Periodic host absence can select for higher or lower parasite transmission rates

Abstract

This paper explores the effect of discontinuous periodic host absence on the evolution of pathogen transmission rates by using R ₀ maximisation techniques. The physiological consequence of an increased transmission rate can be either an increased virulence, i.e. there is a transmission-virulence trade-off or ii) a reduced between season survival, i.e. there is a transmission-survival trade-off. The results reveal that the type of trade-off determines the direction of selection, with relatively longer periods of host absence selecting for higher transmission rates in the presence of a trade-off between transmission and virulence but lower transmission rates in the presence of a trade-off between transmission and between season survival. The fact that for the transmission-virulence trade-off both trade-off parameters operate during host presence whereas for the transmission-survival trade-off one operates during host presence (transmission) and the other (survival) during the period of host absence is the main cause for this difference in selection direction. Moreover, the period of host absence seems to be the key determinant of the pathogen’s transmission rate. Comparing plant patho-systems with contrasting biological features suggests that airborne plant pathogens respond differently to longer periods of host absence than soil-borne plant pathogens.

Appendices

Appendix: Proof for R ₀ maximisation

The proof is an adaptation of the one given, e.g., by Nowak and Sigmund (2002) for the case without seasonality. Given that S(t) = 0 for nT + τ < t < (n + 1)T, the original nonlinear model can also be written as (cf. Eqs. 7–9 in main text)

$$ {\frac{{dZ_{r} }}{dt}} = \beta_{r} Z_{r} S - B_{r} (t)Z_{r} ,\quad\,{\frac{{dZ_{i} }}{dt}} = \beta_{i} Z_{i} S - B_{i} (t)Z_{i} $$

(18)

Here subscripts ‘r’ and ‘i’ represent the resident and invader strain, respectively. Some rearranging and subsequently subtracting the invader equation from the resident equation leads to

$$ {\frac{1}{{\beta_{r} Z_{r} }}}\,{\frac{{dZ_{r} }}{dt}} - {\frac{1}{{\beta_{i} Z_{i} }}}\,{\frac{{dZ_{i} }}{dt}} = - {\frac{{B_{r} (t)}}{{\beta_{r} }}} + {\frac{{B_{i} (t)}}{{\beta_{i} }}}. $$

(19)

Assume $R_{0}^{(i)} > R_{0}^{r} > 1$. Integrating and letting t tend to infinity, we find

$$ \begin{aligned} {\frac{1}{{\beta_{r} }}}\log \left( {{\frac{{Z_{r} (t)}}{{Z_{r} (0)}}}} \right) - {\frac{1}{{\beta_{i} }}}\log \left( {{\frac{{Z_{i} (t)}}{{Z_{i} (0)}}}} \right) = & - {\frac{1}{{\beta_{r} }}}\int\limits_{0}^{t} {B_{r} (u)du} + {\frac{1}{{\beta_{i} }}}\int\limits_{0}^{t} {B_{i} (u)du} \\ & \sim \frac{t}{T}\left( { - {\frac{1}{{\beta_{r} }}}\int\limits_{0}^{t} {B_{r} (u)du} + {\frac{1}{{\beta_{i} }}}\int\limits_{0}^{t} {B_{i} (u)du} } \right), \\ & \sim \frac{t}{T}\left( { - {\frac{1}{{R_{0}^{(r)} }}} + {\frac{1}{{R_{0}^{(i)} }}}} \right)\int\limits_{0}^{\tau } {S^{*} (u)du,} \\ \to - \infty \\ \end{aligned} $$

(20)

So either Z _i (t) → +∞ or Z _r (t) → 0 as t → ∞. The crop population size is constrained by a carrying capacity which means that Z _i (t) → +∞ is impossible, resulting in Z _r (t) → 0 being the only possible solution, leading to “competitive exclusion”. The only strain remaining is the one with the highest value of R ₀.

Appendix: Marginal value theory and the evolution of pathogen transmission rates

In this appendix we show that the R ₀ maximisation problem as described in this paper is susceptible to the marginal value theorem and graphical analysis from evolutionary ecology (Charnov 1976; Stephens and Krebs 1986). To improve comparison with the existing literature we assume that not the transmission rate, but the virulence (in the case of the transmission-virulence trade-off) or the between season pathogen death rate (in the case of the transmission-survival trade-off) evolves. The trade-off parameters are however subject to the same constraint, which means that this redefinition does not affect the results as discussed in the main text.

