Global Stability for Fractional Diffusion Equations in Biological Systems

In recent years, fractional differential equations (FDEs) are used to describe the temporal dynamics of various systems in many fields. .ese equations are the generalization of the classical ordinary differential equations (ODEs). However, fractional partial differential equations (FPDEs) are the generalization of the partial differential equations (PDEs) which can be an effective tool to describe the spatiotemporal dynamics of several phenomena with memory or have hereditary properties. .e construction of Lyapunov functionals to prove the global stability of fractional dynamic systems has attracted the attention of some authors. Aguila-Camacho et al. [1] established a new lemma for fractional derivative in Caputo sense with order α ∈ (0, 1). .ey used this lemma to demonstrate the stability of some fractional-order systems by mean of quadratic Lyapunov functionals. Duarte-Mermoud et al. [2] extended the lemma of [1] to a vector of differentiable functions, and they used Lyapunov functionals containing general quadratic forms in order to analyze the stability of fractional-order model reference adaptive control (FOMRAC) schemes. Vargas-De-León [3] extended the Volterra-type Lyapunov function to fractional-order epidemic systems via an inequality to estimate the Caputo fractional derivative of this function. On the other hand, a study in [4] has been devoted to establish the global stability for some diffusion equations in biology by means of Lyapunov functionals. .e methods mentioned above are applied for particular Lyapunov functionals such as quadratic or Volterra-type. Likewise, the work in [4] is especially applicable for the models formulated by PDEs..erefore, the main goal of this study is to develop a new mathematical method to construct the Lyapunov functionals for FDEs and FPDEs based on those of ODEs. To do this, the next section deals with the description of the method and the last section is devoted to the application of our method to investigate the global stability of some mathematical models in epidemiology as well as in virology.


Introduction
In recent years, fractional differential equations (FDEs) are used to describe the temporal dynamics of various systems in many fields. ese equations are the generalization of the classical ordinary differential equations (ODEs). However, fractional partial differential equations (FPDEs) are the generalization of the partial differential equations (PDEs) which can be an effective tool to describe the spatiotemporal dynamics of several phenomena with memory or have hereditary properties. e construction of Lyapunov functionals to prove the global stability of fractional dynamic systems has attracted the attention of some authors. Aguila-Camacho et al. [1] established a new lemma for fractional derivative in Caputo sense with order α ∈ (0, 1). ey used this lemma to demonstrate the stability of some fractional-order systems by mean of quadratic Lyapunov functionals. Duarte-Mermoud et al. [2] extended the lemma of [1] to a vector of differentiable functions, and they used Lyapunov functionals containing general quadratic forms in order to analyze the stability of fractional-order model reference adaptive control (FOMRAC) schemes. Vargas-De-León [3] extended the Volterra-type Lyapunov function to fractional-order epidemic systems via an inequality to estimate the Caputo fractional derivative of this function. On the other hand, a study in [4] has been devoted to establish the global stability for some diffusion equations in biology by means of Lyapunov functionals. e methods mentioned above are applied for particular Lyapunov functionals such as quadratic or Volterra-type. Likewise, the work in [4] is especially applicable for the models formulated by PDEs. erefore, the main goal of this study is to develop a new mathematical method to construct the Lyapunov functionals for FDEs and FPDEs based on those of ODEs. To do this, the next section deals with the description of the method and the last section is devoted to the application of our method to investigate the global stability of some mathematical models in epidemiology as well as in virology.

Description of the Method
Consider the following FDE: where D α t is the fractional derivative in the Caputo sense of order α ∈ (0, 1], the state variable u is a non-negative vector of concentrations u 1 , . . ., u m , and f: IR m ⟶ IR m is a C 1 function. It is obvious that if α � 1, then (1) becomes the following ordinary differential equation: (2) Let Ω be a bounded domain in IR n with smooth boundary zΩ and D � (d 1 , . . . , d m ) with d i ≥ 0. Assume that u * is a steady state of (1). en, u * is also the steady state of the following fractional diffusion system with homogeneous Neumann boundary condition: where △ � n i�1 z 2 /zx 2 i represents the Laplacian operator and zu/z] denotes the outward normal derivative on the boundary zΩ.
Let V(u) be a C 1 function defined on some domain in R m + and u(t) is a solution of (1). Further, we suppose that the range of u(t) is contained in the domain of V(u) and whose equality holds if α � 1.
We observe that the right-hand side of the above inequality is given by the scalar product of the gradient of the function V(u) and the vector field f(u). Hence, the righthand side is defined without the fact that u(t) is a solution of (1), which is very important for the construction of Lyapunov functionals.
Let u(t, x) be a solution of (3). Denote e fractional time derivative of W along the positive solution of (3) satisfies By applying Green's formula, we find According to (zu/z]) � 0 on zΩ, we have Additionally, we assume that the function V satisfies the following condition: From the above, it is not difficult to obtain the following result.
Theorem 1. Let V be a Lyapunov functional for the ordinary differential equation (2). (4), then V is also a Lyapunov functional for fractional differential equation (1). (4) and (10), then the function W defined by (5) is a Lyapunov functional for fractional diffusion system (3).
In the literature, several researchers constructed the Lyapunov functional in the following form: Corollary 2. If V is a Lyapunov functional for ordinary differential equation (2) of the form given in (11), then V is a Lyapunov functional for fractional differential equation (1). Moreover, the function W defined by (5) is a Lyapunov functional for fractional diffusion system (3).
Proof. We have By applying Lemma 3.1 in [3], we get en, V satisfies the condition (4). It follows from eorem 1 (i) that V is also a Lyapunov functional for fractional differential equation (1).
On the other hand, we have which implies that V satisfies the condition (10). According to eorem 1 (ii), we deduce that the function W given by (4) is also a Lyapunov functional for fractional diffusion system (3). is completes the proof.
e method described above can be used to prove the stability of many fractional systems with and without diffusion. It is very important to recall that the steady state u * is stable if there exists a Lyapunov functional satisfying D α t V(u) ≤ 0. Moreover, if D α t V(u) < 0 for all u ≠ u * , then u * is asymptotically stable. Additionally, according to [5], if D α t V(u) ≤ 0 and the largest invariant set in u | D α t V(u) � 0 is the singleton u * { }, then u * is asymptotically stable. is means that the solution of the system starting from any initial conditions converges to u * .

Applications
is section focuses on the application of the method described in the above section in order to establish the global stability of some fractional diffusion biological models by constructing Lyapunov functionals from those of the corresponding systems which are formulated by ODEs.Example 1. Consider the SIR epidemic model described by the following nonlinear system of FDEs: where S, I, and R are the populations of susceptible, infected, and recovered individuals, respectively. e parameters A, μ, d, and r are, respectively, the recruitment rate, the natural death rate, the death rate due to disease, and the recovery rate. e incidence function of system (15) is described by Hattaf-Yousfi functional response [6] of the form βSI/1 + α 1 S + α 2 I + α 3 SI, where the non-negative constants α i , i � 0, 1, 2, 3, measure the saturation, inhibitory, or psychological effects, and the positive constant β is the infection rate. is functional response covers the most famous forms existing in the literature such as the classical bilinear incidence, the saturated incidence, the Beddington-DeAngelis functional response [7], the Crowley-Martin functional response [8], and the specific functional response introduced in [9]. Further, the fractional models proposed in [10,11] are particular cases of model (15); it suffices to take α 0 � 1 and α 1 � α 2 � α 3 � 0 for [10] and α 0 � 1 for [11].
Since the state variable R does not appear in the two first equations of fractional model (15), we can reduce (15) to the following system: Due to the great mobility of individuals inside or outside a country or region, we consider the following fractional model: S(x, t), I(x, t))I(x, t), where ψ(S, I) � (βS/α 0 + α 1 S + α 2 I + α 3 SI) and η � μ + d + r. e parameters d S and d I are the diffusion coefficients for the susceptible and infected populations, respectively. Also, we consider model (17) with homogeneous Neumann boundary conditions: and initial conditions: For α � 1, model (16) becomes the following nonlinear system of ODEs: Obviously, model (20) has always one disease-free equilibrium E f ((A/μ), 0). By a simple computation, the basic reproduction number is given by When R 0 > 1, model (20) has another equilibrium named endemic equilibrium E * (S * , I * ), where with δ � (β − α 1 η + α 2 μ − α 3 Λ) 2 + 4α 3 μ(ηα 0 + α 2 A). System (20) is a special case of the mathematical model presented in [12]. us, the disease-free equilibrium E f is globally asymptotically stable when R 0 ≤ 1. However, E f becomes unstable and the endemic equilibrium E * is globally asymptotically stable if R 0 > 1.
Let S 0 � A/μ and Φ(z) � z − 1 − ln(z) for z > 0. From [12], the function is a Lyapunov functional for ODE model (20) at E f . Moreover, we have Complexity 3 Additionally, we have en, V 1 satisfies the condition (4). By applying eorem 1 (i), we deduce that V 1 is also a Lyapunov functional for FDE model (16) at E f . Now, we construct the Lyapunov functional for fractional diffusion model (17) at E f as follows: In this case, we have In fact, is implies that V 1 satisfies the condition (10). It follows from eorem 1 (ii) that W 1 is a Lyapunov functional for fractional diffusion systems (17) For the global stability of the endemic equilibrium E * , we consider the following function: which is a Lyapunov functional for ODE model (20) at E * . On the other hand, we have us, V 2 obeys the condition (4) and then V 2 is a Lyapunov functional for FDE model (8) at E * . Denote is a Lyapunov functional for ODE model (37) at Q * when R 0 > 1. Let Since L 2 has the form given in (11), we conclude, by applying Corollary 2, that L 2 is a Lyapunov functional for FDE model (33) and L 2 is a Lyapunov functional for fractional diffusion systems (34)-(36) at Q * when R 0 > 1.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.