The global properties of an age-dependent model involving pathogenic virus release and defence mechanisms for pests

Insect pests are common but undesirable elements in ecosystems and represent thorny problems for most developing countries. To prevent pest outbreaks, growers often resort to insect–pathogenic viruses rather than to pesticides which affect human health and the environment. The purpose of this paper is to investigate a new age-structured pest management model which describes the interaction between susceptible insect pests, infected insect pests, pathogenic viruses and defence immunity mechanisms. A feature of this model is that it accounts for the dependence of the amount of pathogenic viruses released and of the efficiency of the defence mechanisms upon the so-called age of infection. First, the asymptotic behavior of the system is established via a monotonicity argument which makes use of several integral inequalities, being shown that the infection ultimately dies out, while under certain circumstances the susceptible pests also become extinct. By means of the Michailov criterion, one then analyzes the linearized stability of the trivial equilibrium and of the semi-trivial infected pest-free equilibrium. In this regard, it is observed that the defence mechanisms and maximal length of the infective period play important roles in the dynamics of the system. Several pest controls strategies are further investigated by means of numerical simulations, which show that when the dose of pathogenic viruses released initially is larger than a certain amount the profile of the response of defence mechanisms can be modified by changing this dose. Finally, the paper is concluded with a discussion on the biological significance of the mathematical results and framework.