Nontrivial periodic solution of a stochastic seasonal rabies epidemic model

Journal Pre-proof Nontrivial periodic solution of a stochastic seasonal rabies epidemic model Zhongwei Cao, Wei Feng, Xiangdan Wen, Li Zu, Jinyao Gao

PII: DOI: Reference:

S0378-4371(19)31880-1 https://doi.org/10.1016/j.physa.2019.123361 PHYSA 123361

To appear in:

Physica A

Received date : 7 April 2019 Revised date : 26 June 2019 Please cite this article as: Z. Cao, W. Feng, X. Wen et al., Nontrivial periodic solution of a stochastic seasonal rabies epidemic model, Physica A (2019), doi: https://doi.org/10.1016/j.physa.2019.123361. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

*Highlights (for review)

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Highlights ●A stochastic nonautonomous rabies epidemic model is studied. ●We establish sufficient conditions for the existence of nontrivial positive

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periodic solution.

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●Our result does not depend on the positive endemic equilibrium P^{*}.

*Revised Manuscript Click here to view linked References

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Nontrivial periodic solution of a stochastic seasonal rabies epidemic model Zhongwei Caoa , Wei Fengb , Xiangdan Wenc1 , Li Zud , Jinyao Gaoe Department of Applied Mathematics, Jilin University of Finance and Economics,

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Changchun 130117, Jilin Province , P.R. China

Department of Anesthesiology, China-Japan Union Hospital of Jilin University,

Changchun 130033, Jilin Province, P.R. China

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a

Department of Mathematics, Yanbian University,

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Yanji 133002, Jilin Province, P.R. China

College of Mathematics and Statistics, Hainan Normal University,

Haikou 571158, Hainan Province, P.R. China

College of Accounting, Jilin University of Finance and Economics, Changchun 130117,

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Jilin Province , P.R. China

Abstract

In this paper, we consider a stochastic nonautonomous rabies epidemic model. By using Khasminskii’s theory of periodic solutions and constructing a suitable stochastic Lyapunov function, we show that the stochastic nonautonomous epidemic model admits at least one nontrivial positive T -periodic solution under certain condition.

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Keywords: Rabies; Stochastic SEIRS epidemic model; Stochastic periodic solution; Vaccination; Lyapunov function. 1. Introduction

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Rabies, a fatal disease of humans and many other mammals, is caused by a virus which is associated with the bite and virus-containing saliva of an infected host [1]. According to World Health Organization’s report, rabies is an acute, 1 Corresponding author at Department of Mathematics, Yanbian University, Yanji, Jilin 133002, P.R. China. E-mail address: [email protected] (X. Wen).

Preprint submitted to PHYSICA A

June 26, 2019

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progressive encephalitis which kills up to 70,000 people every year worldwide. [2]. It is well known that rabies transmission in dogs can be prevented by vaccination. Treatment of humans after exposure, known as post-exposure prophylaxis (PEP), is successful in preventing the disease if administered promptly. Although it is a vaccine-preventable disease, researchers estimate that 30,000 to 70,000 deaths are attributable to rabies each year, with less developed countries affected more. For example, rabies still remains a untreatable disease in many countries in Africa and Asia where 95% of human deaths occur [3]. As we know, mathematical models have been utilized to investigate the rabies epidemics in dogs and the transmission dynamics of rabies from dogs to humans. For instance, Hampson et al. [4] found rabies epidemics cycles with a period of 3-6 years in dog populations in Africa and established a susceptible, exposed, infectious, and vaccinated model with an intervention response variable to show important synchrony. In [5], Zinsstag et al. divided both dog and human populations into susceptible, exposed, infectious, and immunized classes and proposed a mathematical model for dog-human transmission dynamics and economics of rabies control in an African city. A numerical model for the spread of a lethal infectious disease in wildlife is introduced in [6]. Numerical simulations are carried out for rabies epidemics among raccoons in New York state and a qualitative comparison of numerical results to available data from real-world epidemics is discussed. In [7], authors propose a deterministic mathematical model with human, dog and livestock populations and formulated as a system of ordinary differential equations. According to numerical simulations of their system rabies transmission will increase within and around Addis Ababa, and may peak in 2024 and 2026 in human and livestock populations respectively. The simulation shows that 25% vaccination coverage in livestock populations will reduce the future infection by half. Liu et al. [8] proposed a two-patch SEIRS epidemic model and numerical simulations show that, for the parameter values considered, border restriction does not necessarily always have a positive impact on the overall spread of rabies, only when the border control is properly implemented, then it could contribute to stopping the spatial spread of rabies between patches. On the other hand, many diseases exhibit seasonal fluctuations, such as measles, influenza, mumps, polio, etc (see e.g. [9, 10, 11]). Seasonally effective contact rate, vaccination program and periodic changing in the birth rate are regarded as sources of periodicity (see e.g. [12, 13, 14, 15, 16, 17]). Most recently, Zhang et al. [18] proposed a deterministic SEIRS model with periodic transmission rates to investigate the seasonal rabies epidemics in China which consists of susceptible, 2

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exposed, infectious, and recovered subpopulations of both dogs and humans and is described by the following eight ordinary differential equations  dS    = A + λR + σ(1 − γ)E − mS − β(t)S I − kS ,    dt    dE    = β(t)S I − mE − σ(1 − γ)E − kE − σγE,    dt    dI    = σγE − mI − µI,    dt    dR      dt = k(S + E) − mR − λR, (1.1)   dS 1    = B + λ1 R1 + σ1 (1 − γ1 )E1 − m1 S 1 − β1 (t)S 1 I,    dt    dE1    = β1 (t)S 1 I − m1 E1 − σ1 (1 − γ1 )E1 − k1 E1 − σ1 γ1 E1 ,    dt    dI1    = σ1 γ1 E1 − m1 I1 − µ1 I1 ,    dt    dR1    = k1 E1 − m1 R1 − λ1 R1 , dt

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where they denote the total numbers of dogs and humans by N(t) and N1 (t), respectively, and classify each of them into four subclasses: susceptible, exposed, infectious and recovered, with the numbers of dogs denoted by S (t), E(t), I(t) and R(t), and human sizes denoted by S 1 (t), E1 (t), I1 (t) and R1 (t), respectively. Here all parameters are positive, the interpretations are described as follows, A: Dog birth population; λ: Dog loss rate of immunity; i: Dog incubation period; σ: 1/i; γ: Clinical outcome rate of exposed dogs, 0 < γ < 1; m: Dog natural mortality rate; k: Dog vaccination rate; µ: Dog disease-related death rate; B: Human birth population; λ1 : Human loss rate of immunity; i1 : Human incubation period; σ1 : 1/i1 ; γ1 : Clinical outcome rate of exposed humans; m1 : Human natural mortality rate; k1 : Human vaccination rate; µ1 : Human disease-related death rate; 3

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β(t): The transmission rate between S (t) and I(t); β1 (t): The transmission rate between S 1 (t) and I(t). They evaluated the basic reproduction number R0 , analyzed the dynamical behavior of the model (1.1), and used the model to simulate the monthly data of human rabies cases reported by the Chinese Ministry of Health. They also carried out some sensitivity analysis of the basic reproduction number R0 in terms of various model parameters. In system (1.1), the basic reproduction number is R0 := ρ(L) which determines the epidemic occurs or not, and ρ is the spectral radius of L and L is called the next infection operator. If R0 < 1, system (1.1) has a unique ˆ Sˆ 1 , 0, 0, 0) and it is globally asymptotdisease-free equilibrium P0 = (Sˆ , 0, 0, R, (m+λ)A kA ically stable in the positively invariant set Γ, where Sˆ = m(m+λ+k) , Rˆ = m(m+λ+k) , B Sˆ 1 = m1 and Γ = {(S , E, I, R, S 1 , E1 , I1 , R1 ) : S , E, I, R, S 1 , E1 , I1 , R1 ≥ 0, 0 < S + E + I + R ≤ mA , 0 < S 1 + E1 + I1 + R1 ≤ mB1 }. If R0 > 1, then model (1.1) has at least one positive periodic solution. However, the natural world is full of randomness. Hence it is necessary to consider stochasticity into epidemic models (see e.g. [19, 20, 21, 22]). For these models, the nature of epidemic growth and spread is random due to the unpredictability in person-to-person contacts [23]. Therefore the variability and randomness of the environment is fed through the state of the epidemic [24]. And in epidemic dynamics, stochastic systems may be a more appropriate way of modeling epidemics in many circumstances (see e.g. [25, 26, 27, 28]). For example, stochastic models can take care of randomness of infectious contacts occurring in the latent and infectious periods [29]. It is well known that the existence of nontrivial positive periodic solution to stochastic epidemic models is one of the most interesting issue in mathematical biology, and most recently, many authors have done some work in this field, for instance, by using Khasminskii’s theory of periodic solutions and constructing suitable stochastic Lyapunov functions, Lin et al. [30] established sufficient conditions for the existence of periodic solutions to stochastic SIR epidemic model in recent years. Rifhat et al. investigated the dynamics of a class of periodic stochastic SIS epidemic models with general nonlinear incidence and numerical simulations were given to illustrate the main theoretical results [31]. Liu et al. [19] studied a stochastic non-autonomous SISV epidemic model, they showed that there exists at least one nontrivial positive periodic solution of the system. However, as far as we know, there have been no results related the existence of nontrivial positive periodic solution to the stochastic rabies epidemic model. Motivated by the above discussion, in this paper, we attempt to do some work 4

