American Institute of Mathematical Sciences

Advanced
2013, 2013(special): 653-662. doi: 10.3934/proc.2013.2013.653

Dynamically consistent discrete-time SI and SIS epidemic models

Lih-Ing W. Roeger 1,

1. 

Department of Mathematics and Statistics, Box 41042, Texas Tech University, Lubbock, TX 79409-1042, United States

Received  August 2012 Revised  April 2013 Published  November 2013

Discrete-time $SI$ and $SIS$ epidemic models are constructed by applying the nonstandard finite difference (NSFD) schemes to the differential equation models. The difference equation systems are dynamically consistent with their analog continuous-time models. The basic standard incidence $SI$ and $SIS$ models without births and deaths, with births and deaths, and with immigrations, are considered. The continuous models possess either the conservation law that the total population is a constant or the total population $N$ satisfies $N'(t)=\lambda-\mu N$ and so that $N$ approaches a constant $\lambda/\mu$ as $t$ approaches infinity. The difference equation systems via NSFD schemes preserve all properties including the positivity of solutions, the conservation law, and the local and some of the global stability of the equilibria. They are said to be dynamically consistent with the continuous models with respect to these properties. We show that a simple criterion for choosing a certain NSFD scheme such that the positivity solutions are preserved is usually an indication of an appropriate NSFD scheme.
Keywords: difference equations, NSFD schemes., epidemic models, Discrete-time, $SIS$.
Mathematics Subject Classification: 34-04, 39-04, 92-0.
Citation: Lih-Ing W. Roeger. Dynamically consistent discrete-time SI and SIS epidemic models. Conference Publications, 2013, 2013 (special) : 653-662. doi: 10.3934/proc.2013.2013.653
References:
[1]

L.J.S. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci., 124 (1994), 83-105. Google Scholar

[2]

L.J.S. Allen, "An Introduction to Mathematical Biology," Prentice Hall, New Jersey, 2007. Google Scholar

[3]

R. Anguelov and J.M.S. Lubuma, Contribution to the mathematics of the nonstandard finite difference method and applications, Numer. Methods Par. Diff. Equ., 17 (2001), 518-543.  Google Scholar

[4]

M. Chapwanya, Jean M.-S. Lubuma, and R.E. Mickens, From enzyme kinetics to epidemilogical models with Michaelis-Menten contact rate: Design of nonstandard finite difference schemes, Computers and Mathematics with Applications, (2012), xxx-yyy.  Google Scholar

[5]

S. Elaydi, "An Introduction to Difference Equations," $3^{rd}$ edition, Springer, New York, 2005.  Google Scholar

[6]

H.W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356.  Google Scholar

[7]

H.W. Hethcote, The Mathematics of Infectious Diseases, SIAM Review, 42 (2000), 599-653.  Google Scholar

[8]

S.R.J. Jang, Nonstandard finite difference methods and biological models, in "Advances in the applications of nonstandard finite diffference schemes," edited by R.E. Mickens, World Scientific, New Jersey, 2005.  Google Scholar

[9]

P. Liu and S.N. Elaydi, Discrete Competitive and Cooperative Models of Lotka-Volterra Type, Journal of Computational Analysis and Applications, 3 (2001), 53-73.  Google Scholar

[10]

R.E. Mickens, "Nonstandard Finite Difference Models of Differential Equations," World Scientific, New Jersey, 1994.  Google Scholar

[11]

R.E. Mickens, "Advances in the applications of nonstandard finite difference schemes," World Scientific, New Jersey, 2005.  Google Scholar

[12]

R.E. Mickens, Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Numerical Methods for Partial Differential Equations, 23 (2006), 672-691.  Google Scholar

[13]

R.E. Mickens, Numerical integration of population models satisfying conservation laws: NSF methods, Journal of Biological Dynamics, 1 (2007), 427-436.  Google Scholar

[14]

R.E. Mickens, A SIR-model with square-root dynamics: An NSFD scheme, Journal of Difference Equations and Applications, 16 (2010), 209-216.  Google Scholar

[15]

R.E. Mickens and T.M. Washington, A note on and NSFD scheme for a mathematical model of respiratory virus transmission, J. Difference Equations and Appl., 18 (2012), 525-529.  Google Scholar

