
Citation: Qian Li, Yanni Xiao. Analysis of a mathematical model with nonlinear susceptibles-guided interventions[J]. Mathematical Biosciences and Engineering, 2019, 16(5): 5551-5583. doi: 10.3934/mbe.2019276
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In recent decades, the public health system is severely affected by the outbreak and re-occurrence of infectious diseases, which also causes social turbulence and economic retrogression. Many mathematical models are proposed and analyzed to investigate the dynamics of infectious diseases [1,2,3,4,5,6,7,8,9]. Comprehensive interventions, such as vaccination, treatment and isolation, are estimated to be effective for controlling the spread of infectious diseases [10,11,12,13,14,15,16,17], among which many researches studied the saturated continuous treatment related to limited medical resources [12,13,14]. The SIR model with continuously saturated treatment gives:
{dS(t)dt=A−βSI−δ1S,dI(t)dt=βSI−δ2I−γI−ϵI1+ωI,dR(t)dt=γI−δ1R+ϵI1+ωI, | (1.1) |
where S, I and R are the populations of susceptible, infected, and recovered, respectively. A represents the constant recruitment rate, β is the transmission rate, γ is the recovery rate, δ1 denotes the natural death rate, and δ2 denotes the death rate of class I including both the natural death rate and the disease-related death rate, hence, it is reasonable to assume δ1<δ2. The term ϵI1+ωI represents the saturated treatment. Note that the above model assumes that the recovered individuals cannot be infected again, hence the class R doesn't affect the dynamics of system (1.1). Therefore, one only needs to consider the following reduced model:
{dS(t)dt=A−βSI−δ1S,dI(t)dt=βSI−δ2I−γI−ϵI1+ωI. | (1.2) |
Impulsive differential equations, including fixed-moments and state-dependent impulsive strategies, were widely used and have raised human's concern. Fixed-moments impulsive models assume that measures are carried out at fixed discrete times. Using this type of models [17,18,19,20,21,33,34,36,37], researchers can investigate the existence and stability of the disease-free periodic solution. However, these models described that control measures were implemented every fixed time without knowing the number of infected and susceptible individuals and the prevalence of infectious diseases, which may waste the medicine resources [17,19,20,24]. Therefore, it is more reasonable to propose state-dependent impulsive models, in which the implementation of vaccination and isolation is determined by whether the size of infected or susceptible population reaches the threshold level. Traditional state-dependent impulsive mathematical models [16,31,35] considered the size of infected population as an index to trigger impulsive interventions, in which no disease-free periodic solution is feasible and this strategy is unable to eradicate infectious diseases. Moreover, this makes it challengeable to define the basic (or control) reproduction number for impulsive models.
Therefore, a natural consideration is whether or not the susceptibles-guided impulsive interventions can successfully control and finally eradicate infectious diseases, and how this strategy affects the dynamical behaviors. The novel idea comes from the control of measles infection, in which the number of susceptible individuals (or the level of susceptibility) is higher than or exceeds a certain level, then the vaccination will then be implemented [22,23]. Moreover, there are some researches investigating the effectiveness of the susceptible-triggered interventions and showing that the susceptible-triggered interventions are promising and effective strategies [24,25,26,27]. Particularly, studies [24,25] have considered the susceptibles-triggered impulsive interventions on SIR models. They assumed that the vaccination rate and isolation rate are linearly dependent on the number of susceptible and infected individuals, respectively. However, in reality, vaccination and isolation are often restricted by limited medical resources [28,29], which can be expressed as saturation functions:
p1(t)=pS(t)h1+S(t),q1(t)=qI(t)h2+I(t), |
where p∈(0,1) denotes the maximal vaccination rate of susceptible population, and q∈(0,1) is the maximal isolation rate of infected individuals. h1 and h2 denote the half-saturation constants of susceptible and infected individuals, respectively. Therefore, based on (1.2), we propose the following state-dependent impulsive model with susceptibles-guided comprehensive saturated interventions:
{dS(t)dt=A−βSI−δ1S,dI(t)dt=βSI−δ2I−γI−ϵI1+ωI,}S(t)<ST,S(t+)=(1−pS(t)h1+S(t))S(t),I(t+)=(1−qI(t)h2+I(t))I(t),}S(t)=ST, | (1.3) |
where ST represents the threshold level of the number of susceptible individuals determining whether to implement the impulsive control strategies or not. The main purpose of this study is to analyze the mathematical model describing the susceptibles-guided comprehensive saturated interventions (including impulsive vaccination and isolation, and continuous treatment), and further evaluate the effectiveness of this strategy for controlling the spread of infectious diseases.
The rest of this paper is organized as follows. In the next section, we give some basic definitions of the planer impulsive semi-dynamical system. In Section 3, we discuss the existence and stability of the disease-free periodic solution. Then, in the next two sections, we investigate the dynamic behaviors of our proposed model through discussing the existence and stability of the positive order-1 periodic solution. Specifically, in Section 4, we study the existence and stability of the positive order-1 periodic solutions through investigating the bifurcations near the disease-free periodic solution. In Section 5, we define the impulsive set and phase set of the Poincaré map of our proposed model and further discuss the positive order-1 periodic solutions in a large range of the control parameters by examining the properties of the Poincaré map including monotoniciity, continuity, discontinuity and convexity. In section 6, we finally give some conclusions and discussions.
We describe the generalized planer impulsive semi-dynamical system with state-dependent feedback control as:
{dxdt=P(x,y),dydt=Q(x,y),ifϕ(x,y)≠0,Δx=a(x,y),Δy=b(x,y),ifϕ(x,y)=0. | (2.1) |
Here (x,y)∈R2+={(x,y):x≥0,y≥0}, Δx=x+−x and Δy=y+−y. P,Q,a,b are continuous functions from R2+ to R. The impulsive function ψ:R2+→R2+ can be defined as
ψ(x,y)=(x+,y+)=(x+a(x,y),y+b(x,y)), |
and z+=(x+,y+) is called an impulsive point of z=(x,y). In this study, we focus on the special state-dependent impulsive model (1.3). We start with concluding the main dynamics of the ODE subsystem.
The dynamical behaviors of subsystem (1.2) have been discussed in [14], here we just recall them briefly. Consider the region Ω={(S,I):S+I≤Aδ1,S,I≥0} as a positively invariant set of system (1.2), and denote the basic reproduction number of system (1.2) as:
R0=Aβδ1(δ2+γ+ϵ). | (2.2) |
It is easy to see that system (1.2) always has a disease-free equilibrium E0=(A/δ1,0), which is globally stable if there is no endemic equilibrium. The existence of the endemic equilibrium depends on the solutions of the following equations:
{A−βSI−δ1S=0,βSI−δ2I−γI−ϵI1+ωI=0. |
Solving above equations yields
I2+b1I+b2=0, |
with
b1=(δ2+γ)(β+ωδ1)+βϵ−Aβωβω(δ2+γ),b2=δ1(δ2+γ+ϵ)−Aββω(δ2+γ)=δ1(δ2+γ+ϵ)βω(δ2+γ)(1−R0). |
As we can see, b2≤0 holds true if and only if R0≥1.
Denote
I1=−b1+√Δ2,S1=AβI1+δ1,andI2=−b1−√Δ2,S2=AβI2+δ1,withΔ=b21−4b2, |
and solve Δ=0 in terms of R0, we obtain R0=˜R0 with
˜R0=A(δ2+γ)(δ2+γ+(ω√A+√ϵω)2)(δ2+γ+ϵ)((δ2+γ)(δ2+γω+2(A+ϵω))+ω(A−ϵω)2). |
Therefore, we obtain the following results regarding the existence of the endemic equilibria.
Proposition 2.1. For subsystem (1.2):
(1) When R0>1, there exists a unique endemic equilibrium E1=(S1,I1), as shown in Figure 1;
(2) When b1≥0, subsystem (1.2) can undergo a forward bifurcation at R0=1, and there exists no endemic equilibrium if R0≤1;
(3) When b1<0, subsystem (1.2) undergoes a backward bifurcation at R0=1 with a saddle-node bifurcation happening at R0=˜R0. Specifically, there exist two endemic equilibria E1=(S1,I1) and E2=(S2,I2) if ˜R0<R0<1 while the two equilibria coincide into one endemic equilibrium when R0=˜R0, and there exists no endemic equilibrium if R0<˜R0.
Next, we show the stability and bifurcation phenomenons of the endemic equilibria of subsystem (1.2). The characteristic equation at the endemic equilibria is shown as:
λ2+H(Ii)λ+G(Ii)=0,i=1,2, |
where
H(Ii)=δ1+βIi−ϵωIi(1+ωIi)2,G(Ii)=Aβ2Iiδ1+βIi−(δ1+βIi)εωIi(1+ωIi)2. |
Based on the main conclusions in [14], we obtain that equilibrium E2 is always an unstable saddle point if it exists, and we conclude the results for the stability of equilibrium E1 as follows.
Proposition 2.2. When R0>1 or 1>R0>˜R0 and b1<0, subsystem (1.2) can undergo a Hopf bifurcation around equilibrium E1 at the surface H(I1)=0. Corresponding to the Hopf bifurcation, subsystem (1.2) can either have a stable or an unstable limit cycle, as shown in Figure 1(C) and Figure 1(D). Moreover, the endemic equilibrium E1 of subsystem (1.2) is a stable node (Figure 1(A)) or focus (Figure 1(B)) if H(I1)>0, while E1 is an unstable node or focus if H(I1)<0, and subsystem (1.2) has at least one closed orbit in region Ω.
Therefore, from Proposition 2.2, we obtain that when R0>1 and H(I1)>0, then the endemic equilibrium E1 is stable, while when R0>1 and H(I1)<0, the endemic equilibrium E1 is unstable and there is at least one closed orbit. Particularly, if there is a unique closed orbit, it is stable as shown in Figure 1(C). In order to address the dynamics of system (1.3), we conduct the Poincaré map. Denote the two isolines of subsystem (1.2) as follows:
l1:˙S=A−βSI−δ1S≐P(S,I)=0,l2:˙I=βSI−δ2I−γI−ϵI1+ωI≐Q(S,I)=0. |
Furthermore, we define two sections as:
l3:SST={(S,I)|S=ST,I≥0},l4:SSv={(S,I)|S=(1−pSTh1+ST)ST≐Sv,I≥0}. |
Thus, we can define the impulsive function ψ(S,I) as:
ψ1(S,I)=(1−pS(t)h1+S(t))S(t),ψ2(S,I)=(1−qI(t)h2+I(t))I(t)≐w1(I). |
In the current study, we set the section SSv as a Poincarˊe section. Choose an initial point P+k=(Sv,I+k) on the Poincaré section. If the orbit starting from P+k reaches SST at a finite time, we denote the intersection point as Pk+1=(ST,Ik+1), then after the impulsive intervention, the trajectory will jump to P+k+1=(Sv,I+k+1) on section SSv with I+k+1=w1(Ik+1). Following from the existence and uniqueness of solutions, Ik+1 is uniquely determined by I+k, thus we can define a function g with g(I+k)=Ik+1. Therefore, we can define the Poincarˊe map PM for system (1.3) as:
PM:I+k+1=w1(Ik+1)=w1(g(I+k))≐PM(I+k). |
It is worth noting that the domain and range of Poincaré map PM, which we will give detail analyses in Section 5, are strictly determined by the dynamical behaviors of ODE subsystem (1.2). From the main results in Proposition 2.1 and Proposition 2.2, we can conclude the four cases of the dynamics of subsystem (1.2) as follows:
(C1) R0<1andb1≥0orR0<˜R0 (i.e., there is no endemic equilibrium);
(C2) ˜R0<R0<1andb1<0 (i.e., there are two endemic equilibria);
(C3) R0>1andH(I1)>0 (i.e., there is a unique endemic equilibrium, which is globally stable);
(C4) R0>1andH(I1)<0 (i.e., there is a unique endemic equilibrium, which is unstable. Further, there exists at least one limit cycle).
Then, in the next section, we first investigate the dynamic behaviours of system (1.3) through discussing the existence and stability of the disease-free periodic solution.
