Elsevier

Applied Mathematics and Computation

Volume 331, 15 August 2018, Pages 378-393
Applied Mathematics and Computation

Media coverage campaign in Hepatitis B transmission model

https://doi.org/10.1016/j.amc.2018.03.029Get rights and content

Highlights

  • A mathematical model for Hepatitis B with media coverage campaign is presented.

  • Stability results for the model are analyzed.

  • Optimal control treatment problem with suggested controls are presented.

  • Graphical results with suggested parameters are presented and discussed.

Abstract

In this paper, we consider a transmission model of hepatitis B virus by taking into account media coverage. First, we formulate the model and find the basic reproduction number R0 by using the next generation matrix method. We show that the disease free equilibrium is locally asymptotically stable for R0<1 and unstable if R0>1. We also prove that the system is globally asymptotically stable for R0<1. In order to control the spread of this disease in community, we devise an optimal control problem by introducing three control functions, that is, the educational campaign, vaccination and the media coverage. To do this, we solve analytically the control problem with characterization of control variables using the Portraying’s Maximum Principle. Finally, we present some numerical illustrations.

Introduction

Hepatitis B is a viral infection that attacks the liver and can cause both acute and chronic disease. This virus is transmitted through contact with the blood or other body fluids of an infected person. About 6 million people die every year due to the acute or chronic consequences of hepatitis B. The disease such as SARS, flu and hepatitis B have some distinct properties like as visible symptoms and rapid spread [1]. Communicable diseases spread from one person to another or from an animal to a person. The spread often happens via airborne viruses or bacteria, but hepatitis B spread through blood or other bodily fluid. The terms infectious and contagious are also used to describe communicable disease.

In all health care settings, particularly those in which people are at high risk for exposure to HBV, policies and procedures for HBV control should be developed, reviewed periodically, and evaluated for effectiveness to determine the actions necessary to minimize the risk for transmission of HBV. There are certain measures that can be implemented to control the infection of HBV like administrative measures, environmental controls and media coverage. The role of mass media are greatly acknowledged as a key tool in risk communication [2], [3], but the threat of a spectacle for the audience anxiously beneficiaries have been subjected to criticism [4], [5]. However, the media reporting role is regarded important in crises, management and perception [4].

In order to understand the dynamics of these control measures several researchers used the theory of mathematical modelling. These model represent various outbreaks and sustained oscillations of emerging and re-emerging infectious diseases [1]. Under the influence of media coverage Liu and Cui [6] presented a mathematical model of an infectious disease that give information about the disease spread and control. A pulse and constant vaccination has been incorporated in SIS epidemic models by Li and Cui [7]. Cui et al., [8] proved sufficiently strong media impact and shown that with exponential incidence rate alerts every susceptible individuals in a population [1].

Vaccination is a best tool for eliminating the burden of infectious diseases [9]. Despite its benefits on public health it has received a lot of criticism. People often consider it risky for themselves and thus are inclined to refuse it. As an example, we mention the recent rumors about the vaccine campaign of polio in Nigeria that might cause sterility and HIV [10], and the widespread fear that some vaccines might cause autism[11]. Reports show that those individuals who takes vaccine gaining a positive effects on the disease transmission and also the be havior therapy has a great impression on the course of the disease [12], [13].

In this work, we develop a compartmental mathematical model of hepatitis B by taking into account the media effect on the community. We study the dynamics of hepatitis B in the presence of available vaccine and the role of media to make awareness of vaccination when the infection is in full swing in the community. We find the basic reproduction number and prove that the disease free equilibrium is stable locally as well as globally. The disease free equilibrium, is both locally and globally unstable when R0>1. In order to control the spread of this infection we use three control variables such as educational campaign, available vaccination and control through media coverage. To do this we use optimal control theory to find our optimal problem. Finally, we solve both problems, with and without control, numerically and illustrate the obtained results.