[16]

L.-I.W. Roeger, Nonstandard finite difference schemes for differential equations with $n+1$ distinct fixed-points, Journal of Difference Equations and Applications, 15 (2009), 133-151.  Google Scholar

[17]

L.-I.W. Roeger, Dynamically consistent finite difference schemes for the differential equation $dy/dt=b_ny^n+b_{n-1}y^{n-1}+\cdots+b_1 y+b_0$, Journal of Difference Equations and Applications, 18 (2012), 305-312.  Google Scholar

[18]

L.-I. W. Roeger and G. Lahodny, Jr., Dynamically consistent discrete Lotka-Volterra competition systems, Journal of Difference Equations and Applications, 19 (2013), 191-200.  Google Scholar

show all references

References:
[1]

L.J.S. Allen, Some discrete-time SI, SIR, and SIS epidemic models, Math. Biosci., 124 (1994), 83-105. Google Scholar

[2]

L.J.S. Allen, "An Introduction to Mathematical Biology," Prentice Hall, New Jersey, 2007. Google Scholar

[3]

R. Anguelov and J.M.S. Lubuma, Contribution to the mathematics of the nonstandard finite difference method and applications, Numer. Methods Par. Diff. Equ., 17 (2001), 518-543.  Google Scholar

[4]

M. Chapwanya, Jean M.-S. Lubuma, and R.E. Mickens, From enzyme kinetics to epidemilogical models with Michaelis-Menten contact rate: Design of nonstandard finite difference schemes, Computers and Mathematics with Applications, (2012), xxx-yyy.  Google Scholar

[5]

S. Elaydi, "An Introduction to Difference Equations," $3^{rd}$ edition, Springer, New York, 2005.  Google Scholar

[6]

H.W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356.  Google Scholar

[7]

H.W. Hethcote, The Mathematics of Infectious Diseases, SIAM Review, 42 (2000), 599-653.  Google Scholar

[8]

S.R.J. Jang, Nonstandard finite difference methods and biological models, in "Advances in the applications of nonstandard finite diffference schemes," edited by R.E. Mickens, World Scientific, New Jersey, 2005.  Google Scholar

[9]

P. Liu and S.N. Elaydi, Discrete Competitive and Cooperative Models of Lotka-Volterra Type, Journal of Computational Analysis and Applications, 3 (2001), 53-73.  Google Scholar

[10]

R.E. Mickens, "Nonstandard Finite Difference Models of Differential Equations," World Scientific, New Jersey, 1994.  Google Scholar

[11]

R.E. Mickens, "Advances in the applications of nonstandard finite difference schemes," World Scientific, New Jersey, 2005.  Google Scholar

[12]

R.E. Mickens, Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Numerical Methods for Partial Differential Equations, 23 (2006), 672-691.  Google Scholar

[13]

R.E. Mickens, Numerical integration of population models satisfying conservation laws: NSF methods, Journal of Biological Dynamics, 1 (2007), 427-436.  Google Scholar

[14]

R.E. Mickens, A SIR-model with square-root dynamics: An NSFD scheme, Journal of Difference Equations and Applications, 16 (2010), 209-216.  Google Scholar

[15]

R.E. Mickens and T.M. Washington, A note on and NSFD scheme for a mathematical model of respiratory virus transmission, J. Difference Equations and Appl., 18 (2012), 525-529.  Google Scholar

[16]

L.-I.W. Roeger, Nonstandard finite difference schemes for differential equations with $n+1$ distinct fixed-points, Journal of Difference Equations and Applications, 15 (2009), 133-151.  Google Scholar

[17]

L.-I.W. Roeger, Dynamically consistent finite difference schemes for the differential equation $dy/dt=b_ny^n+b_{n-1}y^{n-1}+\cdots+b_1 y+b_0$, Journal of Difference Equations and Applications, 18 (2012), 305-312.  Google Scholar

[18]

L.-I. W. Roeger and G. Lahodny, Jr., Dynamically consistent discrete Lotka-Volterra competition systems, Journal of Difference Equations and Applications, 19 (2013), 191-200.  Google Scholar

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