Chemotactic effects in reaction-diffusion equations for inflammation

Abstract

Predator–prey systems are used to model time-dependent virus and lymphocyte population during a liver infection and to discuss the influence of chemotactic behavior on the chronification tendency of such infections. Therefore, a model family of reaction-diffusion equations is presented, and the long-term behavior of the solutions is estimated by a critical value containing the reaction strength, the diffusion rate, and the extension of the liver domain. Fourier techniques are applied to evaluate the influence of chemotactic behavior of the immune response to the long-term behavior of locally linearized models. It turns out that the chemotaxis is a subordinated influence with respect to the chronification of liver infections.

Introduction

The different types of hepatitis are the most frequent infective diseases, and hepatitis B and C often chronify, cf. [1]. Until now, the detailed mechanisms of the chronification of hepatitis are unclear, cf. [2]. A possible quantitative reason was presented in [3] by using rather simple mechanisms to describe an abstract interaction between the virus and the immune response. Since the lymphocytes move actively into the direction of the virus, chemotactic behavior of the immune response is a candidate for an enforcement of the chronification tendency of the liver infections, cf. [4]. The present paper tests this idea. Following [3], we extend the modeling framework and discuss mathematical models including chemotaxis. Therefore, we use the reaction-diffusion equation from [3] and develop it to a model family including chemotactic behavior.

There is an enormous amount of medical research on hepatitis and its mechanisms like found in [5–7] or [8]. Mathematical models are less frequent, cf. [9] and [10] for models using ordinary differential equations or [11] using partial ones for particular sub-systems involved in the infection process. Approaches as dynamical systems are followed in [12] and [13] on differently fine resolutions of the sub-types of T cells. A completely theoretical approach is found early in [14] while discussing the question of a hypothetical optimal immune response to eliminate the virus from the organ.

The paper is organized as follows: Section 2 presents the medical preliminaries of liver infections. In Section 3, we present a general reaction diffusion system and a general theorem about the asymptotics of solutions. Moreover, we connect the medical observations with a reaction-diffusion system. This consists of a predator–prey system with the virus u = u(t, x) as prey and the immune response v = v(t, x) as predator. Together with a diffusion or migration term, it forms a reaction-diffusion equation. We formulate assumptions on the reaction function with the aim to describe the medical observations mathematically. In Section 4, we present and shortly discuss linear models for liver infections and extend these models to a nonlinear reaction-diffusion system in Section 5. After the discussion of its long-term behavior, Section 6 introduces chemotaxis into the model. The lymphocytes of the immune response move actively into the direction of the by-products of the virus, cf. [4]. The investigation of the linearization of this model extension by Fourier techniques in Section 6.2 shows that a moderate chemotaxis does not enforce the chronification tendency of the modeled infection. Finally, Section 7 looks back to the hierarchical order of the different models under consideration, and it orders the mechanisms with respect to their influence on the chronification behavior. The paper finishes with a short conclusion.

Medical preliminaries

Here, we shortly summarize basic facts about the course of a liver infection, cf. [15].

The liver is a large organ, which consists of two large liver lobes. The lobes have a ramified small-scale structure. The liver is surrounded by Glisson’s capsule.

After a viral infection in the liver, the virus population grows up to a maximal capacity and meanwhile it spreads out within the liver. Consequently, the immune system in the lymph nodes is activated, and it answers with different types T cells as immune response, which are transported via the blood vessels into the infected organ. In this work, we focus on the description of cytotoxic T cells and name them by T cells. The activation process usually takes a while, which is referred to as incubation time.

The T cells which have arrived in the liver migrate through the liver. Their migration is partly driven by the waste products of the virus and partly undirected so that it behaves like a diffusive motion. Both the virus and the immune response usually do not penetrate the fasciae surrounding the liver.

Within the liver, the immune response attacks the virus, and new T cells are sent out as long as the virus is persistent in the liver. After having done their job, i.e., after having the virus population made extinct or at least diminished, the T cells are degraded by regulatory cells.

The main damage of the liver and thus of the organism does not result from the virus itself but from the inflammation caused by the T cells in the organ, cf. [16] and [17].

There are two courses of liver infections to distinguish. First, an acute course means that the immune response completely conquers the organ and the virus dies out so that we find a healing of the infection and the inflammation. Second, a chronic course means that—after an acute phase—the virus population decreases but does not disappear. Thus, also a small T cell population persists in the liver. The infection chronifies and the predator–prey system of the virus and the immune response stays in equilibrium.

In [3], necessary conditions for the chronification of a liver infection are discussed by hands of a mathematical model. Coinciding with medical observations, this model shows persistent virus populations in the remote parts of the liver and persistent T cell populations near the blood vessels, cf. [18].

Reaction-diffusion equations

In this section, we present general reaction-diffusion equations, a theorem for predicting the asymptotic of their solutions and requirements for modeling liver infections with reaction-diffusion equations.

We consider general reaction-diffusion systems for n different species. With the assumption of a large population, let $\mathbf{q}=\mathbf{q}(t,\mathbf{x})\in {ℝ}^n$ denote the population density at the point x in $\mathrm{\Omega }\subset {ℝ}^m$ and for the time t. We summarize the reaction terms in a smooth function $\mathbf{f}:{ℝ}^n\to {ℝ}^n$ and the diffusion constants in the diagonal elements of the matrix $\mathfrak{D}\in {ℝ}^{n\times n}$. With the Laplacian Δ with respect to the position x, which is used element wise on q, we get the general reaction-diffusion system, cf. [19],

$$\begin{array}{rcl}\dot{\mathbf{q}}(t,\mathbf{x})& =& \mathbf{f}(\mathbf{q}(t,\mathbf{x}))+\mathfrak{D}\mathrm{\Delta }\mathbf{q}(t,\mathbf{x})\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ \mathbf{q}(0,\mathbf{x})& =& {\mathbf{q}}_{\mathbf{0}}(\mathbf{x})\text{for}\mathbf{x}\in \mathrm{\Omega },\\ \nabla \mathbf{q}\cdot \mathbf{n}& =& 0\text{for}\mathbf{x}\in \partial \mathrm{\Omega },t>0.\end{array}$$ 1

The reaction-diffusion system combines reactions f(q) between the species or concerning a single species with spatial diffusion of the species $\mathfrak{D}\mathrm{\Delta }\mathbf{q}$. With the aim to describe liver infections, we can interpret the reaction-diffusion system as a predator–prey system with additional diffusion effects. The predator–prey mechanisms are concluded in the reaction function f.

Reaction-diffusion systems classically use a point-wise dependency of f on q, but they are not restricted to. We start our investigations with point-wise dependencies in (1) and extend them later to more general predator–prey relations.

Asymptotics of solutions

The general formulation (1) allows us to apply a theorem found in [20] about an estimation of the long-term behavior of the solutions of reaction-diffusion equations. The theorem uses the smallest eigenvalue d of the diffusion matrix $\mathfrak{D}$, a parameter M of the maximal reaction strength

$$M=\underset{\mathbf{q}\in \mathrm{\Sigma }}{\max }\parallel \nabla \mathbf{f}(\mathbf{q})\parallel ,$$

where $\mathrm{\Sigma }\subseteq {ℝ}^n$ is a set which the solution q of (1) does not leave, and the smallest positive eigenvalue $\lambda =\underset{k\in \mathbb{N}}{\min }\{{\lambda }_k>0\}$ of the negative Laplacian with Neumann boundary conditions, cf. [21]. In formula, the eigenvalue problem reads

$$\begin{array}{rcl}-\mathrm{\Delta }{u}_k(\mathbf{x})& =& {\lambda }_k{u}_k(\mathbf{x})\text{for}\mathbf{x}\in \mathrm{\Omega },\\ 0& =& \nabla u_k\cdot \mathbf{n}\text{for}\mathbf{x}\in \partial \mathrm{\Omega }\mathrm{.}\end{array}$$ 2

The parameter σ combines the three parameters as σ = λd − M.

