Review Special Issues

Neuropeptidases, Stress, and Memory—A Promising Perspective

  • Received: 18 October 2016 Accepted: 06 December 2016 Published: 16 December 2016
  • Stress has been demonstrated to be a key modulator in learning and memory processes, in which the hippocampus plays a central role. A great number of neuropeptides have been reported to modulate learning and memory under stressful conditions. Neuropeptidases are proteolytic enzymes capable of regulating the function of neuropeptides in the central and peripheral nervous system. In this regard, a number of neuropeptidases, i.e. angiotensinases, oxytocinase, or enkephalinases, have received attention. Their involvement in stress and memory processes is a promising perspective, as it is possible to influence their activities through various activators or inhibitors and, consequently, to pharmacologically modulate the functions of the endogenous substrates that are involved. The present review describes the key findings showing the involvement of neuropeptides and neuropeptidases in stress and memory and highlights the role of the hippocampus in these processes.

    Citation: I. Prieto, A.B. Segarra, M. de Gasparo, M. Ramírez-Sánchez. Neuropeptidases, Stress, and Memory—A Promising Perspective[J]. AIMS Neuroscience, 2016, 3(4): 487-501. doi: 10.3934/Neuroscience.2016.4.487

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  • Stress has been demonstrated to be a key modulator in learning and memory processes, in which the hippocampus plays a central role. A great number of neuropeptides have been reported to modulate learning and memory under stressful conditions. Neuropeptidases are proteolytic enzymes capable of regulating the function of neuropeptides in the central and peripheral nervous system. In this regard, a number of neuropeptidases, i.e. angiotensinases, oxytocinase, or enkephalinases, have received attention. Their involvement in stress and memory processes is a promising perspective, as it is possible to influence their activities through various activators or inhibitors and, consequently, to pharmacologically modulate the functions of the endogenous substrates that are involved. The present review describes the key findings showing the involvement of neuropeptides and neuropeptidases in stress and memory and highlights the role of the hippocampus in these processes.



    The effect of disease on eco-epidemiology system is a significant topic from both mathematical and ecological perspectives. The disease factor usually leads to a more complex and diverting dynamics than those in the disease-free system [1,2]. Within the interactions between predator and prey, the disease could only spread in prey or predator population, also could spread between prey and predator [3,4,5]. Birds (particularly pelicans) infect vibrio and die by preying on vibrio-infected fish (particularly tilapia) at the Salton Sea in the desert of Southern California [3], which is an example of disease spreads amongst the prey. For the disease in predator, taking fox rabies as an example, foxes (Vulpis) infect rabies and transmit to other foxes or their prey rabbits by biting in Europe and North America [6]. More relevant examples could be found in [7]. From the mathematical epidemiology point of view, one needs much more attention in the dynamics of infected predator to observe whether the presence of the prey allows the survival of a part of the predator population [8].

    A variety of diseased predator models have been proposed to study the complex interaction between prey and predator with infected diseases [2,9,10] and the reference therein. Most common epidemic model applied in predator-prey interactions is the SI-type, i.e., the predator population Y(t) is divided into two sub-classes, namely susceptible predator S(t) and infected predator I(t), respectively [10,11,12]. The infection term could be mass-action term (bilinear form) βSI or saturation form βSIS+I [4]. The infected predators usually behave differently with susceptible ones, and suffer an additional death rate. In a epidemic model, the global dynamics are usually determined by the basic reproduction number R0, i.e., the disease will dies out in the population when R01, and the disease will persist in the population when R0>1. However, the basic reproduction number is no longer a threshold parameter determining the global dynamics in diseased predator models, on the contrary, the dynamics are relatively comprehensive and unexpected.

    Predation is the key force in a prey-predator interaction, which could affect the size of prey population by direct hunting [9,13,14,15], and elicit a variety of anti-predator responses [16,17,18]. Consequently, prey tends to alter behaviors in a certain extent, such as change of habitat, foraging activity, vigilance, physiological changes. This anti-predator behaviors accelerate the extinction, evolution and development of prey population in the long run. Under the risk of predation, prey may reduce its foraging activity in order to stay alert, leading to starvation which impacts on population growth [19,20]. Therefore, an immediately result of anti-predator behaviors is the reduction of prey growth rate, which is the cost for prey in prey defense [19,21,22,23,24,25,26].