Letting I(t)=0 for all t≥0, then we consider the following subsystem
{dS(t)dt=A−δ1S,S(t)<ST,S(t+)=(1−pS(t)h1+S(t))S(t),S(t)=ST. | (3.1) |
Solving Eq (3.1) with initial condition S(0)=Sv(i.e., (1−pSTh1+ST)ST), we obtain
S(t)=A−(A−δ1Sv)exp(−δ1t)δ1 |
with period
T=1δ1lnA−δ1SvA−δ1ST. |
This indicates that system (1.3) has a disease-free periodic solution with period T, denoted as (ξ(t),0), with
ξ(t)=A−(A−δ1Sv)exp(−δ1(t−(k−1)T))δ1,(k−1)T<t≤kT,k∈N. | (3.2) |
Then we discuss the stability of the disease-free periodic solution (ξ(t),0). There are
a(S,I)=−pS2(t)h1+S(t),b(S,I)=−qI2(t)h2+I(t),ϕ(S,I)=S−ST,(ξ(T),η(T))=(ST,0),(ξ(T+),η(T+))=(Sv,0). |
Using Lemma A.1 in Appendix A, we obtain
Δ1=P+(∂b∂I∂ϕ∂S−∂b∂S∂ϕ∂I+∂ϕ∂S)+Q+(∂a∂S∂ϕ∂I−∂a∂I∂ϕ∂S+∂ϕ∂I)P∂ϕ∂S+Q∂ϕ∂I=P+(1−qI(2h2+I)(h2+I)2)P=P(ξ(T+),η(T+))(1−qI(2h2+I)(h2+I)2)P(ξ(T),η(T))=(1−qI(2h2+I)(h2+I)2)A−δ1SvA−δ1ST, |
and
exp(∫T0(∂P∂S(ξ(t),η(t))+∂Q∂I(ξ(t),η(t)))dt)=exp(∫T0(−δ1−δ2−γ−ϵ+βξ(t))dt)=exp(∫T0(−δ1−δ2−γ−ϵ+βAδ1−β(A−δ1Sv)exp(−δ1t)δ1)dt)=exp(βA−δ1(δ1+δ2+γ+ϵ)δ21lnA−δ1SvA−δ1ST−βpS2Tδ1(h1+ST))=(A−δ1SvA−δ1ST)βA−δ1(δ1+δ2+γ+ϵ)δ21exp(−βpS2Tδ1(h1+ST)). |
Therefore, there is
μ2=Δ1exp(∫T0(∂P∂S(ξ(t),η(t))+∂Q∂I(ξ(t),η(t)))dt)=(1−∂b∂I|I=0)(A−δ1SvA−δ1ST)βA−δ1(δ2+γ+ϵ)δ21exp(−βpS2Tδ1(h1+ST))≐Rb. | (3.3) |
Note that the relationship between μ2 and 1 determines the stability of the disease-free periodic solution, thus the Floquet multiplier μ2 can be defined as the control reproduction number of the state-dependent impulsive model (1.3), denoted by Rb, which is crucial to study the development of infectious diseases. From Eq (3.3), it is clear to see that A−δ1SvA−δ1ST>1. Furthermore, we can verify that if h2>0, then ∂b∂I|I=0=0 with
Rb=(A−δ1SvA−δ1ST)βA−δ1(δ2+γ+ϵ)δ21exp(−βpS2Tδ1(h1+ST)), |
while if h2=0, then ∂b∂I|I=0=−q with
Rb=(1−q)(A−δ1SvA−δ1ST)βA−δ1(δ2+γ+ϵ)δ21exp(−βpS2Tδ1(h1+ST)). |
For convenient, we denote
J≐βA−δ1(δ2+γ+ϵ)δ21lnA−δ1SvA−δ1ST+β(Sv−ST)δ1=∫STSvβs−δ2−γ−ϵA−δ1sds, |
thus,
Rb={(1−q)∗exp(J),ifh2=0,exp(J),ifh2>0. | (3.4) |
Based on above discussions, we have the following conclusions.
Theorem 3.1. If Rb<1 holds true, then the disease-free periodic solution of system (1.3) is locally stable, while if Rb>1, then the disease-free periodic solution of system (1.3) is unstable. Particularly, for cases (C1) and (C2), inequality Rb<1 always holds true, further, the disease-free periodic solution is globally stable for case (C1). For cases (C3) and (C4), the disease-free periodic solution is locally stable when ST≤¯S. Furthermore, for case (C3), the disease-free periodic solution is globally stable when ST≤min{¯S,S1}.
Proof We have R0<1 for cases (C1) and (C2), then there are βA−δ1(δ2+γ+ϵ)δ21<0 and 0<(A−δ1SvA−δ1ST)βA−δ1(δ2+γ+ϵ)δ21<1. Therefore, Rb<1 holds, which indicates that the disease-free periodic solution is orbitally asymptotically stable. For the global stability, we need to prove that the disease-free periodic solution (ξ(t),0) is globally attractive. It follows from the definition of the Poincarˊe map and the property of subsystem (1.2) that Poincarˊe map PM satisfies PM(I0)<I0 for I0≥0 for case (C1). Therefore, the disease-free periodic solution (ξ(t),0) is globally attractive for case (C1). For cases (C3) and (C4), letting
V(s)=βs−δ2−γ−ϵA−δ1s, |
we obtain
dV(s)ds=βA−δ1(δ2+γ+ϵ)(A−δ1s)2>0. |
Thus, V(s) is increasing for s∈(0,Aδ1) and V(¯S)=0 with ¯S=δ2+γ+ϵβ, which means that V(s)<0 and J<0 always hold for ST≤¯S<Aδ1. Thus, when ST≤¯S, we have Rb<1, correspondingly, the disease-free periodic solution is locally stable for cases (C3) and (C4). In addition, when ST≤min{¯S,S1}, we can similarly verify that the disease-free periodic solution (ξ(t),0) is globally attractive for case (C3). This completes the proof.
In the next two sections, we discuss the existence and stability of the positive order-1 periodic solution from two points of view: through investigating the bifurcations near the disease-free periodic solution and examining the properties of the Poincaré map including monotonicity, continuity, discontinuity and convexity.
Based on the discussions in the last section, for case (C3) or (C4), the sign of J can vary when ST>¯S, which indicates that system (1.3) may undergo bifurcations near the disease-free periodic solution as the parameter values vary. Therefore, we can discuss the bifurcations near the disease-free periodic solution by assuming R0>1 and ST>¯S. Consider subsystem (1.2) in the phase space, we define a scalar differential equation
{dIdS=Q(S,I)P(S,I)≐W(S,I),I(Sv)=I+0. | (4.1) |
For system (4.1), we focus on region
Ω1={(S,I)|S>0,I>0,I<A−δ1SβS}, |
in which function W(S,I) is continuously differentiable. Given initial condition (S0,I0), which belongs to the phase set on the Poincarˊe section, one obtains
I(S;S0,I0)=I0+∫SSvW(s,I(s;Sv,I0))ds. |
Then, PM takes the following form:
PM(I0,α)=w1(I(ST;Sv,I0)), |
where α represents a bifurcation parameter. Through some straightforward calculations, we get
∂I(S;Sv,I0)∂I0=exp(∫SSv∂W(s,I(s;Sv,I0))∂Ids),∂2I(S;Sv,I0)∂I20=∂I(S;Sv,I0)∂I0∫SSv∂2W(s,I(s;Sv,I0))∂I2∂I(s;Sv,I0)∂I0ds. |
Denoting
∂I(ST;Sv,I0)∂I0=∂g(I0;α)∂I0≐g′(I0;α), |
then, we have
∂PM∂I0(0,α)=[(1−qI(ST;Sv,I0)(2h2+I(ST;Sv,I0))(h2+I(ST;Sv,I0))2)g′(I0;α)]|I0=0=w′1(I(ST;Sv,0))g′(0;α)=Rb,∂2PM∂I20(0,α)=[(1−qI(ST;Sv,I0)(2h2+I(ST;Sv,I0))(h2+I(ST;Sv,I0))2)g″(I0;α)−2h22q(h2+I(ST;Sv,I0))3(g′(I0;α))2]|I0=0=w′1(I(ST;Sv,0))g″(0;α)−2h22q(h2+I(ST;Sv,0))3(g′(0;α))2,∂3PM∂I30(0,α)=[(1−qI(ST;Sv,I0)(2h2+I(ST;Sv,I0))(h2+I(ST;Sv,I0))2)g‴(I0;α)−6h22qg′(I0;α)g″(I0;α)(h2+I(ST;Sv,I0))3+6h22q(g′(I0;α))3(h2+I(ST;Sv,I0))4]|I0=0=w′1(I(ST;Sv,0))g‴(0;α)−6h22qg′(0;α)g″(0;α)(h2+I(ST;Sv,0))3+6h22q(g′(0;α))3(h2+I(ST;Sv,0))4, |
where
g′(0;α)=exp(∫STSv∂W(s,I(s;Sv,0))∂Ids)=exp(∫STSvβs−δ2−γ−ϵA−δ1sds)=(A−δ1SvA−δ1ST)βA−δ1(δ2+γ+ϵ)δ21exp(β(Sv−ST)δ1),g″(0;α)=g′(0;α)∫STSv∂2W(s,I(s;Sv,0))∂I2∂I(s;Sv,0)∂I0ds=g′(0;α)∫STSvm(s)∂I(s;Sv,0)∂I0ds,g‴(0;α)=g″(0;α)∫STSvm(s)∂I(s;Sv,0)∂I0ds+g′(0;α)∂∂I0(∫STSvm(s)∂I(s;Sv,0)∂I0ds), |
with
m(s)=∂2W(s,I(s;Sv,0))∂I2=2ωϵ(A−δ1s)+2βs(βs−δ2−γ−ϵ)(A−δ1s)2,∂I(s;Sv,0)∂I0=(A−δ1SvA−δ1s)βA−δ1(δ2+γ+ϵ)δ21exp(β(Sv−s)δ1). |
Based on above calculations, we mainly focus on discussing the transcritical and pitchfork bifurcations near the disease-free periodic solution with respect to the key parameters for h2>0. Note that all of the parameters appearing in the expression of Rb can be chosen as bifurcation parameters. In what follows, we choose control parameters, such as ϵ, p, ST and h1 to investigate the bifurcation near the disease-free periodic solution and the bifurcation with respect to other parameters can be studied by using similar method. Furthermore, the bifurcation near the disease-free periodic solution for h2=0 can be investigated similarly, and we study it by taking the parameter related to impulsive isolation strategy q as an example in such case.
In this subsection, ϵ is chosen as a bifurcation parameter. For h2>0, taking the derivative of Rb(ϵ) with respect to ϵ yields
∂Rb(ϵ)∂ϵ=−Rb(ϵ)δ1∗ln(A−δ1SvA−δ1ST)<0, |
which means that Rb(ϵ) is decreasing for ϵ∈(0,+∞). It is easy to verify that
lim |
Furthermore, if
R_b(0) = \left(\frac{A-\delta_1S_v}{A-\delta_1S_T}\right)^{\frac{\beta A-\delta_1(\delta_2+\gamma)}{\delta_1^2}}\exp\left(-\frac{\beta pS_T^2}{\delta_1\left(h_1+S_T\right)}\right) \gt 1, |
then we have that there is a unique \epsilon^*\in(0, +\infty) such that R_b(\epsilon^*) = 1 and \frac{\partial R_b(\epsilon^*)}{\partial \epsilon} < 0 with \epsilon^* satisfying
\begin{array}{c} \left(\frac{A-\delta_1S_v}{A-\delta_1S_T}\right)^{\frac{\beta A-\delta_1(\delta_2+\gamma+\epsilon^*)}{\delta_1^2}}\exp\left(-\frac{\beta pS_T^2}{\delta_1\left(h_1+S_T\right)}\right) = 1. \end{array} |
Therefore, we have the main results as follows.
Proposition 4.1. Suppose h_2 > 0 , R_0 > 1 and S_T > \overline{S} . If R_b(0) > 1 holds true, then there exists a unique \epsilon^*\in(0, +\infty) such that R_b(\epsilon^*) = 1 with \frac{\partial R_b(\epsilon^*)}{\partial \epsilon} < 0 . And the disease-free periodic solution (\xi(t), 0) of system (1.3) is orbitally asymptotically stable for \epsilon\in(\epsilon^*, +\infty) and unstable for \epsilon\in(0, \epsilon^*) .
