Analysis and Simulations of Dynamic Models of Hepatitis B Virus

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1 Analysis an Simulations of Dynamic Moels of Hepatitis B Virus Xisong Dong (Corresponing author) National Engineering Laboratory for Disaster Backup an Recovery Beijing University of Posts an Telecommunications 10 Xi Tu Cheng Roa Beijing China & The Key Laboratory of Complex Systems an Intelligence Science Institute of Automation Chinese Acaemy of Sciences 95 Zhong Guan Cun East Roa Beijing China Tel: ongcomic04@163.com Cong Wang National Engineering Laboratory for Disaster Backup an Recovery Beijing University of Posts an Telecommunications 10 Xi Tu Cheng Roa Beijing China Tel: wangc@bupt.eu.cn Gang Xiong The Key Laboratory of Complex Systems an Intelligence Science Institute of Automation Chinese Acaemy of Sciences 95 Zhong Guan Cun East Roa Beijing China Tel: gang.xiong@ia.ac.cn The research is finance by Beijing Natural Science Founation China (No ) Abstract A mathematical moel consisting of two systems of Hepatitis B virus (HBV) infection is set up base on the moel of virus ynamics an experimental observation of anti-viral rug therapy for HBV infection patients provie by Nowak. A mathematical analysis of ynamic behaviors shows that the moel has two kins of euilibrium points which represent the patients complete recovery an HBV persistent infection at the en of the treatment with rug lamivuine respectively. Our moel may provie possible uantitative interpretations for the long-time treatments of chronic HBV infections with the rug lamivuine accoring to ifferent moel parameters in particularly it may explain why the plasma virus of Nowak et al s patients turnover the original level after stopping the lamivuine treatment. Keywors: Hepatitis B virus(hbv) Mathematical moel Lamivuine treatment Numerical simulation 1. Introuction Hepatitis B is a potentially life-threatening liver infection cause by the hepatitis B virus (HBV). It is a major global health problem an the most serious type of viral hepatitis. It is a viral infection that attacks the liver an can cause both acute an chronic isease (WHO 00). About billion people worlwie have been infecte with the virus an about 350 million live with chronic infection. An estimate persons ie each year ue to the acute or chronic conseuences of hepatitis B (WHO 010). About 5% of aults who become chronically infecte uring chilhoo later ie from liver cancer or cirrhosis (scarring of the liver) cause by the chronic infection. HBV is 50 to 100 times more infectious than HIV (WHO 010). HBV is an important occupational hazar for health workers an 50 million new cases are iagnose annually (WHO 010). Recently rugs calle interferon or lamivuine have been use to treat patients with chronic hepatitis B. Consiering the nee for various long-term treatments it is necessary to construct a mathematical moel that enables us to stuy the ynamics of HBV (Moskovitz 005; Nowak 000). In (Nowak 000) 95 patients were treate with various ose of lamivuine for 8 ays or 4 weeks respectively. Base on the clinical ata Nowak propose a mathematical moel with three variables: uninfecte cells infecte cells an free 1 ISSN E-ISSN

2 virus (Nowak 000). This moel can escribe some viral ynamics in HBV infection. But the moel i not analyze mathematically an the moel cannot escribe the phenomena that the plasma virus of most patients resurge rapily as soon as the rug was withrawn. In this paper a mathematical moel consisting of two systems is esigne. One system is the moel propose by Nowak which escribes the viral ynamics in HBV infection after withrawing the lamivuine therapy. The other system is a moel with four state variables: uninfecte cells infecte cells free virus an cytotoxic cells which represent the viral ynamics in HBV infection uring the lamivuine therapy. The mathematical analysis of the ynamics of our moel shows that the two systems have two kins of euilibrium points which represent the patient s complete recovery an HBV persistent infection respectively. An the numerical simulation can explain why the plasma virus of Nowak et al s patients turnover the original level after stopping the lamivuine treatment an it also may preict the long-time treatments of chronic HBV infections with the rug lamivuine accoring to ifferent moel parameters.. Immune Moel of HBV Infection an Analysis.1 The HBV Moel we propose the following mathematical moel to escribe the viral ynamics of the anti-hbv infection treatment with lamivuine. During processing the lamivaine therapy we assume that the immune moel of HBV infection has the form: x = λ x bxv y = bxv ay k 3 y v = ky uv k 1 y e = k 1 y k e where the 4 variables-x y v an e represent the numbers of uninfecte cells infecte cells free virus an cytotoxic cells respectively. λ is the rate of reprouce susceptible cells. Uninfecte cells ie at rate x an become infect at rate bxv where b is the rate constant escribing the infection process. Infecte cells are prouce at rate bxv an ie at rate ay. Free virus are prouce from infecte cells at rate ky an remove at rate uv. k 1 k 3 represent the pharmacological effect of the lamivaine to infecte cells an free virus respectively. Cytotoxic cells are prouce at rate k 1 y an remove at rate k e. λ b a k u k 1 k k 3 are positive constant an will be etermine by antiviral immune responses. After withrawing the lamivaine therapy the ynamic moel of HBV infection is still escribe by Nowak s moel at (Nowak 000): E(1) has two euilibrium points: x = λ x bvx y = bvx ay v = ky uv Q 1 = ( λ 0 0 0) Q = (x λ x a + k 3 λ x bx k 1(λ x ) k (a + k 3 ) ). (1) () where x = u(a + k 3) b(k k 1 ). E() has two euilibrium points: Q 1 = ( λ 0 0) Q = ( au bk a (λ au bk ) λ b ( bk au λ )). Clearly the euilibrium points Q 1 Q 1 an Q Q stan for the patient s complete recovery an HBV persistent infection respectively. Publishe by Canaian Center of Science an Eucation 13

