Periodic and almost periodic oscillations in a delay differential equation system with time-varying coefficients

It is extremely difficult to establish the existence of almost periodic solutions for delay differential equations via methods that need the compactness conditions such as Schauder's fixed point theorem. To overcome this difficulty, in this paper, we employ a novel technique to construct a contraction mapping, which enables us to establish the existence of almost periodic solution for a delay differential equation system with time-varying coefficients. When the system's coefficients are periodic, coincide degree theory is used to establish the existence of periodic solutions. Global stability results are also obtained by the method of Liapunov functionals.


Introduction.
Recently there has been increasing attention to migrant workers. The number of migrant workers in many countries has been increasing dramatically. This is particularly true in China, where millions of individuals from the rural area temporarily leave their home seeking for employment in urban cities every year. The number of migrant workers in China was as high as 281.71 million in 2016 according to the national migrant workers monitoring report [25]. Studies have shown that migrant workers are much more vulnerable to infectious diseases such as tuberculosis, HIV and other sexually transmitted infections [18,24]. To study the influence of temporary migration on the transmission of infectious diseases in migrant workers' home residence, Wang and Wang proposed and studied the following model in [27]      (1.1) Here S(t), I(t) and R(t) denote the population sizes at time t of three disjoint compartments, namely, susceptible, infectious and recovered, respectively. The constant Λ > 0 is the recruitment rate of the susceptibles, β > 0 is the infection transmission coefficient, µ S , µ I , µ R are the natural death rates of susceptible, infectious and The special mobility pattern of migrant workers motivates us to generalize model (1.1) to the case with periodic/almost periodic coefficients. More precisely, in this paper, we are concerned with the following delay differential equation system with time-varying coefficients: (1.2) where Λ(t), β(t), m S (t) and m R (t) are almost periodic (including periodic case as a special case).
Mathematically, the analysis of (1.2) is much more challenging, since traditional tools such as Schauder's fixed point theorem require compactness of the family of almost periodic functions, which indeed is extremely difficult to obtain. To overcome this difficulty, we employ a novel technique to construct a contraction mapping and establish the existence result by means of the contraction mapping theorem.
We point out that the history of mathematical modeling on infectious diseases could be tracked back to Bernoulli's work on the spread of smallpox [21]. The classical foundation work is due to Kermack and McKendrick [19]. The SARS outbreak in 2003 [34], the 2009 H1N1 pandemic [31] and the 2014 Ebola virus epidemic in West Africa [2,11] promoted further interest and efforts in disease modeling. Many aspects have been incorporated into disease modeling in order to accurately predict the spread and control of infectious diseases. Of which, one essential aspect has been human mobility pattern. In this aspect, many mathematical models considering immigrations [5,6], travel and dispersal between patches [17], group mixing [32], population mobility between an urban city and its satellite towns [3] have been proposed and studied. For some earlier work on epidemic models with periodic parameters in the literature, we refer to [1,8,23,20,26,30,33].
The rest of this paper is organized as follows. We establish some preliminary results in Section 2, and deal with the almost periodic case in Section 3. The periodic case is considered in Section 4. Then in Section 5 we present some numerical simulations. Finally, in the last section we give a brief summary and discussion.
Since continuous periodic or almost periodic functions are bounded and for a continuous periodic or almost periodic function f , we define Lemma 2.1. For any positive continuous function f in R, the following inequality holds Proof. Since the positive function f is continuous on R, one can get Theorem 2.2. Consider (1.2) with initial conditions S(θ) = ϕ(θ) ≥ 0, I(0) > 0 and R(θ) ≥ 0 for θ ∈ [−τ, 0], where ϕ is continuous on [−τ, 0] with ϕ(0) > 0. Then system (1.2) has a unique solution (S(t), I(t), R(t)) which is positive for t > 0.

This implies that
Note that M 1 is independent of initial conditions. The uniform boundedness of S(t), together with the second equation of (2.4), yields This completes the proof.
