Modeling of biomedical processes using differential equations has become more and more widespread over recent years [6, 12, 29]. Various differential equation models on the use of oncolytic viruses as therapeutic agents against cancer are discussed in . A clinically validated model of tumor-immune cell interactions is considered in . A new mathematical model for the explanation of the failure of cancer chemotherapy treatment is presented in . A mathematical model based on differential equations is used to describe the interactions between Ebola virus and wild-type Vero cells in vitro in .
Beginning with the classical paper by Neumann et al , various differential equation models for the modeling of hepatitis virus infection have been proposed. Global dynamics of a delay differential model of hepatitis B infection evolution are studied in [5, 27]. The transmission of hepatitis C virus (HCV) among injecting drug users is modeled using ordinary differential equations in . A mathematical multi-scale
model of the within-host dynamics of HCV infection is used to study patients under treatment with direct acting antiviral medication in . The authors of  give a review of recent HCV kinetics models.
Reluga et al  present the following model of hepatitis C virus infection that explicitly includes proliferation of infected and uninfected hepatocytes:(1)
where pt is time; represents uninfected hepatocytes; represents infected cells and represents free virus population. The parameters of (1) have the following meaning: β is the rate of infection per free virus per hepatocyte; c is the immune virus clearance rate; p is the free virus production rate per infected cell; dT, dI are death rates for uninfected hepatocytes and infected cells respectively; rT, rI are parameters of the logistic proliferation of T and I respectively; logistic proliferation happens only if T < Tmax; parameters 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 ϵ. Ranges of parameters are given in .(2)
where x, y are dimensionless state variables for uninfected hepatocytes and infected cells respectively; r, b, θ, d, q, s P R are real parameters.
System (2) can be rewritten in a more general form:(3)
where c, u, v, ak , bk P R, k = 1, . . . , 4.
The main objective of this paper is to study soliton-like dynamics of the system (3). Note that since (3) is not a system of nonlinear partial differential equations (PDEs), soliton (or solitary) solutions cannot exist, due to their definition being closely connected to concrete physical phenomena. However, as is demonstrated in the paper, solutions that exhibit analogous dynamics to those observed in solitary solutions, can be constructed for system (3). Since the phase trajectories of these solutions are homoclinic or heteroclinic, we refer to such solutions and homoclinic/heteroclinic solutions.
In the case a4 = b4 = 0, system (3) has already been shown to admit homoclinic/heteroclinic solutions , . Solutions described in  have simple monotonous transitions from two steady states, while those found in  exhibit much more complicated transient effects. Because of this reason, only the latter homoclinic and heteroclinic solutions to (2), (3) are considered.
Using the inverse balancing and generalized differential operator techniques, explicit homoclinic and heteroclinic solution existence conditions are obtained in terms of the parameters of (2). These conditions, together with explicit expressions of such solutions, provide insight not only into HCV model (2), but also other models of nonlinear evolution.
Note that the application of direct techniques to compute the homoclinic/heteroclinic trajectories of (3) is not straightforward. For example, computation of the first integral requires the solution of the following first-order ODE:(4)
While the above ODE can be integrated for some parameter values, there is no general method to determine such cases. Furthermore, the generalized differential operator technique yields not only phase trajectories of (3), but also its general solution and the conditions with respect to a0, . . . , a4; b0, . . . , b4 under which homoclinic/heteroclinic solutions exist.
2.1 Power series and their extensions
In this paper, functions of the following power series form are considered:(5)
where z, aj P C. The coefficients of power series (33) are constructed via generalized differential operator technique, described in the following sections of the paper.
We treat the convergence of series (33) as follows. If (33) converges in some ball |z| < R; R ¡ 0, then it is possible to extend (33) to a wider complex domain (not including the singularities of (33)) via classical extension techniques. Let t P R denote a real argument of this extended function. Inserting t into the extension of (33) yields a real power series f (x) defined for values not necessarily in the radius |t| < R. For the purposes of this paper, we consider f (x) and its power series representation to be congruent.
