Research article Special Issues

Construction of a three commitment points for S phase entry cell cycle model and immune-related ceRNA network to explore novel therapeutic options for psoriasis


  • While competing endogenous RNAs (ceRNAs) play pivotal roles in various diseases, the proliferation and differentiation of keratinocytes are becoming a research focus in psoriasis. Therefore, the three commitment points for S phase entry (CP1–3) cell cycle model has pointed to a new research direction in these areas. However, it is unclear what role ceRNA regulatory mechanisms play in the interaction between keratinocytes and the immune system in psoriasis. In addition, the ceRNA network-based screening of potential therapeutic agents for psoriasis has not been explored. Therefore, we used multiple bioinformatics approaches to construct a ceRNA network for psoriasis, identified CTGF as the hub gene, and constructed a ceRNA subnetwork, after which validation datasets authenticated the results' accuracy. Subsequently, we used multiple online databases and the single-sample gene-set enrichment analysis algorithm, including the CP1–3 cell cycle model, to explore the mechanisms accounting for the increased proliferation and differentiation of keratinocytes and the possible roles of the ceRNA subnetwork in psoriasis. Next, we performed cell cycle and cell trajectory analyses based on a single-cell RNA-seq dataset of psoriatic skin biopsies. We also used weighted gene co-expression network analysis and single-gene batch correlation analysis-based gene set enrichment analysis to explore the functions of CTGF. Finally, we used the Connectivity Map to identify MS-275 (entinostat) as a novel treatment for psoriasis, SwissTargetPrediction to predict drug targets, and molecular docking to investigate the minimum binding energy and binding sites of the drug to target proteins.

    Citation: Jingxi Xu, Jiangtao Li. Construction of a three commitment points for S phase entry cell cycle model and immune-related ceRNA network to explore novel therapeutic options for psoriasis[J]. Mathematical Biosciences and Engineering, 2022, 19(12): 13483-13525. doi: 10.3934/mbe.2022630

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  • While competing endogenous RNAs (ceRNAs) play pivotal roles in various diseases, the proliferation and differentiation of keratinocytes are becoming a research focus in psoriasis. Therefore, the three commitment points for S phase entry (CP1–3) cell cycle model has pointed to a new research direction in these areas. However, it is unclear what role ceRNA regulatory mechanisms play in the interaction between keratinocytes and the immune system in psoriasis. In addition, the ceRNA network-based screening of potential therapeutic agents for psoriasis has not been explored. Therefore, we used multiple bioinformatics approaches to construct a ceRNA network for psoriasis, identified CTGF as the hub gene, and constructed a ceRNA subnetwork, after which validation datasets authenticated the results' accuracy. Subsequently, we used multiple online databases and the single-sample gene-set enrichment analysis algorithm, including the CP1–3 cell cycle model, to explore the mechanisms accounting for the increased proliferation and differentiation of keratinocytes and the possible roles of the ceRNA subnetwork in psoriasis. Next, we performed cell cycle and cell trajectory analyses based on a single-cell RNA-seq dataset of psoriatic skin biopsies. We also used weighted gene co-expression network analysis and single-gene batch correlation analysis-based gene set enrichment analysis to explore the functions of CTGF. Finally, we used the Connectivity Map to identify MS-275 (entinostat) as a novel treatment for psoriasis, SwissTargetPrediction to predict drug targets, and molecular docking to investigate the minimum binding energy and binding sites of the drug to target proteins.



    Mosquito-borne diseases such as dengue, yellow fever, and Zika are threatening more than half the world's population. Due to the lack of vaccines, the primary traditional method in control of these mosquito-borne diseases to suppress the mosquito population density by spraying insecticides. However, this method failed to achieve a sustainable effect on keeping mosquito population density below the critical level of epidemic risk. Even worse, heavy applications of pesticides have led to insecticide resistance and environmental pollution. In recent years, releasing sterile mosquitoes has provided an effective and biologically safe control method for eliminating or reducing mosquito populations and thus to control mosquito-borne diseases. In such a method, male mosquitoes are first sterilized using radiological or chemical techniques, and then released into the field to sterilize wild females. A wild female mosquito that mates with a sterile male will either be non-reproductive, or lay eggs that are not hatchable. Repeating releases of sterile mosquitoes could eventually wipe out wild mosquitoes, or, more realistically, suppress the wild mosquito population [1,2].

