# Existence of Periodic Solutions of Seasonally Forced SEIR Models with Pulse Vaccination

Authors: Lin Wang
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, we are interested in finding the periodic oscillation of seasonally forced SEIR models with pulse vaccination. Many infectious diseases show seasonal patterns of incidence. Pulse vaccination strategy is an effective tool to control the spread of these infectious diseases. Assuming that the seasonally dependent transmission rate is a T-periodic forcing, we obtain the existence of positive T-periodic solutions of seasonally forced SEIR models with pulse vaccination by Mawhin’s coincidence degree method. Some relevant numerical simulations are presented to illustrate the effectiveness of such pulse vaccination strategy.


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## Body

## 1. Introduction

It is a common phenomenon that the incidence of many infectious diseases often changes periodically with the seasonal cycle, such as measles, chickenpox, mumps, rubella, pertussis, and influenza [1–3]. In order to understand the mechanisms responsible for seasonal disease incidence and the epidemiological consequences of seasonality, a large number of mathematical models of infectious diseases with periodic transmission rates have been established [4–7]. Dietz [8] was the first to investigate the effects of one-year periodic contact rate in the classical SIR and SEIR models. Dietz considered a periodical contact rate given by(1)βt=βm1+Acosωt.The periodically forced nonlinear effects in epidemic models have been studied extensively in the mathematical literature [9, 10].Pulse vaccination strategy (PVS) is an effective tool to control the spread of epidemics, for example, control of poliomyelitis and measles in Central [11] and South American [12] and the UK vaccination campaigns against measles in 1994 [13]. The theoretical study on pulse vaccination strategy was firstly presented by Agur et al. [14]. Shulgin et al. [15, 16] incorporated pulse vaccination into the SIR epidemic model. Nokes and Swinton studied the control of childhood viral infections by pulse vaccination [17]. d’Onofrio applied the pulse vaccination method for SIR and SEIR epidemic models [18, 19]. PVS consists of periodical repetitions of impulsive multiage cohort vaccinations in a population [18, 19]. PVS proposes to vaccinate a constant fraction p of the entire susceptible people in a single pulse, which can be formulated as(2)Sti+≔limh⟶0+Sti+h=Sti−pSti,where St is left continuous satisfying Sti=Sti−≔limh⟶0−Sti+h. Pulse vaccination gives life-long immunity to pS susceptibles who are transferred to the “recovered” class of the population, which can be formulated as(3)Rti+=Rti+pSti.This kind of vaccination is called impulsive since all the vaccine doses are applied in a time which is short considering the dynamics of the disease. PVS has been further developed, for example, in [20–23]. A comprehensive introduction on vaccination strategies can be found in [24].Many infectious diseases do not die out, but become endemic. For autonomous epidemic models, the existence of the positive equilibrium plays an important role. A positive periodic solution in the periodic model will play the same role as a positive equilibrium in the autonomous model [25, 26]. Recently, Katriel [27] proved that the seasonally forced SIR model with a T-periodic forcing has a periodic solution with periodic T by Leray–Schauder degree theory provided 1/T∫0Tβtdt>γ+μ. Jódar et al. [26] obtained that a T-periodic solution exists for a more general system by the famous Mawhin’s coincidence degree method, whenever the condition mint∈ℝβt>γ+μ holds. Using Leray–Schauder degree theory, Zu and the author [28] established new results on the existence of at least one positive periodic solution for a seasonally forced SIR model with impact of media coverage. The author [29] proved the existence of positive periodic solutions of seasonally forced SIR models with impulse vaccination at fixed time by Mawhin’s coincidence degree method if the basic reproductive number ℛ0=β¯/γ+μ>1. Coincidence degree theory has been applied to prove the existence of multiple periodic solutions of the epidemic model with seasonal periodic rate [30, 31]. There are some research activities about the existence of periodic solutions of impulsive differential equation [32–36].The aim of this paper is to study the existence of periodic solution of seasonally forced SEIR models with pulse vaccination. The paper is organized as follows. A seasonally forced SEIR model with pulse vaccination is formulated and a suitable region to our problem is chosen in Section2. The existence of periodic solutions of our impulsive systems is established in Section 3. Some numerical simulations are demonstrated to verify the effectiveness of our pulse vaccination strategy in Section 4. The relevant conclusion will be stated in Section 5.


