Delays in Plant Virus Models and Their Stability

Viruses infect humans and animals but also infect plants and cause great economic and ecological damage. In most cases, the virus is transmitted by a vector. After being bitten by an infected vector, the virus takes some time to replicate and spread in the plant. We present two models of the spread of viruses in plants based on ordinary differential equations, and then add either a delay or an exposed plant population. We study two ways of adding the delay. In the first one, a plant infected by a vector changes from susceptible to infective after a time equal to the delay. In the second one, immediately after the contact between a susceptible plant and infective vector, the plant is no longer susceptible, but it takes time equal to the delay for it to turn infective. To better explain the two ways of incorporating the delays, we first introduce them in a simple SIRS model. We analyze the models and study their stability numerically. We conclude by studying the interactions and the conservation of the total plant population that the first way of introducing the delay is better justified.