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

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.


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][2][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][5][6][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 β(t) � β m (1 + A cos(ωt)).
(1) e 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]. e 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 where Pulse vaccination gives life-long immunity to pS susceptibles who are transferred to the "recovered" class of the population, which can be formulated as is 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][21][22][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) T 0 β(t)dt > c + 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 min t∈R β(t) > c + 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. e 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 R 0 � (β/(c + 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].
ere are some research activities about the existence of periodic solutions of impulsive differential equation [32][33][34][35][36]. e aim of this paper is to study the existence of periodic solution of seasonally forced SEIR models with pulse vaccination. e 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 Section 2. e 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. e relevant conclusion will be stated in Section 5.

e 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 △S| t�nT+t i � − J i (S(t), I(t))| t�nT+t i , modeling the spread of infectious diseases with PVS under the following hypotheses: (i) e population is divided into four classes; S(t), E(t), I(t), and R(t) denote, respectively, the fractions of the susceptible, exposed/latent, infective, and recovered population, and S(t) + E(t) + I(t) + V(t) � 1 is invariant with S(t), E(t), I(t), R(t) ≥ 0, for all t ≥ 0. (ii) μ, ϵ, and c denote the birth (death) rate, the rate of latent individuals becoming infectious, and recovery rate, respectively, which are positive constants. (iii) β(t) is the seasonally dependent transmission rate, which is a positive continuous T-periodic function. (iv) e 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: Our vaccination strategies concern the impact of infected population, which can be formulated as where 0 ≤ p i < 1 and the sensitivity coefficient α > 0 is sufficiently large. Denote the basic reproduction number: Obviously, we have (dS(t)/dt) + (dE(t)/dt) + (dI(t)/ dt) + (dR(t)/dt) ≡ 0. Since S(t), E(t), I(t), and R(t) are fractions of the population, we have S(t) + E(t)+ I(t) + R(t) � 1 for all t. Because R(t) does not appear in the first three equations in (4), system (4) reduces to the following 3-dimensional system: Obviously, finding the periodic solution of (8) is equivalent to finding the solutions of the following periodic boundary value problem: 2 Discrete Dynamics in Nature and Society with periodic boundary condition

e Suitable Region to Our Problem.
In order to prove the existence of periodic solutions of (8), we consider the following auxiliary problem: , Let D be an open bounded subset of X (will be denoted in Section 2.4) satisfying (12).

Proposition 1. D is an invariant region with respect to
Proof. First, we will prove that D is an invariant region.
In fact, it follows from model (12) that dS dt Since there is no impulsive motion for E, I, and it is easy to conclude that every possible solution will remain in the region D ultimately. Second, we will prove that the disease-free equilibrium (S 0 , E 0 , I 0 ) ≡ (1, 0, 0) is the unique periodic solution of (12) satisfying (S, E, I) ∈zD.
We assume that (S(t), E(t), I(t)) ∈zD is a solution of (12); this means at least one of the following conditions holds: We now consider each of these four cases: In case of (i), we have us, it is easy to obtain that I(t) < 0 for t < t 0 sufficiently close to t 0 , which contradicts that D is an invariant region.
In case of (ii), we have S(t 0 ) � 0 and S ′ (t 0 ) � λμ > 0. us, it is easy to obtain that S(t) < 0 for t < t 0 sufficiently close to t 0 , which contradicts that D is an invariant region.
In case of (iii), we have E(t 0 ) � 0 and and S(t 0 ) � 0 have been discussed above, we have E ′ (t 0 ) > 0 which again contradicts that D is an invariant region.
In case of (iv), we get Because I(t 0 ) � 0 has been discussed, we only discuss can have a nonconstant periodic solution on zD, 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 set Ω ⊆ D, such that there is no solution (S, E, I) of (12) satisfying (S(t), E(t), I(t)) ∈ zΩ for any λ ∈ (0, 1). Motivated by the idea of Katriel [27], we choose Ω to be the open subset of D given by where δ ∈ (0, 1) is to be fixed.