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in this field to fill the gap. By taking into account the effect of randomly fluctuating environment, we adopt the approach used in Liu et al. [19] and assume that the environmental noise is proportional to the variables of system (1.1). Then the stochastic version corresponding to system (1.1) takes the following form                                   

2. Preliminaries

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dS = [A + λR + σ(1 − γ)E − mS − β(t)S I − kS ]dt + σ1 (t)S dB1 (t), dE = [β(t)S I − mE − σ(1 − γ)E − kE − σγE]dt + σ2 (t)EdB2 (t), dI = [σγE − mI − µI]dt + σ3 (t)IdB3 (t), dR = [k(S + E) − mR − λR]dt + σ4 (t)RdB4 (t), dS 1 = [B + λ1 R1 + σ1 (1 − γ1 )E1 − m1 S 1 − β1 (t)S 1 I]dt + σ5 (t)S 1 dB5 (t), dE1 = [β1 (t)S 1 I − m1 E1 − σ1 (1 − γ1 )E1 − k1 E1 − σ1 γ1 E1 ]dt + σ6 (t)E1 dB6 (t), dI1 = [σ1 γ1 E1 − m1 I1 − µ1 I1 ]dt + σ7 (t)I1 dB7 (t), dR1 = [k1 E1 − m1 R1 − λ1 R1 ]dt + σ8 (t)R1 dB8 (t), (1.2) where the parameter functions β(t), β1 (t) and σi (t) (i = 1, . . . , 8) are all positive and continuous functions of period T and T is a positive constant. Bi (t) (i = 1, . . . , 8) are mutually independent standard Brownian motions, σ2i > 0 are the intensities of the white noise, i = 1, . . . , 8. Other parameters are the same as those in system (1.1). This paper is organized as follows. In Section 2, we introduce some needed results throughout this paper. In Section 3, we show that there is a unique global positive solution of system (1.2) with any positive initial value. In Section 4, we establish sufficient conditions for the existence of nontrivial positive T -periodic solution of system (1.2). Finally, we provide a brief discussion and summarize the main results obtained in this paper.

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Throughout this paper, unless otherwise specified, let (Ω, F , {Ft }t≥0 , P) be a complete probability space with a filtration {Ft }t≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contains all P-null sets), Bi (t) (i = 1, . . . , 8) be defined on the complete probability space, we also let Rd+ = {x ∈ Rd : xi > 0, 1 ≤ i ≤ d}. If f (t) is a bounded function on [0, ∞), define fl = R T inf t∈[0,∞) f (t), f u = supt∈[0,∞) f (t). And for simplicity, define hx(t)iT = T1 0 x(r)dr. Here the notations used in this article are described as follows a ∨ b: the maximum of a and b, a ∧ b: the minimum of a and b, A×B: the Cartesian product of A and B, which means A×B = {(x, y) | x ∈ A, y ∈ B}. 5

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In general, consider the d-dimensional stochastic differential equation dx(t) = f (x(t), t)dt + g(x(t), t)dB(t) f or t ≥ t0 ,

(2.1)

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with the initial value x(0) = x0 ∈ Rd . B(t) denotes an n-dimensional standard Brownian motion defined on the complete probability space (Ω, F , {Ft }t≥0 , P). Denote by C 2,1 (Rd × [t0 , ∞); R+ ) the family of all nonnegative functions V(x, t) defined on Rd × [t0 , ∞) such that they are continuously twice differentiable in x and once in t. The differential operator L of Eq. (2.1) is defined by [32] d d ∂ 1X T ∂2 ∂ X + fi (x, t) + [g (x, t)g(x, t)]i j . ∂t i=1 ∂xi 2 i, j=1 ∂xi ∂x j

If L acts on a function V ∈ C 2,1 (Rd × [t0 , ∞); R+ ), then

where Vt = then

∂V , ∂t

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1 LV(x, t) = Vt (x, t) + V x (x, t) f (x, t) + trace[gT (x, t)V xx (x, t)g(x, t)], 2 2

∂V ∂V V V x = ( ∂x , . . . , ∂x ), V xx = ( ∂x∂i ∂x )d×d . By Itˆo’s formula, if x(t) ∈ Rd , 1 j d

dV(x(t), t) = LV(x(t), t)dt + V x (x(t), t)g(x(t), t)dB(t).

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Next, we shall recall some theory about the existence of the periodic Markov processes. For more details, we refer the readers to [33]. Definition 2.1. A stochastic process ξ(t) = ξ(t, ω) (−∞ < t < ∞) is said to be periodic with period T if for every finite sequence of numbers t1 , . . . , tn the joint distribution of random variables ξ(t1 + h), . . . , ξ(tn + h) is independent of h, where h = kT , k = ±1, ±2, . . . Remark 2.1. In [33], Khasminskii showed that a Markov process x(t) is T periodic if and only if its transition probability function is T -periodic and the function P0 (t, A) = P{X(t) ∈ A} satisfies the equation Z P0 (s, A) = P0 (s, dx)P(s, x, s + T, A) ≡ P0 (s + T, A) Rd

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for every A ∈ B, where B is a Borel σ-algebra. Consider the following equation x(t) = x(t0 ) +

Z

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b(s, x(s))ds +

k Z X r=1

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t t0

σr (s, x(s))dBr (s), x ∈ Rd ,

(2.2)

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where the vectors b(s, x), σ1 (s, x), . . . , σk (s, x) (s ∈ [t0 , T ], x ∈ Rd ) are continuous functions of (s, x) and satisfy the following conditions k k X X |b(s, x)−b(s, x¯)|+ |σr (s, x)−σr (s, x¯)| ≤ B|x− x¯|, |b(s, x)|+ |σr (s, x)| ≤ B(1+|x|), r=1

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(2.3) in every cylinder I × U, where B is a constant and I is a subset of R+ . Let C 2 denote the class of functions on I × U which are twice continuously differentiable with respect to x1 , . . . , xd and continuously differentiable with respect to t. Lemma 2.1. Suppose that the coefficients of (2.2) are T -periodic in t and satisfy the condition (2.3) in every cylinder I × U, and suppose further that there exists a function V(t, x) ∈ C 2 in I × U which is T -periodic in t, and satisfies the following conditions (i) inf V(t, x) → ∞ as R → ∞; (2.4) |x|>R

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(ii) LV(t, x) ≤ −1 outside some compact set.