Next, we consider the bifurcation near the disease-free periodic solution at \epsilon = \epsilon^* . We have that \mathcal{P}_M(0, \epsilon) = 0 always holds, further,
\frac{\partial \mathcal{P}_M}{\partial I_0}(0, \epsilon^*) = 1, \; \; \frac{\partial^2 \mathcal{P}_M}{\partial I_0\partial \epsilon}(0, \epsilon^*) \lt 0, \; \; \frac{\partial^2 \mathcal{P}_M}{\partial I_0^2}(0, \epsilon^*) = g''(0;\epsilon^*)-\frac{2q}{h_2}. |
Note that if g''(0;\epsilon^*)\neq \frac{2q}{h_2} , then \frac{\partial^2 \mathcal{P}_M}{\partial I_0^2}(0, \epsilon^*)\neq 0 . Furthermore, g''(0;\epsilon^*) > \frac{2q}{h_2} indicates \frac{\partial^2 \mathcal{P}_M}{\partial I_0^2}(0, \epsilon^*) > 0 , while g''(0;\epsilon^*) < \frac{2q}{h_2} means \frac{\partial^2 \mathcal{P}_M}{\partial I_0^2}(0, \epsilon^*) < 0 . As for the special condition \frac{\partial^2 \mathcal{P}_M}{\partial I_0^2}(0, \epsilon^*) = 0 \left(\mbox{i.e.}, \; g''(0;\epsilon^*) = \frac{2q}{h_2}\right) , we further consider the sign of \frac{\partial^3 \mathcal{P}_M}{\partial I_0^3}(0, \epsilon^*) . Note that
\frac{\partial^3 \mathcal{P}_M}{\partial I_0^3}(0, \epsilon^*) = g'''(0;\epsilon^*)-\frac{6q(2q-1)}{h_2^2}, |
thus, \frac{\partial^3 \mathcal{P}_M}{\partial I_0^3}(0, \epsilon^*)\neq 0 when g'''(0;\epsilon^*)\neq \frac{6q(2q-1)}{h_2^2}. Based on above discussions and Lemma A.2 and Lemma A.3 presented in Appendix A, we have the following conclusions.
Theorem 4.1. Suppose h_2 > 0 , R_0 > 1 , S_T > \overline{S} and R_b(0) > 1 . We have:
(a) If g''(0;\epsilon^*) > \frac{2q}{h_2} holds true, then the Poinceré map \mathcal{P}_M(I_0, \epsilon) undergoes a transcritical bifurcation at \epsilon = \epsilon^* . Further, an unstable positive fixed point appears when \epsilon passes through \epsilon = \epsilon^* from left to right. Correspondingly, system (1.3) has an unstable positive periodic solution for \epsilon\in(\epsilon^*, \epsilon^*+\varepsilon) with \varepsilon > 0 small enough;
(b) If g''(0;\epsilon^*) < \frac{2q}{h_2} holds true, then a stable positive fixed point appears when \epsilon passes through \epsilon = \epsilon^* from right to left. Correspondingly, system (1.3) has a stable positive periodic solution for \epsilon\in(\epsilon^*-\varepsilon, \epsilon^*) with \varepsilon > 0 small enough;
(c) If g''(0;\epsilon^*) = \frac{2q}{h_2} and g'''(0;\epsilon^*) > \frac{6q(2q-1)}{h_2^2} , then the Poincar \acute{e} map \mathcal{P}_M(I_0, \epsilon) undergoes a pitchfork bifurcation at \epsilon = \epsilon* . Accordingly, system (1.3) has an unstable positive periodic solution for \epsilon\in(\epsilon^*, \epsilon^*+\varepsilon) with \varepsilon > 0 small enough;
(d) If g''(0;\epsilon^*) = \frac{2q}{h_2} and g'''(0;\epsilon^*) < \frac{6q(2q-1)}{h_2^2} , then \mathcal{P}_M(I_0, \epsilon) undergoes a pitchfork bifurcation at \epsilon = \epsilon^* . Accordingly, system (1.3) has a stable positive periodic solution for \epsilon\in(\epsilon^*-\varepsilon, \epsilon^*) with \varepsilon > 0 small enough.
When h_2 > 0 , R_b can be written as a function with respect to parameter p , given as:
R_b(p) = \left(\frac{A-\delta_1S_v}{A-\delta_1S_T}\right)^{\frac{\beta A-\delta_1(\delta_2+\gamma+\epsilon)}{\delta_1^2}}\exp \left(-\frac{\beta pS_T^2}{\delta_1(h_1+S_T)}\right). |
Taking the derivative of R_b(p) with respect to p , we obtain
\begin{array}{c} \frac{\partial R_b(p)}{\partial p} = \frac{R_b(p)S_T^2}{(A-\delta_1S_v)(h_1+S_T)}\left[\beta S_v-(\delta_2+\gamma+\epsilon)\right].\\ \end{array} |
It is clear that \frac{R_b(p)S_T^2}{(A-\delta_1S_v)(h_1+S_T)} > 0 , thus the sign of \frac{\partial R_b(p)}{\partial p} is determined by \beta S_v-\delta_2-\gamma-\epsilon . Solving \frac{\partial R_b(p)}{\partial p} = 0 , we obtain a unique root, denoted by \overline{p} , with
\overline{p} = \left(1+\frac{h_1}{S_T}\right)\left(1-\frac{\overline{S}}{S_T}\right). |
We further assume \frac{h_1S_T}{h_1+S_T}\leq \overline{S} to ensure that \overline{p}\in(0, 1) . As a result, there is a unique \overline{p} such that S_v < \overline{S} and \frac{\partial R_b(p)}{\partial p} < 0 for p > \overline{p} , while S_v > \overline{S} and \frac{\partial R_b(p)}{\partial p} > 0 for p < \overline{p} , which means that R_b(p) is increasing on the interval (0, \overline{p}] and decreasing on the interval [\overline{p}, 1) . Furthermore,
\begin{array}{c} R_b(0) = 1, \; \; R_b(\overline{p}) = \exp\left(\int_{\overline S}^{S_T}\frac{\beta s-\delta_2-\gamma-\epsilon}{A-\delta_1s}ds\right) \gt 1.\\ \end{array} |
Therefore, considering the monotonicity of R_b(p) , we have
(1) If p\in(0, \overline{p}) , then R_b(p) > 1 always holds, which indicates that the disease-free periodic solution ( \xi(t), 0 ) is unstable.
(2) If p\in(\overline{p}, 1) and R_b(1) > 1 , then R_b(p) > 1 for p\in(0, 1) , indicating that ( \xi(t), 0 ) is always unstable.
(3) If p\in(\overline{p}, 1) and R_b(1) < 1 , then there is a unique p^* satisfying R_b(p^*) = 1 . This means that ( \xi(t), 0 ) is unstable for p\in(\overline{p}, p^*) , while ( \xi(t), 0 ) is stable for p\in(p^*, 1) , indicating that the bifurcations could occur at p = p^* .
Proposition 4.2. Suppose h_2 > 0 , R_0 > 1 and S_T > \overline{S} . If R_b(1) > 1 holds true, then the disease-free periodic solution (\xi(t), 0) is always unstable for p\in(0, 1) ; If R_b(1) < 1 holds, then the disease-free periodic solution (\xi(t), 0) is unstable for p\in(0, p^*] and orbitally asymptotically stable for p\in[p^*, 1) .
Based on above discussions, we next consider the bifurcations with respect to p . We have \mathcal{P}_M(0, p) = 0 for all p\in(0, 1) , and it is easy to see that
\begin{array}{l} \frac{\partial \mathcal{P}_M}{\partial I_0}(0, p^*) = R_b(p^*) = 1, \; \; \frac{\partial^2 \mathcal{P}_M}{\partial I_0\partial p}(0, p^*) = \frac{\partial R_b(p^*)}{\partial p} \lt 0. \end{array} |
Moreover, there are
\begin{equation} \begin{array}{lll} g''(0;p^*)& = &g'(0;p^*)\int_{S_{vp^*}}^{S_T}m(s)\frac{\partial I(s;S_v, 0)}{\partial I_0}ds = \int_{S_{vp^*}}^{S_T}m(s)\frac{\partial I(s;S_v, 0)}{\partial I_0}ds, \\ g'''(0;p^*)& = &\frac{4q^2}{h_2^2}+\frac{\partial }{\partial I_0}\left(\int_{S_{vp^*}}^{S_T}m(s)\frac{\partial I(s;S_v, 0)}{\partial I_0}ds\right), \end{array} \end{equation} | (4.2) |
with S_{vp^*} = \left(1-\frac{p^*S_T}{h_1+S_T}\right)S_T . Thus,
\begin{equation} \begin{array}{lll} \frac{\partial^2 \mathcal{P}_M}{\partial I_0^2}(0, p^*)& = &g''(0;p^*)-\frac{2q}{h_2}.\\ \frac{\partial^3 \mathcal{P}_M}{\partial I_0^3}(0, p^*)& = &g'''(0;p^*)-\frac{6q(2q-1)}{h_2^2}.\\ \end{array} \end{equation} | (4.3) |
Based on above discussions and Lemma A.2 and Lemma A.3 presented in Appendix A, we conclude as follows.
Theorem 4.2. Suppose h_2 > 0 , R_0 > 1 , S_T > \overline{S} and R_b(1) < 1 . We have:
(a) If g''(0;p^*) > \frac{2q}{h_2} holds true, then the Poincar \acute{e} map \mathcal{P}_M(I_0, p) undergoes a transcritical bifurcation at p^* . Moreover, an unstable positive fixed point appears when p changes through p = p^* from left to right. Then system (1.3) accordingly has an unstable positive periodic solution if p\in(p^*, p^*+\varepsilon) with \varepsilon > 0 small enough;
(b) If g''(0;p^*) < \frac{2q}{h_2} holds true, then a stable positive fixed point of map \mathcal{P}_M(I_0, p) appears when p changes through p = p^* from right to left. System (1.3) accordingly has a stable positive periodic solution if p\in(p^*-\varepsilon, p^*) with \varepsilon > 0 small enough;
(c) If g''(0;p^*) = \frac{2q}{h_2} and g'''(0;p^*) > \frac{6q(2q-1)}{h_2^2} hold, then the Poincar \acute{e} map \mathcal{P}_M(I_0, p) undergoes a pitchfork bifurcation at p = p^* . Correspondingly, system (1.3) has an unstable positive periodic solution if p\in(p^*, p^*+\varepsilon) with \varepsilon > 0 small enough;
(d) If g''(0;p^*) = \frac{2q}{h_2} and g'''(0;p^*) < \frac{6q(2q-1)}{h_2^2} hold, then \mathcal{P}_M(I_0, p) undergoes a pitchfork bifurcation at p = p^* . Correspondingly, system (1.3) has a stable positive periodic solution if p\in(p^*-\varepsilon, p^*) with \varepsilon > 0 small enough.
In this subsection, we choose S_T as a bifurcation parameter. When h_2 > 0 , we take the derivative of R_b(S_T) with respect to S_T and obtain
\begin{array}{c} \frac{\partial R_b(S_T)}{\partial S_T} = \exp(J(S_T))\frac{\partial J(S_T)}{\partial S_T}, \\ \end{array} |
with \frac{\partial J(S_T)}{\partial S_T} = \frac{\beta S_T-(\delta_2+\gamma+\epsilon)}{A-\delta_1 S_T}-\left(1-\frac{pS_T(2h_2+S_T)}{(h_2+S_T)^2}\right)\frac{\beta S_v-(\delta_2+\gamma+\epsilon)}{A-\delta_1 S_v}. Denote f(x) = \frac{\beta s-(\delta_2+\gamma+\epsilon)}{A-\delta_1 s} , we have
\frac{\partial J(S_T)}{\partial S_T} = f(S_T)-\left(1-\frac{pS_T(2h_2+S_T)}{(h_2+S_T)^2}\right)f(S_v). |
Furthermore, there is
\begin{array}{c} f'(x) = \frac{\beta A-\delta_1(\delta_2+\gamma+\epsilon)}{(A-\delta_1 x)^2} \gt 0.\\ \end{array} |
Thus, f(x) is monotonically increasing with respect to x . In what follows, we discuss the sign of \frac{\partial J(S_T)}{\partial S_T} :
(1) If S_v\leq \overline{S} , then f(S_v)\leq 0 , which indicates that \frac{\partial J(S_T)}{\partial S_T} > 0 always holds;
(2) If S_v > \overline{S} , then f(S_v) > 0 , and one has
\frac{\partial J(S_T)}{\partial S_T} \gt f(S_v)-\left(1-\frac{pS_T(2h_2+S_T)}{(h_2+S_T)^2}\right)f(S_v) = \frac{pS_T(2h_2+S_T)}{(h_2+S_T)^2}f(S_v) \gt 0. |
This means that \frac{\partial J(S_T)}{\partial S_T} > 0 holds under both conditions. Hence, \frac{\partial R_b(S_T)}{\partial S_T} > 0 holds, i.e., R_b(S_T) is monotonically increasing with respect to S_T . Denoting K\doteq \frac{A}{\delta_1} for convenience, then we have
\begin{array}{c} R_b(\overline{S}) \lt 1, \; \; \lim_{S_T\rightarrow K^-}R_b(S_T) = +\infty. \end{array} |
Thus, there is a unique S_T^*\in(\overline{S}, K) such that R_b(S_T^*) = 1 . Based on above discussions, we conclude the following main results.
Proposition 4.3. Suppose h_2 > 0 and R_0 > 1 . There is a unique S_T^*\in(\overline{S}, K) satisfying R_b(S_T^*) = 1 . The disease-free periodic solution (\xi(t), 0) of system (1.3) is orbitally asymptotically stable for S_T\in(\overline{S}, S_T^*) and unstable for S_T\in(S_T^*, K) .