3 . Analysis of Euilibrium Points Lemma 1: a = [a 0 a 1 a n 1 ] p(s a) = s n + a n 1 s n a 1 s + a 0 n = {a R n : p(s a) = 0 Re[s] < 0} 3 = {a R 3 : a 1 a > a 0 a i > 0 i = 0 1 } 4 = {a R 4 : a 1 a a 3 > a 1 + a 0a 3 a i > 0 i = 0 1 3}. Lemma : Denote where Ẋ = f (X) If f (X ) = 0 an f is ifferentiable on X X = (x 1 x x n ) R n f (X) = ( f 1 (X) f (X) f n (X)) T f 1 (X) x 1 A(X ) =. f n (X) x 1 f 1 (X) x n.... f n (X) x n X=X 1. If λ σ(a) Re(λ) < 0 then X is an asymptotically stable euilibrium point;. If λ σ(a) Re(λ) > 0 then X is an unstable euilibrium point. Where σ(a) are the eigenvalues of A(X )...1 Analysis of Euilibrium Points Q 1 Q Case 1: If the parameters satisfy the ineuality: A = λbk < 1 (3) au then the euilibrium point Q 1 is an asymptotically stable euilibrium point because all the eigenvalues of the linearize euation of E() at Q 1 λ 1 = λ = (a + u) + [(a + u) + 4( λbk au)] 1 λ 3 = (a + u) [(a + u) + 4( λbk au)] 1 are less than 0. For another euilibrium point Q it can be prove that the eigenfunction of the Jacobean matrix of the linearize euation of E() at Q has the form p(s a) = s 3 + a u + au + λbk s + au = s 3 + a s + a 1 s + a 0 abkλ + bkuλ s + λbk au au Because a 0 = λbk au < 0 from Lemma 1 p(s a) = 0 has a root with positive real part. This means that Q is unstable. 14 ISSN E-ISSN

4 Case : If the parameters satisfies the ineuality A = λbk au > 1 then the euilibrium point Q 1 is unstable euilibrium point because one of the real parts of the eigenvalues of the Jacobian matrix of the linearize euation of E() at Q 1 is larger than 0. On the other han we can get From Lemma 1 an we have Q is asymptotically stable..3. Analysis of Euilibrium Points Q 1 Q Case 1: If the parameters satisfies the ineuality a i > 0 i = 0 1 a 1 a >λbk > a 0. < 1 (4) (a + k 3 )u then the euilibrium point Q 1 is an asymptotically stable euilibrium point because all the eigenvalues of the Jacobian matrix of the linearize euation of E(1) at Q 1 λ 1 = λ = k λ 3 = (a + u + k 3) + [(a + u + k 3 ) + 4( λb(k k 1) (a + k 3 )u)] 1 λ 4 = (a + u + k 3) [(a + u + k 3 ) + 4( λb(k k 1) (a + k 3 )u)] 1 are less than 0. For another euilibrium point Q it can be prove that the eigeneuation of the Jacobean matrix of the linearize euation of E() at Q has the form p(s a) = s 4 + λk + (u + a + k + k 3 ) + uk (u + k 3 )(λ ) s 3 + (λ(k + u + a) + )(a + u + k 3 ) = s 4 + a 3 s 3 + a s + a 1 s + a 0 s + (λk + u u)(u + k 3 ) s where Because a 0 = uk (u + k 3 )(λ ) = (a + k 3)u b(k k 1 ) = uk (u + k 3 ) (λ (a + k 3)u b(k k 1 ) ) = uk λ(u + k 3 ) (1 1 A ) < 0 then from Lemma 1 an Q is unstable. Case : If the parameters satisfies the ineuality (a + k 3 )u > 1 then the euilibrium point Q 1 is unstable because one of the real parts of the eigenvalues of the Jacobean matrix of the linearize euation of E() at Q 1 is larger than 0. On the other han we can get A > 1 k > k 1 > 0 a i > 0 i = 1 3 λ = λ (a + k 3)u b(k k 1 ) = λ(1 1 A ) > 0 a 0 > 0 Publishe by Canaian Center of Science an Eucation 15

