The Evolution of the Age of Onset of Resistance to Infectious Disease

Abstract

Many organisms experience an increase in disease resistance as they age, but the time of life at which this change occurs varies. Increases in resistance are partially due to prior exposure and physiological constraints, but these cannot fully explain the observed patterns of age-related resistance. An alternative explanation is that developing resistance at an earlier age incurs costs to other life-history traits. Here, we explore how trade-offs with host reproduction or mortality affect the evolution of the onset of resistance, depending on when during the host's life cycle the costs are paid (only when resistance is developing, only when resistant or throughout the lifetime). We find that the timing of the costs is crucial to determining evolutionary outcomes, often making the difference between resistance developing at an early or late age. Accurate modelling of biological systems therefore relies on knowing not only the shape of trade-offs but also when they take effect. We also find that the evolution of the rate of onset of resistance can result in evolutionary branching. This provides an alternative, possible evolutionary history of populations which are dimorphic in disease resistance, where the rate of onset of resistance has diversified rather than the level of resistance.

Keywords: Adult; Juvenile; Parasite; Pathogen; Resistance; Susceptibility.

Fig. 1

a Model schematic for a monomorphic population. b Examples of reproduction trade-off functions (with $a_0=5$). c Examples of mortality trade-off functions (with $b_0=1$). Trade-off strength is controlled by the parameter $c_1^i$; a relatively strong trade-off ($c_1^i=0.5$, red) results in a much larger reduction in the birth rate for a given level of adult resistance than a relatively weak trade-off does ($c_1^i=0.25$, blue). Trade-off shape is controlled by the parameter $c_2^i$; a positive value ($c_2^i=2$, solid) means that the costs of resistance decelerate (increasing returns) as the rate of onset gets faster whereas a negative value ($c_2^i=-1$, dashed) leads to accelerating costs (diminishing returns)

Fig. 2

Evolution of the rate of onset of resistance, ${\zeta }^{\ast }$, in the case of constant costs of resistance. Parameters used are as in Table 1 except for $\beta =1$, with a $1-f=0$, $1-h=0$ and $\alpha =1$, b $1-f=0$, $1-h=0$ and $\delta =0.25$, c $1-f=0.5$, $\alpha =0$ and $\delta =0$ and d $1-h=0.25$, $\alpha =0$ and $\delta =0$

Fig. 3

Simulated trajectories of the evolution of the rate of onset of resistance over time. The red, dashed lines show the analytically determined values of the singular strategies. Parameters used are as in Table 1 except for $\beta =1$, with a $1-f=0$, $\alpha =1$, $1-h=0$ and $\delta =0.25$ in the case of scenario (2) where the host pays a constant mortality cost when resistant and with b $1-f=0.5$, $\alpha =0$, $1-h=0.25$ and $\delta =0$ in the case of scenario (1) where the host pays a constant fecundity cost when resistant

Fig. 4

Heatmaps showing the evolved rate of onset of resistance as mortality and sterility virulence vary and for qualitatively different trade-offs. Contours show the proportion of the host population that is resistant to the disease and so do not correspond to heatmap colours. The rate of onset of resistance is highest (red) when mortality virulence is low and sterility virulence is high (bottom right of each panel). Increasing sterility virulence (left to right within each panel) always causes the rate of onset of resistance to increase. Increasing mortality virulence (bottom to top within each panel) may also cause the rate of onset of resistance to rise and then fall for sufficiently low values of sterility virulence. These patterns are broadly consistent across all six trade-off scenarios (a–f). Parameters used are as in Table 1, except $c_2^a=c_2^b=-2$ (accelerating costs), $h=0.75$ and $\delta =0.25$, where applicable

Fig. 5

The effect of disease transmissibility $\left(\beta \right)$ on the rate of onset of resistance, when either a fecundity cost (blue curves) or a mortality cost (red curves) is paid throughout the lifetime of the host (scenarios 3 and 4). The qualitative effects of transmissibility are the same whether the costs of fast onset of resistance are low ($c_1^a\text{or}{c}_1^b=0.3$; solid) or high ($c_1^a\text{or}{c}_1^b=0.8$; dotted). Parameters used are as in Table 1 with $c_2^a=-2$, $c_2^b=-2$, $f=0.5$ and $\alpha =0$

Fig. 6

Simulated trajectories of the evolution of the rate of onset of resistance over time. a When costs of fast onset of resistance are paid throughout the lifetime of the host, the lower branch often tends towards a low, non-zero value whereas the upper branch goes to infinity. b When costs are paid only before the onset of resistance, the lower branch goes to zero whereas the upper branch goes to infinity. Parameters used are as in Table 1 with $1-f=0.5$ and $\alpha =0$. We also use a $c_2^a=5$ and b $c_2^a=1$ to generate branching points

Fig. 7

The effect of the shape of trade-offs on the incidence of branching points for different timings of costs. Continuously stable strategies are shown in green (solid), repellers are shown in orange (dotted) and branching points are shown in purple (dashed). a When costs of fast onset of resistance are paid throughout the host’s lifetime, costs must be strongly decelerating (high $c_2^a$) for branching to occur. b When costs are paid only before the onset of resistance, they must be weakly decelerating for branching to occur. Parameters used are as in Table 1 with $1-f=0$ and $\alpha =1$