For an almost periodic function f ∈ AP (R, R + ), let Then according to [10,16], the module of f is defined as The following lemma taken from [10, Theorem 4.5] will be used to prove our mains results on existence of almost periodic solutions of system (1.2).
Lemma 2.4. The following statements are equivalent for any f and g ∈ AP (R, R + ): 3. Almost periodic case. In this section, we consider system (1.2) in which the parameters Λ(t), β(t), m S (t) and m R (t) are almost periodic. We first establish the existence and uniqueness of almost periodic solution for (1.2) in the following theorem.
Proof. It follows from the theory of dichotomies [7] that for any given where h(t, s) is defined in (2.5). For any given I(t) ∈ AP (R, R + ), define an operator F : AP (R, R + ) → AP (R, R + ) as follows Then for any S 1 (t),S 2 (t) ∈ AP (R, R + ), we have This shows that F is a contraction mapping. By the contraction mapping principle, for any given I(t) ∈ AP (R, R + ), the first equation of (1.2) has a unique positive almost periodic solution S(t) = S I (t). From the proof of Theorem 2.3, we have Next we define an operator G : Then for any I 1 (t), Thus, we have On the other hand, S(t) is a positive almost periodic solution of the first equation of system (1.2) if and only if (3.5) Then for any I 1 (t), I 2 (t) ∈ AP (R + , (0, M 2 ]), there are corresponding positive almost periodic solutions S I1 (t) and S I2 (t) to the first equation of system (1.2) satisfying This implies that Therefore, the operator G is a contraction mapping and thus the second Define an operator H : AP (R, R + ) → AP (R, R + ) as Then for any R 1 (t), R 2 (t) ∈ AP (R, R + ), it follows from Lemma 2.1 that Since pe −δτ < 1, the operator H is a contraction mapping and there is a unique positive almost periodic solution RĨ (t) =:R(t) to the third equation of system (1.2). Finally, we show that mod(S,Ĩ,R) ⊂ mod(Λ, β, m S , m R ). For any sequence   Proof. Suppose (S(t), I(t), R(t)) is a positive solution of system (1.2). Let x(t) = S(t) −S(t), y(t) = I(t) −Ĩ(t) and R(t) = R(t) −R(t), then x(t), y(t) and z(t) satisfy Then it suffices to show that lim Calculating the upper right derivative of V along the solutions of (3.6) yields It follows from Theorem 2.2 that x(t) and y(t) are bounded on (0, +∞). This implies that V is bounded on (0, +∞). Note from the first two equations in (3.6) that x (t) and y (t) are bounded on (0, +∞) and hence x(t) and y(t) are uniformly continuous on (0, +∞). Consequently |x(t)| + |y(t)| ∈ L 1 ((0, ∞)). 4. Periodic case. In this section, we deal with the periodic case. To apply the coincide degree theory to establish the existence of positive periodic solutions of system (1.2), we need some standard notations. Let X, Z be real Banach spaces, L : DomL ⊂ X → Z be a linear Fredholm mapping of index 0, and N : X → Z be continuous. Let P : X → X, Q : Z → Z be continuous projections such that ImP = KerL, KerQ = ImL and X = KerL KerP, Z = ImL ImQ. Then by Gaines and Mawhin [12]), L : DomL ∩ KerP → ImL is one to one, so its inverse K P : ImL → DomL ∩ KerP exists. J : ImQ → KerL is an isomorphism of ImQ onto KerL. We need the following continuation theorem to prove our existence result. In this section, we denote C ω = {u : u(t) = (x(t), y(t)) T ∈ C(R; R 2 ), u(t) ≡ u(t + ω), ∀t ∈ R} and define u = max{|x| 0 , |y| 0 }, where |x| 0 = max t∈[0,ω] |x(t)| and |y| 0 = max t∈[0,ω] |y(t)|. Then C ω is a Banach space with the norm · . Denotē Assume that S(t) and I(t) are positive periodic solutions of the first two equations of system (1.2), let x(t) = ln S(t) and y(t) = ln I(t), then x(t) and y(t) satisfy and N : X → X as Denote the two