## 2. The Model and Prelimimaries

### 2.1. The Seasonally Forced SEIR Model with Pulse Vaccination

In this paper, we focus on the existence of periodic solution of seasonally forced SEIR models with pulse vaccination; we consider models of the form(4)dStdt=μ−βtStIt−μSt,dEtdt=βtStIt−ϵ+μEt,dItdt=ϵEt−μ+γIt,dRtdt=γIt−μRt,△S|t=nT+ti=−JiSt,It|t=nT+ti,△R|t=nT+ti=JiSt,It|t=nT+ti,modeling the spread of infectious diseases with PVS under the following hypotheses:(i)
The population is divided into four classes;St, Et, It, and Rt denote, respectively, the fractions of the susceptible, exposed/latent, infective, and recovered population, and St+Et+It+Vt=1 is invariant with St,Et,It,Rt≥0, for all t≥0.(ii)
μ, ϵ, and γ denote the birth (death) rate, the rate of latent individuals becoming infectious, and recovery rate, respectively, which are positive constants.(iii)
βt is the seasonally dependent transmission rate, which is a positive continuous T-periodic function.(iv)
The susceptible population can be divided into many groups and all the groups cannot be vaccinated at the same time, and the susceptible population will be vaccinated for several times:(5)△S|t=nT+ti=SnT+ti+−SnT+ti,0≤t1<⋯<tk<T,n∈ℕ.Our vaccination strategies concern the impact of infected population, which can be formulated as(6)JiSt,It|t=nT+ti=pi1−e−αInT+tiSnT+ti,i=1,…,k,where 0≤pi<1 and the sensitivity coefficient α>0 is sufficiently large.Denote the basic reproduction number:(7)ℛ0=mint∈ℝβtϵϵ+μγ+μ.Obviously, we havedSt/dt+dEt/dt+dIt/dt+dRt/dt≡0. Since St, Et,It, and Rt are fractions of the population, we have St+Et+It+Rt=1 for all t. Because Rt does not appear in the first three equations in (4), system (4) reduces to the following 3-dimensional system:(8)dStdt=μ−βtStIt−μSt,dEtdt=βtStIt−ϵ+μEt,dItdt=ϵEt−μ+γIt,△S|t=nT+ti=−JiSt,It|t=nT+ti,with(9)St≥0,Et≥0,It≥0,St+Et+It≤1.Obviously, finding the periodic solution of (8) is equivalent to finding the solutions of the following periodic boundary value problem:(10)dStdt=μ−βtStIt−μSt,dEtdt=βtStIt−ϵ+μEt,dItdt=ϵEt−μ+γIt,△S|t=ti=−JiSt,It|t=ti,with periodic boundary condition(11)S0=ST,E0=ET,I0=IT.


### 2.2. The Suitable Region to Our Problem

In order to prove the existence of periodic solutions of (8), we consider the following auxiliary problem:(12)dStdt=λμ−βtStIt−μSt,dEtdt=λβtStIt−ϵ+μEt,dItdt=λϵEt−μ+γIt,△S|t=ti=−λJiSti,Iti,where λ∈0,1 and t∈0,T.LetD be an open bounded subset of X (will be denoted in Section 2.4) satisfying(13)D=St,Et,It∈X|St>0,Et>0,It>0,St+Et+It<1.Proposition 1.
D¯ is an invariant region with respect to (12). The disease-free equilibrium S0,E0,I0=1,0,0 is the unique periodic solution of (12) satisfying St,Et,It∈∂D, 0<λ≤1.Proof.
First, we will prove thatD¯ is an invariant region.
In fact, it follows from model (12) that(14)dSdt∣S=0=λμ>0,dEdt|E=0≥λβtSI≥0,dIdt|I=0=λϵE≥0,dS+E+Idt|S+E+I=1=−λγI≤0.
Since there is no impulsive motion forE,I, and(15)Sti+=1−λpi1−e−αItiSti,it is easy to conclude that every possible solution will remain in the region D¯ ultimately.
Second, we will prove that the disease-free equilibriumS0,E0,I0≡1,0,0 is the unique periodic solution of (12) satisfying S,E,I∈∂D.
We assume thatSt,Et,It∈∂D is a solution of (12); this means at least one of the following conditions holds:(i)
There existst0∈0,T so that It0=0(ii)
There existst0∈0,T so that St0=0(iii)
There existst0∈0,T so that Et0=0(iv)
There existst0∈0,T so that St0+Et0+It0=1
We now consider each of these four cases:
In case of (i), we haveIt0=0 and I′t0=λϵEt0≥0. If Et0=0, I′t≡0 and △Sti=0,i=1,…,k. Thus, the only possible periodic solution of S′=λμ1−S is S≡1. On the other hand, if Et0≠0, I′t0>0. Thus, it is easy to obtain that It<0 for t<t0 sufficiently close to t0, which contradicts that D¯ is an invariant region.
In case of (ii), we haveSt0=0 and S′t0=λμ>0. Thus, it is easy to obtain that St<0 for t<t0 sufficiently close to t0, which contradicts that D¯ is an invariant region.
In case of (iii), we haveEt0=0 and E′t0=λβt0St0It0≥0. Because the cases It0=0 and St0=0 have been discussed above, we have E′t0>0 which again contradicts that D¯ is an invariant region.
In case of (iv), we get(16)S+E+I′t0=λμ1−St0−Et0−It0−λγIt0=−λγIt0≤0.
BecauseIt0=0 has been discussed, we only discuss St0+Et0+It0=1, S+E+I′t0<0, which contradicts that D¯ is an invariant region.Remark 1.
If the impulsive motion is not influenced byI, which means Sti+=1−λpiSti, the system E≡0,I≡0, S′=μ1−S can have a nonconstant periodic solution on ∂D, which is hard to handle. In fact, if there are no infectious patients, it is often meaningless to vaccinate the susceptible people.
In order to use continuity method, it is necessary to choose an open bounded setΩ⊆D, such that there is no solution S,E,I of (12) satisfying St,Et,It∈∂Ω for any λ∈0,1. Motivated by the idea of Katriel [27], we choose Ω to be the open subset of D given by(17)Ω=St,Et,It∈D|mint∈0,TSt<δ,where δ∈0,1 is to be fixed.Proposition 2.
Letℛ0>1, δ∈1/ℛ0,1. Then, there exists no solution St,Et,It of (12) satisfying St,Et,It∈∂Ω, for any λ∈0,1.Proof.
SupposeSt,Et,It∈∂Ω. Then, either St,Et,It∈∂D|mint∈0,TSt<δ or St,Et,It∈D|mint∈0,TSt=δ.
In the first case,St≡1 and Proposition 1 imply that there is no solution of (12) on S,E,I∈∂D|mint∈0,TSt<δ.
In the second case, we haveIt>0,Et>0 and St≥δ, ∀t∈0,T. Since there is no impulsive motion for I and I0=IT, after integrating the third equation of (12) over 0,T, we obtain(18)μ+γ1T∫0TItdt=ϵ1T∫0TEtdt.
Since there is no impulsive motion forE and E0=ET, after integrating the second equation of (12) over 0,T, we obtain(19)ϵ+μ1T∫0TEtdt=1T∫0TβtStItdt.
With the help of (17)–(19), we conclude that(20)γ+μϵ+μϵ1T∫0TItdt=1T∫0TβtStItdt≥δmint∈ℝβt1T∫0TItdt.
ForIt>0, we get immediately(21)γ+μϵ+μϵ≥δmint∈ℝβt,which is a contradiction to the assumption δ>1/ℛ0.


