American Institute of Mathematical Sciences

Advanced
December  2016, 21(10): 3379-3390. doi: 10.3934/dcdsb.2016102

Dynamics of a networked connectivity model of epidemics

Cristina Cross 1, , Alysse Edwards 2, , Dayna Mercadante 3, and  Jorge Rebaza 4,

1. 

Health Science Center at Houston, University of Texas, Houston, TX 77030, United States
2. 

Department of Mathematical Sciences, Clemson University, Clemson, SC 29634, United States
3. 

Cystic Fibrosis Foundation Therapeutics Lab, Bethesda, MD 20814, United States
4. 

Department of Mathematics, Missouri State University, Springfield, MO 65897

Received  April 2015 Revised  September 2016 Published  November 2016

A networked connectivity model of waterborne disease epidemics on a site of $n$ communities is studied. Existence and local stability analysis for both the disease-free equilibrium and the endemic equilibrium are studied. Using an appropriate Lyapunov function and Lasalle invariance principle, global asymptotic stability of the disease-free equilibrium point is established. Existence of a transcritical bifurcation at the disease outbreak is also proved. This work extends previous research in networked connectivity models of epidemics.
Keywords: dynamical systems, epidemics., Networked connectivity models.
Mathematics Subject Classification: Primary: 37B25, 92D30; Secondary: 92D2.
Citation: Cristina Cross, Alysse Edwards, Dayna Mercadante, Jorge Rebaza. Dynamics of a networked connectivity model of epidemics. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3379-3390. doi: 10.3934/dcdsb.2016102
References:
[1]

F. Albertini and D. D'Alessandro, Asymptotic stability of continuous-time systems with saturation nonlinearities,, Systems & Control Letters, 29 (1996), 175.  doi: 10.1016/S0167-6911(96)00052-7.  Google Scholar

[2]

A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Classics in Applied Mathematics, (1994).  doi: 10.1137/1.9781611971262.  Google Scholar

[3]

C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of $R_0$ and its role on global stability,, Math. Appr. for Emerg. and Reemerg. Infect. Dis, 125 (2002), 229.   Google Scholar

[4]

H. Freedman, S. Ruan and M. Tang, Uniform persistence and flows near a closed positively invariant set,, J. Dynamics & Diff. Equations, 6 (1994), 583.  doi: 10.1007/BF02218848.  Google Scholar

[5]

M. Gatto, L. Mari, E. Bertuzzo, R. Casagrandi, L. Righetto, I. Rodriguez-Iturbe and A. Rinaldo, Generalized reproduction numbers and the prediction of patterns in waterborne disease,, Proceed. Nat. Acad. of Scienc., 109 (2012), 19703.  doi: 10.1073/pnas.1217567109.  Google Scholar

[6]

Z. Hu, Z. Teng and L. Zhang, Stability and bifurcation analysis in a discrete SIR epidemic model,, Mathematics & Computers in Simulation, 97 (2014), 80.  doi: 10.1016/j.matcom.2013.08.008.  Google Scholar

[7]

J. P. LaSalle, The Stability of Dynamical Systems,, SIAM, (1976).  doi: 10.1137/1.9781611970432.  Google Scholar

[8]

M. Li, J. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Mathematical Biosciences, 160 (1999), 191.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[9]

J. Rebaza, Dynamics of prey threshold harvesting and refuge,, Computational & Applied Mathematics, 236 (2012), 1743.  doi: 10.1016/j.cam.2011.10.005.  Google Scholar

[10]

E. Seneta, Nonnegative Matrices and Markov Chains,, Springer-Verlag, (1981).  doi: 10.1007/0-387-32792-4.  Google Scholar

[11]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions,, SIAM J. of Applied Mathematics, 73 (2013), 1513.  doi: 10.1137/120876642.  Google Scholar

[12]

J. R. Silvester, Determinants of block matrices,, Mathematical Gazette, 84 (2000), 460.  doi: 10.2307/3620776.  Google Scholar

