Dynamics of a Hepatitis B Viral Infection Model with Logistic Hepatocyte Growth and Cytotoxic T-Lymphocyte Response

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1 Nonlinear Analysis and Differential Equations, Vol. 4, 16, no. 3, 19-1 HIKARI Ltd, Dynamics of a Hepatitis B Viral Infection Model with Logistic Hepatocyte Growth and Cytotoxic T-Lymphocyte Response Karam Allali 1 Department of Mathematics, Faculty of Sciences and Technologies University Hassan II, P.O. Box 146, Mohammedia, Morocco Adil Meskaf Department of Mathematics, Faculty of Sciences and Technologies University Hassan II, P.O. Box 146, Mohammedia, Morocco Youssef Tabit Department of SEG, Polydisciplinary Faculty University Chouaib Doukkali, El Jadida, Morocco Copyright c 15 Karam Allali, Adil Meskaf and Youssef Tabit. This article is 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 paper, we present and study the mathematical model of HBV dynamics with logistic hepatocyte growth and cytotoxic T-lymphocyte (CTL) response. The positivity and boundedness of solutions for nonnegative initial data are proved. The stability of disease-free equilibrium and endemic equilibrium are analyzed. Numerical simulations are performed and oscillatory convergence is observed. Mathematics Subject Classification: 34D, 37M5, 37N5, 9D3 Keywords: HBV infection, Viral dynamics, CTL response. 1 Corresponding author

2 11 Karam Allali, Adil Meskaf and Youssef Tabit 1 Introduction Hepatitis B virus (HBV) is a worldwide public health problem with more than 5 million deaths annually [4]. This sever disease can be transmitted easily through contact with infected body fluids [5] and it is, for example, 1 times more infectious than HIV (human immunodeficiency virus) [1, 1]. In the last two decades, many mathematical models have been developed to describe and understand the dynamics of HBV infection [7,8,11]. Recently, some works have improved the previous models by replacing the constant infusion of healthy hepatocytes with a logistic growth function [, 3, 6, 9], this change enriches the dynamics of the HBV pathogenesis model. In this paper, we will consider the logistic hepatocyte growth, the standard incidence functions and we will study the role of the cellular immune system in controlling viral replication. For this purpose, the model that we consider is formulated by the following system of equations: with The initial data are: dx T (t) = rx(t)(1 ) β v(t)x(t), dt T m T (t) dy dt = β v(t)x(t) ay(t) py(t)z(t), T (t) dv = ky(t) µv(t), dt dz = cy(t)z(t) bz(t), dt T (t) = x(t) + y(t). x() = x, y() = y, v() = v and z() = z. (1.1) In this model, x, y, v and z denote the concentration of uninfected cells, infected cells, free virus and CTL cells, respectively. The uninfected hepatocytes grow at a rate that depends on the liver size, T m, at a maximum per capita proliferation rate r. The healthy hepatocytes become infected by the virus at a rate β vx, where β is the mass action constant. Infected cells die at a rate ay T and are killed by the CTL response at a rate pyz. Free virus is produced by infected cells at a rate ky and decays in the presence of the immune system at a rate µv. Finally, CTLs expand in response to viral antigen derived from infected cells at a rate cyz and decay in the absence of antigenic stimulation at a rate bz.

3 Dynamics of a hepatitis B viral infection model 111 The paper is organized as follows. The positivity and boundedness of solutions along with the equilibria and their stability is established in the next section. The Section 3 deals with the some numerical simulations of the model. Finally, short conclusions are given in the last section. Analysis of the model.1 Positivity and boundedness of solutions For the problems deal with cell population evolution, the cell densities should remain non-negative and bounded. In this subsection, we will establish the positivity and boundedness of solutions of the model (1.1). First of all, for biological reasons, the parameters x, y, v and z must be larger than or equal to. Hence, we have the following result: Proposition.1. The solutions of the problem (1.1) exist. Moreover, they are bounded and nonnegative. Proof. First, we will show the nonnegative orthant IR 4 + = {(x(t), y(t), v(t), z(t)) IR 4 : x, y, v and z } is a positively invariant region. Indeed, for (x(t), y(t), v(t), z(t)) IR 4 + we have: ẋ x= =, ẏ y= = βvx, v v= = ky and ż z= =. T Therefore, all solutions initiating in IR 4 + are positive. Adding the two first equations in (1.1), we have: Since we obtain: dt (t) dt = rx(t)(1 T (t) T m ) y(t)(a + pz(t)). T (t) T m and x(t) T (t), thus dt (t) dt rt (t)(1 T (t) T m ) rt (t), T (t) T e rt,