(2.5)

Then there exists a solution of (2.2) which is a T -periodic Markov process. The Lemma 2.1 has been proved by Khasminskii in his monograph [33], Chapter 3, page 80. Readers can refer to this book for details. Remark 2.2. According to the proof of Lemma 2.1, the linear growth condition (2.3) is only used to guarantee the existence and uniqueness of the solution to Eq. (2.2). 3. Existence and uniqueness of the positive solution

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To study the dynamical behavior of an epidemic model, the first concern is whether the solution is global and positive. The following result is regarded to the existence and uniqueness of the global positive solution, which is a prerequisite for investigating the long term behavior of model (1.2). Theorem 3.1. For any initial value X0 = (S (0), E(0), I(0), R(0), S 1 (0), E1 (0), I1 (0), R1 (0)) ∈ R8+ , there exists a unique solution X(t) = (S (t), E(t), I(t), R(t), S 1 (t), E1 (t), I1 (t), R1 (t)) of system (1.2) on t ≥ 0 and the solution will remain in R8+ with probability one. Proof. Since the coefficients of system (1.2) satisfy the local Lipschitz condition, then for any initial value X0 = (S (0), E(0), I(0), R(0), S 1 (0), E1 (0), I1 (0), R1 (0)) ∈ R8+ there exists a unique local solution X(t) = (S (t), E(t), I(t), R(t), S 1 (t), E1 (t), I1 (t), R1 (t)) on [0, τe ), where τe denotes the explosion time [32]. To prove this solution is global, we only need to show that τe = ∞ a.s. To this end, let n0 > 0 be sufficiently large such that every component of X0 lying within the interval [ n10 , n0 ]. 7

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For each integer n > n0 , define the following stopping time  1 τn = inf t ∈ [0, τe ) : min{S (t), E(t), I(t), R(t), S 1 (t), E1 (t), I1 (t), R1 (t)} ≤ or n  max{S (t), E(t), I(t), R(t), S 1 (t), E1 (t), I1 (t), R1 (t)} ≥ n ,

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where throughout this paper, we set inf ∅ = ∞ (as usual ∅ denotes the empty set). It is easy to see that τn is increasing as n → ∞. Let τ∞ = limn→∞ τn , then τ∞ ≤ τe a.s. In what follows, we need to verify τ∞ = ∞ a.s. If this assertion is false, there exists a constant T > 0 and an  ∈ (0, 1) such that P{τ∞ ≤ T } > . Hence there is an integer n1 ≥ n0 such that P{τn ≤ T } ≥ , n ≥ n1 .

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Define a C 2 -function V : R8+ → R+ by  S + (E − 1 − ln E) + (I − 1 − ln I) V(S , E, I, R, S 1 , E1 , I1 , R1 ) = S − a − a ln a  S1 +(R − 1 − ln R) + S 1 − b − b ln + (E1 − 1 − ln E1 ) b +(I1 − 1 − ln I1 ) + (R1 − 1 − ln R1 ), where a, b are positive constants to be determined later. The nonnegativity of this function can be seen from

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u − 1 − ln u ≥ 0 f or any u > 0.

Let n ≥ n0 and T > 0 be arbitrary. Making use of Itˆo’s formula to V, we obtain

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dV(S , E, I, R, S 1 , E1 , I1 , R1 ) = LV(S , E, I, R, S 1 , E1 , I1 , R1 )dt + σ1 (t)(S − a)dB1 (t) +σ2 (t)(E − 1)dB2 (t) + σ3 (t)(I − 1)dB3 (t) + σ4 (t)(R − 1)dB4 (t) +σ5 (t)(S 1 − b)dB5 (t) + σ6 (t)(E1 − 1)dB6 (t) +σ7 (t)(I1 − 1)dB7 (t) + σ8 (t)(R1 − 1)dB8 (t),

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where L : R8+ → R+ is defined by   1 a LV = 1 − (A + λR + σ(1 − γ)E − mS − β(t)S I − kS ) + 1 − (β(t)S I − mE − σ(1 − γ)E S E   1 1 −kE − σγE) + 1 − (σγE − mI − µI) + 1 − (k(S + E) − mR − λR) I R 8

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Choose a =

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  b 1 + 1− (B + λ1 R1 + σ1 (1 − γ1 )E1 − m1 S 1 − β1 (t)S 1 I) + 1 − (β1 (t)S 1 I − m1 E1 S1 E1   1 1 −σ1 (1 − γ1 )E1 − k1 E1 − σ1 γ1 E1 ) + 1 − (σ1 γ1 E1 − m1 I1 − µ1 I1 ) + 1 − I1 R1 2 2 2 2 2 2 aσ1 (t) + σ2 (t) + σ3 (t) + σ4 (t) + bσ5 (t) + σ6 (t) + σ27 (t) + σ28 (t) ×(k1 E1 − m1 R1 − λ1 R1 ) + 2 ≤ A + B + a(m + k) + bm1 + 3m + 3m1 + σ + σ1 + k + k1 + µ + µ1 + λ + λ1 Aa −m(S + E + R) − m1 (S 1 + E1 + R1 ) + [αβu + bβu1 − (m + µ)]I − (m1 + µ1 )I1 − S l l Bb aλR λ1 bR1 aσ(1 − γ)E bσ1 (1 − γ1 )E1 β S I β1 S 1 I σγE − − − − − − − − S1 S S1 S S1 E E1 I 2u 2u 2u 2u 2u 2u 2u 2u σ1 γ1 E1 k(S + E) aσ1 + σ2 + σ3 + σ4 + bσ5 + σ6 + σ7 + σ8 − − + I1 R 2 ≤ A + B + a(m + k) + bm1 + 3m + 3m1 + σ + σ1 + k + k1 + µ + µ1 + λ + λ1 2u 2u 2u 2u 2u 2u bσ2u aσ2u 5 + σ6 + σ7 + σ8 1 + σ2 + σ3 + σ4 + . +[αβu + bβu1 − (m + µ)]I + 2 2 such that αβu + bβu1 − (m + µ) = 0, then we obtain

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LV ≤ A + B + a(m + k) + bm1 + 3m + 3m1 + σ + σ1 + k + k1 + µ + µ1 + λ + λ1 2u 2u 2u 2u 2u 2u aσ2u + σ2u 2 + σ3 + σ4 + bσ5 + σ6 + σ7 + σ8 + 1 := M, 2 where M is a suitable constant which is independent of S , E, I, R, S 1 , E1 , I1 , R1 and t. The remainder of the proof is similar to Theorem 3.1 of Mao et al. [34] and hence is omitted. This completes the proof. 4. Existence of nontrivial positive periodic solution to system (1.2)

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In this section, based on the periodic solutions theory of Khasminskii [33], we show that there is at least one nontrivial positive T -periodic solution of system (1.2) which reveals that the disease will persist in the human population. Define a parameter

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RS0

where Se =

:=

1 Seσγhβ 3 i3T

(m + σ + k + 12 hσ22 iT )(m + µ + 12 hσ23 iT )

A(m+λ+ 12 hσ24 iT ) 1 m(m+λ+k)+ 2 (m+λ)hσ21 iT + 12 (m+k+ 12 hσ21 iT )hσ24 iT

Theorem 4.1. Assume that

RS0

,

.

> 1, then system (1.2) has at least one nontrivial 9

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A λR σ(1 − γ)E 1 − + β(t)I − + m + k + σ21 (t) S S S 2

and L(− ln R) = −

kS kE 1 − + m + λ + σ24 (t). R R 2

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L(− ln S ) = −

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positive T -periodic solution. Proof. By Lemma 2.1, we can obtain that in order to verify Theorem 4.1, it suffices to find a C 2 -function V(t, x) which is T -periodic in t and a closed set U ⊂ R3+ such that (2.4) and (2.5) hold. By system (1.2), we have

Then define

V1 (t, S , R) = − ln S − c ln R + ω1 (t),

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where c is a positive constant to be determined later, ω1 (t) is a function defined on [0, ∞) satisfying ω ˙ 1 (t) = 12 hσ21 iT + 2c hσ24 iT − 12 σ21 (t) − 2c σ24 (t) and ω1 (0) = 0. It is easy to see that ω1 (t) is a T -periodic function on [0, ∞). Therefore, we obtain  A λR ckS 1 2 1 2  − + m + k + σ1 (t) + c m + λ + σ4 (t) + ω ˙ 1 (t) + βu I LV1 ≤ − − S S R 2 2  A λR ckS 1 2 1 2  = − − − + m + k + hσ1 iT + c m + λ + hσ4 iT + βu I S S R 2 2  √ 1 2  1 2 A ≤ − − 2 cλk + c m + λ + hσ4 iT + m + k + hσ1 iT + βu I. S 2 2 Let

λk , (m + λ + 12 hσ24 iT )2

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c=

λk A 1 2 − + m + k + hσ iT + βu I. S m + λ + 12 hσ24 iT 2 1

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then

LV1 ≤ −

On the other hand, we have

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L(− ln E) = −

β(t)S I 1 + m + σ + k + σ22 (t) E 2

and

L(− ln I) = −

σγE 1 + m + µ + σ23 (t). I 2 10

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Then we define V2 (t, S , E, I, R) = V1 − c1 ln E − c2 ln I + ω2 (t),