In what follows, we discuss the bifurcation near the disease-free periodic solution at S_T = S_T^* . Similarly, \mathcal{P}_M(0, S_T) = 0 holds for all S_T\in(\overline{S}, K) , and
\begin{array}{c} \frac{\partial \mathcal{P}_M}{\partial I_0}(0, S_T^*) = 1, \; \; \frac{\partial^2 \mathcal{P}_M}{\partial I_0\partial S_T}(0, S_T^*) \gt 0, \\ \frac{\partial^2 \mathcal{P}_M}{\partial I_0^2}(0, S_T^*) = g''(0;S_T^*)-\frac{2q}{h_2}, \; \; \frac{\partial^3 \mathcal{P}_M}{\partial I_0^3}(0, S_T^*) = g'''(0;S_T^*)-\frac{6q(2q-1)}{h_2^2}.\\ \end{array} |
Therefore, we obtain the following results.
Theorem 4.3. Suppose h_2 > 0 and R_0 > 1 . We have:
(a) If g''(0;S_T^*) > \frac{2q}{h_2} holds true, then an unstable positive fixed point appears when S_T goes through S_T = S_T^* from right to left. Correspondingly, system (1.3) has an unstable positive periodic solution if S_T\in(S_T^*-\varepsilon, S_T^*) with \varepsilon > 0 small enough;
(b) If g''(0;S_T^*) < \frac{2q}{h_2} holds true, then a stable positive fixed point appears when S_T goes through S_T = S_T^* from left to right. Correspondingly, system (1.3) has a stable positive periodic solution if S_T\in(S_T^*, S_T^*+\varepsilon) with \varepsilon > 0 small enough.
(c) If g''(0;S_T^*) = \frac{2q}{h_2} and g'''(0;S_T^*) > \frac{6q(2q-1)}{h_2^2} , then system (1.3) has an unstable positive periodic solution if S_T\in(S_T^*-\varepsilon, S_T^*) with \varepsilon > 0 small enough;
(d) If g''(0;S_T^*) = \frac{2q}{h_2} and g'''(0;S_T^*) < \frac{6q(2q-1)}{h_2^2} hold true, then system (1.3) has a stable positive periodic solution if S_T\in(S_T^*, S_T^*+\varepsilon) with \varepsilon > 0 small enough.
In this subsection, we choose h_1 as a bifurcation parameter and consider R_b as a function of h_1 , which can help us to reveal the impact of the saturation phenomenon of state-dependent feedback control on infectious diseases. When h_2 > 0 , we have R_b(h_1) = \exp(J(h_1)) . By simple calculations we have
R_b(0) = \left(\frac{A-\delta_1(1-p)S_T}{A-\delta_1S_T}\right)^{\frac{\beta A-\delta_1(\delta_2+\gamma+\epsilon)}{\delta_1^2}}\exp\left(-\frac{\beta pS_T}{\delta_1}\right), \; \; \; \; \lim\limits_{h_1\rightarrow +\infty}R_b(h_1) = 1. |
Moreover, taking the derivative of R_b(h_1) with respect to h_1 yields
\frac{\partial R_b(h_1)}{\partial h_1} = \frac{pS_T^2R_b(h_1)}{(A-\delta_1S_v)(h_1+S_T)^2)}*(\delta_2+\gamma+\epsilon-\beta S_v). |
Solving \frac{\partial R_b(h_1)}{\partial h_1} = 0 , we obtain a unique root \overline{h}_1 with
\overline{h}_1 = \frac{S_T(\overline{S}-(1-p)S_T)}{S_T-\overline{S}}. |
If h_1 < \overline{h}_1 , then \frac{\partial R_b(h_1)}{\partial h_1} > 0 holds, while if h_1 > \overline{h}_1 holds, then \frac{\partial R_b(h_1)}{\partial h_1} < 0 , indicating that R_b(h_1) is increasing for h_1 < \overline{h}_1 and decreasing for h_1 > \overline{h}_1 . If \overline{S} < (1-p)S_T , then we have \overline{h}_1 < 0 and correspondingly, R_b(h_1) is decreasing on the interval (0, +\infty) . Thus, R_b(h_1) > 1 always holds and the disease-free periodic solution (\xi(t), 0) is unstable and there is no bifurcation near (\xi(t), 0) . If \overline{S} > (1-p)S_T , then we have \overline{h}_1 > 0 . Therefore, R_b(h_1) is increasing on the interval (0, \overline{h}_1] and decreasing on the interval [\overline{h}_1, +\infty) . Under this case, when R_b(0) > 1 , then R_b(h_1) > 1 always holds for h_1\in(0, +\infty) , which means that the disease-free periodic solution (\xi(t), 0) is unstable and there is no bifurcation near (\xi(t), 0) . On the other hand, when R_b(0) < 1 , there is a unique h_1^*\in(0, \overline{h}_1) such that R_b(h_1^*) = 1 with \frac{\partial R_b(h_1^*)}{\partial h_1} > 0 .
Therefore, we have the main conclusions as follows.
Proposition 4.4. Suppose h_2 > 0 , R_0 > 1 and S_T > \overline{S} > (1-p)S_T . If R_b(0) < 1 holds, then there exists a unique h_1^*\in(0, \overline{h}_1) satisfying R_b(h_1^*) = 1 with \frac{\partial R_b(h_1^*)}{\partial h_1} > 0 . Accordingly, the disease-free periodic solution (\xi(t), 0) of system (1.3) is orbitally asymptotically stable for h_1\in(0, h_1^*) and unstable for h_1\in(h_1^*, +\infty) .
As for the bifurcation of the disease-free periodic solution (\xi(t), 0) at h_1^* , we have \mathcal{P}_M(0, h_1) = 0 for all h_1\in(0, +\infty) , and
\begin{array}{c} \frac{\partial \mathcal{P}_M}{\partial I_0}(0, h_1^*) = 1, \; \; \frac{\partial^2 \mathcal{P}_M}{\partial I_0\partial h_1}(0, h_1^*) \gt 0, \\ \frac{\partial^2 \mathcal{P}_M}{\partial I_0^2}(0, h_1^*) = g''(0;h_1^*)-\frac{2q}{h_2}, \; \; \frac{\partial^3 \mathcal{P}_M}{\partial I_0^3}(0, h_1^*) = g'''(0;h_1^*)-\frac{6q(2q-1)}{h_2^2}. \end{array} |
Therefore, we have the following conclusions.
Theorem 4.4. Suppose h_2 > 0 , R_0 > 1 , S_T > \overline{S} and R_b(0) < 1 . We obtain:
(a) If g''(0;h_1^*) > \frac{2q}{h_2} holds, then the Poinceré map \mathcal{P}_M(I_0, h_1) undergoes a transcritical bifurcation at h_1 = h_1^* . Further, an unstable positive fixed point appears when h_1 passes through h_1 = h_1^* from right to left. Accordingly, system (1.3) has an unstable positive periodic solution for h_1\in(h_1^*-\varepsilon, h_1^*) with \varepsilon > 0 small enough;
(b) If g''(0;h_1^*) < \frac{2q}{h_2} holds, then a stable positive fixed point appears when h_1 passes through h_1 = h_1^* from left to right. Then, system (1.3) has a stable positive periodic solution for h_1\in(h_1^*, h_1^*+\varepsilon) with \varepsilon > 0 small enough;
(c) If g''(0;h_1^*) = \frac{2q}{h_2} and g'''(0;h_1^*)\neq \frac{6q(2q-1)}{h_2^2} holds, then the Poinceré map \mathcal{P}_M(I_0, h_1) undergoes a pitchfork bifurcation at h_1 = h_1^* . Accordingly, system (1.3) has a positive periodic solution.
Note that the bifurcation with respect to the demographic parameters, such as the recruitment rate A , can also be studied. Here we only give the main conclusions for the bifurcation with respect to A , and the detailed analyses are given in Appendix B.
Theorem 4.5. Suppose h_2 > 0 , R_0 > 1 , and S_T > \overline{S} > \frac{S_v+S_T}{2} . If g''(0;A^*)\neq \frac{2q}{h_2} holds true, then the Poincar \acute{e} map \mathcal{P}_M(I_0, A) occurs with a transcritical bifurcation at A = A^* . Thus, a positive fixed point appears when A goes through A = A^* , and correspondingly, system (1.3) has a positive periodic solution. However, if g''(0;A^*) = \frac{2q}{h_2} and g'''(0;A^*)\neq\frac{6q(2q-1)}{h_2^2} hold, then the Poincar \acute{e} map \mathcal{P}_M(I_0, A) undergoes a pitchfork bifurcation at A = A^* . Thus, a positive fixed point appears when A passes through A = A^* , and accordingly, system (1.3) has a positive periodic solution.
So far we have discussed the bifurcation with respect to key parameters including \epsilon , p , S_T , h_1 and A for h_2 > 0 . Similarly, we can also investigate the bifurcation with these parameters for h_2 = 0 , and list the main results with respect to parameter q in the following and find details in Appendix B.
Theorem 4.6. Suppose h_2 = 0 , R_0 > 1 , S_T > \overline{S} and J > 0 . If g''(0;q^*)\neq 0 holds true, then the Poincar \acute{e} map \mathcal{P}_M(I_0, q) undergoes a transcritical bifurcation at q = q^* . In fact, if g''(0;q^*) > 0 holds true, then an unstable positive fixed point appears when q goes through q = q^* from left to right. Correspondingly, system (1.3) has an unstable positive periodic solution if q\in(q^*, q^*+\varepsilon) with \varepsilon > 0 small enough. However, if g''(0;q^*) < 0 , then the Poincar \acute{e} map \mathcal{P}_M(I_0, q) has a stable positive fixed point when p passes through q = q^* from right to left. Correspondingly, system (1.3) has a stable positive periodic solution if q\in(q^*-\varepsilon, q^*) with \varepsilon > 0 small enough.
Through numerical simulation, we verify the existence of the transcritical bifurcation with respect to some key parameters. We illustrated the relationships between R_0 and R_b with respect to parameters p, \epsilon, A, q (shown in Figure 2) and parameter S_T (shown in Figure 3(A)). We find that for all these parameters, there exists a threshold value such that R_b = 1 . This confirms the existence of the transcritical bifurcation by choosing these parameters as bifurcation parameters. As shown in Figure 3, the disease-free periodic solution is locally stable for S_T < S_T^* and unstable for S_T > S_T^* . Correspondingly, in Figure 3(D), we choose S_T = 3.6 such that S_T > S_T^* , and show that the disease-free periodic solution is unstable and all the orbits finally tend to the positive equilibrium E_1 . In Figure 3(C), as we decrease the parameter value of S_T to 2.8 such that S_T < S_T^* , the disease-free periodic solution becomes stable, which is bistable with the positive equilibrium E_1 . It follows from Figure 3(C) that an unstable positive order-1 periodic solution appears as well via the transcritical bifurcation. Furthermore, in Figure 3(B), by choosing S_T = 1.7 such that S_T < S_1 , the disease-free periodic solution becomes globally stable.
Similarly, we verified the main theoretical results and showed in Figure 4 that when the ODE subsystem has limit cycles, system (1.3) can also undergo the transcritical bifurcation with an unstable positive order-1 periodic solution appearing. Specifically, when there exists a unique stable limit cycle of subsystem (1.2) (Figure 4(A) and (B)), if we decrease the threshold value S_T from S_T = 3.6 (Figure 4(B)) to S_T = 3.4 (Figure 4(A)), then an unstable positive order-1 periodic solution appears and the limit cycle of subsystem (1.2) is bistable with the disease-free periodic solution, shown in Figure 4(A). Similar phenomenons are illustrated in Figure 4(C) and (D) when there are two limit cycles of subsystem (1.2). Note that in Figure 4, we have chosen the threshold level relatively large such that the impulsive line S = S_T did not intersect with limit cycles. If the impulsive line intersects with the limit cycle, the Poincaré map of the system becomes very complex [30] while the dynamical behaviours are very rich and complicated. In Figure 5, we showed that by changing the parameter value of p , the unstable positive order-1 periodic solution (bifurcated from the disease-free periodic solution) can co-exist with a stable positive order-1 periodic solution (Figure 5(A)), or a stable positive order-2 periodic solution (Figure 5(B)), or a stable positive order-3 periodic solution (Figure 5(C)). For another aspect, the existence of order-3 periodic solution implies the existence of the phenomenon of chaos, which is illustrated in Figure 5(D).
