# State-Dependent Pulse Vaccination and Therapeutic Strategy in an SI Epidemic Model with Nonlinear Incidence Rate

Authors: Kaiyuan Liu; Tongqian Zhang; Lansun Chen
Journal: Computational and Mathematical Methods in Medicine
 (2019)

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## Abstract

In this paper, the state-dependent pulse vaccination and therapeutic strategy are considered in the control of the disease. A pulse system is built to model this process based on an SI ordinary differential equation model. At first, for the system neglecting the impulse effect, we give the classification of singular points. Then for the pulse system, by using the theory of the semicontinuous dynamic system, the dynamics is analyzed. Our analysis shows that the pulse system exhibits rich dynamics and the system has a unique order-1 homoclinic cycle, and by choosingp as the control parameter, the order-1 homoclinic cycle disappears and bifurcates an orbitally asymptotical stable order-1 periodic solution when p changes. Numerical simulations by maple 18 are carried out to illustrate the theoretical results.


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## Body

## 1. Introduction

Infectious diseases are caused by various pathogens that can be transmitted from person to person, animal to animal, or human to animal. The ever-changing changes in the pathogens of ancient infectious diseases and the emergence of new pathogens have brought new challenges to the discovery, diagnosis, and prevention of infectious diseases. According to the 2016 report of the World Health Organization [1], about 36.7 million people have been infected with HIV/AIDS, 1.0 million people died of HIV/AIDS, and more than 18 million people worldwide living with HIV are receiving antiviral drugs. And tuberculosis is currently the biggest “killer” caused by a single infectious pathogen after AIDS in the world. As of the end of 2016, there were 10.4 million new tuberculosis cases [2]. Therefore, the control and elimination of infectious diseases has attracted wide attention of people. Various dynamic models have been proposed by mathematicians to investigate the spread and evolution of infectious diseases [3–14]. In particular, mathematical models of differential equations have been extensively investigated, and among them, the most classical well-known model is SIR model [15] or SIS model [16], which have been widely investigated [17–23].It is well known that vaccination is mostly a medical behavior that can evoke the individual’s natural defense mechanism to prevent possible future diseases. This kind of vaccination is known as prophylactic vaccination. Diphtheria, whooping cough, polio, tetanus, herpes, rubella, and mumps are the most common types of vaccines. There are many types of vaccination, the two common types are continuous vaccination and pulsed vaccination. Continuous vaccination is when people are vaccinated at birth to protect themselves from illness, while pulsed vaccination is when people are vaccinated at a fixed period of time in all age groups which was firstly investigated by Agur et al in [24]. Pulse vaccination strategy (PVS) has been studied by many scholars [25, 26]. For example, Lu et al. [27] studied the pulse epidemic model with bilinear incidence and compared the effectiveness of the continuous and pulsed vaccination strategies. Liu et al. [28] investigated the SIR epidemic model with the saturated transmission rate. However, the strategy is taken at certain fixed times and does not depend on the status of infectious diseases. In general, taking into account the limited medical resources and costs, vaccines to susceptible people according to the number of susceptible people or infected people are more reasonable than continuous vaccination and fixed time pulse vaccination. This control strategy relies on the individual (or susceptible individuals) of the infection state and is called a state-dependent pulse vaccination strategy. Based on this idea, Tang et al. [29], Nie et al. [30], Guo et al. [31], and Qin et al. [32] have considered a state-dependent pulse strategy in SIR model and SIRS model. In fact, using state-dependent feedback control strategies to simulate real-world problems is more reasonable. Therefore, the impulsive state feedback control is also widely applied to the population dynamics model [33–46], chemostat model [47], and turbidostat model [48].Firstly, we consider anSI epidemic model with nonlinear incidence rate βSI2 described by the ordinary differential equations as follows:(1)dSdt=θ−βSI2−γS,dIdt=βSI2−γI,which is a special case in the study of Liu et al. [49] and St and It represents the number of susceptible and infected individuals at time t, respectively. θ is the birth rate, β is the contact rate, and γ is the natural death rate.Motivated by the studies of Tang et al. [29], Nie et al. [30], and Zhang et al. [50], we consider state-dependent pulse vaccination and treatment strategy in model (1) and get the following model:(2)dSdt=θ−βSI2−γS,dIdt=βSI2−γI,I<h1,ΔS=−pS,ΔI=−qI,I<h1,where ΔSt=St+−StandΔIt=It+−It. When the amount of infected reaches the hazardous threshold value h1>0, vaccination and treatment are taken into account, and the number of susceptible and infected suddenly turn to 1−pSt and 1−qIt, respectively, where 0<pandq<1 denote the vaccination rate of susceptible individuals and treatment rate of infected individuals, respectively. By the scaling,(3)S=k1x,I=k2y,t=k3τ,k3k1θ=a,βk22k3=1,k3=1γ,then, model (2) transforms into the following form:(4)dxdτ=a−y2x−x,dydτ=y2x−y,Δy=−qy,Δx=−px,Δy=−qy,y=h2,where h2=h1/k2. In the following, according to the actual condition, we always suppose that h≤y2, and based on practical significance, our research scope is limited to the first quadrant, i.e., R+2=x,yx≥0,y≥0.The purpose of this paper is to study the dynamic behavior under the effect of state-dependent pulse vaccination and treatment strategy. This article is organized as follows. In Section2, we introduce some definitions and notations of the geometric theory of semicontinuous dynamic systems, which will be useful for the latter discussion. In Section 3, we qualitatively analyze the dynamics of model (3). In Section 4, the existence of the homoclinic cycle is studied by using the geometrical theory of semicontinuous dynamical systems. At last, we present some numerical simulations.


