Global Existence of a Virus Infection Model with Saturated Chemotaxis

: In this paper, a virus infection model with saturated chemotaxis is formulated and analyzed, where the chemotactic sensitivity for chemotactic movements of the cells is described. This model contains three state variables namely the population density of uninfected cells, the population density of infected cells and the concentration of virus particles, respectively. By virtue of regularized approximation technique and ﬁxed point theorem, the local solvability of the regularized system corresponding to the original system is established. Then by extracting a suitable sequence along which the respective approximate solutions approach a limit in convenient topologies, with addition of Gagliardo-Nirenberg interpolation inequality as well as L p -estimate techniques, we show that the original system describing the virus infection model exists at least one global weak solution. To illustrate the application of our theoretical results, an optimal control problem of the epidemic system is considered, where the admissible control domain is assumed to be a bounded closed convex subset. With the help of Aubin compactness theorem and lower semicontinuous of the cost functional, the existence of the optimal pair is proved. Our results generalize and improve partial previously known ones, and moreover, we ﬁrst prove that the optimal control problem has at least one optimal pair.


Introduction
As the virological, immunological and mathematical plates become interlocked, mathematics is playing an increasingly important role in biology. Many mathematicians began to use the rigorous theory and methods of partial differential equations to elaborate and forecast some complex biological phenomena. Especially for the virus infection dynamic models with diffusion terms. The vast majority of existing research on evolution of a virus infection model was almost described by ordinary differential equations (ODEs) [7,18], this leads to the ignorance of spatial variations, which means the ODEs are not suitable for obtaining spatial information about the distribution of infected cells. To make up for this deficiency, spatial dependence of the virus infection dynamic models must be taken into consideration [5].
It is well known that some epidemic diseases are awful if no effective measures are taken to control them, such as cholera, tuberculosis and so on. Thus, the optimal control problems of epidemic models have been drawing more and more notice in recent decades. For example, Kirschner et al. studied optimal chemotherapy strategy in an early treatment background which depicted the interaction of the immune system with the human immunodeficiency virus (HIV) by the optimal control theories and methods, where the immune system is governed by ODEs [13]. Chang and Astolfi used the drug scheduling methods to measure the states of the HIV model on the basis of a reduced-order model framework, and presented the corresponding simulation results [4]. Also, with the help of optimal control methods, Xiang and Liu solved the inverse problem of an SIS epidemic model of the ecosystem [34,35]. Meantime, Zhou et al. used the two control treatment, that is, vaccination and therapy to consider the optimal control problem of an epidemic system governed by reaction-diffusion equations [36].
Recent modeling methods and experimental results show that the chemotactic sensitivity is in general a tensor for chemotactic movements of the cells [33]. In this paper, we concern with the following virus infection model with saturated chemotaxis: x ∈ Ω, t > 0, v t = ∆v + uw − v, x ∈ Ω, t > 0, w t = ∆w + v − w, x ∈ Ω, t > 0, (1) where Ω ⊂ R N (N ∈ N) is a bounded domain with smooth boundary ∂Ω and ∂ ∂ν denotes the derivative with respect to the outer normal of ∂Ω. u, v and w denote the population density of uninfected cells, the population density of infected cells and the concentration of virus particles, respectively. Initial data u 0 , v 0 , w 0 are known functions satisfying u 0 ∈ C 0 (Ω), v 0 ∈ W 1,∞ (Ω), w 0 ∈ C 0 (Ω), We suppose that S ∈ C 2 (Ω × [0, ∞) 2 ; R N ×N ) denoting the rotational effect, which is induced by a swimming bias and the bacteria themselves, has the property that there exist S 0 > 0 and α > 0 fulfilling Then we will consider (1) along with the initial conditions: and the boundary conditions: In fact, the dynamics of high HIV seminal loads leading to sporadic infection are difficult to understand biologically and completely, when they are falling outside the scope of usual mathematical modelling of infectious diseases described by simple ODEs. In order to better understand the formation of patterns on the onset of an HIV infection, Stancevic et al. proposed the following mathematical model [26] where u, v and w denote the population density of uninfected cells, the population density of infected cells and the concentration of virus particles, respectively. χ, κ, α, β, D 1 , D 2 are suitable positive constants. The virus is also produced by infected cells and its presence causes healthy cells to be converted into infected cells. Furthermore, healthy cells are produced with a constant rate κ. χ∇ · (u∇v) describes chemotactic response to cytokines emitted by infected cells moving toward high concentration. However, the pioneering work of the chemotaxis model was first introduced by Keller and Segel [12], where aggregation of cellular slime mold toward a higher concentration of a chemical signal was described by: where u denotes the cell density and v is the chemical concentration. The mathematical analysis of (7) and the variant thereof mainly concentrated on the boundedness and blow-up of the solutions [6,10]. In addition to the original model, a large number of variants of the classical form have also been studied, including the systems with the logistic terms [19], chemotaxis-haptotaxis models [20], multispecies chemotaxis systems [1,21,24], attraction-repulsion chemotaxis system [23,25], chemotaxis-fluid model [14,22,30] and so on. During the past four decades, the chemotaxis model has become one of the best study models in mathematical biology. And we refer the reader to the survey [3,8,9], in which we can find further examples to illustrate the significant biological correlation of chemotaxis. It is well known that the cross-diffusive term in (6) is the key contributor to analyze the global existence in mathematics. In order to exclude the possibility of blow-up, motivated by [26,Sec. 8], Hu and Lankeit considered the following system [11] where Ω ⊂ R N , N ∈ N is a bounded domain with smooth boundary ∂Ω and ∂ ∂ν denotes the derivative with respect to the outer normal of ∂Ω. In which, they proved that if hold, then the system (8) existed a global bounded solution. For a related system, such as HBV infection model, see also [28,29].
To the best of our knowledge, the optimal control problem of virus infection models with saturated chemotaxis (1) has not been studied. With the addition of the arguments in previous studies [11,16,17,22,24,30,33], the aim of this paper is to consider a virus infection model with saturated chemotaxis. Under appropriate regularity assumptions on the initial data, via L p -estimate techniques, we show that the epidemic system (1) exists at least one global weak solution. This result generalizes and improves Theorem 1.1 [11]. Moreover, the existence of the optimal pair of system (1) is obtained.
In this paper, we use symbols C i and c i (i = 1, 2, · · · ) as some generic positive constants which may vary from line to line. For simplicity, u(x, t) is written as u, the integral Ω u(x)dx is written as Ω u(x) and t 0 Ω u(x)dxdt is written as t 0 Ω u(x). The contents of the paper are as follows. In Section 2, some basic definitions and main theorems as well as some useful lemmas are presented. In Section 3, some fundamental estimates for the solution of the system (14) are given, and the previously mentioned a priori estimate in the process of limit procedure is discussed and Theorem 2.1 is proved. In Section 4, the optimal control problem of the system (70) is considered and the existence of the optimal pair is obtained.

