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Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent incidence function in a heterogeneous environment

  • Received: 23 June 2019 Accepted: 06 September 2019 Published: 14 October 2019
  • In this paper, we consider a diffusive SIS epidemic reaction-diffusion model with linear source in a heterogeneous environment in which the frequency-dependent incidence function is SI/(c + S + I) with c a positive constant. We first derive the uniform bounds of solutions, and the uniform persistence property if the basic reproduction number R0>1. Then, in some cases we prove that the global attractivity of the disease-free equilibrium and the endemic equilibrium. Lastly, we investigate the asymptotic profile of the endemic equilibrium (when it exists) as the diffusion rate of the susceptible or infected population is small. Compared to the previous results [1, 2] in the case of c = 0, some new dynamical behaviors appear in the model studied here; in particular, R0 is a decreasing function in c∈[0, ∞) and the disease dies out once c is properly large. In addition, our results indicate that the linear source term can enhance the disease persistence.

    Citation: Jinzhe Suo, Bo Li. Analysis on a diffusive SIS epidemic system with linear source and frequency-dependent incidence function in a heterogeneous environment[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 418-441. doi: 10.3934/mbe.2020023

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  • In this paper, we consider a diffusive SIS epidemic reaction-diffusion model with linear source in a heterogeneous environment in which the frequency-dependent incidence function is SI/(c + S + I) with c a positive constant. We first derive the uniform bounds of solutions, and the uniform persistence property if the basic reproduction number R0>1. Then, in some cases we prove that the global attractivity of the disease-free equilibrium and the endemic equilibrium. Lastly, we investigate the asymptotic profile of the endemic equilibrium (when it exists) as the diffusion rate of the susceptible or infected population is small. Compared to the previous results [1, 2] in the case of c = 0, some new dynamical behaviors appear in the model studied here; in particular, R0 is a decreasing function in c∈[0, ∞) and the disease dies out once c is properly large. In addition, our results indicate that the linear source term can enhance the disease persistence.


    Let A denote the class of functions of the form

    f(z)=z+a2z2+a3z3+a4z4+, (1.1)

    which are analytic in the open unit disk D=(z:∣z∣<1) and normalized by f(0)=0 and f(0)=1. Recall that, SA is the univalent function in D=(z:∣z∣<1) and has the star-like and convex functions as its sub-classes which their geometric condition satisfies Re(zf(z)f(z))>0 and Re(1+zf(z)f(z))>0. The two well-known sub-classes have been used to define different subclass of analytical functions in different direction with different perspective and their results are too voluminous in literature.

    Two functions f and g are said to be subordinate to each other, written as fg, if there exists a Schwartz function w(z) such that

    f(z)=g(w(z)),zϵD (1.2)

    where w(0) and w(z)∣<1 for zϵD. Let P denote the class of analytic functions such that p(0)=1 and p(z)1+z1z, zϵD. See [1] for details.

    Goodman [2] proposed the concept of conic domain to generalize convex function which generated the first parabolic region as an image domain of analytic function. The same author studied and introduced the class of uniformly convex functions which satisfy

    UCV=Re{1+(zψ)f(z)f(z)}>0,(z,ψA).

    In recent time, Ma and Minda [3] studied the underneath characterization

    UCV=Re{1+zf(z)f(z)>|zf(z)f(z)|},zϵD. (1.3)

    The characterization studied by [3] gave birth to first parabolic region of the form

    Ω={w;Re(w)>∣w1}, (1.4)

    which was later generalized by Kanas and Wisniowska ([5,6]) to

    Ωk={w;Re(w)>kw1,k0}. (1.5)

    The Ωk represents the right half plane for k=0, hyperbolic region for 0<k<1, parabolic region for k=1 and elliptic region for k>1 [30].

    The generalized conic region (1.5) has been studied by many researchers and their interesting results litter everywhere. Just to mention but a few Malik [7] and Malik et al. [8].

    More so, the conic domain Ω was generalized to domain Ω[A,B], 1B<A1 by Noor and Malik [9] to

    Ω[A,B]={u+iv:[(B21)(U2+V2)2(AB1)u+(A21)]2
    >[2(B+1)(u2v2)+2(A+B+C)u2(A+1)]2+4(AB)2v2}

    and it is called petal type region.