### 2.3. Outline of Mawhin’s Coincidence Degree Method

We introduce a few definitions and recall the continuation theorem which will help us to prove the existence of positive solutions of system (10).Consider the operator equation:(22)Lx=Nx,where L:domL⊂X⟶Z is a linear bounded operator, N:X⟶Z is a continuous operator, and X and Z are Banach spaces.Definition 1.
(see [37]). The linear mapping L is called a Fredholm mapping of index zero if(1)
IndexL=dimKerL−codimImL=0(2)
ImLis closed inZ
IfL:Ω⊆X⟶Z is a Fredholm mapping of index zero, there exist continuous projectors P:X⟶X and Q:Z⟶Z such that(1)
ImP=KerL(2)
KerQ=ImL=ImI−Q(3)
X=KerL⊕KerP(4)
Z=ImL⊕ImQ
It follows thatL|domL∩KerP:I−PX⟶ImL is invertible. We denote the inverse of that map by Kp.Definition 2.
(see [37]). Let Ω be an open bounded subset of X. The mapping N is called L-compact on Ω¯, if(1)
QNΩ¯is bounded(2)
Kp1−QN:Ω¯⟶Xis compact
SinceImQ is isomorphic to KerL, there exists an isomorphism Λ:ImQ⟶KerL.Theorem 1 (see [37]) (Mawhin’s continuation theorem).
LetΩ⊂X be an open bounded set. Let L be a Fredholm mapping of index zero and be L-compact on Ω. Assume that(1)
For eachλ∈0,1, x∈∂Ω∩domL, Lx≠λNx(2)
For eachx∈∂Ω∩KerL, QNx≠0(3)
degΛQN,Ω∩KerL,0≠0
Then, the equationLx=Nx has at least one solution in domL∩Ω¯.Lemma 4 (Arzela–Ascoli) (see [38]).
LetD be a compact subset of Rn and LD,Rn be the linear space of continuous functions which take D into Rn; any uniformly bounded equicontinuous sequence of functions ϕn,n=1,2… in LD,Rn has a subsequence which converges uniformly on D.


