Discrete Dynamics in Nature and Society

Volume 2018, Article ID 2076983, 13 pages

https://doi.org/10.1155/2018/2076983

## Application and Optimal Control for an HBV Model with Vaccination and Treatment

School of Science, Xi’an University of Technology, Xi’an 710048, China

Correspondence should be addressed to Suxia Zhang; moc.361@aixusz

Received 26 April 2018; Accepted 13 August 2018; Published 2 September 2018

Academic Editor: Yong Zhou

Copyright © 2018 Jing Zhang and Suxia Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this study, we formulate a model for hepatitis B virus with control strategies of newborn vaccination and treatment. Mathematical analysis is done theoretically and numerically. The results indicate that the stability of equilibria and persistence of the disease are determined by the basic reproductive number . Using the least squares method, the model is applied to simulate yearly new infected cases of hepatitis B in China from 2004 to 2016. Moreover, optimal control problem with newborn vaccination and treatment appearing as functions of time is analyzed by classical optimal theory. The existence of the solution to optimality system is proved, and the simulations are conducted to show the results when optimal control or current intervention is used.

#### 1. Introduction

It has been over 50 years since the identification of hepatitis B virus (HBV) surface antigen in 1967 [1]. An estimated 257 million people worldwide are living with HBV infection, and in 2015, hepatitis B resulted in 887,000 deaths, mostly from complications, including cirrhosis and hepatocellular carcinoma [2]. HBV infection exhibits an acute infection stage and a chronic liver infection, which is determined by the degree of virus replication and the intensity of host immune response. Infection in newborns, who are incapable of constructing a defective immune response, is more likely to result in chronic infection (≈90%) than that in adults (≈5%), in whom most primary infections are self-limited [3].

Safe and effective vaccine is available for all age groups to prevent HBV infection and development of chronic disease. More than 150 countries have vaccine immunization programs, with routine infant vaccination designated as a high priority in all countries [4]. However, about 4.5 million new infections occur each year, of which a quarter progress to liver disease, and vaccine coverage in developing countries with high endemicity is limited due to high cost and social hurdles [5]. When viral replication is observed in a patient, efficient therapy is needed to control the risk of disease progression. It is estimated that, if left untreated, approximately 15-25% of chronically infected individuals would develop liver cirrhosis and HCC after decades of infection [6]. And there is abundant evidence that antiviral therapy, in patients with long-term virological response, can improve liver histology by providing indirect support and possibly even reversing liver damage [7, 8].

Mathematical models for HBV infection have been developed to study the dynamical properties and control strategies analytically and numerically, with the forms of ordinary differential equations (ODE), partial differential equations (PDE), and discrete equations. Medley et al. [9] observed that the prevalence of HBV is largely determined by a feedback mechanism that relates the rate of transmission, average age at infection, and age-related probability of progressing to chronic infection. Several models were constructed to study the transmission dynamics of this disease and applied to simulate statistical data in some specific regions and countries [10, 11]. An SLICRV compartmental model was proposed to investigate the prevalence of HBV infection in China from 2003 to 2008 [12]. Assuming the immunity acquired from vaccination and recovery being both lifelong, an SEICR model was used for data fitting based on available HBV epidemic data in China [13]. Mann and Robers [14] divided the whole population into five age classes to model the epidemiology of hepatitis B in New Zealand. Considering the important consequences of age for HBV infection, PDE models were developed to capture the transmission characteristics, evaluating the long-term effectiveness of vaccination programmes and analyzing the transmission dynamics [15, 16]. A model with age structure was formulated to study the possible effects of variable infectivity, partitioning the entire infectious period into two stages corresponding to acute and chronic infection, respectively [17]. By a discrete dynamic model, Liang et al. [18] evaluated the independent impact of newborn vaccination on reducing HBV prevalence in China.

In this study, we propose a model for HBV with three infectious compartments in the human population, i.e., two infected classes and a treatment class, which is different from the models in [19–22] and mentioned above. The model we consider is formulated based on HBV transmission among population, compared with the in-host model of delay differential equations in [21]. Due to different motivation in [22], we focus on the transmission of HBV monoinfection and take no consideration of its superinfection with HDV, because the prevalence and new infected cases of HDV are obviously much lower than those of HBV in China [23]. Model application is done to simulate HBV data released by the National Health Commission of China from 2004 to 2016, extending the numerical results in [12, 13, 24] by more epidemiological data. Furthermore, in order to examine the effect of newborn vaccination and treatment on its prevention, the corresponding two parameters are regarded as functions of time, resulting in an optimality system, different from the control strategies with vaccination for the susceptibles in [19, 20]. The existence of the solution to the optimality system is discussed by classical optimal theory, converting the problem of minimizing the objective functional to minimizing the Hamiltonian with respect to the control. Based on the parameter values confirmed in model application, simulations of optimal states, new infected cases, and prevalence of chronic carriers are compared to show the different results with optimal control and with current interventions from initial year 2017 till 2030. Optimal control is also plotted to provide information of implementation as time evolves.

This paper is organized as follows. In Section 2, the descriptions of the parameters and variables, and model formulation are done. In Section 3, we study the stability of steady states and uniform persistence of the disease briefly and conduct sensitivity analysis for input parameters and outcome variable. Model application is done to simulate the epidemiological data in China in Section 4. In Section 5, the optimality system is analyzed theoretically and numerically. We conclude in Section 6 with discussions.

#### 2. Formulation of the Model

When modeling HBV transmission, we divide the whole population into five classes, i.e., susceptible individuals, acute infections, chronic carriers, treated patients, and immunized individuals, which are denoted by , , , , and , respectively. In our model, vertical transmission, i.e., transmission from mother to child, is incorporated, since it has great impact on HBV prevalence, especially in highly endemic areas. The assumptions about vertical transmission can be referred to in [13]. For the newborns, they are assumed to be immunized successfully at rate or unsuccessfully at rate , with being the birth rate and being the successfully immunized proportion. When unsuccessfully immunized, the population of is assumed to enter into the chronic carriers, and the rest stays in the susceptible state, with denoting the probability of children developing to chronic state born to carrier mothers. The virus can also be horizontally transmitted by patients in the classes of , , and , and their different transmission rates are supposed to be , , and , respectively, with the assumption of and , reflecting the fact that the acute patients are the most infectious in the three infected states.

The acute infections are assumed to progress at rate , splitting in to chronic class and to the immunized class due to different immune response in host. In chronic class, the carriers can clear virus and become immunized at rate . We take consideration of treatment for both acute and chronic infection and assume that there are rates of and in the acute and chronic class, respectively, receiving treatment and then moving to the treated class. Once being treated, some patients can be cured and then move to immunized class at rate . However, some people may experience treatment failure and move back to the chronic state at rate . In each class, natural death occurs at rate .

There are some other assumptions in modeling listed in the following.

(1) The recruitment into susceptible population is simplified as new birth, and the birth rate and death rate are assumed to be identical; i.e., .

(2) The infected people who experience treatment failure move back only to chronic state due to the short period of acute infection.

(3) We take no consideration of HBV induced death in view of the efforts of improving liver histology by treatment.

(4) The immunity acquired by transient infection and vaccination is lifelong.

A flow diagram is presented in Figure 1 to show the transitions from one state to another. Then our mathematical model for HBV transmission is given by the following system of ordinary differential equations:Notice that the variable does not appear in other equations in (1); thus the equation of can be ignored, and the reduced system, which is studied in the following, has the same dynamical behavior as the original system. For system (1) without equation , its dynamical behavior remains in the following positively invariant subset of :