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STABILIZATION IN A CHEMOTAXIS MODEL FOR VIRUS INFECTION

. This paper presents a qualitative analysis of a model describing the time and space dynamics of a virus which migrates driven by chemotaxis. The initial-boundary value problem related to applications of the model to a real biological dynamics is studied in detail. The main result consists in the proof of global existence and asymptotic stability.

1. Introduction. A minimal virus model to describe the dynamics of the number densities of a virus particles, interacting with infected and uninfected cells, was proposed by Wei et al. [38], see also [23,25,27]. This model rapidly attracted both applied mathematicians and biologists who devoted a great deal of attention to the mathematical problems generated by the ability of the model to depict emerging behaviors of interest in biology. Examples of further development and applications are given by [5,12,13].
Various modifications have been proposed in the literature with the aim of going beyond the the simple description delivered by population dynamics without internal variables. Therefore, various technical modifications have been proposed, for instance, by introducing models with internal structure [7,37], stochastic interactions [11] and delay terms deemed to account also for the heterogeneity of individual entities of large biological systems [9], chapter 2. The books [9,28] refer this specific model to the existing literature in population dynamics, while the book by Nowak shows how evolutionary game theory can be used to model interactions and virus dynamics [23].
An interesting development consists in introducing a space structure and study the virus dynamics coupled with a diffusion process. Therefore the overall dynamics can be modeled by a system of PDEs, while the qualitative analysis can be focused not only on the well-posedness of the initial-boundary value problem, but also on the study of pattern formation somehow related to the stability properties of the original dynamical system described by ODEs. The selection of the diffusion process can take advantage of recent studies specifically focused on biological tissues [35], which have subsequently motivated analytic studies reviewed in the survey [6].
This paper develops the aforementioned concepts focusing on the qualitative analysis of the initial-boundary value model, where space dynamics is a diffusion process induced by chemotaxis. This idea is founded on a biological interpretation on the dynamics of viruses which diffuse following the stimuli of chemoattractants, as an example they are attracted by concentration gradient [14,22] of cytokines from inflammation at sites of infection. Chemotaxis can be modeled by the celebrated Keller-Segel model [18,19], as well as by suitable modifications that can be obtained either by heuristic developments of the original model at the macroscopic scale [15,33,34], or by micro-macro derivation by asymptotic methods from the underlying description at the microscopic scale [3,26]. A detailed study of modeling of interactions at the microscopic scale can lead to models with nonlinear degenerate diffusion terms [3] whose qualitative analysis has been studied in [4]. A possible development of the contents of our paper might consider new models that include this specific feature.
In more detail, the contents of this paper is proposed through three more sections. Section 2 consists of three parts, where the first one rapidly presents the ODE model and some known results on the initial value problem; the second part motivates the introduction of the diffusion dynamics and presents the related initial-boundary value problem which is object of the qualitative analysis; finally, the third part reports the main results of the qualitative analysis, namely global existence and asymptotic stability. The contents of remaining sections are devoted to proofs. In more detail, Section 3 shows how global existence can be proved, while Section 4 is devoted to stability analysis. The results concerning the qualitative analysis are related to the so called basic reproduction number [21], see Chapter 5 in [9].
2. Mathematical models and results. This section first presents the mathematical models which are object of the qualitative analysis and subsequently reports the main results achieved in this paper, while their proof is given in the next sections. In more detail, Subsection 2.1 rapidly introduces the minimal virus model mentioned in Section 1 and some stability results known in the literature, while the Subsection 2.2 motivates the interest in introducing diffusion dynamics and presents the specific mathematical model and related initial-boundary value problem studied in our paper. Finally, Subsection 2.3 reports the main results which have been achieved, namely global existence and asymptotic stability.