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p ro

of

p3 where ω ˙ 2 (t) is a function defined on [0, ∞) satisfying ω 2 (t) = 3 Aβ(t)σγc1 c2 − q 1 3 3 Ahβ 3 i3T σγc1 c2 − c1 (m + σ + k + 12 σ22 (t)) − c2 (m + µ + 12 σ23 (t)) + c1 (m + σ + k + 1 hσ22 iT ) + c2 (m + µ + 21 hσ23 iT ) and ω2 (0) = 0, c1 , c2 are two positive constants to be 2 determined later. It is easy to check that ω2 (t) is a T -periodic function on [0, ∞). Then   A c1 β(t)S I c2 σγE 1 2  1 2  LV2 ≤ − − − + c1 m + σ + k + σ2 (t) + c2 m + µ + σ3 (t) S E I 2 2 λk 1 2 ˙ 2 (t) + m + k + hσ1 iT + βu I + ω − 2 m + λ + 12 hσ24 iT     p3 1 1 ≤ −3 Aβ(t)σγc1 c2 + c1 m + σ + k + σ22 (t) + c2 m + µ + σ23 (t) 2 2 q p λk 1 2 3 1 3 u − + m + k + hσ1 iT + β I + 3 Aβ(t)σγc1 c2 − 3 Ahβ 3 i3T σγc1 c2 1 2 2 m + λ + 2 hσ4 iT     1 2 1 2  1 2  −c1 m + σ + k + σ2 (t) − c2 m + µ + σ3 (t) + c1 m + σ + k + hσ2 iT 2 2 2  1 2  +c2 m + µ + hσ3 iT 2 q     1 1 3 1 = −3 Ahβ 3 i3T σγc1 c2 + c1 m + σ + k + hσ22 iT + c2 m + µ + hσ23 iT 2 2 1 2 λk + m + k + hσ1 iT + βu I. − 1 2 2 m + λ + 2 hσ4 iT

urn

Let

1   Ahβ 3 i3T σγ 1 2  1 2  c1 m+σ+k+ hσ2 iT = c2 m+µ+ hσ3 iT = . 2 2 (m + σ + k + 12 hσ22 iT )(m + µ + 12 hσ23 iT )

Then

LV2 ≤ −

1

Ahβ 3 i3T σγ

1 hσ22 iT )(m 2

1 hσ23 iT ) 2

Jo

(m + σ + k + +µ+   1 2 λk = − m + k + hσ1 iT − 2 m + λ + 12 hσ24 iT 11

−

λk 1 + m + k + hσ21 iT + βu I 1 2 2 m + λ + 2 hσ4 iT

Journal Pre-proof

 ×

1

A(m + λ + 12 hσ24 iT )σγhβ 3 i3T

m(m + λ + k) + 12 (m + λ)hσ21 iT + 12 (m + k + 12 hσ21 iT )hσ24 iT  1 × − 1 + βu I 1 1 2 2 (m + σ + k + 2 hσ2 iT )(m + µ + 2 hσ3 iT ) m + λ + 12 hσ24 iT

 ×

1

A(m + λ + 12 hσ24 iT )σγhβ 3 i3T

of

= −

m(m + λ + k) + 12 (m + λ)hσ21 iT + 12 (m + k + 12 hσ21 iT )hσ24 iT

Let

p ro

m(m + λ + k) + 12 (m + λ)hσ21 iT + 12 (m + k + 12 hσ21 iT )hσ24 iT  1 × − 1 + βu I. 1 1 2 2 (m + σ + k + 2 hσ2 iT )(m + µ + 2 hσ3 iT )

Pr e-

1 1 1 1 m(m + λ + k) + 2 (m + λ)hσ21 iT + 2 (m + k + 2 hσ21 iT )hσ24 iT = , A(m + λ + 12 hσ24 iT ) Se

then we get

≤ − × = −

m(m + λ + k) + 12 (m + λ)hσ21 iT + 12 (m + k + 21 hσ21 iT )hσ24 iT



m + λ + 12 hσ24 iT

1 Seσγhβ 3 i3T

(m + σ + k + 12 hσ22 iT )(m + µ + 12 hσ23 iT ) m+λ+

u

:= −e λ + β I,

where

− 1 + βu I

1 hσ24 iT 2

m(m + λ + k) + 12 (m + λ)hσ21 iT + 12 (m + k + 12 hσ21 iT )hσ24 iT

urn

e λ=



m(m + λ + k) + 12 (m + λ)hσ21 iT + 12 (m + k + 21 hσ21 iT )hσ24 iT

al

LV2

m+λ+

1 hσ24 iT 2

(RS0 − 1) + βu I (4.1)

(RS0 − 1) > 0. (4.2)

Next, we define

V3 (S , E, R, S 1 , E1 , I1 , R1 ) = − ln S − ln E − ln R − ln S 1 − ln E1 − ln I1 − ln R1 ,

Jo

one can derive that LV3 ≤ −

σ2 (t) B β1 (t)S 1 I σ1 γ1 E1 k1 E1 A β(t)S I kS − − − − − − + m + k + β(t)I + 1 S E R S1 E1 I1 R1 2 12

Journal Pre-proof

p ro

of

σ25 (t) σ22 (t) σ24 (t) +m + σ + k + +m+λ+ + m1 + β1 (t)I + + m1 + σ1 + k1 2 2 2 σ2 (t) σ2 (t) σ2 (t) + 6 + m1 + µ1 + 7 + m1 + λ1 + 8 2 2 2 l l A β S I kS B β1 S 1 I σ1 γ1 E1 k1 E1 ≤ − − − − − − − + 3m + 2k + σ + λ + 4m1 S E R S1 E1 I1 R1 σ2u σ2u σ2u σ2u σ2u σ2u σ2u +σ1 + k1 + µ1 + λ1 + 1 + 2 + 4 + 5 + 6 + 7 + 8 + (βu + βu1 )I.(4.3) 2 2 2 2 2 2 2 Now we define 1 V4 (S , E, I, R, S 1 , E1 , I1 , R1 ) = (S + E + I + R + S 1 + E1 + I1 + R1 )θ+1 , θ+1 1) where θ is a constant satisfying 0 < θ < σ2u ∨σ2u ∨σ2u ∨σ2(m∧m 2u ∨σ2u ∨σ2u ∨σ2u ∨σ2u . The differ1 2 3 4 7 8 5 6 ential operator L acting on the function V4 leads to

urn

al

Pr e-

LV4 = (S + E + I + R + S 1 + E1 + I1 + R1 )θ [A + B − mS − mE − (m + µ)I − mR − m1 S 1 1 −m1 E1 − (m1 + µ1 )I1 − m1 R1 ] + θ(S + E + I + R + S 1 + E1 + I1 + R1 )θ−1 2 2 2 2 2 2 2 ×(σ1 (t)S + σ2 (t)E + σ3 (t)I + σ24 (t)R2 + σ25 (t)S 12 + σ26 (t)E12 + σ27 (t)I12 + σ28 (t)R21 ) ≤ (S + E + I + R + S 1 + E1 + I1 + R1 )θ [A + B − m(S + E + I + R) − m1 (S 1 + E1 + I1 1 2u 2u 2u 2u +R1 )] + θ(S + E + I + R + S 1 + E1 + I1 + R1 )θ+1 (σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 2 2u 2u ∨σ2u ∨ σ 7 ∨ σ8 ) 6 ≤ (S + E + I + R + S 1 + E1 + I1 + R1 )θ [A + B − (m ∧ m1 )(S + E + I + R + S 1 + E1 1 2u 2u 2u +I1 + R1 )] + θ(S + E + I + R + S 1 + E1 + I1 + R1 )θ+1 (σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 2 2u 2u 2u ∨σ2u 5 ∨ σ6 ∨ σ7 ∨ σ8 )  1 θ 2u 2u = (A + B)(S + E + I + R + S 1 + E1 + I1 + R1 ) − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 2 

Jo

2u 2u 2u 2u θ+1 ∨σ2u 4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) (S + E + I + R + S 1 + E 1 + I1 + R1 )  1 1 2u 2u 2u 2u 2u 2u 2u ≤ C − (m ∧ m1 ) − θ(σ2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) (S + E + I 1 2 3 4 7 8 5 6 2 2 +R + S 1 + E1 + I1 + R1 )θ+1  1 1 2u 2u 2u 2u 2u 2u 2u 2u ≤ C − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) (S θ+1 + E θ+1 2 2 +I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 (4.4) 1 ),