## 2. Preliminaries

In this section, we introduce some notations, definitions, and lemmas of the geometric theory of semicontinuous dynamic system, which will be useful for the following discussions. The following definitions and lemmas of semicontinuous dynamic system come from the studies of Chen et al. [51] and Wei and Chen [36].Definition 1.
Consider the following state-dependent impulsive differential system(5)dxdt=Px,y,dydt=Qx,y,x,y∉Mx,y,Δx=αx,y,Δy=βx,y,x,y∈Mx,y.
The solution mapping of system (4) is called the semicontinuous dynamical system denoted by Ω,f,φ,andM, where x,y∈Ω⊂R+2andf=fp;t is the semicontinuous dynamical system mapping with initial point p=x0,y0∉M; the sets M and N are called the impulse set and phase set, which are lines or curves on R+2. The continuous function φ:M⟶N is called impulse mapping.Remark 1.
System (4) constitutes a semicontinuous dynamic system Ω,f,φ,M, where Ω=R+2=x,yx≥0,y≥0,M=x,y∈R+2x≥0,y=h2,φ:x,y∈M⟶1−px,1−qh2∈R+2,N=φM=x,y∈R+2x≥0,y=1−qh2.Definition 2.
If there exists a pointP∈N and T>0 such that fP,T=Q∈M and φQ=φfP,T=P∈N, then fP,t is called order-1 periodic solution.Definition 3.
The trajectoryfP,t combining with impulse line QP is called the order-1 cycle. If the order-1 cycle has a singularity, then the order-1 cycle is called the order-1 singular cycle. Furthermore, if the order-1 cycle only has a saddle, then the order-1 singular cycle is called the order-1 homoclinic cycle.Definition 4.
We assume that G is a bounded closed simple connected region, which has the following properties:(i)
Impulse setM is a simple connected bounded closed straight line segments or curve segments which do not contain closed branch(ii)
The boundariesAD, BC, and AB of region G are nontangent arcs of semicontinuous dynamical system (4). The boundary CD is the impulse set of system (4), and its phase set satisfies φCD⊆AB;(iii)
The orientation of the vector fields of semicontinuous dynamical system (4) on the AD, BC, and AB points of the internal of region G. There are no equilibriums on the boundaries and also in the internal of region G of semicontinuous dynamical system (4).
Then region G is called Bendixson’s region of semicontinuous dynamical system (4).Lemma 1.
(Bendixson theorem of semicontinuous dynamical system.) If region G is Bendixson’s region of semicontinuous dynamical system (4), then there exists at least an order-1 periodic solution in the internal of region G (Figure 1).Figure 1
Bendixon region of the semicontinuous dynamical system. This figure is reproduced from the study of Wei and Chen [36] (under the Creative Commons Attribution License/public domain).Next, we will give the definition of successor function of system (4). Firstly, we define a new number axis in set N. On straight line y=1−qh2, take the origin at point 0,1−qh2 of coordinate axis y and define the positive direction and unit length to be consistent with coordinate axis x, then we obtain a number axis l. For any A∈l, let lA be the coordinate of point A which is defined as the distance between point A and the y-axis, i.e., lA=xA.Definition 5.
Supposeg:N⟶N be a map. Let P∈N be the initial mapping point, for any P∈N, there exists a t1>0 such that FP=fP,t1=P1∈M,P1+=φP1∈N. Then, function gP=lP1+−lP is the successor function of point P, and the point P1+ is called the successor point of P (Figure 2).Figure 2
Successor function.Definition 6.
SupposeΓ=fP,t is an order-1 periodic solution of system (4). If for any ε>0, there must exist δ>0 and t0≥0 such that, for any point P1∈∪P,δ∩N, we have ρfP1,t,Γ<ε for t>t0; then we call the order-1 periodic solution Γ is orbitally asymptotically stable.


