m > such that for any solution (S(n), E(n), I(n), R(n)) of model () with initial condition (), m ≤ lim infn→∞ I(n) ≤ lim supn→∞ I(n) ≤ M. We have the following result. Theorem . Disease I(n) in model () is permanent if and only if R > . Proof The necessity is obvious. In fact, if R ≤ , then by Theorem . the disease-free equilibrium P is globally attractive.
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Now, we prove the sufficiency. When R > , we can choose constants p > and ε > such that δ – (μ + δ) > , p
∂f (S , , , ) – ε p – (μ + γ ) > . ∂I
()
We will use the persistence theory of dynamical systems (see [], Section . in Chapter ) to prove the theorem. Define the sets X = (S, E, I, R) : S > , E ≥ , I ≥ , R ≥ and X = (S, E, I, R) ∈ X : E > , I > ,
∂X = (S, E, I, R) ∈ X : EI = .
Let (S(n), E(n), I(n), R(n)) be the solution of model () with initial condition (S(), E(), I(), R()) = (S , E , I , R ). Define the set M∂ = (S , E , I , R ) ∈ ∂X : S(n), E(n), I(n), R(n) ∈ ∂X , n = , , . . . . It is clear that the solution of model () with initial condition (S(), E(), I(), R()) = (S , , , R ) has the form (S(n), , , R(n)). Hence, we have (S , , , R ) : S > , R ≥ ⊂ M∂ . Suppose that there is (S , E , I , R ) ∈ M∂ such that (S , E , I , R ) ∈/ {(S , , , R ) : S > , R ≥ }. Then, we have E > or I > . Let (S(n), E(n), I(n), R(n)) be the solution of model () with initial condition (S(), E(), I(), R()) = (S , E , I , R ). If E > , then from the second equation of model () we have E(n + ) ≥ E(n) – (μ + δ)E(n + ). Hence, E(n) ≥ E()( +μ +δ )n > for all n ≥ . From the third equation of model () we further have I(n + ) > I(n) – (μ + γ )I(n + ),
n ≥ .
Hence, I(n) > I()( +μ +γ )n ≥ for all n ≥ . This shows that a solution (S(n), E(n), I(n), R(n)) ∈/ ∂X for all n > . If I > , then from third equation of model () we have I(n) ≥ I()( +μ +γ )n > for all n ≥ . Since S(n) > and f (S(n), I(n)) > for all n ≥ , from the second equation of model () we further have E(n) > E()( +μ +δ )n ≥ for all n ≥ . This also shows that the solution (S(n), E(n), I(n), R(n)) ∈/ ∂X for all n > . Hence, (S , E , I , R ) ∈/ M∂ , which leads to a contradiction. Thus, we also have M∂ ⊂ (S , , , R ) : S > , R ≥ . Therefore, M∂ = {(S , , , R ) : S > , R ≥ }.
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It is clear that model () restricted to M∂ has a globally attractive equilibrium P (S , , , ). This shows that {P } in M∂ is isolated invariable and acyclic. Now, we prove that W s (P ) ∩ X = ∅, where W s (P ) =
S(), E(), I(), R() : lim S(n), E(n), I(n), R(n) = P , n→∞
which is said to be a stable set of P . Suppose that there is a point (S(), E(), I(), R()) ∈ X such that limn→∞ (S(n), E(n), I(n), R(n)) = P . Since lim
(S,E,I,R)→P
f (S, E, I, R) ∂f (S , , , ) = , I ∂I
for the above ε > , there is η > such that when |S – S | < η , E < η , I < η , and R < η , we have f (S, E, I, R) ∂f (S , , , ) ≥ – ε . I ∂I We can choose an integer n > such that |S(n) – S | < η , E(n) < η , I(n) < η , and R(n) < η for all n ≥ n . Consider the Lyapunov function V (n) = pE(n) + I(n). We have that, for n > n ,
V (n) = V (n + ) – V (n) = p f S(n + ), E(n + ), I(n + ), R(n + ) – (μ + δ)E(n + ) + δE(n + ) – (μ + γ )I(n + ) ∂f (S , , , ) – ε I(n + ) – (μ + δ)E(n + ) ≥p ∂I + δE(n + ) – (μ + γ )I(n + )
∂f (S , , , ) δ = p – ε – (μ + γ ) I(n + ) + – (μ + δ) pE(n + ) ∂I p ≥ mV (n + ), where ∂f (S , , , ) δ m = min p – ε – (μ + γ ), – (μ + δ) > . ∂I p Hence, we finally have limn→∞ V (n) = ∞, which leads to a contradiction with limn→∞ V (n) = . It follows that W s (P ) ∩ X = ∅. Thus, by the theorems of uniform persistence for dynamical systems given in [], we obtain that model () is permanent. This completes the proof.
