# American Institute of Mathematical Sciences

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

## Dynamically consistent discrete-time SI and SIS epidemic models

 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.
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. [2] L.J.S. Allen, "An Introduction to Mathematical Biology,", Prentice Hall, (2007). [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. [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). [5] S. Elaydi, "An Introduction to Difference Equations,", $3^{rd}$ edition, (2005). [6] H.W. Hethcote, Qualitative analyses of communicable disease models,, Math. Biosci., 28 (1976), 335. [7] H.W. Hethcote, The Mathematics of Infectious Diseases,, SIAM Review, 42 (2000), 599. [8] S.R.J. Jang, Nonstandard finite difference methods and biological models,, in, (2005). [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. [10] R.E. Mickens, "Nonstandard Finite Difference Models of Differential Equations,", World Scientific, (1994). [11] R.E. Mickens, "Advances in the applications of nonstandard finite difference schemes,", World Scientific, (2005). [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. [13] R.E. Mickens, Numerical integration of population models satisfying conservation laws: NSF methods,, Journal of Biological Dynamics, 1 (2007), 427. [14] R.E. Mickens, A SIR-model with square-root dynamics: An NSFD scheme,, Journal of Difference Equations and Applications, 16 (2010), 209. [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. [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. [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. [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.

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##### References:
 [1] L.J.S. Allen, Some discrete-time SI, SIR, and SIS epidemic models,, Math. Biosci., 124 (1994), 83. [2] L.J.S. Allen, "An Introduction to Mathematical Biology,", Prentice Hall, (2007). [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. [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). [5] S. Elaydi, "An Introduction to Difference Equations,", $3^{rd}$ edition, (2005). [6] H.W. Hethcote, Qualitative analyses of communicable disease models,, Math. Biosci., 28 (1976), 335. [7] H.W. Hethcote, The Mathematics of Infectious Diseases,, SIAM Review, 42 (2000), 599. [8] S.R.J. Jang, Nonstandard finite difference methods and biological models,, in, (2005). [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. [10] R.E. Mickens, "Nonstandard Finite Difference Models of Differential Equations,", World Scientific, (1994). [11] R.E. Mickens, "Advances in the applications of nonstandard finite difference schemes,", World Scientific, (2005). [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. [13] R.E. Mickens, Numerical integration of population models satisfying conservation laws: NSF methods,, Journal of Biological Dynamics, 1 (2007), 427. [14] R.E. Mickens, A SIR-model with square-root dynamics: An NSFD scheme,, Journal of Difference Equations and Applications, 16 (2010), 209. [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. [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. [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. [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.
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