    Consider a simple birth-death process of the prey X(t) with the cost of anti-predator behaviors [27]:

    dXdt=[F(k,Y)a]XdX,

    where X,Y represent the density of the prey and predator, respectively. a is the birth rate of prey, d is the natural death rate of prey. F(k,Y) accounts for the cost of anti-predator defence due to fear, the parameter k reflects the level of fear which drives anti-predator behaviors of prey. The fear factor F(k,Y) has some specific assumptions under the ecological motions, for details see [20,27].

    To derive a simple diseased predator model incorporating the anti-predator defence due to fear, we adopted the following fear effect term F(k,Y):

    F(k,Y)=11+kY=11+k(S+I).

    Based on the results in [4,9,11], we can obtain the eco-epidemiological system with cost of anti-predator behaviors as following system of nonlinear differential equations:

    {dXdt=rX1+k(S+I)rX2KaXS1+bX,dSdt=eaXS1+bXd1SβSI,dIdt=βSId2I, (1.1)

    where X,S,I represent the density of prey, susceptible predator and infected predator at time t, respectively. r is the intrinsic growth rate of prey, K is the carrying capacity of the prey, a is the predation coefficient, b is the predators handling time of a prey, e is the biomass conversion constant, β is the transmissibility coefficient. d1 and d2 are the mortality rates of the susceptible predator and infected predator, and naturally d1<d2.

    This paper consists of six sections. In the next section, we prove the positivity and boundedness of the solution of system (1.1). In Section 3, we provide the existence conditions of the equilibria of the model. We analyze the stability of equilibria and show the occurrence of Hopf bifurcation in Section 4. In Section 5, the correctness of the theoretical proof is illustrated by numerical simulation. Finally, we summarize our results with ecological interpretations in Section 6.

    In view of the ecological significance, we only consider the solutions (X(t),S(t),I(t)) of system (1.1) on

    R3+={(X(t),S(t),I(t))R3+:X(t)0,S(t)0,I(t)0}.

    Theorem 2.1. Each solution of system (1.1) with initial value (X(0),S(0),I(0))R3+ is positive and ultimately bounded.

    Proof. Since the right-hand side of system (1.1) is completely continuous and locally Lipschitzian on R3+, the solution (X(t),S(t),I(t)) with initial condition (X(0),S(0),I(0))R3+ exists and is unique on R3+.

    By integrating, it follows from system (1.1) that

    X(t)=X(0)exp{t0(r1+k(S(τ)+I(τ))rX(τ)KaS(τ)1+bX(τ))dτ}0,S(t)=S(0)exp{t0(eaX(τ)1+bX(τ)d1βI(τ))dτ}0,I(t)=I(0)exp{t0(βS(τ)d2)dτ}0.

    Hence, the solution (X(t),S(t),I(t)) of system (1.1) with the initial condition (X(0),S(0),I(0))R3+ remains positive.

    From the first equation of (1.1), we can obtain

    dXdt=rX1+k(S+I)rX2KaXS1+bXrXrX2K=rX(1XK),

    then

    lim suptX(t)K.

    Let N(t)=eX(t)+S(t)+I(t), we can get

    dNdt=erX1+k(S+I)erX2Kd1Sd2IerXerX2Kd1Sd2IerX(1XK)+ed1Xd1NeK(r+d1)24rd1N,

    then

    lim suptN(t)eK(r+d1)24rd1.

    This ends the proof.

    Remark 2.2. From Theorem 2.1, we know that all positive solutions of system (1.1) with initial conditions (X(0),S(0),I(0))R3+ are defined in the following positive bounded invariant:

    Γ:={(X(t),S(t),I(t))R3+:0X(t)K,0eX(t)+S(t)+I(t)eK(r+d1)24rd1}.