### 2.4. Notations

We now prepare the setting to apply Mawhin’s continuation theorem. DefineCj0,T;t1,…,tk=x:0,T⟶|ℝ|xmt exists for t≠t1,…,tk and xmti+, xmti+ exist with xmti≔xmti+, i=1,…,k where j is a non-negative integer and m∈0,…,j, i=1,…,k, m=0,…,jDefine a Banach space(23)X=x1,x2,x3|x1∈C0,T;t1,…,tk,x2,x3∈C0,T,x10=x1T,x20=x2T,x30=x3T,with the norm(24)‖x1,x2,x3‖X=maxt∈0,Tx1t+x2t+x3t.Define another Banach space(25)Z=C0,T;t1,…,tk×C0,T×C0,T×ℝk,with the norm(26)‖z1,z2,z3,C1,…,Ck‖Z=maxt∈0,Tz1t+z2t+z3t+C1+⋯+Ck.Let(27)L:domL⟶Z,S,E,I⟶S′,E′,I′,△St1,…,△Stk,where(28)domL=S,E,I|S∈C10,T;t1,…,tk,E,I∈C10,T,S0=ST,E0=ET,I0=IT.Let(29)N:X⟶Z,S,E,I⟶f1⋅,S,E,I,f2⋅,S,E,I,f3⋅,S,E,I,−J1St1,It1,…,−JkStk,Itk,where(30)f1⋅,S,E,I=μ−βtSI−μS,f2⋅,S,E,I=βtSI−ϵ+μE,f3⋅,S,E,I=ϵE−μ+γI.Obviously, system (10) can be written by LS,E,I=NS,E,I. By a simple calculation, we get(31)KerL=S,E,I|S,E,I≡C∈ℝ3,It is easy to see that(32)dimKerL=codimImL=3.SinceImL is closed, L is Fredholm mapping of index 0. Let P:X⟶X be the projector given by(33)PS,E,I=1T∫0TSτdτ,1T∫0TEτdτ,1T∫0TIτdτ,S,E,I∈X.Obviously,(34)ImP=KerL=ℝ3.LetQ:Z⟶Z be the projector given by(35)Qz1,z2,z3,C1,⋯Ck=1T∫0Tz1τdτ+1T∑i=1kCi,1T∫0Tz2τdτ,1T∫0Tz3τdτ,01×k,Obviously,(36)KerQ=ImL=ImI−Q.Furthermore, the generalized inverse (toL) Kp:ImL⟶KerP∩domL exists given by(37)Kpz1,z2,z3,C1,⋯Ck=∫0tz1τdτ+∑t>tiCi−1T∫0T∫0tz1τdτdt−∑i=1kCi,∫0tz2τdτ−1T∫0T∫0tz2τdτdt,∫0tz3τdτ−1T∫0T∫0tz3τdτdt.Then,QN:⟶XZ has the following form:(38)QNS,E,I=1T∫0Tf1τ,S,E,Idτ−1T∑i=1kJiSti,Iti,1T∫0Tf2τ,S,E,Idτ,1T∫0Tf3τ,S,E,Idτ,01×k.By a direct calculation, we have(39)Kp1−QNS,E,I=Kpf1⋅,S,E,I−1T∫0Tf1τ,S,E,Idτ+1T∑i=1kJiSti,Iti,f1⋅,S,E,I−1T∫0Tf2τ,S,E,Idτ,f3⋅,S,E,I−1T∫0Tf3τ,S,E,Idτ,J1St1,It1,…,JkStk,Itk=∫0tf1dτ−1T∫0T∫0tf1dτdt+12−tT∫0Tf1dτ+∑i=1kJiSti,Iti+∑t>tiJiSti,Iti−∑i=1kJiSti,Iti,∫0tf2dτ−1T∫0T∫0tf2dτdt+12−tT∫0Tf2dτ,∫0tf3dτ−1T∫0T∫0tf3dτdt+12−tT∫0Tf3dτ.


## 2.1. The Seasonally Forced SEIR Model with Pulse Vaccination

In this paper, we focus on the existence of periodic solution of seasonally forced SEIR models with pulse vaccination; we consider models of the form(4)dStdt=μ−βtStIt−μSt,dEtdt=βtStIt−ϵ+μEt,dItdt=ϵEt−μ+γIt,dRtdt=γIt−μRt,△S|t=nT+ti=−JiSt,It|t=nT+ti,△R|t=nT+ti=JiSt,It|t=nT+ti,modeling the spread of infectious diseases with PVS under the following hypotheses:(i)
The population is divided into four classes;St, Et, It, and Rt denote, respectively, the fractions of the susceptible, exposed/latent, infective, and recovered population, and St+Et+It+Vt=1 is invariant with St,Et,It,Rt≥0, for all t≥0.(ii)
μ, ϵ, and γ denote the birth (death) rate, the rate of latent individuals becoming infectious, and recovery rate, respectively, which are positive constants.(iii)
βt is the seasonally dependent transmission rate, which is a positive continuous T-periodic function.(iv)
The susceptible population can be divided into many groups and all the groups cannot be vaccinated at the same time, and the susceptible population will be vaccinated for several times:(5)△S|t=nT+ti=SnT+ti+−SnT+ti,0≤t1<⋯<tk<T,n∈ℕ.Our vaccination strategies concern the impact of infected population, which can be formulated as(6)JiSt,It|t=nT+ti=pi1−e−αInT+tiSnT+ti,i=1,…,k,where 0≤pi<1 and the sensitivity coefficient α>0 is sufficiently large.Denote the basic reproduction number:(7)ℛ0=mint∈ℝβtϵϵ+μγ+μ.Obviously, we havedSt/dt+dEt/dt+dIt/dt+dRt/dt≡0. Since St, Et,It, and Rt are fractions of the population, we have St+Et+It+Rt=1 for all t. Because Rt does not appear in the first three equations in (4), system (4) reduces to the following 3-dimensional system:(8)dStdt=μ−βtStIt−μSt,dEtdt=βtStIt−ϵ+μEt,dItdt=ϵEt−μ+γIt,△S|t=nT+ti=−JiSt,It|t=nT+ti,with(9)St≥0,Et≥0,It≥0,St+Et+It≤1.Obviously, finding the periodic solution of (8) is equivalent to finding the solutions of the following periodic boundary value problem:(10)dStdt=μ−βtStIt−μSt,dEtdt=βtStIt−ϵ+μEt,dItdt=ϵEt−μ+γIt,△S|t=ti=−JiSt,It|t=ti,with periodic boundary condition(11)S0=ST,E0=ET,I0=IT.