2.2 Monotonous and non-monotonous homoclinic/heteroclinic solutions
where η ≠ 0, σ, γ ∈ R are constants; depend on initial conditions u, v.
The biological interpretation of (6), (7) represents the transition from the size of population of cells before therapy to the size of population after therapy. However, this transition is monotonous; the solutions shown in Fig. 1 (a) describe the difference between the sizes of populations before and after therapy, and the transition between the steady states.(8)(9)
where η ≠ 0, σ, γ ∈ R are constants; depend on initial conditions u, v.
Solutions (8), (9) describe much more complex transition processes between the steady states. The size of the population of cells during the transient process exceeds populations both at the beginning and the end of the therapy if only the considered solutions have minimum points (the black line in Fig. 1 (b)) Analogously,
solutions with maximum points describe complex transitions from the population of cells before and after the treatment (the gray line in Fig. 1 (b))
From the biological point of view, transient processes governed by homoclinic and heteroclinic solutions highlight important phenomena. Let us consider the dynamics of uninfected cells (the black line in Fig. 1 (b)) The population of uninfected cells after the therapy becomes lower than the population before the therapy. However, the number of uninfected cells grows during the therapy and exceeds the population of uninfected cells at the beginning of the computational experiment (Fig. 1 (b))
Note that the negative values of cell population x(τ) and y(τ) are a consequence of the non-dimensionalization of system (1).
2.3 Solution transformation
In the following derivations, the standard independent variable transformation will be used:(10)(11)(12)(13)(14)
where λk , μk , ρk , νk , k = 1, 2 are functions of u, v.
2.4 Generalized differential operator technique
In this section, a summary on the generalized differential operator technique for the construction of solutions to ordinary differential equations in presented. More detailed derivations can be found in .
2.4.1 Generalized differential operators
2.4.2 Multiplicative operators
Using (15), the multiplicative operator can be constructed:(19)
where t is an arbitrary real variable. Operator (19) has two important properties:(20)(21)
Let y1 :═ Mu ═ y1(t, u, v), y2 :═ Mv ═ y2(t, u, v), z :═ Mf (u, v) ═ z(t, u, v) and w :═ f (Mu,Mv) ═ f (y1, y2). To prove (21), it needs to be shown that z ═ w for all t, u, v.
Thus, the function z(t, u, v) satisfies the partial differential equation:(23)
with initial condition z(0, u, v) ═ f (u, v) that follows from the definition of z.
Analogously, it is shown that:(24)
with y1(0, u, v) ═ u and y2(0, u, v) ═ v. Using (24) and the definition of w yields:(25)
Note that w satisfies the initial condition w(0, u, v) ═ f (y1(0, u, v), y2(0, u, v)) ═ f (u, v), thus z and w coincide, which results in the proof of (21).
Construction of general solutions to ODEs requires one final operator which is denoted as the generalized multiplicative operator:(26)
Operator G has two properties analogous to (20), (21):(27)(28)(29)
Substituting t for t ═ c yields (28).
2.4.3 Construction of solutions to ODEs
Let us consider the following system of ODEs:(30)(31)(32)
The convention where I is the identity operator, is used.
Identities (32) can be proven using properties (21) and (28) derived in the previous section. Consider operators M, G defined with respect to the generalized differential operator (31). First, let and Property (21) yields:(33)
In the following derivations,(35)
will be used, which transforms (32) into:(36)(37)
Furthermore, coefficients p j , qj satisfy recurrence relations:(38)
3 Existence of homoclinic/heteroclinic solutions in (30)
Analogous derivations with respect to y and ν1 ≠ ν2 result in:(42)(43)(44)(45)(46)(47)(48)
k, l = 1, 2; k ≠ l.
Necessity Let (41) hold true. Taking j = 1, 2 yields:(49)(50)
Solving the above equations for λ1, λ2 results in (44).