[13]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[14]

J. P. Tian, S. Liao and J. Wang, Analyzing the infection dynamics and control strategies of cholera,, Discr. Cont. Dynam. Syst. 2013, (2013), 747.   Google Scholar

[15]

C. Torres Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir,, BMC Infectious Diseases, 1 (2001), 1.  doi: 10.1186/1471-2334-1-1.  Google Scholar

show all references

References:
[1]

F. Albertini and D. D'Alessandro, Asymptotic stability of continuous-time systems with saturation nonlinearities,, Systems & Control Letters, 29 (1996), 175.  doi: 10.1016/S0167-6911(96)00052-7.  Google Scholar

[2]

A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences,, Classics in Applied Mathematics, (1994).  doi: 10.1137/1.9781611971262.  Google Scholar

[3]

C. Castillo-Chavez, Z. Feng and W. Huang, On the computation of $R_0$ and its role on global stability,, Math. Appr. for Emerg. and Reemerg. Infect. Dis, 125 (2002), 229.   Google Scholar

[4]

H. Freedman, S. Ruan and M. Tang, Uniform persistence and flows near a closed positively invariant set,, J. Dynamics & Diff. Equations, 6 (1994), 583.  doi: 10.1007/BF02218848.  Google Scholar

[5]

M. Gatto, L. Mari, E. Bertuzzo, R. Casagrandi, L. Righetto, I. Rodriguez-Iturbe and A. Rinaldo, Generalized reproduction numbers and the prediction of patterns in waterborne disease,, Proceed. Nat. Acad. of Scienc., 109 (2012), 19703.  doi: 10.1073/pnas.1217567109.  Google Scholar

[6]

Z. Hu, Z. Teng and L. Zhang, Stability and bifurcation analysis in a discrete SIR epidemic model,, Mathematics & Computers in Simulation, 97 (2014), 80.  doi: 10.1016/j.matcom.2013.08.008.  Google Scholar

[7]

J. P. LaSalle, The Stability of Dynamical Systems,, SIAM, (1976).  doi: 10.1137/1.9781611970432.  Google Scholar

[8]

M. Li, J. Graef, L. Wang and J. Karsai, Global dynamics of a SEIR model with varying total population size,, Mathematical Biosciences, 160 (1999), 191.  doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[9]

J. Rebaza, Dynamics of prey threshold harvesting and refuge,, Computational & Applied Mathematics, 236 (2012), 1743.  doi: 10.1016/j.cam.2011.10.005.  Google Scholar

[10]

E. Seneta, Nonnegative Matrices and Markov Chains,, Springer-Verlag, (1981).  doi: 10.1007/0-387-32792-4.  Google Scholar

[11]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions,, SIAM J. of Applied Mathematics, 73 (2013), 1513.  doi: 10.1137/120876642.  Google Scholar

[12]

J. R. Silvester, Determinants of block matrices,, Mathematical Gazette, 84 (2000), 460.  doi: 10.2307/3620776.  Google Scholar

[13]

H. L. Smith and P. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition,, Cambridge University Press, (1995).  doi: 10.1017/CBO9780511530043.  Google Scholar

[14]

J. P. Tian, S. Liao and J. Wang, Analyzing the infection dynamics and control strategies of cholera,, Discr. Cont. Dynam. Syst. 2013, (2013), 747.   Google Scholar

[15]

C. Torres Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir,, BMC Infectious Diseases, 1 (2001), 1.  doi: 10.1186/1471-2334-1-1.  Google Scholar

[1]

João Borges de Sousa, Bernardo Maciel, Fernando Lobo Pereira. Sensor systems on networked vehicles. Networks & Heterogeneous Media, 2009, 4 (2) : 223-247. doi: 10.3934/nhm.2009.4.223
[2]

Suoqin Jin, Fang-Xiang Wu, Xiufen Zou. Domain control of nonlinear networked systems and applications to complex disease networks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2169-2206. doi: 10.3934/dcdsb.2017091
[3]

