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Mathematical Models of Host Plant Infection by Helper-Dependent Virus Complexes: Why Are Helper Viruses Always Avirulent?

Interactions between viruses in plants are common, and some viruses depend on such interactions for their survival. Frequently, a virus lacks some essential molecular function that another provides. In “helper-dependent” virus complexes, the helper virus is transmitted independently by a vector, whereas the dependent virus depends on molecular agents associated with the helper virus for transmission by a vector. A general mathematical model was developed of the dynamics of host plant infection by a helper-dependent virus complex. Four categories of host plants were considered: healthy, infected with helper virus alone, infected with dependent virus alone, and infected with both viruses. New planting of the host crop was constrained by a maximum abundance due to limitation of the cropping area. The ratio of infection rate to host loss rate due to infection is proposed as an important epidemiological quantity, A, that can be used as a measure of the mutual adaptation of the virus and host. A number of alternative equilibria of host infection could occur and were determined exclusively by parameter values; it was informative to display their distribution in the parameter plane: (1/A)_helper versus (1/A)_dependent. A simple analysis of the distribution of the final equilibria illustrated that the dependent virus could affect the survival of the helper virus, so facilitation between the two can be reciprocal. The distribution of the final equilibria also indicated that a well-adapted helper virus increases the opportunity for a dependent virus to evolve and survive, and the model, therefore, explains why infection with a helper virus usually causes no or little damage to plants, whereas infection with a dependent virus or mixed infection with both often causes very severe damage.