×

Global attractivity of a discrete SIRS epidemic model with standard incidence rate. (English) Zbl 1278.92032

The authors use the forward Euler scheme for a straightforward discretization of a continuous-time SIRS model with a standard incidence of the form \[ \begin{aligned} \dot{S} & =A-dS-\lambda\frac{SI}{S+I+R}+\sigma R,\\ \dot{I} & =\lambda\frac{SI}{S+I+R}-\left( d+\gamma+\alpha\right) I,\\ \dot{R} & =\gamma I-\left( d+\sigma\right) R \end{aligned}\tag{1} \] obtaining a discrete system \[ \begin{aligned} S_{t+1} & =S_{t}+h\left[ A-dS_{t}-\lambda\frac{S_{t}I_{t}}{S_{t}+I_{t} +R_{t}}+\sigma R_{t}\right] ,\\ I_{t+1} & =I_{t}+h\left[ \lambda\frac{S_{t}I_{t}}{S_{t}+I_{t}+R_{t}}-\left( d+\gamma+\alpha\right) I_{t}\right] ,\\ R_{t+1} & =R_{t}+h\left[ \gamma I_{t}-\left( d+\sigma\right) R_{t}\right] . \end{aligned}\tag{2} \] Stating that “for model (2), from the preceding process of discretization, we can imagine that, when the time step size \(h>0\) is small enough, the properties of stability of model (2) should be accordant with model (1)”, the authors study global attractivity of the endemic equilibrium of model (2) concluding that “the dynamical behavior of the discrete-time epidemic model (2) is more complicated than the corresponding continuous-time epidemic model (1)”.
It should be emphasized at this point that even for a simple logistic equation, a direct discretization leads to an equation whose solutions with the growth of the intrinsic growth rate \(r\) exhibit a completely different dynamics as compared to that of the solutions to the original differential equation. Therefore, it is not clear whether equations (1) and (2) model same epidemiological phenomena.

MSC:

92C60 Medical epidemiology
92D30 Epidemiology
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Anderson, Infectious Diseases of Humans: Dynamics and Control (1991)
[2] Andersson, Stochastic epidemic models and their statistical analysis 151 (2000) · Zbl 0951.92021
[3] Diekmann, Series in Mathematical and Computational Biology, in: Mathematical epidemiology of infectious diseases: model building, analysis and interpretation (2000)
[4] Fraser, Factors that make an infectious disease outbreak controllable, Proceedings of National Academy of Sciences of the United States of America 101 pp 6146– (2004)
[5] Halloran, Antiviral effects on influenza viral transmission and pathogenicity: observations from household-based trials, American Journal of Epidemiology 165 pp 212– (2007)
[6] Zhang, Permanence and extinction for a nonautonomous SIRS epidemic model with time delay, Applied Mathematical Modelling 33 pp 1058– (2009) · Zbl 1168.34358
[7] Zhang, Global behavior and permanence of SIRS epidemic model with time delay, Nonlinear Analysis: Real Worlds Applications 9 pp 1409– (2008) · Zbl 1154.34390
[8] Gakkhar, Pulse vaccination SIRS epidemic model with non monotonic incidence rate, Chaos, Solitons and Fractals 35 pp 626– (2008) · Zbl 1131.92052
[9] Mena-Lorca, Dynamic models of infectious diseases as regulators of population sizes, Journal of Mathematical Biology 30 pp 693– (1992) · Zbl 0748.92012
[10] Allen, Some discrete-time SI, SIR, and SIS epidemic models, Mathematical Biosciences 124 pp 83– (1994) · Zbl 0807.92022
[11] Allen, The basic reproduction number in some discrete-time epidemic models, Journal of Difference Equations and Applications 14 pp 1127– (2008) · Zbl 1147.92032
[12] D’Innocenzo, A numerical investigation of discrete oscillating epidemic models, Physica A 364 pp 497– (2006)
[13] Li, A discrete epidemic model with stage structure, Chaos, Solitons and Fractals 26 pp 947– (2005) · Zbl 1066.92045
[14] Sekiguchi, Global dynamics of a discretized SIRS epidemic model with time delay, Journal of Mathematical Analysis and Applications 371 pp 195– (2010) · Zbl 1193.92081
[15] Allen, Spatial patterns in a discrete-time SIS patch model, Journal of Mathematical Biology 58 pp 339– (2009) · Zbl 1162.92033
[16] Castillo-Chavez, Discrete-time SIS models with complex dynamics, Nonlinear Analysis 47 pp 4753– (2001) · Zbl 1042.37544
[17] Franke, Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models, Journal of Mathematical Biology 57 pp 755– (2008) · Zbl 1161.92046
[18] Li, Some discrete SI and SIS epidemic models, Applied Mathematics and Mechanics (English Edition) 29 pp 113– (2008) · Zbl 1231.39006
[19] Li, Global analysis of discrete-time SI and SIS epidemic models, Mathematical Biosciences and Engineering 4 pp 699– (2007) · Zbl 1142.92038
[20] Mendez, Dynamical evolution of discrete epidemic models, Physica A 284 pp 309– (2000)
[21] Ramani, Oscillating epidemics: a discrete-time model, Physica A 333 pp 278– (2004)
[22] Satsuma, Extending the SIR epidemic model, Physica A 336 pp 369– (2004)
[23] Willoxa, Epidemic dynamics: discrete-time and cellular automaton models, Physica A 328 pp 13– (2003) · Zbl 1026.92042
[24] Zhang, Oscillation and global asymptotic stability in a discrete epidemic model, Journal of Mathematical Analysis and Applications 278 pp 194– (2003) · Zbl 1025.39013
[25] Zhou, A discrete epidemic model for SARS transmission and control in China, Mathematical and Computers Modelling 40 pp 1491– (2004) · Zbl 1066.92046
[26] Hu, Stability analysis in a class of discrete SIRS epidemic models, Nonlinear Analysis: Real Worlds Applications 13 pp 2017– (2012) · Zbl 1254.92082
[27] Kocic, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications (1993) · Zbl 0787.39001
[28] Wang, Ordinary Difference Equations (1989)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.