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

Dynamics of a networked connectivity model of epidemics

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.
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

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