13

Journal Pre-proof

where C =

sup

(S ,E,I,R,S 1 ,E1 ,I1 ,R1 )∈R8+



(A + B)(S + E + I + R + S 1 + E1 + I1 + R1 )θ −

1 (m ∧ m1 ) 2

+I1 + R1 )θ+1 < ∞.

p ro

Define a C 2 -function V : R8+ × R+ → R as follows

of

 1 2u 2u 2u 2u 2u 2u 2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) (S + E + I + R + S 1 + E1 − θ(σ2u 1 2 3 4 7 8 5 6 2  V(t, S , E, I, R, S 1 , E1 , I1 , R1 ) = MV2 + V3 + V4 ,

where M > 0 is a sufficiently large number satisfying the following condition −Me λ + D ≤ −2,

(4.5)

Pr e-

where e λ is defined in (4.2) and  1  1 2u 2u 2u 2u 2u 2u 2u 2u D = sup − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 2 2 (S ,E,I,R,S 1 ,E1 ,I1 ,R1 )∈R8+ u u ×(S θ+1 + E θ+1 + I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + (β + β1 )I + 3m + 2k + σ  σ2u σ2u σ2u σ2u σ2u σ2u σ2u 5 6 8 7 1 2 4 + + + + + + . +λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + 2 2 2 2 2 2 2

Obviously, V(S , E, I, R, S 1 , E1 , I1 , R1 , t) is T -periodic in t and satisfies lim inf

V(S , E, I, R, S 1 , E1 , I1 , R1 , t) = ∞,

al

k→∞,(S ,E,I,R,S 1 ,E1 ,I1 ,R1 )∈R8+ \Uk

Jo

urn

where Uk = ( 1k , k) × ( 1k , k) × ( 1k , k) × ( 1k , k) × ( 1k , k) × ( 1k , k) × ( 1k , k) × ( 1k , k), which shows that the condition (i) in Lemma 2.1 holds. Therefore, by (4.1), (4.3) and (4.4), one can obtain that  1 1 2u 2u 2u 2u 2u 2u 2u 2u u e LV ≤ −M λ + Mβ I − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 2 2 B A βl S I kS ×(S θ+1 + E θ+1 + I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 ) − − − − 1 S E R S1 l β1 S 1 I σ1 γ1 E1 k1 E1 σ2u − − − + 3m + 2k + σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + 1 E1 I1 R1 2 2u 2u 2u 2u 2u 2u σ σ σ σ σ σ + 2 + 4 + 5 + 6 + 7 + 8 + (βu + βu1 )I. 2 2 2 2 2 2 14

Journal Pre-proof

Now we are in the position to construct a compact subset U such that the condition (ii) in Lemma 2.1, LV(t, S , E, I, R, S 1 , E1 , I1 , R1 ) ≤ −1 f or all (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U C ,

p ro

of

holds. Define a bounded closed set  1 1 1 1 U = (S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R8+ :  ≤ S ≤ ,  ≤ I ≤ ,  3 ≤ E ≤ 3 ,  2 ≤ R ≤ 2 ,     1 3 1 4 1 4 1  ≤ S 1 ≤ ,  ≤ E1 ≤ 3 ,  ≤ I1 ≤ 4 ,  ≤ R1 ≤ 4 ,     where 0 <  < 1 is a sufficiently small constant. In the set R8+ \ U, we can choose  sufficiently small such that the following conditions hold A + F ≤ −1, 

Pr e-

−

−Me λ + Mβu  + D ≤ −1, βl + F ≤ −1,  k − + F ≤ −1, 

−

(4.6) (4.7) (4.8) (4.9)

Jo

urn

al

 1 1 1 2u 2u 2u 2u 2u 2u 2u 2u − (m∧m1 )− θ(σ1 ∨σ2 ∨σ3 ∨σ4 ∨σ5 ∨σ6 ∨σ7 ∨σ8 ) θ+1 +G ≤ −1, (4.10) 4 2    1 1 1 2u 2u 2u 2u 2u 2u 2u − (m∧m1 )− θ(σ2u 1 ∨σ2 ∨σ3 ∨σ4 ∨σ5 ∨σ6 ∨σ7 ∨σ8 ) θ+1 +H ≤ −1, (4.11) 4 2    1 1 1 2u 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 3(θ+1) + J ≤ −1, 4 2  (4.12)   1 1 1 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 2(θ+1) + K ≤ −1, 4 2  (4.13) B (4.14) − + F ≤ −1,  βl1 (4.15) − + F ≤ −1,  σ1 γ1 − + F ≤ −1, (4.16)  15

Journal Pre-proof

−

k1 + F ≤ −1, 

(4.17)

Pr e-

p ro

of

 1 1 1 2u 2u 2u 2u 2u 2u 2u 2u − (m∧m1 )− θ(σ1 ∨σ2 ∨σ3 ∨σ4 ∨σ5 ∨σ6 ∨σ7 ∨σ8 ) θ+1 +L ≤ −1, (4.18) 4 2    1 1 1 2u 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 3(θ+1) + N ≤ −1, 4 2  (4.19)  1 1 1 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 4(θ+1) + P ≤ −1, 4 2  (4.20)  1 1 1 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 4(θ+1) + Q ≤ −1, 4 2  (4.21) where F, G, H, J, K, L, N, P and Q are positive constants which can be seen from (4.23), (4.28), (4.30), (4.32), (4.34), (4.40), (4.42), (4.44) and (4.46), respectively. Note that for a sufficiently small , the condition (4.7) holds due to (4.5). In order to analyze LV on R+ × U C , we divide the domain into sixteen domains according to the structure of LV and the number of each subpopulation, U1 = {(S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R8+ : 0 < S < }, U2 = {(S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R8+ : 0 < I < }, U3 = {(S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R8+ : 0 < E <  3 , S ≥ , I ≥ },

al

U4 = {(S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R8+ : 0 < R <  2 , S ≥ },   1 1 8 8 U5 = (S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ : S > , U6 = (S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ : I > ,      1 1 8 8 U7 = (S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ : E > 3 , U8 = (S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ : R > 2 ,   U9 = {(S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R8+ : 0 < S 1 < },

urn

U10 = {(S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R8+ : 0 < E1 <  3 , S 1 ≥ , I ≥ }, U11 = {(S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R8+ : 0 < I1 <  4 , E1 ≥  3 },

Jo

U13

U12 = {(S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R8+ : 0 < R1 <  4 , E1 ≥  3 },   1 1 8 8 = (S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ : S 1 > , U14 = (S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ : E1 > 3 ,      1 1 8 8 = (S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ : I1 > 4 , U16 = (S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ : R1 > 4 .  

U15

16

Journal Pre-proof

of

S S S Clearly, U C = U1 U2 . . . U16 . Next, we will show that LV(t, S , E, I, R, S 1 , E1 , I1 , R1 ) ≤ −1 on R+ × U C , which is equivalent to showing it on the above sixteen domains. Therefore, we discuss the following 16 cases. Case 1. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U1 , (4.6) shows that A 1 1 2u 2u 2u 2u 2u 2u LV ≤ − + Mβu I + (βu + βu1 )I − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 S  2 2 θ+1 + E θ+1 + I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 ∨σ2u 8 ) (S 1 ) + 3m + 2k + σ + λ + 4m1 ≤ −

A A + F ≤ − + F, S 

σ2u σ2u σ2u σ2u σ2u σ2u σ2u 1 + 2 + 4 + 5 + 6 + 7 + 8 2 2 2 2 2 2 2

where sup

(S ,E,I,R,S 1 ,E1 ,I1 ,R1 )∈R8+

u

u

Mβ I + (β +

βu1 )I

(4.22)

1 1 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 2 2

Pr e-

F =



p ro

+σ1 + k1 + µ1 + λ1 + C +

 2u 2u ∨σ2u ∨ σ ∨ σ ) (S θ+1 + E θ+1 + I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 7 8 1 ) + 3m + 2k 6 +σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C +

σ2u σ2u σ2u σ2u σ2u σ2u σ2u  1 + 2 + 4 + 5 + 6 + 7 + 8 .(4.23) 2 2 2 2 2 2 2

By virtue of the condition (4.6), one can obtain that

al

LV ≤ −1 f or any (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U1 .

