Table 1.
Host-pathogen interactions in a mixture composed of 2 resistant varieties.
Each resistant variety (row) corresponds to a single resistance gene (either V1 or V2). There are three possible pathogen genotypes that are able to infect at least one variety (columns): av1/Av2, Av1/av2 and av1/av2 (Av means “avirulent” and av means “virulent”). For instance, av1/av2 means that this pathogen genotype is able to infect both V1 and V2: this is a doubly virulent pathogen. In contrast, av1/Av2 and Av1/av2 cannot infect V2 and V1, respectively, but instead trigger immune priming on V2, and V1, respectively. They are monovirulent pathogen genotypes. We ignore the doubly avirulent (Av1/Av2) pathogen genotype, since it can infect none of the resistant varieties in the mixture, and therefore can not get established in the type of host mixtures considered in this study. The symbol + means infection and the * means priming.
Fig 1.
Flow diagram of model 1.
S1 and S2 are uninfected hosts of varieties V1 and V2, respectively. and
are primed hosts of varieties V1 and V2, respectively. I1 and I2 are hosts of varieties V1 and V2 (respectively) that are infected by the corresponding monovirulent pathogen genotype. J1 and J2 are hosts of varieties V1 and V2 (respectively) that are infected by the doubly virulent pathogen genotype. The square grouping J1 and J2 represents the doubly virulent pathogen genotype. The dashed arrows represent interactions leading to infection or priming. The gray color represents the attenuating effect of priming.
Table 2.
Model parameters and variables.
Table 3.
Summary of model (2) equilibria, with their positiveness and stability conditions.
All conditions are necessary and sufficient conditions except those marked with an exclamation mark, which are only sufficient conditions in general. The expressions of are given by equations S5, S7, S13, S14 in S1 Text, respectively. The exclamation mark means that we have no explicit conditions in general. The meaning of the parameters can be found in Table 2. The “extra” stability conditions are those that are not redundant with the positiveness conditions.
Fig 2.
Prevalence of the disease (P, black line) at equilibrium as a function of the proportion of resistance 2 (p), when varieties epidemiologically interchangeable (c1 = c2).
(A) When priming occurs (ρ = 0.7), the optimal proportion deviates from p = 0.5. (B) In absence of priming (ρ = 0), the disease prevalence is minimized for a range of p values. The colored areas correspond to different genetic compositions of the pathogen population at equilibrium. From left to right: monovirulent 1 only, coexistence of monovirulent 1 and doubly virulent, doubly virulent only, coexistence of monovirulent 2 and doubly virulent, and monovirulent 2 only. Parameter values: R = 7, ν = 1, and c1 = c2 = 0.4. The prevalences are the same at the edges (p = 0 and p = 1, in which one or the other monovirulent genotype is present) or in the middle region (in which only the doubly virulent genotype is present) regardless of whether priming occurs (A) or not (B). Priming only has an effect in the intermediate regions, in which a monovirulent genotype coexists with the doubly virulent one (in the absence of priming).
Fig 3.
Prevalence of the disease (P) at equilibrium as a function of the proportion of resistance 2 (p), when varieties are epidemiologically interchangeable (c1 = c2 < 0.5) for different values of (A) the basic reproductive number R, (B) the priming effect ρ, (C) the removal rate ν, (D) and the virulence costs c1, and c2. Parameter values: (A) ρ = 0.7, ν = 1, and c1 = c2 = 0.35, (B) R = 7, ν = 1, and c1 = c2 = 0.35, (C) R = 7, ρ = 0.8, and c1 = c2 = 0.35, (D) and R = 7, ρ = 0.8, and ν = 1.
Fig 4.
Prevalence of the disease (P, black line) at equilibrium as a function of the proportion of resistance 2 (p), when varieties are not epidemiologically interchangeable (c1 ≠ c2).
(A) When priming occurs (ρ = 0.9), there is a unique optimal proportion biased towards the variety the most likely to be broken, here variety 1 (assumming c1 < c2). (B) In absence of priming (ρ = 0), the disease prevalence is minimized for a range of p values. The colored areas correspond to different genetic compositions of the pathogen population at equilibrium. From left to right: monovirulent 1 only, coexistence of monovirulent 1 and doubly virulent, doubly virulent only, coexistence of monovirulent 2 and doubly virulent, and monovirulent 2 only. Parameter values: R = 7, ν = 1, c1 = 0.1, and c2 = 0.4.
Fig 5.
Prevalence of the disease (P) at equilibrium as a function of the proportion of resistance 2 (p), when varieties are not epidemiologically interchangeable (c1 ≠ c2 < 0.5).
Parameter values: R = 7, ρ = 0.8, and ν = 1.