en, either (S(t), E(t), I(t)) ∈zD |min t∈[0,T] S(t) < δ} or (S(t), E(t), I(t)) ∈ D|min t∈[0,T] S(t) � δ}.
In the first case, S(t) ≡ 1 and Proposition 1 imply that there is no solution of (12) on (S, E, In the second case, we have Since there is no impulsive motion for I and I(0) � I(T), after integrating the third equation of (12) over [0, T], we obtain Since there is no impulsive motion for E and E(0) � E(T), after integrating the second equation of (12) over [0, T], we obtain With the help of (17)- (19), we conclude that For which is a contradiction to the assumption δ > (1/R 0 ). □

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: 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]). e linear mapping L is called a Fredholm mapping of index zero if If L: Ω ⊆ X ⟶ Z is a Fredholm mapping of index zero, there exist continuous projectors P: X ⟶ X and Q: Z ⟶ Z such that It follows that L| do mL∩KerP : (I − P)X ⟶ ImL is invertible. We denote the inverse of that map by K p .
Definition 2 (see [37]). Let Ω be an open bounded subset of X. e mapping N is called L-compact on Ω, if Since ImQ is isomorphic to KerL, there exists an isomorphism Λ: ImQ ⟶ KerL.
Theorem 1 (see [37]) (Mawhin's continuation theorem). Let Ω ⊂ X be an open bounded set. Let L be a Fredholm mapping of index zero and be L-compact on Ω. Assume that Lemma 4 (Arzela-Ascoli) (see [38]). Let D be a compact subset of R n and L(D, R n ) be the linear space of continuous functions which take D into R n ; any uniformly bounded equicontinuous sequence of functions ϕ n , n � 1, 2 . . . in L(D, R n ) has a subsequence which converges uniformly on D.

KerL � (S, E, I)|(S, E, I)
It is easy to see that Since ImL is closed, L is Fredholm mapping of index 0. Let P: X ⟶ X be the projector given by Obviously, Let Q: Z ⟶ Z be the projector given by Obviously, Furthermore, the generalized inverse (to L) K p : ImL ⟶ KerP ∩ domL exists given by en, QN: ⟶ XZ has the following form: By a direct calculation, we have (39) Discrete Dynamics in Nature and Society 5

Results
e following theorem gives the main results of this paper. (1), all of whose components are positive.

Proof
Step 1: N is L-compact on Ω.
First, it is obvious that QN(Ω) is bounded. For any (S, E, I) ∈ Ω, Using Arzela-Ascoli lemma again on [t 1 , t 2 ], we have a uniformly convergent subsequence K p (I − Q) N(S j 2 , E j 2 , I j 2 ) which is also uniformly convergent on [0, t 1 ]. Repeat it again and again, and we can prove that K p (I − Q)N(S j k+1 , E j k+1 , I j k+1 ) is uniformly convergent on [0, T]. In this way, K p (1 − Q)N: Ω ⟶ X is compact.
Simulation 2. Set β � 15c, 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 p i � 0 (the surface at the bottom) and p i � 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. e susceptible population of system (10) with p i � 0.2 has periodic and impulsive properties. us, it is very effective to lower the infective population by PVS. Furthermore, Figure 2 shows the stability of the periodic solution by Newton iteration.
Simulation 3. Set β � 15c, ϵ � 0.9/2π, and μ � 0.8/2π; we simulate system (10) with p i � 0, p i � 0.2, and p i � 0.4 by Newton iteration. Obviously, it is very effective to lower the exposed population and the infective population by PVS in Figure 3.  Discrete Dynamics in Nature and Society Simulation 4. Set ϵ � 0.9/2π, μ � 0.8/2π, and p i � 0.2; we simulate system (10) with β � 12c, 15c, and 18c 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.

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.

Data Availability
e data used to support the findings of this study are included within the article.