# Threshold Dynamics of an SIR Model with Nonlinear Incidence Rate and Age-Dependent Susceptibility

Authors: Junyuan Yang; Xiaoyan Wang
Journal: Complexity
 (2018)

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## Abstract

We propose an SIR epidemic model with different susceptibilities and nonlinear incidence rate. First, we obtain the existence and uniqueness of the system and the regularity of the solution semiflow based on some assumptions for the parameters. Then, we calculate the basic reproduction number, which is the spectral radius of the next-generation operator. Second, we investigate the existence and local stability of the steady states. Finally, we construct suitable Lyapunov functionals to strictly prove the global stability of the system, which are determined by the basic reproduction numberℛ0 and some assumptions for the incidence rate.


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## Body

## 1. Introduction

Mathematical modelling has been paid attention on investigating the intrinsic mechanisms of the disease prevalence. Many factors have been illustrated to affect dynamics of diseases spread. One of them is the susceptibility, which describes the fragility of individuals attacked during their lifetime. The susceptible ability is determined by individuals’ behaviors and the immunity of individuals. As Brucellosis, humans associated with animals are more easy to be transmitted compared with ones apart from these kinds of jobs. There are few literatures [1, 2] studying this factor to affect the dynamics of systems. In this paper, we introduce susceptible age to evaluate the susceptibility for the modelling process.Besides, incidence rates play an important role in the disease spread pattern. Compartment modelling method was firstly proposed by Kermack and McKendrick [3, 4]. The baseline models are SIS-type and SIR-type, respectively, in which S denotes the susceptible individuals, I represents the infectious individuals, and R denotes the recovered individuals (N is the total population). From mathematical points of view, there are two classical forms: one is bilinear denoted and the other is standard incidence denoted by βSI and βSI/N, where β represents the transmission rate. As literatures indicated, the two incidence rates have distinct limitations, that is, the bilinear one is suitable for an initial phase of some disease spread, while the other one is applicable for the final phase. In order to overcome these defaults, Capasso and Serio in [5] introduced a saturated incidence rate βSI/1+αI to connect the above two cases. There is no doubt that it is more realistic and is more meaningful than the old ones. Thereafter, a general case given by fStGIt was introduced into the models (see, e.g., [6, 7]). Furthermore, a general incidence rate written as fS,I in [8] had been used to investigate the global dynamics of SIR and SIRS epidemic models.To our best knowledge, Magal et al. are the pioneers to investigate the global stability of the steady states for age-structured model by constructing Lyapunov functional in [9]. Since then, many researchers [10–13] have extensively studied the global behavior for age-structured epidemic models and bilinear incidence rates and the references therein. Since the complex structure for age-structured models, there are few references to establish global dynamics of epidemic models combining age structure and nonlinear incidence [14–17]. This issue has been becoming a challenging task for lacking well-established tools.As we know, the recovered individuals can gain permanent disease-induced immunity for some diseases such as chickenpox, mumps, and measles. Motivated by what has been discussed, we integrate the susceptible age with a general incidence rate into anSIR model given as follows:
(1)∂st,a∂t+∂st,a∂a=−μst,a−βafst,a,It,st,0=Λ,dItdt=∫0∞βafst,a,Itda−μ+γ+δ1It,dRtdt=γIt−μRt,with initial condition
(2)s0,a=s0a∈L+1,I0=I0≥0,R0=R0≥0,where L+1 is the set of integrable functions from 0,∞ into ℝ+=0,∞. The total population is classified into three classes: susceptible st,a, infected It, and recovered Rt. μ denotes the natural death rate, δ1 denotes the death rate due to disease, and γ is the recovery rate. βa is the transmission rate with respect to susceptible age a; fst,a,It represents the transmission incidence and it concludes many common forms such as βast,aIt, βast,aIt/1+αI, and βast,aIt/1+α1st,a+α2It+α3ast,aIt. All the parameters satisfy the following assumptions:Assumption 1.1.
(A1)Λ and μ are positive; δ1 and γ are nonnegative.
(A2) The functionβa∈CBUℝ+,ℝ+∩L∞ℝ+ where CBUℝ+,ℝ+ is the set of all bounded and uniformly continuous functions from ℝ+ to ℝ+. Denote
(3)β¯=esssupa∈ℝ+βa.
(A3) Fors, I∈ℝ+, f0,I=fs,0=0; ∂fs,I/∂s≥0, for I>0; ∂fs,I/∂s2 is an increasing function for s, while ∂fs,I/∂I≥0 for s>0; ∂f2s,I/∂I2≤0; fs,I/s is a decreasing function with respect to s for I>0.The functionfs,I satisfies the local Lipschitz continuous in s and J, which coincides the assumption in [15]
(4)∫0∞fs1t,⋅,I1t−fs2t,⋅,I2t≤Ls1t,⋅−s2t,⋅1+I1t−I2tℝ.EquationR is separated with other equations of system (1), and it can be omitted. Hence, system (1) can lower as a two-dimensional system:
(5)∂st,a∂t+∂st,a∂a=−μst,a−βafst,a,It,st,0=Λ,dItdt=∫0∞βafst,a,Itda−μ+δIt,where δ=γ+δ1. The goal of this paper is to establish global dynamics for (5). This paper supplements or generalizes many existing results [10, 14] if we take some special incidences or some specific parameters.Note that the total populationNt=∫0∞st,ada+It+Rt satisfies the following equation:
(6)dNtdt=Λ−μNt−δIt≤Λ−μNt,which implies that limsupt→∞Nt≤Λ/μ.


