
Previous Article
Analysis of a mathematical model for jellyfish blooms and the cambric fish invasion
 PROC Home
 This Issue

Next Article
Parameter dependent stability/instability in a human respiratory control system model
Dynamically consistent discretetime SI and SIS epidemic models
1.  Department of Mathematics and Statistics, Box 41042, Texas Tech University, Lubbock, TX 794091042, United States 
References:
[1] 
L.J.S. Allen, Some discretetime SI, SIR, and SIS epidemic models, Math. Biosci., 124 (1994), 83105. 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), 518543. Google Scholar 
[4] 
M. Chapwanya, Jean M.S. Lubuma, and R.E. Mickens, From enzyme kinetics to epidemilogical models with MichaelisMenten contact rate: Design of nonstandard finite difference schemes, Computers and Mathematics with Applications, (2012), xxxyyy. 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), 335356. Google Scholar 
[7] 
H.W. Hethcote, The Mathematics of Infectious Diseases, SIAM Review, 42 (2000), 599653. 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 LotkaVolterra Type, Journal of Computational Analysis and Applications, 3 (2001), 5373. 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), 672691. Google Scholar 
[13] 
R.E. Mickens, Numerical integration of population models satisfying conservation laws: NSF methods, Journal of Biological Dynamics, 1 (2007), 427436. Google Scholar 
[14] 
R.E. Mickens, A SIRmodel with squareroot dynamics: An NSFD scheme, Journal of Difference Equations and Applications, 16 (2010), 209216. 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), 525529. Google Scholar 
[16] 
L.I.W. Roeger, Nonstandard finite difference schemes for differential equations with $n+1$ distinct fixedpoints, Journal of Difference Equations and Applications, 15 (2009), 133151. Google Scholar 
[17] 
L.I.W. Roeger, Dynamically consistent finite difference schemes for the differential equation $dy/dt=b_ny^n+b_{n1}y^{n1}+\cdots+b_1 y+b_0$, Journal of Difference Equations and Applications, 18 (2012), 305312. Google Scholar 
[18] 
L.I. W. Roeger and G. Lahodny, Jr., Dynamically consistent discrete LotkaVolterra competition systems, Journal of Difference Equations and Applications, 19 (2013), 191200. Google Scholar 
show all references
References:
[1] 
L.J.S. Allen, Some discretetime SI, SIR, and SIS epidemic models, Math. Biosci., 124 (1994), 83105. 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), 518543. Google Scholar 
[4] 
M. Chapwanya, Jean M.S. Lubuma, and R.E. Mickens, From enzyme kinetics to epidemilogical models with MichaelisMenten contact rate: Design of nonstandard finite difference schemes, Computers and Mathematics with Applications, (2012), xxxyyy. 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), 335356. Google Scholar 
[7] 
H.W. Hethcote, The Mathematics of Infectious Diseases, SIAM Review, 42 (2000), 599653. 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 LotkaVolterra Type, Journal of Computational Analysis and Applications, 3 (2001), 5373. 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), 672691. Google Scholar 
[13] 
R.E. Mickens, Numerical integration of population models satisfying conservation laws: NSF methods, Journal of Biological Dynamics, 1 (2007), 427436. Google Scholar 
[14] 
R.E. Mickens, A SIRmodel with squareroot dynamics: An NSFD scheme, Journal of Difference Equations and Applications, 16 (2010), 209216. 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), 525529. Google Scholar 
[16] 
L.I.W. Roeger, Nonstandard finite difference schemes for differential equations with $n+1$ distinct fixedpoints, Journal of Difference Equations and Applications, 15 (2009), 133151. Google Scholar 
[17] 
L.I.W. Roeger, Dynamically consistent finite difference schemes for the differential equation $dy/dt=b_ny^n+b_{n1}y^{n1}+\cdots+b_1 y+b_0$, Journal of Difference Equations and Applications, 18 (2012), 305312. Google Scholar 
[18] 
L.I. W. Roeger and G. Lahodny, Jr., Dynamically consistent discrete LotkaVolterra competition systems, Journal of Difference Equations and Applications, 19 (2013), 191200. Google Scholar 
[1] 