4 11 Karam Allali, Adil Meskaf and Youssef Tabit with T = x + y. we conclude that T is bounded, which means also that x and y are bounded. From the third equation of (1.1), we have: therefore v(t) v()e µt + k t y(ξ)e (ξ t)µ dξ, v(t) v() + k µ y (1 e µt ), so v(t) v() + k µ y (1 e µt ). Since (1 e µt ) 1, we conclude that v is bounded. From the last equation of the system (1.1), we have: therefore dz dt + bz =cyz = c ( rx(t)(1 T (t) ) ay p T m c ( rt (t)(1 T (t) ) ay T p T m z ( c p T () + z())e bt + t Since T is bounded, thus z is also bounded. ) T ), ( c p (r + b)t )eb(ξ t) dξ.. The equilibrium points In this subsection, we will study the stability of the disease-free equilibrium and the endemic equilibrium points. System (1.1) has an infection-free equilibrium E f = (T m,,, ), representing the healthy and disease free liver. By a simple calculation, the endemic equilibrium points of the problem (1.1) are:

5 Dynamics of a hepatitis B viral infection model (rt mµc rbµ+ 1 (rt mµc rbµ ( Tm a(µr kβ + aµ) E 1 =, T m(µr kβ + aµ)( kβ + aµ), rkβ rkβµ T ) m(µr kβ + aµ)( kβ + aµ),, rβµ ( 1 rt m µc rbµ + r E = Tmµ c 4µrβkbT m c rµc r Tmµ c 4µrβkbT mc)βk + 1 µr pµ( 1 rt mµc rbµ+, b c, kb cµ, (rt mµc rbµ+ (r Tmµ c 4µrβkbT mc))a r r T mµ c 4µrβkbT mc µr + b) ( 1 rt m µc rbµ r E 3 = Tmµ c 4µrβkbT m c rµc and r Tmµ c 4µrβkbT mc)βk + 1 µr pµ( 1 rt mµc rbµ, b c, kb cµ, (rt mµc rbµ (r Tmµ c 4µrβkbT mc))a r r T mµ c 4µrβkbT mc µr + b) E 4 = (, b c, kb cµ, a p ), E 5 = (,,, ). + abµ + abµ The three first points E 1, E and E 3 symbolize the persistent and chronic HBV infection. The point E 4 represents the complete liver failure and no healthy cell remains, the evolution of the tissue in this case means that the patient dies; The case E 5 is not possible biologically because its fourth component is a strictly negative number; however the number of CTL cells should be always nonnegative. Since these two last points do not present any biological interest, we will not study their stability..3 The stability analysis First, the jacobian matrix of the system (1.1) is given by: ), ), J = r(1 x+y T m ) βvy λ rx (x+y) T m + βvx βx (x+y) x+y βvy βvx βx a λ pz (x+y) (x+y) x+y k µ λ cz cy b λ (.1)

6 114 Karam Allali, Adil Meskaf and Youssef Tabit by simple calculation the basic reproduction number is given by: R = kβ aµ Proposition.. The free equilibrium point E f is locally asymptotically stable when R < 1 and unstable when R > 1. Proof. The characteristic polynomial of the jacobian matrix (.1) at E f given by: is P Ef (λ) = (λ + r)(λ + b)(λ + (a + µ)λ + (aµ kβ)), then the eigenvalues of the jacobian matrix at E f are: r, b, 1 (a+µ+ (a + µ) + 4aµ(R 1)) and 1 (a+µ (a + µ) + 4aµ(R 1)). The three first eigenvalues are negative while the fourth is negative when R < 1. We conclude that the free-equilibrium point E f is locally asymptotically stable when R < 1 and unstable when R > 1. Proposition.3. The endemic equilibrium point E 1 exists when R R > 1 and R > 1. Moreover, The endemic equilibrium point E 1 is locally asymptotically stable when δ > σ and D < 1. Here and R = a + r, r δ = a ( R 1)(R 1) + a (R R R 1) ar (R 1) + aµ( R 1), R R σ = (a µr + a 3 R )(R R )(R 1) R (µr ar ) D = ct ma br (R 1)(R 1). R Proof. First the endemic equilibrium point E 1 can be rewritten as follows: ( Tm a E 1 = r (R with 1), T ma R r (R 1)(R 1), T ) mka R µr (R 1)(R 1),, R