    Various mathematical models have been developed to study the interactive dynamics of wild and sterile or Wolbachia-infected mosquitoes, including ordinary differential equation models [3,4,5,6,7,8,9], delay differential equation models [10,11,12,13,14], partial differential equation models [15,16,17,18], and stochastic dynamical equation models [19,20], to cite only a few. Recently, Li in [7] formulated a simple model with constant release rate of sterile mosquitoes

    {dSν(t)dt=aSν(t)Sν(t)+Sg(t)(1ξνSν(t))Sν(t)μ1Sν(t),dSg(t)dt=bμ1Sg(t), (1.1)

    where Sν(t) and Sg(t) are the numbers of wild and sterile mosquitoes at time t respectively, a is the number of offspring produced per individual female adult per unit of time, b is the constant release rate of sterile mosquitoes, ξν is the carrying capacity parameter such that 1ξνSν describes the effect of density dependence, and μ1 is the death rate of wild or sterile mosquitoes. In (1.1), it was assumed implicitly that the mating of sterile male mosquitoes with wild females has an instant impact on the reproduction such that the model is based on ordinary differential equations without time-delay. It does not incorporate the development or maturation of mosquitoes that undergo four distinct life stages.

    Including the maturation process, we let τ be the average waiting duration from the eggs to the eclosion of adults in the next generation. We then extend model (1.1) to the following model with time delay

    {dSν(t)dt=aeμ0τSν(tτ)Sν(tτ)+Sg(tτ)(1ξνSν(tτ))Sν(tτ)μ1Sν(t),dSg(t)dt=bμ2Sg(t). (1.2)

    In this new model eμ0τ is the survival rate of the immature mosquitoes that were born at time tτ and are still alive at the time t, and μ1,μ2 are the death rates of wild and sterile mosquitoes, respectively.

    The initial condition for (1.2) is given as

    ϕ(t)=(ϕν(t),ϕg(t))C1([t0τ,t0],(0,1ξν)×(0,)), (1.3)

    where t0τ. For convenience, we write

    M=maxt[t0τ,t0]ϕg(t),M0=max{b/μ2,M},m=mint[t0τ,t0]ϕg(t),m0=min{b/μ2,m}. (1.4)

    It is well-known that the oscillation phenomena may appear when time delays are included in differential equations, as shown, for example, in recent studies of mosquito population models with time delays [10,12]. We aim to offer an accurate description of the oscillation phenomena of (1.2), and study how the oscillatory properties change with respect to the release of sterile mosquitoes. Specifically to our model (1.2)-(1.3), we define the oscillations as follows.

    Definition 1.1. Let (Sν(t),Sg(t)) be a non-constant positive solution of (1.2)-(1.3). We say that (Sν(t),Sg(t)) is non-oscillatory about its equilibrium (¯Sν,¯Sg) if Sν(t)¯Sν and Sg(t)¯Sg are positive or negative eventually. Otherwise, it is said to be oscillatory about (¯Sν,¯Sg). We say that (1.2) is oscillatory about (¯Sν,¯Sg) if every non-constant solution of (1.2) is oscillatory. Otherwise, (1.2) is said to be non-oscillatory about (¯Sν,¯Sg).

    In this paper, we study the oscillatory properties of (1.2). The paper is organized as follows. In Section 2, we establish useful lemmas and find that, (1.2) is oscillatory about the unique non-zero equilibrium when no sterile mosquitoes are released. In Section 3, we determine an oscillation threshold, denoted by ˆb, for the constant release rate b of sterile mosquitoes, such that all non-trivial positive solutions oscillate when b<ˆb, and the oscillation disappears when b>ˆb. In Section 4, numerical simulations are provided to demonstrate our new findings. Concluding remarks are finally given in Section 5.

