On threshold dynamics for periodic and time-delayed impulsive systems and application to a periodic disease model

Abstract: The basic reproduction ratioR0 of more general periodic and time-delayed impulsive model which the period of model coefficients is different from that of fixed impulsive moments, is developed. That R0 is the threshold parameter for the stability of the zero solution of the associated linear system is also shown. The developed theory is further applied to a swine parasitic disease model with pulse therapy. Threshold results on its global dynamics in terms of R0 are obtained. Some numerical simulation results are also given to support our main results.


Introduction
The basic reproduction number (ratio) R 0 is an important parameter in epidemiology and withinhost pathogen dynamics. R 0 is defined as the expected number of secondary cases produced by a single (typical) infection in a completely susceptible population in epidemiology. R 0 determines whether an infectious disease will break out usually. If R 0 > 1, it means that an individual can produce more than one individuals on average, then the population persists, if R 0 < 1, an individual can produce less than one individual on average, thus the population becomes extinct. Diekmann et al. [1] first introduced the next generation matrix approach to R 0 and Van den Driessche et al. [2] established the theory of R 0 for autonomous compartmental epidemic models. These two works have been widely used in various infectious disease models. For epidemic models with periodic coefficients, Bacaër et al. [3] presented a general definition of R 0 , that is, R 0 is the spectral radius of an integral operator on the space of continuous periodic functions. Since then, there has been a lot of research on R 0 in epidemic models. Wang et al. [4] proved that R 0 is a threshold parameter for the local stability of the disease-free periodic solution of periodic compartmental ODE models. Liang et al. [5] and Zhao [6], proved similar results for a class of time-delay evolution equations, respectively. Bai et al. [7] generalized the above results to impulse differential equations. Further, Thieme [8] characterized the relationship between spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. Bacaër et al. [9] obtained a more biological interpretation of R 0 for periodic models, and Inaba [10] used the concept of a generation evolution operator to give a new definition of R 0 in heterogeneous environments.
In population biology and epidemiology, more and more people adopt impulsive delay differential equations. For instance, Huo et al. [11] studied the global attractivity of positive periodic solutions for an scalar impulsive differential equations. Gourley et al. [12] evaluated the effectiveness of the agestructure culling strategies for controlling the vector-borne diseases by pulse model with two delays. Shen et al. [13] studied the positive periodic solutions of a predator-prey model with impulsive harvest and time delays. Motivated by the recent works of Liang et al. [5], Zhao [6], especially Bai et al. [7], by using a method similar to Bai et al. [7], we show that R 0 − 1 determines the stability of the zero solution of the associated periodic linear impulsive system and there are no special requirements for the period of the model coefficients and pulse time of the model coefficients.
This paper is organized as follows. In Section 2, we introduce the definition of the basic reproduction ratio R 0 for impulse delay periodic compartmental model and prove a stability equivalence result (Theorem 2.1). In Section 3, we apply the developed theory to an impulsive model of parasitic diseases in aquaculture with time delay, and establish a threshold type result in terms of R 0 . A brief discussion section then completes the paper.

The model formulation
Let τ ≥ 0 be a given number, denote , ∀t ∈ (−τ, 0], φ(t + ) exists for t ∈ [−τ, 0) and φ(t + ) = φ(t) for all but at most finite number of points t ∈ [−τ, 0)}, where φ(t + ) = lim s→t + φ(s) and φ(t − ) = lim s→t − φ(s). We equip the linear space PC([−τ, 0], R m ) with the norm · τ defined by φ τ = sup −τ≤θ≤0 |φ(θ)|, where | · | is any convenient norm on R m (for example the Euclidean norm). For a left continuous function u : R → R m and u(t + ) = u(t) for all but at most finite number of points within a limited interval, define u t ∈ PC([−τ, 0], R m ) by Assume that F(t) and V(t) are ω−periodic in t for some real number ω > 0, and for each fixed φ, F(t)φ is a measurable function of t, continuous in φ for each fixed t and there exists M > 0 such that We consider a linear impulsive and periodic functional differential system where P n is m × m matrix such that det(P n + I) 0 (n ∈ N), I is the identity matrix, t k : N → R + and there exists an integer q ≥ 1 such that t n+q = t n + ω, P n+q = P n , n ∈ N. LetJ = [−τ, 0) ∩ {t i + nω : n ∈ Z and i = 1, ..., q} (possibly empty), and J = [−τ, 0)\J. Denote the space PC J by It is easy to see that PC J ⊂ PC([−τ, 0], R m ), and PC J is a Banach space endowed with the norm · τ . Given φ ∈ PC J we consider the following initial condition A function is said to be a solution of (2.1) with initial conditions (2.2) if the following condition are satisfied: By the general theory of impulsive delay differential equations [14], it follows that for any φ ∈ PC J , system (2.1) has a unique solution The internal evolution of individuals in the infectious compartments (e.g., natural and diseaseinduced deaths, movements among compartments and recovery) is governed by the linear impulsive ordinary differential system: By the theory of impulsive differential equations [15, Sect.1.2], we denote by U n (t, s)(n ∈ Z, t, s ∈ (t n−1 , t n ]) the fundamental matrix for the linear equation Then the evolution matrices associated with (2.3) can be written as A straightforward verification shows that In order to introduce the basic reproduction ratio for system (2.1), we need the following assumptions.
