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Research article Special Issues

New numerical dynamics of the fractional monkeypox virus model transmission pertaining to nonsingular kernels


  • Received: 03 July 2022 Revised: 22 September 2022 Accepted: 25 September 2022 Published: 09 October 2022
  • Monkeypox (MPX) is a zoonotic illness that is analogous to smallpox. Monkeypox infections have moved across the forests of Central Africa, where they were first discovered, to other parts of the world. It is transmitted by the monkeypox virus, which is a member of the Poxviridae species and belongs to the Orthopoxvirus genus. In this article, the monkeypox virus is investigated using a deterministic mathematical framework within the Atangana-Baleanu fractional derivative that depends on the generalized Mittag-Leffler (GML) kernel. The system's equilibrium conditions are investigated and examined for robustness. The global stability of the endemic equilibrium is addressed using Jacobian matrix techniques and the Routh-Hurwitz threshold. Furthermore, we also identify a criterion wherein the system's disease-free equilibrium is globally asymptotically stable. Also, we employ a new approach by combining the two-step Lagrange polynomial and the fundamental concept of fractional calculus. The numerical simulations for multiple fractional orders reveal that as the fractional order reduces from 1, the virus's transmission declines. The analysis results show that the proposed strategy is successful at reducing the number of occurrences in multiple groups. It is evident that the findings suggest that isolating affected people from the general community can assist in limiting the transmission of pathogens.

    Citation: Maysaa Al Qurashi, Saima Rashid, Ahmed M. Alshehri, Fahd Jarad, Farhat Safdar. New numerical dynamics of the fractional monkeypox virus model transmission pertaining to nonsingular kernels[J]. Mathematical Biosciences and Engineering, 2023, 20(1): 402-436. doi: 10.3934/mbe.2023019

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  • Monkeypox (MPX) is a zoonotic illness that is analogous to smallpox. Monkeypox infections have moved across the forests of Central Africa, where they were first discovered, to other parts of the world. It is transmitted by the monkeypox virus, which is a member of the Poxviridae species and belongs to the Orthopoxvirus genus. In this article, the monkeypox virus is investigated using a deterministic mathematical framework within the Atangana-Baleanu fractional derivative that depends on the generalized Mittag-Leffler (GML) kernel. The system's equilibrium conditions are investigated and examined for robustness. The global stability of the endemic equilibrium is addressed using Jacobian matrix techniques and the Routh-Hurwitz threshold. Furthermore, we also identify a criterion wherein the system's disease-free equilibrium is globally asymptotically stable. Also, we employ a new approach by combining the two-step Lagrange polynomial and the fundamental concept of fractional calculus. The numerical simulations for multiple fractional orders reveal that as the fractional order reduces from 1, the virus's transmission declines. The analysis results show that the proposed strategy is successful at reducing the number of occurrences in multiple groups. It is evident that the findings suggest that isolating affected people from the general community can assist in limiting the transmission of pathogens.



    As we know, one of the most common ways to study the asymptotic stability for a system of delay differential equations (DDEs) is the Lyapunov functional method. For DDEs, the Lyapunov-LaSalle theorem (see [6,Theorem 5.3.1] or [11,Theorem 2.5.3]) is often used as a criterion for the asymptotic stability of an autonomous (possibly nonlinear) delay differential system. It can be applied to analyse the dynamics properties for lots of biomathematical models described by DDEs, for example, virus infection models (see, e.g., [2,3,10,14]), microorganism flocculation models (see, e.g., [4,5,18]), wastewater treatment models (see, e.g., [16]), etc.

    In the Lyapunov-LaSalle theorem, a Lyapunov functional plays an important role. But how to construct an appropriate Lyapunov functional to investigate the asymptotic stability of DDEs, is still a very profound and challenging topic.

    To state our purpose, we take the following microorganism flocculation model with time delay in [4] as example:

    {˙x(t)=1x(t)h1x(t)y(t),˙y(t)=rx(tτ)y(tτ)y(t)h2y(t)z(t),˙z(t)=1z(t)h3y(t)z(t), (1.1)

    where x(t), y(t), z(t) are the concentrations of nutrient, microorganisms and flocculant at time t, respectively. The positive constants h1, r, h2 and h3 represent the consumption rate of nutrient, the growth rate of microorganisms, the flocculating rate of microorganisms and the consumption rate of flocculant, respectively. The phase space of model (1.1) is given by

    G={ϕ=(ϕ1,ϕ2,ϕ3)TC+:=C([τ,0],R3+) : ϕ11, ϕ31}.