Jianquan Li, Zhien Ma, Fred Brauer. Global analysis of discretetime SI and SIS epidemic models. Mathematical Biosciences & Engineering, 2007, 4 (4) : 699710. doi: 10.3934/mbe.2007.4.699 
[2] 
John E. Franke, AbdulAziz Yakubu. Periodically forced discretetime SIS epidemic model with disease induced mortality. Mathematical Biosciences & Engineering, 2011, 8 (2) : 385408. doi: 10.3934/mbe.2011.8.385 
[3] 
Francisco de la Hoz, Anna Doubova, Fernando Vadillo. Persistencetime estimation for some stochastic SIS epidemic models. Discrete & Continuous Dynamical Systems  B, 2015, 20 (9) : 29332947. doi: 10.3934/dcdsb.2015.20.2933 
[4] 
LihIng W. Roeger, Razvan Gelca. Dynamically consistent discretetime LotkaVolterra competition models. Conference Publications, 2009, 2009 (Special) : 650658. doi: 10.3934/proc.2009.2009.650 
[5] 
Abhyudai Singh, Roger M. Nisbet. Variation in risk in singlespecies discretetime models. Mathematical Biosciences & Engineering, 2008, 5 (4) : 859875. doi: 10.3934/mbe.2008.5.859 
[6] 
LihIng W. Roeger. Dynamically consistent discrete LotkaVolterra competition models derived from nonstandard finitedifference schemes. Discrete & Continuous Dynamical Systems  B, 2008, 9 (2) : 415429. doi: 10.3934/dcdsb.2008.9.415 
[7] 
Carlos M. HernándezSuárez, Carlos CastilloChavez, Osval Montesinos López, Karla HernándezCuevas. An application of queuing theory to SIS and SEIS epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (4) : 809823. doi: 10.3934/mbe.2010.7.809 
[8] 
Fred Brauer, Zhilan Feng, Carlos CastilloChávez. Discrete epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (1) : 115. doi: 10.3934/mbe.2010.7.1 
[9] 
Carlos Lizama, Marina MurilloArcila. Discrete maximal regularity for volterra equations and nonlocal timestepping schemes. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 509528. doi: 10.3934/dcds.2020020 
[10] 
Yicang Zhou, Paolo Fergola. Dynamics of a discrete agestructured SIS models. Discrete & Continuous Dynamical Systems  B, 2004, 4 (3) : 841850. doi: 10.3934/dcdsb.2004.4.841 
[11] 
Jianquan Li, Zhien Ma. Stability analysis for SIS epidemic models with vaccination and constant population size. Discrete & Continuous Dynamical Systems  B, 2004, 4 (3) : 635642. doi: 10.3934/dcdsb.2004.4.635 
[12] 
Angelica Pachon, Federico Polito, Costantino Ricciuti. On discretetime semiMarkov processes. Discrete & Continuous Dynamical Systems  B, 2021, 26 (3) : 14991529. doi: 10.3934/dcdsb.2020170 
[13] 
Guangjun Shen, Xueying Wu, Xiuwei Yin. Stabilization of stochastic differential equations driven by GLévy process with discretetime feedback control. Discrete & Continuous Dynamical Systems  B, 2021, 26 (2) : 755774. doi: 10.3934/dcdsb.2020133 
[14] 
Yijun Lou, XiaoQiang Zhao. Threshold dynamics in a timedelayed periodic SIS epidemic model. Discrete & Continuous Dynamical Systems  B, 2009, 12 (1) : 169186. doi: 10.3934/dcdsb.2009.12.169 
[15] 
Yicang Zhou, Zhien Ma. Global stability of a class of discrete agestructured SIS models with immigration. Mathematical Biosciences & Engineering, 2009, 6 (2) : 409425. doi: 10.3934/mbe.2009.6.409 
[16] 
Yoichi Enatsu, Yukihiko Nakata, Yoshiaki Muroya. Global stability for a class of discrete SIR epidemic models. Mathematical Biosciences & Engineering, 2010, 7 (2) : 347361. doi: 10.3934/mbe.2010.7.347 
[17] 
Yuyun Zhao, Yi Zhang, Tao Xu, Ling Bai, Qian Zhang. pth moment exponential stability of hybrid stochastic functional differential equations by feedback control based on discretetime state observations. Discrete & Continuous Dynamical Systems  B, 2017, 22 (1) : 209226. doi: 10.3934/dcdsb.2017011 
[18] 
Eduardo Liz. A new flexible discretetime model for stable populations. Discrete & Continuous Dynamical Systems  B, 2018, 23 (6) : 24872498. doi: 10.3934/dcdsb.2018066 
[19] 
Ming Chen, Hao Wang. Dynamics of a discretetime stoichiometric optimal foraging model. Discrete & Continuous Dynamical Systems  B, 2021, 26 (1) : 107120. doi: 10.3934/dcdsb.2020264 
[20] 
Ciprian Preda. Discretetime theorems for the dichotomy of oneparameter semigroups. Communications on Pure & Applied Analysis, 2008, 7 (2) : 457463. doi: 10.3934/cpaa.2008.7.457 
Impact Factor:
Tools
Metrics
Other articles
by authors
[Back to Top]