5 a 1 a a 3 (a 1 + a 0a 3 ) = (λ (k 3 + u + a) + λ(a + u + (a + k 3 )u) + (a + k 3 )u)((a + k 3 )(k + k λ + u(λ )) + k u + k3 + k λ) 3 > (λ (k 3 + u + a) + λ(a + u + (a + k 3 )u) + (a + k 3 )u)((a + k 3 )(k + k λ) + k u + k3 + k λ) 3 > 0 From Lemma 1 an we can get Q is asymptotically stable. Because A = λbk au < 1 A = λb(k k 1) (a + k 3 )u < 1 then when A = λbk au < 1 Q 1 Q 1 are both asymptotically stable euilibrium points; because (a + k 3 )u > 1 A = λbk au > 1 then when (a + k 3 )u > 1 Q Q are both unstable. 3. Numeric simulation Now let us use our moel (E(1) an ()) to simulate the antiviral treatment of chronic HBV infection with rug lamivaine reporte by Nowak(Nowak 000) using MATLAB program. Take {a u k 1 λ} = { } an ifferent values of other parameters an initial values the simulation result is isplaye in Figure 1. From Figure 1 it can be conclue that our moel may provie possible uantitative interpretations for the treatment of chronic HBV infection with the rug lamivuine reporte by Nowak(Nowak 000) in particularly explain why the plasma virus of patients turnover the original level after stopping the lamivuine therapy. If we prolong the treatment time we can get Figure. The values of patient 1-6 for criteria of the stabilities of euilibrium points are liste in Table 1: From Table 1 we can see that for the patient 135 Q 1 Q 1 is unstable euilibrium point an Q Q is stable one an for the patient 46 Q Q is stable an Q 1 Q 1 is unstable. So if we prolong treatment time Patient 46 may recovere completely an Patient 135 may harly be cure even we prolong the time of therapy. In summary we can obtain following conclusions: (1). If ineuality (3) an (4) hols prolonging treatment time makes v(t ) < 0 at some time t then the patient can recovere completely. In fact it has been foun that if prolonging patient 4 s treatment time from 8 ays to about 00 ays the virus can be elete (see Figure ). (). If ineualities (3) or (4) o not hol then the necessity of treatment with rug lamivaine seems to be uestionable. Although the population of infecte cells an virus are reuce an the population of uninfecte cells is increase remarkably prolonging treatment time may not make the patient recovere. As soon as withrawn treatment of rug the population of infecte cells uninfecte cells an free virus will return the level before therapy (see Figure 1). 4. Conclusion In this paper a mathematical moel consisting of two systems of Hepatitis B virus (HBV) infection is set up which has two kins of euilibrium points representing the patients complete recovery an HBV persistent infection respectively. Our moel can explain why the plasma virus of Nowak et al s patients turnover the original level after stopping the lamivuine treatment an it may provie possible uantitative interpretations for the long-time treatments of chronic HBV infections with the rug lamivuine accoring to ifferent moel parameters. Some parameters of moels of HBV infection are ifficult (if not impossible) to be obtaine irectly. However theoretical analysis may be helpful for figure out them. Practically the ynamic behaviors of HBV infection are very complex an puzzling. Our research might provie a possible interpretation for some of them. More an accurate experimental 16 ISSN E-ISSN

6 ata are neee for moeling ynamics of HBV infection. It seems that new treatment approaches are expecte to treat patients. The uantitative unerstaning of HBV ynamics will make it possible to evise optimal treatment strategies for iniviual patients. Further research for HBV ynamics is promising. References Lee W. (1997). Hepatitis B virus infection New Eng. J. Me Moskovitz N. D. Osiowy C. & Giles E. et al. (005). Response to long-term lamivuine treatment(up to 5 years)in patients with severe chronic hepatitis Brole of genotype an rug resistance J.of Viral Hepatitis Nowak M. A. & Robert M. (000). Virus ynamics mathematical principles of immunology an virology. Oxfor university press Worl Health Organization. (00). Department of Communicable Disease Surveillance an Response Hepatitis B. [Online] Available: whocscsrlyo00.pf. WHO010. (010). Hepatitis B. [Online] Available: (June ). Table 1. The ynamic values of patients Patient 1 Patient Patient 3 Patient 4 Patient 5 Patient 6 A A Figure 1. Dynamic routs of improve HBV immune moel(the circles are the numbers reporte by Nowak Publishe by Canaian Center of Science an Eucation 17

7 Figure. Long-Time ynamic routs of improve HBV immune moel 18 ISSN E-ISSN