    In model (1.1), there exists a forward bifurcation or backward bifurcation under some conditions [4]. Thus, it is difficult to use the research methods that some virus models used to study the dynamics of such model.

    Clearly, (1.1) always has a microorganism-free equilibrium E0=(1,0,1)T. For considering the global stability of E0 in G, we define the functional L as follows,

    L(ϕ)=ϕ2(0)+r0τϕ1(θ)ϕ2(θ)dθ, ϕG. (1.2)

    The derivative of L along a solution ut (defined by ut(θ)=u(t+θ) for θ[τ,0]) of (1.1) with any ϕG is taken as

    ˙L(ut)=(rx(t)1h2z(t))y(t)(r1h2z(t))y(t). (1.3)

    Obviously, if r1, L is a Lyapunov functional on G since ˙L0 on G.

    However, we can not get r1h2z(t)<0 for all t0 and r>1. From this and some conditions on bifurcations of equilibria, L is not a Lyapunov functional on G. But by (3.5), we know that

    lim inftz(t)h1h1+rh3. (1.4)

    If r<1+h1h2/(h1+rh3), then there can be found an ε>1 such that r1h1h2/ε(h1+rh3)<0. Hence, there exists a T=T(ϕ)>0 such that z(t)h1/ε(h1+rh3) for all tT. By (1.3), we have

    ˙V(ut)[r1h1h2ε(h1+rh3)]y(t)0, tT.

    Obviously, for all t0, the inequality ˙V(ut)0 may not hold since we only can obtain ˙V(ut)0, tT for some T>0. Therefore, it does not meet the conditions of Lyapunov-LaSalle theorem. But we can give the global stability of E0 if r<1+h1h2/(h1+rh3) by using the developed theory in Section 3.

    In this paper, we will expand the view of constructing Lyapunov functionals, namely, we first give a new understanding of Lyapunov-LaSalle theorem (including its modified version [9,15,19]), and based on it establish some global stability criteria for an autonomous delay differential system.

    Let C:=C([τ,0],Rn) be the Banach space with the norm defined as ϕ=maxθ[τ,0]|ϕ(θ)|, where ϕ=(ϕ1,ϕ2,,ϕn. We will consider the dynamics of the following system of autonomous DDEs

    ˙u(t)=g(ut), t0, (2.1)

    where ˙u(t) indicates the right-hand derivative of u(t), utC, g:CRn is completely continuous, and solutions of system (2.1) which continuously depend on the initial data are existent and unique. For a continuous functional L:CR, we define the derivative of L along a solution of system (2.1) in the same way as [6,11] by

    ˙L(ϕ)=˙L(ϕ)|(2.1)=lim sups0+L(us(ϕ))L(ϕ)s.

    Let X be a nonempty subset of C and ¯X be the closure of X. Let

    u(t)=u(t,ϕ):=(u1(t,ϕ),u2(t,ϕ),,un(t,ϕ))T

    denote a solution of system (2.1) satisfying u0=ϕX. If the solution u(t) with any ϕX is existent on [0,) and X is a positive invariant set of system (2.1), and ut(ϕ):=(u1t(ϕ),u2t(ϕ),,unt(ϕ))T, we define the solution semiflow of system (2.1):

    U(t):=ut():XX (which also satisfies U(t):¯X¯X),

    and for ϕX, T0, we also define

    OT(ϕ):={ut(ϕ):tT}.

    Let ω(ϕ) be the ω-limit set of ϕ for U(t) and Im the set {1,2,,m} for mN+.

    The following Definition 2.1 and Theorem 2.1 (see, e.g., [6,Theorem 5.3.1], [11,Theorem 2.5.3]) can be utilized in dynamics analysis of lots of biomathematical models in the form of system (2.1).

    Definition 2.1. We call L:CR is a Lyapunov functional on X for system (2.1) if

    (ⅰ) L is continuous on ¯X,

    (ⅱ) ˙L0 on X.