    System (1.1) possesses at most three boundary equilibria:

    (i) Trivial equilibrium: E0=(0,0,0);

    (ii) Axial equilibrium: E1=(K,0,0);

    (iii) Planar equilibrium: E2=(X2,S2,0) exists if eabd1>0 and K>d1eabd1, where

    X2=d1eabd1,S2=[K(eabd1)2+rd1ke]+[K(eabd1)2rd1ke]2+4K2kre(eabd1)32Kk(eabd1)2. (3.1)

    For epidemic models, the most critical problem is the threshold property for the extinction and persistence of the disease, which is generally governed by the basic reproduction number R0. The basic reproduction number can be interpreted as the expected number of secondary cases produced, in a completely susceptible population, by a typical infected individual during its entire period of infectiousness. Following [28], we define the basic reproduction number for the predator population in the system (1.1) by

    R0:=βS2d2,

    where S2 is given by (3.1).

    Next, we mainly focus on the existence of positive equilibrium E3=(X3,S3,I3) of system (1.1). The coordinates X3,S3,I3 are positive solutions to the following system of equilibrium equations:

    {r1+k(S3+I3)rX3KaS31+bX3=0,eaX31+bX3d1βI3=0,βS3d2=0.

    Thus,

    S3=d2β,I3=X3(eabd1)d1(bX3+1)β,

    and X3 is the positive root of (3.2) in (X2,+):

    Q(X)=m3X3+m2X2+m1X+m0=0, (3.2)

    where

    m3:=bβr(k(eabd1)+b(kd2+β)),m2:=βr(Kb2βk(eabd1)2b(kd2+β)+bkd1),m1:=(kd2a(eabd1)b(akd22+aβd22β2r))Kβr(kd1+kd2+β),m0:=K(ad2(kd1kd2β)β2r).

    If eabd1>0 and r>ad2(β+k(d2d1))β2, we have

    m3<0,m0>0.

    By Descartes' rule of signs, system (1.1) has at least one positive equilibrium E3.

    Hence, we have the following results on the existence of the positive equilibrium. It is worthy to note that the positive equilibrium is not unique due to the impact of fear effect k.

    Theorem 3.1. If eabd1>0 and r>ad2(β+k(d2d1))β2, then system (1.1) has at least one positive equilibrium E3=(X3,S3,I3), where S3=S2R0, I3=X3(eabd1)d1(bX3+1)β and X3 is the positive root of (3.2) in (X2,+).

    Regarding the local stability of trivial equilibrium E0 and axial equilibrium E1, we have the following results. The proof is standard, so we omit it here.

    Theorem 4.1. For system (1.1),

    (i) The trivial equilibrium E0=(0,0,0) is unstable;

    (ii) If one of the following inequalities holds:

    (ii-1) eabd1<0;

    (ii-2) eabd1>0 and K<d1eabd1,

    then the axial equilibrium E1=(K,0,0) is stable; while E1=(K,0,0) is unstable if eabd1>0 and K>d1eabd1.

    Secondly, we will show the local stability of the planar equilibrium E2 of system (1.1). For convenience, set

    r1:=d2(eabd1)βe,r2:=d2(eabd1)(ea+bd1)aβe2,K1:=ea+bd1b(eabd1),K2:=βerd1(eabd1)(βerd2(eabd1)),k1:=Kb(eabd1)2(Kb(eabd1)(ea+bd1))ae2r(ea+bd1),k2:=β((eabd1)(d2(eabd1)βer)K+βerd1)(Kd2(eabd1)2+βerd1)d2. (4.1)

    Theorem 4.2. For system (1.1), assume that eabd1>0. If one of the following inequalities holds:

    (Ⅰ) rr1 and one of the following inequalities holds:

    (Ⅰ-1) d1eabd1<KK1;

    (Ⅰ-2) K>K1 and k>k1;

    (Ⅱ) r1<r<r2 and one of the following inequalities holds:

    (Ⅱ-1) d1eabd1<KK1;

    (Ⅱ-2) K1<K and k>max{k1,k2};

    (Ⅲ) r>r2 and one of the following inequalities holds:

    (Ⅲ-1) d1eabd1<KK2;

    (Ⅲ-2) K2<K and k>max{k1,k2},

    then equilibrium E2 is stable; otherwise, it is unstable.