## 2.2. The Suitable Region to Our Problem

In order to prove the existence of periodic solutions of (8), we consider the following auxiliary problem:(12)dStdt=λμ−βtStIt−μSt,dEtdt=λβtStIt−ϵ+μEt,dItdt=λϵEt−μ+γIt,△S|t=ti=−λJiSti,Iti,where λ∈0,1 and t∈0,T.LetD be an open bounded subset of X (will be denoted in Section 2.4) satisfying(13)D=St,Et,It∈X|St>0,Et>0,It>0,St+Et+It<1.Proposition 1.
D¯ is an invariant region with respect to (12). The disease-free equilibrium S0,E0,I0=1,0,0 is the unique periodic solution of (12) satisfying St,Et,It∈∂D, 0<λ≤1.Proof.
First, we will prove thatD¯ is an invariant region.
In fact, it follows from model (12) that(14)dSdt∣S=0=λμ>0,dEdt|E=0≥λβtSI≥0,dIdt|I=0=λϵE≥0,dS+E+Idt|S+E+I=1=−λγI≤0.
Since there is no impulsive motion forE,I, and(15)Sti+=1−λpi1−e−αItiSti,it is easy to conclude that every possible solution will remain in the region D¯ ultimately.
Second, we will prove that the disease-free equilibriumS0,E0,I0≡1,0,0 is the unique periodic solution of (12) satisfying S,E,I∈∂D.
We assume thatSt,Et,It∈∂D is a solution of (12); this means at least one of the following conditions holds:(i)
There existst0∈0,T so that It0=0(ii)
There existst0∈0,T so that St0=0(iii)
There existst0∈0,T so that Et0=0(iv)
There existst0∈0,T so that St0+Et0+It0=1
We now consider each of these four cases:
In case of (i), we haveIt0=0 and I′t0=λϵEt0≥0. If Et0=0, I′t≡0 and △Sti=0,i=1,…,k. Thus, the only possible periodic solution of S′=λμ1−S is S≡1. On the other hand, if Et0≠0, I′t0>0. Thus, it is easy to obtain that It<0 for t<t0 sufficiently close to t0, which contradicts that D¯ is an invariant region.
In case of (ii), we haveSt0=0 and S′t0=λμ>0. Thus, it is easy to obtain that St<0 for t<t0 sufficiently close to t0, which contradicts that D¯ is an invariant region.
In case of (iii), we haveEt0=0 and E′t0=λβt0St0It0≥0. Because the cases It0=0 and St0=0 have been discussed above, we have E′t0>0 which again contradicts that D¯ is an invariant region.
In case of (iv), we get(16)S+E+I′t0=λμ1−St0−Et0−It0−λγIt0=−λγIt0≤0.
BecauseIt0=0 has been discussed, we only discuss St0+Et0+It0=1, S+E+I′t0<0, which contradicts that D¯ is an invariant region.Remark 1.
If the impulsive motion is not influenced byI, which means Sti+=1−λpiSti, the system E≡0,I≡0, S′=μ1−S can have a nonconstant periodic solution on ∂D, which is hard to handle. In fact, if there are no infectious patients, it is often meaningless to vaccinate the susceptible people.
In order to use continuity method, it is necessary to choose an open bounded setΩ⊆D, such that there is no solution S,E,I of (12) satisfying St,Et,It∈∂Ω for any λ∈0,1. Motivated by the idea of Katriel [27], we choose Ω to be the open subset of D given by(17)Ω=St,Et,It∈D|mint∈0,TSt<δ,where δ∈0,1 is to be fixed.Proposition 2.
Letℛ0>1, δ∈1/ℛ0,1. Then, there exists no solution St,Et,It of (12) satisfying St,Et,It∈∂Ω, for any λ∈0,1.Proof.
SupposeSt,Et,It∈∂Ω. Then, either St,Et,It∈∂D|mint∈0,TSt<δ or St,Et,It∈D|mint∈0,TSt=δ.
In the first case,St≡1 and Proposition 1 imply that there is no solution of (12) on S,E,I∈∂D|mint∈0,TSt<δ.
In the second case, we haveIt>0,Et>0 and St≥δ, ∀t∈0,T. Since there is no impulsive motion for I and I0=IT, after integrating the third equation of (12) over 0,T, we obtain(18)μ+γ1T∫0TItdt=ϵ1T∫0TEtdt.
Since there is no impulsive motion forE and E0=ET, after integrating the second equation of (12) over 0,T, we obtain(19)ϵ+μ1T∫0TEtdt=1T∫0TβtStItdt.
With the help of (17)–(19), we conclude that(20)γ+μϵ+μϵ1T∫0TItdt=1T∫0TβtStItdt≥δmint∈ℝβt1T∫0TItdt.
ForIt>0, we get immediately(21)γ+μϵ+μϵ≥δmint∈ℝβt,which is a contradiction to the assumption δ>1/ℛ0.


