Stability analysis and optimal control of pine wilt disease with horizontal transmission in vector population

Abstract

In this paper, we have proposed and mathematically modeled an epidemic problem with vector-borne disease. We have taken three different classes for the trees, namely susceptible, exposed and infected, and two different classes for the vector population, namely susceptible and infected. In the first part of our paper, we rigorously analyze our model using the dynamical systems approach. Global stability of equilibria is resolved by using Lyapunov functional. In the second part, the model is reformulated as an optimal control problem in order to determine the significance of certain control measures on the model. We apply four control parameters, namely the tree injection control to the trees, deforestation of infected trees, eradication effort of aerial insecticide spraying and the effort of restrain of mating. Both numerical and analytical methods are employed to ascertain the existence of cost effective control measures.

Introduction

Infectious diseases are caused by pathogens such as bacteria, protozoa, and viruses. These pathogens may reach the healthy organism by disease causing biological agents, known as vectors (e.g insects and ticks), who carry the pathogen from infected individuals and transmit it to the uninfected ones. Infectious diseases which spread by means of vectors are known as vector-borne diseases, and these effect all living organisms, including plants. Examples of vector-borne infections in trees include the pine wilt disease and the red ring disease in palms which are caused by nematodes (roundworms), and are vectored by insects [1]. Bursaphelenchus xylophilus is the nematode which causes the pine wilt disease (PWD) [2], [3]. The pine sawyer beetle (or Monochamus alternatus) is the vector for PWD. This beetle scatters the nematode, from space, over healthy pine trees. However, direct transmission in vector population also occurs during mating [4], [5]. Reproductively mature beetles use twig bark of healthy trees for feeding purpose, and concentrate on diseased trees for copulation and oviposition [6]. Horizontal transmissions of Bursaphelenchus xylophilus between heterosexual vectors enhances the level of multiple infections. The first epidemic of PWD occurred in Japan in 1905. Except for the northern districts of the country, this disease had spread in all of Japan by the 1970s [7]. After another ten years, the PWD epidemic had spread to many parts of Asia, such as China, Taiwan, Hong Kong and Korea. In 1999, the disease hit Europe (Portugal) [8]. Today, the PWD is one of the leading threats to forest ecosystems throughout the world.

Mathematical modeling is useful in understanding the process of transmission of a disease, and determining the different factors that influence the spread of disease. In this way, different control strategies can be developed to limit the spread of infection. Lately, some mathematical models have been formulated on pest-tree dynamics, that have given successful results [9], such as a PWD transmission model was investigated by Lee and Kim [10]. This model incorporated non linear incidence rates. In our paper, we have proposed a mathematical model which describes the host-vector relationship between pine trees and pine sawyer beetles (carrying nematode) by means of ordinary differential equations with bilinear incidence rate. That is, a vector-host epidemic model with horizontal transmission in vector population is presented, where the dynamics of the host pine trees and vector beetles are described by SEI and SI models, respectively. The research work done previously in this area contains some good papers on the vector diseases with direct transmission in host population [11], [12], [13]. Some mathematical work about the transmission dynamics of vector-borne disease have been published recently [14], [15], [16]. Few researchers have also applied optimal control methods to limit the virus transmission [17]. Moulay et al. [18] investigated the impact of three time dependent control variables, namely destruction of breeding sites, prevention, and treatment efforts. Okusun et al. [19] used vaccination and treatment control of a malaria disease transmission model.

The purpose of this paper is two fold. The first is to carry out the stability analysis of the proposed model. Detailed study of the model reveals that there exist two equilibria; the disease-free equilibria and the endemic equilibria. In order to prove the global asymptotical stability, we have used the Lyapunov function theory. It is proved that the global dynamics are completely determined by the basic reproduction number. The second aim is to achieve awareness about the most desirable technique for minimizing the transmission of PWD with horizontal transmission in vector population using the optimal control theory. To do this, we consider an optimal control model with four time-dependent controls: tree-injection $u_1$, deforestation of infected trees $u_2$, aerial pesticide spraying $u_3$ and the effort of restrain of mating $u_4$. We carry out further analysis and deduce those conditions where it is optimal to totally eliminate, and not just control, the PWD. In those cases where eradication is unattainable, we have given the necessary conditions for optimal control of the PWD using Pontryagin’s Maximum Principle.

The paper is organized as follows. The model is formulated in Section 2. Existence and global stability of the equilibria of the model are investigated in Section 3. In Section 4, the model is extended and an optimal control problem is formulated. In Section 5, the existence of an optimal control is examined, then, the optimality system is derived, which characterizes the optimal control. Section 6 provides numerical simulations for the optimal control model. Lastly, we give a brief discussion of our results in Section 7.