    Proof. The Jacobian matrix of system (1.1) at E2 is given by

    J2=(a11a12a13a210βS200βS2d2),

    where

    a11:=X2(rK+abS2(1+bX2)2),a12:=krX2(1+kS2)2aX21+bX2,a13:=krX2(1+kS2)2,a21:=eaS2(1+bX2)2.

    Hence, the characteristic equation of J2 is given as

    f(λ)(λβS2+d2)=0, (4.2)

    where

    f(λ):=λ2a11λa12a21.

    Clearly, one can see that J2 has three eigenvalues λ1, λ2 and λ3=βS2d2. Since a12<0,a21>0, then a12a21>0.

    From (3.1), we can obtain

    a11=X2(rK+abS2(1+bX2)2)=X2Φ2Kka2e2,

    where

    Φ:=ab(K(eabd1)2rd1ke)2+4K2kre(eabd1)3ab(K(eabd1)2+rd1ke)2rka2e2.

    Note that the sign of Φ depends on

    ˜Φ:=a2b2((K(eabd1)2rd1ke)2+4K2kre(eabd1)3)(ab(K(aebd1)2+rd1ke)+2rka2e2)2=4a2kerP(k),

    where

    P(k):=ae2r(ea+bd1)k+Kb(eabd1)2(Kb(eabd1)(ea+bd1)).

    One can obtain that P(k) is decreasing with respect to k. If Kea+bd1b(eabd1) holds, we have P(0)0, which means that P(k)<0 for all k>0; if K>ea+bd1b(eabd1) and k>k1 hold, we can get P(k)<0. Therefore, when one of the following inequalities holds:

    (i) Kea+bd1b(eabd1);

    (ii) K>ea+bd1b(eabd1) and k>k1,

    we can obtain a11<0, which implies that the real parts of λ1 and λ2 are all negative.

    It follows from system (3.1) that

    βS2d2=β[K(eabd1)2+rd1ke]+β[K(eabd1)2rd1ke]2+4K2kre(eabd1)32Kk(eabd1)2d2=Θ2Kk(eabd1)2,

    where

    Θ:=K(eabd1)2(2kd2+β)βekrd1+β(K(eabd1)2rd1ke)2+4K2kre(eabd1)3.

    Note that the sign of Θ depends on

    ˜Θ:=β2(K(eabd1)2rd1ke)2+4β2K2kre(eabd1)3(K(eabd1)2(2kd2+β)+βekrd1)2=4Kk(aebd1)2[(Kd2(eabd1)2+βerd1)d2k+β((eabd1)(d2(eabd1)βer)K+βerd1)].

    Then if one of the following inequalities holds:

    (Ⅰ) eabd1>0 and rd2(eabd1)βe;

    (Ⅱ) eabd1>0, r>d2(eabd1)βe and one of the following inequalities:

    (Ⅱ-1) Kβerd1(eabd1)(βerd2(eabd1));

    (Ⅱ-2) K>βerd1(eabd1)(βerd2(eabd1)) and k>k2:=β((eabd1)(d2(eabd1)βer)K+βerd1)(Kd2(eabd1)2+βerd1)d2,

    we have λ3=βS2d2<0.

    Thus, we can arrive at the conclusion.

    It should be pointed out that another way to state Theorem 4.2 is as follows.

    Remark 4.3. For system (1.1), assume that eabd1>0 and R0<1. If one of the following inequalities:

    (Ⅰ) d1eabd1<KK1;

    (Ⅱ) K>K1 and k>k1

    holds, then the planar equilibrium E2 is stable; otherwise, it is unstable.

    Next, we will show the local stability of the positive equilibrium E3 of system (1.1).