    A function p(z) is said to be in the class UP[A,B], if and only if

    p(z)(A+1)˜p(z)(A1)(B+1)˜p(z)(B1), (1.6)

    where ˜p(z)=1+2π2(log1+z1z)2.

    Taking A=1 and B=1 in (1.8), the usual classes of functions studied by Goodman [1] and Kanas ([5,6]) will be obtained.

    Furthermore, the classes UCV[A,B] and ST[A,B] are uniformly Janoski convex and Starlike functions satisfies

    Re((B1)(zf(z))f(z)(A1)(B+1)(zf(z))f(z)(A+1))>|(B1)(zf(z))f(z)(A1)(B+1)(zf(z))f(z)(A+1)1| (1.7)

    and

    Re((B1)zf(z)f(z)(A1)(B+1)zf(z)f(z)(A+1))>|(B1)zf(z)f(z)(A1)(B+1)zf(z)f(z)(A+1)1|, (1.8)

    or equivalently

    (zf(z))f(z)UP[A,B]

    and

    zf(z)f(z)UP[A,B].

    Setting A=1 and B=1 in (1.7) and (1.8), we obtained the classes of functions investigated by Goodman [2] and Ronning [10].

    The relevant connection to Fekete-Szegö problem is a way of maximizing the non-linear functional |a3λa22| for various subclasses of univalent function theory. To know much of history, we refer the reader to [11,12,13,14] and so on.

    The error function was defined because of the normal curve, and shows up anywhere the normal curve appears. Error function occurs in diffusion which is a part of transport phenomena. It is also useful in biology, mass flow, chemistry, physics and thermomechanics. According to the information at hand, Abramowitz [15] expanded the error function into Maclaurin series of the form

    Erf(z)=2πz0et2dt=2πn=0(1)nz2n+1(2n+1)n! (1.9)

    The properties and inequalities of error function were studied by [16] and [4] while the zeros of complementary error function of the form

    erfc(z)=1erf(z)=2πzet2dt, (1.10)

    was investigated by [17], see for more details in [18,19] and so on. In recent time, [20,21,22] and [23] applied error functions in numerical analysis and their results are flying in the air.

    For f given by [15] and g with the form g(z)=z+b2z2+b3z3+ their Hadamard product (convolution) by fg and at is defined as:

    (fg)(z)=z+n=2anbnzn (1.11)

    Let Erf be a normalized analytical function which is obtained from (1.9) and given by

    Erf=πz2erf(z)=z+n=2(1)n1zn(2n1)(n1)! (1.12)

    Therefore, applying a notation (1.11) to (1.1) and (1.12) we obtain

    ϵ=AErf={F:F(z)=(fErf)(z)=z+n=2(1)n1anzn(2n1)(n1)!,fA}, (1.13)

    where Erf is the class that consists of a single function or Erf. See concept in Kanas et al. [18] and Ramachandran et al. [19].

    Babalola [24] introduced and studied the class of λpseudo starlike function of order β(0β1) which satisfy the condition

    Re(z(f(z))λf(z))>β, (1.14)

    where λ1(zD) and denoted by λ(β). We observed from (1.14) that putting λ=2, the geometric condition gives the product combination of bounded turning point and starlike function which satisfy

    Ref(z)(z(f(z))f(z))>β

    Olatunji [25] extended the class λ(β) to βλ(s,t,Φ) which the geometric condition satisfy

    Re((st)z(f(z))λf(sz)f(tz))>β,

    where s,tC,st,λ1,0β<1,zD and Φ(z) is the modified sigmoid function. The initial coefficient bounds were obtained and the relevant connection to Fekete-Szegö inequalities were generated. The contributions of authors like Altinkaya and Özkan [26] and Murugusundaramoorthy and Janani [27] and Murugusundaramoorthy et al. [28] can not be ignored when we are talking on λ-pseudo starlike functions.

    Inspired by earlier work by [18,19,29]. In this work, the authors employed the approach of [13] to study the coefficient inequalities for pseudo certain subclasses of analytical functions related to petal type region defined by error function. The first few coefficient bounds and the relevant connection to Fekete-Szegö inequalities were obtained for the classes of functions defined. Also note that, the results obtained here has not been in literature and varying of parameters involved will give birth to corollaries.