## 2.3. Outline of Mawhin’s Coincidence Degree Method

We introduce a few definitions and recall the continuation theorem which will help us to prove the existence of positive solutions of system (10).Consider the operator equation:(22)Lx=Nx,where L:domL⊂X⟶Z is a linear bounded operator, N:X⟶Z is a continuous operator, and X and Z are Banach spaces.Definition 1.
(see [37]). The linear mapping L is called a Fredholm mapping of index zero if(1)
IndexL=dimKerL−codimImL=0(2)
ImLis closed inZ
IfL:Ω⊆X⟶Z is a Fredholm mapping of index zero, there exist continuous projectors P:X⟶X and Q:Z⟶Z such that(1)
ImP=KerL(2)
KerQ=ImL=ImI−Q(3)
X=KerL⊕KerP(4)
Z=ImL⊕ImQ
It follows thatL|domL∩KerP:I−PX⟶ImL is invertible. We denote the inverse of that map by Kp.Definition 2.
(see [37]). Let Ω be an open bounded subset of X. The mapping N is called L-compact on Ω¯, if(1)
QNΩ¯is bounded(2)
Kp1−QN:Ω¯⟶Xis compact
SinceImQ is isomorphic to KerL, there exists an isomorphism Λ:ImQ⟶KerL.Theorem 1 (see [37]) (Mawhin’s continuation theorem).
LetΩ⊂X be an open bounded set. Let L be a Fredholm mapping of index zero and be L-compact on Ω. Assume that(1)
For eachλ∈0,1, x∈∂Ω∩domL, Lx≠λNx(2)
For eachx∈∂Ω∩KerL, QNx≠0(3)
degΛQN,Ω∩KerL,0≠0
Then, the equationLx=Nx has at least one solution in domL∩Ω¯.Lemma 4 (Arzela–Ascoli) (see [38]).
LetD be a compact subset of Rn and LD,Rn be the linear space of continuous functions which take D into Rn; any uniformly bounded equicontinuous sequence of functions ϕn,n=1,2… in LD,Rn has a subsequence which converges uniformly on D.


## 2.4. Notations

We now prepare the setting to apply Mawhin’s continuation theorem. DefineCj0,T;t1,…,tk=x:0,T⟶|ℝ|xmt exists for t≠t1,…,tk and xmti+, xmti+ exist with xmti≔xmti+, i=1,…,k where j is a non-negative integer and m∈0,…,j, i=1,…,k, m=0,…,jDefine a Banach space(23)X=x1,x2,x3|x1∈C0,T;t1,…,tk,x2,x3∈C0,T,x10=x1T,x20=x2T,x30=x3T,with the norm(24)‖x1,x2,x3‖X=maxt∈0,Tx1t+x2t+x3t.Define another Banach space(25)Z=C0,T;t1,…,tk×C0,T×C0,T×ℝk,with the norm(26)‖z1,z2,z3,C1,…,Ck‖Z=maxt∈0,Tz1t+z2t+z3t+C1+⋯+Ck.Let(27)L:domL⟶Z,S,E,I⟶S′,E′,I′,△St1,…,△Stk,where(28)domL=S,E,I|S∈C10,T;t1,…,tk,E,I∈C10,T,S0=ST,E0=ET,I0=IT.Let(29)N:X⟶Z,S,E,I⟶f1⋅,S,E,I,f2⋅,S,E,I,f3⋅,S,E,I,−J1St1,It1,…,−JkStk,Itk,where(30)f1⋅,S,E,I=μ−βtSI−μS,f2⋅,S,E,I=βtSI−ϵ+μE,f3⋅,S,E,I=ϵE−μ+γI.Obviously, system (10) can be written by LS,E,I=NS,E,I. By a simple calculation, we get(31)KerL=S,E,I|S,E,I≡C∈ℝ3,It is easy to see that(32)dimKerL=codimImL=3.SinceImL is closed, L is Fredholm mapping of index 0. Let P:X⟶X be the projector given by(33)PS,E,I=1T∫0TSτdτ,1T∫0TEτdτ,1T∫0TIτdτ,S,E,I∈X.Obviously,(34)ImP=KerL=ℝ3.LetQ:Z⟶Z be the projector given by(35)Qz1,z2,z3,C1,⋯Ck=1T∫0Tz1τdτ+1T∑i=1kCi,1T∫0Tz2τdτ,1T∫0Tz3τdτ,01×k,Obviously,(36)KerQ=ImL=ImI−Q.Furthermore, the generalized inverse (toL) Kp:ImL⟶KerP∩domL exists given by(37)Kpz1,z2,z3,C1,⋯Ck=∫0tz1τdτ+∑t>tiCi−1T∫0T∫0tz1τdτdt−∑i=1kCi,∫0tz2τdτ−1T∫0T∫0tz2τdτdt,∫0tz3τdτ−1T∫0T∫0tz3τdτdt.Then,QN:⟶XZ has the following form:(38)QNS,E,I=1T∫0Tf1τ,S,E,Idτ−1T∑i=1kJiSti,Iti,1T∫0Tf2τ,S,E,Idτ,1T∫0Tf3τ,S,E,Idτ,01×k.By a direct calculation, we have(39)Kp1−QNS,E,I=Kpf1⋅,S,E,I−1T∫0Tf1τ,S,E,Idτ+1T∑i=1kJiSti,Iti,f1⋅,S,E,I−1T∫0Tf2τ,S,E,Idτ,f3⋅,S,E,I−1T∫0Tf3τ,S,E,Idτ,J1St1,It1,…,JkStk,Itk=∫0tf1dτ−1T∫0T∫0tf1dτdt+12−tT∫0Tf1dτ+∑i=1kJiSti,Iti+∑t>tiJiSti,Iti−∑i=1kJiSti,Iti,∫0tf2dτ−1T∫0T∫0tf2dτdt+12−tT∫0Tf2dτ,∫0tf3dτ−1T∫0T∫0tf3dτdt+12−tT∫0Tf3dτ.