    In this section, we first establish useful lemmas that help us to prove the main results of this paper. In addition, we show that every non-trivial solution of (1.2) oscillates with respect to its unique non-zero equilibrium when no sterile mosquitoes are released. We first determine the monotonicity of the following birth-progression function.

    Lemma 2.1. Consider the birth-progression function

    g(x,z)=aeμ0τ(1ξνx)x2x+z,0<x<1/ξν,z>0,

    and let

    C(z)=13ξνz+(13ξνz)2+16ξνz4ξν.

    Then

    gx(x,m0){>0,0<x<C(m0),=0,x=C(m0),<0,C(m0)<x<1ξν.

    Proof. We omit the detail of the proof since the result can be obtained by directly taking the derivative of g(x,z) with respect to x.

    We next discuss the boundedness and positivity of solutions of (1.2).

    Lemma 2.2. Suppose μ1>ξνg(C(m0),m0). Then system (1.2)-(1.3) has a unique solution (Sν,Sg) which is bounded and positive for all t[t0,).

    Proof. The existence and uniqueness of the solution of (1.2) follows from the standard results in the theory of delay differential equations [21]. From the second equation of (1.2), it is easy to see that 0<m0Sg(t)M0 for all tt0, where m0 and M0 are defined in (1.4). It remains to verify the positivity and boundedness of Sν in (1.2).

    We first confirm that Sν(t)<1/ξν holds for all tt0 when μ1>ξνg(C(m0),m0). Otherwise, there exists t1>t0 such that Sν(t1)=1/ξν, and Sν(t)<1/ξν for all t[t0τ,t1). Hence Sν(t1)0. However, from (1.4) and the first equation of (1.2), we have

    Sν(t1)μ1Sν(t1)+g(Sν(t1τ),m0)1ξν(μ1ξνg(C(m0),m0))<0,

    which is a contradiction to Sν(t1)0.

    Next, we prove the positivity of Sν(t). If not, there exists t2>t0 such that Sν(t2)=0 and 0<Sν(t)<1/ξν for all t[t0τ,t2). Hence Sν(t2)0, which contradicts

    Sν(t2)=aeμ0τ(1ξνSν(t2τ))S2ν(t2τ)Sν(t2τ))+Sg(t2τ))>0.

    The proof is complete.

    Remark 1. We assume that the condition μ1>ξνg(C(m0),m0) holds throughout the rest of this paper.

    Define the intrinsic growth rate of wild mosquito population by

    r0=aeμ0τμ1.

    The following lemma clarifies the existence of equilibria of (1.2).

    Lemma 2.3. Define

    b:=μ2(r01)24ξνr0. (2.1)

    If 0<b<b, then (1.2) has three nonnegative equilibria: N0:=(0,bμ2) and N=(Sν,bμ2), where

    Sν:=(r01)μ2(r01)2μ224bξνr0μ22ξνr0μ2;

    When b=b, (1.2) has two nonnegative equilibria: N0 and N:=(r012r0ξν,bμ2); When b>b, N0 is the only nonnegative equilibrium of (1.2).

    We next investigate the oscillation of (1.2) with respect to the unique non-zero equilibrium when b=0. In this case, (1.2) becomes

    {dSν(t)dt=aeμ0τSν(tτ)Sν(tτ)+Sg(tτ)(1ξνSν(tτ))Sν(tτ)μ1Sν(t),dSg(t)dt=μ2Sg(t). (2.2)

    It is clear that limtSg(t)=0. Therefore, the oscillation of (2.2) is exactly the same as that of the next equation

    dSν(t)dt=aeμ0τSν(tτ)(1ξνSν(tτ))μ1Sν(t), (2.3)

    which has a unique positive equilibrium S(1)ν=(r01)/r0ξν.

    We now study the oscillation of (2.3) with respect to S(1)ν when r0>2. Let y(t)=Sν(t)S(1)ν. Then instead of considering the oscillation of Sν(t) with respect to S(1)ν in (2.3), we consider the oscillation of y(t) about y(t)=0 for the following equation

    y(t)=μ1y(t)qy(tτ)μ1r0ξνy2(tτ), (2.4)

    where q=μ1(r02)>0.