    Theorem 2.1 (Lyapunov-LaSalle theorem [11]). Let L be a Lyapunov functional on X and ut(ϕ) be a bounded solution of system (2.1) that stays in X, then ω(ϕ)M, where M be the largest invariant set for system (2.1) in E={ϕ¯X.

    In Theorem 2.1, a Lyapunov functional L on X occupies a decisive position, usually, X is positively invariant with respect to system (2.1). If the condition (ⅰ) in Definition 2.1 is changed into the conditions that the functional L is continuous on X and L is not continuous at φ¯X implies limψφ,ψXL(ψ)=, then the conclusion of Theorem 2.1 still holds, refer to the slightly modified version of Lyapunov-LaSalle theorem (see [9,15,19]). In fact, for analysing the dynamics of many mathematical models with biological background, such functional L in the modified Lyapunov-LaSalle theorem is often used and has become more and more popular. For instance, the functional L constructed with a logarithm (such as ϕi(0)1lnϕi(0) for some iIn), is defined on

    X={ϕ=(ϕ1,ϕ2,,ϕn)TC:ϕi(0)>0}, (2.2)

    which can ensure ui(t,ϕ) is persistent, that is, lim inftui(t,ϕ)>σϕ, where σϕ is some positive constant (see, e.g., [8,12]%). Clearly, this functional L is not continuous on ¯X since limφi(0)0+,φXL(φ)=. Thus, this does not meet the conditions of Theorem 2.1, but it satisfies the conditions of the modified Lyapunov-LaSalle theorem.

    However, we will assume that L is defined on a much smaller subset of X. On the phase space X, we can not ensure that ˙L(ut)0 for all t0. Thus, for each ϕX, we construct such a bounded subset OT(ϕ):={ut(ϕ):tT} of X for sufficiently large T=T(ϕ). Now, we are in position to give the following Corollary 2.1 equivalent to Theorem 2.1.

    Corollary 2.1. Let the solution ut(ϕ) of system (2.1) with ϕX be bounded (if and only if the set O is precompact). If there exists T=T(ϕ)0 such that L is a Lyapunov functional on OT(ϕ), then ˙L=0 on ω(ϕ).

    Proof. It is clear that if O is precompact, ut(ϕ) is bounded. Suppose that ut(ϕ)<Uϕ, t0. Clearly, O0(ϕ) is uniformly bounded. Since g is completely continuous, ˙u(t) is bounded. Hence, u(t) is uniformly continuous on [τ,), from which, it follows that O0(ϕ) is equi-continuous. By Arzelà-Ascoli theorem, we thus have that O0(ϕ) is precompact. Due to L is a Lyapunov functional on OT(ϕ) and ¯OT(ϕ) is compact, L(ut(ϕ)) is decreasing and bounded on [T,). Thus, there exists some k< such that limtL(ut(ϕ))=L(φ)=k for all φω(ϕ). Thus, ˙L=0 on ω(ϕ).

    Remark 2.1. It is not difficult to find that in the modified Lyapunov-LaSalle theorem (see, e.g., [9,15,19]), if L is not continuous on ¯X, then ¯OT(ϕ)X, thus, Corollary 2.1 is an extension of Theorem 2.1 and its modified version.

    Remark 2.2. In fact, we can see that a bounded OT(ϕ) is positively invariant for system (2.1), L is a Lyapunov functional on it, and then ˙L=0 on ω(ϕ) follows from Theorem 2.1. Hence, Corollary 2.1 is also an equivalent variant of Theorem 2.1.

    From Corollary 2.1, we may consider the global properties of system (2.1) on the larger space than X. For example, for such functional L constructed with a logarithm, we can always think about the global properties of the corresponding model in more larger pace X={ϕC:ϕi (povided that ui(t,ϕ)>0 for t>0 and ϕX) than (2.2).

    Let X be positively invariant for system (2.1) and E denote the point (E1,E2,En)TRn. Then we have the following results for the global stability for system (2.1), which are the implementations of Corollary 2.1.

