A fractional order model for the transmission dynamics of hepatitis B virus with two-age structure in the presence of vaccination

Abstract In this study, we proposed a fractional order model of hepatitis B virus transmission dynamics with two-age structure under vaccination. A qualitative analysis of the model is performed. Basic reproduction number of the model is determined. Local stability conditions of disease-free equilibrium point are proven by using fractional Ruth-Hurwitz conditions. Global stability conditions of both disease-free and endemic equilibrium points are shown by constructing appropriate Lyapunov functions. Sensitivity analysis was done by using normalized forward sensitivity index approach. Numerical simulation is performed to investigate the effect of memory on hepatitis B disease dynamics by varying order of derivatives and to simulate the effects of vaccinating newborns immediately after birth, vaccinating children and adult vaccination. Then, we compared their effects on hepatitis B disease dynamics in the sense of control and elimination. It is observed that the number of infective individuals decreases faster and even falls to zero over a long run for the model with memory than memory-less model. Comparing results between vaccination of different ages show that increasing newborn vaccination immediately after birth has the highest effect on hepatitis B disease control.


Introduction
Hepatitis is an inflammation of liver and mostly caused by viruses (Moorman et al., 2015;Rehermann & Nascimbeni, 2005). Hepatitis viral infection is a serious public health problem worldwide. Hepatitis B (HB) disease is liver infection caused by hepatitis virus called hepatitis B virus (HBV). It can be acute and resolve without treatment. However, some forms can cause chronic hepatitis, meaning it lasts more than six months. Having chronic hepatitis B increases your risk of developing liver failure, liver cancer or cirrhosis, (Rehermann & Nascimbeni, 2005;World Health Organization, 2020). Hepatitis B virus is one of the most serious and prevalent health problems, affecting more than 2 billion people worldwide. Although highly effective vaccines against hepatitis B virus have been available since 1982, there are still more than 350 million chronic carriers, 75% of whom reside in the Asia Pacific region (Liaw & Chu, 2009). It is estimated that 1.4 million deaths per year are caused by viral hepatitis, which is higher than death caused by HIV and compared with death caused by tuberculosis. Of those deaths approximately 47% are caused by HBV (Rehermann & Nascimbeni, 2005;World Health Organization and others, 2017). It is estimated that 240 million people are chronically infected with hepatitis B. The largest number of people with chronic HBV lives in the African region (over 75 million) next to the Western Pacific region (over 95 million) (Umare, Seyoum, Gobena, & Haile Mariyam, 2016).
HBV infection can be transmitted through vertical transmission (from mother to child at birth), horizontal transmission (exposure to infected blood) and sexual transmission (unprotected sex), and by using needles with infected persons or inapparent percutaneous or permucosal exposure to infected blood or other body fluids (Liaw & Chu, 2009;World Health Organization and others, 2012). Although symptoms not appear on most people when newly infected, some people have acute illness with symptoms, including yellowing of the skin and eyes (jaundice), dark urine, extreme fatigue, nausea, vomiting and abdominal pain. HB infection becomes chronic depends on the age at which a person infected. 80%À90% of infants infected during the first year of life, 30%À50% of children infected before the age of 6 years and less than 5% of healthy persons who are infected as adults will develop chronic infections (Dan, Moses-Eisenstein, & Valdiserri, 2015).
Vaccination of susceptible individuals especially newborns is the most important way to reduce the incidence of HBV infection (Lai, Ratziu, Yuen, & Poynard, 2003;Liang, Zu, & Zhuang, 2018). Despite the success availability of vaccine, problem of HBV infection still remains. Therefore, it is important to predict the future trends in HBV prevalence and provide full information for public health. One of the possible methods to predict the prevalence of infectious disease is to use mathematical model (Liang et al., 2018). As it is directly mentioned in Liaw and Chu (2009) since the introduction of the hepatitis B vaccine and other preventive measures, the worldwide prevalence of hepatitis B infection has fallen. However, chronic infection remains a challenging global health problem. Thus, an improved understanding of hepatitis B virology, immunology, and the natural course of chronic infection is important.
Mathematical model can be helpful in both performing qualitative and quantitative evaluation, forecasting for both injecting drug use and epidemiological information on the incidence of and prevalence of infection, and evaluate the impact of interventions aimed at secondary prevention or harm reduction (Thomson, 2005). There are several types of mathematical models have been used to describe the spread of disease including hepatitis B. These are the deterministic models, stochastic models, fractional order models, etc.
