\beta^{*}-\alpha(\beta^{*})-b\sigma}\left[(1-\alpha^{\prime}(\beta^{*}))\underbrace{\left(-1+\frac{b(1-\sigma)(1-\xi(\beta^{*}))}{\beta^{*}-\alpha(\beta^{*})-b\sigma}\right)}_{-\hat{i}(\beta^{*})}+\hat{i}(\beta^{*})\right] \\ &=& \frac{\alpha^{\prime}(\beta^{*})\hat{i}(\beta^{*})}{\beta^{*}-\alpha(\beta^{*})-b\sigma}. \end{array} $$

So, it holds that

$$\begin{array}{@{}rcl@{}} M = \left.\frac{\partial^{2}s_{\beta}(\beta_{mut})}{\partial\beta\partial\beta_{mut}}\right|_{\beta_{mut}=\beta=\beta^{*}}&=& -\hat{i}^{\prime}(\beta^{*})\\&& = -\frac{\alpha^{\prime}(\beta^{*})\hat{i}(\beta^{*})}{\beta^{*}-\alpha(\beta^{*})-b\sigma}.\\ \end{array} $$

(18)

Since $0<\hat {i}(\beta ^{*})<1$ implies β ^∗−α(β ^∗)−b σ>0 (Proposition 1 in Appendix in Online Resource 1) and since we assume that the trade-off function for virulence α(β) is increasing in β, i.e., α ^′(β)>0, then α ^′(β ^∗)>0 and hence M<0. This implies that two strains in the neighborhood of an evolutionary singularity can invade each other. Moreover, if the condition (10) for convergence stability of the singularity is met then if E<0 the singularity is an evolutionary attractor, and if E>0 the singularity is a branching point. On the other hand, if the condition (10) is not met then if E>0, the singularity is a repellor. The Garden of Eden configuration is not possible for model (1) since it requires E+M>0 and E<0 which is never fulfilled.

Appendix: Appendix B: Evolution of pure sexual transmission for z≤1

We are able to prove that for z≤1 evolution may go to β _max. The selection gradient in this case is

$$\begin{array}{@{}rcl@{}} s_{\beta}(\beta_{mut}) &=& b(1-\sigma)\left(\xi(\beta_{mut})-\xi(\beta)\right)\\ &+& (1-\hat{i}(\beta))(\beta_{mut}-\beta) - (\alpha(\beta_{mut})-\alpha(\beta)). \end{array} $$

(19)

Since we assume evolution to proceed in small steps, β _{m u t} ≈β and we can approximate both ξ(β _{m u t} ) and α(β _{m u t} ) by the first two terms of the respective Taylor series as

$$ \begin{array}{l} \xi(\beta_{mut}) = \xi(\beta) + \xi^{\prime}(\beta)(\beta_{mut}-\beta), \\ \alpha(\beta_{mut}) = \alpha(\beta) + \alpha^{\prime}(\beta)(\beta_{mut}-\beta). \end{array} $$

(20)

Incorporating Eq. 20 into Eq. 19, we get

$$\begin{array}{@{}rcl@{}} &&{}s_{\beta}(\beta_{mut}) = (\beta_{mut}-\beta)\left[ b(1-\sigma)\xi^{\prime}(\beta)\right.\\&& \left.{\kern8pc}+ (1-\hat{i}(\beta)) - \alpha^{\prime}(\beta) \right] \end{array} $$

(21)

Note that the term in square parentheses is actually the selection gradient D(β). Substituting for $\hat {i}(\beta )$ the formula (A.1) from Appendix in Online Resource 1, we get

$$\begin{array}{@{}rcl@{}} &&{}s_{\beta}(\beta_{mut}) = (\beta_{mut}-\beta)\\&&{\kern3pc}\times\left[ b(1-\sigma)\xi^{\prime}(\beta) + \frac{b(1-\sigma)(1-\xi(\beta))}{\beta-\alpha(\beta)-b\sigma} - \alpha^{\prime}(\beta) \right].\\ \end{array} $$

(22)

Let z=1 and β=β _max. Then, ξ ^′(β _max)=−1/β _max, ξ(β _max)=0, and α ^′(β _max)>0 (we assume β _α >β _max). With these expressions, the term in square parentheses of Eq. 22 will equal

$$\begin{array}{@{}rcl@{}} -\frac{b(1-\sigma)}{\beta_{\max}} + \frac{b(1-\sigma)}{\beta_{\max}-\alpha(\beta_{\max})-b\sigma} - \alpha^{\prime}(\beta_{\max}).\\ \end{array} $$