7 Dynamics of a hepatitis B viral infection model 115 R = a + r r The characteristic polynomial of the jacobian matrix (.1) at E 1 is given by: with P E1 = (λ + b ct ma ( R 1)(R 1))(λ 3 + a λ + a 1 λ + a ), r R a = µ + a R R, a 1 = a ( R 1)(R 1) + a (R R R 1) ar (R 1) + aµ( R 1), R R and a = a µ( R R 1)(R 1). It is clear that the endemic point E 1 exists when a > and a >. We have also that a a 1 > a when δ > σ with and δ = a ( R 1)(R 1) + a (R R R 1) ar (R 1) + aµ( R 1) R R σ = (a µr + a 3 R )(R R )(R 1). R (µr ar ) From Routh-Hurwitz criterion, we conclude that the endemic-point E 1 is locally stable when δ > σ and D < 1, with D = ct ma br (R 1)(R 1). R Proposition.4. If H Z > 1 or R < 1, the endemic point E = (x, y, v, z ) does not exist. The free equilibrium point E is locally asymptotically stable when R > 1 and H Z < 1. Here H Z = 4 βkb rt m µc.

8 116 Karam Allali, Adil Meskaf and Youssef Tabit Moreover, we have the fraction ration of the infected cells verifies: b ct m < y x + y < b ct m. Proof. The endemic equilibrium point E can be rewritten as follows: with E = ( A µc, b c, kb cµ, Aa(R ) 1) abµ, (.) pa + pµb A = 1 rt m µc rbµ + r Tmµ c 4µrβkbT m c r From (.), it is clear that the endemic-point E does not exist if R < 1 or 4 βkb > 1. rt mµc The jacobian matrix of the system (1.1) at E is given by: with J E = P My λ rx T m + Mx N My Mx a λ N pz k µ λ cz λ, P = r(1 x + y T m ) >, M = βv βx > and N = (x + y ) x + y >. The characteristic equation associated to J E is given by: P E (λ) = λ 4 + b 1 λ 3 + b λ + b 3 λ + b 4 =, (.3) where b 1 = µ + a P + Mx + My, b = kn + µa µp ap + c(z ) p + µmx + µmy + My x r/t m Mx P + amy, b 3 = µap + c(z ) pµ + knp + c(z ) pmy µmx P + µamy + c(z ) pp + µmy x r/t m, b 4 = c(z ) pµmy + c(z ) pµp, From the Routh-Hurwitz theorem applied to the fourth order polynomial, the eigenvalues of the jacobian matrix J E have negative real parts since we have b 1 b > b 3 and b 1 b b 3 > b 3 + b 1b 4. Consequently, we obtain the asymptotic local stability of the endemic point E.

9 Dynamics of a hepatitis B viral infection model 117 Finally, the fraction of infected hepatocytes during this chronic steady state is given by from which, we deduce y x + y = b ct m < b, ct m ( βkb ) rt mµc y x + y < b ct m. The stability analysis of the point E 3 is omitted since it is similar to the study of E. 3 Numerical simulations The numerical simulations are performed by using the explicit Euler finitedifference scheme method. Attention will be focused on the numerical stability of the endemic-equilibrium E = (x, y, v, z ), corresponding to non vanishing CTL response. In our study, all the parameters of the simulations are the same as in [3], except the three new parameters to the problem (1.1); p, c and b are chosen adequately. Figure 1 shows the evolution of the infection during the 15 first days. It is interesting to point out that with cellular immune response the oscillations are damped much faster and tend quickly to its equilibrium comparing with the previous model without this response [3], in which the oscillations are maintained for nearly 8 days. Also, we remark that the CTL response reduces virus replication, decreases the value of the infected cells and increases the healthy cells. Figure shows the oscillatory convergence to the endemic point E = (1.999e+11, 6.666, e+3, e+) in the parametric plans with CTL response in the x-axis; the three plots confirm the previous results and show that E is a locally asymptotic stable point. These numerical results are in good agreements with the theoretical result proved in the Proposition (.4); since with our given parameters we have R = 9.3 > 1 and H Z = e-11 < 1. Finally and surprisingly, the lower and upper bound y x +y of the fraction of infected hepatocytes in this chronic state = 3.33e-11 depend only on the parameters describing the CTL dynamics and on the total liver size. This interesting result proves that the fraction of infected hepatocytes is controlled by the CTL immune response.