Equation (41) yields the following determinant equality:(51)
Expanding the left side of (51) yields:(52)
Solving (52) for ρk results in: k(56)
Since ρ1 ≠ ρ2, the discriminant which results in condition (47).
Denoting and applying operator to (56) results in:(57)
Using recursion (38) it can be obtained that:(58)(59)(60)(61)(62)(63)(64)(65)
Sufficiency Condition (44) yields:(71)
Applying operator to (71) results in:(72)
Continuing by induction yields (41).
The proof for parameters of y is analogous.
Corollary 3.1 If conditions of Theorem 3.1 hold true, then the third and higher order Hankel determinants of sequences are equal to zero(73)(74)
n = 3, 4, . . .
4 Necessary homoclinic/heteroclinic solution existence conditions in (3)
The inverse balancing technique can be used to determine necessary existence conditions of solutions (8), (9) to (3). The main principle of this technique is to insert the solution ansatz into the considered equations and obtain a system of equations linear in system parameters ak , b k , k = 0, . . . , 4. The inverse balancing technique has been successfully used to obtain necessary solution existence conditions in a variety of nonlinear ordinary and partial differential equations [10, 15, 18]. Note that the inverse balancing technique does not possess the drawbacks associated with various solution construction (or direct ansatz) methods, which have attracted a significant amount of criticism [1, 8, 9, 17, 24].
4.1 Transformation of (3)
Equation (78) results in:(82)(83)(84)(85)(86)(87)(88)(89)(90)
where T(t) := (t = (1) (t = t2).
Analogous computations with respect to (92) result in:(97)(98)(99)(100)(101)(102)(103)(104)(105)(106)
Note that there are 10 parameters in (75) and (91), (92) yields a non-degenerate system of 10 linear balancing equations, thus no constraints on the parameters of solution (90) needs to be imposed. However, as shown by (99), (100) conditions a3 = b2 and b3 = a2 must hold if (75) admits solution (90).
The results of this section are summarized in the following Lemma.(107)(108)
5 Construction of homoclinic/heteroclinic solutions to (3)
5.1 Derivation of parameter η
Parameter η is derived using Corollary 3.1. Consider the following Hankel determinants:(109)
Parameter η must be chosen to satisfy(110)
Furthermore, η can only depend on coefficients a0, . . . , a4; b0, . . . , b4, otherwise Theorem 3.1 does not hold true and obtained solutions would not be valid for all initial conditions.
It can be observed that:(111)(112)(113)
where F is a polynomial in u, v.
Since the roots η must not depend on initial conditions, any values of u, v can be chosen and inserted into (111). Let(115)
then A6 = A4 = 0 and using (111), η2 can be expressed as:(116)
The numerator and denominator of (116) depend linearly on u:(117)
where αk , βk are functions of a0, . . . , a4; b0, . . . , b4.
Analogous computations with respect to lead to:(118)(119)
Parameter η does not depend on u, v only if:(120)(121)
which leads to the following sufficient existence condition for homoclinic/heteroclinic solutions to (3):(123)
5.2 Necessary and sufficient existence conditions for homoclinic/heteroclinic solutions to (3)
Theorem 3.1, Lemma 4.1 and condition (123) together with computer algebra computations result in the following theorem.
Parameters σ,γ read:(132)
Note that (117) yields two values for η, however, it is sufficient to consider only the positive or negative root of (117) to obtain the general solution to (3) when Theorem 5.1 holds true, because the sign of η can be interchanged:(133)
As demonstrated in , the value does not depend on initial conditions, which proves that changing the sign of η does not yield new solutions.
6 Homoclinic/heteroclinic solutions to hepatitis C model (2)
6.1 Existence conditions
Using Theorem 5.1 conditions for the existence of homoclinic/heteroclinic solutions to (2) can be derived. Note that only homoclinic/heteroclinic solutions with can be considered. Inserting (134), (135) into (108) yields two congruent equations:(136)
Both equations are satisfied if parameter r reads:(137)(138)
Computer algebra computations prove that when Corollary (6.1) holds true, parameters y1 = y2 = 0.