### 1.1. Integrated Semigroup Formula

In this subsection, we employ the approach proposed by Thieme [18] to formulate system (5) as an abstract Cauchy problem. Now, we define the state space as follows:
(7)X≔ℝ×L1ℝ+;ℝ×ℝ,X+=ℝ+×L+1ℝ+;ℝ+×ℝ+,X0=0×L1ℝ+;ℝ×ℝ,X0,+=0×L+1ℝ+;ℝ+×ℝ+,with the norm form given by
(8)ϕ,ψ·,ψ1tX=ϕ+ψ·1+ψ1=ϕ+∫0∞ψada+ψ1.In order to formulate system (5) as an abstract Cauchy problem, we define the linear operator A on X0 as follows:
(9)A0ψψ1≔−ψ0−dψada−μψa−μ+δψ1,0ψψ1∈DA,where DA=0×W1,1ℝ;ℝ×ℝ and W1,1ℝ;ℝ denote the Sobolev space. The nonlinear operator F on X0 is given.
(10)F0ψψ1≔Λ−βafψa,ψ1∫0∞βafψ+a,ψ1da,0ψψ1∈DA.Note thatDA⊂X0 and DA¯=X0. Denote xt=0,st,·,ItT and x0=0,s0·,I0T, we can rewrite system (5) as an abstract Cauchy problem.
(11)dxtdt=Axt+Fxt,t≥0,x0=x0∈DA¯∩X0,+.Next, we will show the existence and uniqueness of system (11) by the results in [19]. First, we will give the following two lemmas.Lemma 1.1.
There exists a real constantM≥1 and ω∈ℝ such that ω,+∞⊂ρA, and
(12)λI−A−n≤Mλ−wn,foralln∈ℕ,λ>ω,where ρA denotes the resolvent set of A and I denotes the identity operator. Furthermore, for all λ>ω, λI−A−1X+⊂X+.Proof 1.
For any0,ψ,ψ1T∈X and λ>−µ, it follows from the definitions of the linear operator A that
(13)λI−A−1ϕ^ψ^aψ^1=0ψaψ1,which implies that
(14)ϕ^ψ^aψ^1=ψ0ddaψa+μψaμ+δψ1.
We directly calculate and obtain(15)ψa=ϕ^e−λ+μa+∫0aψ^ηe−λ+μa−ηdη,ψ1=ψ^1λ+μ+δ.
So that we have the following estimation.(16)λI−A−1ϕ^ψ^aψ^1X=0+∫0∞ψada+ψ1=∫0∞ϕ^e−λ+μada+∫0∞∫0aψ^ηe−λ+μa−ηdηda+ϕ^λ+μ+δ≤ϕ^λ+μ+ψ^1λ+μ+ψ^λ+μ=1λ+μϕ^ψ^aψ^1X.
Hence,ω=−µ and M=1 are the values that we are looking for in the presentation of the theorem.DefineϒL0=ξ∈X0∣ξϒ≤L. Under Assumption 1.1, we have the following lemma for the nonlinear operator F on X0.Lemma 1.2.
Suppose Assumption1.1 holds. For all L>0, there exists ΔL>0 such that
(17)Fξ−Fξ^X≤ΔLξ−ξ^X,for any ξ=0,ψa,ψ1T∈X0∩ϒL0 and ξ^=0,ψ^a,ψ^1T∈X0∩ΥL0.Proof 2.
For anyξ, ξ^∈X0∩ϒL0, we have
(18)Fξ−Fξ^=0−βafψa,ψ1+βafψ^a,ψ^1∫0∞βafψa,ψ1da−∫0∞βafψ^a,ψ^1daX≤2∫0∞βafψa,ψ1−fψ^a,ψ^1da≤2β¯Lψ−ψ^1+ψ1−ψ^1≕ΔLξ−ξ^X,where Δ=2β¯L.Lemmas1.1 and 1.2 imply that Assumption 3.1 of [19] holds. It follows from Lemma 3.1 in [19] that the existence and uniqueness of system (11) are given by the following theorem.Theorem 1.3.