    The Jacobian matrix of system (1.1) at E3 is given by

    J3=(b11b12b13b210d20βI30),

    where

    b11=X3(rK+abS3(1+bX3)2),b12=krX3(1+k(S3+I3))2aX31+bX3<0,b13=krX3(1+k(S3+I3))2<0,b21=eaS3(1+bX3)2>0. (4.3)

    The characteristic equation of J3 is given as

    λ3+A1λ2+A2λ+A3=0, (4.4)

    where

    A1=b11,A2=βd2I3b12b21,A3=b11βd2I3b13b21βI3. (4.5)

    Note that if A1>0 holds, then b11<0, which means that A3>0. According to Routh-Hurwitz criterion, the positive equilibrium E3 is locally asymptotically stable when A1>0 and A1A2A3>0.

    Therefore, we can establish the following statement.

    Theorem 4.4. Assume that eabd1>0 and r>ad2(β+k(d2d1))β2 hold. The positive equilibrium E3 of system (1.1) is locally asymptotically stable if A1>0 and A1A2A3>0, where Ai,i=1,2,3 is defined as in (4.5). Otherwise, it is unstable.

    Remark 4.5. Theorem 4.4 gives a sufficient condition about the stability of the positive equilibrium E3 for system (1.1). However, the complexity of model (1.1) leads to the failure to theoretically demonstrate how the fear factor affects the stability of the positive equilibrium. This will be discussed later through numerical simulations.

    In this subsection, we take k as the bifurcation parameter. The characteristic equation of system (1.1) at E3 is (4.4), and Ai(k),i=1,2,3 are defined as (4.5).

    Theorem 4.6. Hopf bifurcation near the positive equilibrium E3 for system (1.1) occurs whenever the critical parameter k attains the value k=kh in the domain:

    Ω={khR+:Δ(kh):=[A1(k)A2(k)A3(k)]|k=kh=0withA2(kh)>0,[dΔ(k)dk]|k=kh0}.

    Proof. If k=kh, the characteristic Eq (4.4) equals

    λ3+A1(kh)λ2+A2(kh)λ+A3(kh)=0, (4.6)

    then (4.6) can be factorized as

    (λ2+A2(kh))(λ+A1(kh))=0. (4.7)

    Clearly, (4.7) has three roots: λ1=iA2(kh), λ2=iA2(kh) and λ3=A1(kh). The roots are of the form λ1=p1(k)+ip2(k), λ2=p1(k)ip2(k) and λ3=p3(k), where pi(k)(i=1,2,3) are real numbers.

    From the characteristic Eq (4.4), we can get

    dλdk=λ2A1+λA2+A33λ2+2A1λ+A2, (4.8)

    where =ddk. Substituting λ=iA2 into (4.8), we obtain that

    A3A2A1+iA2A22(A2iA1A2)=dΔ(k)dk2(A21+A2)+i[A2A22A2A1A2dΔ(k)dk2A2(A21+A2)],

    which implies that

    [dRe(λ)dk]|k=kh=dΔ(k)dk2(A21+A2)|k=kh.

    By using monotonicity condition in the real part of the complex root dRe(λ)dk|k=kh0, the transversality condition dΔ(k)dk|k=kh0 can be obtained to ensure the existence of Hopf bifurcation.

    Results from numerical simulations are provided in this section to demonstrate our theoretical results. As we will show, the observations shed lights on the impact of fear factor. We choose the parameters of system (1.1) as follows:

    r=0.8,a=0.2,b=0.1,e=0.9,d1=0.05,β=0.1,d2=0.053. (5.1)

    Then we have

    eabd1=0.175>0,d1eabd1=0.286,r1=d2(eabd1)βe=0.103,r2=d2(eabd1)(ea+bd1)aβe2=0.106,K1=ea+bd1b(eabd1)=10.571,K2=βerd1(eabd1)(βerd2(eabd1))=0.328.

    Example 5.1 (The stability of E1).

