Dynamical analysis of an epidemic model with saturated incidence rate and vaccination

Authors

  • Amos Ogunsola Ladoke Akintola University of Technology, Ogbomoso, Nigeria
  • Olukayode Adebimpe Landmark University, Omuaran, Nigeria
  • Bolaji Popoola Federal College of Education, Osiele, Nigeria

DOI:

https://doi.org/10.14419/ijams.v2i3.3363

Published:

2014-12-02

Keywords:

Basic Reproduction Number, Dulac’s Criterion, Epidemic Model, Lyapunov Function, Poincare- Bendixson Theorem, Vaccination.

Abstract

An epidemic model with saturated incidence rate and vaccination is investigated. The model exhibits two equilibria namely disease-free and endemic equilibria. It is shown that if the basic reproduction number (R0) is less than unity, the disease-free equilibrium is locally asymptotically stable and in such case, the endemic equilibrium does not exist. Also, it is shown that if R0 > 1, the disease is persistent and the unique endemic equilibrium of the system with saturation incidence is locally asymptotically stable. Lyapunov function and Dulac’s criterion plus Poincare-Bendixson theorem are applied to prove the global stability of the disease-free and endemic equilibria respectively. The effect of vaccine in the model is critically looked into.

References

C. M.Kribs-Zaleta, J. X.Velasco-Hernadez, A simple vaccination model with multiple endemic states. Math Biosci 164 (2000) 183-201. http://dx.doi.org/10.1016/S0025-5564 (00)00003-1.

J.Arino, C. C.McCluskey, P.van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation. SIAM J. Appl. Math. Vol.64 No.1 (2003) pp. 260-276. http://dx.doi.org/10.1137/S0036139902413829.

J. Li, Z. Ma, Z., Y. Zhou, Global Analysis of SIS epidemic model with a simple vaccination and multiple endemic equilibrium. Acta Mathematica Scientia B (1): (2006) 83-93.

F. Brauer, Backward bifurcations in simple vaccination models, J. Math. Anl. Appl. 298, (2004) pp. 418-431. http://dx.doi.org/10.1016/j.jmaa.2004.05.045.

J. Li, Z. Ma, Qualitative analyses of SIS epidemic model with vaccination and varying total population size. Math. Comp. Model 35, (2002) 1235-1243. http://dx.doi.org/10.1016/S0895-7177 (02)00082-1.

J. Li, Z. Ma, Global Analysis of SIS epidemic models with variable total population size. Math Comput Model 39 (2004) 1231-1242. http://dx.doi.org/10.1016/j.mcm.2004.06.004.

O. Adebimpe, Stability Analysis of a SEIV Epidemic Model with Saturated incidence Rate, British Journal of Mathematics and Computer Science 4(23): (2014), pp. 3358-3368 http://dx.doi.org/10.9734/BJMCS/2014/2758.

O. Adebimpe, B. O. Moses, O. J. Okoro, Global Stability Analysis of a SEIR Epidemic Model with Saturated Incidence Rate, International Journal of Mathematical Sciences, vol.34, Issue 1, (2014) 1504-1512.

R. Ullah, G. Zahman,S. Islam, I. Ahmad, Dynamical features and vaccination Strategies in an SEIR epidemic model, Research Journal of Recent Sciences, vol.2(10), (2013),pp. 48-56

Md. Saiful Islam, Md. Asaduzzaman, Md. Nazrul Islam Mondal, Stability Analysis of DFE of an Epidemic Model in the presence of a Preventive Vaccine, IOSR Journal of Mathematics, volume 3, issue 2, (2012) pp. 25-31.

View Full Article: