# Global Stability for Fractional Diffusion Equations in Biological Systems

Authors: Khalid Hattaf; Noura Yousfi
Journal: Complexity
 (2020)

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## Abstract

This paper proposes a new method of construction of Lyapunov functionals for the dynamical systems described by fractional differential equations and fractional partial differential equations. The proposed method is rigorously presented. Furthermore, the method is applied to establish the global stability of some fractional biological models with and without diffusion.


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## Body

## 1. Introduction

In recent years, fractional differential equations (FDEs) are used to describe the temporal dynamics of various systems in many fields. These 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.The 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. They 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.The 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. Therefore, 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.


## 2. Description of the Method

Consider the following FDE:(1)Dtαu=fu,where Dtα is the fractional derivative in the Caputo sense of order α∈0,1, the state variable u is a non-negative vector of concentrations u1, …, um, and f: IRm⟶IRm is a C1 function. It is obvious that if α=1, then (1) becomes the following ordinary differential equation:(2)u˙=fu.LetΩ be a bounded domain in IRn with smooth boundary ∂Ω and D=d1,…,dm with di≥0. Assume that u∗ is a steady state of (1). Then, u∗ is also the steady state of the following fractional diffusion system with homogeneous Neumann boundary condition:(3)∂tαu=DΔu+fuinΩ×0,+∞,∂u∂ν=0on∂Ω×0,+∞,ux,0=u0xinΩ¯,where △=∑i=1n∂2/∂xi2 represents the Laplacian operator and ∂u/∂ν denotes the outward normal derivative on the boundary ∂Ω.LetVu be a C1 function defined on some domain in R+m and ut is a solution of (1). Further, we suppose that the range of ut is contained in the domain of Vu and(4)DtαVut≤∇Vu⋅fu,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 functionVu and the vector field fu. Hence, the right-hand side is defined without the fact that ut is a solution of (1), which is very important for the construction of Lyapunov functionals.Letut,x be a solution of (3). Denote(5)W=∫ΩVut,xdx.The fractional time derivative ofW along the positive solution of (3) satisfies(6)DtαW≤∫Ω∇Vu⋅DΔu+fudx=∫Ω∇Vu⋅fudx+∫Ω∇Vu.DΔudx=∫Ω∇Vu⋅fudx+∑i=1mdi∫Ω∂V∂uiΔuidx.By applying Green’s formula, we find(7)∫Ω∂V∂uiΔuidx=∫∂Ω∂V∂ui∂ui∂νdσ−∫Ω∇ui⋅∇∂V∂uidx.According to∂u/∂ν=0 on ∂Ω, we have(8)∫Ω∂V∂uiΔuidx=−∫Ω∇ui⋅∇∂V∂uidx.Therefore,(9)dWdt≤∫Ω∇Vu⋅fudx−∑i=1mdi∫Ω∇ui⋅∇∂V∂uidx.Additionally, we assume that the functionV satisfies the following condition:(10)di∫Ω∇ui⋅∇∂V∂uidx≥0,for alli=1,…,m.From the above, it is not difficult to obtain the following result.Theorem 1.
LetV be a Lyapunov functional for the ordinary differential equation (2).(i)
IfV satisfies the condition (4), then V is also a Lyapunov functional for fractional differential equation (1).(ii)
IfV satisfies the conditions (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:(11)V=∑i=1maiui−ui∗−ui∗lnuiui∗.Corollary 2.
IfV 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(12)DtαV=∑i=1maiDtαui−ui∗−ui∗lnuiui∗.
By applying Lemma 3.1 in [3], we get(13)DtαV≤∑i=1mai1−ui∗uiDtαui=∇Vu⋅fu.
Then,V satisfies the condition (4). It follows from Theorem 1 (i) that V is also a Lyapunov functional for fractional differential equation (1).
On the other hand, we have(14)∫Ω∇ui⋅∇∂V∂uidx=aiui∗∫Ω∇ui2ui2dx≥0,which implies that V satisfies the condition (10). According to Theorem 1 (ii), we deduce that the function W given by (4) is also a Lyapunov functional for fractional diffusion system (3). This completes the proof.Remark 3.
The 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 stateu∗ is stable if there exists a Lyapunov functional satisfying DtαVu≤0. Moreover, if DtαVu<0 for all u≠u∗, then u∗ is asymptotically stable. Additionally, according to [5], if DtαVu≤0 and the largest invariant set in uDtαVu=0 is the singleton u∗, then u∗ is asymptotically stable. This means that the solution of the system starting from any initial conditions converges to u∗.


## 3. Applications

This 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:(15)DtαS=A−μS−βSIα0+α1S+α2I+α3SI,DtαI=βSIα0+α1S+α2I+α3SI−μ+d+rI,DtαR=rI−μR,where S, I, and R are the populations of susceptible, infected, and recovered individuals, respectively. The 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. The incidence function of system (15) is described by Hattaf–Yousfi functional response [6] of the form βSI/1+α1S+α2I+α3SI, where the non-negative constants αi, i=0,1,2,3, measure the saturation, inhibitory, or psychological effects, and the positive constant β is the infection rate. This 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 α0=1 and α1=α2=α3=0 for [10] and α0=1 for [11].Since the state variableR does not appear in the two first equations of fractional model (15), we can reduce (15) to the following system:(16)DtαS=A−μS−βSIα0+α1S+α2I+α3SI,DtαI=βSIα0+α1S+α2I+α3SI−μ+d+rI.Due to the great mobility of individuals inside or outside a country or region, we consider the following fractional model:(17)∂tαS=dSΔS+A−ψSx,t,Ix,tIx,t,∂tαI=dIΔI+ψSx,t,Ix,tIx,t−ηIx,t,where ψS,I=βS/α0+α1S+α2I+α3SI and η=μ+d+r. The parameters dS and dI are the diffusion coefficients for the susceptible and infected populations, respectively. Also, we consider model (17) with homogeneous Neumann boundary conditions:(18)∂S∂ν=∂I∂ν=0,on∂Ω×0+∞,and initial conditions:(19)Sx,0=S0x≥0,Ix,0=I0x≥0,x∈Ω¯.Forα=1, model (16) becomes the following nonlinear system of ODEs:(20)S˙=A−μS−ψS,II,I˙=ψS,II−ηI.Obviously, model (20) has always one disease-free equilibrium EfA/μ,0. By a simple computation, the basic reproduction number is given by(21)ℛ0=βAα0μ+α1Aμ+d+r.Whenℛ0>1, model (20) has another equilibrium named endemic equilibrium E∗S∗,I∗, where(22)S∗=2ηα0+α2Aβ−α1η+α2μ−α3A+δ,I∗=Λ−μS∗η,with δ=β−α1η+α2μ−α3Λ2+4α3μηα0+α2A.System (20) is a special case of the mathematical model presented in [12]. Thus, the disease-free equilibrium Ef is globally asymptotically stable when ℛ0≤1. However, Ef becomes unstable and the endemic equilibrium E∗ is globally asymptotically stable if ℛ0>1.LetS0=A/μ and Φz=z−1−lnz for z>0. From [12], the function(23)V1u=α0S0α0+α1S0ΦSS0+Iis a Lyapunov functional for ODE model (20) at Ef. Moreover, we have(24)∇V1u⋅fu=α0α0+α1S01−S0SA−μS−ψS,II+ψS,II−ηI=−α0μS−S02α0+α1S0S+ηIψS,IψS,0ℛ0−1≤−α0μS−S02α0+α1S0S+ηIℛ0−1.Additionally, we have(25)DtαV1u≤∇V1u⋅fu.Then,V1 satisfies the condition (4). By applying Theorem 1 (i), we deduce that V1 is also a Lyapunov functional for FDE model (16) at Ef.Now, we construct the Lyapunov functional for fractional diffusion model (17) at Ef as follows:(26)W1=∫ΩV1ux,tdx.In this case, we have(27)di∫Ω∇ui⋅∇∂V1∂uidx≥0,for alli=1,2.In fact,(28)dS∫Ω∇S⋅∇∂V1∂Sdx=α0dSS0α0+α1S0∫Ω∇S2S2dx≥0,dI∫Ω∇I⋅∇∂V1∂Idx=0.This implies thatV1 satisfies the condition (10). It follows from Theorem 1 (ii) that W1 is a Lyapunov functional for fractional diffusion systems (17)–(19) at Ef when ℛ0≤1.For the global stability of the endemic equilibriumE∗, we consider the following function:(29)V2u=S−S∗−∫S∗SψS∗,I∗ψX,I∗dX+I∗ΦII∗,which is a Lyapunov functional for ODE model (20) at E∗. On the other hand, we have(30)∇V2u⋅fu=1−ψS∗,I∗ψS,I∗A−μS−ψS,II+1−I∗IψS,II−ηI,DtαV1u≤∇V1u⋅fu.Thus,V2 obeys the condition (4) and then V2 is a Lyapunov functional for FDE model (8) at E∗.Denote(31)W2=∫ΩV2ux,tdx.It is not hard to show that(32)dS∫Ω∇S⋅∇∂V2∂Sdx=α0+α2I∗dSS∗α0+α1S∗+α2I∗+α3S∗I∗∫Ω∇S2S2dx≥0,dI∫Ω∇I⋅∇∂V2∂Idx=dII∗∫Ω∇I2I2dx≥0.Then, condition (10) holds. Therefore, W2 is a Lyapunov functional for fractional diffusion systems (17)–(19) at E∗ when ℛ0>1.Consequently, we have the following:(i)