urn

Case 2. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U2 , we have  1 1 2u 2u 2u 2u 2u 2u 2u LV ≤ −Me λ + Mβu I − (m ∧ m1 ) − θ(σ2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) 1 2 3 4 7 8 5 6 2 2 u u ×(S θ+1 + E θ+1 + I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + (β + β1 )I + 3m + 2k + σ σ2u σ2u σ2u σ2u σ2u σ2u σ2u 5 6 7 1 2 4 +λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + + + + + + + 8 2 2 2 2 2 2 2 u u e e ≤ −M λ + Mβ I + D ≤ −M λ + Mβ  + D. (4.24)

Jo

By the condition (4.7), we can get that for a sufficiently small ,

LV ≤ −1 f or any (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U2 . 17

Journal Pre-proof

+λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + βl S I βl + F ≤ − + F. E 

σ2u σ2u σ2u σ2u σ2u σ2u σ2u 1 + 2 + 4 + 5 + 6 + 7 + 8 2 2 2 2 2 2 2

p ro

≤ −

of

Case 3. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U3 , one can see that βl S I 1 1 u u u 2u 2u 2u 2u 2u LV ≤ − + Mβ I + (β + β1 )I − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 E 2 2  2u 2u θ+1 θ+1 θ+1 θ+1 θ+1 ∨σ7 ∨ σ8 ) (S + E + I + R + S 1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + 3m + 2k + σ

In view of the condition (4.8), we have

(4.25)

LV ≤ −1 f or any (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U3 .

Pr e-

Case 4. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U4 , we have 1 kS 1 2u 2u 2u 2u 2u + Mβu I + (βu + βu1 )I − (m ∧ m1 ) − θ(σ2u LV ≤ − 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 R 2 2  2u 2u ∨σ7 ∨ σ8 ) (S θ+1 + E θ+1 + I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + 3m + 2k + σ σ2u σ2u σ2u σ2u σ2u σ2u σ2u 5 6 7 1 2 4 +λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + + + + + + + 8 2 2 2 2 2 2 2 kS k ≤ − + F ≤ − + F. (4.26) R 

It follows from the condition (4.9) that

al

k LV ≤ − + F ≤ −1 on R+ × U4 . 

Jo

urn

Case 5. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U5 , one can get that  1 1 2u 2u 2u 2u 2u 2u 2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) LV ≤ − (m ∧ m1 ) − θ(σ2u S θ+1 1 2 3 4 7 8 5 6 4 2  1 1 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) S θ+1 1 2 3 4 7 8 5 6 4 2  1 1 2u 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) (E θ+1 + I θ+1 2 2 u u u +Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + Mβ I + (β + β1 )I + 3m + 2k + σ + λ + 4m1 σ2u σ2u σ2u σ2u σ2u σ2u σ2u 5 6 7 2 4 1 + + + + + + 8 +σ1 + k1 + µ1 + λ1 + C + 2 2 2 2 2 2 2 18

Journal Pre-proof

1 ≤ − (m ∧ m1 ) − 4 1 ≤ − (m ∧ m1 ) − 4

where



of

1 1 2u 2u 2u 2u 2u 2u (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 4 2 8 (S ,E,I,R,S 1 ,E1 ,I1 ,R1 )∈R+   1 1 2u θ+1 2u 2u 2u 2u 2u 2u 2u 2u ∨σ8 ) S − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 2 2 θ+1 θ+1 θ+1 θ+1 u u u ×(E + I + R + S 1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + Mβ I + (β + β1 )I + 3m + 2k σ2u σ2u σ2u σ2u σ2u σ2u +σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + 1 + 2 + 4 + 5 + 6 + 7 2 2 2 2 2 2 2u  σ (4.28) + 8 . 2 sup

−

Pr e-

p ro

G =

 1 2u 2u 2u 2u 2u 2u 2u 2u θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) S θ+1 + G 2  1 1 2u 2u 2u 2u 2u 2u 2u 2u θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) θ+1 + G, (4.27) 2 

where

1 1 2u 2u 2u 2u 2u 2u (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 4 2 8 (S ,E,I,R,S 1 ,E1 ,I1 ,R1 )∈R+   1 1 2u 2u θ+1 2u 2u 2u 2u 2u 2u 2u ∨σ8 ) I − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 2 2 sup

Jo

H =

urn

al

By the condition (4.10), we can deduce that LV ≤ −1 on R+ × U5 . Case 6. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U6 , we obtain  1 1 2u 2u 2u 2u 2u 2u 2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) I θ+1 LV ≤ − (m ∧ m1 ) − θ(σ2u 1 2 3 4 7 8 5 6 4 2  1 1 2u 2u 2u 2u 2u 2u 2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) − (m ∧ m1 ) − θ(σ2u I θ+1 1 2 3 4 7 8 5 6 4 2  1 1 2u 2u 2u 2u 2u 2u 2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) (S θ+1 + E θ+1 − (m ∧ m1 ) − θ(σ2u 1 2 3 4 7 8 5 6 2 2 u u u +Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + Mβ I + (β + β1 )I + 3m + 2k + σ + λ + 4m1 σ2u σ2u σ2u σ2u σ2u σ2u σ2u 5 6 7 1 2 4 +σ1 + k1 + µ1 + λ1 + C + + + + + + + 8 2 2 2 2 2 2 2  1 1 2u 2u 2u 2u 2u 2u 2u θ+1 ≤ − (m ∧ m1 ) − θ(σ2u +H 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) I 4 2   1 1 1 2u 2u 2u 2u 2u 2u 2u ≤ − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) θ+1 + H, (4.29) 4 2  

−

19

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Jo

urn

al

Pr e-

p ro

of

u u u ×(S θ+1 + E θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + Mβ I + (β + β1 )I + 3m + 2k σ2u σ2u σ2u σ2u σ2u σ2u +σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + 1 + 2 + 4 + 5 + 6 + 7 2 2 2 2 2 2 2u  σ + 8 , (4.30) 2 which together with the condition (4.11), we can conclude that LV ≤ −1 on R+ × U6 . Case 7. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U7 , we can derive that  1 1 2u 2u 2u 2u 2u 2u 2u ) E θ+1 ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ LV ≤ − (m ∧ m1 ) − θ(σ2u 7 8 3 4 2 1 6 5 4 2  1 1 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ2u ∨ σ ∨ σ ) E θ+1 ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ 8 7 4 1 2 3 5 6 4 2  1 1 2u 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) (S θ+1 + I θ+1 2 2 u u u +Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + Mβ I + (β + β1 )I + 3m + 2k + σ + λ + 4m1 σ2u σ2u σ2u σ2u σ2u σ2u σ2u 5 6 7 2 4 1 + + + + + + 8 +σ1 + k1 + µ1 + λ1 + C + 2 2 2 2 2 2 2  1 1 2u 2u 2u 2u 2u 2u 2u θ+1 ≤ − (m ∧ m1 ) − θ(σ2u +J 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) E 4 2  1 1 1 2u 2u 2u 2u 2u 2u 2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) ≤ − (m ∧ m1 ) − θ(σ2u + J, (4.31) 1 2 3 4 7 8 5 6 4 2  3(θ+1) where  1 1 2u 2u 2u 2u 2u 2u J = sup − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 4 2 (S ,E,I,R,S 1 ,E1 ,I1 ,R1 )∈R8+    1 1 θ+1 2u 2u 2u 2u 2u 2u 2u 2u ∨σ2u ) E − (m ∧ m ) − θ(σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) 1 8 1 2 3 4 7 8 5 6 2 2 u u u ×(S θ+1 + I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + Mβ I + (β + β1 )I + 3m + 2k σ2u σ2u σ2u σ2u σ2u σ2u +σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + 1 + 2 + 4 + 5 + 6 + 7 2 2 2 2 2 2 2u  σ + 8 . (4.32) 2 According to the condition (4.12), one can see that LV ≤ −1 on R+ × U7 . Case 8. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U8 , we obtain  1 1 2u 2u 2u 2u 2u 2u 2u 2u LV ≤ − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) Rθ+1 4 2