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Remark . When f (S, E, I, R) = ease in model () is permanent.
SI , N
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by Theorem ., if R =
βδ (μ +γ )(μ +δ)
> , then the dis-
Remark . Theorem . only obtains the permanence of the disease for model (). However, whether we can also prove that an endemic equilibrium P∗ is globally attractive for model () when R > ? In the following section, we will give a partial positive answer. We will prove that, for special case σ = of model (), an endemic equilibrium P∗ is globally attractive only when R > .
5 Global attractivity of endemic equilibrium in a particular case In this section, we consider a particular case of model (), that is, f (S, E, I, R) = f (S, E, I) and σ = in model (). Model () becomes of the form ⎧ S(n + ) = S(n) + – f (S(n + ), E(n + ), I(n + )) – μ S(n + ), ⎪ ⎪ ⎪ ⎨E(n + ) = E(n) + f (S(n + ), E(n + ), I(n + )) – (μ + δ)E(n + ), ⎪ + γ )I(n + ), I(n + ) = I(n) + δE(n + ) – (μ ⎪ ⎪ ⎩ R(n + ) = R(n) + γ I(n + ) – μ R(n + ).
()
Because R(n) does not appear in the first three equations of model (), we only need to consider the equivalent system ⎧ ⎪ ⎨S(n + ) = S(n) + – f (S(n + ), E(n + ), I(n + )) – μ S(n + ), E(n + ) = E(n) + f (S(n + ), E(n + ), I(n + )) – (μ + δ)E(n + ), ⎪ ⎩ I(n + ) = I(n) + δE(n + ) – (μ + γ )I(n + ).
()
We have the following result on the global attractivity of the endemic equilibrium for model (). Theorem . If R > , then the endemic equilibrium P∗ of model () is globally attractive. Proof Let P∗ (S∗ , E∗ , I∗ ) be an endemic equilibrium of model (). Then ⎧ ⎪ ⎨ – f (S∗ , E∗ , I∗ ) – μ S∗ = , f (S∗ , E∗ , I∗ ) – (μ + δ)E∗ = , ⎪ ⎩ δE∗ – (μ + γ )I∗ = . Let (S(n), E(n), I(n)) be any positive solution of system (). Define the functions V S(n) = S(n) – S∗ –
S(n)
S∗
f (S∗ , E∗ , I∗ ) dη, f (η, E∗ , I∗ )
E(n) V E(n) = E(n) – E∗ – E∗ ln , E∗ and I(n) . V I(n) = I(n) – I∗ – I∗ ln I∗
()
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From (H) we easily obtain that when S(n) = S∗ , V S(n) > S(n) – S∗ –
S(n)
f (S∗ , E∗ , I∗ ) dη = . f (S∗ , E∗ , I∗ )
S∗
Since g(x) = x – – ln x > for x > and x = , we obtain that when E(n) = E∗ and I(n) = I∗ , V (E(n)) > and V (I(n)) > . Computing V (n) = V (S(n + )) – V (S(n)), we have
V (n) = S(n + ) – S(n) –
S(n+)
S(n)
f (S∗ , E∗ , I∗ ) dη. f (η, E∗ , I∗ )
From (H) it follows that, for any η between S(n) and S(n + ), –
f (S∗ , E∗ , I∗ ) f (S∗ , E∗ , I∗ ) ≤– f (η, E∗ , I∗ ) f (S(n + ), E∗ , I∗ )
if S(n + ) ≥ S(n),
–
f (S∗ , E∗ , I∗ ) f (S∗ , E∗ , I∗ ) ≥– f (η, E∗ , I∗ ) f (S(n + ), E∗ , I∗ )
if S(n + ) ≤ S(n).