We refer the named theorem from [20] and use the notations

$${\mathbf{q}}_{\text{sum}}={\int }_{\mathrm{\Omega }}\mathbf{q}\mathrm{d}\mathbf{x}\text{and}\overline{\mathbf{q}}={\mathbf{q}}_{\text{mean}}=\frac{1}{|\mathrm{\Omega }|}{\mathbf{q}}_{\text{sum}}$$ 3

with the volume |Ω| of the domain Ω.

Theorem 1

The system (1) may have an invariant solution spaceΣ, i.e.,q₀(x) ∈Σ for allx ∈Ω impliesq(t, x) ∈Σ for allt > 0 and allx ∈Ω, and the initialconditions may fulfillq₀(x) ∈Σ for allx ∈Ω. Thematrix$\mathfrak{D}$may bepositive definite andfmay be smooth enough. Then there are constantsc_i > 0, i ∈{1,2,3} so thatfort > 0 the following estimates yields

1. $\parallel \nabla \mathbf{q}(t,\cdot ){\parallel }_{L_2(\mathrm{\Omega })}\le c_1{\mathrm{e}}^{-\mathrm{\sigma t}}$ ,

2. $\parallel \mathbf{q}(t,\cdot )-\overline{\mathbf{q}}(t){\parallel }_{L_2(\mathrm{\Omega })}\le c_2{\mathrm{e}}^{-\mathrm{\sigma t}}$ , and

3. the mean $\overline{\mathbf{q}}(t)$ obeys $\dot{\overline{\mathbf{q}}}=\mathbf{f}(\overline{\mathbf{q}})+\mathbf{g}(t)$ with a function g with $\parallel \mathbf{g}(t){\parallel }_{L_2(\mathrm{\Omega })}\le c_3{\mathrm{e}}^{-\mathrm{\sigma t}}$ .

See [20, Theorem 14.17, p. 233] for a proof.

Under the conditions of Theorem 1 and σ > 0, we get solutions q(t, x) which asymptotically approach their mean $\overline{\mathbf{q}}(t)$, i.e., they tend to position independent solutions. That means that σ > 0 prevents system (1) from having spatially inhomogeneous stationary solutions. Spatially inhomogeneous stationary solutions, which are interpreted as chronic infection courses, are only possible if σ is negative. That is the reason why this Theorem 1 helps to decide under which conditions chronic infections may occur.

Assumptions for modeling liver infections

In this section, we connect the general reaction-diffusion system in (1) and the medical preliminaries of Section 2. Based on this, we develop assumptions on the reaction function f according to the medical observations.

The simply connected, bounded, and open domain $\mathrm{\Omega }\subset {ℝ}^m$ with m ∈{2,3}, or in exceptional cases even m ∈{1,2,3}, may describe the position and extension of a part of a liver. Of course, a realistic liver lies in a space with m = 3, but we use m = 2 in the numerical simulations and illustrations. We assume that the boundary ∂Ω is sufficiently smooth and that the outer normal n exists almost everywhere on ∂Ω. So, the normal derivative of a sufficiently smooth function u defined in $\overline{\mathrm{\Omega }}$ is ∇u ⋅n on ∂Ω. Of course, a realistic liver has a very complicated form and a complicated inner structure. Therefore, we regard areas that are simply connected separately. We interpret the domain Ω as a part of the liver, which lies inside a single lobe and which can be seen as simply connected in the mathematical sense of the word. Each of the parts is separated from the others by tissue like fascia so that the assumption of homogeneous Neumann boundary conditions holds. By dividing the liver into different parts, we get different parameters λ for each part of the liver. Hence, we regard exemplarily one part of the liver and name this part Ω. On this mesoscopic scale, we regard virus and T cell densities u = u(t, x) and v = v(t, x), see below.

As discussed in [3], the dynamics of a liver infection is modeled by hands of two populations. The first population is the virus, and the second population is the immune reaction, which we summarize as general T cells without any distinction of the different sub-types. The virus population is modeled by the density u = u(t, x) depending on the time t > 0 and the position x ∈Ω. Analogously, the density of the T cells is described by v = v(t, x).

Referring to Lotka–Volterra models, we regard the virus population u as prey and the T cells v as predator because the T cells in the immune response attack the virus and the T cell population profits from the virus via the enhancement by the immune systems [22]. We consider these two effects as main effects of a predator prey system. Let q(t, x) = (u(t, x),v(t, x))^T be the population vector in (1). The first entry of the reaction function f(q) = (f₁(q),f₂(q))^T describes the effect of interactions on the virus population. As we regard the virus as the prey for the T cells, the derivative of f₁(q) with respect to the T cells v is negative, so

$$\frac{\partial }{\mathrm{\partial v}}{f}_1(\mathbf{q})=f_{1,v}<0.$$ 4

This describes the observations that T cells fight against the virus and a larger T cell population leads to a larger decrease of the virus. In the same way, the T cells are predators for the virus and therefore a larger virus population leads to a higher inflow of T cells. In formula, this influence is

$$\frac{\partial }{\mathrm{\partial u}}{f}_2(\mathbf{q})=f_{2,u}>0.$$ 5

In Section 2, we described the decrease of T cells after having done their job. That includes two effects we distinguish. First, we find a natural decay of the T cells which can be seen as a natural dying process. We formulate this observation as a decay

$$\frac{\partial }{\mathrm{\partial v}}{f}_2(\mathbf{q})=f_{2,v}<0$$ 6

of the T cells depending on their population size. Second, we consider the addition in the formulation of the decay “after having done their job”. This means, that the decay of the T cells depend on the left amount of the virus population. We cannot formulate the effect of decaying “after having done their job” in a pure monotonicity behavior. For this reason, we regard (6) as the dominant but not exclusive behavior. The monotonicity behavior, c.f. (4), (5) and (6), describes the persistent direction of interaction, and is independent of the particular choice of the kinetics.

The diffusion term in (1) describes the spreading of the two populations in the liver from parts with high population to low populations. Combining the reaction functions f₁ and f₂ with the diffusion terms, we get a reaction-diffusion system like (1) for describing liver infections.

Due to the connective tissue around the liver, cf. Glisson’s copsule, we consider homogeneous Neumann boundary conditions as seen in (1). Those boundary conditions describe that there is no inflow or outflow through the boundary of the liver in the neighbored tissue.

The initial values q₀ are the initial data at the measurement start, which we chose after the incubation time with the start of the immune reaction.

Finally, we mention possible interpretation of the solution behavior of (10) with respect to the course of the liver infection. The infection follows an acute course with a healing process, if

$$\underset{t\to \infty }{\lim }u(t,\mathbf{x})=\underset{t\to \infty }{\lim }v(t,\mathbf{x})=0$$ 7

is satisfied. That means the virus is eliminated by the T cells and the immune reaction fades out. Infections with a chronic course are characterized by a surviving virus population in remote areas of the liver and a persisting but decreased immune reaction. Therefore, we interpret stationary spatial inhomogeneous solutions as chronic infections, cf. [3]. Further solution behavior like e.g., oscillating solutions which could represent a recurrent infection course, are conceivable but they will not be of special interest here.

Linear models

In a first modeling step, we convert the general assumptions of Section 3.2 into linear functions.