20

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p ro

of

 1 1 2u 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) Rθ+1 4 2  1 1 2u 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) (S θ+1 + E θ+1 2 2 θ+1 θ+1 +I + S 1 + E1θ+1 + I1θ+1 + R1θ+1 ) + Mβu I + (βu + βu1 )I + 3m + 2k + σ + λ + 4m1 σ2u σ2u σ2u σ2u σ2u σ2u σ2u +σ1 + k1 + µ1 + λ1 + C + 1 + 2 + 4 + 5 + 6 + 7 + 8 2 2 2 2 2 2 2  1 1 2u 2u 2u 2u 2u 2u 2u θ+1 ≤ − (m ∧ m1 ) − θ(σ2u +K 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) R 4 2  1 1 1 2u 2u 2u 2u 2u 2u 2u + K, (4.33) ≤ − (m ∧ m1 ) − θ(σ2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) 1 2 3 4 7 8 5 6 4 2  2(θ+1)

where

1 1 2u 2u 2u 2u 2u 2u (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 4 2 8 (S ,E,I,R,S 1 ,E1 ,I1 ,R1 )∈R+   1 1 2u θ+1 2u 2u 2u 2u 2u 2u 2u 2u ∨σ8 ) R − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 2 2 u u u θ+1 θ+1 θ+1 θ+1 ×(S + E + I + S 1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + Mβ I + (β + β1 )I + 3m + 2k σ2u σ2u σ2u σ2u σ2u σ2u +σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + 1 + 2 + 4 + 5 + 6 + 7 2 2 2 2 2 2 2u  σ (4.34) + 8 . 2 sup



−

Pr e-

K =

urn

al

Combining with the condition (4.13) leads to LV ≤ −1 on R+ × U8 . Case 9. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U9 , we get B 1 1 u u u 2u 2u 2u 2u 2u LV ≤ − + Mβ I + (β + β1 )I − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 S1 2 2  2u 2u ∨σ7 ∨ σ8 ) (S θ+1 + E θ+1 + I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + 3m + 2k

Jo

σ2u σ2u σ2u σ2u σ2u σ2u σ2u 5 6 7 1 2 4 +σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + + + + + + + 8 2 2 2 2 2 2 2 B B ≤ − + F ≤ − + F. (4.35) S1 

It follows from the condition (4.14) that LV ≤ −1 f or any (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U9 . 21

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Case 10. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U10 , one can see that

βl1 S 1 I 1 1 2u 2u 2u 2u + Mβu I + (βu + βu1 )I − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 E1 2 2  2u 2u ∨σ2u ∨ σ ∨ σ ) (S θ+1 + E θ+1 + I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 7 8 1 ) + 3m 6

of

LV ≤ −

+2k + σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C +

σ2u σ2u 7 + 8 2 2 l β1 S 1 I βl1 + F ≤ − + F. ≤ − E1 

p ro

+

σ2u σ2u σ2u σ2u σ2u 1 + 2 + 4 + 5 + 6 2 2 2 2 2

(4.36)

According to the condition (4.15), one can obtain that

Pr e-

LV ≤ −1 f or any (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U10 . Case 11. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U11 , then 1 1 σ 1 γ1 E 1 2u 2u 2u 2u + Mβu I + (βu + βu1 )I − (m ∧ m1 ) − θ(σ2u LV ≤ − 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 I1 2 2  2u 2u ∨σ2u ∨ σ ∨ σ ) (S θ+1 + E θ+1 + I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 7 8 1 ) + 3m 6 +2k + σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C +

urn

al

σ2u σ2u 7 + 8 2 2 σ1 γ1 σ 1 γ1 E 1 ≤ − +F ≤− + F. I1  +

σ2u σ2u σ2u σ2u σ2u 1 + 2 + 4 + 5 + 6 2 2 2 2 2

(4.37)

By the condition (4.16), one can obtain that LV ≤ −1 on R+ × U11 .

Jo

Case 12. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U12 , one can derive that 1 k1 E1 1 u u u 2u 2u 2u 2u + Mβ I + (β + β1 )I − (m ∧ m1 ) − θ(σ2u LV ≤ − 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 R1 2 2  2u 2u θ+1 ∨σ2u + E θ+1 + I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + 3m 6 ∨ σ7 ∨ σ8 ) (S 22

Journal Pre-proof

According to the condition (4.17), one can see that

of

σ2u σ2u σ2u σ2u σ2u 5 1 2 4 +2k + σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + + + + + 6 2 2 2 2 2 2u σ σ2u + 7 + 8 2 2 k1 E1 k1 ≤ − + F ≤ − + F. (4.38) R1 

p ro

LV ≤ −1 f or any (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U12 .

al

Pr e-

Case 13. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U13 , one can get that  1 1 2u 2u 2u 2u 2u 2u 2u LV ≤ − (m ∧ m1 ) − θ(σ2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) S 1θ+1 1 2 3 4 7 8 5 6 4 2  1 1 2u 2u 2u 2u 2u 2u 2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) S 1θ+1 − (m ∧ m1 ) − θ(σ2u 1 2 3 4 7 8 5 6 4 2  1 1 2u 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) (S θ+1 + E θ+1 2 2 u u u +I θ+1 + Rθ+1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + Mβ I + (β + β1 )I + 3m + 2k + σ + λ + 4m1 σ2u σ2u σ2u σ2u σ2u σ2u σ2u 5 6 7 1 2 4 +σ1 + k1 + µ1 + λ1 + C + + + + + + + 8 2 2 2 2 2 2 2  1 1 2u 2u 2u 2u 2u 2u 2u θ+1 ≤ − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) S 1 + L 4 2  1 1 1 2u 2u 2u 2u 2u 2u 2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) + L, (4.39) ≤ − (m ∧ m1 ) − θ(σ2u 1 2 3 4 7 8 5 6 4 2  θ+1

where

1 1 2u 2u 2u 2u 2u 2u L = sup − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 4 2 8 (S ,E,I,R,S 1 ,E1 ,I1 ,R1 )∈R+   1 1 θ+1 2u 2u 2u 2u 2u 2u 2u 2u ∨σ2u ) S − (m ∧ m ) − θ(σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) 1 8 1 2 3 4 7 8 1 5 6 2 2 u u u ×(S θ+1 + E θ+1 + I θ+1 + Rθ+1 + E1θ+1 + I1θ+1 + Rθ+1 1 ) + Mβ I + (β + β1 )I + 3m + 2k σ2u σ2u σ2u σ2u σ2u σ2u 5 6 2 4 1 + + + + + 7 +σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + 2 2 2 2 2 2  σ2u + 8 . (4.40) 2

Jo

urn



In view of the condition (4.18), we can conclude that LV ≤ −1 on R+ × U13 . 23

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al

Pr e-

p ro

of

Case 14. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U14 , we get  1 1 2u 2u 2u 2u 2u 2u 2u 2u LV ≤ − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) E1θ+1 4 2  1 1 2u 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) E1θ+1 4 2  1 1 2u 2u 2u 2u 2u 2u 2u ) (S θ+1 + E θ+1 ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ − (m ∧ m1 ) − θ(σ2u 8 7 4 3 2 1 6 5 2 2 u u u +I θ+1 + Rθ+1 + S 1θ+1 + I1θ+1 + Rθ+1 1 ) + Mβ I + (β + β1 )I + 3m + 2k + σ + λ + 4m1 σ2u σ2u σ2u σ2u σ2u σ2u σ2u +σ1 + k1 + µ1 + λ1 + C + 1 + 2 + 4 + 5 + 6 + 7 + 8 2 2 2 2 2 2 2  1 1 2u 2u 2u 2u 2u 2u 2u θ+1 ≤ − (m ∧ m1 ) − θ(σ2u +N 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) E 1 4 2  1 1 1 2u 2u 2u 2u 2u 2u 2u 2u ≤ − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 3(θ+1) + N, (4.41) 4 2  where  1 1 2u 2u 2u 2u 2u 2u N = sup − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 4 2 (S ,E,I,R,S 1 ,E1 ,I1 ,R1 )∈R8+    1 1 θ+1 2u 2u 2u 2u 2u 2u 2u 2u ∨σ2u ) E − θ(σ ∨ σ ∨ σ ∨ σ ∨ σ (m ∧ m ) − ∨ σ ∨ σ ∨ σ ) 1 8 1 2 3 4 7 8 1 5 6 2 2 u u u ×(S θ+1 + E θ+1 + I θ+1 + Rθ+1 + S 1θ+1 + I1θ+1 + Rθ+1 1 ) + Mβ I + (β + β1 )I + 3m + 2k σ2u σ2u σ2u σ2u σ2u σ2u +σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + 1 + 2 + 4 + 5 + 6 + 7 2 2 2 2 2 2  σ2u (4.42) + 8 . 2

