# zbMATH — the first resource for mathematics

Global analysis for an epidemical model of vector-borne plant viruses with disease resistance and nonlinear incidence. (English) Zbl 1460.92194
Summary: Vector-borne disease models play an important role in understanding the mechanism of plant disease transmission. In this paper, we study a vector-borne model with plant disease resistance, disease exposed period and nonlinear incidence. We compute the basic reproduction number, determine the implicit locations of equilibria and then investigate their global stability by generalizing a classic geometric approach to higher dimensional systems. Higher dimensions cause greater difficulties such as the construction of the transformation matrix and the estimate of the Lozinskiĩ measure in this geometric approach. For a complete control of vector-borne diseases, a quantitative way is provided by the given expression of the basic reproduction number, from which we need not only increasing plant disease resistance but also decreasing the contact rate between infected plants and susceptible vectors instead of a single one of them.
##### MSC:
 92D30 Epidemiology 34C60 Qualitative investigation and simulation of ordinary differential equation models 34D05 Asymptotic properties of solutions to ordinary differential equations 34D20 Stability of solutions to ordinary differential equations
Full Text:
##### References:
 [1] L. Boiteux and L. Giordano,Genetic basis of resistance against two Tospovirus species in tomato (Lycopersicon esculentum), Euphytica, 1993, 71(1), 151-154. [2] C. Brittlebank,Tomato diseases, J. Depar. Agri. Vict., 1919, 17, 1348-1352. [3] G. Butler, H. Freedman and P. Waltman,Uniformly persistent systems, Proc. Amer. Math. Soc., 1986, 96(3), 425-430. · Zbl 0603.34043 [4] K. Chen, Z. Xu, L. Yan and G. Wang,Characterization of a new strain of Capsicum chlorosis virus from peanut (Arachis hypogaea L.) in China, J. Phyt., 2007, 155(3), 178-181. [5] W. A. Coppel,Stability and Asymptotic Behavior of Differential Equations, Heath, Boston, 1965. · Zbl 0154.09301 [6] N. Cunniffe and C. Gilligan,Invasion, persistence and control in epidemic models for plant pathogens: the effect of host demography, J. Roy. Soc. Inter., 2010, 7(44), 439-451. [7] A. Czech, M. Szklarczyk, Z. Gajewski, et a1,Selection of tomato plants resistant to a local Polish isolate of tomato spotted wilt virus (TSWV), J. Appl. Genet., 2003, 44(4), 473-480. [8] O. Diekmann, J. Heesterbeek and J. Metz,On the definition and the computation of the basic reproduction ratioR0in models for infectious diseases in heterogeneous populations, J. Math. Biol., 1990, 28(4), 365-382. · Zbl 0726.92018 [9] M. Ding, Y. Luo, Q. Fang, Z. Zhang and Z. Zhao,First report of Groundnut yellow spot virus infecting Capsicum annuum in China, J. Plant Path., 2007, 89(2), 305. [10] P. Driessche and J. Watmough,Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Bios., 2002, 180(1), 29-48. · Zbl 1015.92036 [11] X. Feng, S. Ruan, Z. Teng and K. Wang,Stability and backward bifurcation in a malaria transmission model with applications to the control of malaria in China, Math. Bios., 2015, 266, 52-64. · Zbl 1356.92081 [12] L. Ferrand, M. M. S. Almeida, et al,Biological and molecular characterization of tomato spotted wilt virus (TSWV) resistance-breaking isolates from Argentina, Plant Pathology, 2019, 68(9), 1587-1601. [13] F. R. Gantmacher,The theory of matrices, Amer. Math. Soc., New York, 1959. 