We have
S(n+)
– S(n)
f (S∗ , E∗ , I∗ ) f (S∗ , E∗ , I∗ ) S(n + ) – S(n) . dη ≤ – f (η, E∗ , I∗ ) f (S(n + ), E∗ , I∗ )
Therefore, from () we obtain
f (S∗ , E∗ , I∗ ) S(n + ) – S(n) f (S(n + ), E∗ , I∗ )
f (S∗ , E∗ , I∗ ) – f S(n + ), E(n + ), I(n + ) = – f (S(n + ), E∗ , I∗ ) – μ S(n + )
f (S∗ , E∗ , I∗ ) S(n + ) – S∗ = – μ – f (S(n + ), E∗ , I∗ ) + f (S∗ , E∗ , I∗ ) – f S(n + ), E(n + ), I(n + )
V (n) ≤ –
–
f (S∗ , E∗ , I∗ ) f (S∗ , E∗ , I∗ ) f (S(n + ), E∗ , I∗ )
+
f (S∗ , E∗ , I∗ ) f S(n + ), E(n + ), I(n + ) . f (S(n + ), E∗ , I∗ )
Calculating V (n) = V (E(n + )) – V (E(n)), we obtain V (n) = E(n + ) – E(n) – I∗ ln
E(n + ) . E(n)
Using the inequality ln( – x) ≤ –x for x < , we have – ln
E(n) E(n) E(n + ) = ln – – ≤– – . E(n) E(n + ) E(n + )
()
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Therefore,
V (n) ≤ –
E∗ E(n + ) – E(n) E(n + )
E∗ f S(n + ), E(n + ), I(n + ) – (k + μ )E(n + ) = – E(n + ) = f S(n + ), E(n + ), I(n + ) – (k + μ )E(n + ) –
E∗ f S(n + ), E(n + ), I(n + ) + (k + μ )E∗ . E(n + )
()
Similarly, calculating V (n) = V (I(n + )) – V (I(n)), we obtain
V (n) ≤ δE(n + ) – (μ + γ )I(n + ) –
I∗ δE(n + ) + (μ + γ )I∗ . I(n + )
()
Choose the Lyapunov function f (S∗ , E∗ , I∗ ) V I(n) . V (n) = V S(n) + V E(n) + (μ + γ )I∗ For convenience of calculations, we denote S = S(n + ), E = E(n + ), and I = I(n + ). Computing V (n) = V (n + ) – V (n), from ()-() we obtain
f (S∗ , E∗ , I∗ ) f (S∗ , E∗ , I∗ ) (S – S∗ ) + f (S∗ , E∗ , I∗ ) –
V (n) ≤ –μ – f (S, E∗ , I∗ ) f (S, E∗ , I∗ )
E∗ f (S, E, I) I I∗ E f (S, E, I) – – – + f (S, E∗ , I∗ ) Ef (S∗ , E∗ , I∗ ) I∗ IE∗
I∗ E I∗ E I∗ E = f (S∗ , E∗ , I∗ ) – ∗ + ln ∗ – f (S∗ , E∗ , I∗ ) ln ∗ IE IE IE
∗ f (S, E, I)E∗ f (S, E, I)E + ln + f (S∗ , E∗ , I∗ ) – f (S∗ , E∗ , I∗ )E f (S∗ , E∗ , I∗ )E – f (S∗ , E∗ , I∗ ) ln
f (S, E, I)E∗ f (S∗ , E∗ , I∗ )E
f (S∗ , E∗ , I∗ ) f (S∗ , E∗ , I∗ ) + f (S∗ , E∗ , I∗ ) – + ln f (S, E∗ , I∗ ) f (S, E∗ , I∗ )
f (S∗ , E∗ , I∗ ) f (S, E∗ , I∗ )
If (S, E∗ , I∗ ) If (S, E∗ , I∗ ) f (S∗ , E∗ , I∗ )f (S, E, I) – + ln + f (S, E∗ , I∗ ) I∗ f (S, E, I) I∗ f (S, E, I) – f (S∗ , E∗ , I∗ ) ln
–
f (S∗ , E∗ , I∗ )f (S, E, I) If (S, E∗ , I∗ ) ln f (S, E∗ , I∗ ) I∗ f (S, E, I)
I∗ f (S, E, I) f (S∗ , E∗ , I∗ )f (S, E, I) If (S, E∗ , I∗ ) – ln If (S, E∗ , I∗ ) f (S, E∗ , I∗ ) I∗ f (S, E, I)
f (S, E∗ , I∗ ) f (S, E, I) f (S∗ , E∗ , I∗ ) f (S, E∗ , I∗ ) – f (S, E, I) ln . – ln = f (S, E, I) I∗ I
≤ –f (S∗ , E∗ , I∗ ) ln
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Figure 1 Time series of S(n), E(n), I(n), and R(n) in Example 6.1
From (H) we obtain V (n) ≤ for any n ≥ , and V (n) ≡ implies I(n) ≡ I∗ for all n ≥ . From I(n) ≡ I∗ and the third equation of model () it follows that E(n) ≡ E∗ for all n ≥ . Furthermore, from the second equation of model () we obtain that S(n) ≡ S∗ for all n ≥ . Therefore, using the theorems of stability of difference equations, we finally obtain that the endemic equilibrium P∗ of model () is globally attractive. This completes the proof. Remark . In Remark ., we indicated that for SEIRS-type model (), an important problem is to prove that the endemic equilibrium is globally attractive only when R > . From Theorem . we see that only for the particular case σ = of model (), that is, SEIRtype model, we get a positive answer. Therefore, an interesting open problem for general SEIRS model () is whether the endemic equilibrium is also globally attractive only when
R > . Remark . From the proofs of Theorem ., Theorem ., and Theorem . we easily see that the condition μ ≤ min{μ , μ , μ } is not used. In fact, this condition is only used in Theorem . to obtain the positivity of solutions of model (). Therefore, an interesting question is whether the condition μ ≤ min{μ , μ , μ } can be taken out in the proof of the positivity of solutions of model ().