Jo

urn

Combining with the condition (4.19), we can obtain that LV ≤ −1 on R+ × U14 . Case 15. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U15 , we have  1 1 2u 2u 2u 2u 2u 2u 2u LV ≤ − (m ∧ m1 ) − θ(σ2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) I1θ+1 1 2 3 4 7 8 5 6 4 2  1 1 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) I1θ+1 1 2 3 4 7 8 5 6 4 2  1 1 2u 2u 2u 2u 2u 2u 2u 2u − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) (S θ+1 + E θ+1 2 2 +I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + R1θ+1 ) + Mβu I + (βu + βu1 )I + 3m + 2k + σ + λ + 4m1 σ2u σ2u σ2u σ2u σ2u σ2u σ2u 5 6 7 2 4 1 + + + + + + 8 +σ1 + k1 + µ1 + λ1 + C + 2 2 2 2 2 2 2 24

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1 ≤ − (m ∧ m1 ) − 4 1 ≤ − (m ∧ m1 ) − 4

where



of

1 1 2u 2u 2u 2u 2u 2u (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 4 2 8 (S ,E,I,R,S 1 ,E1 ,I1 ,R1 )∈R+   1 1 2u θ+1 2u 2u 2u 2u 2u 2u 2u 2u ∨σ8 ) I1 − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 2 2 θ+1 θ+1 θ+1 θ+1 u u u ×(S + E + I + R + S 1θ+1 + E1θ+1 + Rθ+1 1 ) + Mβ I + (β + β1 )I + 3m + 2k σ2u σ2u σ2u σ2u σ2u σ2u +σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + 1 + 2 + 4 + 5 + 6 + 7 2 2 2 2 2 2 2u  σ (4.44) + 8 . 2 sup

−

Pr e-

p ro

P =

 1 2u 2u 2u 2u 2u 2u 2u 2u θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) I1θ+1 + P 2  1 1 2u 2u 2u 2u 2u 2u 2u 2u θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 4(θ+1) + P, (4.43) 2 

where

1 1 2u 2u 2u 2u 2u 2u (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 4 2 8 (S ,E,I,R,S 1 ,E1 ,I1 ,R1 )∈R+   1 1 2u 2u θ+1 2u 2u 2u 2u 2u 2u 2u ∨σ8 ) R1 − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 2 2 sup

Jo

Q =

urn

al

According to the condition (4.20), one can get that LV ≤ −1 on R+ × U15 . Case 16. If (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U16 , we obtain  1 1 2u 2u 2u 2u 2u 2u 2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) Rθ+1 LV ≤ − (m ∧ m1 ) − θ(σ2u 1 2 3 4 7 8 1 5 6 4 2   1 1 2u 2u 2u 2u 2u 2u 2u θ+1 − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) R1 4 2  1 1 2u 2u 2u 2u 2u 2u 2u ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ∨ σ ) (S θ+1 + E θ+1 − (m ∧ m1 ) − θ(σ2u 1 2 3 4 7 8 5 6 2 2 +I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 ) + Mβu I + (βu + βu1 )I + 3m + 2k + σ + λ + 4m1 σ2u σ2u σ2u σ2u σ2u σ2u σ2u 5 6 7 1 2 4 +σ1 + k1 + µ1 + λ1 + C + + + + + + + 8 2 2 2 2 2 2 2  1 1 2u 2u 2u 2u 2u 2u 2u θ+1 ≤ − (m ∧ m1 ) − θ(σ2u 1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) R1 + Q 4 2  1 1 1 2u 2u 2u 2u 2u 2u 2u 2u ≤ − (m ∧ m1 ) − θ(σ1 ∨ σ2 ∨ σ3 ∨ σ4 ∨ σ5 ∨ σ6 ∨ σ7 ∨ σ8 ) 4(θ+1) + Q, (4.45) 4 2  

−

25

Journal Pre-proof

of

×(S θ+1 + E θ+1 + I θ+1 + Rθ+1 + S 1θ+1 + E1θ+1 + I1θ+1 ) + Mβu I + (βu + βu1 )I + 3m + 2k σ2u σ2u σ2u σ2u σ2u σ2u +σ + λ + 4m1 + σ1 + k1 + µ1 + λ1 + C + 1 + 2 + 4 + 5 + 6 + 7 2 2 2 2 2 2 2u  σ + 8 . (4.46) 2

p ro

Combining with the condition (4.21) yields LV ≤ −1 on R+ × U16 . Obviously, from (4.22), (4.24), (4.25), (4.26), (4.27), (4.29), (4.31), (4.33), (4.35), (4.36), (4.37), (4.38), (4.39), (4.41), (4.43) and (4.45), one can derive that for a sufficiently small ,

LV(t, S , E, I, R, S 1 , E1 , I1 , R1 ) ≤ −1 f or all (t, S , E, I, R, S 1 , E1 , I1 , R1 ) ∈ R+ × U C ,

urn

al

Pr e-

which shows that the condition (ii) in Lemma 2.1 holds. Hence in view of Lemma 2.1, we obtain that system (1.2) has a nontrivial positive T -periodic solution. Furthermore, one can see that for any initial value (S (0), E(0), I(0), R(0), S 1 (0), E1 (0), I1 (0), R1 (0)) ∈ R8+ , system (1.2) has a unique global positive solution and hence system (1.2) has at least one nontrivial positive T -periodic solution. This completes the proof. Remark 4.1. By Lemma 2.1, the proof of Theorem 4.1 has two key points: finding a C 2 -function V(t, x) which is T -periodic in t and a closed set U ⊂ R3+ such that (2.4) and (2.5) hold. First, we construct a suitable Lyapunov function V(t, x) satisfying (2.4) and a compact subset U. Next, in order to analyze LV on R+ × U C easier and more convenient, we divide R+ × U C into sixteen domains and show LV ≤ −1 on the sixteen cases respectively. Remark 4.2. Due to the seasonal variation, individuals lifecycle, hunting, food supplies, mating habits, harvesting and so on, the birth rate, the transmission rate between susceptible and infectious will not remain constant, but exhibit a more-or-less periodicity. In order to derive sustained oscillatory, investigating the effect of seasonal variation and randomness is of great significance. According to Theorem 4.1, we can find this physical phenomenon: under the disturbance of white noise, if R0s > 1, system (1.2) has at least one nontrivial positive T-periodic solution in the sense of distribution. Theorem 4.1 also reveals that the disease will persist in the human population. 5. Concluding remarks

Jo

This paper is concerned with a stochastic nonautonomous rabies epidemic model. By using Khasminskii’s theory of periodic solutions and constructing a suitable stochastic Lyapunov function, we show that system (1.2) has at least one 26

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nontrivial positive T -periodic solution under the condition RS0 = A(m+λ+ 12 hσ24 iT ) 1 m(m+λ+k)+ 2 (m+λ)hσ21 iT + 12 (m+k+ 12 hσ21 iT )hσ24 iT

1

Seσγhβ 3 i3T

(m+σ+k+ 21 hσ22 iT )(m+µ+ 12 hσ23 iT )

References

Pr e-

p ro

of

1, where Se = . To the best of our knowledge, this paper is the first attempt to study the existence of nontrivial positive T -periodic solution to a stochastic seasonal rabies epidemic model. Some interesting topics deserve further consideration. On the one hand, one may propose some more realistic but complex models, such as considering the effects of impulsive perturbations on system (1.2). On the other hand, it is necessary to reveal that the methods used in this paper can be also applied to investigate other interesting epidemic models, such as SIRS model, SIQR model and so on. We leave these cases for our future work. Acknowledgments This work was supported by the National Natural Science Foundation of P.R. China (No.11701209), Science and Technology Research Project of Jilin Provincial Department of Education of China (No.JJKH20180462KJ), the Education Department of Hainan province (No.Hnky2017ZD-14), and Hainan Provincial Natural Science Foundation of China (No.119QN205). Competing interests statement The authors declare that there is no conflict of interest regarding the publication of this paper.

al

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