    Theorem 3.1. Suppose that the following conditions hold:

    (ⅰ) Let ut(ϕ) be a bounded solution of system (2.1) with any ϕX. Then there exists T=T(ϕ)0 such that L is continuous on ¯OT(ϕ)X and for any φ¯OT(ϕ),

    ˙L(φ)w(φ)b(φ), (3.1)

    where wT=(w1,w2,,wk)TC(¯O,Rk), b=(b1,b2,,bk)TC(¯OT(ϕ),Rk+), k1.

    (ⅱ) There exist ki=ki(ϕ)=(ki1 such that for any φω(ϕ), there hold

    k1φk2, w(φ)(w01,w02,,w0k)w0=w0(k1,k2)0,

    and w0b(φ)=0 implies that for each jIn, φj(θ)=Ej for some θ=θ(j)[τ,0].

    Then E is globally attractive for U(t).

    Proof. To obtain E is globally attractive for U(t) in X, we only need to prove ω(ϕ)={E} for any ϕX. Since ut(ϕ) is bounded, w(ut(ϕ)) is also bounded. Let wi(ut(ϕ))=fi(t) for each iIk; then there exist sequences {tim}R+ such that

    lim inftw(ut(ϕ)):=(lim inftw1(ut(ϕ)),lim inftw2(ut(ϕ)),,lim inftwk(ut(ϕ)))=(limmf1(t1m),limmf2(t2m),,limmfk(tkm)).

    For each sequence {tim}, {utim(ϕ)} contains a convergent subsequence; for simplicity of notation let us assume that {utim(ϕ)} is this subsequence and let limmutim. Since wi (iIk) is continuous on ¯OT(ϕ),

    lim inftwi(ut(ϕ))=limmwi(utim(ϕ))=wi(ϕi).

    By the condition (ⅱ), wi(ϕi)w0i>0, iIk. Thus, lim inftw(ut(ϕ))w00. Hence, there exists T1=T1(ϕ)T such that w(ut(ϕ))w0/2 for all tT1. Accordingly, for any φOT1(ϕ),

    ˙L(φ)w(φ)b(φ)w0b(φ)20.

    Hence, L is a Lyapunov functional on OT1(ϕ). By Corollary 2.1, we have that ˙L=0 on ω(ϕ).

    Next, we show that ω(ϕ)={E}. Let ut(ψ) be a solution of system (2.1) with any ψω(ϕ)¯OT(ϕ). Then, from (ⅰ) and the invariance of ω(ϕ), it follows

    ˙L(ut(ψ))w(ut(ψ))b(ut(ψ)), t0.

    By (ⅱ), ˙L(ut(ψ))w0b(ut(ψ))0, t0. Furthermore, uτ(ψ)=E for any ψω(ϕ), and then ω(ϕ)={E}. Thus E is globally attractive for U(t).

    Remark 3.1. By ω(ϕ)={E}, it is clear that E is an equilibrium of system (2.1). In theorem 3.1, a Lyapunov functional L on w(ϕ) implies a Lyapunov functional L on OT(ϕ) for some T=T(ϕ).

    Next, we will give an illustration for Theorem 3.1. Now, we reconsider the global stability for the infection-free equilibrium E0=(x0 (x0=s/d) of the following virus infection model with inhibitory effect proposed in [1],

    {˙x(t)=sdx(t)cx(t)y(t)βx(t)v(t),˙y(t)=eμτβx(tτ)v(tτ)py(t),˙v(t)=ky(t)uv(t), (3.2)

    where x(t), y(t), and v(t) denote the population of uninfected cells, infected cells and viruses at time t, respectively. The positive constant c is the apoptosis rate at which infected cells induce uninfected cells. All other parameters in model (3.2) have the same biological meanings as that in the model of [7].

    In [1], we know E0 is globally asymptotically stable if the basic reproductive number of (3.2) R0=eμτkβx0/pu<1 in the positive invariant set

    G={ϕC([τ,0],R3+):ϕ1x0}C+:=C([τ,0],R3+).

    Indeed, by Theorem 3.1, we can extend the result of [1] to the larger set C+. Thus, we have

    Corollary 3.1. If R0<1, then E0 is globally asymptotically stable in C+.

