| Citation: |
Xiaoyu Jiang, Yang Yu, Fanwei Meng, Yancong Xu. MODELLING THE DYNAMICS OF AVIAN INFLUENZA WITH NONLINEAR RECOVERY RATE AND PSYCHOLOGICAL EFFECT[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 1170-1192. doi: 10.11948/20190253
|
MODELLING THE DYNAMICS OF AVIAN INFLUENZA WITH NONLINEAR RECOVERY RATE AND PSYCHOLOGICAL EFFECT
-
Abstract
In this paper, a SI-SEIR type avian influenza epidemic model with psychological effect, nonlinear recovery rate and saturation inhibition effect is formulated to study the transmission and control of avian influenza virus. By setting the basic reproductive number as the threshold parameter and constructing Lyapunov function, Dulac function and using the Li-Muldowney's geometry approach, we prove the local and global stability of disease-free equilibria and endemic equilibrium. Theoretical analysis are carried out to show the role of the saturation inhibition effect, psychological effect and effective medical resources in this model, and numerical simulations are also given to verify the results. -
-
References
[1] R. M. Anderson, B. Anderson, R. M. May, Infectious diseases of humans: dynamics and control, Oxford University Press, 1992. [2] T. Y. Aditama, G. Samaan, R. Kusriastuti, et al., Avian influenza H5N1 transmission in households, Indonesia, PloS One, 2012, 7(1), e29971. doi: 10.1371/journal.pone.0029971 [3] A. Abdelrazec, J. Belair, C. Shan, et al., Modeling the spread and control of dengue with limited public health resources, Mathematical Biosciences, 2016,271,136-145. doi: 10.1016/j.mbs.2015.11.004 [4] C. T. Bauch, A. P. Galvani, Social factors in epidemiology, Science, 2013,342(6154), 47-49. doi: 10.1126/science.1244492 [5] Y. Chen, Y. Wen, Global dynamic analysis of a H7N9 avian-human influenza model in an outbreak region, Journal of Theoretical Biology, 2015,367,180-188. doi: 10.1016/j.jtbi.2014.12.002 [6] V. Capasso, G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 1978, 42(1-2), 43-61. [7] Centers for Disease Control and Prevention (CDC), Bird Flu Basics, https://www.cdc.gov/flu/avianflu/bird-flu-basics.htm. [8] Centers for Disease Control and Prevention (CDC), Influenza Type A Viruses, http://www.cdc.gov/flu/avianflu/influenza-avirus-subtypes.htm. [9] C. Castillo-Chavez, et al., Mathematical approaches for emerging and reemerging infectious diseases: an introduction, Springer Science & Business Media, 2002. [10] O. Diekmann, J. A. P. Heesterbeek, M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, Journal of the Royal Society Interface, 2009, 7(47), 873-885. [11] P. Van den Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 2002,180(1-2), 29-48. [12] I. ElMasry, H. Elshiekh, A. Abdlenabi, et al., Avian influenza H5N1 surveillance and its dynamics in poultry in live bird markets, Egypt, Transboundary and emerging diseases, 2017, 64(3), 805-814. doi: 10.1111/tbed.12440 [13] S. Funk, E. Gilad, C. Watkins, et al., The spread of awareness and its impact on epidemic outbreaks, Proceedings of the National Academy of Sciences, 2009,106(16), 6872-6877. doi: 10.1073/pnas.0810762106 [14] S. Funk, M. SalathšŠ, V. A. A. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: a review, Journal of the Royal Society Interface, 2010, 7(50), 1247-1256. doi: 10.1098/rsif.2010.0142 [15] S. Funk, S. Bansal, C. T. Bauch, et al., Nine challenges in incorporating the dynamics of behaviour in infectious diseases models, Epidemics, 2015, 10, 21-25. doi: 10.1016/j.epidem.2014.09.005 [16] H. I. Freedman, S. Ruan, M. Tang, Uniform persistence and flows near a closed positively invariant set, Journal of Dynamics and Differential Equations, 1994, 6(4), 583-600. [17] D. J. Gubler, Dengue and dengue hemorrhagic fever, Clinical microbiology reviews, 1998, 11(3), 480-496. [18] S. Iwami, Y. Takeuchi, X. Liu, Avian Chuman influenza epidemic model, Mathematical Biosciences, 2007,207(1), 1-25. [19] M. J. Keeling, P. Rohani, Modeling infectious diseases in humans and animals, Princeton University Press, 2011. [20] M. A. Khan, M. Farhan, S. Islam, et al., Modeling the transmission dynamics of avian influenza with saturation and psychological effect, Discrete & Continuous Dynamical Systems-S, 2019, 12(3), 455-474. [21] S. Liu, L. Pang, S. Ruan, et al., Global dynamics of avian influenza epidemic models with psychological effect, Computational and mathematical methods in medicine, 2015, 