## 3. Qualitative Analysis of System Neglecting the Impulse Effect

First, we consider the classification of singular points of the system neglecting the impulse effect. Neglecting the impulse effects, system (3) reduces to(6)dxdτ=a−y2x−x,dydτ=y2x−y,system (5) always has one equilibrium E0a,0. Denote R0=a2/4; if R0>1, the system (5) has two positive equilibria E1x1,y1andE2x2,y2, where x1=a+a2−4/2,y1=a−a2−4/2 and x2=a−a2−4/2,y2=a+a2−4/2.Next, we will analysis of the stability of the equilibria of system (5). For equilibrium E0, we have(7)JE0=−100−1,obviously E0 is a stable node.ForE1, we have(8)JE1=−1−y12−2x1y1y12−1+2x1y1,and the characteristic equation is λ2+y12λ+y12−1=0. Let λ1 and λ2 be the two characteristic roots of the characteristic equation, then we have(9)λ1+λ2=−y12<0,λ1λ2=y12−1=y1x1−1=−2a2−4a+a2−4<0,obviously E1 is a saddle.ForE2, we have(10)JE2=−1−y22−2x2y2y22−1+2x2y2,and by calculations, we get(11)λ1+λ2=−y22<0,λ1λ2=y22−1=y2x2−1=2a2−4a−a2−4>0,obviously E2 is a stable node (Figure 3).Figure 3
Phase diagram of system (3) with θ=2.5,β=1,andγ=1.Lemma 2.
System (5) is uniformly bounded.Proof.
Firstly, if we have the isoclinesL1:dx/dt=0andL2:dy/dt=0 (Figure 4) and the straight line l1:x−a=0, then we get dl1/dt=dx/dt|x=a=−ay2<0; thus, according to the qualitative theory of ordinary differential equations, the trajectory of the system (5) passes through l1 and goes from the right side of l1 to the left side of l1. Consider the straight line l2:x+y−M=0, where M is large enough and 0≤x≤a. Then, we obtain dl2/dt|l2=0=a−M<0, and thus, the straight line l2 is nontangent; then, according to the qualitative theory of ordinary differential equations, the trajectory of the system (5) passes through l2 and goes from the upper right side of l2 to the lower left side of l2. Let us denote the intersections of l2 and L2 be HxH,yH and consider the straight line l3:y−yH=0, obviously, we have dy/dt|l3=0<0, and then, the trajectory of model (5) passes through l3 and goes from the top side of l3 to the bottom side of l3. Thus, the model (5) is uniformly upper bounded. This completes the proof.Figure 4
System (3) is the uniformly bounded.