    Proof. It is not difficult to obtain E0 is locally asymptotically stable. Thus, we only need to prove that E0 is globally attractive in C+. With the aid of the technique of constructing Lyapunov functional in [3], we define the following functional:

    L(ϕ)=ϕ1(0)x0x0lnϕ1(0)x0+a1ϕ2(0)+a1eμτ0τβϕ1(θ)ϕ3(θ)dθ+a2ϕ3(0), (3.3)

    where

    a1=2(kβx0+ucx0)pueμτkβx0,a2=2(pβx0+eμτcβx20)pueμτkβx0.

    Let ut=(xt,yt,vt)T be the solution of (3.2) with any ϕC+. From [1,Lemma 2.1] and its proof, it follows ω(ϕ)G. Thus, for φω(ϕ), we have

    w(φ)(dφ1(0),a1pa2kcx0,a2ua1eμτβφ1(0)βx0)(dx0,a1pa2kcx0,a2ua1eμτβx0βx0)=(dx0,cx0,βx0)w00,

    where c,β>0 and φ1(0)x0 are used. Let b(φ)((x0φ1(0))2,φ2(0),φ3(0))T. Then w0b(φ)=0 implies φ(0)=E0.

    The derivative of L1 along this solution ut of (3.2) for tτ is given as

    ˙L1(ut)=d(x0x(t))(1x0x(t))+x0(cy(t)+βv(t))x(t)(cy(t)+βv(t))+a1eμτβx(t)v(t)a1py(t)+a2ky(t)a2uv(t)dx(t)(x0x(t))2(a1pa2kcx0)y(t)(a2ua1eμτβx(t)βx0)v(t)=w(ut)b(ut).

    Therefore, it follows from Theorem 3.1 that E0 is globally attractive in C+.

    In [3,Theorem 3.1], the infection-free equilibrium E0 is globally asymptotically stable in G under some conditions. By Theorem 3.1, we can show that E0 is globally asymptotically stable in C+. The global dynamical properties of the model in [10] have been discussed in the positive invariant set ΩC+, thus we can discuss them in the larger set C+.

    Theorem 3.2. In the condition (ii) of Theorem 3.1, if the condition that w0b(φ)=0 implies that for each jIn, φj(θ)=Ej for some θ=θ(j)[τ,0] is replaced by the condition that w0b(φ)=0 implies that φj(θ)=EjR for some jIn and some θ=θ(j)[τ,0], then limtujt(ϕ)=Ej for any ϕX.

    Proof. In the foundation of the similar argument as in the proof of Theorem 3.1, we have that ˙L=0 on ω(ϕ). Let ut(ψ) be a solution of system (2.1) with any ψω(ϕ). Then it follows from the invariance of ω(ϕ) that ut(ψ)ω(ϕ) for any tR, and

    ˙L(ut(ψ))w0b(ut(ψ))0.

    Hence, ujt(ψ)=Ej for tR, and then ψj Therefore, limt for any ϕX.

    Next, by using Theorem 3.2, we will give the global stability of the equilibrium E0 of (1.1) under certain conditions. By (1.3),

    ˙L(ut)w(ut)b(ut), (3.4)

    where

    w(ut)=1+h2zt(0)r=1+h2z(t)r,b(ut)=yt(0)=y(t).

    Let

    p(t)=rh1xt(τ)+yt(0)=rh1x(tτ)+y(t), tτ.

    Then we have ˙p(t)r/h1p(t), and it holds lim supty(t)r/h1. Hence, it follows from the first and the third equations of (1.1) that

    lim inftx(t)1r+1, lim inftz(t)h1h1+rh3. (3.5)

    Thus, for any φω(ϕ), there hold

    (1/(r+1),0,h1/(h1+rh3))Tφ(1,r/h1,1)T,w(φ)=1+h2φ3(0)r1+h1h2/(h1+rh3)rw0>0,

    and w0b(φ)=w0φ2(0)=0 implies φ2(0)=0. Therefore, it follows from Theorem 3.2 limtu2t(ϕ)=limtyt=0. The first and the third equations of (1.1) together with the invariance of w(ϕ) yield that w(ϕ)={E0} for any ϕG. Thus, E0 is globally stable with the local stability of E0 if 1+h1h2/(h1+rh3)>r.