In order to further discuss the existence and stability of the positive order-1 periodic solution of system (1.3), we initially define the impulsive set and phase set of the Poincaré map for various cases. For case ( C_1 ), the disease-free equilibrium E_0\left(\frac{A}{\delta_1}, 0\right) is globally asymptotically stable. As shown in Figure 6(A), depending on the properties of the vector fields of subsystem (1.2), it is easily verified that there is an orbit \Gamma_1 tangent to S_{S_v} at point Q_{S_v} = \left(S_v, I_{S_v}\right) with I_{S_v} = \frac{A-\delta_1S_v}{\beta S_v} . The intersection point of \Gamma_1 to S_{S_T} can be denoted as
\begin{array}{c} Q^* = \left(S_T, I^*\right) = \left(S_T, I\left(S_T;S_v, I_{S_v}\right)\right). \end{array} |
Then the impulsive set is
M_1 = \{(S, I)|S = S_T, I\in[0, I^*]\}, |
and the phase set can be defined as:
N_1 = \{(S^+, I^+)|S^+ = S_v, I^+\in[0, w_1(I^*)]\}. |
For case (C_2) , due to the complex trajectories of subsystem (1.2), we cannot determine the exact domains of the impulsive set and phase set. Under scenario (C_3) , there exists a unique endemic equilibrium E_1(S_1, I_1) which is globally stable. In what follows, we consider \Delta < 0 implying E_1 is a focus. If S_T < S_1 holds, denoted as case (C_{31}) , we can define the definitions of impulsive set and phase set of system (1.3) as M_1 and N_1 , respectively, which is similar to case ( C_1 ).
When S_T > S_1 , there is an orbit \Gamma_2 tangent to section S_{S_T} at point Q_{S_T} = (S_T, I_{S_T}) with I_{S_T} = \frac{A-\delta_1S_T}{\beta S_T} and \Gamma_2 intersects with line l_1 at point L(S_l, I_l) , as shown in Figure 6(B-C). Then we consider the following two subcases:
(C_{32})\; \; \; \; S_v \lt S_l\; \; \; \; \mbox{and}\; \; \; \; (C_{33})\; \; \; \; S_v\geq S_l. |
For subcase (C_{32}) , the impulsive set and phase set are M_1 and N_1 , respectively, through similar methods used for case (C_1) . Note that for (C_{33}) , the orbit \Gamma_2 intersects with line l_4 at two points B_1(S_v, I_{b_1}) and B_2(S_v, I_{b_2}) with I_{b_1} < I_{b_2} , shown in Figure 6(C). Moreover, the orbit \Gamma_2 will reach line l_4 at Q_{S_v} = (S_v, I_{S_v}) with I(S_T; S_v, I_{S_v}) = I_{S_T} . This indicates that any solution of system (1.3) with initial value (S_v, I_0^+) , where I_0^+\in(0, I_{S_v}) , will reach l_4 in a finite time. Thus, we can define the impulsive set and the phase set of system (1.3) as:
\begin{array}{c} M_2 = \left\{(S, I)|S = S_T, I\in[0, I_{S_T}]\right\}, \end{array} |
and
\begin{array}{c} N_2 = \left\{(S^+, I^+)|S = S_v, I^+\in[0, w_1(I_{S_T})]\cap [0, I_{b_1}] \right\}. \end{array} |
For case (C_4) , there exists at least one limit cycle. Assuming that E_1 is an unstable focus and there is a unique stable limit cycle of subsystem (1.2), shown in Figure 1(C), then we discuss the impulsive set and the phase set for the Poincaré map \mathcal{P}_M of system (1.3). In this circumstance, the limit cycle intersects with line l_1 at two points T_1(S_{t_1}, I_{t_1}) and T_2(S_{t_2}, I_{t_2}) with S_{t_1} < S_{t_2} . Depending on the positions between S_T , S_1 and S_{t_2} , we consider three subcases as follows:
(C_{41})\; \; \; \; S_T\leq S_1, \; \; \; \; (C_{42})\; \; \; \; S_1 \lt S_T \lt S_{t_2}, \; \; \; \; \mbox{and}\; \; \; \; (C_{43})\; \; \; \; S_T\geq S_{t_2}. |
When (C_{41}) holds true, by using similar methods for case (C_1) , it is clear that the impulsive set and the phase set are M_1 and N_1 , respectively. When S_1 < S_T < S_{t_2} (i.e., subcase (C_{42}) ), we consider:
(C_{42}^a)\; \; \; \; S_v\leq S_{t_1}, \; \; \; \; (C_{42}^b)\; \; \; \; S_{t_1} \lt S_v \lt S_1, \; \; \; \; \mbox{and}\; \; \; \; (C_{42}^c)\; \; \; \; S_1\leq S_v \lt S_T. |
If S_v\leq S_{t_1} (i.e., (C_{42}^a) ) holds, the impulsive set and the phase set can also be defined as M_1 and N_1 , respectively. For subcase (C_{42}^b) , there are two possible cases depending on whether orbit \Gamma_2 crosses line l_4 before it is tangents to line l_3 at point Q_{S_T} . If \Gamma_2 crosses line l_4 before it is tangents to line l_3 and \Gamma_2 intersects with line l_4 at two points \gamma_1(S_{\gamma_1}, I_{\gamma_1}) and \gamma_2(S_{\gamma_2}, I_{\gamma_2}) with I_{\gamma_1} < I_{\gamma_2} , denoted as case C_{42}^{b_1} , the impulsive set is defined as M_2 and the phase set is
N_3 = \left\{(S^+, I^+)|S = S_v, I^+\in[0, w_1(I_{S_T})] \right\}. |
However, if \Gamma_2 crosses line l_4 after it is tangents to line l_3 , denoted as case C_{42}^{b_2} , then the impulsive set and the phase set are M_1 and N_1 , respectively.
For subcase (C_{42}^c) , the impulsive set and the phase set can be similarly defined as those for subcase (C_{42}^b) with M_2 and N_3 , respectively.
When S_T\geq S_{t_2} (i.e., (C_{43}) ), depending on the position between S_v and S_{t_1} , we consider the following two subcases:
(C_{43}^a)\; \; \; \; S_v \lt S_{t_1}\; \; \; \; \mbox{and}\; \; \; \; (C_{43}^b)\; \; \; \; S_v\geq S_{t_1}. |
Under scenario (C_{43}^a) , the impulsive set and the phase set are defined as M_1 and N_1 , respectively. However, for subcase (C_{43}^b) , the limit cycle intersects with line l_4 at two points C_1(S_{c_1}, I_{c_1}) and C_2(S_{c_2}, I_{c_2}) with I_{c_1} < I_{c_2} and it is clear that the impulsive set and the phase set are M_2 and N_3 , respectively.
In this subsection, based on above discussions of the impulsive set and the phase set of the Poincaré map, we further discuss the existence and stability of the positive order-1 periodic solution of system (1.3) through analyzing the properties of the Poincaré map. As we mentioned above, based on various ODE dynamical behaviors, the definition of \mathcal{P}_M , especially for the domain and the range of it, could be various. Thus, we also consider the properties of the Poincaré map in different cases of the dynamics of the ODE subsystem. Due to the complex trajectories of subsystem (1.2) for case (C_2) , we cannot determine the exact domains of the impulsive set and the phase set, indicating that it is difficult to study the properties of the Poincaré map for case (C_2) . Therefore, we focus on investigating the properties of the Poincaré map for cases (C_1) , (C_3) and (C_4) . For case (C_1) , we have the following results.
Theorem 5.1. For case (C_1) , the Poincar \acute{\mathrm{e}} map \mathcal{P}_M of system (1.3) satisfies the following properties.
(1) The domain and range of \mathcal{P}_M are [0, +\infty) and [0, w_1(I^*)] , respectively. \mathcal{P}_M is increasing on [0, I_{S_v}] and decreasing on [I_{S_v}, +\infty) ;
(2) \mathcal{P}_M is continuously differentiable on its domain and convex on [0, I_{S_v}] provided that \frac{\partial^2 \mathcal{P}_M(I_0)}{\partial I_0^2} > 0 for all I_0\in[0, I_{S_v}] ;
(3) There exists no positive fixed point for \mathcal{P}_M .
Proof (1) The vector field of system (1.3) without impulsive strategies implies that the domain of \mathcal{P}_M is [0, +\infty) . For any I_{k1}^+, I_{k2}^+\in [0, I_{S_v}] with I_{k1}^+ < I_{k2}^+ , it is clear that g(I_{k1}^+) < g(I_{k2}^+) , and consequently, \mathcal{P}_M(I_{k1}^+) < \mathcal{P}_M(I_{k2}^+) . For any I_{k1}^+, I_{k2}^+\in [I_{S_v}, +\infty) with I_{k1}^+ < I_{k2}^+ , the orbits initiating from (S_v, I_{k1}^+) and (S_v, I_{k2}^+) will cross line l_4 before they hit line l_3 . Denoting the vertical coordinates of the two orbits intersecting with line l_4 as I_{q1}^+ and I_{q2}^+ , we note that I_{q1}^+ > I_{q2}^+ . Similarly, we have g(I_{q1}^+) > g(I_{q2}^+) and \mathcal{P}_M(I_{k1}^+) = \mathcal{P}_M(I_{q1}^+) > \mathcal{P}_M(I_{q2}^+) = \mathcal{P}_M(I_{k2}^+) . Therefore, \mathcal{P}_M is increasing on the interval [0, I_{S_v}] and decreasing on the interval [I_{S_v}, +\infty) . Meanwhile, The range of \mathcal{P}_M is [0, \mathcal{P}_M(I_{S_v})] (i.e., [0, w_1(I^*)] ).
(2) It follows from (4.1) that
\begin{array}{lll} \frac{\partial W(S, I)}{\partial I}& = &\frac{(A-\delta_1 S)\left(\beta S-\delta_2-\gamma-\frac{\epsilon}{(1+\omega I)^2}\right)}{(A-\beta SI-\delta_1 S)^2}, \\ \frac{\partial^2 W(S, I)}{\partial I^2}& = &\frac{(A-\delta_1 S)\left(\frac{\epsilon\omega(A-\beta SI-\delta_2S)}{(1+\omega I)^3}+2\beta S\left(\beta S-\delta_2-\gamma-\frac{\epsilon}{(1+\omega I)^2}\right)\right)}{(A-\beta SI-\delta_1 S)^3}. \end{array} |
According to the theorem of Cauchy and Lipschitz with parameters on the scalar differential equation, we obtain
\begin{array}{lll} \frac{\partial I(s, I_0)}{\partial I_0}& = &\exp\left(\int_{S_v}^{s}\frac{\partial}{\partial I}W(z, I(z, I_0))dz\right) \gt 0, \\ \end{array} |
and
\begin{array}{lll} \frac{\partial^2 I(s, I_0)}{\partial I_0^2}& = &\frac{\partial I(s, I_0)}{\partial I_0}\exp \int_{S_v}^{s}\frac{\partial^2}{\partial I^2}W(z, I(z, I_0))\frac{\partial I(z, I_0)}{\partial I_0}dz.\\ \end{array} |
Following from the definition of function \mathcal{P}_M(I_0) = I(S_T, I_0)\left(1-\frac{qI(S_T, I_0)}{h_2+I(S_T, I_0)}\right) , we have
\begin{array}{lll} \frac{\partial \mathcal{P}_M(I_0)}{\partial I_0}& = &\frac{\partial I(S_T, I_0)}{\partial I_0}\left(1-\frac{qI(S_T, I_0)(2h_2+I(S_T, I_0))}{(h_2+I(S_T, I_0))^2}\right), \\ \end{array} |
and
\begin{array}{lll} \frac{\partial ^2 \mathcal{P}_M(I_0)}{\partial I_0^2}& = &\frac{\partial^2 I(S_T, I_0)}{\partial I_0^2}\left(1-\frac{qI(S_T, I_0)(2h_2+I(S_T, I_0))}{(h_2+I(S_T, I_0))^2}\right)+\left(\frac{\partial I(S_T, I_0)}{\partial I_0}\right)^2\frac{2qh_2^2}{(h_2+I(S_T, I_0))^3}.\\ \end{array} |
Based on above discussions, we conclude that \frac{\partial \mathcal{P}_M(I_0)}{\partial I_0} > 0 while the sign of \frac{\partial^2 \mathcal{P}_M(I_0)}{\partial I_0^2} is not determined. Therefore, if \frac{\partial^2 \mathcal{P}_M(I_0)}{\partial I_0^2} > 0 holds true on the interval [0, I_{S_v}] , \mathcal{P}_M is convex on the interval [0, I_{S_v}] .
