The evolution of stage‐specific virulence: Differential selection of parasites in juveniles

Abstract The impact of infectious disease is often very different in juveniles and adults, but theory has focused on the drivers of stage‐dependent defense in hosts rather than the potential for stage‐dependent virulence evolution in parasites. Stage structure has the potential to be important to the evolution of pathogens because it exposes parasites to heterogeneous environments in terms of both host characteristics and transmission pathways. We develop a stage‐structured (juvenile–adult) epidemiological model and examine the evolutionary outcomes of stage‐specific virulence under the classic assumption of a transmission‐virulence trade‐off. We show that selection on virulence against adults remains consistent with the classic theory. However, the evolution of juvenile virulence is sensitive to both demography and transmission pathway with higher virulence against juveniles being favored either when the transmission pathway is assortative (juveniles preferentially interact together) and the juvenile stage is long, or in contrast when the transmission pathway is disassortative and the juvenile stage is short. These results highlight the potentially profound effects of host stage structure on determining parasite virulence in nature. This new perspective may have broad implications for both understanding and managing disease severity.


A Invasion analysis
The epidemiological dynamics is given by: (A.1) with the notation explained in the main text; here, for the sake of generality, we incorporated recovery J , A , which Noting that 0 ≤ A ≤ 1, we have: (A.8) Integrating with respect to firstly and then integrating with respect to A , we have: as shown in the main text.

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Note that if + J = A , then A is of the form "0∕0". As such A is interpreted as the limit lim which is the probability of maturation ( S ) times the conditional expectation of the fraction of sub-lifespan as an 23 adult (given that a sampled adult host has matured into an adult). This calculation is obtained by setting exp( ) ∶= 24 A ∕( + J ) and using the Taylor expansion exp( ) = 1 + + 2 2 + ( 3 ) where () represents the Landau's big- 25 for → +0. Exact computation including the evaluation of integral is shown in a Mathematica-code (SI Fig 1). 27 Hereafter, without special remarks, we will assume that ⪇ 1 (i.e., transmission can occur between classes).

A.3 Mutant dynamics
28 When = 1, as shown in Osnas & Dobson (2011), a special treatment is needed. 29 Evaluating the stage-period requires variable-transformation, but Mathematica can skip this task. (A.10) Here, Elementary algebra of matrices gives the matrix-product form of ′ in the main text. 35 The dominant eigenvalue of ′ (denoted Λ[ ′ ]) is given by: (A.14) Here note that under weak selection (i.e., when | ′ − | is negligibly small) and the continuity of An elementary calculation (using the endemic condition for the ODE, It is only the final factor that can change its sign (see Footnote 2 in Appendix A.7). To obtain the selection gradient 54 for juvenile virulence, more tedious work is needed. As such, we will use Fisher's reproductive value (Fisher 1958;55 Taylor 1990; Frank 1998; Caswell 2001).

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A.6 Reproductive values 57 We here provide the reproductive value-based approach. Note that the case = 1 violates this approach.
then, we can get: Since the eigenvalue of • is unity, premultiplying the left eigenvector Although it is possible to analytically solve ( • J , • A ), it does not lead to a transparent expression. Therefore, we (where is the identity matrix), which explicitly (in elements) reads:

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Under weak selection, where (| ′ − |) represents the Landau's big- (for | ′ − | → 0) such that the latter two terms both tend towards In terms of , and I , the invasion fitness reads: Specifically, the fitness subcomponents involving ′ J amount to 4 :  For the completeness, we can similarly get: Note that the first multiplicative term is always positive (see Footnote 2 and eqn (A.28)). 82 and thus * A = √ A ∕ A in the absence of recovery ( A = 0).

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A.8 Graph-theoretical approach 85 We here employ the graph-theoretical approach

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The premise of the approach is to decompose fecundity output and state-transitions as in the next-generation 90 theorem. The Jacobian around the endemic equilibrium reads: .
( Rule A: Self-loop elimination (trivialization) To reduce the loop ′ XX (which is < 0) to −1 at node X, every arc 94 entering X has weight divided by − ′ XX  ′ m (SI Fig 2A).
the weight given as the sum of the two weights (SI Fig 2B).

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Rule C: trivial node elimination For a trivialized node Y on a path X → Y → Z, the two arcs are replaced by a 98 single arc X → Z with weight equal to ′ XY times ′ YZ . Weights on multiple arcs X → Z are added. If there 99 are no more paths through the trivial node Y, then it can be disregarded (SI Fig 2C).

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Applying these rules, we can obtain the invasion condition, of: is a matter of preference, all giving the same result for selection gradients and stability 104 analyses. Taking advantage of deriving  ′ m (Eqn (A.35)), we can simplify the stability analysis (see the next 105 subsection).

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Here we outline the stability analyses for the evolutionary dynamics. Since the invasion fitness is not explicitly 108 dependent on wild type strategy, the evolutionary stability and attainability conditions necessarily coincide (Otto 109 & Day 2007). For this reason, we need only work on the evolutionary stability condition.

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From Eqn (A.35), the invasion fitness is, in a product form, given by: from which we can say that: indicating that the Hessian matrix  of  ′ m at SS be given as a diagonal matrix; indeed: The "original" diagram depicting the pathways of reproductive success. ′ J and ′ A both have self-loop, so we will apply "self-loop" elimination rule. In addition, we apply parallel path elimination rule (by summing the transition, , and the reproductive success of parasites infecting juveniles to adults through transmission, ′ AJ ), obtaining (E): besides two trivial edges ("−1"), two nodes loop mutually and we apply node elimination rule, ending up with (F): the reproductive success of parasites infecting adults, the total number of "secondary" infection by mutant parasites, with all possible transmission-pathways included.
which with straightforward calculations gives: as desired; note that if = 1 then this second derivative is always null at the SS, meaning that any mutants in A 118 are selectively neutral at the SS.
which completes the proof of the statement Eqn (A.39).