Therefore we start with the reaction function of the virus, named f₁(u, v). In (4), we formulated the decay of the virus u caused by T cells v as a negative derivate of f₁ with respect to the T cell population v. As a linear function, we get the decaying term − v and weight the influence of this effect on the total change of the virus population by a constant parameter $\tilde{\gamma }\in ℝ$. The second influence on the change of the virus population is an ongoing growth of the virus depending on the virus population itself. This neglects the effect of a maximal capacity of the virus, described in Section 2. Expressed in a linear term, the growth is considered as u. Scaling the time scale, we can scale the growth factor to 1. For the reaction function of the virus, we get in total

$$f_1(u,v)=u-\tilde{\gamma }v\mathrm{.}$$

The reaction of the T cells is influenced by two main effects. First, the T cell amount depends on the total virus population u_sum, see (3), in the liver. The larger the virus population is, the stronger the immune system reacts. The T cells reach the liver through the blood vessels and we write this dependency of the inflow by a function J(x). We formulate this relation in a general linear term by u_sumJ(x) in the reaction function f₂ and weight the term with a parameter δ > 0. Later, we term the inflow by j[u], which we specify in different variants, e.g., in (12) or (13). The second effect influencing the reaction function of the T cells is the decay of T cells in absence of virus. In Section 3.2, we discussed two different mechanisms in this decaying process. In a linear model, we find two different ways for formulating the decay, concentrating on only one of the mechanisms. On the one hand, we find the natural decay as − v weighted with a parameter η₁. On the other hand, we can focus on the decay in absence of virus and find therefore a term − (1 − u), where we consider a limited virus population with a maximum population size of 1. We weight this influence by a parameter η₂.

For the reaction function of the T cells, we get two different functions

$$f_2^{(1)}(u,v)=\delta u_{\text{sum}}J(\mathbf{x})-{\eta }_{1}v$$

and

$$f_2^{(2)}(u,v)=\delta u_{\text{sum}}J(\mathbf{x})-{\eta }_2(1-u)\mathrm{.}$$

The first function $f_2^{(1)}$ neglects the dependence of the decay on the absence of the virus. The second version $f_2^{(2)}$ neglects the current amount of T cells for the decay. Both mechanisms describe only part of the observations.

Right now, we cannot decide which function is more likely to describe the long-term behavior. We formulate two linear models differentiating in the reaction function f₂ of the T cells.

Adding the diffusion terms like in (1), we get

$$\begin{array}{rcl}\dot{u}& =& u-\tilde{\gamma }v+\alpha \mathrm{\Delta u}\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ \dot{v}& =& \mathrm{\delta J}(x)u_{\text{sum}}-{\eta }_{1}v+\beta \mathrm{\Delta v}\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ u(0,\mathbf{x})& =& u_0(\mathbf{x}),v(0,\mathbf{x})=v_0(\mathbf{x})\text{for}\mathbf{x}\in \mathrm{\Omega },\\ 0& =& \nabla u\cdot \mathbf{n}=\nabla v\cdot \mathbf{n}\text{for}\mathbf{x}\in \partial \mathrm{\Omega },t>0\end{array}$$ 8

and

$$\begin{array}{rcl}\dot{u}& =& u-\tilde{\gamma }v+\alpha \mathrm{\Delta u}\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ \dot{v}& =& \mathrm{\delta J}(x)u_{\text{sum}}-{\eta }_2(1-u)+\beta \mathrm{\Delta v}\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ u(0,\mathbf{x})& =& u_0(\mathbf{x}),v(0,\mathbf{x})=v_0(\mathbf{x})\text{for}\mathbf{x}\in \mathrm{\Omega },\\ 0& =& \nabla u\cdot \mathbf{n}=\nabla v\cdot \mathbf{n}\text{for}\mathbf{x}\in \partial \mathrm{\Omega },t>0\end{array}$$ 9

as two linear models.

Both models allow an unrestricted growth of virus and T cells. This is not realistic due to the limited space in the liver. Besides, in both models, a negative virus population is possible, which does not make sense and is outside of the scope of model validity.

The linear models cannot describe the whole mechanism of liver infection. Nevertheless, we want to gain some insight in the linear models and the solutions they provide because they can be seen later as linearization of nonlinear models and will be used for predictions around stationary solutions.

Simulations like shown in Fig. 1 show that the linear models provide acceptable solutions for short time intervals and remote from the blood vessels, where the inflow takes place. We find very high T cell populations in the inflow area and due to the decay term of the virus a negative virus population.

Fig. 1.

Comparison of the solutions obtained by the two linearized models (light grey mesh) and the model in (10) (dark mesh) for a short time interval and for $\mathrm{\Omega }=(0,1)\subset ℝ$. The solutions for the virus u emerge in all three models from the same initial conditions, just as the solutions for the T cells v are emerging from the same initial conditions. a Comparison of the model in (10) and the linearized model in (8). b Comparison of the model in (10) and the linearized model in (9)

So, the linear models do not provide acceptable solutions for discussing the long-term behavior but allow insight in short-term variations around medium population sizes.

A nonlinear reaction-diffusion model for liver infections

Starting with the results of Section 3.2, we extend the linear models to nonlinear models. Therefore, we use the model presented in [3] as a basic model.

Model set-up

First, we generalize the growth of the virus by a growth rate w(u). The growth $\dot{u}$ of the virus depends on the virus population u itself and on the availability of host cells. Consequently, a logistic growth, comp. Equation (11), with the available host cells as capacity, is a nearby choice. The proportional growth in (8) and (9) is a linearization and thus an approximation of the logistic growth. A possible local immune reaction is regarded as Allee effect in (11).

The decay of the virus caused by the T cells depends on the virus population as well as on the T cell population. Referring to the modeling of chemical reaction, we model the decay by the term − γuv. With this term, we show consideration for the probability for T cells reaching the virus. We write the inflow of T cells through the blood vessels as a general function j[u], depending somehow on the virus population. Later, we will specify this function.

Now, the interaction between virus und T cells in the liver Ω is modeled by the system of reaction-diffusion equations

$$\begin{array}{rcl}\dot{u}& =& \mathrm{uw}(u)-\mathrm{\gamma uv}+\alpha \mathrm{\Delta u}\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ \dot{v}& =& j[u]-\eta (1-u)v+\beta \mathrm{\Delta v}\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ u(0,\mathbf{x})& =& u_0(\mathbf{x}),v(0,\mathbf{x})=v_0(\mathbf{x})\text{for}\mathbf{x}\in \mathrm{\Omega },\\ 0& =& \nabla u\cdot \mathbf{n}=\nabla v\cdot \mathbf{n}\text{for}\mathbf{x}\in \partial \mathrm{\Omega },t>0,\end{array}$$ 10

where j[u] is the inflow of T cells and w(u) is the growth rate of the virus depending on u itself. The virus population is normalized with respect to its capacity, which is u_max = 1 after the normalization. The parameter γ describes the rate of predation upon the prey. The decrease of T cells in absence of a virus is modeled by the term − ηv(u_max − u) = −ηv(1 − u), cf. Section 2. The normalization of u is already included in (10). Please note that the predator–prey mechanism in (10) generalizes the classical Lotka–Volterra model. It contains two ways of enhancing the predator, i.e., the T cell population—once by the inflow j and once by the term ηuv.

Here, the migration of the virus and the T cells within the liver is assumed to be a pure diffusion with the diffusion parameters α ≥ 0 and β ≥ 0, respectively, i.e., we assume the spreading to be directed by the negative gradient and the liver to be rather homogeneous. A directed migration is added later in Section 6 by chemotactical effects.

The homogeneous Neumann boundary conditions in (10) model the impenetrable fasciae. Furthermore, the initial conditions are given by u₀(x) and v₀(x). A simple choice for the initial conditions at the end of the incubation phase is u₀(x) = 1 for x ∈Ω describing a grown virus population and v₀(x) = 0 for x ∈Ω describing a not yet delivered immune response.