    We adopt K=0.2,k=0.01, then system (1.1) has trivial equilibrium E0=(0,0,0) and axial equilibrium E1=(0.2,0,0). In this case, one can know that the conditions of Theorem 4.1 are satisfied, which means that E1 is locally asymptotically stable. The numerical results are shown in Figure 1.

    Figure 1.  Population dynamics of X(t), S(t) and I(t) of system (1.1) with K=0.2,k=0.01.

    Example 5.2 (The impacts of K and k on the stability of E2).

    In this example, we will choose three values of carrying capacity K for numerical experiments. We conclude that the carrying capacity and fear effect are other key factors related to the extinction of infected predators, in addition to the basic reproduction number R0.

    Firstly, we take K=0.3<K2, then we have k=0.1 which yields that R0=0.263<1. In this case, system (1.1) has trivial equilibrium E0=(0,0,0), axial equilibrium E1=(0.3,0,0), and planar equilibrium E2=(0.286,0.139,0). By Theorem 4.5, E2 is locally asymptotically stable, see Figure 2(a). Thus, when the carrying capacity of the prey K is small, no matter what the level of fear k is, the small size of prey population will lead to the extinction of infected predators.

    Figure 2.  The impacts of K and k on the stability of E2.

    Secondly, for comparison, we take K2<K=15, then

    k1=Kb(eabd1)2(Kb(eabd1)(ea+bd1))ae2r(ea+bd1)=0.149,k2=β((eabd1)(d2(eabd1)βer)K+βerd1)(Kd2(eabd1)2+βerd1)d2=10.873.

    Choosing k=0.1<max{k1,k2} which yields R0=5.792>1, then we have

    A1=0.25791>0,A1A2A3=0.00523>0.

    In this case, system (1.1) has trivial equilibrium E0=(0,0,0), axial equilibrium E1=(15,0,0), planar equilibrium E2=(0.286,3.070,0), and positive equilibrium E3=(7.351,0.530,7.125). By Theorem 4.5, E2=(0.286,3.070,0) is unstable. On the contrary, E3=(7.351,0.530,7.125) is locally asymptotically stable. The numerical simulation is shown in Figure 2(b).

    Finally, we take K2<K=60, then we have

    k1=Kb(eabd1)2(Kb(eabd1)(ea+bd1))ae2r(ea+bd1)=6.629,k2=β((eabd1)(d2(eabd1)βer)K+βerd1)(Kd2(eabd1)2+βerd1)d2=12.238.

    Choosing k=30>max{k1,k2} which yields that R0=0.649<1, system (1.1) has trivial equilibrium E0=(0,0,0), axial equilibrium E1=(60,0,0), and planar equilibrium E2=(0.286,0.344,0). By Theorem 4.5, E2 is locally asymptotically stable, see Figure 2(c). Thus, when the carrying capacity of the prey K is relatively large, a high level of fear k will lead to the extinction of infected predators.

    Example 5.3 (The impact of k on the stability of E3). We adopt K=60, then we have kh=0.26. In the next, we will choose three values of k, corresponding to the local stability of E3, Hopf bifurcation, and instability of E3, to illustrate the impact of fear factor on the population dynamics.

    Firstly, we take k=0.1<kh which yields that R0=5.881>1, then system (1.1) has trivial equilibrium E0=(0,0,0), axial equilibrium E1=(60,0,0), planar equilibrium E2=(0.286,3.117,0) and a unique positive equilibrium E3=(24.047,0.530,12.213). In this case, we obtain that

    A1=0.29863>0,A1A2A3=0.00065>0,

    which means that E3 is local asymptotically stable. The numerical results are shown in Figure 3.

    Figure 3.  Population dynamics of X(t), S(t) and I(t) of system (1.1) with K=60,k=0.1<kh.

