Hopf bifurcation and global dynamics of time delayed Dengue model

Zain Ul Abadin Zafar; Nigar Ali; Zahir Shah; Gul Zaman; Prosun Roy; Wejdan Deebani

doi:10.1016/j.cmpb.2020.105530

Hopf bifurcation and global dynamics of time delayed Dengue model

Comput Methods Programs Biomed. 2020 Oct:195:105530. doi: 10.1016/j.cmpb.2020.105530. Epub 2020 May 22.

Authors

Zain Ul Abadin Zafar¹, Nigar Ali², Zahir Shah³, Gul Zaman², Prosun Roy⁴, Wejdan Deebani⁵

Affiliations

¹ Faculty of Information Technology, University of Central Punjab, Lahore, Pakistan.
² Department of Mathematics, University of Malakand, Chakdara, Dir(Lower),Khyber Pakhtunkhwa, Pakistan.
³ Center of Excellence in Theoretical and Computational Science (TaCS-CoE), SCL 802 Fixed Point Laboratory, Science Laboratory Building, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thrung Khru, Bangkok 10140, Thailand. Electronic address: Zahir1987@yahoo.com.
⁴ Department of Mechanical Engineering, University of Wisconsin, Milwaukee, WI, USA.
⁵ Department of Mathematics, College of Science & Arts, King Abdulaziz University, P.O. Box 344, Rabigh 21911, Saudi Arabia.

PMID: 32531718
DOI: 10.1016/j.cmpb.2020.105530

Abstract

Background and objective: Dengue viral infections are a standout amongst the supreme critical mosquito-borne illnesses nowadays. They create problems like dengue fever (DF), dengue stun disorder (DSS) and dengue hemorrhagic fever (DHF). Lately, the frequency of DHF has expanded considerably. Dengue may be caused by one of serotypes DEN-1 to DEN-4. For the most part, septicity with one serotype presents upcoming defensive resistance against that specific serotype yet not against different serotypes. When anyone is infected for a second time with different serotypes, a serious ailment will occur. The proposed model focused on the dynamic interaction between susceptible cells and free virus cells. The ailment free steady states of the specimen are determined. The steadiness of the steady states has been examined by using Laplace transform.

Methods: We introduce an appropriate numerical technique based on an Adams Bash-forth Moulton method for non-integer order delay differential equations. The numerical simulations validate the accuracy and efficacy of the numerical method.

Results: In this paper, we study a non-integer order model with temporal delay to elaborate the dynamics of Dengue internal transmission dynamics. The temporal delay is presented in the susceptible cell and free virus cell. Centered on non-integer Laplace transform, some environs on firmness and Hopf bifurcation are derived for the model. Beside these global stability analysis is also done. Lastly, the imitative theoretical results are justified by few numerical simulations.

Conclusion: The study spectacles that the non-integer order with temporal-delay can successfully enhance the dynamics and rejuvenate the steadiness terms of non-integer order septicity prototypes. Both the ailment free equilibrium (AFE) node and ailment persistent equilibrium (APE) node are steady for the given system. We deduce a recipe that regulates the critical value at threshold.

Keywords: Hopf Bifurcation, Dengue Internal; Laplace Transform; Time Delay; Transmission Model, Fractional Differential Equation.

MeSH terms

Animals
Dengue*
Time Factors