Epidemiology and disease-control under gene-for-gene plant-pathogen interaction.

An introduction of disease-resistant variety of a crop plant often leads to the development of a virulent race in pathogen species that restores the pathogenicity to the resistant crop. This often makes disease control of crop plants extremely difficult. In this paper, we theoretically explore the optimal 'multiline' control, which makes use of several different resistant varieties, that minimizes the expected degree of crop damages caused by epidemic outbreaks of the pathogen. We examine both single-locus and two-locus gene-for-gene (GFG) systems for the compatibility relationship between host genotypes and pathogen genotypes, in which host haplotype has either susceptible or resistant allele in each resistance locus, and the pathogen haplotype has either avirulent or virulent allele in the corresponding virulence locus. We then study the optimal planting strategy of host resistant genotypes based on standard epidemiological dynamics with pathogen spore stages. The most striking result of our single-locus GFG model is that there exists an intermediate optimum mixing ratio for the susceptible and resistant crops that maximizes the final yield, in spite of the fact that the susceptible crop has no use to fight against either avirulent or virulent race of the pathogen. The intermediate mixture is optimum except when the initial pathogen spore population in the season consists exclusively of the virulent race. The optimal proportion of resistant crops is approximately 1/R(0), where R(0) is the basic reproductive ratio of pathogen--the rest (the vast majority if R(0) is large) of crops should be the susceptible genotype. By mixing susceptible and resistant crops, we can force the pathogen races to compete with each other for their available hosts. This competition between avirulent and virulent races prevents the fatal outbreak of the virulent race (the super-race) that can infect all the host genotypes. In the two-locus GFG control, there again exists the optimal mixing ratio for the fraction of universally susceptible genotype and the total fraction of various resistant genotypes, with the ratio close to 1/R(0).