Modeling and control of the hepatitis B virus spreading using an epidemic model

Khan, Tahir; Ullah, Zakir; Ali, Nigar; Zaman, Gul

doi:10.1016/j.chaos.2019.04.033

Cited by 53 publications

(43 citation statements)

References 25 publications

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“…One of the very applied and effective analysis to design the control programme for infectious diseases is optimal control theory, see for detail [12] , [20] , [22] , [23] . The optimal control theory will be utilized to design a control mechanism for COVID-19 on the basis of local sensitivity analysis.…”

Section: Optimal Control Analysismentioning

confidence: 99%

Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China

Din

Khan

et al. 2020

Chaos, Solitons & Fractals

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115

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Number of well-known contagious diseases exist around the world that mainly include HIV, Hepatitis B, influenzas etc., among these, a recently contested coronavirus (COVID-19) is a serious class of such transmissible syndromes. Abundant scientific evidence the wild animals are believed to be the primary hosts of the virus. Majority of such cases are considered to be human-to-human transmission, while a few are due to wild animals-to-human transmission and substantial burdens on healthcare system following this spread. To understand the dynamical behavior such diseases, we fitted a susceptible-infectious-quarantined model for human cases with constant proportions. We proposed a model that provide better constraints on understanding the climaxes of such unseen disastrous spread, relevant consequences, and suggesting future imperative strategies need to be adopted. The main features of the work include the positivity, boundedness, existence and uniqueness of solution of the model. The conditions were derived under which the COVID-19 may extinct or persist in the population. Sensitivity and estimation of those important parameters have been carried out that plays key role in the transmission mechanism. To optimize the spread of such disease, we present a control problem for further analysis using two control measures. The necessary conditions have been derived using the Pontryagin’s maximum principle. Parameter values have been estimated from the real data and experimental numerical simulations are presented for comparison as well as verification of theoretical results. The obtained numerical results also present the verification, accuracy, validation, and robustness of the proposed scheme.

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Section: Optimal Control Analysismentioning

confidence: 99%

Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China

Din

Khan

et al. 2020

Chaos, Solitons & Fractals

Self Cite

115

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show abstract

“…Several mathematical models for HBV have been developed to analytically and numerically study this disease's transmission dynamics in the form of differential equations (see [6,26,38,39,42] and references therein). If the differential models are properly constructed and analyzed, then we can propose effective strategies to prevent and control the transmission of HBV and this will eventually lead to eradication of the disease.…”

Section: Introductionmentioning

confidence: 99%

“…Very recently, a hepatitis B epidemic model with saturated incidence rate was proposed in [26]. To facilitate the tracking, we recall from the original work [26] the formulation of this HBV model as follows. The model formulation is based on HBV transmission characteristics.…”

Section: Introductionmentioning

confidence: 99%

On the global asymptotic stability of a hepatitis B epidemic model and its solutions by nonstandard numerical schemes

Hoang

Egbelowo

2020

Bol. Soc. Mat. Mex.

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Very recently, a hepatitis B epidemic model with saturated incidence rate has been proposed and analyzed in Khan et al. (Chaos Solitons Fractals 124:1-9, 2019). The local asymptotic stability of the disease endemic equilibrium (DEE) point of model has been established theoretically but its global asymptotic stability has not been studied. In this paper, we present a mathematically rigorous analysis for the global asymptotic stability of the DEE point of the model. More precisely, we prove that if the DEE point exists, then it is not only locally asymptotically stable but also globally asymptotically stable. Furthermore, we present an alternative proof for the global stability of the disease-free equilibrium point. The main result is that we obtain the complete global stability of the hepatitis B virus model. Besides, we construct nonstandard finite difference (NSFD) schemes preserving the essential qualitative properties of the continuous model. These properties include the positivity of solutions and the stability of the model. Dynamical properties of the proposed schemes are rigorously investigated by mathematical analyses and numerical simulations. Finally, numerical simulations are performed to confirm the validity of the theoretical results as well as the advantages of the NSFD schemes. Numerical simulations also indicate that the NSFD schemes are appropriate and effective to solve the continuous model. Meanwhile, the standard finite difference schemes such as the Euler scheme, the classical fourth-order Runge-Kutta scheme cannot preserve the essential properties of the continuous model. Consequently, they can generate numerical approximations which are completely different from the solutions of the model.

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“…In this model, X 1 , X 2 , X 3 : R + ∪ {0} → R are sufficiently smooth functions that represent the total of susceptible, infected and recovered individuals, respectively. This model was proposed recently in [29] as a mathematical model of transmission of hepatitis B, and its inhibition effects are the most notable physical features. Here, α and λ represent, respectively, the transmission and birth rates.…”

Section: Introductionmentioning

confidence: 99%

“…In turn, the global stability of the model can be discussed using Lypanov function theory. Moreover, the local sensitivity analysis was discussed in [29] along with the development of an optimal control mechanism. Furthermore, this model has been validated for large-scale scenarios, and the feasibility of its optimization has been thoroughly established.…”

Section: Introductionmentioning

confidence: 99%

Analysis and Nonstandard Numerical Design of a Discrete Three-Dimensional Hepatitis B Epidemic Model

2019

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In this work, we numerically investigate a three-dimensional nonlinear reaction-diffusion susceptible-infected-recovered hepatitis B epidemic model. To that end, the stability and bifurcation analyses of the mathematical model are rigorously discussed using the Routh-Hurwitz condition. Numerically, an efficient structure-preserving nonstandard finite-difference time-splitting method is proposed to approximate the solutions of the hepatitis B model. The dynamical consistency of the splitting method is verified mathematically and graphically. Moreover, we perform a mathematical study of the stability of the proposed scheme. The properties of consistency, stability and convergence of our technique are thoroughly analyzed in this work. Some comparisons are provided against existing standard techniques in order to validate the efficacy of our scheme. Our computational results show a superior performance of the present approach when compared against existing methods available in the literature.

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Modeling and control of the hepatitis B virus spreading using an epidemic model

Cited by 53 publications

References 25 publications

Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China

Mathematical analysis of spread and control of the novel corona virus (COVID-19) in China

On the global asymptotic stability of a hepatitis B epidemic model and its solutions by nonstandard numerical schemes

Analysis and Nonstandard Numerical Design of a Discrete Three-Dimensional Hepatitis B Epidemic Model

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