There exists a unique semiflowΦ:ℝ+×X0,+→X0,+ for system (11). For all t≥0∫0tΦtx0ds∈DA and for any x0∈X0,+,
(19)Φtx0=x0+A∫0tΦηx0dη+∫0tFΦηx0dη.In what follows, we will give the regularity of the semiflowΦtx0. By the proof process of Theorem 6.3 in [19], we define the set as
(20)DB=φ∈DA∣Aφ+Fφ∈DA¯.Assumption 1.2.
For any0,ψa,ψ1∈X0, the following conditions hold.
(21)∂fψa,ψ1∂ψaψa∈L1ℝ+,∂fψa,ψ1∂ψ1∈L1ℝ+.Obviously,DB∩X0,+=X0,+¯. Based on Assumptions 1.1 and 1.2, the nonlinear operator F on X0 is differentiable with respect to t, and the differentiation of F at ξ^=0,ψ^,ψ^1T∈X0 is given by
(22)F′ξ^0ψψ1=0−βa∂fψ^a,ψ^1∂ψ^aψa−βa∂fψ^a,ψ^1∂ψ^1ψ1∫0∞βa∂fψ^a,ψ^1∂ψaψada+∫0∞βa∂fψ^a,ψ^1∂ψ1ψ1da,0ψψ∈X0.From what has been discussed, we obtain the regularity of the semiflowΦt on X0.Theorem 1.4.
For anyx0∈DB∩X0,+, then the semiflow Φtx0 is continuously differentiable with respect to t, and xt=Φtx0 is the global classical solution of the abstract Cauchy problem (11).Based on Assumption1.1, we have the following priori estimations.Lemma 1.5.
For0,s,IT∈X0,+ and t≥0, the following estimations hold:
(23)∫0∞st,ada+It≤max∫0∞s0ada+I0,Λμ≔M1,and
(24)Jt≔∫0∞βafst,a,Itda≤β¯∫0∞fμM1πa,M1da.
Furthermore,(25)limsupt→∞∫0∞st,ada+It≤Λμ,and
(26)limsupt→∞Jt≤β¯∫0∞fΛπa,Λμ.Before ending this section, we define a set(27)Γ=0,s0⋅,I0T∈X0:s00=Λ,N0≤Λμ.Then, it follows from Lemma1.5 that Γ is positively invariant. This indicates that Φt,x0∈Γ, ∀t≥0, and x0∈Γ. In what follows, for convenience, we denote 0,s,IT as s,IT.The paper is devoted to study the global stability of the steady states of the system by employing the Lyapunov functionals. The rest of this paper is organized as follows: Section2 gives the asymptotic smoothness of the solution semiflow Φtx0. In Section 3, we calculate the basic reproduction number by renewal equation and discuss the existence and local stability of steady states of system (5). In Section 4, the global dynamics of system (5) is established by considering the persistence of system (5) and constructing suitable Lyapunov functionals.