Examples for the growth rate w(u) are w_log in a pure logistic growth and w_Allee in a logistic growth including the Allee effect [23] with

$$w_{\log }(u)=1-u\text{and}{w}_{\text{Allee}}(u)=(1-u)\cdot A(u)\text{with}A(u)=\frac{u-\epsilon }{u+\kappa }\mathrm{.}$$ 11

The Allee effect concerns the observation that a very small population density u < ε leads to the extinction of the population. In our example, a very small virus population is eliminated by local immune reactions. The parameters ε > 0 and κ > 0 shall ensure that w_Allee(u) ≈ w_log(u) for typical surviving population densities u ≫ ε. The factor A(u) in (11) is zero for the small virus population u = ε, so that w_Allee(ε) = 0 and w_Allee(u) < 0 for 0 < u < ε. Nevertheless, larger u ≫ ε lead to $A(u)=1-\frac{\epsilon +\kappa }{u+\kappa }\approx 1$ and thus w_Allee(u) ≈ w_log(u).

The two growth functions, resulting from the two growth rates w_log and w_Allee are displayed in Fig. 2. The parameters ε and κ are rather small with e.g., ε = 0.05 and κ = 0.01 here.

Fig. 2.

Growth functions g(u) = uw(u) for $w_{\log }$, solid, and w_Allee, dashed, with ε = 0.05 and κ = 0.01. For large u, we find a comparable growth for the logistic growth with and without including the Allee effect. For small u < ε, we find w_Allee(u) < 0

Equation (11) uses the normed carrying capacity 1 for both examples of growth functions. This idealization means that the cell tissue is approximated by a solvent consisting of a constant number of cells which might be inhabited by the virus in different concentrations u. After having been attacked by the immune response, the regeneration is assumed to be rather fast. Of course, a more detailed modeling of the cell death in apoptosis and of the regeneration of the cell would require a position- and time-depending carrying capacity in (11).

Now, we remember examples for inflow functions j[u] = j[u](t, x) of the immune response into the liver. Since the immune response is mediated in the lymph nodes, the strength of the immune response, modeled by the amount of inflowing T cells, depends on the total virus population u_sum(t) at this time instant t. Realistically, it takes some time to activate the lymph nodes, cf. the rather long incubation time. Once activated, the reaction time of the lymph nodes is short compared to the duration of a hepatitis course, and we abstract from the reaction time. The domain Θ ⊆Ω may indicate the area of the incoming blood vessels. With the indicator function χ_Θ(x) of the area Θ, we get

$$j[u](t,\mathbf{x})=\delta \cdot u_{\text{sum}}(t)\cdot \frac{{\chi }_{\mathrm{\Theta }}(\mathbf{x})}{|\mathrm{\Theta }|}\mathrm{.}$$ 12

It is easily checked that the total inflow j[u]_sum(t) = δ ⋅ u_sum(t) is independent of the area Θ because of χ_Θ,sum = |Θ|. The parameter δ contains the strength of the immune response. The larger δ is, the more T cells enter instantaneously the infected liver.

Since (12) makes (10) to be an integro-differential equation, the simplification

$$j{[u]}_{\text{ubi}}(t,\mathbf{x})=\delta \cdot u(t,\mathbf{x})$$ 13

of a point-wise proportional inflow is used in some later considerations. Equation (13) abstracts from the local dependence of j[u]. Also here,

$$j{[u]}_{\text{ubi},\text{sum}}(t)=\delta \cdot u_{\text{sum}}(t)$$

holds true.

Analysis of the long-term behavior of solutions

In the case that the immune response enters the liver modeled in system (10) via blood vessels in Θ ⊂Ω, which are not ubiquitous in whole Ω, non-vanishing position independent solutions with j[u] > 0 on Θ≠Ω are not possible. Thus σ > 0 implies solutions with u → 0 and v → 0 for t →∞, which is (7) for an acute course of the infection.

Unfortunately, system (10) with the simplified inflow j[u] = j[u]_ubi = δu from (13) and logistic growth w_log(u) = 1 − u does not fulfill all conditions of Theorem 1, particularly not the existence of an invariant solution space Σ. First, we see that no rectangular domain Σ₀ = [0,u₀] × [0,v₀] is invariant, because we could choose constant initial conditions with u₀ and v₀, respectively, and, like in all predator–prey systems, we find points (u, v) ∈ Σ with a slope $(\dot{u},\dot{v})$ directed outwards Σ, see Fig. 3.

Fig. 3.

Solutions of system (10) leave every rectangular initial area Σ₀ = [0,u₀] × [0,v₀] in the phase space, even in the absence of diffusion. Here, an example with u₀ = 1 and v₀ = 0.7 is shown. Independently of the T cell population v₀, it holds $\dot{v}(0,\cdot )>0$ and v increases due to the presence of a saturated virus population

Even worse, the a priori assumption of the existence of an invariant set Σ leads to contradictions for every non-trivial system (1). We see that by considering a bounded set Σ with an almost everywhere smooth boundary. Every non-corner point q ∈ ∂Σ with the outer normal m of Σ fulfills q + εv ∈Σ for every direction v with v ⋅m < 0 and sufficiently small ε > 0. Now, we can choose initial conditions which have a very steep dip or peak near q ∈ ∂Σ in each component of q so that the local diffusion term $\mathfrak{D}\mathrm{\Delta q}$ dominates all reaction terms, which are bounded by M. The dominance of the diffusion terms can be increased by steeper dips or peaks and $\dot{\mathbf{q}}$ gets arbitrarily strong in any possible direction. Therefore initial conditions can always be chosen in a way so that $\dot{\mathbf{q}}$ directs outwards of Σ.

Of course, such initial values fade out quickly if system (1) does not allow Turing pattern, cf. [22], which we discuss in Section 5.2.1. Consequently, Theorem 1 gives an a posteriori estimate for the solution behavior when the set Σ is chosen so that it contains a—in some sense—known solution q = q(t, x), and this estimate indicates a tendency of the system behavior if initial conditions with a small C¹-norm are used.

Estimation of the solution space

Here, first we prove the existence of an invariant solution space Σ for the system (10) with logistic growth w_log and j[u]_ubi = δu but without diffusion, i.e., for α = 0 and β = 0. Therefore, we show that a Lyapunov function exists in a certain part of the phase space. Afterwards we give reasons why the system with diffusion reacts in a comparable manner.

A dynamical system $\dot{\mathbf{q}}=\mathbf{f}(\mathbf{q})$ has a Lyapunov function L = L(q) if the trajectories q = q(t) remain inside a level line described by L(q) = c if they are once inside. A necessary condition is that L(q(t)) is non-increasing with t. That can be written as

$$0\ge \frac{\mathrm{d}}{\mathrm{d}t}L(\mathbf{q}(t))=\nabla L(\mathbf{q}(t))\cdot \dot{\mathbf{q}}(t)=\nabla L(\mathbf{q})\cdot \mathbf{f}(\mathbf{q})$$ 14

at least for the points of the trajectory. Oftentimes, one tries to find a Lyapunov function L that fulfills (14) for all relevant q. Then, it can be used to discuss the stability of a dynamical system, cf. [24].

Here, we use the idea of a Lyapunov function to prove that a Σ containing the whole trajectory exists in dependence of the initial values of the above system (10) without diffusion, i. e., of the position-independent variant of system (10).

Theorem 2

For initial values (u₀,v₀) of system (10) withw_log,j[u]_ubi = δuandα = β = 0,there area > 0 andb ≥ 0 so that (u(t),v(t)) ∈Σ = {(u, v) : u ∈ [0,max{1,u₀}],L(u, v) = bu + v ≤ a} holds true for allt ≥ 0.

Proof

We investigate the boundary ∂Σ in the phase plan, and we have to show that $(\dot{u},\dot{v})$ directs inwards Σ at ∂Σ. At the axes, v = 0 implies $\dot{v}\ge 0$, and u = 0 implies $\dot{u}=0$. Thus, the population densities never become negative. Furthermore, u ≥ 1 implies $\dot{u}\le 0$ because of w_log(u) = 1 − u ≤ 0.