    Secondly, we take k=0.26=kh which yields that R0=4.682>1, then system (1.1) has trivial equilibrium E0=(0,0,0), axial equilibrium E1=(60,0,0), planar equilibrium E2=(0.286,2.481,0) and a unique positive equilibrium E3=(12.975,0.530,9.665). In this case, we obtain that

    A1=0.14694>0,A1A2A3=0,

    which means that system (1.1) undergoes a Hopf bifurcation and there is a limit cycle around E3. The numerical results and the bifurcation diagrams of system (1.1) with respect to the parameter k are shown in Figures 4 and 5, respectively. Comparing Figures 3 and 4(a), one can see that there are two different implications induced by the fear factor k: the first is that the stability of E3 converts from stable into unstable, and the second is the decrease of values of X3 and I3 of E3.

    Figure 4.  Population dynamics of X(t), S(t) and I(t) of system (1.1) with K=60,k=0.26=kh. (a) Time-series plots; (b) Phase portraits in 3-dimensional space.
    Figure 5.  Bifurcation diagram of the system (1.1) with respect to the parameter k. Here K=60, other parameters are taken as in (5.1).

    Finally, we take k=0.5>kh which yields that R0=3.821>1, then system (1.1) has trivial equilibrium E0=(0,0,0), axial equilibrium E1=(60,0,0), planar equilibrium E2=(0.286,2.025,0) and a unique positive equilibrium E3=(7.685,0.530,7.322). In this case, we can obtain that

    A1=0.07642>0,A1A2A3=0.00051<0,

    which means that E3 is unstable. The numerical results are shown in Figure 6. One can find that the difference between Figures 4 and 6 is the decrease of values of E3 from (12.975,0.530,9.665) to (7.685,0.530,7.322), which is induced by the impact of the feat factor.

    Figure 6.  Population dynamics of X(t), S(t) and I(t) of system (1.1) with K=60,k=0.5>kh. (a) Time-series plots; (b) Phase portraits in 3-dimensional space.

    In this paper, we explored a predator-prey model that incorporates infectious disease in predator population and the cost of anti-predator behaviors. The cost of anti-predator behaviors is measured by a fear effect k leading to an reduction of prey's birth rate. We fulfill a complete stability analysis of equilibria for system (1.1) and show that the system (1.1) exhibits the Hopf bifurcation. Biologically, we focus on the impact of fear effect on the population dynamics. As we will see later, the cost of a high level of fear effect is disastrously. The main findings are summarized in the following.

    1) Small size of prey population leads to the extinction of infected predators.

    If the carrying capacity K is relatively small, the planar equilibrium E2 is stable, see Figure 2(a). Thus, no matter what the level of fear effect k is, a small size of prey population will lead to the extinction of infected predators.

    2) Low level of the fear effect doesn't impact on the population dynamics.

    If the level of fear effect k<kh, the positive (coexistence) equilibrium E3 is stable, see Figure 3. Hence, we conclude that a small fear effect k is not the key disturbance and does not change the coexistence dynamics of system (1.1). However, the densities of the prey and infected predator gradually decrease as k increasing.

    3) Certain medium level of the fear effect lead to periodic oscillation.

    If k=kh, the fear effect can destabilize the stability of E3 and will benefit the occurrence of periodic oscillation. In other words, system (1.1)undergoes a limit cycle, see Figures 4 and 5.

    4) High level of the fear effect leads to complex dynamics and the infected predator can go to extinction.

    If k>kh, E3 is unstable, see Figure 6. Therefore, a large fear effect k persistently and dramatically influence the population dynamics of prey and predator. Furthermore, if the level of the fear factor k is extremely high, the planar equilibrium E2 is stable, see Figure 2(c). The prey will respond to perceived predation risk and show a variety of anti-predator responses, dramatically decreasing the recruitment of susceptible predator, which will lead to an extinction of infected predator.

    The authors would like to thank the editor and the referees for their helpful comments. This research was supported by the National Natural Science Foundation of China (Grant No. 12171192, 12031020 and 12071173), the Science and Technology Research Projects of the Education Office of Jilin Province, China (JJKH20211033KJ), the Technology Development Program of Jilin Province, China (20210508024RQ) and Huaian Key Laboratory for Infectious Diseases Control and Prevention, China (HAP201704).

    The authors declare that there are no conflicts of interest regarding the publication of this paper.

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