A system of ordinary differential equations is the most used mathematical expression in modelling the transmission dynamics of hepatitis B (Liang et al., 2018). When ordinary differential equation is used the state of the systems at each time does not depend on the previous history of the systems. It is memory-less. However, the evolution and control of epidemic processes in human population can not be considered without memory effect (Pimenov et al., 2012). If people know the history of particular disease in their region, they use different preventive measures, such as isolation of infected individuals and vaccination, when possible (Saeedian, Khalighi, Azimi-Tafreshi, Jafari, & Ausloos, 2017). Since memory indicates the dependence of the system not only on the current state of the system but also on previous history of the system, it enables the prediction of the future on the basis of experience (Kumar, Ghosh, Samet, & Goufo, 2020;Pimenov et al., 2012;Tarasov, 2018). Fractional calculus is a powerfull tool to describe the effect of memory. Fractional derivatives and non-integer orders can be used to describe systems with memory. Due to its memory properties, fractional differential equations have been used in different areas such as physics, engineering, biological systems and financial systems (Pimenov et al., 2012;Saeedian et al., 2017;Tarasov, 2018).
In the modelling of hepatitis B diseases, one of the most important characteristics is considering age (Liang et al., 2018). In the last 10 years, many researchers have been using mathematical model to study the dynamics of HBV transmission. For example, the authors (Adu, Aidoo, Darko, & Osei-Frimpong, 2014;Akbari, Kamyad, & Heydari, 2016;Aniji, Kavitha, & Balamuralitharan, 2019;2014;Khan & Zaman, 2016;Kimbir, Aboiyar, Abu, & Onah, 2014;Nana-Kyere, Ackora-Prah, Okyere, Marmah, & Afram, 2017;Nayeem & Podder, 2014;Seyoum Desta & Koya, 2019;Shaban & Manoza, 2016;Shi, Lu, & Wang, 2019;Wiah, Makinde, & Adetunde, 2015;Zhang, Guo, & Smith, 2018;Zou, Zhang, & Ruan, 2010) propose mathematical model of HBV transmission dynamics. However, most these works did not consider both the effect of memory using fractional order derivatives and age structure model simultaneously. It is widely accepted that the fractional order models have played substantial role in the enhancement of results associated with the existing mathematical model based on classical system of ordinary differential equations. A large number of researchers have modelled many real processes with the aid of fractional calculus. Due to the applicability of differential equations with fractional order in science and engineering, more applied researchers have been attracted to it (Carvalho, Pinto, & Baleanu, 2018;Ghanbari, Kumar, & Kumar, 2020;Kazem, Abbasbandy, & Kumar, 2013;Khan, Hammouch, & Baleanu, 2019;Khodabakhshi, Vaezpour, & Baleanu, 2017;Kumar, 2014;Kumar, Ghosh, et al., 2020;Kumar, Kumar, & Baleanu, 2016;Qureshi & Atangana, 2019;Rashidi, Hosseini, Pop, Kumar, & Freidoonimehr, 2014), and the references there in, just to mention a few. Generally speaking, fractional calculus is a generalization of classical differentiation and integration to arbitrary (non-integer) order. Zhang, Wang, and Zhang, (2015) consider age structure model with out memory effect. Wiah et al. (2015) and Shi et al. (2019) consider memory effect but they did not consider age structure. To our knowledge, no works have been done to model the transmission dynamics of hepatitis B virus infection by considering both age structure and effect of memory. Thus, our main contribution is to formulate a mathematical model that considers both memory effect (fractional order derivatives) and structure at the same time. Web has also performed stability analysis, sensitivity analysis and numerical simulations and such model is used to investigate the effect of varying parameters related to the memory and age structure. Motivated by this, in this paper we proposed a compartmental model of HBV transmission dynamics with two age structures by including memory effect. Fractional calculus involves various types of fractional order derivatives such as Riemann-Liouville (RL), Caputo (C), Hamdard (H), Caputo Febrizo (CF), Atangana-Baleanu (AB) and Atangana-Baleanu-Caputo (ABC). The three most commonly used fractional order differential operators, especially in modelling, are the Caputo, the Caputo-Fabrizio and the Atangana-Baleanu-Caputo operators (Qureshi & Atangana, 2019). However, in this paper, we use the fractional order derivative in the sense of the Capito. More details and definition of these fractional order derivatives can be obtained (Podlubny, 1998;Qureshi & Atangana, 2019) and the references therein.
The rest of the paper is arranged as follows: in Section 2, we formulate age-structured compartmental model represented by system of ordinary differential equations and we extend to fractional order model represented by system of fractional differential equations. In Section 3, qualitative analysis of fractional order model is analysed. In Section 4, we perform sensitivity analysis of parameters. In Section 5, we present numerical simulation. Summary and conclusions of our study are presented in Section 6.