## 4. Homoclinic Cycle of Model about Parameterp

In this section, we will discuss the existence of order-1 homoclinic cycle of model (3) by choosing p as the control parameter.Theorem 1.
IfR0>1, then there exists p′∈0,1 such that model (3) has an order-1 homoclinic cycle.Proof.
In model (3), since the point E1 is a saddle point, then there exist two manifolds which will enter or leave the saddle point E1, one is the unstable manifold (ΓA) and another is the stable manifold ΓB. According to Lemma 2 and the qualitative theory of ordinary differential equations, ΓA and impulse set M must intersect, and the intersection is denoted as AxA,yA. If we denote the intersection of impulse set M and the isocline L1 as point CxC,yC, the intersection of image set N and isoclines L1 as point DxD,yD, and the intersection of image set N and ΓB as point BxB,yB, by the qualitative theory of ordinary differential equations, the unstable manifold ΓA is above of the isoclines L1, and the stable manifold ΓB is below the isoclines L2dy/dt=0 (Figure 5). Because the monotonicity of the impulse function φx,p=1−px with respect to x and p, there must exist p′∈0,1 such that φxA,p′=1−p′xA=xB, and then the stable manifold ΓB starting form the point B, the unstable manifold ΓA starting form the point E1, and the impulse line AB formed a homoclinic cycle.Figure 5
The existence of the order-1 homoclinic cycle.Remark 2.
Ifp>p′, according to the theory of differential equations, the trajectory tends to E0, and in a biological sense, the disease eventually extincts. However, the relatively high vaccination rate will waste medical resources. So, we always assume that p<p′ in the following theorem.Theorem 2.
IfR0>1, p<p′, and p′−p≪1, then the homoclinic cycle of model (3) disappears and bifurcates an unique order-1 periodic solution.Proof.
By Theorem 1, ifR0>1, then there exists p′∈0,1 such that model (3) has an order-1 homoclinic cycle, i.e., the stable manifold ΓB starting form the point B, the unstable manifold ΓA starting form the point E1, and the impulse line AB formed a homoclinic cycle. Now, we consider whether there will be a periodic solution that bifurcates out of the homoclinic cycle when p changes. In fact, consider the unstable manifold ΓA starting form the point E1, when ΓA touches the impulsive set M (the intersection is denoted as A), then a pulse happens and then the impulsive function transfers the point A into D1 and the point C into B1, and according the definition of impulsive function, we have φxA,p=1−pxA=xD1,φxC,p=1−pxC=xB1. If p<p′, we obtain xD1>xB. Since xB≤φxC,p=xB1andxD≥φxA,p=xD1, then we get xD≥xD1≥xB1≥xB, and then by the definition of Bendixon region of semicontinuous dynamics system, AC,CD (part of isoclinal L1), DB (B1D1⊂BD), BE1 (part of the ΓB), and E1A (part of the ΓA) constitute the Bendixon region G of the system (3). According to Lemma 1 and Lemma 2 in [36], system (3) must exist an order-1 periodic solution, initial point of which is between B1 and D1 in image set N (Figure 6).
Next, we show the order-1 periodic solution of system (3) is unique if it exists. The idea of the proof comes from the study of Wei and Chen [36]. Select two points I and J in phase set B1D1 arbitrarily, where xB1<xJ<xI<xD1. Let FI=I1∈MandFJ=J1∈M, after that due to the impulsive effects, points I1 and J1 jump to I1+,J1+∈N. For xJ<xI, we have xI1<xJ1 and xI1+=1−pxI1, xJ1+=1−pxJ1; hence, we have xI1+<xJ1+. Using Definition 5 and Definition 6 in [36], we obtain gI=xI1+−xI and gJ=xJ1+−xJ. Hence, we have gI−gJ=xI1+−xI−xJ1+−xJ=xJ−xI+xI1+−xJ1+<0; that is, the successor function gp is monotonic in B1D1. Therefore, there is an unique point H such that gH=0; thus, system (3) has an unique order-1 periodic solution (Figure 7).Figure 6
The existence of order-1 periodic solution of system (3).Figure 7
The monotonicity of the successor function G in the segmentA1B1.Theorem 3.
IfR0>1, p<p′, and p′−p≪1, then the order-1 periodic solution of model (3) is orbitally asymptotically stable.Proof.
By Theorem 2, we have that the order-1 periodic solution in system (3) is unique. Let the initial point of the order-1 periodic solution is H∈B1D1, where xB1<xH<xD1. Set FD1=C1∈M, then due to the impulsive effects C1 jumps to C1+ which is the successor point of D1 (Figure 8). We have xB1<xC1+<xH. Set FC1+=C2∈M. Owing to trajectories do not intersect, we have xH<xC2+<xD1 and xC1<xF<xC2<xA, where F is the impulsive point of the order-1 periodic solution.
Similar to the above method, letFC2+=C3∈M, we have xC1+<xC3+<xH<xC2+ and xC1<xC3<xF<xC2. We can repeat the above steps and have a sequence Ckk=1,2,… of impulse set M and a sequence Ck+k=1,2,… of image set N satisfying FCk+=Ck+1, xC2k−1+<xC2k+1+<xH<xC2k+, xC2k−1+<xH<xC2k+<xC2k−2+. That is, we have(12)xB1<xC1+<xC3+<⋯<xC2k−1+<xC2k+1+<⋯<xH,xD1>xC2+>xC4+>⋯>xC2k+>xC2k+1+>⋯>xH.
Thus, seriesxC2k−1+,k=1,2,…, is monotonically increasing, and xC2k+,k=1,2,…, is monotonically decreasing; xC2k+⟶xH, as k⟶∞, and xC2k−1+⟶xH, as k⟶∞. Select a point H0∈C1+D1 that is different from point H. If xH<xH0<xD1 (otherwise, xC1+<xH0<xH, the discussion is similar to xH<xH0<xD1), there must be an integer k such that xC2k+1+<xH0<xC2k+. The orbit starting from point H0 will also experience an infinite number of impulsive effects. Let the phase point be Hl,l=0,1,2,… which is after the lth impulsive effect. Then for any l, we have xC2k+l+1+<xH2l<xC2k+l+ and xC2k+l+1+<xH2l+1<xC2k+l+1+1+. Hence, xH2l,l=0,1,2,…, is monotonically decreasing, and xH2l+1,i=0,1,2,…, is monotonically increasing. Thus, after the pulse effects the successor points are attracted to the point H, which means that the order-1 periodic solution of the system (3) is orbitally asymptotically stable.Figure 8
The stability of order-1 periodic solution of system (3).


