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Mathematical Biosciences and Engineering

2020, Volume 17, Issue 5: 5234-5249. doi: 10.3934/mbe.2020283
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Research article

Global stability of a pseudorabies virus model with vertical transmission

  • 1.
    College of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, PRC
  • 2.
    Center for Applied Mathematics, Guangzhou University, Guangzhou, 510006, PRC
  • Received: 11 June 2020 Accepted: 23 July 2020 Published: 05 August 2020

  • Porcine pseudorabies infection is an acute infectious disease caused by pseudorabies virus. In this paper, we formulate a mathematical susceptible-incubating-infected-treated (SEIT) model with vertical transmission. The existence and stability of the equilibrium points of the model are characterized by the basic reproduction number $\Re_0$. When $\Re_0<1$, we show that the disease free equilibrium is unique and globally asymptotically stable. When $\Re_0>1$ and $p_{1}\geq\max\{\beta,b\}$, using the Lyapunov function method and the theory of competitive system, we obtain the global asymptotical stability of a unique disease endemic equilibrium.

    Citation: Yuhua Long, Yining Chen. Global stability of a pseudorabies virus model with vertical transmission[J]. Mathematical Biosciences and Engineering, 2020, 17(5): 5234-5249. doi: 10.3934/mbe.2020283

    Related Papers:

  • Porcine pseudorabies infection is an acute infectious disease caused by pseudorabies virus. In this paper, we formulate a mathematical susceptible-incubating-infected-treated (SEIT) model with vertical transmission. The existence and stability of the equilibrium points of the model are characterized by the basic reproduction number $\Re_0$. When $\Re_0<1$, we show that the disease free equilibrium is unique and globally asymptotically stable. When $\Re_0>1$ and $p_{1}\geq\max\{\beta,b\}$, using the Lyapunov function method and the theory of competitive system, we obtain the global asymptotical stability of a unique disease endemic equilibrium.


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    [1] N. Ketusing, A. Reeves, K. Portacci, T. Yano, F. Olea-Popelka, T Keefe, et al., Evaluation of strategies for the eradication of pseudorabies virus (Aujeszky's disease) in commercial swine farms in Chiang-Mai and Lampoon Provinces, Thailand, Using a Simulation Disease Spread Model, Transbound. Emerg. Dis., 61(2014), 169-176.
    [2] G. Wittmann, H. Rziha, Aujeszky's Disease (Pseudorabies) in Pigs, Springer, Boston, MA, 1989.
    [3] J. Zimmerman, L. Karriker, A. Ramirez, K. Schwartz, G. Stevenson, Diseases of Swine 10th Edition, Wiley, 2019.
    [4] S. Brogger, Systems analysis in tuberculosis control: A model, Am. Rev. Respir. Dis., 95(1967), 419-434.
    [5] S. Z. Jin, L. J. Sun, Q. P. Tong, Z. W. Tong, Clinical diagnosis and ccontrol of Swine Pseudorabies (in Chinese), Shanghai Animal Husbandry and Veterinary Communication, 000(005)(2013), 90- 91.
    [6] A. C. Aujeszky, Ueber eine new Infektions krankheit bei Haustieren, Zentbl Bakt ParasitKde, 32(1902), 353-357.
    [7] R. P. Hanson, The history of pseudorabies in the United States, J. Am. Vet. Med. Assoc., 124(1954), 259-261.
    [8] T. Mettenleiter, Aujeszky, Disease (Pseudorabies) virus: The Virus and molecular pathogenesisState of the Art, June 1999, Vet. Res., 31(2000), 99.
    [9] T. Q. An, J. M. Peng, Z. J. Tian, H. Y. Zhao, N. Li, Y. M. Liu, et al., Pseudorabies virus variant in Bartha-K61-vaccinated pigs, China, 2012, Emerg. Infect. Dis., 19(2013), 1749-1755.
    [10] N. Denzin, F. J. Conraths, T. C. Mettenleiter, C. M. Freuling, M. Thomas, Monitoring of pseudorabies in wild boar of Germany: A spatiotemporal snalysis, Pathogens, 9(2020).
    [11] X. F. Zhai, W. Zhao, K. M. Li, C. Zhang, Genome characteristics and evolution of pseudorabies virus strains in eastern China from 2017 to 2019, Virol. Sinica, 34(2019), 601-609.
    [12] H. J. Wu, D. C. Ye, Epidemiological characteristics, purification techniques and application demonstration of pseudorabies in pigs, China Animal Husbandry and Veterinary Digest, 34(2018), 148-148.
    [13] C. Freuling, T. Müller, T. Mettenleiter, Vaccines against pseudorabies virus (PrV), Vet. Microbiol., 206(2017), 3-9.
    [14] Y. H. Long, L. Wang, Global Dynamics of a delayed Two-patch discrete SIR disease model, Commun. Nonlinear Sci. Numer. Simul., 83(2020), 105-117.
    [15] C. H. Li, Y. Wang, F. Y. Jiang, J. H. Hu, Research progress of swine pseudorabies (in Chinese), Progress Animal Med., 29(2008), 68-72.
    [16] F. Rao, P. Mandal, Y. Kang, Complicated endemics of an SIRS model with a generalized incidence under preventive vaccination and treatment controls, Appl. Math. Model., 67(2019), 38-61.
    [17] O. Diekmann, J. A. P. Heesterbeek, J. A. J. Metz, On the definition and the computation of the basic reproduction ratio ℜ<0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28(1990), 365-382.
    [18] P. Van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180(2002), 29-48.
    [19] J. P. LaSalle, The stability of dynamical systems, Regional Conference Series in Applied Mathematics, Siam, Philadelphia, 1976.
    [20] J. Li, Z. Ma, F. Zhang, Stability analysis for an epidemic model with stage structure, Nonlin. Anal., 9(2008), 1672-1679.
    [21] Z. E. Ma, Y. C. Zhou, Qualitative and stability methods of ordinary differential equations (in Chinese), Science Press, 2015.
    [22] M. Y Li, J. S. Muldowney, Global stability for the SEIR model in epidemiology, Math. Biosci., 125(1995), 155-164.
    [23] M. Y. Li, H. L. Smith, L. C. Wang, Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J. Appl. Math., 62(2001), 58-69.
    [24] M. Y. Li, J. R. Graef, L. C. Wang, J. Karsai, Global dynamics of a SEIR model with varying total population size, Math. Biosci., 160(1999), 191-213.
    [25] H. L. Smith, Systems of Ordinary Differential Equations Which Generate an Order Preserving Flow, A Survey of Results, Society for Industrial and Applied Mathematics, 1988.
    [26] M. W. Hirsch, Systems of differential equations which are competitive or cooperative: I. limit sets, SIAM J. Math. Anal., 13(1982), 167-179.
    [27] J. S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain J. Math., 20(1990), 857-872.
    [28] Jr. R. H. Martin, Logarithmic norms and projections applied to linear differential systems, J. Appl. Math Anal. Appl, 45(1974), 432-454.
    [29] J. S. Muldowney, Dichotomies and asymptotic behaviour for linear differential systems, Trans. Amer. Math. Soc., 283(1984), 465-484.
    [30] L. Cesari, Asymptotic behavior and stability problems in ordinary differential equations, Springer Science & Business Media, 2012.
    [31] M. Y. Li, J. S. Muldowney, A geometric approach to Global-stability problems, SIAM J. Math. Anal., 27(1996), 1070-1083.
    [32] G. Butler, P. Waltman, Persistence in dynamical systems, J. Differ. Equat., 63(1986), 255-263.
    [33] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42(2000), 599-653.
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  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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