Modeling Plant Virus Propagation and an Optimal Control
Abstract
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
which are caused by viruses. These viruses often kill the plant. As a result, billions of dollars are
lost every year because of virus related crop loss. Most of the time, virus propagation is done by
a vector, usually insects that bite infected plants, get themselves infected and then bite
susceptible plants. To combat the vectors, and ultimately the viruses, pesticides are often used as
a control. Unfortunately, chemicals in pesticides can have a harmful effect on their environment.
An alternative method to control the insect population is to introduce a natural predator of the
insect. These predators may be more expensive than insecticides, but they are more
environmentally friendly. To understand the dynamics, a system of ordinary and delay
differential equations modeling interactions between insects and plants is considered and
analyzed. To analyze the system, the basic reproductive number is used along with numerical
simulations to find bifurcations. Then, a predator is introduced to the model, and the dynamics
are studied in a similar fashion. Because of the seasonality of insects, active in the warm months
and almost dormant in the cooler ones, the model is then analyzed with periodic coefficients. To
study this model, the basic reproductive number is used, but calculated in a couple of different
ways: a time average approach and a linear operator one. Finally an optimal control problem is
studied. In this problem, the goal is to minimize the cost of the insecticide, predator, and cost of
an infected plant. To solve this problem, two approaches are taken: an indirect approach using
Pontryagin maximum principle and a direct approach used in the BOCOP software package.