Whenℛ0≤1, the disease-free equilibrium Ef of (17)–(19) is globally asymptotically stable.(ii)
Whenℛ0>1, the endemic equilibrium E∗ of (17)–(19) is globally asymptotically stable.Example 2. Consider the following fractional viral infection model:(33)DtαH=λ−μH−β1HV−β2HI,DtαI=β1HV+β2HI−δI,DtαV=pI−cV,where healthy cells (H) are produced at rate λ, die at rate μH, and become infected by contact with virus at rate β1HV and by contact with infected cells at rate β2HI. Infected cells (I) die at rate δI. Free viral particles (V) are released from infected cells at rate pI and decay at rate cV. Note that when α=1 and β1=0, we obtain the basic model which was used to study some viral infections such as HIV, HBV, and HCV [13–16].Now, we extend the above fractional model (33) by taking into account the mobility of the virus as well as the cells. Then, model (33) becomes(34)∂tαH=dHΔH+μ−μHx,t−β1Hx,tVx,t−β2Hx,tIx,t,∂tαI=dIΔI+β1Hx,tVx,t+β2Hx,tIx,t−δIx,t,∂tαV=dVΔV+pIx,t−cVx,t,where Hx,t, Ix,t, and Vx,t denote the concentrations of healthy cells, infected cells, and free viral particles at position x and at time t, respectively. The parameters dH, dI, and dV represent, respectively, the diffusion coefficients of healthy cells, infected cells, and free viral particles.The cells and virus are biological quantities which should be non-negative, and they also should not cross the boundary of the domain. Then, we consider system (34) with the homogeneous Neumann boundary conditions:(35)∂H∂ν=∂I∂ν=∂V∂ν=0,on∂Ω×0,+∞,and initial conditions:(36)Hx,0=H0x≥0,Ix,0=I0x≥0,Vx,0=V0x≥0,∀x∈Ω¯.The corresponding ordinary system of the above models is given by(37)H˙=λ−μH−β1HV−β2HI,I˙=β1HV+β2HI−δI,V˙=pI−cV.This system admits a unique infection-free equilibriumQfH0,0,0, where H0=λ/μ. Further, the basic reproduction number is given by(38)ℛ0=λpβ1+cβ2cδμ.This basic reproduction number is the sum of two basic reproduction numbers due to virus-to-cell and cell-to-cell transmissions. Ifℛ0>1, ODE model (37) has another biological steady state called the infection chronic which is labeled by Q∗H∗,I∗,V∗, where(39)H∗=λdℛ0,I∗=cμR0−1pβ1+cβ2,V∗=pcμR0−1cpβ1+cβ2.These two equilibria are also steady states of both models (33) and (34). First, we consider a Lyapunov functional for ODE model (37) at Qf as follows:(40)L1u=H0ΦHH0+I+β1H0cV.Calculating the time derivative ofL1 along the solution of (37), we get(41)dL1dt=∇L1u⋅fu=1−H0Hλ−μH−β1HV−β2HI+β1HV+β2HI−δI+β1H0cpI−cV=−μHH−H02+δIℛ0−1.Ifℛ0≤1, then dL1/dt≤0 with equality if and only if H=H0, I=0 and V=0. Therefore, the infection-free equilibrium Qf is globally asymptotically stable when ℛ0≤1. Since(42)DtαL1u≤∇L1u⋅fu,it follows from Theorem 1 (i) that L1 is also a Lyapunov functional for FDE model (33) at Qf.Next, we construct a Lyapunov functional for fractional diffusion model (34) at Qf as follows:(43)ℒ1=∫ΩL1ux,tdx.We have(44)dH∫Ω∇H⋅∇∂L1∂Hdx=dHH0∫Ω∇H2H2dx≥0,dI∫Ω∇I⋅∇∂L1∂Idx=0,dV∫Ω∇V⋅∇∂L1∂Vdx=0.Then,L1 satisfies the condition (10). Using Theorem 1 (ii), we deduce that ℒ1 is a Lyapunov functional for fractional diffusion systems (34)–(36) at Qf if ℛ0≤1.Based on the results in [17], the function(45)L2u=H−H∗−H∗lnTT∗+I−I∗−I∗lnII∗+β1H∗V∗pV∗V−V∗−V∗lnVV∗is a Lyapunov functional for ODE model (37) at Q∗ when ℛ0>1. Let(46)ℒ2=∫ΩL2ux,tdx.SinceL2 has the form given in (11), we conclude, by applying Corollary 2, that L2 is a Lyapunov functional for FDE model (33) and ℒ2 is a Lyapunov functional for fractional diffusion systems (34)–(36) at Q∗ when ℛ0>1.



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