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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 and 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), 175-180. doi: 10.1016/S0167-6911(96)00052-7.

[2]

A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. doi: 10.1137/1.9781611971262.

[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), 229-250.

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H. Freedman, S. Ruan and M. Tang, Uniform persistence and flows near a closed positively invariant set, J. Dynamics & Diff. Equations, 6 (1994), 583-600. doi: 10.1007/BF02218848.

[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), 19703-19708. doi: 10.1073/pnas.1217567109.

[6]

Z. Hu, Z. Teng and L. Zhang, Stability and bifurcation analysis in a discrete SIR epidemic model, Mathematics & Computers in Simulation, 97 (2014), 80-93. doi: 10.1016/j.matcom.2013.08.008.

[7]

J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. doi: 10.1137/1.9781611970432.

[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), 191-213. doi: 10.1016/S0025-5564(99)00030-9.

[9]

J. Rebaza, Dynamics of prey threshold harvesting and refuge, Computational & Applied Mathematics, 236 (2012), 1743-1752. doi: 10.1016/j.cam.2011.10.005.

[10]

E. Seneta, Nonnegative Matrices and Markov Chains, Springer-Verlag, New York, 1981. doi: 10.1007/0-387-32792-4.

[11]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. of Applied Mathematics, 73 (2013), 1513-1532. doi: 10.1137/120876642.

[12]

J. R. Silvester, Determinants of block matrices, Mathematical Gazette, 84 (2000), 460-467. doi: 10.2307/3620776.

[13]

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

[14]

J. P. Tian, S. Liao and J. Wang, Analyzing the infection dynamics and control strategies of cholera, Discr. Cont. Dynam. Syst. 2013, Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., (2013), 747-757.

[15]

C. Torres Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infectious Diseases, 1 (2001), 1-14. doi: 10.1186/1471-2334-1-1.

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), 175-180. doi: 10.1016/S0167-6911(96)00052-7.

[2]

A. Berman and R. Plemmons, Nonnegative Matrices in the Mathematical Sciences, Classics in Applied Mathematics, 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. doi: 10.1137/1.9781611971262.

[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), 229-250.

[4]

H. Freedman, S. Ruan and M. Tang, Uniform persistence and flows near a closed positively invariant set, J. Dynamics & Diff. Equations, 6 (1994), 583-600. doi: 10.1007/BF02218848.

[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), 19703-19708. doi: 10.1073/pnas.1217567109.

[6]

Z. Hu, Z. Teng and L. Zhang, Stability and bifurcation analysis in a discrete SIR epidemic model, Mathematics & Computers in Simulation, 97 (2014), 80-93. doi: 10.1016/j.matcom.2013.08.008.

[7]

J. P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. doi: 10.1137/1.9781611970432.

[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), 191-213. doi: 10.1016/S0025-5564(99)00030-9.

[9]

J. Rebaza, Dynamics of prey threshold harvesting and refuge, Computational & Applied Mathematics, 236 (2012), 1743-1752. doi: 10.1016/j.cam.2011.10.005.

[10]

E. Seneta, Nonnegative Matrices and Markov Chains, Springer-Verlag, New York, 1981. doi: 10.1007/0-387-32792-4.

[11]

Z. Shuai and P. van den Driessche, Global stability of infectious disease models using Lyapunov functions, SIAM J. of Applied Mathematics, 73 (2013), 1513-1532. doi: 10.1137/120876642.

[12]

J. R. Silvester, Determinants of block matrices, Mathematical Gazette, 84 (2000), 460-467. doi: 10.2307/3620776.

[13]

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

[14]

J. P. Tian, S. Liao and J. Wang, Analyzing the infection dynamics and control strategies of cholera, Discr. Cont. Dynam. Syst. 2013, Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., (2013), 747-757.

[15]

C. Torres Codeço, Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir, BMC Infectious Diseases, 1 (2001), 1-14. doi: 10.1186/1471-2334-1-1.

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