The upper boundary of Σ is the only non-trivial one. There, we show condition (14), which reads now

$$\frac{\mathrm{\partial L}}{\mathrm{\partial u}}\cdot \dot{u}+\frac{\mathrm{\partial L}}{\mathrm{\partial v}}\cdot \dot{v}=b\cdot [u(1-u)-\mathrm{\gamma uv}]+[\mathrm{\delta u}-\eta (1-u)v]\le 0\text{for}\mathrm{bu}+v=a\mathrm{.}$$

Using v = a − bu, we get the condition

$$u^2\left(\gamma b^2-b-\mathrm{\eta b}\right)+u(b-\mathrm{\gamma ab}+\delta +\mathrm{\eta a}+\mathrm{\eta b})-\mathrm{\eta a}\le 0,$$

which is fulfilled for all u ≥ 0 if and only if the coefficient of the parabola obey

$$\gamma b^2-(1+\eta )b\le 0\text{and}b(1+\eta )-a(\mathrm{\gamma b}-\eta )+\delta \le 0.$$ 15

A proper choice of the parameters a and b, namely

$$\eta <\mathrm{\gamma b}<1+\eta \text{and}a\ge \frac{b(1+\eta )+\delta }{\mathrm{\gamma b}-\eta }+b{u}_0+v_0,$$ 16

ensures the condition in (15). Since it is required that the initial values (u₀,v₀) are inside Σ, we have augmented a by bu₀ + v₀.

Indeed, every η > 0 and γ > 0 allows to find a constant b in the first condition in (16), and then a sufficiently large a can be found for all δ > 0 in the second one because of γb − η being already positive. Hence, a required upper boundary of Σ described by L(u, v) = bu + v = a exists with b > 0 and a > 0. □

Let us remark, that an exemplary choice is

$$b_{\exp }=\frac{1+2\eta }{2\gamma }\text{and}{a}_{\exp }=\frac{(1+\eta )(1+2\eta )}{\gamma }+2\delta +b{u}_0+v_0,$$

what leads to the maximal predator population v_max = a_exp in dependence of the initial values. We can resume that a trajectory (u(t),v(t)) starting in Σ₀ = [0,u₀] × [0,v₀] stays in Σ ⊆ [0,max{1,u₀}] × [0,v_max]. An example of Σ is shown in Fig. 4. With the existence of an invariant space Σ, the conditions of Theorem 1 are fulfilled for model (10) without diffusion.

Fig. 4.

Gradient field and upper boundary $L(u,v)=b_{\exp }u+v=a_{\exp }$ for the position independent predator-prey system (10) with $w_{\log }$. The solutions (u(t),v(t)) are bounded by the particular level line of the Lyapunov function L in the phase space

Diffusion is an additional mechanism in model (10). In general, diffusion is a balancing process. As an exception, Turing patterns are characteristic for some reaction diffusion equations, cf. [22], and can be mistaken with solutions, which we interpret as chronic infections.

The condition that including diffusion provokes a non-dissipative wave round a stable stationary point of the position-independent system is often referred to as the third condition [22] for the existence of Turing patterns. It reads here

$$\beta \cdot \frac{\partial }{\mathrm{\partial u}}[u(1-u)-\mathrm{\gamma uv}]+\alpha \frac{\partial }{\mathrm{\partial v}}[\mathrm{\delta u}-\eta (1-u)v]=\beta (1-2u-\mathrm{\gamma v})-\mathrm{\alpha \eta }(1-u)>0.$$

It is not satisfied at the non-trivial stationary point with u(1 − u) − γuv = u(1 − u − γv) = 0 because of − βu_stat − αηγv_stat > 0 for all parameter choices. As a result, there are no Turing patterns in our model of liver infections.

The argumentation for the system without diffusion is transferable to the system with diffusion. Diffusion may raise the population of T cells at a certain point in a time interval if there is a much larger T cell population right next to this point, cf. the above argumentation about the dips and peaks in population densities. This local growth of the population is a temporary process, which can exceed the boundary of the solution space. In this special case, the new local maximum can be used as a new initial value for the estimation of the boundaries. Diffusion is balancing so that the process of extending the boundaries of the solution space is time-limited. As a result, we get a solution space Σ(t), that is bounded for all time instants t.

Extended version of Theorem 1

We formulate a slightly extended version of Theorem 1 which can be proven in a completely analogous manner just by considering the time dependency of Σ(t).

Theorem 3

If system (1) has a solution spaceΣ(t), which isbounded for allt ≥ 0,withq(t, x) ∈Σ(t) forallx ∈Ω andt ≤ 0,if$\mathfrak{D}$is positive definiteandfis smooth, thenthere are constantsc_i > 0, i ∈{1,2,3} so that fort > 0 withσ(t) = λd − M(t) where

$$M(t)=\underset{\mathbf{q}\in \mathrm{\Sigma }(t)}{\max }\parallel \nabla \mathbf{f}(\mathbf{q})\parallel ,\text{and with}S(t)={\int }_{t_0}^t\sigma (\tau )\mathrm{d}\tau$$

it follows that

1. $\parallel \nabla \mathbf{q}(t,\cdot ){\parallel }_{L_2(\mathrm{\Omega })}\le c_1{\mathrm{e}}^{-S(t)}$,

2. $\parallel \mathbf{q}(t,\cdot )-\overline{\mathbf{q}}(t){\parallel }_{L_2(\mathrm{\Omega })}\le c_2{\mathrm{e}}^{-S(t)}$, and

3. the mean $\overline{\mathbf{q}}(t)$ obeys $\dot{\overline{\mathbf{q}}}=\mathbf{f}(\overline{\mathbf{q}})+\mathbf{g}(t)$ with a function g with $\parallel \mathbf{g}(t){\parallel }_{L_2(\mathrm{\Omega })}\le c_3{\mathrm{e}}^{-S(t)}$.

If σ is positive, then there are decaying solutions in the time. The infection is healing out. With Theorem 3 we get a time-dependent σ(t) = λd − M(t), because the parameter M(t) of the reaction strength depends on the solution space Σ(t). If σ(t) is positive for all time t, the solutions decay. If M(t) is growing in time, the sign of σ(t) may change from positive to negative. For negative σ(t) the estimations of Theorem 3 provide a growing e^−S(t) in time. Therefore, we cannot state a certain solution behavior. A growing solution space allows chronic infections but there is no evidence whether they really occur.

Figure 5 shows an example of a spatially inhomogeneous stationary solution on a rectangular domain Ω with a fissure, i.e., Ω = (0,1) × (0,1)∖{(x₁,x₂)^T : x₁ ∈ (0,0.9), x₂ = 0.5}. So neither the virus nor the T cells can pass x₂ = 0.5 in the range x₁ ∈ (0,0.9). Only a small connection allows the interaction between both halves. We interpret this solution as chronic infection course, because the two populations are separated in the liver. The T cell population is high near to the blood vessel whereas the virus population is high remote from the vessel.

Fig. 5.

Non-trivial stationary and stable solution of the virus and the T cell populations in a domain Ω with a fissure where the virus may persist behind during a chronic course of the liver infection, cf. end of Section 5.2.2 for Ω

Modeling chemotactical effects

Chemotaxis is a mechanism that describes the active search of the immune response after the virus. So, chemotaxis leads to a directed or active motion of the immune response in addition to the pure and not directed diffusion of the T cells. It seems plausible that an active search of the virus could influence the course of the infection and its tendency to chronify.

