The awareness effect of the dynamical behavior of SIS epidemic model with Crowley-Martin incidence rate and holling type III treatment function

Document Type : Research Paper

Authors

1 Ministry of Education, Rusafa/1, Baghdad, Iraq

2 Department of Mathematics, College of Science, Baghdad University, Baghdad, Iraq

3 Centre Regional desMetiers de l’Education et de la Formation (CRMEF), 20340 Derb Ghalef, Casablanca, Morocco, & LAMS, Faculty of Sciences Ben M’sik, Hassan II University of Casablanca, P.O Box 7955 Sidi Othman, Casablanca, Morocco

Abstract

This article deals with the dynamical behaviors for a biological model of epidemic diseases with holling type III treatment function. A Crowley-Martin formula to transmission of disease with coverage media programs effect on the population are introduced and investigated. Through some basic analyses, an explicit formula for the basic reproduction number of the model is calculated, and some results such as the stability analysis and instability of all equilibrium points for the model are established. The local bifurcation occurs near all equilibrium points for the model under some special cases that are studied. The numerical simulations are executed to confirm the theoretical results.

Keywords

[1] J. Adnani, K. Hattaf and N. Yousfi, Stability analysis of a stochastic SIR epidemic model with specific nonlinear incidence rate, (2013)1-4 article ID 431257.
[2] R. M. Anderson, R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, New York (1992).
[3] N. T. Bailey, etal., The mathematical theory of infectious diseases and its applications, Charles Griffin & Company Ltd, 5a Crendon Street, High Wycombe, Bucks HP13 6LE.(1975).
[4] P. Dubey, B. Dubey and S. Dubey, An SIR model with nonlinear incidence rate and holling type III treatment rate, Applied Analysis in Bioloical and Physical Sciences, 186 (2016) 63-81.
[5] I. M. Foppa, A historical introduction to mathematical modeling of infectious diseases, Elsevier Inc., (2017).
[6] J. M. Hyman and J. Li, Modeling the effectiveness of isolation strategies in preventing STD epidemics, SIAM J. Appl. Math. 3 (1998) 912-925.
[7] M. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press. (2008).
[8] R. Kumar, A. Kumar, S. Kumari and P. Roy, Dynamic of an SEIR epidemic model with nonlinear incidence and treatment rates, Nonlinear Dyn., 96 (2019) 2351-2368.
[9] A. Kumar and Nilam, Dynamic Behavior of an SIR Epidemic Model along with Time Delay; Crowley-Martin Type Incidence Rate and Holling Type II Treatment Rate, International Journal of Nonlinear Sciences and Numerical Simulation, 20 (2019) 757-771.
[10] L. Li, Y. Bai, Z. Jin, Periodic solutions of an epidemic model with saturated treatment, Nonlinear Dyn. 2 (2014) 1099-1108.
[11] M. Li, An introduction to Mathematical Modeling of Infectious Diseases, Springer International publishing. (2018).
[12] M. Ibrahim, M. Kamran, M. Naeem, S. Kim and H. Jung, Impact of awareness to control malaria disease: a mathematical modeling approach, Complexity, Article ID 8657410. (2020) 1-13.
[13] Z. Ma and J. Li, Dynamical Modeling and Analysis of Epidemics, World scientific publishing. (2009).
[14] A. Mohsen, Bifurcation analysis in simple SIS epidemic model involving immigrations with treatment, Appl. Math. Inf. Sci. Lett. 3 (2015) 97-102.
[15] A. Mohsen, H. AL-Husseiny, X. Zhou and K. Hattaf, Global stability of COVID-19 model involving the quarantine strategy and media coverage effects, AIMS Public Health, 3 (2020) 587-605. doi: 10.3934/publichealth.2020047.
[16] R. Naji and A. Mohsen, Stability analysis with bifurcation of an SVIR epidemic model involving immigrants, Iraqi journal of Science. 54 (2013) 397-408.
[17] https://www.cdc.gov/vaccines/vac-gen/imz-basics.htm.
[18] L. I. Wu and Z. Feng, Homoclinic bifurcation in an SIQR model for childhood diseases, J. Differ. Equ. 1 (2000)150-167.
[19] R. Xu and Z. Ma, Global stability of a delayed SEIRS epidemic model with saturation incidence rate. Nonlinear Dyn, 61 (2010) 229-239.
[20] L. Yang and F. Wei, Analysis of an epidemic model with Crowley-Martin incidence rate and holling type II treatment, Ann. Of Appl. Math., 36 (2020) 204-220.
[21] X. Zhou, J. Cui, Analysis of stability and bifurcation for an SEIV epidemic model with vaccination and nonlinear incidence rate, Nonlinear Dyn. 4 (2011) 639-653.
Volume 12, Issue 2
November 2021
Pages 1083-1097
  • Receive Date: 08 May 2021
  • Revise Date: 19 June 2021
  • Accept Date: 26 June 2021