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A.10 Condition for parasite persistence 128 In the absence of diseases, from which we can get: Parasites attempting to invade such a disease-free, stage-structured host population can establish only if: In this case, obtaining the selection gradient is not needed. Instead, we can directly see that the evolutionary stability 137 condition reads: for any ′ ≠ . This is thus obtained by jointly maximizing two functions ′ giving the CSS as ) .

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In the main text, we have assumed: • Fecundity is the same for susceptible and infected adults.

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Here we will check the robustness of our prediction against these variants. Specifically, we will work on the We used relatively small values of ( J , A ) in the ODE, because high recovery can readily result in parasite 162 extinction. We again numerically obtained the CSS virulence and plotted them on the ( , A )-plane. We can see that 163 our prediction is qualitatively robust against this variant. Quantitative differences are that recovery can in general 164 favour fast exploitation, which is obvious from the CSS for adult virulence, In the numerical examble, A = 0.25, A = A = 1 yields * A = √ 5∕2 ≈ 1.118. As for juvenile virulence * J , the 166 general trend is unchanged (SI Fig 5).

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As recovery increases, evolutionary suicide is more readily to occur (white zone). This is so because parasites  Overall, the effects of recovery are similar to those of mortality (see Figure 2 in the main text).

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We here introduce a difference in 's, which corresponds to the situation where juveniles and adults show 173 quantitatively different transmission-blocking mechanisms. This does not affect the results critically; a difference 174 is that evolutionary suicide is more likely to occur with smaller 's.

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Tolerance, or reduced negative impacts of the disease on hosts, can affect the tradeoff through X . For 177 simplicity, we assume that X is constant (see next section). To incorporate tolerance, we further decompose parasite-induced mortality into X = ( 1 − X ) X , where X tunes tolerance and X represents exploitation.
whereas a derivative is given by: which is a constant for each X (with X = J or A). Higher tolerance (larger X ) leads to larger X ∕(1 − X ).

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Marginal value theorem (Charnov 1976) shows that SS solves: Hence SS for A is smaller with tolerance. To look at the consequences for J , 184 we again solved the equations, observing that the results are qualitatively unchanged.

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We assess the effects of varying X . Obviously, increasing X results in higher transmission but does not affect 187 the SS for adult virulence (SI Fig 6).

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Because the densities would be of greater importance to the force of infection with this assumption, we used a 190 smaller value of J = A = 0.13. We found quantitatively similar outcomes (SI Fig 7). 192 We here explore the effects of fecundity shifts on evolution of virulence, looking at the possibility that parasites 193 deprive some amounts of resource of infected hosts that would have been otherwise available to the hosts for 194 reproduction. We do so by considering two models: in the first model, we assume that the fecundity shift in 195 adults, denoted ℎ, is a constant (ℎ can be negative). We consequently found that the results are robust.  We further impose more constraints. In the first case, we assume JJ = 1 − AJ and AA = 1 − JA (normalized 211 pathway) and varying JJ and AA ; the second is to fix AA and vary JJ and JA = AJ (symmetric pathway). We conducted several literature searches in Google Scholar combining the terms "age-related" or 223 "age-dependent" or "stage-dependent" or "juvenile" + "susceptibility" or "resistance" or "tolerance" or 224 "immunocompetence" + "infection" or "infectious disease". From these searches, we collected data from papers 225 where the parasite could be judged to be adapted to its host (i.e., not a recent host shift and without significant multi-226 species transmission) and where differences in virulence across life stages could be distinguished from age-related 227 trends in additional mortality due to increasing adaptive immunity with age due to previous exposure and increased 228 mortality of poor-condition hosts during the juvenile stages. Therefore, we collected data from papers for host-229 pathogen systems where adaptive immunity to the pathogen was not significant or infection-related mortality was 230 measured in naïve juveniles and adults in either a natural population or in an experimental lab population. From 231 the papers that we found, we also searched their citations and papers that cited them for other publications that we 232 may have missed in the first search. After we had found papers with reliable data on age-biased virulence, we 233 searched for "host" and "life history" or "age at reproduction" to find data on the host's maturation rate. Finally, we 234 searched for transmission assortativity data for each selected system by searching the terms "host"+ "transmission" 235 or "contact network"+ "age" or "stage" or "juvenile". We used estimated values of J versus A . The extracted 236 data are plotted against a ( , A )-plane.

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Concerning the data on asian elephants (Elephas maximus), we assessed the relative virulence J ∕ A from the

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We propose that future studies quantifying stage-dependent parasite prevalence is greatly promising to test our Due to the tolerance, the number of infected adults increase with ℎ A . Overall, the qualitative trend is unchanged.   Figure 6: Effects of varying infectiousness. Different infectiousness can lead to higher virulence for juveniles; note that * A = √ A ∕ A is independent of A and J . Overall, the qualitative trend is unchanged, but the disease prevalence among juveniles is dramatically lower with assortativity. SI Figure 7: Effects of density-dependence. SI Figure 8: Effects of constant fecundity virulence. The factor 1 − ℎ measuring the fecundity of infected adults. The resulting difference is minor, as fecundity reduction acts only via ecological feedback without any direct effects on the invasion fitness. Also note that in panel (A), the fecundity is higher for infected than for susceptible adults.  Figure 9: Effects of the normalized pathway structure, with JA = 1 − AA and AJ = 1 − JJ . Orthogonal dashed line, which satisfies = JJ + AA − 1 = 0, gives * J = * A . Note, we fixed J = A = 1, and thus A is a function of (e.g., = 1 gives A = 0.306853).