We extend model (10) by a chemotaxis term. We model the active search of the T cells as a negative diffusion depending on the virus population. This leads to the model

$$\begin{array}{rcl}\dot{u}& =& \mathrm{uw}(u)-\mathrm{\gamma uv}+\alpha \mathrm{\Delta u}\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ \dot{v}& =& j[u]-\eta (1-u)v+\beta \mathrm{\Delta v}-\mu \mathrm{\Delta u}\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ u(0,\mathbf{x})& =& u_0(\mathbf{x}),v(0,\mathbf{x})=v_0(\mathbf{x})\text{for}\mathbf{x}\in \mathrm{\Omega },\\ 0& =& \nabla u\cdot \mathbf{n}=\nabla v\cdot \mathbf{n}\text{for}\mathbf{x}\in \partial \mathrm{\Omega },t>0.\end{array}$$ 17

Equation (17) appears as an extension of the system in (10), we started with, by the chemotaxis term − μΔu. It is of the form of (1) with a non-diagonal matrix $\mathfrak{D}$. We note that the chemotaxis term − μΔu is a term related to inverse heat conduction. Already due to the relation to inverse heat equation, we expect a rather crazy behavior of the solution if μ is large. We will investigate the behavior of the solution and in particular its tendency to blow up in Sections 6.1 and 6.2.

The chemotaxis term − μΔu seems to have a strong influence on the T cell population v, especially if v is rather small. Later, we analyze the effect of this chemotaxis term on model (10). However, before we start with the analysis, we have a look at the origin of model (17). Equation (17) can be seen as a reduction of a model including three components, the virus u, the T cells v and a signal s. The T cells direct their movement after waste products of the virus; see Section 2. The new component s is the signal caused by the waste products.

The introduction of a chemotaxis term C and the signal s = s(t, x) exposed by the virus expands the system in (10) to q = (u, v, s)^T and the system

$$\begin{array}{rcl}\dot{u}& =& u(1-u)-\mathrm{\gamma uv}+\alpha \mathrm{\Delta u}\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ \dot{v}& =& j[u]-\eta (1-u)v+\beta \mathrm{\Delta v}+C\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ \dot{s}& =& \mathrm{\tau u}-\mathrm{𝜗s}+\kappa \mathrm{\Delta s}\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ u(0,\mathbf{x})& =& u_0(\mathbf{x}),v(0,\mathbf{x})=v_0(\mathbf{x}),s(0,\mathbf{x})=s_0(\mathbf{x})\text{for}\mathbf{x}\in \mathrm{\Omega },\\ 0& =& \nabla u\cdot \mathbf{n}=\nabla v\cdot \mathbf{n}=\nabla s\cdot \mathbf{n}\text{for}\mathbf{x}\in \partial \mathrm{\Omega },t>0\end{array}$$ 18

with the notations from above as well as the production rate τ of the signal in dependency on the virus, the decay rate 𝜗 of the signal and the diffusion constant κ of the signal.

Since the migration flux of the immune response caused by chemotaxis is assumed to be proportional to the concentration of the immune response itself, the chemotaxis term is usually modeled by

$$C=-\tilde{\mu }\nabla \cdot [v(\mathbf{x})\psi \nabla s]$$

with a chemotaxis coefficient $\tilde{\mu }$, cf. [25], and a factor ψ = ψ(v, s) decreasing in s like in the Keller-Segel model. For studying the basic behavior, let us abstract to ψ = 1. This form ensures v ≥ 0 for all initial conditions, but it introduces a next non-linearity into the system in (18).

The decomposition $v(t,\mathbf{x})=v_{\text{mean}}(t)+\tilde{v}(t,\mathbf{x})$ with the mean immune response v_mean leads to the rather strong simplification

$$C=-\tilde{\mu }{v}_{\text{mean}}\mathrm{\Delta s}-\tilde{\mu }\nabla \cdot [\tilde{v}(\mathbf{x})\nabla s]\approx -\mu \mathrm{\Delta s},$$

the availability of which would need a small C¹-norm of $\tilde{v}(x)$ and the time-independence of v_mean. Nevertheless, the simplified system

$$\begin{array}{rcl}\dot{u}& =& \mathrm{uw}(u)-\mathrm{\gamma uv}+\alpha \mathrm{\Delta u}\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ \dot{v}& =& j[u]-\eta (1-u)v+\beta \mathrm{\Delta v}-\mu \mathrm{\Delta s}\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ \dot{s}& =& \mathrm{\tau u}-\mathrm{𝜗s}+\kappa \mathrm{\Delta s}\text{for}\mathbf{x}\in \mathrm{\Omega },t>0,\\ u(0,\mathbf{x})& =& u_0(\mathbf{x}),v(0,\mathbf{x})=v_0(\mathbf{x}),s(0,\mathbf{x})=s_0(\mathbf{x})\text{for}\mathbf{x}\in \mathrm{\Omega },\\ 0& =& \nabla u\cdot \mathbf{n}=\nabla v\cdot \mathbf{n}=\nabla s\cdot \mathbf{n}\text{for}\mathbf{x}\in \partial \mathrm{\Omega },t>0,\end{array}$$ 19

conserves the fundamental behavior of the differential operators, and it is more appropriate for analytical investigations than (18).

A next modeling approach is found if the role of the signal or trace s of the virus is integrated again. We can assume that the chemotaxis acts quickly or at least sufficiently fast to neglect the time-dependent decay of the signal. As a consequence, the immune response does not follow the signal alone but the virus itself. On the level of (19), we model this situation by τ ∼ 𝜗 →∞ after having calibrated the value of the signal to τ = 𝜗. Then, the form

$$\dot{s}=\tau (u-s)+\kappa \mathrm{\Delta s}$$

shows that any difference between the virus concentration u and the signal strength s fades out while the first term becomes dominant with τ →∞. The limit transition gives

$$\underset{\tau \to \infty }{\lim }s=u,$$

and we get the reduced system (17).

The models in (19) and (18) include three components and are hence more expensive to analyze analytically than (17). Therefore, (17) is used to give first ideas about the influence of the chemotaxis term to the chronification behavior of liver infections.

Influence of small chemotactical effects on the long-term solution

Theorem 3 requires a positive definite diffusion matrix $\mathfrak{D}$. Writing the model in (17) in the general form of (1), gives

$$\mathfrak{D}=\left(\begin{array}{cc}\alpha & 0\\ -\mu & \beta \end{array}\right)$$

as combined diffusion and chemotaxis matrix. The definiteness of matrix $\mathfrak{D}$ depends on the parameters. Only if 4αβ > μ² holds true, then $\mathfrak{D}$ is positive definite.

Since Theorem 3 shall be applied for all parameter, the matrix $\mathfrak{D}$ is diagonalized by the matrix S of eigenvectors and the matrix Λ containing the eigenvalues of $\mathfrak{D}$. We get $\mathfrak{D}=S\mathrm{\Lambda }{S}^{-1}$ with

$$\mathrm{\Lambda }=\left(\begin{array}{cc}\alpha & 0\\ 0& \beta \end{array}\right),S=\frac{1}{\nu }\left(\begin{array}{cc}\beta -\alpha & 0\\ \mu & \nu \end{array}\right),$$

and the normalization $\nu =\sqrt{{(\beta -\alpha )}^2+{\mu }^2}$. The co-ordinate transformation $\mathbf{q}\mapsto \tilde{\mathbf{q}}$ with $\tilde{\mathbf{q}}=S^{-1}\mathbf{q}$ leads to the modified reaction-diffusion equation

$$\dot{\tilde{\mathbf{q}}}=S^{-1}\mathbf{f}(S\tilde{\mathbf{q}})+\mathrm{\Lambda }\mathrm{\Delta }\tilde{\mathbf{q}},$$ 20

with the diagonal matrix Λ. The reaction terms can be abbreviated by $\tilde{\mathbf{f}}(\tilde{\mathbf{q}})=S^{-1}F(S\tilde{\mathbf{q}})$ with the modified reaction function $\tilde{\mathbf{f}}$. The conditions for Turing pattern of this transformed system are not fulfilled as well; see Section 5.2.1.