The transmission-virulence trade-off

The optimal virulence, given the trade-off constraints

$$ \beta = g(\alpha ),\,{\frac{dg}{d\alpha }} > 0\quad{\text{and}}\quad{\frac{{d^{2} g}}{{d\alpha^{2} }}} < 0 $$

(21)

can be calculated from dR ₀/dα = 0, leading to

$$ g(\alpha ) = {\frac{dg}{d\alpha }}[\alpha + d - \mu + \mu T/\tau - \log (\theta_{1} \theta_{2} )/\tau $$

(22)

In analogy to the graphic solutions from optimal foraging theory (Charnov 1976; Stephens and Krebs 1986) this expression can be depicted as illustrated in Fig. 2a. The curve represents the trade-off following the constraints as set out in (21) and the straight line represents the solution of (22) and is the tangent of trade-off curve. The optimal strategy is given by the point where these two lines intersect. From this graph it can immediately be seen that $\left( {d + \mu {\frac{T - \tau }{\tau }} - {\frac{{\log (\theta_{1} \theta_{2} )}}{\tau }}} \right)$ and hence the ESS virulence value, α_ESS, increases when the host growing period, τ, decreases.

Fig. 2

Graphical representation of the optimal strategy for a hypothetical trade-off relation between a transmission and virulence and b transmission and between season pathogen death rate. The tangent is given by the solution of dR ₀/dα = 0 and dR ₀/dμ = 0 for the transmission-virulence and the transmission-survival trade-off, respectively. The optimal strategy is given by the point where the two lines intersect

The ESS condition can also be written in a marginal value form

$$ {\frac{d[\tau /T]g(\alpha )}{d[\tau /T]\alpha }} = {\frac{{[{\tau \mathord{\left/ {\vphantom {\tau T}} \right. \kern-\nulldelimiterspace} T}]g(\alpha )}}{{[{\tau \mathord{\left/ {\vphantom {\tau T}} \right. \kern-\nulldelimiterspace} T}](\alpha + d) + [(T - \tau )/T]\mu - \log (\theta_{1} \theta_{2} )}}} $$

(23)

which states that the increment in year averaged infectivity per increment in year averaged death rate should match the year averaged infectivity divided by the year averaged death rate. From this it can directly be derived that decreasing the host growing season, τ, means that the relative contribution of virulence, α, to the year averaged death rate decreases, which allows for the selection of higher virulence and therewith higher transmission rates.

The transmission-survival trade-off: The optimal between season pathogen death rate, given the trade-off constraints

$$ \beta = g(\mu ),\,{\frac{dg}{d\mu }} > 0\,\quad{\text{and}}\,\quad{\frac{{d^{2} g}}{{d\mu^{2} }}} < 0 $$

(24)

can be calculated from dR _o/dμ = 0, leading to

$$ g(\mu ) = {\frac{dg}{d\mu }}\left[ {(\alpha + d){\frac{\tau }{T - \tau }} + \mu - {\frac{{\log (\theta_{1} \theta_{2} )}}{T - \tau }}} \right] $$

(25)

Figure 2b shows how the optimal between season pathogen death rate can be derived graphically in analogy to the graphic solutions from optimal foraging theory (Charnov 1976; Stephens and Krebs 1986). From this graph it can immediately be seen that ${\frac{(\alpha + d)\tau }{T - \tau }} - {\frac{{\log (\theta_{1} \theta_{2} )}}{T - \tau }}$ and hence the ESS between season pathogen death value, μ_ESS, decreases when the host growing period, τ, decreases.

The ESS condition can also be written in a marginal value form

$$ {\frac{{d[{\tau \mathord{\left/ {\vphantom {\tau T}} \right. \kern-\nulldelimiterspace} T}]g(\mu )}}{{d[{\tau \mathord{\left/ {\vphantom {\tau T}} \right. \kern-\nulldelimiterspace} T}]\mu }}} = {\frac{{[{\tau \mathord{\left/ {\vphantom {\tau T}} \right. \kern-\nulldelimiterspace} T}]g(\mu )}}{{[{\tau \mathord{\left/ {\vphantom {\tau T}} \right. \kern-\nulldelimiterspace} T}](\alpha + d) + [{{(T - \tau )} \mathord{\left/ {\vphantom {{(T - \tau )} T}} \right. \kern-\nulldelimiterspace} T}]\mu - \log (\theta_{1} \theta_{2} )}}} $$

(26)

which states that the increment in year averaged infectivity per increment in year averaged death rate should match the year averaged infectivity divided by the year averaged death rate. From this it can directly be derived that decreasing the host growing season, τ, means that the relative contribution of the between season pathogen death rate, μ, to the year averaged death rate increases, which suggests the pathogen should invest in a better survival strategy leading to the selection of lower between season pathogen death rates and therewith lower transmission rates.