(H1) For any given t ∈ R, operator F(t) : For any given t ∈ R, matrix −V(t) is cooperative, and r(W(ω, 0)) < 1, (H3) P n + I is nonnegative for every n ∈ N, which means all entries of P n + I are nonnegative, where r(W(ω, 0)) represents the spectral radius of W(ω, 0). By the theory of impulsive delay differential equations [15,Remark 3.5] with (H2), we can conclude that there exists K ≥ 1 and α > 0 such that In view of the periodic environment, we suppose that v(t), ω−periodic in t, is the initial distribution of infectious individuals. For any t ≥ s, F(t − s)v t−s is the distribution of newly infected individuals at time t − s, which is produced by the infectious individuals who were introduced over the time interval [t − s − τ, t − s]. Then W(t, t − s)F(t − s)v t−s is the distribution of those infected individuals who were newly infected at time t − s and remain in the infected compartments at time t. It follows that is the distribution of accumulative new infections at time t produced by all those infectious individuals introduced at all previous times to t.
By the definition of W(t, s), for any given s ≥ 0, W(t, t − s)v t−s is the distribution of those infectious individuals who were introduced at time t − s and remain in the infected compartments at time t, then w(t) := ∞ 0 W(t, t − s)v t−s ds is the distribution of accumulative infectious individuals who were introduced at all previous times to t and remain in the infected compartments at time t. Thus, the distribution of newly infected individuals at time t is Let the Banach space X ω be given by where | · | is any convenient norm on R m and the positive cone According to the above analysis, we define two linear operators on X ω by Moreover, let A and B be two bounded linear operators on X ω defined by It then follows that L = A • B andL = B • A, and hence, L andL have the same spectral radius. Motivated by the concept of next generation operators in [7], we define the spectral radius of L andL as the basic reproduction ratio R 0 := r(L) = r(L) for periodic system (2.1).
In view of the previous assumptions and conclusions, one can verify that L andL are well defined. To see this, we introduce the following operator.
For any given λ ∈ R, let E λ be a linear operator on It is easy to see that E λ ≤ max{1, e −λτ } for any λ ∈ R. Then we introduce a family of linear operators L λ on X ω : Obviously, L 0 = L, and L λ is well defined for all λ > −α. Further, we can prove the following assertion.
Proof. Let λ > −α be given. Clearly, E λ is a positive linear operator on PC([−τ, 0], R m ). By hypothesis (H1), (H2) and (H3), F(t) and W(t, s)(t ≥ s) are positive linear operators. Then it is easy to see that L λ is positive on X ω . In view of Thus, That means L λ is bounded, and hence, continuous on X ω Let v ∈ X ω and ∀t, t ∈ (0, ω] with t < t. Note that In order to study the properties of the solution of the equation (2.1), we need the following auxiliary system. For any given λ ∈ [0, +∞), consider the linear impulsive system: a.e. t > 0, t t n , For any φ ∈ PC + J , [14, Corollary 3.1] implies that system (2.6) admits a unique solution u λ (t, φ) on [0, +∞) satisfying u λ 0 = φ. In addition, the following statements are valid.
Lemma 2.4. For each n > τ ω , U n is positive and compact on PC J . Proof. Let u(t, φ) be the unique solution of (2.1) with u 0 = φ ∈ PC J . By Lemma 2.2(i), U n , n ∈ N is positive on PC J . Under the assumption that nω > τ, U n φ can be written as follows where t k is the impulsive point on [0, nω + θ). Let B be the bounded subset of PC J . Obviously, for any φ ∈ B, u(s, φ) is bounded for s ∈ [0, nω], and hence, U n (B) is bounded.
Next, we prove that the set U n (B) is quasiequicontinuous whenever nω > τ. Then, with [15, Lemma 2.4], we can deduce that U n is compact for nω > τ.
In fact, we have proved the main theorem of this section.
The following result is about the numerical calculation of R 0 (see, e.g., [7]). For completeness, here we list the main steps.