Compared to the model without chemotaxis, we still have the same parameter d = min{α, β} and the same parameter λ of the extent of Ω. In contrast, the parameter of the diffusion strength is modified by the transformation, and so is M as the maximum of the Jacobian of $\tilde{\mathbf{f}}(\tilde{\mathbf{q}})$.

Instead of

$$\frac{\partial }{\partial \mathbf{q}}\mathbf{f}(\mathbf{q})=\left(\begin{array}{cc}1-2u-\mathrm{\gamma v}& -\mathrm{\gamma u}\\ \delta +\mathrm{\eta v}& -\eta (1-u)\end{array}\right),$$

with $\tilde{\mathbf{q}}=\left(\begin{array}{c}U\\ V\end{array}\right)$, we get the expression

$$\frac{\partial }{\partial \tilde{\mathbf{q}}}\tilde{\mathbf{f}}(\tilde{\mathbf{q}})=\left(\begin{array}{cc}1-2\frac{\beta -\alpha +\mathrm{\gamma \mu }}{\nu }U-\mathrm{\gamma V}& -\mathrm{\gamma U}\\ Z& \frac{\mathrm{\mu \gamma }+\eta (\beta -\alpha )}{\nu }U-\eta \end{array}\right)$$ 21

with

$$Z=\frac{\delta (\beta -\alpha )-(1+\eta )\mu }{\nu }+\frac{2\mu ((\beta -\alpha )(1+\eta )+\mathrm{\gamma \mu })}{{\nu }^2}U+\frac{\eta (\beta -\alpha )+\mathrm{\gamma \nu }}{\nu }V\mathrm{.}$$

We find

$$B=\left(\begin{array}{cc}1-\gamma (2U+V)& -\mathrm{\gamma U}\\ -1-\eta +\gamma (2U+V)& -\eta +\mathrm{\gamma U}\end{array}\right)\text{with}\underset{\mu \to \infty }{\lim }\frac{1}{\mu }\left[\frac{\partial }{\partial \tilde{\mathbf{q}}}\tilde{\mathbf{f}}(\tilde{\mathbf{q}})-B\right]=0\in {ℝ}^{2\times 2}\mathrm{.}$$ 22

Of course, the limit in (22) exceeds the condition 4αβ > μ² of a small chemotactical effect for every given α and β. Nevertheless, the limit shows the mathematical behavior of the matrix B, and therefore, we can use B as an approximation of $\frac{\partial }{\partial \tilde{\mathbf{q}}}\tilde{\mathbf{f}}(\tilde{\mathbf{q}})$ for large μ. The new state variables U and V depend through

$$U=\frac{\nu }{\beta -\alpha }u\text{and}V=-\frac{\mu }{\beta -\alpha }u+v$$

on the parameter μ. Therefore a growing μ leads to growing U and V. The entries B in (22) grow consequently with μ. Hence, the upper Lipschitz boundary M ∼ μ grows with the chemotaxis coefficient.

The parameter σ(t) in Theorem 3 becomes negative with a growing chemotaxis coefficient. Therefore, chronification gets more probable with stronger chemotactic effects. Again, the theorem gives a necessary condition for chronifications but this is not identical with the occurrence of chronification itself. Let us mention the important point that only the estimate gets weaker with increasing chemotaxis coefficient, and there is no statement about the real occurrence of non-trivial stable stationary solutions. We will see later that there is a non-monotonous behavior about the occurrence of chronic course in dependence of the chemotaxis coefficient μ.

For small chemotactical effects, we observe a similar estimation from Theorem 3 as for model (10) without chemotactic effects because model (17) is an extension of model (10). In the next section, we study large the influence of strong chemotactic effects on the chronification tendency.

Influence of strong chemotactical effects on the long-term solution

We have seen in the previous section that small chemotactic effects do not lead to a higher chronification tendency of model (17). Now we study the influence of strong chemotactic effects on the chronification tendency. The model in (17) is rather complex to analyze analytically. Therefore we use the linear models from Section 4 extended by the chemotaxis term − μΔu in the dynamics of the T cells. The qualitative behavior of the linear models makes assertions about the existence and stability of such solutions. We remember that chronic courses are associated with non-constant stationary solutions where the virus and the lymphocytes share the domain Ω in a kind of armistice. The linear models provide acceptable solutions around fixed values of the populations. This limitation is not a real restriction because we want to study the local tendency towards chronification.

We use a Fourier approach for approximating the solutions around a mean value. The Fourier series in the eigenfunctions u_k, cf. (2), give

$$u(t,\mathbf{x})=u_{\text{mean}}(t)+\sum _{k=0}^{\infty }{a}_k(t)u_k(\mathbf{x})\text{and}v(t,\mathbf{x})=v_{\text{mean}}(t)+\sum _{k=0}^{\infty }{b}_k(t)u_k(\mathbf{x})$$

analogously. Furthermore, we decompose

$$J(\mathbf{x})=J_{\text{mean}}+\sum _{k=0}^{\infty }{c}_k{u}_k(\mathbf{x})\mathrm{.}$$

Now, the frequencies in the linear predator–prey systems separate. Equation (8) transforms into the position-independent predator–prey system for the mean populations

$$\begin{array}{rcl}{\dot{u}}_{\text{mean}}& =& u_{\text{mean}}-\tilde{\gamma }{v}_{\text{mean}},\\ {\dot{v}}_{\text{mean}}& =& \delta |\mathrm{\Omega }|J_{\text{mean}}{u}_{\text{mean}}-{\eta }_1{v}_{\text{mean}}\end{array}$$

and the system of ordinary differential equations for the frequencies with the indices k = 0,1,…

$$\begin{array}{rcl}{\dot{a}}_k& =& a_k-\tilde{\gamma }{b}_k-\alpha {\lambda }_k{a}_k,\\ {\dot{b}}_k& =& \delta |\mathrm{\Omega }|c_k{u}_{\text{mean}}-{\eta }_1{b}_k-\beta {\lambda }_k{b}_k+\mu {\lambda }_k{a}_k\mathrm{.}\end{array}$$

We see that the stationary solution with vanishing time derivatives of the frequency-resolved system is completely zero with the exception of very particular parameter choices with singular system matrices. Such particular parameter choices are not regarded as evolutionary stable and no analysis should be based on parameters, a very tiny deviation of which would totally change the system behavior. Of course, as always in predator–prey systems, the trivial vanishing stationary solution is not stable, because a small perturbation of u_mean = 0 into u > 0 theoretically would generate an infinite growth of the prey population. This case is not realistic for the present situation because the immune response would be generated even if it is not represented in (9). So, this theoretical exponential growth can be interpreted as an unrealistic property of the simplified model. Finally, the chemotaxis coefficient μ has no influence on the stationary solution and hence, (8) is not appropriate to discuss the influence of chemotaxis on the chronification.

