A model of biological control of plant virus propagation with delays

Plants are a food source for man and many species. They also are sources of medicines, fibers for clothes and are essential for a healthy environment. But plants are subject to diseases. Many of these are caused by viruses. In plants most of the virus propagation is done by a vector, usually insects that bite infected plants and the susceptible plants. Chemical insecticides are commonly used to control the insects, but unfortunately these chemicals have toxic effects on humans, animals and the environment in general. An alternative is to introduce a new species, or just increase the number of a naturally present one, to prey on the insects and thus control their number. The different steps in the virus propagation process take time. In this paper we add a predator to a plant-virus propagation model with delays. The model is based on delay differential equations. There are five populations: susceptible and infected plants, susceptible and infected vectors, and predators. The infected vectors have no ill effects from the virus so they do not fight the virus and therefore they do not recover. The predators do not get sick from the virus. The total number of plants is assumed to be constant since for crops the framers will replace the death plants and one of the plant populations can be determined using the total constant plant population. Therefore we have a system of five delay ordinary differential equations in five unknowns. We assume that predators are introduced at the initial time and after their growth rate has a Holling 2 type dependence on the number of vectors. We consider two delays: the time it takes from the moment a plant is bitten by an infected vector to the moment the plant is infected; and a second smaller delay for the time from the moment a vector bites an infected plant to the moment the vector becomes infected. We determine the non-negative steady state solutions of the system of delay differential equations. We analyze their stability by calculating the eigenvalues of the linearized system about each equilibrium point. We do numerical simulations for certain values of the parameters. Finally we search for bifurcation points numerically using the software package biftool. Conditions are presented for which the disease is controlled or even eliminated. From the practical side, the model can be used to determine the amount of introduced predators necessary to eliminate the epidemic under different conditions.