An application
Mathematical methods have been used to study infectious diseases for a long time and certain results have been obtained. People have added various factors to the model by analyzing the actual situation to better reflect the actual problem. For example, Sharma et al. [19] considered the incubation period of the disease and used the time-delayed model, Gupta et al. [20] used a time-delay SEIRD model to analyze the spread of COVID-19 infection in a population. Church et al. [21] considered the vaccination situation and used the impulse model, etc.
The non-impulsive system has been studied in [6] and non-delayed system has been studied in [22], In this section, inspired by the above work, we combine a variety of factors and apply the theory developed in the previous section to an impulsive model of parasitic diseases in pork farming with time delay. Parasites can cause a decrease in feed conversion rate and weight gain, and there is a risk of pig-to-human transmission in pork and various organs after slaughter. This has caused huge economic losses to the global pork industry [23]. Concentrated breeding makes the treatment of parasitic diseases easier to achieve. The purpose of treatment is achieved by adding to the feed and spraying the corresponding drugs on the body surface. The timely removal of feces and other potential sources of infection can effectively reduce the infection rate of parasitic diseases. However, there are few researches on the above strategies in mathematics. The following simulation of the above methods by applying the method of infectious disease dynamics has some reference significance for when and how to control parasitic diseases in reality.
Let S (t), E(t) and I(t) be the total numbers at time t of the susceptible, exposed, and infective populations, respectively. For simplicity, we assume that the latent period of the disease is τ, and the incidence rate function f (t, S , I) depends on time t and variables S and I. Let µ(t) be the natural death rate of the population. It then follows that the rate of entry into the infective class from the exposed one at time t is As discussed in [24], E(t) can be represented as It is easy to show that Then, we obtain the following non-autonomous SEI model with impulses: a.e. t > 0, t t n , (3.1) Here δ(t) is the recruitment rate, θ represents the treatment rate of parasite drugs.
According to [25], we need to impose the following compatibility condition: Assume that f (t, S , I) and all these time-dependent coefficients are ω-periodic in t for some real number ω > 0. Obviously the function is also ω-periodic, and hence, model (3.1) is an ω-periodic and time-delayed system with impulses. To study the dynamic behavior of system (3.1), we make the following assumptions, which are natural considering the biological background of the system (3.1): (A1) δ(t) and µ(t) are all non-negative and continuous functions with δ(t) > 0 and ω 0 µ(t)dt > 0.
(A2) f (t, S , I) is a C 1 -function with the following properties: ∂I I for all (t, S , I) ∈ R × R 2 + . By virtue of (A1), the scalar linear periodic equation has a unique positive ω-periodic solution S * (t), which is globally stable in R. Linearizing system (3.1) at its disease-free periodic solution (S * (t), 0, 0), we obtain the following periodic linear impulsive differential equation for the infective variable I: a.e. t > 0, t t n , , and V(t) = µ(t). Then system (2.1) becomes system (3.4), and W(t, s) = e − t s µ(r)dr From the definition of the basic reproduction number, we get R 0 = r(L) for the system (3.4) By the general theory of impulsive delay differential equations [14], it follows that for any φ ∈ PC J , system (3.1) has a unique solution u(t, φ) = (S (t), E(t), I(t)) on [0, +∞) with u 0 = φ ∈ D.
where u(t, φ) is the unique solution of (3.1) satisfying u 0 = φ ∈ PC J . Then U := U(ω) is the Poincaré (period) map associated with (3.1) on PC J . Let r(U) be the spectral radius of U. Then the following assertion holds true.
ω . Then there exists a positive and ω-periodic function v * (t) such that e µt v * (t) is a solution of (3.1).
due to Theorem 2.1. Let φ ∈ PC + J \{0}. We assert that there exists t * ∈ [0, τ] such that I(t * ) > 0. If that's not correct, it means I(t) = 0 for all t ∈ [0, τ]. Then the first equation of (3.4) implies but a(t) > 0, which contradicts the assumption that φ > 0. In view of a.e. t > 0, t t n , we have This means that P n is strongly positive on PC J whenever nω ≥ 2τ. By a method similar to the proof of Lemma 2.4, we can show that P n is compact when nω ≥ 2τ. Since P n = P(nω), [26,Lemma 3.1] implies that r(P) is a simple eigenvalue of P with a strongly positive eigenvector, and the modulus of any other eigenvalue is less than r(P).