    To proceed, we first establish the following three lemmas to reach the conclusion of Theorem 1.

    Lemma 2.4. If r0>2, then every non-oscillatory solution of (2.4) about y(t)=0 converges to zero as t.

    Proof. Let y(t) be an arbitrary non-oscillatory solution of (2.4) about y(t)=0. Then it is positive or negative eventually. It suffices to prove that

    limty(t)=0. (2.5)

    (ⅰ) Suppose that y(t) is eventually positive. Then there exists ¯t1>t0 large enough such that y(t)>0 for all t>¯t1+τ. In the meantime, based on (2.4), we have

    y(t)=μ1y(t)qy(tτ)μ1r0ξνy2(tτ)<0.

    Thus limty(t)=L holds and we claim that L is non-negative. If L>0, then

    limty(t)=μ1LqLμ1r0ξνL2:=ϱ<0.

    Thus we have y(t)ϱ<0 for sufficiently large t, and hence limty(t)=, which is a contradiction. Therefore, limty(t)=L=0.

    (ⅱ) Suppose that y(t) is eventually negative. To prove (2.5), we let

    ¯y=lim supty(t),y_=lim infty(t).

    It is easy to see that (2.5) is true if and only if y_=0. Otherwise, y_<0. There are only two cases to consider: y_=¯y, and y_<¯y.

    If y_=¯y<0, then limny(t)=y_=¯y exists. Taking limits in (2.4) on both sides gives

    limty(t)=y_(μ1+q+μ1r0ξνy_).

    Notice that y(t) is bounded, then we derive μ1+q+μ1r0ξνy_=0 and y_=S(1)ν. Hence there exists a sequence {tn} large enough such that y(tn)0, y(tn)S(1)ν, and y(tn)=minttny(t). Then it follows from (2.4) that

    0y(tn)=μ1y(tn)qy(tnτ)μ1r0ξνy2(tnτ),

    which leads to

    μ1y(tn)y(tnτ)[q+μ1r0ξνy(tnτ)],

    and

    μ1y(tnτ)y(tn)[q+μ1r0ξνy(tnτ)][q+μ1r0ξνy(tnτ)],

    Solving the inequality above, we obtain y(tnτ)S(1)ν, which is a contradiction to y(tnτ)>S(1)ν.

    If y_<¯y<0, then there exists a sequence {sn} large enough such that y(sn)=0, y(sn)y_. From (2.4), we have

    μ1y(sn)=qy(snτ)μ1r0ξνy2(snτ).

    Notice that y(snτ) is bounded in the interval (S(1)ν,0). Then there exists a convergent subsequence, denoted by {sn} again, such that

    μ1y_=qy1μ1r0ξνy21,

    where y_y1:=limny(snτ)¯y. Hence

    μ1=y1y_[q+μ1r0ξνy1][q+μ1r0ξνy1],

    and y1S(1)ν. Meanwhile, y1y_S(1)ν, thus y1=y_=S(1)ν.

    Take t>t0 large enough such that y(t)=0, and y(t)=mintty(t). Then it follows from (2.4) that

    μ1y(t)=qy(tτ)μ1r0ξνy2(tτ),

    which yields

    μ1=y(tτ)y(t)[q+μ1r0ξνy1][q+μ1r0ξνy(tτ)],

    and y(tτ)S(1)ν, a contradiction to y(tτ)>S(1)ν. The proof is complete.

    To show the oscillatory behavior of solutions of (2.4) about y(t)=0, we linearize (2.4) at y(t)=0 which leads to

    y(t)=μ1y(t)qy(tτ). (2.6)

    Based on the results in [22], we immediately have the following necessary and sufficient condition for the oscillation of solutions of (2.6) with respect to y(t)=0.

    Lemma 2.5. [22] Every non-trivial solution of (2.6) is oscillatory about y(t)=0 if and only if

    qτeμ1τ>1e, (2.7)

    where q=μ1(r02)>0.