Ahmadreza Argha, Steven W. Su, Lin Ye, Branko G. Celler. Optimal sparse output feedback for networked systems with parametric uncertainties. Numerical Algebra, Control & Optimization, 2019, 9 (3) : 283-295. doi: 10.3934/naco.2019019
[4]

Ying Wu, Zhaohui Yuan, Yanpeng Wu. Optimal tracking control for networked control systems with random time delays and packet dropouts. Journal of Industrial & Management Optimization, 2015, 11 (4) : 1343-1354. doi: 10.3934/jimo.2015.11.1343
[5]

Carlos Castillo-Chavez, Baojun Song. Dynamical Models of Tuberculosis and Their Applications. Mathematical Biosciences & Engineering, 2004, 1 (2) : 361-404. doi: 10.3934/mbe.2004.1.361
[6]

Glenn F. Webb. Individual based models and differential equations models of nosocomial epidemics in hospital intensive care units. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 1145-1166. doi: 10.3934/dcdsb.2017056
[7]

H. W. Broer, Renato Vitolo. Dynamical systems modeling of low-frequency variability in low-order atmospheric models. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 401-419. doi: 10.3934/dcdsb.2008.10.401
[8]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785
[9]

Nicolas Bacaër, Xamxinur Abdurahman, Jianli Ye, Pierre Auger. On the basic reproduction number $R_0$ in sexual activity models for HIV/AIDS epidemics: Example from Yunnan, China. Mathematical Biosciences & Engineering, 2007, 4 (4) : 595-607. doi: 10.3934/mbe.2007.4.595
[10]

Vincenzo Capasso, Sebastian AniȚa. The interplay between models and public health policies: Regional control for a class of spatially structured epidemics (think globally, act locally). Mathematical Biosciences & Engineering, 2018, 15 (1) : 1-20. doi: 10.3934/mbe.2018001
[11]

El Houcein El Abdalaoui, Sylvain Bonnot, Ali Messaoudi, Olivier Sester. On the Fibonacci complex dynamical systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2449-2471. doi: 10.3934/dcds.2016.36.2449
[12]

Lianfa He, Hongwen Zheng, Yujun Zhu. Shadowing in random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 355-362. doi: 10.3934/dcds.2005.12.355
[13]

Fritz Colonius, Marco Spadini. Fundamental semigroups for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 447-463. doi: 10.3934/dcds.2006.14.447
[14]

John Erik Fornæss. Sustainable dynamical systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1361-1386. doi: 10.3934/dcds.2003.9.1361
[15]

Vieri Benci, C. Bonanno, Stefano Galatolo, G. Menconi, M. Virgilio. Dynamical systems and computable information. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 935-960. doi: 10.3934/dcdsb.2004.4.935
[16]

Mădălina Roxana Buneci. Morphisms of discrete dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 91-107. doi: 10.3934/dcds.2011.29.91
[17]

Philippe Marie, Jérôme Rousseau. Recurrence for random dynamical systems. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 1-16. doi: 10.3934/dcds.2011.30.1
[18]

Tobias Wichtrey. Harmonic limits of dynamical systems. Conference Publications, 2011, 2011 (Special) : 1432-1439. doi: 10.3934/proc.2011.2011.1432
[19]

Josiney A. Souza, Tiago A. Pacifico, Hélio V. M. Tozatti. A note on parallelizable dynamical systems. Electronic Research Announcements, 2017, 24: 64-67. doi: 10.3934/era.2017.24.007
[20]

Birol Yüceoǧlu, ş. ilker Birbil, özgür Gürbüz. Dispersion with connectivity in wireless mesh networks. Journal of Industrial & Management Optimization, 2018, 14 (2) : 759-784. doi: 10.3934/jimo.2017074

2019 Impact Factor: 1.27

Tools

Metrics

  • PDF downloads (39)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences