Analysis and fitting of an SIR model with host response to infection load for a plant disease

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Abstract
We reformulate a model for botanical epidemics into an SIR form for susceptible ( S ), infected ( I ) and removed ( R ) plant organs, in order to examine the effects of different models for the effect of host responses to the load of infection on the production of susceptible tissue. The new formulation also allows for a decline in host susceptibility with age. The model is analysed and tested for the stem canker disease of potatoes, caused by the soil–borne fungus, Rhizoctonia solani . Using a combination of model fitting to field data and analysis of model behaviour, we show that a function for host response to the amount (load) of parasite infection is critical in the description of the temporal dynamics of susceptible and infected stems in epidemics of R. solani . Several different types of host response to infection are compared including two that allow for stimulation of the plant to produce more susceptible tissue at low levels of disease and inhibition at higher levels. We show that when the force of infection decays with time, due to increasing resistance of the host, the equilibrium density of susceptible stems depends on the parameters and initial conditions. The models differ in sensitivity to small changes in disease transmission with some showing marked qualitative changes leading to a flush of susceptible stems at low levels of disease transmission. We conclude that there is no evidence to reject an SIR model with a simpler linear term for the effect of infection load on the production of healthy tissue, even though biological considerations suggest greater complexity in the relationship between disease and growth. We show that reduction in initial inoculum density, and hence in the force of infection, is effective in controlling disase when the simple model applies.

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