Since R 0 < 1, we have r(P) < 1. Let P be the Poincaré map of the following perturbed impulsive delay differential system a.e. t > 0, t t n , (3.6) Note that lim →0 r(P ) = r(P) < 1, we can fix a sufficiently small number > 0 such that r(P ) < 1. By Lemma 3.1, there is positive ω-periodic function v (t) such that u (t) = e µ t v (t) is a positive solution of (3.6), where µ = lnr(P ) ω < 0. For any given φ ∈ D, let v(t, φ) = (S (t), E(t), I(t)). Due to (3.5) and the global stability of S * (t) for (3.3), there exists a sufficiently large integer n 1 > 0 such that n 1 ω ≥ τ and S (t) < S * + , ∀t ≥ n 1 ω − τ. In view of assumption (A2), we have a.e. t > 0, t t n , for all t ≥ n 1 ω. Choose a sufficiently large number K > 0 such that Thus, the same method as in [7,Theorem 3] shows that and hence, lim t→∞ I(t) = 0. Thus, for any given κ ∈ R, by the continuity of f (t, S , I) and f (t, S , 0) = 0, there is T 1 > 0 such that f (t, S (t), I(t)) − e − t t−τ µ(t)dr f (t − τ, S (t − τ), I(t − τ)) < κ, By the second and fifth equations of (3.1), we have a.e. t > 0, t t n , Consider the following impulsive differential equation a.e. t > 0, t t n , By virtue of (A1), we see that the system (3.7) has a unique positive ω-periodic solution x * which is globally stable in R.
x * (t) = ce − t 0 µ(s)ds Let κ → 0 and t → ∞, then x * (t) → 0. Thus the comparison theorem for impulsive differential equations and the global stability of x * (t) for (3.7) implies that E(t) → 0 as t → ∞. Note that S (t) + E(t) + I(t) = S 1 (t), where S 1 (t) is the solution of (3.1) satisfying S 1 (0) = φ 1 (0) + φ 2 (0) + φ 3 (0) It is easy to see that lim t→∞ (S 1 (t) − S * (t)) = 0 then lim t→∞ (S (t) − S * (t)) = 0. Proof. Let P δ be the Poincaré map of a.e. t > 0, t t n , on PC J . Since lim δ→0 r(P δ ) = r(P) > 1, we can fix a small number δ > 0 such that r(P δ ) > 1. It then follows that there is a small number η 0 > 0 such that We claim that lim sup t→∞ I(t) > η 0 , ∀φ ∈ D. Suppose, by contradiction, that there is at > 0 such that I(t) ≤ η 0 for all t ≥t. Then system (3.6) becomes a.e. t > 0, t t n , Choose a sufficiently small real numberk such that Note that µ δ > 0, let t → ∞, then I(t) → ∞, which is a contradiction. Then there are only two possibilities: (i) for all large t, I(t) ≥ η 0 ; (ii) I(t) oscillates around η 0 . Obviously, we only need to consider the second case. Put t and t are sufficiently large such that If t − t ≤ τ, then the system (3.1) implies that Because µ(t) is ω-periodic and t n+q = t n + ω, it is obvious that e − t+τ t µ(s)ds and k:t k ∈[ t, t+τ ) , for any given τ, have a minimum value which is independent oft and t, denoted by c. Then For t ∈ (t + τ, t ], we consider the following comparison equation a.e. t > 0, t t n , Note that v(t) is ω-periodic, let m, M be the minimum and maximum of v(t), respectively, then choose a sufficiently small real number k > 0 such that I(t) ≥ q = ke µ δ (t+τ) M ≥ ke µ δ t v δ (t) ≥ ke µ δ t m ≥ ke µ δ t m = q e µτ M = q , t ∈ [t, t + τ ], where q is a constant independent of t and k. Then combined with the principle of comparison, we conclude that due to I(t) > ke µ δ t v δ (t) > q . Consequently, we get I(t) > η for t ∈ [ t, t ], where η = min{q, q }. Since this kind of interval [ t , t ] is chosen arbitrarily, we get lim inf t→∞ I(t) ≥ η.

Discussion
In this paper, we develop the basic reproduction ratio R 0 of more general periodic and time-delayed impulsive model. Note that the period of model coefficients is not same as that of fixed impulsive moments. We extend some known results of Bai et al. [7]. We show that R 0 is the threshold parameter for the stability of the zero solution of the associated linear system.Furthermore, We use the developed theory to a swine parasitic disease model with pulse therapy and obtain threshold results on its global dynamics in terms of R 0 . We hope that our method can also be applied to more general population models or epidemic model with impulse.
In the early stage of the disease outbreak, the corresponding measures have not been implemented. If the time of the outbreak of the infectious disease can be known, preparing in advance is of great significance for controlling the spread of the infectious disease. Turkyilmazoglu [28] proposed a explicit formulae for the peak time of SIR model. Kröger et al. [29] compared different forecasting methods and this is a very meaningful but challenging task for non-autonomous impulsive systems.