(3) Note that \frac{dI}{dt} < 0 always holds due to the assumption that S_T < \frac{A}{\delta_1} . Therefore, for any initial point (S_v, I_0) on line l_4 , there is g(I_0) < I_0 . Furthermore, there is \mathcal{P}_M(I_0) = w_1(g(I_0)) . Thus, we have \mathcal{P}_M(I_0) < I_0 for I_0\in[0, +\infty) . This means that there is no positive fixed point for the Poincaré map \mathcal{P}_M . This completes the proof.
According to the third property in Theorem 5.1, we obtain that there is no positive order-1 periodic solution of system (1.3) for case (C_1) . Furthermore, it is clear that for case (C_{31}) , the properties are the same as those shown in Theorem 5.1. Correspondingly, there exists no positive order-1 periodic solution of system (1.3) for case (C_{31}) . In what follows, we initially investigate the existence and stability of the positive order-1 periodic solutions under case (C_{32}) . Similar to the properties proposed in Theorem 5.1, we can conclude that the domain and range of \mathcal{P}_M are [0, +\infty] and [0, w_1(I^*)] , respectively, and \mathcal{P}_M is increasing on the interval [0, I_{S_v}] and decreasing on the interval [I_{S_v}, +\infty) . Furthermore, \mathcal{P}_M is convex on [0, I_{S_v}] provided that \frac{\partial^2 \mathcal{P}_M(I_0)}{\partial I_0^2} > 0 for all I_0\in[0, I_{S_v}] . It is easy to see that I^* < I_{S_T} and I_{S_T} < I_{S_v} . Thus, the relationship between I^* and I_{S_v} is \mathcal{P}_M(I_{S_v}) = w_1(I^*) < I^* < I_{S_v} . Combining with \frac{\partial^2 \mathcal{P}_M(I_0)}{\partial I_0^2} > 0 for all I_0\in[0, I_{S_v}] , we have that \mathcal{P}_M(I_0) < I_0 holds for all I_0\in[0, I_{S_v}] and there is no positive fixed point of \mathcal{P}_M . Accordingly, there is no positive order-1 periodic solution of system (1.3). Therefore, we have the following conclusion:
Theorem 5.2. For case (C_{31}) , there is no fixed point of the Poincaré map, hence no positive order-1 periodic solution is feasible for system (1.3). For case (C_{32}) , if \frac{\partial^2 \mathcal{P}_M(I_0)}{\partial I_0^2} > 0 holds true for all I_0\in[0, I_{S_v}] , there exists no positive periodic solution of system (1.3), shown in Figure 7(A).
As for case (C_{33}) , we get the main properties of the Poinceré map \mathcal{P}_M as follows.
Theorem 5.3. For case (C_{33}) , we obtain the following results of the Poinceré map \mathcal{P}_M :
(1) The domain and range of the Poinceré map \mathcal{P}_M are [0, I_{b_1}]\cup[I_{b_2, +\infty}) and [0, \omega_1(I_{S_T})] , respectively;
(2) \mathcal{P}_M is continuous on the two intervals [0, I_{b_1}] and [I_{b_2}, +\infty) . Moreover, it is increasing on the interval [0, I_{b_1}] and decreasing on the interval [I_{b_2}, +\infty) ;
(3) Suppose \frac{\partial^2 \mathcal{P}_M(I_0)}{\partial I_0^2} > 0 holds true for all I_0\in[0, I_{b_1}] . If \mathcal{P}_M(I_{b_1}) < I_{b_1} , then there is no positive fixed point of \mathcal{P}_M , shown in Figure 7 (B). Accordingly, there is no positive order-1 periodic solution of system (1.3). If \mathcal{P}_M(I_{b_1}) > I_{b_1} holds, there exists a unique fixed points belonging to [0, I_{b_1}] , shown in Figure 7 (C). Then, system (1.3) has a unique positive order-1 periodic solution.
Proof The methods of the proof of properties (1) and (2) are similar to the proof of Theorem 5.1. Thus, in the following we focus on proving the existence of the positive order-1 periodic solution in property (3). We know that if \frac{\partial^2 \mathcal{P}_M(I_0)}{\partial I_0^2} > 0 for all I_0\in[0, I_{b_1}] , then \mathcal{P}_M is convex on the interval [0, I_{b_1}] . Then there must be an interval (0, \delta]\in[0, I_{b_1}] such that \mathcal{P}_M(I_0) < I_0 for all I_0\in(0, \delta] . When \mathcal{P}_M(I_{b_1}) < I_{b_1} holds, it is clear that \mathcal{P}_M(I_0) < I_0 for I_0\in[0, I_{b_1}] . Therefore, there is no fixed point belonging to [0, I_{b_1}] . Moreover, as a result of I_{b_2} > I_l > I_{S_T} , we have \mathcal{P}_M(I_{b_2}) < \mathcal{P}_M(I_{b_1}) < I_{S_T} < I_{b_2} . Then \mathcal{P}_M(I_0) < \mathcal{P}_M(I_{b_2}) < I_{b_2} < I_0 for all I_0\in(I_{b_2}, +\infty) . Thus, there exists no fixed point belonging to [I_{b_2}, +\infty] . Then we conclude that there exists no positive fixed point of \mathcal{P}_M and there is no positive order-1 periodic solution of system (1.3). However, if \mathcal{P}_M(I_{b_1}) > I_{b_1} holds, there is a unique fixed point \overline{I}\in(\delta, I_{b_1}) satisfying \mathcal{P}_M(\overline{I}) = \overline{I} due to the continuity and convexity of \mathcal{P}_M . As mentioned above, there is no fixed point on the interval [I_{b_2}, +\infty] . Therefore, there exists a unique fixed point \overline{I}\in(\delta, I_{b_1}) of \mathcal{P}_M . Correspondingly, system (1.3) has a unique positive order-1 periodic solution. The proof is completed.
Remark 1. Note that if \frac{\partial^2 \mathcal{P}_M(I_0)}{\partial I_0^2} > 0 for all I_0\in[0, I_{b_1}] and \mathcal{P}_M(I_{b_1}) > I_{b_1} hold, then 0 < \frac{\partial \mathcal{P}_M(I_0)}{\partial I_0} < 1 holds true on the interval [0, \overline{I}] . Therefore, we obtain |\mu_2| < 1 . According to the properties of the Poinceré map \mathcal{P}_M , the unique positive order-1 periodic solution of system (1.3) is unstable, which matches the conclusions shown in the study of the bifurcations near the disease-free periodic solution of system (1.3).
When there is a unique stable limit cycle of subsystem (1.2), we mainly consider the most complicated subcase, i.e., case (C_{42}^c) . Although the domain of the Poinceré map \mathcal{P}_M is [0, +\infty) for case (C_{42}^c) , the continuity and monotonicity of \mathcal{P}_M can be much more complex. Therefore, we further discuss the properties of \mathcal{P}_M for case (C_{42}^c) in more details. When orbit \Gamma_2 intersects with line l_4 (i.e., the line S = S_v ) at a unique point P(S_v, I_{p}) before it is tangents to line l_3 (i.e., the line S = S_T ), shown in Figure 8 (A), we have the following conculsions.
Theorem 5.4. For case (C_{42}^c) , if there exists a unique discontinuous point P , then the Poinceré map \mathcal{P}_M satisfies the following properties:
(1) The domain and range of the Poinceré map \mathcal{P}_M are [0, \infty) and [0, w_1(I_{S_T})] , respectively;
(2) \mathcal{P}_M is continuous on the intervals [0, I_{p}] , (I_{p}, I_{S_v}] and [I_{S_v}, +\infty) . Moreover, it is increasing on the intervals [0, I_{p}] and (I_{p}, I_{S_v}] and decreasing on the interval [I_{S_v}, +\infty) ;
(3) Suppose \frac{\partial^2 \mathcal{P}_M(I_0)}{\partial I_0^2} > 0 holds true for all I_0\in[0, I_{p}] . If \mathcal{P}_M(I_{p}) < I_{p} , then there is no positive fixed point of \mathcal{P}_M and no positive periodic solution of system (1.3). If \mathcal{P}_M(I_{p}) > I_{p} holds, then there may exist one or two positive fixed points, shown in Figure 8 (B-C). Accordingly, system (1.3) has one or two positive order-1 periodic solutions.
Proof The first two results can be similarly proved as before. As for the existence of the positive periodic solution of system (1.3), we give the proof as follows. When \frac{\partial^2 \mathcal{P}_M(I_0)}{\partial I_0^2} > 0 for all I_0\in[0, I_{p}] and \mathcal{P}_M(I_{p}) < I_{p} , we have \mathcal{P}_M(I_0) < I_0 for I_0\in[0, I_{p}] . In addition, it is clear that \mathcal{P}_M(I_0) < I_0 for I_0\in(I_{p}, +\infty) . Therefore, there is no positive fixed point of \mathcal{P}_M . However, if \mathcal{P}_M(I_{p}) > I_{p} , there is a unique positive fixed point \overline{I}_{1} \in(0, I_{p}) . Moreover, if there exists \overline{\delta} > 0 small enough such that \mathcal{P}_M(I_{p}+\overline{\delta}) > I_{p}+\overline{\delta} , combining with \mathcal{P}_M(I_{S_v}) < I_{S_T} < I_{S_v} and the monotonicity of \mathcal{P}_M , we obtain that there is another fixed point \overline{I}_{2} \in(I_{p}, I_{S_v}) . Due to the monotonically decrease of \mathcal{P}_M on the interval [I_{S_v}, +\infty) , we have that \mathcal{P}_M(I_0) < I_0 for all I_0\in[I_{S_v}, +\infty) . Thus, there are two positive fixed points of \mathcal{P}_M and two positive periodic solutions of system (1.3). On the contrary, when there exists no \overline{\delta} satisfying \mathcal{P}_M(I_{p}+\overline{\delta}) > I_{p}+\overline{\delta} , if \frac{\partial^2 \mathcal{P}_M(I_0)}{\partial I_0^2} > 0 holds for all I_0\in(I_{p}, I_{S_v}] , then \mathcal{P}_M(I_0) < I_0 for I_0\in(I_{S_v}, +\infty) . Then there is only one positive fixed point of \mathcal{P}_M and a unique positive periodic solution of system (1.3). This completes the proof.
Next, we consider the case that orbit \Gamma_2 intersects with line l_4 at three points P_1(S_v, I_{p_1}) , P_2(S_v, I_{p_2}) and P_3(S_v, I_{p_3}) before it is tangents to line l_3 with I_{p_1} < I_{p_3} < I_{p_2} , shown in Figure 8 (A). Therefore, the domain of \mathcal{P}_M can be divided into:
[0, I_{p_1}], \; \; (I_{p_1}, I_{p_3}], \; \; (I_{p_3}, I_{S_v}], \; \; [I_{S_V}, I_{p_2}), \; \; [I_{p_2}, +\infty). |
Based on above discussions, we have the main conclusions as follows.
Theorem 5.5. For case (C_{42}^c) , if there are three discontinuous points P_1 , P_2 and P_3 , then the Poinceré map \mathcal{P}_M satisfies the following properties:
(1) The domain and range of the Poinceré map \mathcal{P}_M are [0, \infty) and [0, w_1(I_{S_T})] , respectively;
(2) \mathcal{P}_M is continuous on the five intervals [0, I_{p_1}] , (I_{p_1}, I_{p_3}] , (I_{p_3}, I_{S_v}] , [I_{S_V}, I_{p_2}) and [I_{p_2}, +\infty) . Moreover, it is increasing on the intervals [0, I_{p_1}] , (I_{p_1}, I_{p_3}] and (I_{p_3}, I_{S_v}] and decreasing on the intervals [I_{S_V}, I_{p_2}) and [I_{p_2}, +\infty) ;
(3) Suppose \frac{\partial^2 \mathcal{P}_M(I_0)}{\partial I_0^2} > 0 holds true for all I_0\in[0, I_{p_1}] . If \mathcal{P}_M(I_{p_1}) < I_{p_1} , then there exists no positive fixed point of \mathcal{P}_M and no positive periodic solution of system (1.3). If \mathcal{P}_M(I_{p_1}) > I_{p_1} holds, there may exist one, two or three positive fixed points of \mathcal{P}_M , shown in Figure 8 (D-F). Correspondingly, there may be one, two or three positive order-1 periodic solutions of system (1.3).
The properties given by Theorem 5.5 can be similarly proved by using the methods in Theorem 5.4, and we omit the details. For convenience, we just considered two conditions for (C_{42}^c) (i.e., there is one discontinuous point P or three discontinuous points P_1 , P_2 and P_3 ) to discuss the existence of the positive periodic solution of system (1.3). It is worth noting that for case (C_{42}^c) , before orbit \Gamma_2 reaches line S = S_T , it may intersect with line S = S_v 2n+1 times, and n is increasing as S_v tend to the equilibrium E_1 . Thus, the number of discontinuous points could be infinitely countable, which indicates that system (1.3) may exist an infinite number of positive order-1 periodic solutions.