2.1. Basic virus dynamics model. Let us start from a model of virus dynamics which is considered the prototype of this type of biological competition: where u, v and w correspond, respectively, to the population, namely the number, of healthy uninfected cells, of infected cells, and of viruses. This model reflects the following dynamics: 1. Healthy cells are constantly produced by the body at a rate h, die at a rate d 1 u proportional to the current population, and become infected at a rate βuw proportional to their interaction with the virus population w; 2. Infected cells are produced at the rate βuw and die at rate d 2 v; 3. New virus is produced at a rate kv proportional to the number of infected cells, and die at a rate d 3 w. This model has been studied by several authors, for instance [5,7,17,24,25,27,38], who have studied the qualitative behavior of the solutions to the initial value problem enlightening a broad variety of asymptotic behaviors, related to the parameters of the model, which appear to be consistent with empirical data. An important role is acted by the so called basic reproduction number introduced by [21] In more detail, it can be rapidly shown that the system always possesses the infection-free equilibrium Q 0 := ( h d1 , 0, 0) := (u , 0, 0), while whenever R 0 > 1, the system has a coexistence equilibrium In addition, the following holds true (cf. [21] for instance): It is plain that, in spite of its popularity, this model contains various simplifications of the real underlying dynamics of viral infections. In particular, the specific features of structured population models that include delay terms and internal variables are not considered. In addition, the model is derived at the scale of population dynamics, where pattern formation is not accounted for. Therefore some modeling development is needed to depict this important feature by which a virus can diffuse in tissues. Indeed, the formation of space patterns can be studied by taking advantage of the immediate ability of this simple model to lead equilibrium configurations and their stability properties. Some specific models are proposed in the next subsection looking ahead to a qualitative analysis of mathematical problems.

Chemotaxis models for virus dynamics.
Space patterns related to virus dynamics can be studied by models, where diffusion is induced by chemotaxis phenomena, see Chapter 5 of [28] and the two surveys [33,34]. In fact, target cells are attracted by the concentration gradient of cytokines from inflammations at sites of infection see [14,22] and more in detail [30], where the following chemotaxis model was proposed: where, in addition to a linear diffusion term, a chemotaxis term is inserted in the first equation to account for the aforementioned gradient's attraction. The authors after having written the model by dimensionless variables develop a linear stability analysis and subsequently an interesting numerical analysis which illuminates some specific features of pattern formation such as the onset of hot spots. This is an important feature considering that their localization can contribute to detect early HIV infections.
In addition, numerical work in [30] shows that the model exhibits a blow up of solutions in finite time for sufficiently large χ. Therefore a specific assumption has been proposed in [30] to control this distinct feature. In more detail, the authors assume that the term χ decay with u. An alternative to this assumption can be inspired by the Beddington-DeAngelis model [2,8] on the functional response β u w which grows with u and w, but saturates for large values of this variables. More precisely, the following assumption has been proposed in [36]: while the interested reader is referred to [15] for further heuristic regularized models and to [3] for derivation of macroscopic models from the underlying description at the scale of cells. Additional phenomenological studies [20] on the biological phenomena under consideration suggest to introduce a space structure on the terms h, β and k, namely h → h(x), β → β(x) and h → h(x); as well as a nonlinear cross diffusion in the diffusion term D w , namely D w → D w (v), which corresponds to repulsion dynamics of superinfecting virions by infected cells [10,36].
Let now u := u(x, t), v := v(x, t) and w := w(x, t) denote the densities of uninfected cells, infected cells and virus, at location x and time t, respectively. To slightly reduce the complexity of the model in [36], we assume that the infected cells are unable to recover into healthy state and that a possible absorption effect [1] of virus is neglected. Then, a simplified version of the mathematical model in [36] for virus infection dynamics reads as follows: where Ω ⊂ R n , n ≥ 1, is a bounded domain with smooth boundary, D u , D v , χ, d 1 , d 2 , d 3 and a are positive constants, b is a nonnegative parameter, and h(x), k(x) and β(x) are given positive functions.
Throughout this paper we shall assume that h(x), k(x), β(x) ∈ C α (Ω) with some α ∈ (0, 1) (2.6) and where D 0 stands for random diffusion rate of free virions and g ∈ C 2 (R + , R + ) fulfilling g(0) = 0 is a prescribed increasing function of v, addressing the repulsion of super-infecting virions by infected cells.
such that u, v and w are nonnegative inΩ × (0, ∞), and such that (u, v, w) is bounded in the sense that with some C > 0.
A crucial step in the proof of Theorem 2.1 is to establish the coupled L p estimates for v and w (see Lemma 3.3 and Lemma 3.4 below).
When the parameter functions h(x), k(x) and β(x) are spatially homogeneous, the model (2.5) takes the following simplified version x ∈ Ω, (2.9) where h, k and β are positive constants. Obviously, this simplified model always has an infection-free steady state E 0 := ( h d1 , 0, 0) := (u , 0, 0). Only the local linear stability of E 0 was discussed in [36]. Under suitable parameter conditions, the present work further asserts the global asymptotic stability of this infection-free steady state when R 0 < 1. More precisely: Let Ω ⊂ R n , n ≥ 1, be a bounded domain with smooth boundary, and assume that the prescribed function D w (·) fulfills the assumption (2.7), that the parameters D u , D v , χ, d 1 , d 2 , d 3 , a, h, k and β are positive, and that b is nonnegative.
