TRANSMISSION DYNAMICS OF HEPATITIS B: ANALY-SIS AND CONTROL

Abstract

The infection of hepatitis B attacks the liver and can produce acute and chronic diseases, while it is a major health problem and life-threatening around the globe. The control of this infection is a difficult task due to several reasons such as variation of human behavior, proper medication, vaccination, and existence of a large number of carries, etc., but understanding the dynamics of the infection helps to design appropriate control strategies. Thus, a proper complex dynamical system is needed to find the stability conditions and propose intervention strategies for forecasting the control of hepatitis B virus transmission. We formulate a model that will be helpful to investigate the temporal dynamics and suggest control strategies for hepatitis B infection. The well-posedness of the proposed model will be shown, and used to find the threshold parameter to analyze the model equilibria and its stability. We also perform the sensitive analysis of the threshold quantity to quantify the most sensitive epidemic parameters. Based on the temporal dynamics and sensitivity, we investigate effective methods to minimize the infection of hepatitis B, and develop the algorithms to support the theoretical results with the help of numerical simulations.

1. INTRODUCTION

Hepatitis means liver inflammation produced by the virus's bacterial infections and continuous exposure to alcohol or drugs (Ganem & Prince, 2004) . Hepatitis B infection is caused by a non-cytopathic virus called the hepatitis B virus (HBV). The viruses are in vaginal fluids, blood, and semen, and are transferred in multiple ways from one individual to other. Major routes of the transmissions are blood, sharing razors, toothbrushes, and unprotected sexual contacts (Chang, 2007) . Another source of HBV transmission is maternal transmission (i.e., from an infected mother to her child). However, this virus can not be transmitted from causal contact (McMahon, 2005) . The infectious hepatitis B has multiple infection phases: acute and chronic carries. The first phase refers to the initial three months after someone becomes infected with hepatitis B. The immune system can recover in this stage without taking any treatment/hospital care usually, however, may also lead to a long-term infection indicating the chronic phase for someone. The individual with chronic HBV stage often has no acute history of infection. The chronic hepatitis B phase is a severe stage and causes many complications, for example, liver scarring, failure of the liver, and liver cancer (Ringehan et al., 2017) . In the chronic hepatitis B phase, there is a need for treatment with medicines, containing oral antiviral agents. The proper treatment and hospital care will be necessary for the rest of their lives, because this can slow cirrhosis progression and reduce the liver cancer incidence with an improvement in long-term survival. In 2021, WHO recommended oral treatments for the hepatitis B virus as the most potent drugs. The symptom of hepatitis B includes skin yellowing, abdominal pain, urine darkness, fever, and loss of appetite, etc. But for every individual, this is not common to suffer from hepatitis B because 40% of the acute individuals have no symptoms in the acute stage of the infection. Although the vaccine for hepatitis B is available, new cases still have been reported and the infection of hepatitis B is one of the main public health issues that produce a high mortality rate around the globe (Shepard et al., 2006) . Worldwide two billion of the population are infected with hepatitis B, among which the number of chronic hepatitis B is 360 million (Libbus & Phillips, 2009) . Viral hepatitis B is a leading source of death among other diseases while producing 820000 deaths in 2019, with 1.5 million newly infected. Mathematical modeling is a useful tool and has been extensively used to test various theories related to the dynamics of infectious diseases. Various mathematicians, researchers, and biologists used mathematical models to study the communication of transmittible diseases (Rao & Kumar, 2015; Lessler et al., 2016; Ndeffo-Mbah et al., 2018; Li, 2018) . The temporal dynamics of hepatitis B also have a rich literature, and a variety of epidemic models have been used frequently to know the dynamics of the disease. For example, a model has been reported to analyze the dynamics of hepatitis B in Ghana by Dontwi et al. (2014) . The dynamics of the hepatitis B virus with fractional derivatives have been studied by Ullah et al. (2018a; b) . Similarly, many more studies have been reported to discuss the dynamics of hepatitis B (Koonprasert et al., 2016; Khan et al., 2019) . However, the recent models are aimed to analyze the temporal dynamics of the infection using simple disease transmission rate rather than probability-based transmission, which is not realistic especially for a disease having multiple infection phases. Since the contagious infection of hepatitis B has multiple phases and it would be better to use probability-based transmission in the inflow and outflow of individuals from one compartment to another compartment. We try to fill the gap by formulating a model using probability-based rate of transmission according to the characteristics of hepatitis B virus transmission. The model will be developed using the idea of classical susceptible-infected-recovered (SIR) model and probability-based transmission of individuals. We then discuss the existence of the model solution using the Cauchy abstract equation and calculate the threshold parameter of the model to analyze the sensitivity of important parameters. The threshold parameter and its sensitivity are discussed with the help of the next-generation matrix method and sensitivity index, respectively. We also investigate the qualitative behaviors of the model to derive the stability conditions using linear stability analysis. Based on sensitivity and stability conditions, we use optimal control theories to formulate effective methods for hepatitis B eradication. Finally, algorithms for the proposed model will be developed to present the numerical simulations and verify the theoretical analysis.

2. PRELIMINARIES

In this section, we recall some essential concepts, preliminaries, definitions, and methods for the analysis of proposed model that will be helpful for getting our results. Let R n and R n + respectively represent the space of n-tuple and the space of n-tuple with nonnegative entries. The system of non-linear autonomous differential equation then looks like d⃗ u dt = ψ(⃗ u(t)), ⃗ u(t 0 ) = ⃗ u 0 , where ⃗ u ∈ R n and the function ψ implicitly depends only on the dependent variable ⃗ u and not on the independent variable t, for example, du dt = (a -bu(t)) u(t). Let J be the variational matrix of Eq.( 1 (2) Theorem 1 (Yassen, 2005) Let us assume that P(x) is a polynomial of degree n, such that P(x) = x n + α 1 x n-1 + . . . + α n , where α 1 , α 2 , . . . , α n , are constants. The n-Hurwitz matrices for the above polynomial are defined by  H 1 = α 1 , H 2 = α 1 1 α 3 α 2 , H 3 = α 1 1 0 α 3 α 2 α 1 α 4 α 3 α 3 , . . .



. . 0 α 3 α 2 α 1 1 . . . 0 . . . . . . . . . . . . . . .