## 1.1. Integrated Semigroup Formula

In this subsection, we employ the approach proposed by Thieme [18] to formulate system (5) as an abstract Cauchy problem. Now, we define the state space as follows:
(7)X≔ℝ×L1ℝ+;ℝ×ℝ,X+=ℝ+×L+1ℝ+;ℝ+×ℝ+,X0=0×L1ℝ+;ℝ×ℝ,X0,+=0×L+1ℝ+;ℝ+×ℝ+,with the norm form given by
(8)ϕ,ψ·,ψ1tX=ϕ+ψ·1+ψ1=ϕ+∫0∞ψada+ψ1.In order to formulate system (5) as an abstract Cauchy problem, we define the linear operator A on X0 as follows:
(9)A0ψψ1≔−ψ0−dψada−μψa−μ+δψ1,0ψψ1∈DA,where DA=0×W1,1ℝ;ℝ×ℝ and W1,1ℝ;ℝ denote the Sobolev space. The nonlinear operator F on X0 is given.
(10)F0ψψ1≔Λ−βafψa,ψ1∫0∞βafψ+a,ψ1da,0ψψ1∈DA.Note thatDA⊂X0 and DA¯=X0. Denote xt=0,st,·,ItT and x0=0,s0·,I0T, we can rewrite system (5) as an abstract Cauchy problem.
(11)dxtdt=Axt+Fxt,t≥0,x0=x0∈DA¯∩X0,+.Next, we will show the existence and uniqueness of system (11) by the results in [19]. First, we will give the following two lemmas.Lemma 1.1.
There exists a real constantM≥1 and ω∈ℝ such that ω,+∞⊂ρA, and
(12)λI−A−n≤Mλ−wn,foralln∈ℕ,λ>ω,where ρA denotes the resolvent set of A and I denotes the identity operator. Furthermore, for all λ>ω, λI−A−1X+⊂X+.Proof 1.
For any0,ψ,ψ1T∈X and λ>−µ, it follows from the definitions of the linear operator A that
(13)λI−A−1ϕ^ψ^aψ^1=0ψaψ1,which implies that
(14)ϕ^ψ^aψ^1=ψ0ddaψa+μψaμ+δψ1.
We directly calculate and obtain(15)ψa=ϕ^e−λ+μa+∫0aψ^ηe−λ+μa−ηdη,ψ1=ψ^1λ+μ+δ.
So that we have the following estimation.(16)λI−A−1ϕ^ψ^aψ^1X=0+∫0∞ψada+ψ1=∫0∞ϕ^e−λ+μada+∫0∞∫0aψ^ηe−λ+μa−ηdηda+ϕ^λ+μ+δ≤ϕ^λ+μ+ψ^1λ+μ+ψ^λ+μ=1λ+μϕ^ψ^aψ^1X.
Hence,ω=−µ and M=1 are the values that we are looking for in the presentation of the theorem.DefineϒL0=ξ∈X0∣ξϒ≤L. Under Assumption 1.1, we have the following lemma for the nonlinear operator F on X0.Lemma 1.2.
Suppose Assumption1.1 holds. For all L>0, there exists ΔL>0 such that
(17)Fξ−Fξ^X≤ΔLξ−ξ^X,for any ξ=0,ψa,ψ1T∈X0∩ϒL0 and ξ^=0,ψ^a,ψ^1T∈X0∩ΥL0.Proof 2.
For anyξ, ξ^∈X0∩ϒL0, we have
(18)Fξ−Fξ^=0−βafψa,ψ1+βafψ^a,ψ^1∫0∞βafψa,ψ1da−∫0∞βafψ^a,ψ^1daX≤2∫0∞βafψa,ψ1−fψ^a,ψ^1da≤2β¯Lψ−ψ^1+ψ1−ψ^1≕ΔLξ−ξ^X,where Δ=2β¯L.Lemmas1.1 and 1.2 imply that Assumption 3.1 of [19] holds. It follows from Lemma 3.1 in [19] that the existence and uniqueness of system (11) are given by the following theorem.Theorem 1.3.