10 118 Karam Allali, Adil Meskaf and Youssef Tabit Uninfected Cells HBV Virus x Time (days) x Time (days) 1 15 CTL Cells Infected Cells Time (days) Time (days) 1 15 Figure 1: The time series solution of a chronic infection under the following parameter values, r = 1, T m = e11, β =.14, k = 3, a =.693, µ =.693, p =.1, c =.3 and b =.. Uninfected Cells x CTL Cells 3 35 Infected Cells CTL Cells 3 35 HBV Virus.5 x CTL Cells 3 35 Figure : Oscillatory convergence to the endemic point E under the following parameter values, r = 1, T m = e11, β =.14, k = 3, a =.693, µ =.693, p =.1, c =.3 and b =.. 4 Conclusion In this work, we have studied the model describing the dynamics of the hepatitis B viral infection model with logistic hepatocyte growth and CTL response. The model includes four equations illustrating the interaction between the uninfected cells, infected cells, HBV virus and CTL cells. The positiveness and the boundedness of solutions are established. Furthermore, we have studied the stability of both disease-free equilibrium and endemic equilibrium. Finally, numerical simulations are performed in order to show the behavior of infection during time. It was shown that in the presence of CTL immune response, the infection dies out more faster comparing with the previous model without this

11 Dynamics of a hepatitis B viral infection model 119 response and the fraction of infected hepatocytes is controlled by the parameters describing this immune response. Furthermore, the results of this work confirm that the cellular immunity may control viral replication and reduce the infection. References [1] M.J. Alter, Epidemiology of viral hepatitis and HIV co-infection, Journal of Hepatology, 44 (6), S6 S9. [] K. Hattaf, N. Yousfi, Hepatitis B virus infection model with logistic hepatocyte growth and cure rate, Appl. Math. Sci., 5 (11), [3] S. Hews, S. Eikenberry, J.D. Nagy, Y. Kuang, Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth, J. Math. Biol., 6 (1), [4] A. Kane, J. Lloyd, M. Zaffran, L. Simonsen, M. Kane, Transmission of hepatitis B, hepatitis C and human immunodeficiency viruses through unsafe injections in the developing world: model-based regional estimates, Bull. World Health Organ., 77 (1999), [5] D. Lavanchy, Hepatitis B virus epidemiology, disease burden, treatment, and current and emerging prevention and control measures, Journal of Viral Hepatitis, 11 (4), [6] J. Li, K. Wang, Y. Yang, Dynamical behaviors of an HBV infection model with logistic hepatocyte growth, Mathematical and Computer Modelling, 54 (11), [7] M.A. Nowak, S. Bonhoeffer, A.M. Hill, R. Boehme, H.C. Thomas, H. Mc- Dade, Viral dynamics in hepatitis B virus infection, Proc. Natl. Acad. Sci. U.S.A., 93 (1996), [8] M.A. Nowak, R.M. May, Virus Dynamics: Mathematical Principles of Immunology and Virology: Mathematical Principles of Immunology and Virology, Oxford University Press,. [9] A. Packer, J. Forde, S. Hews, Y. Kuang, Mathematical models of the interrelated dynamics of hepatitis D and B, Mathematical Biosciences, 47 (14),

12 1 Karam Allali, Adil Meskaf and Youssef Tabit [1] A. Wasley, M. Alter, Epidemiology of hepatitis C: geographic differences and temporal trends, Semin. Liver Dis., (), [11] N. Yousfi, K. Hattaf, A. Tridane, Modeling the adaptive immune response in HBV infection, Journal of Mathematical Biology, 63 (11), Received: October 4, 15; Published: January 3, 16