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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 & Continuous Dynamical Systems - B, 2016, 21 (10) : 3379-3390. doi: 10.3934/dcdsb.2016102
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show all references

References:
[1]

Systems & Control Letters, 29 (1996), 175-180. doi: 10.1016/S0167-6911(96)00052-7.  Google Scholar

[2]

Classics in Applied Mathematics, 9. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1994. doi: 10.1137/1.9781611971262.  Google Scholar

[3]

Math. Appr. for Emerg. and Reemerg. Infect. Dis, 125 (2002), 229-250.  Google Scholar

[4]

J. Dynamics & Diff. Equations, 6 (1994), 583-600. doi: 10.1007/BF02218848.  Google Scholar

[5]

Proceed. Nat. Acad. of Scienc., 109 (2012), 19703-19708. doi: 10.1073/pnas.1217567109.  Google Scholar

[6]

Mathematics & Computers in Simulation, 97 (2014), 80-93. doi: 10.1016/j.matcom.2013.08.008.  Google Scholar

[7]

SIAM, Philadelphia, 1976. doi: 10.1137/1.9781611970432.  Google Scholar

[8]

Mathematical Biosciences, 160 (1999), 191-213. doi: 10.1016/S0025-5564(99)00030-9.  Google Scholar

[9]

Computational & Applied Mathematics, 236 (2012), 1743-1752. doi: 10.1016/j.cam.2011.10.005.  Google Scholar

[10]

Springer-Verlag, New York, 1981. doi: 10.1007/0-387-32792-4.  Google Scholar

[11]

SIAM J. of Applied Mathematics, 73 (2013), 1513-1532. doi: 10.1137/120876642.  Google Scholar

[12]

Mathematical Gazette, 84 (2000), 460-467. doi: 10.2307/3620776.  Google Scholar

[13]

Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[14]

Discr. Cont. Dynam. Syst. 2013, Dynamical Systems, Differential Equations and Applications. 9th AIMS Conference. Suppl., (2013), 747-757.  Google Scholar

[15]

BMC Infectious Diseases, 1 (2001), 1-14. doi: 10.1186/1471-2334-1-1.  Google Scholar

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