Model description
We divided the total population into eight classes where S c represent susceptible children (children not infected with hepatitis B virus (HBV) and not vaccinated), R c children vaccinated immediately after birth, I c children infected with HBV infection at birth, S a susceptible adult (uninfected adult), E a infected adult with no symptoms (in latent incubation period), I a acutely infected adult with symptoms, C a chronically infected adults and R a recovered adults. New born to infected mother become infected with ð1ÀkÞ probability and those children infected at birth does not go through the acute phase. Newborn vaccinated immediately after birth join R c with ðxÞ being effectiveness of vaccine. Susceptible children S c join recovered children R c with (v) rate when they got vaccine. Susceptible adult become infected with HB infection through horizontal transmission (any means of blood to blood contact with infected classes) with ðbÞ rate of transmission and join exposed adult E a . Exposed adult E a join acutely infected adult I a with ðrÞ rate. Individuals in acutely infected adult I a naturally recovered and join recovered children R a with ðcÞ rate and also join chronically infected adults C a with ðhÞ rate.
The model is based on the following main assumptions: The death rate is assumed to be constant for all classes, and the total death is balanced by total birth (the population constant). We assume that latently infected ðE a Þ, acutely infected ðI a Þ and chronically infected ðC a Þ by horizontal transmission. Children born from exposed, infected and chronically infected mothers can be infected at birth and newborn infected at birth does not go through the acute phase. The susceptible who vaccinated become recovered and got permanent immunity. Individuals in acutely infected adults naturally recovered and not infected again.
Children infected with HB infection at birth are asymptomatic and do not go through the acute phase of HB infection and they progress to chronic phase of HB infection after 20-30 years (Goyal & Murray, 2014). For simplicity, we assume that children infected at birth progress to chronic phase after approximately 14 years. We consider two classes, children (aged 14 years) and adult (aged > 14 years). The model parameters are defined in Table 1.
We present all information about the model description in the flow diagram given in Figure 1 below.
From the above flow diagram, the model without memory can be represented by the following system of nonlinear ordinary differential equations where the total population at any time t ! 0 is given by, NðtÞ ¼ S a ðtÞ þ E a ðtÞ þ I a ðtÞ þ C a ðtÞ þ R a ðtÞ þ S c ðtÞ þR c ðtÞ þ I c ðtÞ and given initial conditions are S a ð0Þ ¼ To include the effect of memory we rewrite systems of ordinary differential equations (2.1) using time-dependent integrals as given below, where d 1 ðtÀxÞ used as time-dependent kernel defined by the following power law correlation function.
where 0 < a 1 and CðxÞ represents Gamma function.
ðtÀxÞ aÀ2 ðmR c ðxÞ þ pS a ðxÞ þ cI a ðxÞÀdR a ðxÞÞdx: (2.4) The right-hand side of system of Equation (2.4) is fractional integral of order ðaÀ1Þ, 0 a 1 on the interval ½t 0 , t, which is represented by t 0 D ÀðaÀ1Þ t : Since fractional derivatives are left inverse of fractional integrals, we apply Caputo fractional derivatives of order ðaÀ1Þ on both sides of Equation (2.4). Then, the following system of fractional differential equations in Caputo is sense obtained. (2.5) Since the total population NðtÞ ¼ S a ðtÞ þ E a ðtÞ þ I a ðtÞ þ C a ðtÞ þ R a ðtÞ þ S c ðtÞ þ R c ðtÞ þ I c ðtÞ, we normalize by dividing both sides by (N(t)) as 1 ¼ S a þ E a þ I a þ C a þ R a þ S c þ R c þ I c , where S a , E a , I a , C a , R a , S c , R c , I c are fraction of each classes in the total population. Since, we omit the last equation of (2.5) and replace for R a in the second and third equations of (2.5), then we obtain where D ¼ d dt and D a represents Caputo fractional derivatives of order 0 < a 1:

Basic mathematical properties of the model
Here, we show the solution to model Equation (2.6) is non-negative. Since we are dealing with the number of population which cannot be negative number, we show there is non-negative solution to our model. Solution to our fractional order model system (2.6) is non-negative, where, S a0 ! 0, S c0 ! 0, R c0 ! o, I c0 ! 0, E a0 ! 0, I a0 ! 0, and C a0 ! 0: Proof. We prove non-negativity of the solution to system of Equation (2.6) as follows Let us take the first equation of (2.6) By taking the Laplace transform of (3.1), we get By applying the inverse Laplace transform to (3.2), we obtain Since S a0 ! 0 and 0 E a, 1 ðÀðbh 1 þ p þ dÞt a Þ 1, S a ! 0, then S a ðtÞ ! 0, 8t ! 0: Let us take the second equation of (2.6): Taking Laplace transform of (3.3), we get Let us take third equations of (2.6) Taking Laplace transform of (3.7), we get By taking inverse Laplace transform of (3.8), we obtain R c ! R c0 E a, 1 ðÀðm þ dÞt a Þ ! 0: Let us take fourth equation of (2.6) By taking Laplace transform of (3.9), we obtain L I c f g ! s aÀ1 I c0 s a þmþd , by applying inverse Laplace transform we get I c ! I c0 E a, 1 ðÀðm þ dÞt a Þ ! 0: Let us take the fifth equation of (2.6) (3.10) Solving (3.10) by method of Laplace transform, we obtain E a ! E a0 E a, 1 ðÀðr þ dÞt a Þ ! 0: Let us take sixth equation of (2.6) ) D a I a ! Àðc þ h þ dÞI a : (3.11) When solving (3.11) by method of Laplace transform we obtain I a ! I a0 E a, 1 ðÀðc þ h þ dÞt a Þ ! 0: Let us take seventh equation of (2.6) D a C a ¼ hI a þ mI c Àðd þ dÞC a : So, ) D a C a ! Àðd þ dÞC a : (3.12) Solving (3.12) by method of inverse of Laplace transform, we obtain C a ! C a0 E a, 1 ðÀðd þ dÞt a Þ ! 0: Therefore, solution to system of fractional differential equation (2.6) is non-negative provided that initial data are non-negative.
þE a þ I a þ C a 1g is positively invariant region given that the initial data is non-negative.
Proof. Suppose 0 S c0 þ R c0 þ I c0 þ S a0 þ E a0 þ I a0 þ C a0 1: Then adding all equations of (2.6), we get where L is Laplace transform operator. By applying inverse Laplace transform we obtain Z bt a E a, aþ1 ðÀdt a Þ þ E a, 1 ðÀdt a Þ: Using the relation E a, b ðzÞ ¼ zE a, aþb ðzÞ þ 1 CðbÞ we have t a E a, aþ1 ðÀdt a Þ ¼ 1 d ð1ÀE a, 1 ðÀdt a ÞÞ: Then Z b d ð1ÀE a, 1 ðÀdt a ÞÞ þ E a, 1 ðÀdt a Þ: Since 0 E a, 1 ðÀdt a Þ 1, we have 0 Z 1 which implies that, 0 ðS c þ R c þ I c þ S a þ E a þ I a þ C a Þ 1: Therefore, region X ¼ fðS a , S c , R c , I c , E a , I a , C a Þ 2 R 7 þ j0 S c þ R c þ I c þ S a þ E a þ I a þ C a 1g is positively invariant region.