There exists a unique semiflowΦ:ℝ+×X0,+→X0,+ for system (11). For all t≥0∫0tΦtx0ds∈DA and for any x0∈X0,+,
(19)Φtx0=x0+A∫0tΦηx0dη+∫0tFΦηx0dη.In what follows, we will give the regularity of the semiflowΦtx0. By the proof process of Theorem 6.3 in [19], we define the set as
(20)DB=φ∈DA∣Aφ+Fφ∈DA¯.Assumption 1.2.
For any0,ψa,ψ1∈X0, the following conditions hold.
(21)∂fψa,ψ1∂ψaψa∈L1ℝ+,∂fψa,ψ1∂ψ1∈L1ℝ+.Obviously,DB∩X0,+=X0,+¯. Based on Assumptions 1.1 and 1.2, the nonlinear operator F on X0 is differentiable with respect to t, and the differentiation of F at ξ^=0,ψ^,ψ^1T∈X0 is given by
(22)F′ξ^0ψψ1=0−βa∂fψ^a,ψ^1∂ψ^aψa−βa∂fψ^a,ψ^1∂ψ^1ψ1∫0∞βa∂fψ^a,ψ^1∂ψaψada+∫0∞βa∂fψ^a,ψ^1∂ψ1ψ1da,0ψψ∈X0.From what has been discussed, we obtain the regularity of the semiflowΦt on X0.Theorem 1.4.
For anyx0∈DB∩X0,+, then the semiflow Φtx0 is continuously differentiable with respect to t, and xt=Φtx0 is the global classical solution of the abstract Cauchy problem (11).Based on Assumption1.1, we have the following priori estimations.Lemma 1.5.
For0,s,IT∈X0,+ and t≥0, the following estimations hold:
(23)∫0∞st,ada+It≤max∫0∞s0ada+I0,Λμ≔M1,and
(24)Jt≔∫0∞βafst,a,Itda≤β¯∫0∞fμM1πa,M1da.
Furthermore,(25)limsupt→∞∫0∞st,ada+It≤Λμ,and
(26)limsupt→∞Jt≤β¯∫0∞fΛπa,Λμ.Before ending this section, we define a set(27)Γ=0,s0⋅,I0T∈X0:s00=Λ,N0≤Λμ.Then, it follows from Lemma1.5 that Γ is positively invariant. This indicates that Φt,x0∈Γ, ∀t≥0, and x0∈Γ. In what follows, for convenience, we denote 0,s,IT as s,IT.The paper is devoted to study the global stability of the steady states of the system by employing the Lyapunov functionals. The rest of this paper is organized as follows: Section2 gives the asymptotic smoothness of the solution semiflow Φtx0. In Section 3, we calculate the basic reproduction number by renewal equation and discuss the existence and local stability of steady states of system (5). In Section 4, the global dynamics of system (5) is established by considering the persistence of system (5) and constructing suitable Lyapunov functionals.