    For the purpose of the main results, the following lemmas and definitions are very necessary.

    Lemma 1.1. If p(z)=1+p1z+p2z2+ is a function with positive real part in D, then, for any complex μ,

    |p2μp21|2max{1,|2μ1|}

    and the result is sharp for the functions

    p0(z)=1+z1zorp(z)=1+z21z2(zD).

    Lemma 1.2. [29] Let pUP[A,B],1B<A1 and of the form p(z)=1+n=1pnzn. Then, for a complex number μ, we have

    |p2μp21|4π2(AB)max(1,|4π2(B+1)23+4μ(ABπ2)|). (1.15)

    The result is sharp and the equality in (1.15) holds for the functions

    p1(z)=2(A+1)π2(log1+z1z)2+22(B+1)π2(log1+z1z)2+2

    or

    p2(z)=2(A+1)π2(log1+z1z)2+22(B+1)π2(log1+z1z)2+2.

    Proof. For hP and of the form h(z)=1+n=1cnzn, we consider

    h(z)=1+w(z)1w(z)

    where w(z) is such that w(0)=0 and |w(z)|<1. It follows easily that

    w(z)=h(z)1h(z)+1=12z+(c22c214)z2+(c32c2c12+c318)z3+ (1.16)

    Now, if ˜p(z)=1+R1z+R2z2+, then from (1.16), one may have,

    ˜p(w(z))=1+R1w(z)+R2(w(z))2+R3(w(z))3 (1.17)

    where R1=8π2,R2=163π2, and R3=18445π2, see [30]. Substitute R1,R2 and R3 into (1.17) to obtain

    ˜p(w(z))=1+4c1π2z+4π2(c2c216)z2+4π2(c3c1c23+2c3145)z3+ (1.18)

    Since pUP[A,B], so from relations (1.16), (1.17) and (1.18), one may have,

    p(z)=(A+1)˜p(w(z))(A1)(B+1)˜p(w(z))(B1)=2+(A+1)4π2c1z+(A+1)4π2(c2c216)z2+2+(B+1)4π2c1z+(B+1)4π2(c2c216)z2+

    This implies that,

    p(z)=1+2(AB)c1π2z+2(AB)π2(c2c2162(B1)c21π2)z2+8(AB)π2[((B+1)2π4+B+16π2190)c21(B+1π2+112)c1c2+c34]z3+ (1.19)

    If p(z)=1+n=1pnzn, then equating coefficients of z and z2, one may have,

    p1=2π2(AB)c1

    and

    p2=2π2(AB)(c2c2162(B1)c21π2).

    Now for a complex number μ, consider

    p2μp21=2(AB)π2[c2c21(16+2(B+1)π2+2μ(AB)π2)]

    This implies that

    |p2μp21|=2(AB)π2|c2c21(16+2(B+1)π2+2μ(AB)π2)|.

    Using Lemma 1.1, one may have

    |p2μp21|=4(AB)π2max{1,|2v1|},

    where v=16+2(B+1)π2+2μ(AB)π2, which completes the proof of the Lemma.

    Definition 1.3. A function FϵA is said to be in the class UCV[λ,A,B], 1B<A1, if and only if,

    Re((B1)(z(F(z)λ))F(z)(A1)(B+1)(z(F(z)λ))F(z)(A+1))>|(B1)(z(F(z)λ))F(z)(A1)(B+1)(z(F(z)λ))F(z)(A+1)1|, (1.20)

    where λ1ϵR or equivalently (z(F(z)λ))F(z)ϵUP[A,B].

    Definition 1.4. A function FϵA is said to be in the class US[λ,A,B], 1B<A1, if and only if,

    Re((B1)z(F(z)λ)F(z)(A1)(B+1)z(F(z)λ)F(z)(A+1))>|(B1)z(F(z)λ)F(z)(A1)(B+1)z(F(z)λ)F(z)(A+1)1|, (1.21)

    where λ1ϵR or equivalently z(F(z)λ)F(z)ϵUP[A,B].