(23)

We need to determine the sign of the expression (23). Adding up the first two terms and noting that R ₀(β _max)>1⇔β _max−α(β _max)>b we can proceed as follows:

$$\begin{array}{@{}rcl@{}} &&{}-\frac{b(1-\sigma)}{\beta_{\max}} + \frac{b(1-\sigma)}{\beta_{\max}-\alpha(\beta_{\max})-b\sigma}\\&& {\kern1.5pc}-\alpha^{\prime}(\beta_{\max}) < \frac{b(1-\sigma)(\alpha(\beta_{\max})+b\sigma)}{b(1-\sigma)\beta_{\max}}-\alpha^{\prime}(\beta_{\max})\\&& {}=\frac{\alpha(\beta_{\max})+b\sigma}{\beta_{\max}}-\frac{\alpha(\beta_{\max})}{\beta_{\max}}\frac{\beta_{\alpha}}{\beta_{\alpha}-\beta_{\max}}\\ &&{\kern2pc}= \frac{1}{\beta_{\max}}\left[-\alpha(\beta_{\max})\frac{\beta_{\max}}{\beta_{\alpha}-\beta_{\max}}+b\sigma\right]. \end{array} $$

(24)

If σ=0, the term in the square parenthesis of Eq. 24 is always negative. The selection gradient is therefore always negative and the invasion fitness is positive if and only if β _{m u t} <β _max. Hence, β _max will be repelling. On the other hand, if σ>0 the term in the square parenthesis of Eq. 24 will be positive if a will be sufficiently small and/or β _α ≫β _max. Therefore, the invasion fitness will be positive for β _{m u t} >β and β _max will be attracting.

If z<1 then ξ ^′(β _max)=0 and the term in the square parenthesis of Eq. 22 will be equal

$$ \frac{b(1-\sigma)}{\beta_{\max}-\alpha(\beta_{\max})-b\sigma} - \alpha^{\prime}(\beta_{\max}). $$

(25)

Again, since R ₀(β _max)>1⇔β _max−α(β _max)>b we have

$$\begin{array}{@{}rcl@{}} \frac{b(1-\sigma)}{\beta_{\max}-\alpha(\beta_{\max})-b\sigma}{\kern-1.5pt}&-&{\kern-1.5pt} \alpha^{\prime}(\beta_{\max}) < \frac{b(1-\sigma)}{b(1-\sigma)} - \alpha^{\prime}(\beta_{\max})\\ &=& 1 - \alpha^{\prime}(\beta_{\max}). \end{array} $$

(26)

Here, the sign of the selection gradient does not depend on σ. Positivity of the selection gradient is ensured whenever the expression (26) is positive, i.e., whenever α ^′(β _max)<1. This is fulfilled if a is sufficiently small and/or β _α ≫β _max. Then the invasion fitness is positive for β _{m u t} >β and β _max will be attracting.

Finally, if z>1, then ξ ^′(β)→−∞ as β→β _max−, the invasion fitness will be positive for β _{m u t} <β _max and β _max will not attract β from its neighborhood.

This implies that the pure sexual transmission can be an evolutionary outcome only if z<1 or if z=1 and σ>0.

Appendix: Appendix C: Impossibility of evolutionary branching for concave and linear vertical-sexual trade-offs

Here, we show that for concave vertical-sexual trade-offs ξ(β) with z≥1 evolutionary branching is impossible to occur. This statement directly follows from the fact that z>1 if and only if ξ ^′′(β)<0 for all 0≤β≤β _max and hence ξ ^′′(β ^∗)<0. This implies

$$E = b(1-\sigma)\xi^{\prime\prime}(\beta^{*})-\alpha^{\prime\prime}(\beta^{*})<0 $$

since α ^′′(β)>0 for all 0≤β≤β _α and therefore α ^′′(β ^∗)>0. Consequently, the condition for convergent stability E+M<0 is always satisfied since M<0 (see Appendix A).

The second part of the statement follows from the fact that z=1 if and only if ξ ^′′(β)=0 for all 0≤β≤β _max and hence ξ ^′′(β ^∗)=0. This implies

$$E = b(1-\sigma)\xi^{\prime\prime}(\beta^{*})-\alpha^{\prime\prime}(\beta^{*}) = -\alpha^{\prime\prime}(\beta^{*})<0 $$

since α ^′′(β)>0 for all 0≤β≤β _α and therefore α ^′′(β ^∗)>0. Consequently, the condition for convergent stability E+M<0 is always satisfied since M<0.