KINK SOLITARY SOLUTIONS TO A HEPATITIS C EVOLUTION MODEL

The standard nonlinear hepatitis C evolution model described in (Reluga et al. 2009) is considered in this paper. The generalized differential operator technique is used to construct analytical kink solitary solutions to the governing equations coupled with multiplicative and diffusive terms. Conditions for the existence of kink solitary solutions are derived. It appears that kink solitary solutions are either in a linear or in a hyperbolic relationship. Thus, a large perturbation in the population of hepatitis infected cells does not necessarily lead to a large change in uninfected cells. Computational experiments are used to illustrate the evolution of transient solitary solutions in the hepatitis C model. 2010 Mathematics Subject Classification. Primary: 34A34; Secondary: 35C05, 35C07.


(Communicated by Peter E. Kloeden)
Abstract. The standard nonlinear hepatitis C evolution model described in (Reluga et al. 2009) is considered in this paper. The generalized differential operator technique is used to construct analytical kink solitary solutions to the governing equations coupled with multiplicative and diffusive terms. Conditions for the existence of kink solitary solutions are derived. It appears that kink solitary solutions are either in a linear or in a hyperbolic relationship. Thus, a large perturbation in the population of hepatitis infected cells does not necessarily lead to a large change in uninfected cells. Computational experiments are used to illustrate the evolution of transient solitary solutions in the hepatitis C model.

1.
Introduction. Recent developments in computer hardware and software enable to use powerful symbolic computation techniques for the construction of nonlinear wave solutions to high-dimensional nonlinear evolution equations in mathematical physics. Solitary (or soliton) solutions represent solitary wave packets that do not change their shape when propagating at constant velocities [24]. Due to their unique properties, construction of solitary solutions is an important problem in nonlinear science [26,4,1].
A short overview of typical examples illustrating the discovery of solitary solutions in nonlinear evolution problems is presented below. The dynamic pressure of an irrotational solitary wave propagating at the surface of water over a flat bed is studied in [6]. Properties of bright and dark solitary solutions in strongly magnetized warm plasmas are considered in [5]. Solitary wave solutions to a system of coupled complex Newell-Segel-Whitehead equations are constructed in [9]. Gray/dark solitons in nonlocal nonlinear media are analytically studied using the symmetry reduction method in [8]. A closed-form analytical solution, including bright and dark solitons, to the driven nonlinear Schrödinger equation is constructed in [22].
Solitary solutions are often encountered in coupled differential equations. Next we mention several typical examples. Dark-bright soliton solutions to a coupled Schrödinger system with equal, repulsive cubic interactions are considered in [2]. In [23], exact bright one-and two-soliton solutions to a particular type of coherently coupled Schrödinger equations are constructed using the non-standard Hirota's bilinearization method. Three families of analytical solitary wave solutions of generalized coupled cubic-quintic Ginzburg-Landau equations are obtained in [27].
Even though kink solitary solutions are the simplest type of solitons, their construction and analysis is far from being trivial [21]. Kinks and bell-type soliton solutions to a differential equation describing the dynamics of microtubules are constructed in [28]. The interaction of kink-type solutions of harmonic map equations is studied in [7]. The kink solutions to the negative-order KdV equation are constructed using the Lax pair in [18]. Kink solutions to models of transport phenomena and mathematical biology are considered in [25].
The main objective of this paper is to seek kink solitary solutions in a hepatitis C virus infection model [20] that explicitly includes proliferation of infected and uninfected hepatocytes. The mathematical equations of the model are: where t is time; T t represents uninfected hepatocytes; I t represents infected cells and V t represents free virus population. The parameters of (1) have the following meaning: p is the free virus production rate per infected cell; c is the immune virus clearance rate; d T , d I are death rates for uninfected hepatocytes and infected cells respectively; r T , r I are parameters of the logistic proliferation of T and I respectively; logistic proliferation happens only if T < T max ; β is the rate of infection per free virus per hepatocyte; parameters s and q represent the increase rate of uninfected hepatocytes through immigration and spontaneous cure by noncytolytic process respectively; finally the effect of antiviral treatment reduces the infection rate by a fraction η and the viral production rate by a fraction . Ranges of parameters are given in [20]. The third equation of system (1) can be explicitly solved for V if patients are in a steady state before treatment. The introduction of dimensionless state variables x and y for uninfected and infected cells respectively reduces (1) to: Note that (2) can be rearranged in the general form: where c, u, v, a k , b k ∈ R, k = 1, . . . , 4. System (3) is comprised of Riccati equations [19] coupled with both multiplicative and diffusive terms. It appears that models (2) and (3) are natural extensions of the competing species evolution model with the Allee effect [12,3]. Furthermore, similar evolution models are at the forefront of the analysis of population dynamics in many fields of research [11]. Thus, insight into the evolutionary dynamics of (3) would be valuable for understanding of transient processes in the hepatitis C model as well.
It has already been demonstrated that in the case a 4 = b 4 = 0, system (3) does admit both kink and bright/dark solitary solutions [14,16]. The aim of this paper is to construct kink solitary solutions to (3) when parameters a 4 , b 4 are nonzero. Existence conditions in the space of the system parameters and explicit expressions of kink solutions are obtained using the generalized differential operator method.