continuous projection operators P and Q by and L is a Fredholm mapping of index 0. Thus P and Q satisfy ImP = KerL, KerQ = ImL = Im(Ǐ − Q) and L admits an inverse K P : ImL → DomL ∩ KerP with By a direct calculation, we obtain Proof. We only need to show that QN and K P (Ǐ − Q)N are relatively compact onΩ. QN is relatively compact onΩ following directly from (4.3). Note that K P (Ǐ − Q)N is uniformly bounded onΩ, by the Ascoli-Arzela Theorem, we then only need to show the function family K P (Ǐ − Q)N (Ω) is equi-continuous. From (4.4), for any u ∈Ω, we have Then there is a positive constantM such that d dt [K P (Ǐ − Q)N (u)(t)] ≤M , ∀u ∈ Ω. This implies that the function family K P (Ǐ − Q)N (Ω) is equi-continuous. Therefore, K P (Ǐ − Q)N is relatively compact onΩ and N is L-compact onΩ. Proof. In fact, for any λ ∈ (0, 1), if u(t) is an arbitrary periodic solution of Lu = λN u, then u(t) satisfies Let z 1 = e x(t) , z 2 (t) = e y(t) , then the above system can be rewritten as By using the same argument as in the proofs of Theorems 2.2 and 2.3, we can obtain z 1 < M 1 and z 2 < M 2 . Thus the proof is complete.
From Lemma 4.1 and thus system (4.1) has at least one ω-periodic solutionũ(t) = (x(t),ỹ(t)) T . That is, system (1.2) has at least one positive ω-periodic solution (S(t),Ĩ(t)) T = (ex (t) , eỹ (t) ) T . A similar argument can be applied to show the following equation has a positive periodic solutionR(t). Theorems 3.2 and 4.4 immediately give the following result.

Numerical simulations.
In this section, we carry out numerical simulations to illustrate our theoretical results. Throughout this section, we assume µ S = µ R and µ I = µ S + 0.0001 and take Λ(t) = λ 0 (1 − λ 1 (cos(ω 1 t) + sin(ω 2 t))), β(t) = β 0 (1 − β 1 (cos(ω 1 t) + sin(ω 2 t))), m S (t) = m R (t) = m 0 (1 − m 1 (cos(ω 1 t) + sin(ω 2 t))). We should point out that the conditions given in Theorem 3.1 and Theorem 3.2 are only sufficient and not necessary. If we keep the same parameter values except that we take λ 0 = 1.5 and m 1 = 0.2, then in this case we find q ≈ 1.01 > 1, w ≈ −3.23 < 0, µ S + m S * − m * S e −δτ ≈ −0.30 < 0 and γ + µ I − β * M 1 ≈ −0.007 < 0. Theorem 3.1 and Theorem 3.2 do not apply. However, numerical simulations presented in Figure 3 indicate that system (1.2) still admits a globally stable positive almost periodic solution. 6. Summary and discussion. In this paper, motivated by the mobility patterns of migrant workers in China, we have considered a delay differential equation system with time-vary parameters. More specifically we have considered the almost periodic case and periodic case. For the almost periodic case, by a novel technique, we applied the the contraction mapping theorem to establish the existence and uniqueness of almost periodic solution. In addition, for the periodic case, the standard coincidence degree theory has been employed to establish the existence of periodic solutions. For both cases, global stability results are obtained via the method of Liapunov functional. Since there is not standard procedure to construct feasible Liapunov functionals, the stability conditions we have obtained clearly are not the sharpest. Numerical simulations suggest that system (1.2) may still admit a globally stable (almost) periodic solution even if the conditions given in Theorems 3.1 and 4.5 are not satisfied. Based on our extensive simulations, we propose the following conjecture: Conjecture If the time-varying coefficients in (1.2) are positive almost periodic (periodic), then (1.2) has a unique positive almost periodic (periodic), which is globally stable with respect to the nonnegative initial conditions.