The structure of the paper is as follows: In Section 2, we formulate our proposed model. Section 3 is devoted to the basic properties of the model. The stability analysis of the model is presented in Section 4. In Section 5, we present the optimal control treatment. The existence of an optimal control problem is discussed in Section 6. We show numerical simulation of both the models in Section 7. Finally, we give some conclusions.

Section snippets

Model framework

We divided the total population into five subclasses: susceptible S(t), exposed E(t), acute A(t), carrier C(t), vaccinated individuals V(t) and recovered individuals R(t). This model based on the characteristics of hepatitis B virus transmission with media coverage. The population of susceptible individual is increased by birth or immigration, due to loss of immunity or natural infection. The susceptible move to vaccinated class after vaccination. Similarly the complete transmission in each

Invariant region

Consider the biologically feasible region Ω={(S,E,A,C,V,R)R+6:S+E+A+C+V+Rδμ}.Adding the terms on the right-hand sides of the equations in system (1) gives dNdt=δμ(S+E+A+C+V+R),δμ(S+E+A+C+V+R),=δμN.So, dNdt<0 if N>δμ. Since dNdtδμN, it follows also from a standard comparison theorem [15], that N(t)N(0)eμt+δμ(1eμt).If N(0)δμ, then N(t)δμ. All the initial solutions in Ω remain in Ω for all t > 0. Hence, the set Ω is positively invariant.

Disease free equilibrium:

In this subsection, we find the disease free

Stability analysis

In this section, we present the local and global stability of the disease free equilibrium. First, we establish the local stability of the disease free equilibrium (DFE) with some sufficient condition when R01. After this, we use the method of Castillo-Chavez et al., [18] to find the global stability of the (DFE) equilibrium. In the following, we show the local stability of disease free equilibrium.

Theorem 4.1

If R01, then the disease free equilibrium (DFE) of the system (1) at E¯ is locally

Optimal control treatment

In this section, we implement three control variables to the model (1). Our control variable ue represents education campaign, uv represents vaccination and um represents the media coverage. Our goal is to minimize the total number of susceptible, exposed, acute, carrier and to maximize the population of recovered and vaccinated. dS(t)dt=δπ(1ηC(t))((1uv)θ+μ)S(t)βo(A(t)+κC(t))S(t)(1ue)+δoV(t)ν1S(t)+β2A(t)nA(1um)+A(t)S(t)A(t),dE(t)dt=βo(A(t)+κC(t))S(t)(1ue)μE(t)+δπηC(t)γ1E(t)ν2E(t),dA(t

Existence of control problem

In this section, we investigate the existence of an optimal control problem by considering system (9) with appropriate initial conditions (2) at t=0. Note that, for the bounded Lebesgue measurable controls and nonnegative initial conditions, nonnegative bounded solutions to the state system exist (see [20]). To prove the existence of the solution of system (9), we can write Yt=BY+F(Y),where Y=(SEACVR),B=((μ+ν1+(1uv)θ)00δπ(1η)δo00(μ+γ1+ν2)0δπη000γ1(μ+αo)00000αo(μ+γ3)00(1uv)θ00γ3(δo+μ+ν3)0

Numerical results

In this section, we present the numerical simulation of systems (1) and (9), with the non-negative initial conditions (2). For the numerical simulation, we use the Runge–Kutta fourth order backward scheme. First, we solve the model (1), and then find the solution of system (9), together with adjoint equations and characterizations of the optimal control problem. Numerical results are presented in Fig. 2, Fig. 3, Fig. 4, Fig. 5. In this simulation, we show the control system with dashed line

Conclusion

We have successfully presented the media coverage campaign in hepatitis B transmission model. Initially, we formulated the model of HBV with media coverage, and then modified the model by incorporating three control variables, that is ue, uv and um, respectively represent the educational campaign, vaccination, and media coverage campaign. The stability results have been derived and discussed for system (1). System (1) is stable locally as well as globally when the basic reproduction number less

Acknowledgment

This study has been partly supported by Higher Education Commission (HEC) of Pakistan under the project No. 201983/RandD/HEC/11.

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