Fig 1.
The transmission process for NPT viruses, and how it can be manipulated by viruses changing infected plant attractiveness (v) and acceptability (ε).
The virus can be acquired by the aphid when it probes and does not feed on an infected plant. Inoculation can occur when the aphid then probes/feeds on a susceptible plant. Virus transmission requires both acquisition and inoculation. Potential plant phenotypes caused by viruses, and the stage of transmission process they affect, are reflected by the coloured arrows, and are split into effects on infected plant attractiveness (v) and acceptability (ε). These effects combine to create the phenotypes ‘attract-and-deter’ (blue and green) and ‘attract-and-retain’ (blue and pink). I = infected plant. S = susceptible plant. Note, in reality, NPT viruses are retained on an aphid’s stylet (mouthparts), not within its body.
Table 1.
Features and limitations of our representative previous models, in terms of aphid transmission of NPT viruses.
Our MIP-BAR model combines features of previous models for a richer representation of transmission.
Table 2.
Table showing model parameters and their default values across the MIP, BAR and MIP-BAR models.
Not all parameters are common to all models, as indicated by the ticks and crosses.
Fig 2.
Multiple Infective Probes (MIP), Behaviour-based Aphid Rated (BAR) and MIP-BAR models.
(a) MIP and MIP-BAR model structures. The MIP model (Eqs 1–4 in main text) has the purple parameters (ϕ, aphid dispersal rate and τ, aphid infectivity loss rate) as constants. The MIP-BAR model (Eqs 24 and 25 in main text) uses the aphid behaviour-based expressions and
as shown in the purple boxes. (b & c) BAR model structure. (b) Compartmental model structure (Eqs 11 and 12 in main text). (c) Aphid feeding dispersal Markov chain as introduced by Donnelly et al. [17], the expected value of which,
, is fed into the model structure in (b). For all panels, red arrows = virus acquisition by uninfective aphids, blue arrows = plant inoculation by infective aphids. Virus transmission consists of acquisition and inoculation.
= the proportion of infected plants weighted by the infected plant attractiveness parameter, v. For definitions of all parameters, see Table 2.
Fig 3.
Probability tree of aphid landing, probing and feeding behaviour upon which the expression for , the rate of aphid infectivity loss, in the MIP-BAR model is based.
It gives probabilities for each stage of the process (from left to right) from when an infective aphid disperses (with rate , Eq 14) through (a) landing, (b) probing and (c) feeding on or rejecting a new plant, which results in the aphid either maintaining or losing the virus (i.e. its infectivity). Red arrows = paths that lead to the aphid losing infectivity (which when combined create
, Eq 20). Green arrows = paths that lead to infectivity retention by the aphid. S = number of susceptible plants. I = number of infected plants. ρ = probability of infectivity loss from probing. a = probability of virus acquisition from probing an infected plant. ω = probability of feeding on a plant. v = degree of virus-induced plant attractiveness. ε = degree of virus-induced plant acceptability.
Fig 4.
MIP-BAR model is more similar to the BAR than the MIP model.
(a) Epidemic trajectories for MIP, BAR and MIP-BAR models using the default parameterisation (Table 2). Note it is difficult to distinguish the BAR and MIP-BAR models for this parameterisation and for the default initial condition. (b) Comparison of BAR trajectory to MIP-BAR trajectories at the default parameterisation, varying Z(0) (the initial number of infective aphids) in the MIP-BAR model. The inset shows behaviour for small values of t (up to 1 day). (c) I-Z phase plane for MIP-BAR model, showing I cline (dI/dt = 0 in Eq 24; blue) and Z cline (dZ/dt = 0 in Eq 25; green), and epidemic trajectories from (b). The inset shows a zoomed in section of the graph for small values of Z/A and I/H, to make it clearer where the model trajectories hit the Z cline.
Fig 5.
The MIP and MIP-BAR model predictions vary differently across parameter values, driven by variability of behaviour-based aphid rates and
in the MIP-BAR model.
Each column has the same plot for: ω (aphid feeding probability; top row), a (aphid virus acquisition rate; middle row) and Γ (plant death/replanting rate; bottom row) parameters, respectively. Left (a)/(d)/(g): parameter versus MIP-BAR/MIP model aphid infectivity loss rate (rate at equilibrium for MIP-BAR model, calculated as in Eq 20); Middle (b)/(e)/(h): parameter versus MIP-BAR/MIP model aphid dispersal rate (rate at equilibrium for MIP-BAR model, calculated as in Eq 14); Right (c)/(f)/(i): parameter versus disease incidence (equilibrium I/H) for MIP, MIP-BAR and BAR models. For all plots, the red point signifies the default parameterisation. Apart from the parameter being altered in each graph, all parameters are at their default values in all plots (Table 2). Note that in all cases the disease incidence for the BAR model and the MIP-BAR model are identical; note further that the parameters and
are not defined for the BAR model.