    We then show the equivalence of the oscillations between (2.4) and (2.6) about y=0.

    Lemma 2.6. Equation (2.4) is oscillatory about y=0 if and only if (2.6) is oscillatory about y=0.

    Proof. Assume that (2.4) is oscillatory about y=0. We confirm that (2.6) is also oscillatory about y=0. Otherwise, there exists a solution y(t) of (2.6) that is non-oscillatory about y(t)=0. Without loss of generality, suppose that y(t) is eventually negative. Then there exist δ>0 and t1>t0 such that δ<y(t)<0, for all t>t1. Let Y(t) be a solution of (2.4) with the same initial value condition as that of y(t) in (2.6). Notice that

    μ1y(t)qy(tτ)μ1ξνr0y2(tτ)μ1y(t)qy(tτ).

    By the comparison principle, we have Y(t)y(t)<0, for t>t1, which is a contradiction to the assumption that (2.4) is oscillatory about y(t)=0.

    On the other hand, we show that (2.4) oscillates about y=0 if (2.6) oscillates about y=0. If not, then there exists a solution y1(t) of (2.4), which is either eventually positive or eventually negative. There are two cases to consider.

    (ⅰ) If y1(t) is eventually positive, then there exists t2>t0 such that y1(t)>0, for all t>t2. Let Y1(t) be the solution of (2.6) with the same initial condition as that of y1(t) in (2.3). Again, by the comparison principle, we have Y1(t)y1(t)>0, for all t>t2, a contradiction to the oscillation of (2.6) about y(t)=0.

    (ⅱ) If y1(t) is eventually negative, then there exists t3>t0 such that y1(tτ)<0, y1(t)<0, for all t>t3. From Lemma 4, we have limty1(t)=0. Since (2.7) holds, there exists a positive constant ε0<1 such that

    ε0qτeμ1τ>1e,

    which suggests that the equation

    y(t)+μ1y(t)+ε0qy(tτ)=0

    is oscillatory about y(t)=0. Moreover, since

    limtq+μ1ξνr0y(tτ)q=1,

    there exists t4>t3 such that

    q+μ1ξνr0y(tτ)>ε0q,for allt>t4.

    Then

    (q+μ1ξνr0y(tτ))(y(tτ))>ε0q(y(tτ)),for allt>t4,

    and thus

    μ1y(t)qy(tτ)μ1ξνr0y2(tτ)>μ1y(t)ε0qy(tτ).

    Let ¯Y(t) be a solution of the equation y(t)+μ1y(t)+ε0qy(tτ)=0 with the same initial condition as the solution y1(t) of (2.4). Then ¯Y(t)<y1(t)<0 for all t>t4, which is a contradiction to the oscillation of the equation

    y(t)+μ1y(t)+ε0qy(tτ)=0

    about y(t)=0. Therefore, equation (2.4) is oscillatory about y=0 if and only if (2.6) is oscillatory about y=0. The proof is complete.

    We note that the oscillations of Sν about S(1)ν between system (2.2) and (2.3) are exactly the same. Based on Lemmas 2.3, 2.4, and 2.5 above, a sufficient and necessary condition for the oscillations of solutions of equation (2.2) can be summarized as follows.

    Theorem 1. If r0>2, then every non-trivial positive solution of (2.2) is oscillatory about S(1)ν if and only if (2.7) holds.

    Remark 2. Theorem 1 provides a sufficient and necessary condition for the oscillations of non-trivial positive solutions about the unique non-zero equilibrium in equation (1.2) when no sterile mosquitoes are released. In fact, from the estimation of parameters of (1.2) in Table 1, we find that (2.7) is always true and so (2.2) is oscillatory about S(1)ν, which is consistent with the oscillation phenomenon of the annual abundance of wild mosquitoes in the wild. These results are important for the discussion of the main results in Section 3.