Preliminaries
In order to consider the optimal control of virus infection model with saturated chemotaxis, it is necessary to first discuss the well-posedness of the system (1). Inspired by [33], see also [30], the concept of the solution is presented.
N +2 , then for any choice of the initial data (u 0 , v 0 , w 0 ) satisfy (4), system (1)  In order to construct such weak solutions by an approximation procedure, we fix and that and For ε ∈ (0, 1), we then define We will construct solutions of (1) as limits of solutions to relevant regularized approximate problems, and give some basic estimates for the solutions to the regularized system. For any such ε, the regularized problems According to the well-established fixed point arguments, the local solvability of (14) can be obtained, the proof is similar to [10,32], so here we omit the proof.
Proof. The proof is similar to [  (Ω), due to Sobolev embedding theorem, we see that W m,2
Proof. For fixed t > 0 and arbitrary φ ∈ W m,2

Passing to the Limit
Now, we are capable of extracting a suitable sequence of ε along which the respective solutions approach a limit in convenient topologies.
Lemma 3.6. Let α > N +1 N +2 and assume that (u 0 , v 0 , w 0 ) satisfy (4). Then there exist (ε j ) j∈N ⊂ (0, 1) such that ε j 0 as j → ∞ and functions and as well as with some limit functions u, v and w which are such that Lemma 3.7. Let α > N +1 N +2 and assume that (u 0 , v 0 , w 0 ) satisfy (4). Furthermore, let u, v, w denote the limit function provided by Lemma 3.7. Then and v and w fulfill the weak solution properties of (11) and (12), respectively.

Application to Optimal Control
In this section, to apply the existence results to prove the existence of the optimal control pair, we rewrite system (1), (4) and (5) into the following form and give the discussion of optimal control problem.
p 0 ∈ L 2 (0, T ; H) and B ∈ L(V, H), where V = (H 1 0 (Ω)) 3 ∩ H, here B := l 1 I. Denote by the symbol · the norm of the space V, which is defined by and by the symbol | · | the norm of R 3 and (L 2 (Ω)) 3 . We endow the space H with the norm of (L 2 (Ω)) 3 , and presented by ·, · the scalar product of H, ·, · (V,V ) the paring between V and its dual V with the norm · V . Let A ∈ L(V, V ) and trilinear function be defined by: If there is no confuse, we present also by ·, · the dual product between V and its dual V . We define the operators B : V → V by B(p), y = b(p, p, y) ∀y ∈ V.
Let f (t) = P f 0 (t) and D ∈ L(W, H) is given by D = P I, where I ∈ L(W; (L 2 (Ω)) 3 ) is a unit matrix, P : (L 2 (Ω)) 3 → H is the projection on H. Then we may rewrite the optimal control problem (P ) as where λ(t, x) denote control variable. We assume U ad is a closed, bounded, and convex subset of U. Then, we have the following existence theorem. Lemma 4.1. The optimal control problem (P ) has at least one optimal pair (p,Ū ). Proof. We denote J(p, U ) = 1 2 T 0 |B(p(t) − p 0 (t))| 2 dxdt + T 0 h(U (t))dt, Then there exist (p n , U n ) ∈ F w such that γ ≤ J(p n , U n ) ≤ γ + 1 N .