1-12. [22] W. Liu, S. A. Levin, Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, Journal of Mathematical Biology, 1986, 23(2), 187-204. [23] S. Liu, S. Ruan, X. Zhang, Nonlinear dynamics of avian influenza epidemic models, Mathematical biosciences, 2017,283,118-135. doi: 10.1016/j.mbs.2016.11.014 [24] M. Y. Li, J. S. Muldowney, A geometric approach to global-stability problems, SIAM Journal on Mathematical Analysis, 1996, 27(4), 1070-1083. [25] M. Y. Li, J. R. Graef, L. Wang, et al., Global dynamics of a SEIR model with varying total population size, Mathematical Biosciences, 1999,160(2), 191-213. [26] J. Lucchetti, M. Roy, M. Martcheva, An avian influenza model and its fit to human avian influenza cases, Advances in Disease Epidemiology, 2009, 1-30. [27] R. Mu, Y. Yang, Global dynamics of an avian influenza A (H7N9) epidemic model with latent period and nonlinear recovery rate, Computational and mathematical methods in medicine, 2018, 1-11. [28] J. R. H. Martin, Logarithmic norms and projections applied to linear differential systems, Journal of Mathematical Analysis and Applications, 1974, 45(2), . 432-454. [29] L. Perko, Differential Equations and Dynamical Systems, Springer, 2001. [30] A. S. R. S. Rao, Modeling the rapid spread of avian influenza (H5N1) in India, Mathematical Biosciences & Engineering, 2008, 5(3), 523-537. [31] S. Ruan, W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, Journal of Differential Equations, 2003,188(1), 135-163. [32] C. Shan, H. Zhu, Bifurcations and complex dynamics of an SIR model with the impact of the number of hospital beds, Journal of Differential Equations, 2014,257(5), 1662-1688. doi: 10.1016/j.jde.2014.05.030 [33] N. Tuncer, M. Martcheva, Modeling seasonality in avian influenza H5N1, Journal of Biological Systems, 2013, 21(04), 1-30. [34] L. Wang, B. J. Cowling, P. Wu, et al., Human exposure to live poultry and psychological and behavioral responses to influenza A (H7N9), China, Emerging infectious diseases, 2014, 20(8), 1296-1305. [35] P. Wu, L. Wang, B. J. Cowling, et al., Live poultry exposure and public response to influenza A(H7N9) in urban and rural China during two epidemic waves in 2013-2014, PLoS One, 2015, 10. [36] Y. Xiao, X. Sun, S. Tang, et al., Transmission potential of the novel avian influenza A (H7N9) infection in mainland China, Journal of Theoretical Biology, 2014,352, 1-5. doi: 10.1016/j.jtbi.2014.02.038 [37] N. Xiang, Y. Shi, J. Wu, et al., Knowledge, attitudes and practices (KAP) relating to avian influenza in urban and rural areas of China, BMC Infectious Diseases, 2010, 10(1), 34. [38] F. Zhang, X. Zhang, The threshold of a stochastic avian-human influenza epidemic model with psychological effect, Physica A, 2018,492,485-495. doi: 10.1016/j.physa.2017.10.043 -
-
Figures(5) / Tables(1)
Export File
Citation
Xiaoyu Jiang, Yang Yu, Fanwei Meng, Yancong Xu. MODELLING THE DYNAMICS OF AVIAN INFLUENZA WITH NONLINEAR RECOVERY RATE AND PSYCHOLOGICAL EFFECT[J]. Journal of Applied Analysis & Computation, 2020, 10(3): 1170-1192. doi: 10.11948/20190253
Format
Content
DownLoad:
-
Figure 1.
$ (a) $ :$ I_{h}(t) $ converges to the disease-free equilibrium when$ \beta_{a} = 8\times10^{-9} (R_{0}<1) $ ,$ (b) $ :$ I_{h}(t) $ is asymptotically stable and converges to the endemic equilibrium when$ \beta_{a} = 3.5\times10^{-8}(R_{0}>1) $ -
Figure 2. The change curve of
$ I_{h} $ .$ (a) $ : Initial value is$ [100000,\ 200,\ 1000,\ 30,\ 5,\ 0] $ ,$ \beta_{a} = 8\times10^{-9}(R_{0}<1) $ ;$ (b) $ : Initial value is$ [80000,\ 100,\ 10000,\ 100,\ 2,\ 0] $ ,$ \beta_{a} = 3.5\times10^{-9}(R_{0}>1) $ -
Figure 3. The change curve of
$ I_{h} $ .$ (a) $ : Initial value is$ [100000,\ 200,\ 10000,\ 30,\ 5,\ 0] $ ,$ \mu_{1} = 0.1 $ ,$ \beta_{a} = 8\times10^{-9}(R_{0}<1) $ ;$ (b) $ : Initial value is$ [200000,\ 1000,\ 10000,\ 150,\ 10,\ 0] $ ,$ \mu_{1} = 0.25 $ ,$ \beta_{a} = 1.6\times10^{-8}(R_{0}>1) $ -
Figure 4. The change curve of
$ I_{h} $ .$ (a) $ : Initial value is$ [100000,\ 200,\ 10000,\ 30,\ 5,\ 0] $ ,$ \beta_{a} = 8\times10^{-9}(R_{0}<1) $ ;$ (b) $ : Initial value is$ [80000,\ 100,\ 10000,\ 100,\ 2,\ 0] $ ,$ \beta_{a} = 3.5\times10^{-8}(R_{0}>1) $ -
Figure 5. The change curve of
$ I_{h} $ .$ (a) $ : Initial value is$ [100000,\ 200,\ 10000,\ 30,\ 5,\ 0] $ ,$ \beta_{a} = 8\times10^{-9}(R_{0}<1) $ ;$ (b) $ : Initial value is$ [80000,\ 100,\ 10000,\ 100,\ 2,\ 0] $ ,$ \beta_{a} = 3.5\times10^{-8}>\beta_{a}^{*}(R_{0}>1 $ )