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Figure 2 Time series of S(n), E(n), I(n), and R(n) in Example 6.2
6 Numerical examples Now, we give numerical examples to show that for SEIRS-type model (), the endemic equilibrium may be globally attractive for different incidence function f (S, E, I, R), which satisfies (H) only when the basic reproduction number R > . βSI , = ., μ = ., μ = ., μ = Example . In model (), we take f (S, E, I, R) = +αI+ωS ., β = ., δ = ., ω = ., and γ = .. The parameters μ , α, and σ will be chosen later.
By calculating we have the basic reproduction number R = . > . We further take μ = ., α = ., and σ = .. Then the endemic equilibrium P∗ = (., ., ., .). From the numerical simulations (see Figure ) we obtain that P∗ may be globally attractive. βSI Example . In model (), we take f (S, E, I, R) = +αI , = ., μ = ., μ = ., μ = ., β = ., δ = ., and γ = .. The parameters μ , α, and σ will be chosen later.
By calculating we have the basic reproduction number R = . > . We further take μ = ., α = ., and σ = .. Then the endemic equilibrium P∗ = (., ., ., .). By numerical simulations (see Figure ) we obtain that P∗ may be globally attractive.
βS I , = , μ = ., μ = ., Example . In model (), we take f (S, E, I, R) = (+ωS)(+αI) μ = ., β = ., δ = ., ω = ., and γ = .. The parameters μ , α, and σ will be chosen later.
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Figure 3 Time series of S(n), E(n), I(n), and R(n) in Example 6.3
By calculating we have the basic reproduction number R = . > . We further take μ = ., α = ., and σ = .. Then the endemic equilibrium P∗ = (., ., ., .). By numerical simulations (see Figure ) we obtain that P∗ may be globally attractive.
βS I Example . In model (), we take f (S, E, I, R) = (+ωS)(+αI ) , = ., μ = ., μ = ., μ = ., β = ., δ = ., ω = ., and γ = .. The parameters μ , α, and σ will be chosen later.
By calculating we have the basic reproduction number R = . > . We further take μ = ., α = ., and σ = .. Then the endemic equilibrium P∗ = (., ., ., .). By numerical simulations (see Figure ) we obtain that P∗ may be globally attractive. All these examples of numerical simulations show that when R > , no matter sufficiently greater than one or closer to one but still greater than one, we always obtain that the endemic equilibrium P∗ is globally attractive, which may offer an affirmative conjecture to the open problem given in Remark ., that is, for the general SEIRS model () the endemic equilibrium P∗ is globally attractive only when R > . Therefore, in our future work, we expect to obtain the corresponding theoretical results for this open problem.