2.2.
Operator expression of solutions to systems of nonlinear ODEs. Let P, Q be trivariate analytic functions. Consider the following system of differential equations: The generalized differential operator with respect to system (11) is defined as [14]: where D c , D u , D v are partial differentiation operators with respect to the indexed variables. Note that standard properties of differential operators [13] do hold for (12). The general solution to Eq.(11), [14,13], takes the following form: Note that D 0 cuv = I, where I is the identity operator.
Corollary 2. Let (20) hold true. Then: Equivalently, (22) results in: 4. Construction of kink solutions to (3). Results of Theorem 3.1 are used to determine existence conditions of kink solutions to (3) in this section. Furthermore, explicit expressions of kink solutions are given in terms of the system's parameters.
4.1. Transformation of (3). Applying transformation (6) to (3) results in the following system: System (38) is subject to initial conditions: 4.2. Derivation of existence conditions of kink solutions to (3). The generalized differential operator D cuv with respect to (38) reads: The conditions of Theorem 3.1 can only be satisfied for special values of parameters η; a 0 , . . . , a 4 ; b 0 , . . . , b 4 . Derivation of these parameter values is given in the following subsection.

4.2.1.
Computation of parameter η. Coefficients p j , q j ; j = 1, 2, 3 are computed using (14) and (40). Note that to satisfy Theorem 3.1, η must also satisfy Corollary 2. Equations (36), (37) result in the following system: Values of η can be computed from (41) and (42) using computer algebra. First it can be noted that (41) and (42) have the following structure: where A p , A q , B p , B q are known functions of u, v. Equation (43) yields: If the parameter η is not a function of c, u, v, then the kink solution definition holds true. Furthermore, if η does not depend on c, u, v sufficient conditions of Theorem 3.1 hold true. Long division of (45) results in: Analogously, (44) yields: Parameter η does not depend on u, v only if a 0 , . . . , a 4 ; b 0 , . . . , b 4 are chosen in such a way that S p = S q are independent of u, v and 4.2.2. Solution of (48). Remainder R p has the following structure: where coefficients c kj depend only on a 0 , . . . , a 4 ; b 0 , . . . , b 4 . Equations (48) and (49) result in the following system of 19 algebraic equations: Solution to (50) reads: Inserting (51)-(54) into R q results in R q = 0, thus (48) is satisfied.
Note that cρ x = cρ y does not depend on c, thus (85)-(88) depend only on initial conditions u, v. Also note that cases with positive and negative signs of η are interchangeable, because: Analogous rearrangements also hold true for y. Thus in further computations only one sign of η can be considered.
Corollary 6 shows that all phase plane trajectories of system (3) Note that (111) describes infinitely many equilibria that lie on a straight line.
Kink solutions x, y to (113) are depicted in Fig. 1.

5.2.2.
Construction of kink solutions. Consider the following system: x τ = 3 + 2x + 2xy + 3y; Using transformation (6) on (112) results in: ηt x t = 3 + 2 x + 2 x y + 3 y; ηt y t = 3 + y − 2 y 2 ; Kink solutions to (132) are depicted in Fig. 7. Corollary 5 yields that the above system has kink solutions (3) with the following parameters: Kink solutions to (131) are depicted in Fig. 8. The phase plane of (131) can be seen in Fig. 9. Note that by Corollary 5, all solutions to (131) correspond to kink solutions in hyperbolic relationship:  Note that if one kink solution is perturbed by an infinitesimal amount, the other solution can exhibit non-infinitesimal changes. For example, in Fig. 9, the variable representing infected cells y has been decreased by 5.19 from point A to B, which results in an increase of 0.39 in the variable representing uninfected cells x. Such instability under perturbations is observed for all systems (3) that satisfy the conditions of Theorem 4.2. 6. Concluding remarks. Kink solitary solutions to a generalized system of hepatitis C evolution equations (3) coupled with both diffusive and multiplicative terms have been constructed in this paper. The generalized differential operator enabled the derivation of explicit existence conditions for kink solutions in terms of the system's parameters and the construction of general solutions to (3). It has been shown that kink solutions to (3) hold for all initial conditions and can be in either linear or hyperbolic relationship. If kink solutions are in linear relationship, an infinitesimal perturbation of infected cell population results in an infinitesimal perturbation of uninfected cell population. However, if solutions are in hyperbolic relationship the former statement does not hold true -a large perturbation of infected cell population can lead to a infinitesimal alteration of uninfected cell population or vice versa. Such perturbation effects provide valuable insight into hepatitis C and other population evolution models.