## 3. Results

The following theorem gives the main results of this paper.Theorem 2.
Letℛ0>1; there exists at least one T-periodic solution St,Et,It,Rt of (1), all of whose components are positive.Remark 2.
Whenpi≡0, i=1,…,k, system (4) is the usual seasonally forced SEIR model without pulse vaccination.
The proof of Theorem2 will be divided into 4 steps.Proof.
Step 1:N is L-compact on Ω¯.First, it is obvious thatQNΩ¯ is bounded.For anyS,E,I∈Ω¯,(40)‖QNS,E,I‖Z=maxt∈0,T1T∫0Tf1τ,S,E,Idτ−1T∑i=1kJiti,Iti|+|1T∫0Tf2τ,S,E,Idτ|+|1T∫0Tf3τ,S,E,Idτ≤3μ+2β¯+2ϵ+γ+1T∑i=1kpi<+∞Then, it is easy to check thatKpI−QN:Ω¯⟶X is uniform bounded and equicontinuous on each ti,ti+1.Assume thatSj,Ej,Ijj=1∞⊂Ω¯. Using Lemma 4, there exists a uniformly convergent subsequence denoted by KpI−QNSj1,Ej1,Ij1 on 0,t1. Using Arzela–Ascoli lemma again on t1,t2, we have a uniformly convergent subsequence KpI−QNSj2,Ej2,Ij2 which is also uniformly convergent on 0,t1. Repeat it again and again, and we can prove that KpI−QNSjk+1,Ejk+1,Ijk+1 is uniformly convergent on 0,T. In this way, Kp1−QN:Ω¯⟶X is compact.Step 2: For eachλ∈0,1, S,E,I∈∂Ω∩domL, LS,E,I≠λNS,E,I, which has been proved by Proposition 2.Step 3: For eachS,E,I∈∂Ω∩KerL, QNS,E,I≠0.IfQNS1,E1,I1=0, we have(41)1T∫0Tf1τ,S1,E1,I1dτ−1T∑i=1kJiS1ti,I1ti=0,1T∫0Tf2τ,S1,E1,I1dτ=0,1T∫0Tf3τ,S1,E1,I1dτ=0.AssumeS1,E1,I1∈KerL; we know that S1,E1,I1 is a constant vector in ℝ3. Thus, (40) is equivalent to(42)μ−β¯S1I1−μS1−1T∑i=1kpi⋅1−e−αI1S1=0,β¯S1I1−ϵ+μE1=0,ϵE1−γ+μI1=0.We claim that there are exactly two solutions:1,0,0 and S∗,E∗,I∗ in KerL satisfying(43)S∗=ϵ+μγ+μβ¯ϵ,E∗=γ+μϵI∗,μβ¯ϵϵ+μγ+μ−1−β¯I∗−1T∑i=1kpi1−e−αI∗=0.According to the definition ofΩ, we know that 1,0,0∉Ω¯. Denote(44)GI=μβ¯ϵϵ+μγ+μ−1−β¯I−1T∑i=1kpi1−e−αI.We have(45)G0=μβ¯ϵϵ+μγ+μ−1>0,G1=−β¯ϵγ+μγ+μ2ϵ+μγ+μ−μ−1T∑i=1kpi1−e−α<0.Since(46)∂GI∂I=−β¯−1T∑i=1kpiαe−αI<0,GI is a monotonous function with respect to I. Thus, S∗,E∗,I∗ is the unique solution of QNS,E,I=0 in KerL∩Ω¯. Since(47)S∗=ϵ+μγ+μβ¯ϵ<ϵ+μγ+μmint∈ℝβtϵ=1ℛ0<δ,we haveS∗,E∗,I∗∉KerL∩∂Ω.Consequently, for eachS,E,I∈∂Ω∩KerL, QNS,E,I≠0.Step 4. There exists an isomorphismΛ:ImQ⟶KerL such that(48)ΛQz1,z2,C1,⋯,Ck=Λ1T∫0Tz1τdτ+1T∑i=1kCi,1T∫0Tz2τdτ,1T∫0Tz3τdτ,01×k=1T∫0Tz1τdτ+1T∑i=1kCi,1T∫0Tz2τdτ,1T∫0Tz3τdτ.
We will prove thatdegΛQN,Ω∩KerL,0≠0. On account of the discussion in step 3, we know that S∗,E∗,I∗ is the unique solution of ΛQNS,E,I=0 in Ω∩KerL.
A direct calculation shows that(49)degΛQNS,E,I,Ω∩KerL,0=degμ−β¯SI−μS−1T∑ikpi1−e−αIS,β¯SI−ϵ+μE,ϵE−γ+μI,Ω∩KerL,0=Sign−β¯I∗−μ−1T∑i=1kpi1−e−αI∗0−β¯S∗−1T∑i=1kpiαe−αI∗S∗β¯I∗−ϵ+μβ¯S∗0ϵ−γ+μ=−1≠0.
Thus, we conclude from Theorem1 that the equation LS,E,I=NS,E,I has at least one solution on domL∩Ω¯.
From Proposition2, we deduce that for each λ=1, S,E,I∈domL∩∂Ω, LS,E,I≠NS,E,I. It follows that LS,E,I=NS,E,I has at least one solution in domL∩Ω.


