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
F. Takasu, “Individual-based modeling of the spread of pine wilt disease: vector beetle dispersal and the Allee effect,” Population Ecology, vol. 51, no. 3, pp. 399–409, 2009.
M. M. Mota, H. Braasch, M. A. Bravo et al., “First report of Bursaphelenchus xylophilus in Portugal and in Europe,” Nematology, vol. 1, no. 7-8, pp. 727–734, 1999.
K. S. Lee and D. Kim, “Global dynamics of a pine wilt disease transmission model with nonlinear incidence rates,” Applied Mathematical Modelling, vol. 37, no. 6, pp. 4561–4569, 2013.
X. Shi and G. Song, “Analysis of the mathematical model for the spread of pine wilt disease,” Journal of Applied Mathematics, vol. 2013, Article ID 184054, 10 pages, 2013.
V. Capasso and G. Serio, “A generalization of the Kermack-McKendrick deterministic epidemic model,” Mathematical Biosciences, vol. 42, no. 1-2, pp. 43–61, 1978.
L. Cai, S. Guo, X. Li, and M. Ghosh, “Global dynamics of a dengue epidemic mathematical model,” Chaos, Solitons & Fractals, vol. 42, no. 4, pp. 2297–2304, 2009.
L.-M. Cai and X.-Z. Li, “Global analysis of a vector-host epidemic model with nonlinear incidences,” Applied Mathematics and Computation, vol. 217, no. 7, pp. 3531–3541, 2010.
W. M. Liu, H. W. Hethcote, and S. A. Levin, “Dynamical behavior of epidemiological models with nonlinear incidence rates,” Journal of Mathematical Biology, vol. 25, no. 4, pp. 359–380, 1987.
M. Ozair, A. A. Lashari, I. H. Jung, and K. O. Okosun, “Stability analysis and optimal control of a vector-borne disease with nonlinear incidence,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 595487, 21 pages, 2012.
L. Esteva and C. Vargas, “A model for dengue disease with variable human population,” Journal of Mathematical Biology, vol. 38, no. 3, pp. 220–240, 1999.
D. Xiao and S. Ruan, “Global analysis of an epidemic model with nonmonotone incidence rate,” Journal of Differential Equations, vol. 188, pp. 135–163, 2003.
S. Ruan and W. Wang, “Dynamical behavior of an epidemic model with a nonlinear incidence rate,” Journal of Differential Equations, vol. 188, no. 1, pp. 135–163, 2003.
A. Korobeinikov and P. K. Maini, “Non-linear incidence and stability of infectious disease models,” Mathematical Medicine and Biology, vol. 22, no. 2, pp. 113–128, 2005.
B. Buonomo and S. Rionero, “On the Lyapunov stability for SIRS epidemic models with general nonlinear incidence rate,” Applied Mathematics and Computation, vol. 217, no. 8, pp. 4010–4016, 2010.
M. Y. Li and J. S. Muldowney, “A geometric approach to global-stability problems,” SIAM Journal on Mathematical Analysis, vol. 27, no. 4, pp. 1070–1083, 1996.
V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, vol. 125 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1989.
H. R. Thieme, “Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations,” Journal of Mathematical Biology, vol. 30, no. 7, pp. 755–763, 1992.
P. van den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Mathematical Biosciences, vol. 180, pp. 29–48, 2002, John A. Jacquez memorial volume.
J. S. Muldowney, “Compound matrices and ordinary differential equations,” The Rocky Mountain Journal of Mathematics, vol. 20, no. 4, pp. 857–872, 1990.
C. C. McCluskey and P. van den Driessche, “Global analysis of two tuberculosis models,” Journal of Dynamics and Differential Equations, vol. 16, no. 1, pp. 139–166, 2004.
B. Buonomo and C. Vargas-De-León, “Stability and bifurcation analysis of a vector-bias model of malaria transmission,” Mathematical Biosciences, vol. 242, no. 1, pp. 59–67, 2013.
J. Tumwiine, J. Y. T. Mugisha, and L. S. Luboobi, “A host-vector model for malaria with infective immigrants,” Journal of Mathematical Analysis and Applications, vol. 361, no. 1, pp. 139–149, 2010.