Disease-free equilibrium(DFE) point and basic reproduction number (R 0 )
We begin by determining the disease-free equilibrium (DFE) point when there is no infection in the population. So DFE point of model system (2.6) is obtained by making all equations of (2.6) equal to zero. That is Since, at DFE point there is no infection we have I c ¼ 0, E a ¼ 0, I a ¼ 0, and C a ¼ 0, when we substitute I c ¼ 0, E a ¼ 0, I a ¼ 0 and C a ¼ 0 into system of Equation (3.13), we obtain (3.14) Therefore, DFE point is obtained by solving for system of Equation (3.14). From first equation of (3.14), we have From the second equation of (3.14), we have From the third equation of (3.14), we have Equate (3.16) and (3.17) and solve for S c , we obtain Substituting (3.18) into (3.15), we get Then substituting (3.18) into (3.16), we get Therefore, DFE point is given by, , 0, 0, 0, 0 : Next, we find threshold parameter known as basic reproduction number ðR 0 Þ: Basic reproduction number is defined as the average number of secondary infective produced by the presence of one-infective individuals in totally susceptible populations. To compute R 0 , we use method of next-generation matrix method and R 0 is spectral radius of next-generation matrix (Van den Driessche & Watmough, 2002). Let us consider from (2.6) only an equation of infective classes only as follows Let f be the rate of introduction of new infective and v be transmission after new infection and transfer of individual out of compartment, then The inverse of V is given by The characteristic polynomial of FV À1 is given by pðkÞ ¼ ðkÞ 3 ðkÀa 1 b 5 Àa 2 ðb 2 þ b 3 þ b 6 ÞÞ: The eigenvalues of FV À1 are roots of characteristic polynomials PðkÞ: Roots of characteristic polynomial are k 1 ¼ k 2 ¼ k 3 ¼ 0 and k 4 ¼ a 1 b 5 þ a 2 ðb 2 þ b 3 þ b 6 Þ: When we substitute for a 1 , b 5 , a 2 , b 2 , b 3 and b 6 , we get k 4 Since the largest eigenvalue is k 4 , then the basic reproduction number is given as Proof. To show that the DFE point of our model system ð2:6Þ is locally asymptotically stable, we need to show that all the eigenvalues of Jacobian matrix J of system of Equation (2.6) evaluated at DFE point satisfies the condition jargðk i Þj > a p 2 : Let us find the Jacobian matrix J of model system of Equation (2.6) at DFE point. Jacobian matrix J of (2.6) evaluated at DFE point is given by ÀbS 0 a 0 a 1 Àbð1ÀxÞ Àbð1ÀxÞ a 2 a 2 a 2 0 Àbx þ v a 3 Àbx 0 0 0 0 0 0 Àðm þ dÞ bð1ÀxÞ bð1ÀxÞ bð1ÀxÞ where Àbð1ÀxÞÀðv þ m þ dÞ ¼ a 1 , Àbð1ÀxÞð1ÀkÞ ¼ a 2 , ÀbxÀðm þ dÞ ¼ a 3 : The characteristic polynomial PðkÞ of J E 0 is given by The eigenvalues of J E 0 are the roots of PðkÞ: Roots of characteristic polynomial are k 1 ¼ Àðp þ dÞ, k 2 ¼ Àðv þ m þ dÞ, k 3 ¼ Àðb þ m þ dÞ and k 4 þ Ak 3 þ Dk 2 þ Hk þ Q ¼ 0: Let us set The discriminant D(p) of the polynomial pðkÞ is given by All roots of polynomial (3.19) satisfies jargðk i Þj > a p 2 if the following conditions holds If DðpÞ > 0, A ! 0, D ! 0, H ! 0, Q > 0, a < 1 2 , then all roots of polynomial (3.19) satisfies jargðk i Þj > a p 2 , which implies E 0 is locally asymptotically stable. If DðpÞ > 0, A < 0, D < 0, H < 0, a > 1 2 , then all roots of polynomial (3.19) are not satisfies jargðk i Þj > a p 2 , which implies E 0 is unstable. The necessary condition that all roots of (3.19) satisfy jðk i Þj > a p 2 is Q > 0, and Q > 0, if and only if R 0 < 1: Theorem 3.4. (Global Stability of DFE). The disease free equilibrium point is globally asymptotically stable if R 0 < 1: Proof. To proof this theorem we construct a Lyapunov function given below Then, we find Caputo fractional derivatives of V, to obtain (3.20) Then substitute for D a I c , D a E a , D a I a , and D a C a from system (2.6) into (3.20), we get D a V ¼ m mþd ðbð1ÀxÞð1ÀkÞðE a þ I a þ C a ÞÀðm þ dÞ I c Þ þ ðdþdÞðrþcþhþdÞþhr ðrþdÞðcþhþdÞ h i ðbS a ðE a þ I a þ C a ÞÀðr þ dÞE a Þ þ hþdþd cþhþd ðrE a Àðc þ h þ dÞI a Þ þ ðhI a þ mI c Àðd þ dÞC a Þ:

This implies
By collecting like term we obtain is Lyapunov function on the feasible region and the largest compact set in the feasible region.
fðS a , S c , R c , I c , E a , I a , C a Þ 2 R 7 þ , D a V ¼ 0g is the singleton ðS 0 a , S 0 c , R 0 c , 0, 0, 0, 0Þ: Hence, the DFE point of our model is globally asymptotically stable if and only if R 0 < 1:

Endemic equilibrium (EE) point
Endemic equilibrium point occurs when the disease exist in the population. EE point of our model is obtained by making right hand side of all equations of system (2.6) equal to zero. EE point of our model is obtained by solving the following system of equations ( 3.22) From fourth equation of (3.22) we have E a þ I a þ C a ¼ ðm þ dÞ bð1 À xÞð1 À kÞ I c : (3.23) From sixth equation of (3.22) we have From seventh equation of (3.22) we have Plugging (3.24) into (3.25) Substituting (3.24) and (3.26) into (3.23) and solving for E a in terms of I c , we obtain From the fifth equation of (3.22), we have bS a ðE a þ I a þ C a Þ ¼ ðr þ dÞE a : (3.28) Put (3.23) and (3.27) into (3.28) and solve for S a Substitute (3.23) into the second equation of (3.22) and solve for R c in terms of I C and S c , we get From third equation of (3.22), we have (3.31) By equate (3.30) and (3.31) and solve for S c in terms of I c , we get (3.32) Substituting (3.28) into the first equation of (3.22) yields mS c Àðr þ dÞE a Àðp þ dÞS a ¼ 0 (3.33) Substituting (3.27), (3.29) and (3.32) into (3.33) gives Hence, we have After some calculation, we obtain where M ¼ bð1ÀxÞð1ÀkÞðvþmþdÞðpþdÞðdþdÞðrþdÞðcþhþdÞ by and y ¼ mbð1ÀxÞ ð1ÀkÞðm þ dÞððdþ dÞðr þ c þ h þ dÞ þhrÞ þ ðr þ dÞ ðc þ h þ dÞðv þ m þ dÞððmþ dÞðd þ dÞ Àmbð1ÀxÞð1ÀkÞÞ: Substituting (3.34) into (3.27), we obtain (3.37) Then substituting (3.34) into (3.37), we obtain Hence, if R 0 > 1, then model (2.6) has unique endemic equilibrium point given by In this section, we prove global stability of endemic equilibrium (EE) point ðE ? Þ: Theorem 3.5. (Global Stability of Endemic Equilibrium(EE) Point). Let a 2 ð0, 1Þ and R 0 > 1,then unique endemic equilibrium point E ? is globally asymptotically stable.

