Local Dynamics of an SVIR Epidemic Model with Logistic Growth

Joko Harianto, Inda Puspita Sari


Discussion of local stability analysis of SVIR models in this article is included in the scope of applied mathematics. The purpose of this discussion was to provide results of local stability analysis that had not been discussed in some articles related to the SVIR model. The SVIR models discussed in this article involve logistics growth in the vaccinated compartment. The results obtained, i.e. if the basic reproduction number less than one and m is positive, then there is one equilibrium point i.e. E0 is locally asymptotically stable. In the field of epidemiology, this means that the disease will disappear from the population. However, if the basic reproduction number more than one and b1 more than b, then there are two equilibrium points i.e. disease-free equilibrium point denoted by E0 and the endemic equilibrium point denoted by E1*. In this case the endemic equilibrium point E1* is locally asymptotically stable. In the field of epidemiology, this means that the disease will remain in the population. The numerical simulation supports these results.


Local Dynamics ; SVIR Model ; Logistic Growth

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W. O. Kermack and A. G. McKendrick, “Contributions to the mathematical theory of epidemics-I,” Bull. Math. Biol., 1991.

V. Capasso and G. Serio, “A generalization of the Kermack-McKendrick deterministic epidemic model,” Math. Biosci., 1978.

C. M. Kribs-Zaleta and J. X. Velasco-Hernández, “A simple vaccination model with multiple endemic states,” Math. Biosci., 2000.

J. Arino, C. C. Mccluskey, and P. V. Van Den Driessche, “Global results for an epidemic model with vaccination that exhibits backward bifurcation,” SIAM J. Appl. Math., 2003.

M. E. Alexander, C. Bowman, S. M. Moghadas, R. Summers, A. B. Gumel, and B. M. Sahai, “A vaccination model for transmission dynamics of influenza,” SIAM J. Appl. Dyn. Syst., 2004.

E. Shim, “A note on epidemic models with infective immigrants and vaccination,” Mathematical Biosciences and Engineering. 2006.

X. Liu, Y. Takeuchi, and S. Iwami, “SVIR epidemic models with vaccination strategies,” J. Theor. Biol., 2008.

M. A. Khan et al., “Stability analysis of an SVIR epidemic model with non-linear saturated incidence rate,” Appl. Math. Sci., 2015.

S. Islam, “Equilibriums and Stability of an SVIR Epidemic Model,” BEST Int. J. Humanit. Arts, Med. Sci. (BEST IJHAMS) , 2015.

J. Harianto, “Local Stability Analysis of an SVIR Epidemic Model,” CAUCHY, 2017.

J. Harianto and T. Suparwati, “SVIR Epidemic Model with Non Constant Population,” CAUCHY, 2018.

X. Z. Li, W. S. Li, and M. Ghosh, “Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment,” Appl. Math. Comput., 2009.

J. J. Wang, J. Z. Zhang, and Z. Jin, “Analysis of an SIR model with bilinear incidence rate,” Nonlinear Anal. Real World Appl., 2010.

Q. Hu, Z. Hu, and F. Liao, “Stability and Hopf bifurcation in a HIV-1 infection model with delays and logistic growth,” Math. Comput. Simul., 2016.

Á. G. C. Pérez, E. Avila-Vales, and G. E. Garciá-Almeida, “Bifurcation analysis of an SIR model with logistic growth, nonlinear incidence, and saturated treatment,” Complexity, 2019.

E. Avila-Vales and Á. G. C. Pérez, “Dynamics of a time-delayed SIR epidemic model with logistic growth and saturated treatment,” Chaos, Solitons and Fractals, 2019.

X. Lai and X. Zou, “Modeling cell-to-cell spread of HIV-1 with logistic target cell growth,” J. Math. Anal. Appl., 2015.

R. Xu, Z. Wang, and F. Zhang, “Global stability and Hopf bifurcations of an SEIR epidemiological model with logistic growth and time delay,” Appl. Math. Comput., 2015.

L. Perko, Differential Equations and Dynamical Systems, 3th ed. New York: Springer, 2001.

P. Van Den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Math. Biosci., 2002.

P. van den Driessche, “Reproduction numbers of infectious disease models,” Infect. Dis. Model., 2017.

N. Chitnis, J. M. Hyman, and J. M. Cushing, “Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model,” Bull. Math. Biol., 2008.

H. S. Rodrigues, M. T. T. Monteiro, and D. F. M. Torres, “Sensitivity Analysis in a Dengue Epidemiological Model,” Conf. Pap. Math., 2013.

P. J. Witbooi, G. E. Muller, and G. J. Van Schalkwyk, “Vaccination Control in a Stochastic SVIR Epidemic Model,” Comput. Math. Methods Med., 2015.

DOI: https://doi.org/10.18860/ca.v6i3.9891


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