    Thus, we only need to obtain the solutions of a system are bounded and then may establish the upper- and lower-bound estimates of ω-limit set of this system. Thereupon, for ϕX, we can identify its ω-limit set ω(ϕ) by considering a Lyapunov functional L on the orbit through ϕ after some large time T=T(ϕ). In consequence, by Theorem 3.2, the global stability result of the equilibrium E0 of (1.1) in G can also be extended to the larger set C+.

    Corollary 3.2. Let a:RnR+ be continuous and a(s)(|s|), and let OT(ϕ):={ut(ϕ):t[T,εϕ)}, T[0,εϕ), where [0,εϕ)(εϕ>τ) is the maximal interval of existence of u(t,ϕ). If there exists T=T(ϕ)(0,εϕ) such that L is continuous on OT(ϕ), and for any φOT(ϕ),

    a(φ(0))L(φ), ˙L(φ)w0b(φ), 0wT0Rk, (3.6)

    where bC(OT(ϕ),Rk+), then ut(ϕ) is a bounded solution of system (2.1) with any ϕX. In addition, if L is continuous on ¯OT(ϕ)X, (3.6) holds for any φ¯OT(ϕ), and w0b(φ)=0 implies that for each jIn, φj(θ)=Ej for some θ=θ(j)[τ,0], then E is globally attractive for U(t).

    Proof. Since

    a(u(t,ϕ))L(ut(ϕ))L(uT(ϕ)), t[T,εϕ),

    and the fact that a(s) (|s|), ut(ϕ) is bounded on [0,εϕ), and then εϕ=. Thus, Corollary 3.2 follows from Theorem 3.1.

    Corollary 3.3. Assume that E is an equilibrium of system (2.1). Let a:R+R+ be continuous and increasing, a(s)>0 for s>0, a(0)=0, and limsa(s), and let OT(ϕ)={ut, where [0,εϕ) (εϕ>τ) is the maximal interval of existence of u(t,ϕ). If there exists T(0,εϕ) which is independent of ϕ such that L is continuous on OT(ϕ) and uT(X), respectively, and for any φOT(ϕ),

    a(|φ(0)E|)L(φ), ˙L(φ)w0b(φ), 0wT0Rk, (3.7)

    where bC(OT(ϕ),Rk+) and L(E)=0, then ut(ϕ) is a bounded solution of system (2.1) with any ϕX and E is uniformly stable. In addition, if L is continuous on ¯OT(ϕ)X, (3.7) holds for any φ¯OT(ϕ), and w0b(φ)=0 implies that for each jIn, φj(θ)=EjR for some θ=θ(j)[τ,0], then E is globally asymptotically stable for U(t).

    Proof. It follows from Corollary 3.2 that the boundedness of ut(ϕ) and the global attractivity of E are immediate. Thus, we only need to prove E is uniformly stable. By the continuity of solutions with respect to the initial data for compact intervals, for any ε>0, there exists δ1>0 such that

    ut(ϕ)B(ut(E),ε)=B(E,ε),

    where ϕB(E,δ1)X and t[0,T]. Hence, |u(t,ϕ)E|<ε for any t[0,T]. Since L is continuous on uT(X) and L(E)=0, there exists δ2>0 such that for any ϕB(E,δ2)X, L(uT(ϕ))<a(ε). Accordingly, from (3.7), it follows

    a(|u(t,ϕ)E|)L(ut(ϕ))L(uT(ϕ))<a(ε),

    which yields |u(t,ϕ)E|<ε for any tT. Let δ=min{δ1,δ2}. Then for any ϕB(E,δ)X, it follows |u(t,ϕ)E|<ε for any t0. Therefore, E is uniformly stable.

    Lemma 3.1. ([13,Lemma 1.4.2]) For any infinite positive definite function cC(R with respect to origin, there must exist a radially unbounded class K (Kamke) function aC(R+,R+) such that a(|x|)c(x).

    By Lemma 3.1, we have the following remark.

    Remark 3.2. If there exists an infinite positive definite function cC(R with respect to E such that c(φ(0))L(φ), then there is such function a in Corollary 3.3, or rather there is a radially unbounded class K function aC(R+,R+) such that a(|φ(0)E|)c(φ(0))L(φ).

