Global dynamics of a mathematical model for propagation of infection diseases with saturated incidence rate

Document Type : Original Paper


1 Department of Mathematics, Faculty of Mohajer, Isfahan Branch, Technical and Vocational University, Isfahan, Iran

2 Department of Mathematics, Faculty of Science, University of Zabol,, Zabol, Iran


An epidemic model is described and introduced in which a vaccination program has been included. The model considers disease-caused death in addition to natural death, and the total population size is variable. The equilibria of the model, the disease-free equilibrium and the endemic equilibrium, are obtained and the global dynamics of the model are stated via the basic reproduction number using proper Lyapunov functions. The disease-free equilibrium is asymptotically globally stable when this quantity is less than or equal to unity and when it is greater than unity, the endemic equilibrium is asymptotically globally stable.


Main Subjects

[1] Brauer F. and Castillo-Chavez C., Mathematical models in population biology and epidemiology, Springer, 2012.
[2] Bailey N. T., The mathematical theory of infectious diseases and its applications, Charles Griffin & Company Ltd., 1975.
[3] Chalub F. A. and Souza M. O., Discrete and continuous SIS epidemic models: A unifying approach, Ecological complexity, 18 (2014), 83–95.
[4] Chitnis N., Hyman J. M. and Cushing J. M., Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bulletin of mathematical biology, 70(5) (2008), 1272.
[5] Diekmann O. and Heesterbeek J.A.P., Mathematical epidemiology of infectious diseases: model building, analysis and interpretation, John Wiley & Sons, 2000.
[6] Gamboa M. and Lopez-Herrero M. J., On the Number of Periodic Inspections During Outbreaks of Discrete-Time Stochastic SIS Epidemic Models, Mathematics, 6(8) (2018), 128.
[7] Gumel A. B. and Moghadas S. M., A qualitative study of a vaccination model with non-linear incidence, Applied Mathematics and Computation, 143(2-3) (2003), 409–419.
[8] Hassouna M., Ouhadan A. and El Kinani E., On the solution of fractional order SIS epidemic model, Chaos, Solitons & Fractals, 117 (2018), 168–174.
[9] Hethcote H. W., The mathematics of infectious diseases, SIAM review, 42(4) (2000), 599–653.
[10] Hethcote H., Zhien M. and Shengbing L., Effects of quarantine in six endemic models for infectious diseases, Mathematical Biosciences, 180(1-2) (2002), 141–160.
[11] Jian-quan L., Juan Z. and Zhi-en M., Global analysis of some epidemic models with general contact rate and constant immigration, Applied Mathematics and Mechanics, 25(4) (2004), 396–404.
[12] Jianquan L. and Zhien M., Global analysis of SIS epidemic models with variable total population size, Mathematical and computer modelling, 39(11-12) (2004), 1231-1242.
[13] Khan H., Mohapatra R. N., Vajravelu K. and Liao S., The explicit series solution of SIR and SIS epidemic models, Applied Mathematics and Computation, 215(2) (2009), 653–669.
[14] Khan M. A., Khan Y., Badshah Q. and Islam S., Global stability of SEIVR epidemic model with generalized incidence and preventive vaccination, International Journal of Biomathematics, 8(6) (2015), 1550082.
[15] Khan T., Ullah Z., Ali N. and Zaman G., (2019) Modeling and control of the hepatitis B virus spreading using an epidemic model, Chaos, Solitons & Fractals, 124 (2019), 1–9.
[16] Kribs-Zaleta C. M. and Velasco-Hernández J. X., A simple vaccination model with multiple endemic states, Mathematical biosciences, 164(2) (2000), 183–201.
[17] La Salle J. and Lefschetz S., Stability by Liapunov’s Direct Method with Applications, Elsevier, 2012.
[18] Lahrouz A., Omari L., Kiouach D. and Belmaâti A., Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination, Applied Mathematics and Computation, 218(11) (2012), 6519–6525.
[19] McCluskey C. C. and Van den Driessche P., Global analysis of two tuberculosis models, Journal of Dynamics and Differential Equations, 16(1) (2004), 139–166.
[20] Murray J. D., Mathematical biology: I. An Introduction (interdisciplinary applied mathematics)(Pt. 1), New York, Springer, 2007.
[21] Parsamanesh M., Global stability analysis of a VEISV model for network worm attack, University Politehnica of Bucharest Scientific Bulletin-Series A-Applied Mathematics and Physics, 79(4) (2017), 179–188.
[22] Parsamanesh M., The role of vaccination in controlling the outbreak of infectious diseases: a mathematical approach, Vaccine Research, 5(1) (2018), 32–40.
[23] Parsamanesh M., Global dynamics of an SIVS epidemic model with bilinear incidence rate, Italan Journal of Pure Applied Mathematics, 40 (2018), 544–557.
[24] Parsamanesh M. and Erfanian M., Global dynamics of an epidemic model with standard incidence rate and vaccination strategy, Chaos, Solitons & Fractals, 117 (2018), 192–199.
[25] Parsamanesh M. and Farnoosh R., On the global stability of the endemic state in an epidemic model with vaccination, Mathematical Sciences, 12(4) (2018), 313–320.
[26] Parsamanesh M. and Mehrshad S., Stability of the equilibria in a discrete-time SIVS epidemic model with standard incidence, Filomat, 33(8) (2019), 2393–2408.
[27] Safan M. and Rihan F.A., Mathematical analysis of an SIS model with imperfect vaccination and backward bifurcation, Mathematics and Computers in Simulation, 96 (2014), 195–206.
[28] Sun G. Q., Xie J. H., Huang S. H., Jin Z., Li M. T. and Liu L., Transmission dynamics of cholera: Mathematical modeling and control strategies, Communications in Nonlinear Science and Numerical Simulation, 45 (2017), 235–244.
[29] Van den Driessche P. and Yakubu A., Disease extinction versus persistence in discrete-time epidemic models, Bulletin of mathematical biology, 81(11) (2019), 4412–4446.
[30] Van den Driessche P. and Watmough J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical biosciences, 180(1–2) (2002), 29–48.
[31] Vargas-De-León C., On the global stability of SIS, SIR and SIRS epidemic models with standard incidence, Chaos, Solitons & Fractals, 44(12) (2011), 1106–1110.
[32] Zhang X. and Liu X., Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment, Nonlinear Analysis: Real World Applications, 10(2) (2009), 565–575.