If Then for arbitrary nonnegative initial data The stability condition in Theorem 2.2 is independent of the chemotactic coefficient χ and the parameter b, but it strongly depends on the parameter a. We observe that the condition (2.11) holds whenever a is a suitably large. Moreover, it seems that the condition (2.11) is necessary for the validity of (2.12). To figure this out, when the system (2.9) with b = 0 has a coexistence equilibrium E * = (u * , v * , w * ), where Complex patterns might occur around (u * , v * , w * ) for small a and χ > 0; see [30] for the numerical verifications for the case when a = 0 and χ > 0.
Our derivation of the stabilization result in Theorem 2.2 will be based on investing the corresponding Lyapunov functionals for suitably chosen γ > 0, η > 0 and τ > 0.
3. Global existence for (2.5). Proof of Theorem 2.1. As a consequence of the assumption (2.6) we can find some constant L > 0 such that We begin with the local existence, which can be proven by a straightforward fixedpoint argument (cf. [31], for instance).

Lemma 3.1.
Let Ω ⊂ R n , n ≥ 1, be a bounded domain with smooth boundary, suppose that the parameters D u , D v , χ, d 1 , d 2 , d 3 and a are positive, the parameter b is nonnegative, the parameter functions h(x), k(x) and β(x) are positive and satisfy (2.6) and the prescribed function D w (·) complies with the assumption (2.7), and assume that u 0 , v 0 and w 0 are nonnegative functions from C 0 (Ω), W 1,∞ (Ω) and W 1,∞ (Ω), respectively. Then there exist T max ∈ (0, ∞] and a classical solution (u, v, w) of (2.5) in Ω × (0, T max ), satisfying such that u, v and w are nonnegative in Ω × (0, T max ), and such that either T max = ∞, or lim t Tmax sup u(·, t) L ∞ (Ω) = ∞. (3.14) The following mass properties immediately results from simple integrations of equations in (2.5) with respect to x ∈ Ω.
Lemma 3.2. The solution of (2.5) satisfies Ω u(·, t) ≤ m := max Proof. Adding the second equation to the first one in (2.5) and integrating the resulting equation we obtain d dt Upon an ODE comparison, this entails (3.15) and (3.16). Integrating the third equation in (2.5) and invoking (3.16), we prove (3.17). The following energy-type inequalities can be regarded as a corner stone for establishing a priori L p estimates.
Proof. We multiply the second equation in (2.5) by v p−1 and integrate by parts to find on using (3.13) that 1 p for all t ∈ (0, T max ), this implies (3.18), whereas (3.19) can be proven similarly. By a coupled estimate argument together with an Ehrling-type lemma we obtain the following. and Ω w p (·, t) ≤ C for all t ∈ (0, T max ). (3.21) Proof. Adding (3.19) to (3.18) yields for all t ∈ (0, T max ), where c 1 := min{d 2 , d 3 } · p. Here the Young inequality entails with c 2 := pL(1 + 1 a ), and we use an Ehrling-type lemma along with (3.16) to find c 3 > 0 such that Similarly, we can obtain some c 4 > 0 satisfying This immediately leads to the boundedness property of ∇v in L ∞ (Ω).
Corollary 3.5. There exists C > 0 such that the solution of (2.5) satisfies Proof. We rewrite the second equation in the form Since Lemma 3.4 along with (3.13) guarantees that for any p > n and all t ∈ (0, T max ), for all t > 0.
Proof. Since u is positive inΩ × (0, ∞) by the maximum principle and the assumption that h > 0, we may test the first equation in (2.9) by (1 − u u ) to obtain using integration by parts that for all t > 0. As by Young's inequality along with (4.27) we can estimate Proof. From an integration of the second and the third equations in (2.9) we immediately obtain (4.29) and (4.30). We multiply the second equation in (2.9) by v to see using integration by parts that Since we can estimate β Ω v · uw 1 + au + bw ≤ β a Ω vw · au 1 + au + bw ≤ β a Ω vw for all t > 0, this entails (4.31). Finally, (4.32) can be proven in quite a similar manner.
Proof. In view of the assumption that R 0 ≤ 1 we can fix γ > 0 satisfying and according to the hypothesis (2.11) we can also fix ε > 0 sufficiently small such that ad 2 β > 1 d3 k − ε · (4.37)