    Table 1.  Parameter values adapted to Aedes albopictus population suppression in subtropical monsoon climate, especially in Guangzhou.
    Para. Definitions Ranges References
    τ The average waiting duration from eggs to the eclosion [16, 66] [25]
    of adults in the next generation (day)
    a The number of offsprings produced per individual, [3.15, 12.81] [26,28]
    per unit of time
    eμ0τ The survival rate of the immature mosquitoes (day1) 0.05 [27]
    ξν The carrying capacity parameter of wild mosquitoes 0.0025 Given
    μ1 The death rate of wild mosquitoes (day1) [0.0231, 0.0693] [26]
    μ2 The death rate of sterile mosquitoes (day1) 1/7 [23,24]

     | Show Table
    DownLoad: CSV

    Given that (1.2) is oscillatory about the unique non-zero equilibrium in the absence of sterile mosquitoes, it becomes interesting to ask how the oscillatory property of (1.2) with the increase of the releases of sterile mosquitoes. Interestingly, we find an oscillation threshold, denoted by ˆb, for the release rate of the sterile mosquitoes, and show that this oscillation phenomenon will be maintained when the release rate b<ˆb, whereas it will disappear when b>ˆb. The result of oscillation threshold is described as follows.

    Theorem 2. Let ˆb be the unique solution of

    q(b)τeμ1τ=1e,

    where

    q(b)=μ1μ2S+ν(3r0ξνS+ν+12r0)μ2S+ν+b>0.

    Then ˆb is the oscillation threshold of the release rate b, below which the oscillation phenomenon maintain, and above which it will disappear.

    Proof. By using the similar argument as that in Theorem 1, we conclude that every non-trivial positive solution of (1.2) is oscillatory about the non-zero equilibrium N+ if and only if

    q(b)τeμ1τ>1e. (3.1)

    The proof is nearly the same as that of Theorem 1, so we omit it.

    Next, we claim that q(b) decreases with the increase of b. It follows from the definition of S+ν in Lemma 2.3 that S+ν(b) is monotonously decreasing. Hence 3r0ξνS+ν2r0+1 is positive and monotonously decreasing with respect to b. Moreover, By taking derivative, we obtain

    ddb(S+νμ2S+ν+b)<0,

    thus S+ν/(μ2S+ν+b) is monotonously decreasing. Consequently, q(b) is monotonously decreasing.

    Notice that (1.2) is oscillatory about N+ when b=0. According to Theorem 1, we have q(0)τeμ1τ>1/e. Meanwhile, when b=b, we have

    q(b)=μ1μ2S+ν(3r0ξνS+ν+12r0)μ2S+ν+b=(12r012)μ1μ2S+νμ2S+ν+b<0.

    Thus, there exists unique ˆb(0,b) such that q(ˆb)τeμ1τ=1/e. Notice that (1.2) is oscillatory about N+ if and only if (3.1) holds, we conclude that system (1.2) is oscillatory about N+ when 0b<ˆb, and non-oscillatory when ˆbbb. The proof is complete.

    In this section, numerical simulations are given to validate the main theoretical results in Section 3.

    In Table 1 below, we list important parameter values for Aedes albopictus and sterile mosquitoes, most of which are taken from earlier experimental data in the literature [23,24,25,26,27,28,29]. Since the parameter values are affected by many factors such as temperature and rainfall, we can not determine their exact values but take reasonable estimations of the ranges. Using a similar method for the parameter estimations as in [12], we estimate τ[16,66]. By using the measured half-lives, we have μ10.0277 from [29], and μ1[0.0231,0.0693] from [26], hence the life span of adult mosquitoes is T1=1/μ136.10, and T1[14.43,43.29]. Furthermore, it takes 5 days on average for a female mosquito to lay eggs after the eclosion, and the total number of eggs laid by each female is 120.76 on average [26]. Consequently, the egg-laying phase of a female mosquito is T2[9.43,38.29], and a[3.15,12.81]. From [23,24], we find that sterile mosquitoes will die in about 7 days after releasing to the wild field. Thus we estimate that the death rate of sterile mosquitoes is about 1/7. Besides, according to [27], we obtain that the survival probability of the average maturation time of wild mosquitoes is about 0.05. We note that ξν is usually proportional to the area size. From simulations, it shows that different values of ξν produce similar dynamics, which allows us to take ξν=0.0025 as a representative value.