## 5. Numerical Simulations

In this section, we give some numerical simulations to illustrate the theoretical results we previously obtained. First, we consider the system neglecting state-dependent pulse strategy, letθ=2.5,β=1,γ=1,andh=1.6, and simple calculations show R0=1.5625; then system (3) has three equilibria, i.e., E0=2.5,0,E1=2,0.5, and E2=0.5,2 (Figure 3), and among them, E0 is a stable node, E1 is a saddle, and E2 is a stable node.Then, we consider the state-dependent pulse control strategy in system (3). First, we take more moderate preventive and therapeutic measures, and let p=0.496,q=0.5, then system (3) has a homoclinic cycle composed of the unstable manifold (ΓA), the stable manifold (ΓB), and the pulse straight line (Figure 9, the initial value is S0=0.15andI0=0.895). And from the time series diagrams, we can see that x and y show periodic oscillations over time (Figures 10 and 11). If we maintain a certain intensity of treatment (fix parameter q=0.5) and reduce the intensity of prevention, for example, let p=0.3, by Theorem 2, the order-1 homoclinic cycle disappears and bifurcates a order-1 periodic solution, which is shown in Figures 12–14. And if we take more stringent preventive measures which means a larger vaccination rate, for example, let p=0.65, then the disease will become extinct, which is shown in Figures 15–17.Figure 9
Order-1 homoclinic cycle of system (3) with p=0.496andq=0.5.Figure 10
Time series diagram ofxt of system (3) with p=0.496andq=0.5.Figure 11
Time series diagram ofyt of system (3) with p=0.496andq=0.5.Figure 12
Order-1 homoclinic bifurcation of system (3) with p=0.3.Figure 13
Time series diagram ofxt of system (3) with p=0.3.Figure 14
Time series diagram ofyt of system (3) with p=0.3.Figure 15
Phase diagram of system (3) with p=0.65.Figure 16
Time series diagram ofxt of system (3) with p=0.65.Figure 17
Time series diagram ofyt of system (3) with p=0.65.


## 6. Conclusion

In this paper, a different strategy from tradition, i.e., the state-dependent pulse vaccination and therapeutic strategy, is considered in the control of the disease. A pulse system is built to model this process based on an SI ordinary differential equation model. By using the theory of semicontinuous dynamic system, the dynamics of the pulse system is analyzed. Our results show the pulse system exhibits rich dynamics; for example, the system has a unique order-1 homoclinic cycle, and by choosingp as the control parameter, we prove that when p changes, the order-1 homoclinic cycle disappears and bifurcates an orbitally asymptotical stable order-1 periodic solution. However, it should be pointed out here that, in this work, we focused on the theoretical framework and realistic parameters can be incorporated into our model. State-dependent impulsive vaccination strategy may be used a supplementary control measure besides routine vaccination, or it may be used in the situation when vaccine stockpile is limited (for example, the yellow fever outbreaks in Nigeria and Congo in 2017 [52]). The realistic approach in childhood infection and other infections will be conducted in future work.



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