Note that the properties of the Poincaré map for other subcases of case (C_4) can be discussed similarly. Specifically, we can obtain the increasing and decreasing intervals through using the same methods mentioned in above theorems. Moreover, as for the existence of the positive order-1 periodic solution, it can be verified that there may be no positive order-1 periodic solution, which is similar to the results shown in Theorem 5.1 and there may be a finite number of the positive order-1 periodic solutions which is similar to the results shown in Theorem 5.4 and Theorem 5.5, and we give the main properties of the Poincaré map for other subcases of case (C_4) in Table 1.
Cases | Domain and range of \mathcal{P}_M | Monotonicity of \mathcal{P}_M | The number of PPS of system (1.3) |
C_{41} | [0, +\infty) and [0, w_1(I^*)] | \mathcal{P}_M increases on [0, I_{S_v}] and decreases on [I_{S_v}, +\infty) | No PPS |
C_{42}^{a} | [0, +\infty) and [0, w_1(I^*)] | \mathcal{P}_M increases on [0, I_{S_v}] and decreases on [I_{S_v}, +\infty) | No PPS |
C_{42}^{b_1} | [0, +\infty) and [0, w_1(I_{S_T})] | \mathcal{P}_M increases on [0, I_{\gamma_1}] and (I_{\gamma_1}, I_{S_v}] and decreases on [I_{S_v}, I_{\gamma_2}) and [I_{\gamma_2}, +\infty) | At most four PPSs |
C_{42}^{b_2} | [0, +\infty) and [0, w_1(I_{S_T})] | \mathcal{P}_M increases on [0, I_{S_v}] and decreases on [I_{S_v}, +\infty) | Zero or two PPSs |
C_{43}^{a} | [0, +\infty) and [0, w_1(I^*)] | \mathcal{P}_M increases on [0, I_{S_v}] and decreases on [I_{S_v}, +\infty) | Zero or two PPSs |
C_{43}^{b} | [0, I_{c_1}]\cup[I_{c_2, +\infty}) and [0, w_1(I_{S_T})] | \mathcal{P}_M increases on [0, I_{c_1}] and decreases on [I_{c_2}, +\infty) | At most one PPS |
Note: ’PPS’ represents ’ The positive order-1 periodic solution’. |
Many mathematical models have assumed that there is a threshold level of the infected population determining the implementation of control methods. Unfortunately, under this assumption, no disease-free periodic solution is feasible or the control reproduction number of the state-dependent impulsive model cannot be defined. Thus, recent studies [24,25] proposed mathematical models with susceptibles-guided linear impulsive control. In the current study, considering the limitation of resources, we introduced the comprehensive saturated control strategies (including saturated impulsive vaccination and isolation, and saturated continuous treatment), and proposed a state-dependent impulsive model with comprehensive saturation interventions.
We first briefly concluded the main dynamics of the ODE subsystem. Based on the dynamics of the ODE subsystem, we investigated the dynamical behaviours of system (1.3). We find that under the susceptibles-guided impulsive control strategy, there always exists the disease-free periodic solution. Further, by discussing the stability of the disease-free periodic solution, we defined the control reproduction number R_b of the state-dependent feedback control system, that is, the disease-free periodic solution is locally stable when R_b is less than 1 and unstable otherwise.
Furthermore, we studied the existence and stability of the positive order-1 periodic solution through analyzing the bifurcation phenomenon near the disease-free periodic solution and discussing the properties of the Poincaré map. We proved that the system can undergo the transcritical bifurcation and the pitchfork bifurcation with respect to the key parameters, including the control parameters such as the maximal vaccination rate p , the threshold level S_T and the parameter \epsilon related to saturated continuous treatment. Accordingly, it can be shown that by changing key parameter values, a stable or an unstable positive order-1 periodic solution can bifurcate from the disease-free periodic solution. On the other hand, based on the complexity of the definitions of the domain of the Poincaré map for different cases, there will be a finite number of discontinuous points or an infinitely countable number of discontinuous points for the Poincaré map. Consequently, there may exist multiple positive order-1 periodic solution of system (1.3). Comparing with the analysis of the linear susceptibles-guided impulsive control strategy in [25], our current model considered both continuous saturated treatment and nonlinear impulsive interventions, and we investigated the existence of finite or infinite countable positive order-1 periodic solutions through studying the properties of the Poincaré map. Moreover, through discussing the bifurcations near the disease-free periodic solution with respect to the half-saturation constant of susceptible individuals h_1 , we concluded that the disease-free periodic solution is stable when h_1 < h_1^* . This implies that the saturation phenomenon of the impulsive control strategy greatly influences the spread of infectious diseases, and large half-saturation constant of susceptibles induces diseases eradication less likely.
Comparing with the model with continuous treatment (i.e., the ODE subsystem (1.2)), we proved that the disease-free periodic solution is stable provided that S_T \leq \bar{S} even if R_0 > 1 for subsystem (1.2), implying that the susceptibles-guided impulsive strategy can eradicate infectious diseases successfully with choosing proper threshold level of susceptible population even if R_0 > 1 for subsystem (1.2). Moreover, comparing with the modeling approaches of the infected individuals-triggered impulsive control, there always exists the disease-free periodic solution, especially, we can also define the control reproduction number for our state-dependent impulsive model. Therefore, for our proposed model, it is essential to emphasize that the susceptibles-triggered impulsive intervention strategy leads to interesting biological implications, which is helpful to design an optimal treatment strategy. It follows from Figures 2 and 3(A) that selecting proper parameter values plays a crucial effect on controlling infectious diseases. As shown in Figure 2(A), (B) and (D), R_b decreases with respect to q , A and \epsilon , which means that enhancing the maximal isolation rate or the continuous treatment is always beneficial to the control of infectious diseases. In addition, large recruitment rate is also helpful to eradicate infectious diseases. As for another key parameter p , we find that when the chosen value of p is large enough, increasing p results in the decrease of R_b , however, for a quite low level of p , R_b increases with respect to p , shown in Figure 2(B), which means that enhancing maximal vaccination rate may be a disadvantage of controlling infectious disease. These results indicate that it is important to choose proper maximal vaccination rate and we should choose relatively large vaccination rate in order to avoid this kind of paradoxical effects. Meanwhile, it is revealed that relatively large threshold level S_T is not beneficial to eradicate infectious diseases, shown in Figure 3(A). Another interesting result shown in Figure 3 reveals that if we choose a properly small threshold value S_T , infectious diseases can be eventually eradicated, which plays a significant role in mitigating the spread of infectious diseases. Therefore, we should take account of these key parameters in order to develop effective and optimal susceptibles-triggered impulsive control strategies.
This work is supported by the National Natural Science Foundation of China (NSFCs 11631012, 11571273).
The authors declare there is no conflict of interest.
The following lemma shows the local stability of an order-k periodic solution.
Lemma A.1 The order-k periodic solution (x, y) = (\xi(t), \eta(t)) with period T of (1.3) is orbitally asymptotically stable if the Floquet multiplier \mu_2 satisfies |\mu_2| < 1, where
\mu_2 = \prod\limits_{k = 1}^q\Delta_k\exp\left[\int_0^T\left(\frac{\partial P}{\partial x}(\xi(t), \eta(t))+\frac{\partial Q}{\partial y}(\xi(t), \eta(t))\right)dt\right], |
with
\Delta_k = \frac{P_+\left(\frac{\partial b}{\partial y}\frac{\partial \phi}{\partial x}-\frac{\partial b}{\partial x}\frac{\partial \phi}{\partial y}+\frac{\partial \phi}{\partial x}\right)+Q_+\left(\frac{\partial a}{\partial y}\frac{\partial \phi}{\partial y}-\frac{\partial a}{\partial y}\frac{\partial \phi}{\partial x}+\frac{\partial \phi}{\partial y}\right)}{P\frac{\partial \phi}{\partial x}+Q\frac{\partial \phi}{\partial y}}, |
and P, Q, \frac{\partial a}{\partial x}, \frac{\partial a}{\partial y}, \frac{\partial b}{\partial x}, \frac{\partial b}{\partial y}, \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y} are calculated at the point (\xi(\tau_k), \eta(\tau_k)), and P_+ = P(\xi(\tau_k^+), \eta(\tau_k^+)), Q_+ = Q(\xi(\tau_k^+), \eta(\tau_k^+)) with \tau_k(k\in N) denoting the time of the k-th jump. Here, \phi(x, y) is a sufficiently smooth function such that grad\phi(x, y)\neq 0.
Then, we give two lemmas of the transcritical bifurcation and the pitchfork bifurcation of the discrete one-parameter family of maps [32].
Lemma A.2 (Transcritical bifurcation). Let G:U\times I\rightarrow R define a one-parameter family of maps, where G is C^r with r\geq 2, and U, I are open intervals of the real line containing 0. Assume
\begin{array}{ll} (1)~G(0, \alpha) = 0~for~all~\alpha;&(2)~\frac{\partial G}{\partial x}(0, 0) = 1;\\ (3)~\frac{\partial^2 G}{\partial x\partial \alpha}(0, 0) \gt 0;~~&(4)~\frac{\partial^2 G}{\partial x^2}(0, 0) \gt 0. \end{array} |
Then there are \alpha_1 < 0 < \alpha_2 and \zeta>0 such that
(1) If \alpha_1 < \alpha < 0, then G_{\alpha} has two fixed points, 0 and x_{1\alpha}>0 in (-\zeta, \zeta). The origin is asymptotically stable, while the other fixed point is unstable.
(2) If 0 < \alpha < \alpha_2, then G_{\alpha} has two fixed points, 0 and x_{1\alpha} < 0 in (-\zeta, \zeta). The origin is unstable, while the other fixed point is asymptotically stable.
Similarly, note that making the change of parameter \alpha\rightarrow -\alpha, we can handle \frac{\partial^2 G}{\partial x^2}(0, 0) < 0.
Lemma A.3 (Supercritical pitchfork bifurcation). Let G:U\times I\rightarrow R define a one-parameter family of maps as in Lemma A.2, except that G is C^r with r\geq 3, \frac{\partial^2 G}{\partial x^2}(0, 0) = 0 and \frac{\partial^3 G}{\partial x^3}(0, 0) < 0. Then there are \alpha_1 < 0 < \alpha_2 and \zeta>0 such that
(1) If \alpha_1 < \alpha\leq 0, then G_{\alpha} has a unique fixed point, x = 0. And it is asymptotically stable.
(2) If 0 < \alpha < \alpha_2, then G_{\alpha} has three fixed points, 0 and x_{1\alpha} < 0 < x_{2\alpha} in (-\zeta, \zeta). The origin is unstable, while the other two fixed points are asymptotically stable.
Note that the for the case \frac{\partial^3 G}{\partial x^3}(0, 0)>0, we can make the change of parameter \alpha\rightarrow -\alpha, which is called the subcritical pitchfork bifurcation.
(A) The bifurcation near the disease-free periodic solution with respect to A for h_2>0.
Firstly, we investigate the existence of A^*\in(\delta_1S_T, +\infty) such that R_b(A^*) = 1. There are
\begin{equation} \label{mu} \begin{array}{c} \lim_{A\rightarrow \delta_1S_T^+}R_b(A) = +\infty, ~~\lim_{A\rightarrow +\infty}R_b(A) = \lim\limits_{A\rightarrow +\infty}\exp(J(A)) = 1.\\ \end{array} \end{equation} | (6.1) |
Taking the derivative of R_b(A) with respect to A, one obtains
\begin{array}{c} \frac{\partial R_b(A)}{\partial A} = R_b(A)\frac{\partial J(A)}{\partial A}, \end{array} |
with
\begin{array}{c} \frac{\partial J(A)}{\partial A} = \frac{\beta}{\delta_1^2}\left(\ln \frac{A-\delta_1S_v}{A-\delta_1S_T}+\frac{\delta_1(A-\delta_1\overline{S})(S_v-S_T)}{(A-\delta_1S_v)(A-\delta_1S_T)}\right). \end{array} |
Denoting W_1(A) = \ln \frac{A-\delta_1S_v}{A-\delta_1S_T}+\frac{\delta_1(A-\delta_1\overline{S})(S_v-S_T)}{(A-\delta_1S_v)(A-\delta_1S_T)} and taking the derivative of W_1(A) with respect to A, we get
\begin{array}{c} \frac{\partial W_1(A)}{\partial A} = \frac{\delta_1^2(S_v-S_T)}{(A-\delta_1S_v)^2(A-\delta_1S_T)^2}\left((2\overline{S}-S_v-S_T)A+\delta_1(2S_vS_T-\overline{S}(S_v+S_T))\right).\\ \end{array} |
Note that if 2\overline{S} = S_v+S_T, we have
\begin{array}{c} \frac{\partial W_1(A)}{\partial A} = \frac{\delta_1^3(S_v-S_T)}{(A-\delta_1S_v)^2(A-\delta_1S_T)^2}\left(2S_vS_T-\frac{(S_v+S_T)^2)}{2}\right).\\ \end{array} |
As a result of 4S_vS_T < (S_v+S_T)^2, then \frac{\partial W_1(A)}{\partial A}>0. This indicates that W_1(A) is monotonically increasing for A\in(\delta_1S_T, +\infty). Combining with \lim_{A\rightarrow +\infty}W_1(A) = 0, we yield \frac{\partial J(A)}{\partial A} < 0 holds for all A\in(\delta_1S_T, +\infty), i.e., \frac{\partial R_b(A)}{\partial A} < 0, which means that R_b(A) is monotonically decreasing. Therefore, R_b(A)>1 for all A\in(\delta_1S_T, +\infty). Under this situation, the disease-free periodic solution is unstable and no bifurcation occurs with respect to parameter A.