2102L. Fei, L. Zou & X. Chen · Zbl 0085.01001 [14] R. Gupta1, S. Kwon and S. Kim,An insight into the tomato spotted wilt virus (TSWV), tomato and thrips interaction, Plant Biot. Repo., 2018, 12(3), 157- 163. [15] J. Hale,Asymptotic behavior of dissipative systems, Bull. Amer. Math. Soc., 1990, 22, 175-183. [16] M. Jeger, F. Van den Bosch and N. McRoberts,Modelling transmission characteristics and epidemic development of the tospovirus-thrip interaction, Arth.Plant Interactions, 2015, 9(2), 107-120. [17] J. Jia and J. Xiao,Stability analysis of a disease resistance SEIRS model with nonlinear incidence rate, Adv. Diff. Equa., 2018, DOI: 10.1186/s13662-018- 1494-1. · Zbl 1445.92273 [18] Y. Kuang, D. Ben-Arieh, S. Zhao and C. Wu,Using spatial games to model and simulate tomato spotted wilt virus-western flowers thrip dynamic system, Int. J. Mode. Simu., 2018, 38(4), 243-253. [19] M. Li and J. Muldowney,On R.A. Smith’s autonomous convergence theorem, Rock. moun. J. math., 1995, 25(1), 365-379. · Zbl 0841.34052 [20] M. Li and J. Muldowney,A geometric approach to global stability problems, SIAM J. Math. Anal., 1996, 27(4), 1070-1083. · Zbl 0873.34041 [21] Y. Li, Z. Zhang, S. Guan, H. Pen, J. Li and Y. Zou,Kinds of tobacco viral pathogens and the infection cycle at Binchuan county, J. Yunnan Agri. Univ., 1997, 12(4), 263-268. [22] S. Morsello, A. Beaudoin, R. Groves, et al,The influence of temperature and precipitation on spring dispersal of Frankliniella fusca changes as the season progresses, Ento. Expe. Appl., 2010, 134(3), 260-271. [23] R. Olatinwo, J. Paz, S. Brown, et al,A predictive model for spotted wilt epidemics in peanut based on local weather conditions and the tomato spotted wilt virus risk index, Ecol. Epid., 2008, 98(10), 1066-1074. [24] O. Pamella, P. Dany and P. Hans-Michael,Predictive Models for Tomato Spotted Wilt Virus Spread Dynamics, Considering Frankliniella occidentalis Specific Life Processes as Influenced by the Virus, PLoS ONE, 2016, 11(5), e0154533, (20 pages). [25] F. Riesz,Sur les Fonctions Subharmoniques et Leur Rapport ‘a la Th¡äeorie du Potentiel, Acta. Math., 1930, 54(1), 321-360. · JFM 56.0426.01 [26] H. Robert and J. Martin,Logarithmic norms and projections applied to linear differential systems, J. Math. Anal. Appl., (1974), 45(2), 432-454. · Zbl 0293.34018 [27] G. Samuel, J. Bald and H. Pittman,Investigations on ’spotted wilt’ of tomatoes, Aust. Coun. Scie. Indu. Rese. Bull., 1930, 44, 1-64. [28] A. Shaw, M. Igoe, et al,Modeling Approach Influences Dynamics of a VectorBorne Pathogen System, Bull. Math. Biol., 2019, 81(6), 2011-2028. · Zbl 1415.92193 [29] R. Shi, H. Zhao and S. Tang,Global dynamic analysis of a vector-borne plant disease model, Adv. Diff. Equa., 2014, DOI: 10.1186/1687-1847-2014-59. · Zbl 1344.92180 [30] H. Smith and P. Waltman,The Theory of the Chemostat: Dynamics of Microbial Competition, Cambridge University Press, Cambridge, 1995. · Zbl 0860.92031 [31] D. Su, X. Yuan, Y. Xie, S. Wang and H. Ding,Tomato spotted wilt virus in tomato in Chengdu and Dukou, Acta. Phyt. Sini., 1987, 17(4), 255-256. [32] H. Thieme,Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal., 1993, 24(2), 407-435. · Zbl 0774.34030 [33] R. Varga,Iterative analysis, Prentice-Hall, Englewood Cliffs, NJ, 1962. · Zbl 0133.08602 [34] A. Whitfield, D. Ullman and T. German,Tospovirus-thrips interactions, Annu. Revi. Phyt., 2005, 43, 459-489. [35] C. Wu, S. Zhao, Y. Kuang, et al,New mathematical models for vector-borne disease: transmission of tomato spotted wilt virus, Bridging research and good practices towards patient welfare, Taipei: CRC Press, 2014, 32, 259-268. [36] L. Xia, S. Gao, Q. Zou and J. Wang,Analysis of a nonautonomous plant disease model with latent period, Appl. Math. Comp., 2013, 223, 147-159. · Zbl 1329.92080 [37] J. Ye, Y. Gong and R. Fang,Research progress and perspective of tripartite interaction of virus-vector plant in vector-borne viral diseases, Bull. Chin. Acad. Sci., 2017, 32(8), 845-855.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.