    Corollary 3.4. In Corollary 3.2, if the condition w0b(φ)=0 implies that for each jIn, φj(θ)=EjR for some θ=θ(j)[τ,0] is replaced by the condition w0b(φ)=0 implies that φj(θ)=Ej for some jIn and some θ=θ(j)[τ,0], then limtujt(ϕ)=Ej for any ϕX.

    For a dissipative system (2.1), we will give the upper- and lower-bound estimates of M:=ϕXω(ϕ). To this end, we need the following Lemma 3.2.

    Lemma 3.2. Let Q¯X be a precompact invariant set. Then ¯Q is a compact invariant set.

    Proof. For any ϕ¯Q, there is a sequence {ϕn}Q such that limnϕn=ϕ. Consider that ut(ϕn)Q, and ut(ϕ) is continuous in ϕ,t for ϕ¯X and t0. Thus, for any t0, limnut(ϕn)=ut ¯Q. Consequently, ut(¯Q)¯Q for all t0. Since for any t0, there can be found ψnQ such that ut(ψn)=ϕn, ψnQ. By the compactness of ¯Q (since Q is precompact), the sequence {ψn} contains a convergent subsequence; in order not to complicate the notation, we still assume that {ψn} is this subsequence and let limnψn=ψ¯Q. Thus, limnut(ψn)=ut(ψ)=ϕut(¯Q), which shows that ¯Qut(¯Q). Therefore, ¯Q=ut(¯Q) for any t0.

    Theorem 3.3. Suppose that there exist k1,k2Rn such that

    k1lim inftut(ϕ)(θ)lim suptut(ϕ)(θ)k2, ϕX, θ[τ,0], (3.8)

    where

    lim inftut(ϕ)(θ):=(lim inftu1t(ϕ)(θ),,lim inftunt(ϕ)(θ))T,lim suptut(ϕ)(θ):=(lim suptu1t(ϕ)(θ),,lim suptunt(ϕ)(θ))T.

    Then ¯M¯X is compact, and ¯M

    Proof. Clearly, ¯M{φ¯X:k1φk2}. The fact O0(ϕ) is precompact for any ϕX follows from (3.8) and Corollary 2.1. Hence, for any φM, k1. Obviously, M is uniformly bounded. Since g is completely continuous and ut(M)=M, there exists M1>0 such that

    |˙u(t,φ)|M1, t0, φM.

    It follows from the invariance of M that for any φM and any tτ, there exists ψM such that ut(ψ)=φ, it can be shown that M is a precompact invariant set. Accordingly, ¯M is a compact invariant set from Lemma 3.2. Thus ¯M is compact in ¯X.

    In this paper, we first give a variant of Theorem 2.1, see Corollary 2.1. In fact, the modified version of Lyapunov-LaSalle theorem (see, e.g., [9,15,19]) is to expand the condition (ⅰ) of Definition 2.1, while Corollary 2.1 is mainly to expand the condition (ⅱ) of Definition 2.1. More specifically, we assume that L is defined on a much smaller subset of X, indeed, we only need L is a Lyapunov functional on OT(ϕ), where ut(ϕ) is bounded and T=T(ϕ) is a nonnegative constant. In such a way, we may consider the global properties of system (2.1) on the larger space than X and even consider one of system (2.1) on the much lager C (or the nonnegative cone C+ of C).

    As a result, the criteria for the global attractivity of equilibria of system (2.1) are given in Theorem 3.1 and Theorem 3.2, respectively. As direct consequences, we also give the corresponding particular cases of Theorem 3.1 and Theorem 3.2, see Corollaries 3.2, 3.3 and 3.4, respectively. The developed theory can be utilized in many models (see, e.g., [2,3,9,10,14]). The compactness and the upper- and lower-bound estimates of M for a dissipative system (2.1) are given in Theorem 3.3. Further, by employing the results of this paper, the recent research results in [3,10,16,17,18,20,21] can be extended and some techniques in [4,5] can be simplified.

    This work was supported in part by the General Program of Science and Technology Development Project of Beijing Municipal Education Commission (No. KM201910016001), the Fundamental Research Funds for Beijing Universities (Nos. X18006, X18080 and X18017), the National Natural Science Foundation of China (Nos. 11871093 and 11471034). The authors would like to thank Prof. Xiao-Qiang Zhao for his valuable suggestions.

    The authors declare there is no conflict of interest in this paper.



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