    To make the oscillation behavior be clearer, we employ simulation examples with different release rates to testify the result of oscillation threshold given in Theorem 2.

    Suppose that the model parameters of (1.2) are given as

    a=4,τ=36,eμ0τ=0.05,μ1=0.05,μ2=1/7,ξν=0.0025. (4.1)

    All the parameter values in (4.1) are within the estimation ranges given in Table 1. By substituting these parameter values into (2.1), we derive b32.143. Notice that the oscillation threshold ˆb is the unique solution of the equation q(b)τeμ1τ=1/e, we have ˆb21.7754.

    We select three different release rates b=0, b=3 and b=6 from the interval [0,ˆb). For these three cases, the number of wild mosquitoes Sν(t) oscillates around S+ν, as shown in Figure 1. This testifies the first part of Theorem 2, that is, the non-trivial positive solutions of (1.2) are oscillatory about N+ when b<ˆb. When b=0, there is no sterile mosquitoes released into the wild, the number of mosquitoes oscillates with a larger amplitude, as shown in Figure 1 (A). As the release rate increases, the amplitude is reduced. When b=6, the number of mosquitoes oscillates in a narrow range around S+ν, as shown in Figure 1 (C).

    Figure 1.  The oscillatory behavior of (1.2) with respect to N+ when the release rates b=0,3,6 are smaller than the oscillation threshold ˆb21.7754.

    We increase the release rate of sterile mosquitoes further, and let ˆb<b<b. For instance, we take b=24, b=27 and b=30 separately from the interval [ˆb,b]. As shown in Figure 2, we find that the number Sν(t) is less than S+ν, and then gradually increases and approaches to S+ν. This phenomenon testifies the second part of Theorem 2, that is, the solutions (Sν,Sg) are non-oscillatory about N+.

    Figure 2.  The release rates b=24, b=27 and b=30, respectively, are larger than the oscillation threshold ˆb21.7754. The corresponding solutions (Sν,Sg) of (1.2) are non-oscillatory about N+.

    Figure 2 also shows that the wild mosquito population is first suppressed to a low level by releasing sterile mosquitoes with higher rates, then gradually tends to a stable level S+ν. Figure 1 and 2 both show that the stable level of wild mosquitoes is reduced when the release rate increases from 0 to b.

    To control the life-threatening mosquito-borne diseases such as dengue, yellow fever, and Zika, one of the biologically safe methods is to release male sterile mosquitoes into the field to suppress the wild mosquito population. We, in this paper, consider the oscillatory properties of a delayed mosquito population suppression model with a constant release rate of sterile mosquitoes. We find that every non-trivial positive solution of the model oscillates with respect to its unique non-zero equilibrium when no sterile mosquitoes are released, i.e., b=0. We then study the oscillatory behavior of the model with the releases of sterile mosquitoes, and establish an oscillation threshold, denoted by ˆb, for the constant release rate b of sterile mosquitoes such that the oscillation of solutions maintains when b<ˆb, whereas the oscillation disappears when b>ˆb. Furthermore, through numerical examples, we show that the amplitudes of the oscillation become smaller and smaller and then the oscillation disappears as release rates of sterile mosquitoes increase. Oscillatory phenomenon is common for solutions of delay differential equations. However, to the best of our knowledge, an establishment of the oscillation threshold which determines the existence or disappearance of solution oscillations is one of the first in the field of mosquito population dynamics.

    This work was supported by the National Natural Science Foundation of China (No. 11631005), the Program for Changjiang Scholars and Innovative Research Team in University (No. IRT-16R16), the Guangzhou Postdoctoral International Training Program Funding Project, the Science and Technology Program of Guangzhou (No. 201707010337) and the Foundation of Yunnan Educational Committee (No. 2017ZDX027). The authors thank Professor Jianshe Yu for his help in improving the original Lemma 2.4.

    The authors have declared that no competing interests exist.



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