Numerical study for epidemic model of hepatitis-B virus

Tahira Sumbal Shaikh; Nimra Fayyaz; Nauman Ahmed; Naveed Shahid; Muhammad Rafiq; Ilyas Khan; Kottakkaran Sooppy Nisar

doi:10.1140/epjp/s13360-021-01248-8

2023 Impact factor 2.8

EPJ PLUS - Condensed Matter and Complex Systems

Eur. Phys. J. Plus (2021) 136: 367
https://doi.org/10.1140/epjp/s13360-021-01248-8

Regular Article

Numerical study for epidemic model of hepatitis-B virus

Tahira Sumbal Shaikh¹, Nimra Fayyaz¹, Nauman Ahmed², Naveed Shahid²^,3, Muhammad Rafiq⁴, Ilyas Khan⁵^f and Kottakkaran Sooppy Nisar⁶

¹ Department of Mathematics, Lahore College for Women University, Lahore, Pakistan
² Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
³ Department of Mathematics, University of Management and Technology, Lahore, Pakistan
⁴ Department of Mathematics, Faculty of Sciences, University of Central Punjab, Lahore, Pakistan
⁵ Department of Mathematics, College of Science Al-Zulfi, Majmaah University, 11952, Al Majma’ah, Saudi Arabia
⁶ Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, Prince Sattam Bin Abdulaziz University, Wadi-al-dawser, Saudi Arabia

^f i.said@mu.edu.sa

Received: 31 December 2020
Accepted: 17 February 2021
Published online: 7 April 2021

Abstract

Hepatitis-B is a highly infectious disease and causes many worldwide deaths. In this article, a mathematical model of hepatitis-B virus is described. The optimal existence and optimality conditions for the existence of solutions of the concerned model are established. The numerical behavior of the continuous system, susceptible–exposed–acute–carrier–hospitalized–recovered, of hepatitis-B virus is also discussed. Stability of the equilibrium points is studied using Routh–Hurwitz stability criteria. To study this model numerically, a nonstandard finite difference (NSFD) scheme is designed which has the most important features, for instance, unconditional stability, positivity and convergence of the continuous system. Basic reproduction number $R_{0}$ $$R_0$$ that imparts a decisive role in disease prediction as well as stability analysis is described. It is observed that disease vanishes when the value of $R_{0}$ $$R_0$$ is less then unity and it persists for $R_{0} > 1$ $$R_0>1$$ . Stability analysis is carried out, and two bench mark results for the system to be locally asymptotically stable are established. Comparison of the NSFD scheme with two renowned techniques, RK-4 and Forward Euler’s method, is shown in this study. Numerical simulations show that proposed NSFD scheme is convergent to the steady states at all step sizes. It shows that the proposed scheme is independent of the step size while the other two existing schemes do not observe this property. Also, the other methods do not preserve some of the main conditions of the continuous system at certain time steps.