## 2. Asymptotic Smoothness

In view of system (5), we note that it has one integrate term. Hence, it is necessary to investigate the existence of a compact attractor of all bounded sets of X. In order to establish this goal, we need the results of Theorem 2.33 and Theorem 2.46 in [20]. To apply them, we decompose the solution semiflow Φ=Φ^+Φ~, where
(28)Φ^t,s0⋅,I0=s^t,⋅,0,Φ~t,s0⋅,I0=s~t,⋅,It,where
(29)s~t,a=st,a,for0≤a≤t0,fort≤a=Λπa−∫0aβa−ηfst−η,a−η,It−ηe−μηdη,for0≤a≤t,0,fort≤a.and
(30)s^t,a=st,a−s~t,a=0,for0≤a≤t,s0a−te−μt−∫0tβa−ηfst−η,a−η,It−ηe−μtdη,fort≤a.Theorem 2.1.
The semiflowΦ:ℛ+×X0→X0 is asymptotically smooth if there are two maps Φ^ and Φ~:ℝ+×X0→X0 such that Φtx0=Φ^tx0+Φ~tx0, and for any bounded set K, the following statements hold:
(1)
For anyx0∈K, there exists a function δ:ℝ+×ℝ+→ℝ+ such that for any r>0limt→+∞δt,r=0 with x0K≤r, then Φ^tx0≤δt,r.
(2)
There existsatK>0 such that Φ~tx0 has a compact closure for each t≥tK.Lemma 2.2.
For anyr>0, there exists a function δ:ℝ+×ℝ+ such that for any r>0,
(31)limt→+∞δt,r=0,and
(32)Φ^tx0K≤δt,r,∀x0∈K,t≥0.Proof 3.
This lemma follows the process as Theorem 2.46 in [21]. Let K be a bounded subset of s0·,I0∈K⊂Γ. It is easy to see that s^, I^ and s~, I~ are nonnegative. Using (30) leads to
(33)Φ^t,s0·,I0K=s^t,·1=∫t∞s0a−te−µtda−∫t∞∫0aβa−ηfsa−η,t−η,It−ηe−μηdηda≤e−μt∫t∞s0a−tda=e−µt∫0∞s0ada=e−μts01=e−μts0⋅,I0K≤δt,r,where δt,r=e−µtr. Hence, the assumption in (1) of Theorem 2.1 holds.Next, we prove that condition (2) of Theorem2.1 is also satisfied. To achieve such goal, we employ the Fréchet-Kolmogorov theorem for the compactness of sets in L1 [21].Theorem 2.3
(Fréchet-Kolmogorov theorem, [21]).
LetK be a subset of L1R+. Then, K has compact closure if and only if the following conditions hold:
(i)
supf∈K∫0∞fada<∞,
(ii)
limr→∞∫r∞fada=0, uniformly in f∈K,
(iii)
limh→0+∫0∞fa+h−fada=0, uniformly in f∈K,
(iv)
limh→0+∫0hfada=0, uniformly in f∈K.Lemma 2.4.
Φ~t maps any bounded subsets of K into sets with compact closure in X0.Proof 4.
For any fixedt∈ℝ+ and any bounded set K⊆Γ0, the set Kt≜Φ~t,s0,I0:s0,I0∈K is precompact. With a notice, It is bounded by the virtue of fx,y and Lemma 1.5. This implies that Kt,s=s~t:s~t,·,It∈Kt is precompact. It is enough to show that Kt,s=s~t,·:s~t,·,It∈Kt is precompact. This can be achieved by applying Fréchet-Kolmogorov Theorem 2.3 in [21]. First, st,a≤Λπa and ∫0∞πada=1/μ indicate that Kt,s is bounded. This implies that condition (i) of Fréchet-Kolmogorov theorem holds. Second, it is an immediate result that ∫r∞s~t,ada approaches zero as r goes to infinity. By the boundedness of st,a, condition (iv) of Fréchet-Kolmogorov theorem is well established. Finally, condition (iii) of the Fréchet-Kolmogorov theorem should be verified. In order to prove it, we should show that Φ~ is uniformly continuous on the bounded set Kt,s or
(34)limh→0+s~t,⋅−s~t,⋅+h1=0,uniformlyinKt,s.
Equation (34) holds obviously when t=0 since s~0,⋅=s0a. So, we need to show that the result holds for t>0. Since we are concerned with the limit as h tends to 0+, we assume that h∈0,t. Then,
(35)s~t,⋅−s~t,⋅h1=∫0∞s~t,a−s~t,a+hda=∫0t−h∣∫0a+hβa+h−ηfst−η,a+h−η,It−ηπη−∫0aβa−ηfst−η,a−η,It−ηπη∣dηda+∫t−ht∫0aβa−ηfst−η,a−η,It−ηπηdηda+Λ∫0∞πa+h−πada≤∫0t−h∫aa+hβa+h−ηfst−η,a+h−η,It−ηπηdηda+∫0t−h∫0aβa+h−η−βa−ηfst−η,a+h−η,It−ηπηdηda+∫0t−h∫0aβa−ηfst−η,a+h−η,It−η−fst−η,a−η,It−ηπηdηda+β¯fΛ,Λμh+Λt−hh≤fΛ,Λμt−hh+fΛ,Λμ1+1μt−hLβh+2Λt−hh+β¯fΛ,Λμh+Λt−hh≤fΛ,Λμ1+Lβ1+1μ+3Λt−h+β¯fΛ,Λμh,since st,a≤Λπa, It≤Λ/µ for t, a∈ℝ+, and πa+h−πa≤1−e−∫aa+hμsds≤μh. This estimate immediately yields (34).With the help of Lemmas2.2 and 2.4, together with Φtx0X≤Λ/µ, we conclude that Φt has compact closure in X for any x0∈K. Theorem 2.1 ensures that the semiflow Φt is asymptotically smooth.Proposition 2.5.
Let Assumption1.1 hold, and then, the solution semiflow Φ defined by (19) on Γ0 is asymptotically smooth.