Analogously, (9) transforms into

$$\begin{array}{rcl}{\dot{u}}_{\text{mean}}& =& u_{\text{mean}}-\tilde{\gamma }{v}_{\text{mean}},\\ {\dot{v}}_{\text{mean}}& =& \delta |\mathrm{\Omega }|J_{\text{mean}}{u}_{\text{mean}}-{\eta }_2(1-u_{\text{mean}})\end{array}$$

and the system of ordinary differential equations for the frequencies with the indices k = 0,1,…

$$\begin{array}{rcl}{\dot{a}}_k& =& a_k-\tilde{\gamma }{b}_k-\alpha {\lambda }_k{a}_k,\\ {\dot{b}}_k& =& \delta |\mathrm{\Omega }|c_k{u}_{\text{mean}}+{\eta }_2{a}_k-\beta {\lambda }_k{b}_k+\mu {\lambda }_k{a}_k\mathrm{.}\end{array}$$

Now, the stationary mean immune response is not vanishing anymore, and we find independently of the chemotaxis coefficient

$$u_{\text{mean}}^{\ast }=\frac{{\eta }_2}{\delta |\mathrm{\Omega }|J_{\text{mean}}+{\eta }_2}>0\text{and}{v}_{\text{mean}}^{\ast }=\frac{{\eta }_2}{\gamma (\delta |\mathrm{\Omega }|J_{\text{mean}}+{\eta }_2)}>0$$

what leads to

$$\left(\begin{array}{cc}1-\alpha {\lambda }_k& -\tilde{\gamma }\\ {\eta }_2+\mu {\lambda }_k& -\beta {\lambda }_k\end{array}\right)\left(\begin{array}{c}{a}_k^{\ast }\\ b_k^{\ast }\end{array}\right)=\left(\begin{array}{c}0\\ -\delta |\mathrm{\Omega }|c_k{u}_{\text{mean}}\end{array}\right),$$

and therefore to the stationary solutions $(a_k^{\ast },b_k^{\ast })$ with

$$a_k^{\ast }=\frac{-\delta |\mathrm{\Omega }|c_k\tilde{\gamma }{u}_{\text{mean}}}{({\eta }_2+\mu {\lambda }_k)\tilde{\gamma }-\beta {\lambda }_k(1-\alpha {\lambda }_k)}\text{and}{b}_k^{\ast }=\frac{-\delta |\mathrm{\Omega }|c_k\tilde{\gamma }{u}_{\text{mean}}(1-\alpha {\lambda }_k)}{({\eta }_2+\mu {\lambda }_k)\tilde{\gamma }-\beta {\lambda }_k(1-\alpha {\lambda }_k)}\mathrm{.}$$ 23

Now, a simple calculation shows that large frequencies and thus large k leading to large eigenvalues λ_k produce small amplitudes $a_k^{\ast }$ and $b_k^{\ast }$, i. e., the larger the frequencies the smaller the amplitudes. Furthermore, we find

$$\underset{\mu \to \infty }{\lim }{a}_k^{\ast }=\underset{\mu \to \infty }{\lim }{b}_k^{\ast }=0.$$

This limit transition shows that a strong chemotaxis effect leads to stationary solutions, which are locally constant and which could not persist in a more general modeling framework. Thus, strong chemotaxis would prevent chronification within this model framework. Unfortunately, it says nothing about the influence of a small chemotaxis parameter to the tendency of chronification.

Similarly, the stability of the stationary solution depends on the chemotaxis parameter μ. Calculating the system matrix of the frequency-resolved system shows that large chemotaxis coefficients μ make the stationary solution stable, whereas small μ might provoke unstable solutions.

The doubling of this non-monotonous result in the stationary solution on one hand and in its stability on the other hand supports the reliability of this simplified linear approach.

Surely, Fourier techniques are applicable in a nonlinear setting, too, but the form of the Fourier series of a non-linear image of a function is often very sophisticated to get. Here, we can apply the idea given above for the product terms omitted in (8) and (9). That leads to longer sums in the frequency-resolved system. Furthermore, the partial differential equations do not separate anymore into the single frequencies. So, we should apply the results of the linear Fourier analysis only for small deviations of an assumed solution.

Hierarchical model family

The basal model in (10) is modified in different manners. First, there are some specifications concerning the inflow term in (12) and (13). Similarly, (19) specifies (18) by choosing a particular chemotaxis term.

The opposite of a model specification is an extension to a more general model. Of course, the more general model can be adapted to a wider range of observations or measurements by a suitable parameter choice. It reproduces a wider class of observations. Thus, qualitative results about the model behavior descend to model specifications but not necessarily vice versa.

Different model extensions of (10) are given in (19) and (17). Here, (17) is an extension of (10) by just the chemotaxis term − μΔu, and setting the chemotaxis coefficient μ = 0 specifies (17) to (10) again. On the other side, (19) is an extension by an additional component, here the signal s governed by mechanisms like its excitation τu by the virus, its decay in − 𝜗s and its diffusion in κΔs. The consideration of the signal necessarily requires the inclusion of mechanisms belonging to this component. Also, the chemotaxis term enters (19) which is here − μΔs.

Now, (19) is not a pure specification of (17) or vice versa, because (17) is a limit of (19) for the limit transition τ ∼ 𝜗 →∞. The conservation of qualitative properties during the limit transition is a critical and deeply mathematical question in general and in particular here, where the dimension of the system of ordinary differential equations changes. This change can be compared to the change of the degrees of freedom in mechanical systems found in the transition from stiff mechanical systems to mechanical systems with constraints, where non-continuous transitions of the quantitative properties are possible.

This effect gets more remarkable if we consider the inverse τ^′ = 1/τ = 1/𝜗 instead the parameters τ = 𝜗 itself. That makes (17) to be a specification of (19) by setting τ^′ = 0. Nevertheless, this setting is theoretical because it hides the limit transition τ →∞. It is not clear, how to distinguish between a model specification and a limit transition in general, compare the theory of singular perturbations; see [26].

Here, the three models in (10), (17) and (19) form a model family where (17) and (19) are extensions of (10) see Fig. 6. They both inherit the qualitative property of a possible chronification or a possible non-trivial stable stationary state and extent the model in (10) by the hierarchically sub-ordinated influence of the chemotaxis.

Fig. 6.

Hierarchy of the models. Above: model refinements; below: model specifications or simplifications

Parallel model families are found e. g., by different linearization as used in (8) and (9). The observation that the linearized model in (9) supports the numerical observations of (17) gives more confidence in the reliability of the modeling approach. The chronification property is robust against the modifications of the model and against a wide range of possible parameters.

Conclusions

We have investigated model extensions of a predator–prey model with local resolution for the interaction between the virus and the immune response during a liver infection with special regard to the influence of chemotaxis effects to the chronification of the infection. Chronic courses of liver infections were associated with non-trivial stable stationary solutions of the resulting reaction-diffusion equations. Such stationary solutions, which are locally non-constant, represent a kind of armistice between virus and immune response where both share the liver domain and concentrate in different parts. The inflammation of liver persists near the incoming blood vessels, i.e., the T cell population concentrates near the incoming blood vessels modeled by the sub-domain Θ. Thus, the model reproduces the observations mentioned in Section 2.

The theoretical estimation of the long-term behavior of the solution is accompanied by numerical experiments. A critical value defines the exponential growth or decay of upper bounds of the solution. The critical value depends on the reaction strength, the minimal migration speed, and the geometry of the domain. In particular, the extension of the domain influences the tendency to chronify. So, fissures in the domain decrease the minimal eigenvalue of the related Neumann eigenvalue problem, and thus they increase distances inside the domain. Hence, the extension of the domain grows. This extension is related to the geometry of a realistic liver with its lobes and sub-structures. So, the tendency of chronification can be explained as quantitative geometric property of the liver domain.

We have found that strong chemotaxis diminishes the tendency of the infection to chronify. Even if the influence of the chemotaxis to the chronification is non-monotonous, we can carefully conclude that chemotaxis is not the one basal mechanism causing chronic courses but an additional mechanism, which modifies slightly the chronification tendency of the model. The tendency of chronification of the chemotaxis-free model is robust to model extensions. Only a very strong chemotaxis dominates the system behavior in a qualitative manner but does not lead to the feared crazy behavior. We might assume that the theoretical framework for predictions can be useful in the development of therapeutical approaches.

Compliance with Ethical Standards

Conflict of interests

The authors declare that they have no conflict of interest.