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Global Stability of a Host-Vector Model for Pine Wilt Disease with Nonlinear Incidence Rate

DOI: 10.1155/2014/219173

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Based on classical epidemic models, this paper considers a deterministic epidemic model for the spread of the pine wilt disease which has vector mediated transmission. The analysis of the model shows that its dynamics are completely determined by the basic reproduction number . Using a Lyapunov function and a LaSalle's invariant set theorem, we proved the global asymptotical stability of the disease-free equilibrium. We find that if , the disease free equilibrium is globally asymptotically stable, and the disease will be eliminated. If , a unique endemic equilibrium exists and is shown to be globally asymptotically stable, under certain restrictions on the parameter values, using the geometric approach method for global stability, due to Li and Muldowney and the disease persists at the endemic equilibrium state if it initially exists. 1. Introduction Pine wilt disease (PWD) is caused by the pinewood nematode Bursaphelenchus xylophilus Nickle, which is vectored by the Japanese pine sawyer beetle Monochamus alternatus. The first epidemic of PWD was recorded in 1905 in Japan [1]. Since PWD was found in Japan, the pinewood nematode has spread to Korea, Taiwan, and China and has devastated pine forests in East Asia. Furthermore, it was also found in Portugal in 1999 [2]. The greatest losses to pine wilt have occurred in Japan. During the 20th century, the disease spread through highly susceptible Japanese black (P. thunbergiana) and Japanese red (P. densiflora) pine forests with devastating impact. Iowa, Illinois, Missouri, Kentucky, eastern Kansas, and southeastern Nebraska have experienced heavy losses of Scots pine. Thus, PWD has become the most serious threat to forest worldwide [3]. Mathematical modeling is useful in understanding the process of transmission of a disease, and determining the different factors that influence the spread of the 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, such as PWD transmission model which was investigated by Lee and Kim [4] and Shi and Song [5]. The incidence rate of the transmission of the disease plays an important role in the study of mathematical epidemiology. In classical epidemiological models, the incidence rate is assumed to be bilinear given by , where is the probability of transmission per contact rate, is susceptible, and is infective populations, respectively. However, actual data and evidence observed for many diseases show that dynamics of disease transmission are not always

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