Sensitivity analysis
In this section, we do sensitivity analysis of model parameters. For this, we do sensitivity analysis of basic reproduction number R 0 by computing the sensitivity index of each parameter. Sensitivity index allows to measure the relative change in basic reproduction number when parameter value changes. For this, we use normalized forward sensitivity index of basic reproduction number with respect to a given parameter using similar approach used (Tilahun, Makinde, & Malonza, 2017;Utoyo & Sa'adah, 2018). Probability that disease transmit to new born at birth Definition 4.1. (Utoyo & Sa'adah, 2018) Normalized forward sensitivity index of basic reproduction number (R 0 ) with respect to the given parameter k is defined as Y R 0 k ¼ @R o @k k R 0 : Using the above formula, we obtained the sign of the sensitivity index for each parameter in basic reproduction number, as summarized in Table 2.
Next, we present the interpretation of sensitivity indices. From Table 2, those parameters with positive indices include b, b, h, m, and 1Àk and those parameters with negative indices include v, p, x, d, d, r, and c. Those parameters with positive indices have great role in the expansion of the disease in the population as their value increases because when their value increases, basic reproduction number increases. Those parameters with negative indices have great role in eliminating the disease from the population as their value increases. Therefore, to eliminate the diseases from the population it is important to increase the value of those parameters with negative indices.

Numerical simulation
Under this section, we performed numerical simulation of the proposed fractional order model. For simulation parameters value from Table 3 is used. To obtain the numerical results of fractional order model we used Euler fractional method. We simulate effects of memory (order of derivatives)(a) on number of HB-infected individuals. We also simulated the effects of newborn vaccination immediately after birth(x), vaccination of children(v) and adult vaccination(p) and their effects compared on the number of HB-infected individuals when the order of derivative is (a ¼ 0:85).  Table 2. Parameter symbols and their sensitivity indices.

Effect of memory on HB infection
Here, we simulate the effects of memory (order of derivatives)(a) on the number of HB-infected individuals. In simulation, we compare the number of infected individuals for memory-less(integer order) model that is when (a ¼ 1) and model with memory that is when (a ¼ 0:85 and a ¼ 0:65). We display the effects of memory(order of derivatives)(a) on the number of HB-infected individuals on Figure 2a-d.
As we observe in Figure 2a, the number of I c is larger in system with out memory(memory-less model)(a ¼ 1) compared with systems with memory (a ¼ 0:85 and a ¼ 0:65).
Similarly as we observe from Figure 2b,c, the number of E a and I a is larger in system with out memory(memory-less model)(a ¼ 1) compared with systems with memory (a ¼ 0:85 and a ¼ 0:65). From Figure 2d, the number of C a increases initially, this is due to movement from I c to C a by maturation and progression form I a to C a , but through time the number of C a becomes larger in system without memory(memory-less model)(a ¼ 1) compared with systems with memory (a ¼ 0:85 and a ¼ 0:65). In other words, as we observe from Figure 2a-d as memory decreases (order of derivatives(a) increases) the number of infected individuals decreases slowly through time.

Effect of newborn vaccine immediately after birth(x)
Here, we simulate the effect of newborn vaccine (x) on HB infection by comparing the number of HBinfected individuals when no vaccine implemented(x ¼ 0, v ¼ 0 and p ¼ 0) and when only newborn vaccine immediately after birth (x) implemented(x 6 ¼ 0, v ¼ 0 and p ¼ 0). On Figure 3a-d, we simulate the effects of newborn vaccine immediately after birth(x) on the number of HB-infected individuals order of derivative is (a ¼ 0:85).
From Figure 3a, the number of I c decreases as value of (x) increases and also from Figure 3b,c, the number of E a and I a decreases and even falls to zero over times as value of (x) increases. From Figure 3d, increasing value of (x) decreases the number of C a : Therefore, as we observe from Figure 3a-d, increasing newborn vaccine (x) decreases the number of infected individuals.

Effect of children vaccination(v) on HB infection
Here, we simulate effect of children vaccination(v) on HB infection by comparing the number of HBinfected individuals when no vaccine implemented(x ¼ 0, v ¼ 0 and p ¼ 0) and when only children vaccination(v) implemented(x ¼ 0,v 6 ¼ 0 and p ¼ 0).
On Figure 4a-d, we simulate the effects of children vaccination(v) on number of HB-infected individuals when order of derivative is (a ¼ 0:85).
From Figure 4a,d, increasing value of children vaccination(v) reduces the number of I c and C a : From Figure 4b,c, the number of E a and I a decreases and even falls to zero over time as children vaccination(v) increases. In general as we observe from Figure  4a-d, increasing children vaccination(v) reduces HB infection from the population.

