Chaos in Applied Sciences and Engineering 2021View this Special Issue
The Transmission Dynamics of Hepatitis B Virus via the Fractional-Order Epidemiological Model
We investigate and analyze the dynamics of hepatitis B with various infection phases and multiple routes of transmission. We formulate the model and then fractionalize it using the concept of fractional calculus. For the purpose of fractionalizing, we use the Caputo–Fabrizio operator. Once we develop the model under consideration, existence and uniqueness analysis will be discussed. We use fixed point theory for the existence and uniqueness analysis. We also prove that the model under consideration possesses a bounded and positive solution. We then find the basic reproductive number to perform the steady-state analysis and to show that the fractional-order epidemiological model is locally and globally asymptotically stable under certain conditions. For the local and global analysis, we use linearization, mean value theorem, and fractional Barbalat’s lemma, respectively. Finally, we perform some numerical findings to support the analytical work with the help of graphical representations.
Hepatitis B virus causes inflammation of the liver. It results from a noncytopathic virus which is called the hepatitis B virus (HBV). Characteristic of HBV is its high tissue and species specificity, as well as a unique genomic organization and replication mechanism. The infection of HBV has multiple phases: acute and chronic. The acute one refers to the first six months whenever there is an exposure of some one to the virus. Usually, in this period, the immune system has the capability to vanish the infection, while for some severe cases, it may also lead to the serious stage and so results in the lifelong illness. This is also known as the chronic stage. It could be noted that whenever HBsAg is positive for a person for a period of more than 6 months, it shows that it has a chronic illness. In case of the chronic stage, often, the individual has no history of the acute stage. This infection may also lead to the scarring of the liver, become liver failure, and produce liver cancer . Hepatitis B virus is transferred by many ways: blood (razors, sharing of blades, tooth brushes, etc.) and semen and vaginal [2–5]. One of the other key sources of transmission is from the infected mother to her child called vertical transmission . Worldwide, there are millions of infected population according to the WHO, in which only 93 millions are infected in China [7, 8]. Vaccines are available to immunize from the HBV which are very effective and almost provide permanent immunity [9, 10].
Mathematical modeling of infectious diseases has a vast field and has a rich literature, which plays a significant role to explore the dynamics and suggest the control mechanism. Since hepatitis B is one of the life-threatening and leading causes of death, it obtained the attention of various researches, and consequently, many epidemiological models were developed (see [11–15]). Anderson and May presented a study in the form of a simple model to investigate the influence of carriers on the transmission of hepatitis B . Williams et al. presented and analyzed the hepatitis B dynamics in the United Kingdom . Moreover, a model was presented by Medley et al. to forecast a mechanism for eliminating hepatitis B from New Zealand . In a similar way, a model that evaluates the effectiveness of the vaccination programme with the effect of age in China was presented by Zhao et al. . Bakare et al. proposed the analysis of control by using an SIR epidemic model . More epidemic models were investigated with control strategies by Kamyad et al. . Onyango developed a model to study the multiple endemic solutions . Similarly, Zhang et al. studied the dynamics of hepatitis B in Xinjiang . Very recently, Khan et al. [24, 25] and Nana-Kyere et al.  formulated some epidemiological models to study different parameters’ influences on the disease transmission and to suggest some control measures for the elimination of the infection. The study of fractional calculus obtained the attention of researchers and is growing rapidly. This analysis has been used to capture the axioms of inherited and the memory of various natural and physical phenomena occurring in different fields of science and technology. Numerous classical models have been proved with less accuracy in case of predicting the future dynamics of a system. However, models having fractional order are more useful to allocate and detain the missing information [27, 28]. It could also be stated that the classical derivative does not provide the dynamics between two different points [29, 30].
It is noted that hepatitis B virus transmission is influenced by different factors, i.e., various phases, routes of transmission, etc. Especially, the carriers are significant. The chronic carriers have no symptoms while transmitting the infection. Moreover, it could also be noted that the increased development of fractional calculus and fractional-order epidemiological models are more suitable than the classical order epidemic models and complex dynamics of hepatitis B; we therefore investigate a hepatitis B virus transmission epidemic model with various infection phases and multiple routes of transmission. Moreover, we also use the fractional calculus to fractionalize the model under consideration which has not yet been studied to the best of our knowledge. Once we formulate the model, we then study the existence analysis as well as uniqueness to prove the well-posedness and biological feasibility of the problem under consideration. For this analysis, the fixed point theory will be used. We also prove that the solutions of the proposed system are bounded and positive. We then discuss the steady state of the proposed model and investigate that the model under consideration is locally and globally asymptotically stable. For local stability analysis, we use the method of linearization, mean value theorem, and fractional Barbalat’s lemma. Finally, some numerical simulations will be performed to support the analytical work and show the difference between the classical and fractional order.