7 Discussion In this paper, we proposed a discrete SEIRS epidemic model () with general nonlinear incidence, which is described by the backward difference scheme. By our discussions presented in this paper, necessary and sufficient conditions for the global attractivity of the
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Figure 4 Time series of S(n), E(n), I(n), and R(n) in Example 6.4
disease-free equilibrium and the permanence of the disease are established, that is, if the basic reproduction number R ≤ , then the disease-free equilibrium is globally attractive, and if R > , then the disease is permanent. Furthermore, when the model degenerates into SEIR model, it is proved that when R > , the model has a unique globally attractive endemic equilibrium. Unfortunately, for SEIRS model (), when the basic reproduction number is greater than one, we do not obtain the local asymptotic stability and global attractivity of the endemic equilibrium. But the numerical examples given in Section show that the endemic equilibrium for general SEIRS model () may be globally attractive. Therefore, it is still an important and interesting open problem how to apply the linearization method to establish the local asymptotic stability of the endemic equilibrium and how to construct the discrete analogue Lyapunov functions to study the global attractivity of the endemic equilibrium for general SEIRS model (). In addition, the dynamical behaviors for the nonautonomous discrete SEIRS epidemic models, discrete SEIRS epidemic models with vaccination, stage-structured discrete SEIRS epidemic models, and delayed discrete SEIRS epidemic models with nonlinear incidence described by the backward difference scheme are rarely considered. Whether similar results on the permanence and extinction of the disease and the global attractivity of the disease-free equilibrium for these models can be obtained is also an interesting open question. On the other hand, corresponding to continuous model (), we also have the following discrete SEIRS epidemic models with general nonlinear incidence described by the for-
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ward difference scheme or Micken’s nonstandard finite difference scheme: ⎧ S(n + ) – S(n) = – f (X(n)) – μ S(n) + σ R(n), ⎪ ⎪ ⎪ ⎨ E(n + ) – E(n) = f (X(n)) – (μ + δ)E(n), ⎪I(n + ) – I(n) = δE(n) – (μ + γ )I(n), ⎪ ⎪ ⎩ R(n + ) – R(n) = γ I(n) – (μ + σ )R(n), where X(n) = (S(n), E(n), I(n), R(n)), and ⎧ S(n+)–S(n) ⎪ = – f (S(n + ), I(n)) – μ S(n + ) + σ R(n + ), ⎪ φ(h) ⎪ ⎪ ⎨ E(n+)–E(n) = f (S(n + ), I(n)) – (μ + δ)E(n + ),
φ(h)
I(n+)–I(n) ⎪ = δE(n + ) – (μ + γ )I(n + ), ⎪ φ(h) ⎪ ⎪ ⎩ R(n+)–R(n) = γ I(n + ) – (μ + σ )R(n + ), φ(h)
μ h
with the denominator function φ(h) = e μ – , and h > is the time-step size. An important open problem is whether the results obtained in this paper for model () can also be extended to these models.
Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 College of Mathematics and System Sciences, Xinjiang University, Urumqi, 830046, People’s Republic of China. 2 Department of Basic Education, Xinjiang Institute of Engineering, Urumqi, 830091, People’s Republic of China. 3 Department of Medical Engineering and Technology, Xinjiang Medical University, Urumqi, 830054, People’s Republic of China. Acknowledgements This work is supported by the Doctoral Subject Science Foundation (Grant No. 20136501110001), and the National Natural Science Foundation of China (Grant Nos. 11271312, 11401512, 11261056, 11301451, 61473244, 11402223). Received: 29 October 2015 Accepted: 25 April 2016 References 1. Bai, Z, Zhou, Y: Global dynamics of an SEIRS epidemic model with periodic vaccination and seasonal contact rate. Nonlinear Anal., Real World Appl. 13, 1060-1068 (2012) 2. Gao, S, Chen, L, Teng, Z: Impulsive vaccination of an SEIRS model with time delay and varying total population size. Bull. Math. Biol. 68, 731-745 (2007) 3. Korobeinikov, A: Lyapunov functions and global properties for SEIR and SEIS epidemic models. Math. Med. Biol. 21, 75-83 (2004) 4. Korobeinikov, A: Global properties of infectious disease models with nonlinear incidence. Bull. Math. Biol. 69, 1871-1886 (2007) 5. Korobeinikov, A, Maini, P: Non-linear incidence and stability of infectious disease models. Math. Med. Biol. 22, 113-128 (2005) 6. Kuniya, T, Nakata, Y: Permanence and extinction for a nonautonomous SEIRS epidemic model. Appl. Math. Comput. 218, 9321-9331 (2012) 7. Melesse, DY, Gumel, AB: Global asymptotic properties of an SEIRS model with multiple infectious stages. J. Math. Anal. Appl. 366, 202-217 (2010) 8. Nakata, Y, Kuniya, T: Global dynamics of a class of SEIRS epidemic models in a periodic environment. J. Math. Anal. Appl. 363, 230-237 (2010) 9. Zhang, T, Teng, Z: On a nonautonomous SEIRS model in epidemiology. Bull. Math. Biol. 69, 2537-2559 (2007) 10. Mickens, RE: Application of Nonstandard Finite Difference Scheme. World Scientific, Singapore (2000) 11. Mickens, RE: Dynamics consistency: a fundamental principle for constructing nonstandard finite difference scheme for differential equation. J. Differ. Equ. Appl. 9, 1037-1051 (2003) 12. Mickens, RE: Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition. Numer. Methods Partial Differ. Equ. 23, 672-691 (2007) 13. Allen, LJS, Driessche, P: The basic reproduction number in some discrete-time epidemic models. J. Differ. Equ. Appl. 14, 1127-1147 (2008) 14. Allen, LJS, Lou, Y, Nevai, AL: Spatial patterns in a discrete-time SIS patch model. J. Math. Biol. 58, 339-375 (2009)
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