6.3 Computational experiment
Let us consider the following system:(145)(146)
The above system corresponds to (2) with the following parameters:(147)
Note that parameters (147) satisfy the guidelines given in  for biologically significant systems. Further more, conditions of Corollary 6.1 are satisfied, thus homoclinic/heteroclinic solutions to (145), (146) do exist.
Equation (117) yields:(148)
Theorem 3.1 yields the following parameters of homoclinic/heteroclinic solutions:(149)(150)(151)
Derivations given in Subsection 5.2 result in:(153)(154)(155)
Solutions with parameters (153)–(155) are pictured in Fig. 2. Note that there are three types of solutions – non-singular solutions (a), (b); solutions with one singularity (c) and solutions with two singularities (d).
The phase plane of (145), (146) can be seen in Fig. 3. Note that labels (a), (b), (c), (d) on the phase plane correspond to respectively labeled solutions pictured in Fig. 2. System(145), (146) has the following equilibria - stable node; - unstable node; - saddle point.(156)
Solution Fig. 3 (a) corresponds to an elliptic trajectory, while the remaining (b), (c), (d) have hyperbolic trajectories. Furthermore, there is a single solution that satisfies the parabola equation:(157)
Stable and unstable manifolds of the saddle point are obtained by setting the numerator and denominator of τ1,2 to zero . This yields that the stable manifold of the saddle point is the x-axis, while the unstable manifold lies on the straight line Manifolds of the saddle point correspond to dashed gray lines in Fig. 3.
7 Concluding remarks
Homoclinic and heteroclinic solutions to hepatitis C evolution model (2) have been constructed in this paper. Inverse balancing and generalized differential operator techniques have enabled the derivation of explicit necessary and sufficient homoclinic and heteroclinic solution existence conditions with respect to the parameters of system (2). Furthermore, it has been shown that these existence conditions are satisfied when (2) described a biologically significant system of HCV evolution.
It has been demonstrated that transient processes of the derived solutions to (2) reveal important phenomena for understanding hepatitis C virus infection dynamics. Even though antiviral therapy reduces the number of infected cells (comparing the beginning to the end of treatment), due to the transient processes during the therapy, population size of infected cells is higher than before or after therapy – if only the considered solutions are heteroclinic with maxima. Analogous biological interpretations can be made for heteroclinic solutions with minima. The population of healthy cells is lower than before or after treatment during antiviral therapy – if the number of uninfected hepatocytes is described by a heteroclinic solution possessing minima.
The main mathematical advancements of this paper can be characterized by new applications of inverse balancing technique and the development of generalized differential operator method for the solution of coupled differential equations with multiplicative and diffusive terms. As noted in Section 4, direct balancing techniques may yield wrong solutions; inverse balancing of such a complex system of nonlinear differential equations poses a number of technical problems. On the other hand, derivation of closed-form homoclinic/heteroclinic solutions and explicit conditions of their existence poses serious mathematical challenges. One of the main contributions of this paper are the necessary and sufficient conditions for the existence of these solutions in the hepatitis C evolution model.
Comparing the results of this paper with  it can be concluded that system (3) (and, by extension (2)) is structurally stable in the topological sense – when a4, b4 tend to zero, the phase plane continuously converges to the phase plane described in . Moreover, structural stability can also be observed in homo-clinic/heteroclinic solution existence condition (123) – in the case a4, b4 → 0, such solutions also exist and the condition (123) is maintained. Since such effects are observed in systems with biological significance, they provide valuable insight not only into (2) but also other nonlinear evolution models.
This research was funded by a grant (No. MIP078/2015) from the Research Council of Lithuania. This research was also funded by Jiangsu Provincial Recruitment Program of Foreign Experts (Type B, Grant 172 no. JSB2017007).
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About the article
Published Online: 2018-12-31
Citation Information: Open Mathematics, Volume 16, Issue 1, Pages 1537–1555, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2018-0130.
© 2018 Telksnys et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0