Here, we describe the fundamental concepts related to the fractional calculus which are helpful to obtain our results.
Definition 1 (see ). Let us assume a function such that ; if and , , then the Caputo and Caputo–Fabrizio derivative of the fractional order are defined, respectively, asandIn equations (1) and (2), C and CF represent, respectively, Caputo and Caputo–Fabrizio, while and represent the normalization function such that .
Definition 2 (see ). If and varies with time , then the Riemann–Liouville integral of order () is defined aswhile the integral of order () in the Caputo–Fabrizio-Caputo (CF) sense is defined by
3. Model Formulation
We formulate the model keeping in view the characteristics of hepatitis B virus and so distribute the total population symbolized by into different compartmental population sizes, i.e., susceptible , acute , chronic , recovered/immune , and vaccinated . We also define some constraints for the proposed problem: : all the variables (, , , , and ) and the parameters (, , , , , , , , , , and ) are nonnegative in the epidemic problem that is under consideration. : the successfully vaccinated portion of the susceptible individuals goes to the recovered class. : the contact of susceptible with acute infected as well as with chronically infected is, respectively, denoted by and , which lead to the acute portion with probability and go to chronic with probability , where this assumption is based on the hypothesis that some of the individuals have no history of acute illness. : since some of the individuals got recovery in the acute stage and it leads to the chronic stage for some severe cases, therefore, a natural recovery with probability has been proposed, while leads to the chronic stage. : the recovery under treatment () is taken of the chronic population. : the disease-induced death rate () occurs in the chronic stage only. : the newborn rate is and assumed to be susceptible, while getting successful vaccination leads to the vaccinated class.
In light of these assumptions, we develop a model as presented in the following:with initial population sizeswhere is the proportion of successful vaccination individuals and is the newborn rate. Similarly, the transmission rate of hepatitis B is denoted by , while the reduced transmission rate is . Moreover, and are, respectively, the natural death rate and permanent recovered individuals’ rate. We also symbolize the recovery rate of acute and chronic hepatitis B individuals by and , respectively. The disease-induced death rate is represented by , while those individuals who lose their immunity are represented by .
We extend the reported model by equation (5) to the associated fractional-order () version by taking into account the Caputo–Fabrizio-Caputo (CF) operator. We therefore replace the derivatives in the problem under consideration with a fractional derivative to maintain the dimension of both sides of the equations of the proposed model taking the power of each parameter which becomes
We now discuss the existence and uniqueness of the above fractional-order epidemiological model (7) in the following section.
4. Existence and Uniqueness
This section is devoted to the existence and uniqueness analysis of the solution of fractional-order epidemiological model (7). We use the concept of fixed point theory and prove the solution existence and uniqueness. For this analysis, transforming the proposed system into an integral equation, we obtain
Taking the CF fractional integral of both sides of the above system leads to the assertions as given in the following:
Let , , , , and be the kernels, and they are defined by
Theorem 1. The above kernels , , , , and satisfy axioms of Lipschitz conditions.
Proof. Let us assume that and , and , and , and , and and are, respectively, the two functions for the kernels , , , , and , so we establish the following system:Cauchy’s inequality application leads to the following system:Recursively, we obtainThe norm application with majorizing and the difference between successive terms implywhereSince the kernels satisfy the Lipschitz conditions,
Theorem 2. The solution of fractional-order epidemiological model (7) exists.
Proof. The use of equation (15) with the recursive scheme impliesWe investigate that equation (17) is the solution of model (7); therefore, we make the following substitutions:where , , , , and denote the remainder terms of the series. So,Applying norm on both sides and the Lipschitz axiom,Taking as approaches , we getwhich proves the conclusion that the solution of the proposed epidemiological model as reported by equation (7) exists.
Theorem 3. The proposed epidemiological model described by equation (7) possesess a unique solution.
Proof. On the contradiction basis, we assume that is another solution of model (7); then,Majorizing the above equations, we obtainUsing Theorems 1 and 2, one may obtainThe inequalities as reported by equation (24) hold for every value of ; thus, we obtainWe now discuss the positivity as well as the boundedness of model (7) to show the well-posedness of the problem. Furthermore, we define a certain region for the dynamics of the proposed problem which is positively invariant. For this, the following lemmas have been explored.
Lemma 1. Since are the solutions of model (7), let us consider that the model possesses nonnegative initial conditions; then, the solutions are nonnegative for all .
Proof. We assume a general fractional-order model of system (7) aswhere represents the fractional-order operator under consideration and is the order. So, equation (26) becomes