On the Well-Posedness of a Fractional Model of HIV Infection

The human immunodeficiency virus (HIV) is one of the world’s leading infectious diseases. A big number of people have died around the globe due to this disease. HIV infects vital cells in the human immune system, such as CD4+ T cells. In this way, the body becomes progressively more susceptible to opportunistic infections, leading to the development of AIDS (acquired immunodeficiency syndrome) (see, e.g., [1, 2]). Mathematical models play an important role in understanding the dynamics of HIV infection. The dynamical models for HIV usually consist of systems of ordinary differential equations which range from two-component models (see, e.g., [3, 4]) to three-component (see e.g. [5–7]) and four-component models (see, e.g., [8]). In particular, in [3], the following two-cell model was proposed to describe the HIV infection:


Introduction
The human immunodeficiency virus (HIV) is one of the world's leading infectious diseases. A big number of people have died around the globe due to this disease. HIV infects vital cells in the human immune system, such as CD4+ T cells. In this way, the body becomes progressively more susceptible to opportunistic infections, leading to the development of AIDS (acquired immunodeficiency syndrome) (see, e.g., [1,2]).
Mathematical models play an important role in understanding the dynamics of HIV infection. The dynamical models for HIV usually consist of systems of ordinary differential equations which range from two-component models (see, e.g., [3,4]) to three-component (see e.g. [5][6][7]) and four-component models (see, e.g., [8]). In particular, in [3], the following two-cell model was proposed to describe the HIV infection: where u is the density of uninfected CD4+ T cells, v is the density of virus-producing cells, κ is the rate of production of CD4+ T cells, σ is their per capita death rate, δ is the rate of infection of CD4+ T cells, and τ is the rate of disappearance of infected cells.
The rest of the paper is organized as follows. In Section 2, we recall some notions on fractional calculus and Perov's fixed point theorem. In Section 3, we state and prove our main results. In Section 4, some special cases are discussed.
Remark 5. Using the change of variable z = ψðsÞ, one deduces from ((11)) that Using Lemma 1 and Remark 5, one deduces the following result.
Lemma 10. Let f ∈ Cð½0, TÞ, 0 < θ < 1 and ψ ∈ Ψ . Then, Journal of Function Spaces Now, we recall some concepts on fixed point theory that will be used later. Let n be a positive natural number and define the partial order°n in ℝ n by for all y, z ∈ ℝ n . We denote by 0 ℝ n the zero vector in ℝ n , i.e., Let X be a nonempty set and d : X × X → ℝ n be a given mapping. We say that d is a vector-valued metric on X (see, e.g., [19]), if for all x, y, z ∈ X, In this case, the pair ðX, dÞ is called a generalized metric space. In such spaces, the notions of convergent sequence, Cauchy sequence, and completeness are similar to those for usual metric spaces.
Let us denote by M n ðℝ + Þ the set of square matrices of size n with nonnegative coefficients. Given M ∈ M n ðℝ + Þ, we denote by ρðMÞ its spectral radius.
Lemma 11 (Perov's fixed point theorem). Let ðX, dÞ be a complete generalized metric space and F : X → X be a given mapping. Suppose that there exists M ∈ M n ðℝ + Þ with ρðMÞ < 1 such that for all x, y ∈ X. Then, (i) the mapping F admits a unique fixed point in X, say x * (ii) for all x 0 ∈ X, the sequence fx m g ⊂ X defined by x m+1 = Fðx m Þ converges to x * We end this section by recalling the Grönwall's lemma.

Main Results
Problem (2) and (3) is investigated under the following assumptions: (i) T > 0 and ψ ∈ Ψ, where Ψ is the functional space defined by (10). (2) and (3). Let Suppose that ðu, vÞ ∈ W is a solution to problem (2) and (3). Using the first equation in (2), one obtains Hence, by Lemma 6 and Lemma 9, it holds that Using the initial conditions (3), one obtains Similarly, using the second equation in (2), one obtains 3 Journal of Function Spaces By the initial conditions (3) Therefore, one deduces that, if ðu, vÞ ∈ W is a solution to problem (2) and (3), then ðu, vÞ ∈ V is a solution to the system of integral equations for all 0 ≤ t ≤ T. Conversely, suppose that ðu, vÞ ∈ V is a solution to (32). By assumptions (iii) and (iv), one deduces that ðu, vÞ ∈ W. Moreover, by (32), one has uð0Þ = u 0 and vð0Þ = v 0 . On the other hand, using the first equation in (32), Lemma 6, and Lemma 10, one obtains Similarly, using the second equation in (32), one obtains Therefore, one deduces that, if ðu, vÞ ∈ V is a solution to the system of integral equation (32), then ðu, vÞ ∈ W is a solution to problem (2) and (3).
From the above study, the following result holds.
(II) ðu, vÞ ∈ V is a solution to the system of integral equations (32).
By the above lemma, the study of problem (2) and (3) in W reduces to the study of the system of integral equation (32) in V.

3.2.
Uniqueness. In this part, using Grönwall's lemma, we shall prove that the system of integral equations (32) admits at most one solution ðu, vÞ ∈ V. Proposition 14. Suppose that the assumptions (i)-(iv) are satisfied. Then the system of integral equation ((32)) admits at most one solution ðu, vÞ ∈ V.
Consider now the self-mapping, By the definition of F, one observes that, if ðu, vÞ ∈ V r is a fixed point of F; then, ðu, vÞ ∈ V is a solution to the system of integral equation (32). In order to prove that F admits a fixed point in V r , we shall use Perov's fixed point theorem (see Lemma 11). Namely, we define the vector-valued metric d : Notice that ðV r , dÞ is a complete generalized metric space. On the other hand, for all ðu 1 , v 1 Þ, ðu 2 , v 2 Þ ∈ V r and 0 ≤ t ≤ T, one has which yields