Effect of adult vaccination(p) on HB infection
Here, we simulate the effect of adult vaccination(p) on HB infection by comparing the number of HBinfected individuals when no vaccine was implemented(x ¼ 0, v ¼ 0 and p ¼ 0) and when only adult vaccination(p) is implemented(x ¼ 0,v ¼ 0 and p 6 ¼ 0). On Figure 5a-d, we simulate the effects of adult vaccination(p) on the number of HB-infected individuals when order of derivative is (a ¼ 0:85).
From Figure 5a,d, the number of I c and C a increases as adult vaccination(p) decreases. From Figure 5b,c, the number of E a and I a decreases as adult vaccination(p) increases. Therefore, from Figure  5a-d, we conclude that increasing adult vaccination reduces the number of HB-infected individuals.

Comparing the effects of newborn vaccine immediately after birth(x), children vaccination(v) and adult vaccination(p) on HB infection
Here, we compare the effects of newborn vaccine(x), children vaccination(v), and adult vaccination(p) on HB infection by comparing number of infected individuals when no vaccine implemented (x ¼ 0, v ¼ 0 and p ¼ 0), only newborn vaccine immediately after birth (x) implemented (x 6 ¼ 0, v ¼ 0 and p ¼ 0), only children vaccination(v) implemented (x ¼ 0,v 6 ¼ 0 and p ¼ 0) and only adult vaccination(p) implemented (x ¼ 0, v ¼ 0 and p 6 ¼ 0) when order of derivative is (a ¼ 0:85). On Figure 6a-d, we compare the effects of newborn vaccine (x), children vaccination(v), and adult vaccination(p) on the number of HB-infected individuals when order of derivative is (a ¼ 0:85).
From Figures 6a,d, we observe that increasing newborn vaccine (x) decreases the number of I c and C a than increasing children vaccination(v) and adult vaccination(p). From Figure 6b,c, we observe that increasing children vaccination(v) decreases the number of E a and I a than increasing newborn vaccine (x) and adult vaccination(p), and also increasing newborn vaccine (x) decreases the number of E a and I a than increasing adult vaccination(p). Therefore, as we observe from Figure 6a-d, increasing newborn vaccine (x) is preferable to eliminate HB infection from the population than increasing children vaccination(v) and adult vaccination(p).

Summary and conclusions
In this study, we formulated mathematical model of HB disease using fractional order to study the dynamics of HB disease in the population taking into consideration memory effect. In Section 2, we described the model assumptions and formulated the deterministic model represented by systems of ordinary differential equations and the model extended to fractional order. In Section 3, model analysis was done by proving positively invariant region in which the solution to the fractional order model is bounded and non-negative, finding equilibrium points and basic reproduction number. Local and global stability analyses of both disease-free and endemic equilibrium points are presented. In Section 4, local normalized sensitivity analysis is performed. In Section 5, numerical simulation was performed to investigate the effect of memory on HB disease dynamics and also to study the effects of newborn vaccine immediately after birth, children vaccination and adult vaccination and to compare their effects on HB disease dynamics. We used MATLAB to perform numerical simulationa.
Using numerical simulations, we investigated the impact of memory on the number of HB-infected individuals, effect of new born vaccination immediately after birth, vaccination of children and adult vaccination by using different values for the order of fractional derivative. Generally, our result shows that memory has great influence on disease dynamics and the result of the comparison between vaccination shows that increasing newborn vaccination is better to eliminate HB disease. The dynamics of the HVB is complicated and needs further research both biologically and mathematically. Although the modern fractional order models developed in this research study could produce better results in the comparison of existing classical models, we strongly believe that this research analysis can further be enhanced. However, we notice that the models with fractional derivatives are more complicated than the ones with classical derivatives.