    Definition 1.5. A function FϵA is said to be in the class UMα[λ,A,B], 1B<A1, if and only if,

    Re((B1)[(1α)z(F(z)λ)F(z)+α(z(F(z)λ))F(z)](A1)(B+1)[(1α)z(F(z)λ)F(z)+α(z(F(z)λ))F(z)](A+1))>|(B1)[(1α)z(F(z)λ)F(z)+α(z(F(z)λ))F(z)](A1)(B+1)[(1α)z(F(z)λ)F(z)+α(z(F(z)λ))F(z)](A+1)1|,

    where α0 and λ1ϵR or equivalently (1α)z(F(z)λ)f(z)+α(z(f(z)λ))f(z)UP[A,B].

    In this section, we shall state and prove the main results, and several corollaries can easily be deduced under various conditions.

    Theorem 2.1. Let FUS[λ,A,B], 1B<A1, and of the form (1.13). Then, for a real number μ, we have

    |a3μa22|40(AB)|13λ|π2max{1,|4(B+1)π2132(AB)(12λ)2π2(2(2λ24λ+1)9μ(13λ)5)|}.

    Proof. If FUS[λ,A,B], 1B<A1, the it follows from relations (1.18), (1.19), and (1.20),

    z(F(z)λ)F(z)=(A+1)˜p(w(z))(A1)(B+1)˜p(w(z))(B1),

    where w(z) is such that w(0)=0 and w(z)∣<1. The right hand side of the above expression get its series form from (1.13) and reduces to

    z(F(z)λ)F(z)=1+2(AB)c1π2z+2(AB)π2(c2c2162(B1)c21π2)z2
    +8(AB)π2[((B+1)2π4+B+16π2190)c21(B+1π2+112)c1c2+c34]z3+. (2.1)

    If F(z)=z+n=2(1)n1anzn(2n1)(n1)!, then one may have

    z(F(z)λ)F(z)=1+12λ3a2z+(2λ24λ+19a2213λ10a3)z2+ (2.2)

    From (2.1) and (2.2), comparison of coefficient of z and z2 gives,

    a2=6(AB)(12λ)π2c1 (2.3)

    and

    2λ24λ+19a2213λ10a3=2(AB)π2(c216c212(B+1)π2c21).

    This implies, by using (2.3), that

    a3=20(AB)(13λ)π2[c216c212(B+1)π2c212(2λ24λ+1)(AB)(12λ)2π2c21].

    Now, for a real number μ consider

    |a3μa22|=
    |20(AB)(13λ)π2(c216c212(B+1)π2c21)+40(AB)2(2λ24λ+1)(12λ)2(13λ)π436μ(AB)2c21(12λ)2π4|
    =20(AB)(13λ)π2|c2c21(16+2(B+1)π22(AB)(2λ24λ+1)(12λ)2π2+9μ(AB)(13λ)5(12λ)2π2)|
    =20(AB)(13λ)π2|c2vc21|

    where v=16+2(B+1)π2(AB)(12λ)2π2(2(2λ24λ+1)9μ(13λ)5).

    Theorem 2.2. Let FUCV[λ,A,B], 1B<A1, and of the form (1.13). Then, for a real number μ, we have

    |a3μa22|40(AB)3|1+3λ|π2max{1,|4(B+1)π2132(1+3λ)(AB)(1+2λ)2π2(λ27μ20)|}

    Proof. If FUCV[λ,A,B], 1B<A1, then it follows from relations (1.18), (1.19), and (1.21),

    (zF(z)λ)F(z)=(A+1)˜p(w(z))(A1)(B+1)˜p(w(z))(B+1),

    where w(z) is such that w(0)=0 and w(z)∣<1. The right hand side of the above expression get its series form from (1.13) and reduces to,

    (zF(z)λ)F(z)=1+2(AB)c1π2z+2(AB)π2(c2c2162(B+1)π2c21)z2+8(AB)π2[(B+1π4+B+16π2+190)c31(B+1π2+112)c1c2+c34]z3+ (2.4)

    If F(z)=z+(1)n1anzn(2n1)(n1)!, then we have,

    (zF(z)λ)F(z)=12(1+2λ)3a2z+(1+3λ)(3a310+2λ9a22)z2+ (2.5)

    From (2.4) and (2.5), comparison of coefficients of z and z2 gives,

    a2=3(AB)c1(1+2λ)π2 (2.6)

    and

    (1+3λ)(3a310+2λ9a22)=2(AB)π2(c2c2162(B+1)c21π2)

    This implies, by using (2.6), that

    a3=103[2(AB)(1+3λ)π2(c2c2162(B+1)c21π2)+2λ(AB)2c21(1+2λ)2π4].