Fig 6.
MIP-BAR and MIP models respond differently to virus manipulation parameters v (infected plant attractiveness) and ε (infected plant acceptability), leading to different predictions for effects of ‘attract-and-deter’ and ‘attract-and-retain’ phenotypes.
(a-f) Columns represent the same plots for the two virus manipulation parameters, ε (infected plant acceptability) and v (infected plant attractiveness). Left (a)/(d): parameter versus MIP-BAR/MIP model aphid infectivity loss rate (rate at equilibrium for MIP-BAR model, calculated as in Eq 20); Middle (b)/(e): parameter versus MIP-BAR/MIP model aphid dispersal rate (rate at equilibrium for MIP-BAR model, calculated as in Eq 14); Right (c)/(f): parameter versus disease incidence (equilibrium I/H) for MIP, MIP-BAR and BAR models. For all plots, the red dot signifies the default parameterisation. Apart from the parameter being altered in each graph, all parameters are at their default values in all plots (Table 2). Note that in all cases the disease incidence for the BAR model and the MIP-BAR model are identical; note further that the parameters and
are not defined for the BAR model. (g-i) Heatmaps across v-ε parameter space of: (g) MIP model disease incidence, (h) MIP-BAR model disease incidence, (i) Difference in disease incidence between MIP-BAR and MIP models (MIP-BAR—MIP). Grey = areas of multiple stable I/H equilibria (or where difference cannot be calculated due to this, for (i)). Dotted lines are at v = 1 and ε = 1, lines of no virus manipulation. To the right of the line of v = 1 represents the ‘attract’ phenotype. Above the line ε = 1 is the ‘retain’ phenotype, and below is ‘deter’. All parameters except v and ε are at their default values in all plots (Table 2).
Fig 7.
Relaxing the assumption of aphids always losing infectivity from probing (ρ = 1) increases epidemic size in the MIP-BAR model.
(a) Epidemic trajectories (proportion of infected plants over time) for different values of ρ, under otherwise default parameterisation. Purple line = default parameterisation (ρ = 1. equivalent to the BAR model). (b) Relationship between ρ and disease incidence at equilibrium. (c) Effect of ρ on the probability of virus retention by the aphid after probing, for both infected and susceptible plants, for different values of a (probability of aphid virus acquisition from an infected plant). When probing infected plants, the retention probability = (1 − ρ) + ρa, where a is the probability of virus (re-)acquisition from the plant from probing. For susceptible plants, the retention probability is always just 1 − ρ.
Fig 8.
The effects of the ‘attract-and-deter’ and ‘attract-and-retain’ phenotypes on epidemic size are driven by the changes they induce in aphid dispersal, , and infectivity loss,
, rates, and vary to differing degrees with decreasing ρ.
Line colours represent virus-induced plant phenotypes: blue = no virus manipulation (v = ε = 1), red = ‘attract-and-deter’ scenario (v = 1.5, ε = 0.5), green = ‘attract-and-retain’ scenario (v = 1.5, ε = 2). Linetype represents the value of ρ, the probability of an aphid losing infectivity (the virus) when probing, for (a)-(f). (a) Model trajectories under each plant phenotype, for ρ = 1. (b) Model trajectories under each plant phenotype, for ρ = 1 and ρ = 0.5. (c) Rate of aphid infectivity loss, (Eq 20), across proportion of plants infected (I/H) for each phenotype. (d) Aphid dispersal rate,
(Eq 14), across proportion of plants infected (I/H), for each phenotype. (e) Aphid rate of virus acquisition (
, the positive term in dZ/dt, Eq 22, but without multiplication by X in order to make it a per-aphid rate) across proportion of plants infected (I/H), for each phenotype. (f) Rate of aphid infectivity loss,
(Eq 20), across proportion of plants infected (I/H) for each phenotype, for ρ = 1 and ρ = 0.5. (g) Disease incidence (equilibrium I/H) across values of ρ for each phenotype, as well as for an additional ‘retain’ phenotype (v = 1, ε = 2). (h) Difference in disease incidence from its value at ρ = 1 (i.e. difference is 0 at ρ = 1) across values of ρ, for each phenotype, including the aforementioned additional ‘retain’ phenotype. All other parameters are at their default values.