However, if 2\overline{S}\neq S_v+S_T, we denote
W_2(A) = (2\overline{S}-S_v-S_T)A+\delta_1(2S_vS_T-\overline{S}(S_v+S_T))\doteq a_1A+a_2. |
Then, we have
a_1 \gt 0\Leftrightarrow\overline{S} \gt \frac{S_v+S_T}{2}, ~~a_2 \gt 0\Leftrightarrow\overline{S} \lt \frac{2S_vS_T}{S_v+S_T}. |
Moreover, there is a unique \overline{A} = -\frac{a_2}{a_1} such that W_2(\overline{A}) = 0. In what follows, we focus on discussing the bifurcation related to parameter A by considering the following cases:
(1) If a_1>0, it is clear that a_2 < 0 holds, thus, \overline{A}>0. Then we consider two subcases as follows:
(a) If \overline{A}\leq \delta_1S_T, we obtain W_2(A)>0, i.e., \frac{\partial W_1(A)}{\partial A} < 0 holds for all A\in(\delta_1S_T, +\infty). Thus, W_1(A) is monotonically decreasing on the interval (\delta_1S_T, +\infty) and \lim_{A\rightarrow +\infty}W_1(A) = 0, which indicates that W_1(A)>0 for all A\in(\delta_1S_T, +\infty). Therefore, R_b(A) is monotonically increasing on the interval (\delta_1S_T, +\infty). However, this result contradicts equations (6.1), indicating that \overline{A}>\delta_1S_T always holds.
(b) In the following, we consider the condition \overline{A}>\delta_1S_T. Under this scenario, we have W_2(A) < 0 for A\in(\delta_1S_T, \overline{A}) and W_2(A)>0 for A\in(\overline{A}, +\infty). Therefore, \frac{\partial W_1(A)}{\partial A}>0 for A\in(\delta_1S_T, \overline{A}) and \frac{\partial W_1(A)}{\partial A} < 0 for A\in(\overline{A}, +\infty), which means that W_1(A) is monotonically increasing on the interval (\delta_1S_T, \overline{A}) and monotonically decreasing on the interval (\overline{A}, +\infty). According to \lim_{A\rightarrow +\infty}W_1(A) = 0, we have W_1(A)>0 for all A\in(\overline{A}, +\infty), and consequently, R_b(A) is monotonically increasing on the interval (\overline{A}, +\infty). It is easy to verify that there is a unique A'\in(\delta_1S_T, \overline{A}) satisfying W_1(A') = 0. In fact, if W_1(A)>0 always holds for A\in(\delta_1S_T, \overline{A}), then R_b(A) is monotonically increasing on the interval (\delta_1S_T, +\infty), which contradicts equations (6.1). Thus, W_1(A) < 0 for A\in(\delta_1S_T, A') and W_1(A)>0 for A\in(A', +\infty). Correspondingly, R_b(A) is monotonically decreasing on the interval (\delta_1S_T, A') and increasing on the interval (A', +\infty). According to equations (6.1), there must be a unique A^*\in(\delta_1S_T, A') such that R_b(A^*) = 1 with \frac{\partial R_b(A^*)}{\partial A} < 0.
(2) If a_1 < 0 and a_2>0, we have \overline{A}>0. Then we consider the following subcases:
(a) If \overline{A}\leq \delta_1S_T, then we have W_2(A) < 0, i.e., \frac{\partial W_1(A)}{\partial A}>0 holds for all A\in(\delta_1S_T, +\infty). Therefore, W_1(A) is monotonically increasing on the interval (\delta_1S_T, +\infty) with \lim_{A\rightarrow +\infty}W_1(A) = 0, which indicates that W_1(A) < 0 holds true for all A\in(\delta_1S_T, +\infty). Correspondingly, R_b(A) is monotonically decreasing on the interval (\delta_1S_T, +\infty). According to \lim_{A\rightarrow+\infty}R_b(A) = 1, we have R_b(A)>1 is true for A\in(\delta_1S_T, +\infty). These results show that the disease-free periodic solution is unstable and there is no bifurcation near the disease-free periodic solution.
(b) If \overline{A}>\delta_1S_T, we have W_2(A)>0 for A\in(\delta_1S_T, \overline{A}) and W_2(A) < 0 for A\in(\overline{A}, +\infty). Consequently, W_1(A) is monotonically decreasing on the interval (\delta_1S_T, \overline{A}) and monotonically increasing on the interval (\overline{A}, +\infty). According to \lim_{A\rightarrow\infty}W_1(A) = 0, we have that W_1(A) < 0 for all A\in(\overline{A}, +\infty) and R_b(A) is monotonically decreasing on the interval (\overline{A}, +\infty). As for A\in(\delta_1S_T, \overline{A}), if there exists a A'' such that W_1(A'') = 0, then W_1(A)>0 for A\in(\delta_1S_T, A'') and W_1(A) < 0 for A\in(A'', +\infty), which contradicts Eq (6.1). Therefore, W_1(A) < 0 holds for A\in(\delta_1S_T, +\infty) and R_b(A) is monotonically decreasing on the interval (\delta_1S_T, +\infty). Similar to above discussions for subcase (a), we know that R_b(A)>1 always holds true. Therefore, the disease-free periodic solution is unstable and there is no bifurcation near the disease-free periodic solution.
(3) If a_1 < 0 and a_2 < 0, then we have \overline{A} < 0. Under this scenario, W_2(A) < 0, i.e., \frac{\partial W_1(A)}{\partial A}>0 holds for all A\in(\delta_1S_T, +\infty). Therefore, W_1(A) is monotonically increasing on the interval (\delta_1S_T, +\infty) with \lim_{A\rightarrow +\infty}W_1(A) = 0. Therefore, W_1(A) < 0 always holds. Accordingly, R_b(A) is monotonically decreasing on the interval (\delta_1S_T, +\infty). Combining with equations (6.1), we have that R_b(A)>1 holds true for A\in(\delta_1S_T, +\infty), meaning that the disease-free periodic solution is unstable and there is no bifurcation near the disease-free periodic solution. Based on above discussions, we have conclusions as follows.
Proposition B.1 Assume R_0>1. If S_T>\overline{S}>\frac{S_v+S_T}{2} holds, then there exists a unique A^*\in(\delta_1S_T, A') satisfying R_b(A^*) = 1 with \frac{\partial R_b(A^*)}{\partial A} < 0. And the disease-free periodic solution (\xi(t), 0) of system (1.3) is orbitally asymptotically stable when A\in(A^*, +\infty) and unstable when A\in(\delta_1S_T, A^*).
As for the bifurcation of the disease-free periodic solution at A^*, we have that \mathcal{P}_M(0, A) = 0 always holds for A\in(\delta_1S_T, +\infty), and
\begin{array}{c} \frac{\partial \mathcal{P}_M}{\partial I_0}(0, A^*) = 1, ~~\frac{\partial^2 \mathcal{P}_M}{\partial I_0\partial A}(0, A^*) \lt 0, \\ \frac{\partial^2 \mathcal{P}_M}{\partial I_0^2}(0, A^*) = g''(0;A^*)-\frac{2q}{h_2}, ~~\frac{\partial^3 \mathcal{P}_M}{\partial I_0^3}(0, A^*) = g'''(0;A^*)-\frac{6q(2q-1)}{h_2^2}.\\ \end{array} |
Therefore, we can conclude the main results for the bifurcation near the disease-free periodic solution with respect to A in Theorem 4.5.
(B) The bifurcation near the disease-free periodic solution with respect to q for h_2 = 0.
When h_2 = 0, the bifurcation near the disease-free periodic solution can be similarly studied. It is clear that R_b(q) = (1-q)\exp(J) when h_2 = 0. Thus, q can be chosen as a bifurcation parameter. It is easily obtained that R_b(1) = 0. When J>0 holds, then there is a unique q^*\in(0, 1) such that R_b(q^*) = 1 with q^* = 1-\exp(-J), which is equal to \frac{\partial \mathcal{P}_M}{\partial I_0}(0, q^*) = 1. Note that \mathcal{P}_M(0, q) = 0 always holds, and \frac{\partial^2 \mathcal{P}_M}{\partial I_0\partial q}(0, q^*) = -\exp(J) < 0. Moreover, there is
\begin{array}{c} \frac{\partial^2 \mathcal{P}_M}{\partial I_0^2}(0, q^*) = (1-q^*)g''(0;q^*).\\ \end{array} |
Therefore, we can obtain the conclusions given in Theorem 4.6.
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Cases | Domain and range of \mathcal{P}_M | Monotonicity of \mathcal{P}_M | The number of PPS of system (1.3) |
C_{41} | [0, +\infty) and [0, w_1(I^*)] | \mathcal{P}_M increases on [0, I_{S_v}] and decreases on [I_{S_v}, +\infty) | No PPS |
C_{42}^{a} | [0, +\infty) and [0, w_1(I^*)] | \mathcal{P}_M increases on [0, I_{S_v}] and decreases on [I_{S_v}, +\infty) | No PPS |
C_{42}^{b_1} | [0, +\infty) and [0, w_1(I_{S_T})] | \mathcal{P}_M increases on [0, I_{\gamma_1}] and (I_{\gamma_1}, I_{S_v}] and decreases on [I_{S_v}, I_{\gamma_2}) and [I_{\gamma_2}, +\infty) | At most four PPSs |
C_{42}^{b_2} | [0, +\infty) and [0, w_1(I_{S_T})] | \mathcal{P}_M increases on [0, I_{S_v}] and decreases on [I_{S_v}, +\infty) | Zero or two PPSs |
C_{43}^{a} | [0, +\infty) and [0, w_1(I^*)] | \mathcal{P}_M increases on [0, I_{S_v}] and decreases on [I_{S_v}, +\infty) | Zero or two PPSs |
C_{43}^{b} | [0, I_{c_1}]\cup[I_{c_2, +\infty}) and [0, w_1(I_{S_T})] | \mathcal{P}_M increases on [0, I_{c_1}] and decreases on [I_{c_2}, +\infty) | At most one PPS |
Note: ’PPS’ represents ’ The positive order-1 periodic solution’. |
Cases | Domain and range of \mathcal{P}_M | Monotonicity of \mathcal{P}_M | The number of PPS of system (1.3) |
C_{41} | [0, +\infty) and [0, w_1(I^*)] | \mathcal{P}_M increases on [0, I_{S_v}] and decreases on [I_{S_v}, +\infty) | No PPS |
C_{42}^{a} | [0, +\infty) and [0, w_1(I^*)] | \mathcal{P}_M increases on [0, I_{S_v}] and decreases on [I_{S_v}, +\infty) | No PPS |
C_{42}^{b_1} | [0, +\infty) and [0, w_1(I_{S_T})] | \mathcal{P}_M increases on [0, I_{\gamma_1}] and (I_{\gamma_1}, I_{S_v}] and decreases on [I_{S_v}, I_{\gamma_2}) and [I_{\gamma_2}, +\infty) | At most four PPSs |
C_{42}^{b_2} | [0, +\infty) and [0, w_1(I_{S_T})] | \mathcal{P}_M increases on [0, I_{S_v}] and decreases on [I_{S_v}, +\infty) | Zero or two PPSs |
C_{43}^{a} | [0, +\infty) and [0, w_1(I^*)] | \mathcal{P}_M increases on [0, I_{S_v}] and decreases on [I_{S_v}, +\infty) | Zero or two PPSs |
C_{43}^{b} | [0, I_{c_1}]\cup[I_{c_2, +\infty}) and [0, w_1(I_{S_T})] | \mathcal{P}_M increases on [0, I_{c_1}] and decreases on [I_{c_2}, +\infty) | At most one PPS |
Note: ’PPS’ represents ’ The positive order-1 periodic solution’. |