## 4. Simulation

In this section, some relevant numerical simulations about the T-periodic solution of the seasonally forced SEIR models are presented to show the effectiveness of PVS. Furthermore, we will compare the effects of different parameters on the solutions of the seasonally forced SEIR models with PVS.With the periodT=2π of the forcing representing one year, we take γ=142π/365 corresponding to a 2-week infectious period. We set βt=β¯1+0.6cost. Assume that there are three impulsive points at fixed time π/2, 2π/2, and 3π/2 with pi, i=1,2,3. Let 0,2π be divided into k=100 intervals equally. Given the initial point S∗∗=ϵ+μγ+μ/β¯ϵ,E∗∗=μ1/μ+ϵ−μ+γ/ϵβ¯,I∗∗=μϵ/μ+ϵμ+γ−1/β¯, which is the endemic equilibrium of the SEIR model without periodic transmission rate and pulse vaccination. The periodic solutions of system (10) can be solved by the Newton iteration method in which we set Sti+2π/100−Sti=pi1−e−αItiSti at fixed time π/2, 2π/2, and 3π/2.Simulation 1.
Setβ¯=15γ, ϵ=0.9/2π, μ=0.8/2π,andpi=0.2; we get the approximate susceptible population, exposed population, infective population, and recovered population of system (10) by Newton iteration. In Figure 1, we compare the T-periodic solution of the seasonally forced SEIR models with PVS in 1 year and 10 years. Figure 1 shows periodicity of the positive solution of the seasonally forced SEIR models with PVS.Figure 1
Positive periodic solutions of seasonally forced SEIR with pulse vaccination in 1 year and 10 years.
(a)(b)Simulation 2.
Setβ¯=15γ, ϵ=0.9/2π, and μ=0.8/2π; we make 8 steps of Newton iteration to get the approximate infective population of system (10) with both pi=0 (the surface at the bottom) and pi=0.2 (the surface at the top). Obviously, the infective population of system (10) with pulse is lower than the infective population of system (10) without pulse in Figure 2. The susceptible population of system (10) with pi=0.2 has periodic and impulsive properties. Thus, it is very effective to lower the infective population by PVS. Furthermore, Figure 2 shows the stability of the periodic solution by Newton iteration.Figure 2
Infective and susceptible population withpi=0.0 and pi=0.2.
(a)(b)Simulation 3.
Setβ¯=15γ, ϵ=0.9/2π, and μ=0.8/2π; we simulate system (10) with pi=0, pi=0.2, and pi=0.4 by Newton iteration. Obviously, it is very effective to lower the exposed population and the infective population by PVS in Figure 3.Figure 3
Exposed population, infective population, recovered population, and susceptible population withpi=0.0,pi=0.2, and pi=0.4.
(a)(b)Simulation 4.
Setϵ=0.9/2π, μ=0.8/2π, and pi=0.2; we simulate system (10) with β¯=12γ, 15γ, and 18γ by Newton iteration. Figure 4 shows the impact on exposed population, infective population, recovered population, and susceptible population by different transmission rates. As β¯ increases, the infective population increases while susceptible population decreases.Figure 4
Exposed population, infective population, recovered population, and susceptible population withβ¯=12γ,β¯=15γ, and β¯=18γ.
(a)(b)Simulation 5.
Setβ¯=15γ, ϵ=0.9/2π, and pi=0.2; we simulate system (10) with μ=0.4/2π, 0.6/2π, and 0.8/2π by Newton iteration. Figure 5 shows the impact on exposed population, infective population, recovered population, and susceptible population by different birth rates. As μ increases, the infective population and susceptible population increase, respectively.Figure 5
Exposed population, infective population, recovered population, and susceptible population withμ=0.8/2π,μ=0.6/2π, and μ=0.4/2π.
(a)(b)Simulation 6.
Setβ¯=15γ, μ=0.8/2π, and pi=0.2; we simulate system (10) with ϵ=0.7/2π, 0.8/2π, and 0.9/2π by Newton iteration. Figure 6 shows the impact on systems by the different rates of latent individuals becoming infectious. As ϵ increases, the infective population increases while susceptible population decreases.Figure 6
Exposed population, infective population, recovered population, and susceptible population withϵ=0.9/2π,ϵ=0.8/2π, and ϵ=0.7/2π.
(a)(b)


## 5. Conclusion

We obtain the existence of positive T-periodic solutions of seasonally forced SEIR models with pulse vaccination by Mawhin’s coincidence degree method. Some relevant numerical simulations are presented to show the T-periodic solution of the seasonally forced epidemiological models and to illustrate the effectiveness of such pulse vaccination strategy.



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