    Now, for a real number μ, consider

    |a3μa22|=|20(AB)3(1+3λ)π2(c216c12(B+1)π2c21)+20(AB)2c213(1+2λ)π49μ(AB)2c21(1+2λ)2π4|
    =20(AB)3(1+3λ)π2|c2c21(16+2(B+1)π2λ(1+3λ)(AB)(1+2λ)2π2+27μ(AB)(1+3λ)20(1+2λ)2π2)|
    =20(AB)3(1+3λ)π2|c2vc21|,

    where

    v=16+2(B+1)π2(1+3λ)(AB)(1+2λ)2π2(λ27μ20).

    Theorem 2.3. FMα[λ,A,B], 1B<A1, α0 and of the form (1.13). Then, for a real number μ, we have

    |a3μa22|40(AB)π2|3(λ+α+2αλ)+α1|max{1,|4(B+1)π2134(AB)[12λα(3+2λ)]2π2(2λ2(1+2α)+2λ(3α2)+1α9μ(3(λ+α+2αλ)+α1)10)|}.

    Proof. Let FMα[λ,A,B], 1B<A1, α0 and of the form (1.13). Then, for a real number μ, we have

    (1α)z(F(z))λF(z)+α(z(F(z))λ)F(z)=(A+1)˜p(w(z))(A1)(B+1)˜p(w(z))(B1), (2.7)

    where w(z) is such that w(z0)=0 and |w(z)|<1. The right hand side of the above expression get its series form from (2.7) and reduces to

    (1α)z(F(z))λF(z)+α(z(F(z))λ)F(z)=1+2(AB)Gπ2z+2(AB)π2(c2c2162(B+1)π2c21)z2+... (2.8)

    If F(z)=z+n=2(1)n1anzn(2n1)(n1)!, then one may have

    (1α)z(F(z))λF(z)+α(z(F(z))λ)F(z)=(1α)[1+12λ3a2z+(2λ24λ+19a2213λ10a3)z2+...]+α[12(1+2λ)3a2z+(1+3λ)(3a310+2λ9a22)z2+...] (2.9)

    from (2.8) and (2.9), comparison of coefficients of z and z2 gives

    a2=6(AB)c1[12λα(3+2λ)]π2 (2.10)

    and

    3(λ+α+2αλ)+α110a32λ2(1+2λ)+α19a22=2(AB)π2(c2c2162(B+1)π2c21)

    This implies, by using (2.10), that

    a3=103(λ+α+2αλ)+α1[2(AB)π2(c2c2162(B+1)π2c21)+4(AB)2[2λ2(1+2λ)+2λ(3α2)+1α][12λα(3+2λ)]2π4c21]

    Now, for a real number μ, consider

    |a3μa22|=|103(λ+α+2αλ)+α1[2(AB)π2(c2c2162(B+1)π2c21)+4(AB)2[2λ2(1+2λ)+2λ(3α2)+1α][12λα(3+2λ)]2π4c21]36(AB)2μG2[12λα(3+2λ)]2π4|
    =|20(AB)π(3(λ+α+2αλ)+α1)|c2c21[16+2(B+1)π22(AB)[2λ2(1+2α)+2λ(3α2)+1α](12λα(3+2λ))2π2+18μ(AB)[3(λ+α+2αλ)+α1]10[12λα(3+2λ)]2π2
    =20(AB)π(3(λ+α+2αλ)+α1)|c2vc21|,

    where

    v=16+2(B+1)π22(AB)[2λ2(1+2α)+2λ(3α2)+1α](12λα(3+2λ))2π2+18μ(AB)[3(λ+α+2αλ)+α1]10[12λα(3+2λ)]2π2.

    The force applied on certain subclasses of analytical functions associated with petal type domain defined by error function has played a vital role in this work. The results obtained are new and varying the parameters involved in the classes of function defined, these will bring new more results that has not been in existence.

    The authors would like to thank the referees for their valuable comments and suggestions.

    The authors declare that they have no conflict of interests.



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