## 3. The Basic Reproduction Number, Existence, and Local Stability of Steady States

In this section, we give the general approach for calculating the basic reproduction number, which is the spectral radius of the next-generation operator. The basic reproduction number is the most important value for characterizing the renewal process in structured population. An epoch-making method was proposed by Diekmann et al. in [22] and then developed by Inaba in [23]. Moreover, Yang and Xu in [24] extend this method to calculate the basic reproduction of disease models on complex networks. Let s¯·,I¯ be a solution of system (5), and then, it satisfies the following equations:
(36)ds¯ada=−μs¯a−βafs¯a,I¯,s¯0=Λ,0=∫0∞βafs¯a,I¯da−μ+δI¯.Obviously, system (36) has a unique disease-free steady state E0=Λπa,0. Now, we are in position to define the basic reproduction number by analyzing the renewal equation. The linear equation in disease-free invasion phase is given by
(37)dItdt=∫0∞βa∂fΛπa,0∂IdaIt−μ+δIt.Solving (37) yields
(38)It=I0e−μ+δt+∫0∞βa∂fΛπa,0∂Ida∫0tIse−μ+δt−sds=I0e−μ+δt+∫0∞βa∂fΛπa,0∂Ida∫0tIt−se−μ+δsds.Hence, the basic reproduction number is calculated as(39)ℛ0=∫0∞βa∂fΛπa,0∂Ida∫0∞e−μ+δsds=1μ+δ∫0∞βa∂fΛπa,0∂Ida.In epidemiology, the basic reproduction numberℛ0 gives the average number of cases that a typical infectious individual generates, if introduced into a susceptible population, over the whole infectious period.Next, we look for the solutions¯a,I¯/E0. It follows from the third equation of (36) that
(40)μ+δI¯=∫0∞βafs¯a,I¯da.Solving the first equation of system (36), we have
(41)s¯a=Λπa−∫0aβτfs¯τ,I¯dτ.Then, we integrate (41) to obtain
(42)S¯≔∫0∞s¯ada=Λμ−∫0∞∫0aβτfsτ,I¯πaπτdτda=Λμ−∫0∞βτfs¯τ,I¯eμτ∫s∞e−μadadτ=Λμ−1μ∫0∞βafs¯a,I¯da.Moreover, we replace∫0∞βafs¯a,I¯da in (42) by (40) and obtain
(43)I¯=μμ+δΛμ−S¯.Combining (40) and (43), we have
(44)I¯=μμ+δΛμ−S¯,μ+δI¯=∫0∞βafs¯a,I¯da.Borrowing the first equation of system (44), we obtain the following equation consisting of one variable
(45)gS¯≔∫0∞βafs¯a,μμ+δΛμ−S¯da−μΛμ−S¯=0.With a notice,(46)g′S¯≔∫0∞βa∂fs¯a,μ/μ+δΛ/μ−S¯∂s¯−μμ+δ∂fs¯a,μ/μ+δΛ/μ−S¯∂I¯da+μ=0.Theng′Λ/µ=µ1−ℛ0<0 if ℛ0>1. Besides, we note that gΛ/µ=0 and g0=−Λ<0. Therefore, there exists at least one positive solution of (45) for S¯∈0,Λ/μ.Next, we will show the uniqueness of the endemic steady state of system (44). By way of contradiction, we assume that there exist two positive solutions of gx=0, denoted by s¯ and s^, respectively. They satisfy that s¯a>s^a. It follows from the relations between s¯a and I¯ that I¯<I^, that is, I^=qI¯ with q>1. By the second equation of (44) and the property of fx,y with respect to x, we obtain
(47)I^=1μ+δ∫0∞βafs^a,I^da≤1μ+δ∫0∞βafs¯a,I^da=1μ+δ∫0∞βafs¯a,qI¯da<1μ+δ∫0∞βafs¯a,I¯I¯qI¯da=qI¯.This is a contradiction with assumptionqI¯=I^. From what has been discussed, we have the following result concerning the existence of steady states.Theorem 3.1.
Consider system (5) with ℛ0 defined in (39). If ℛ0≤1, then there is a unique steady state, which is the disease-free steady state E0; while if ℛ0≤1, then there are two steady states, the disease-free steady state E0 and the endemic steady state E∗.Secondly, we study the local stability of the steady states obtained in Theorem3.1. In order to do this, we linearize system (5) at the endemic steady state E∗ and obtain
(48)∂xt,a∂t+∂xt,a∂a=−μxt−βa∂fs¯a,I¯∂s¯xt,a−βa∂fs¯a,I¯∂I¯yt,xt,0=0,dytdt=∫0∞βa∂fs¯a,I¯∂s¯axt,ada+∫0∞βa∂fs¯a,I¯∂I¯dayt−μ+δyt.Now, we focus on system (48) and obtain the local stability of the steady states.Theorem 3.2.
Letℛ0 define in (39). If ℛ0<1, then the disease-free steady state E0 is locally asymptotically stable and if ℛ0<1, the unique endemic steady state E∗ is locally asymptotically stable.Proof 5.
Letxt,a=xaeλt and yt=y0eλt be a solution of (48). Then, system (48) can be formally written as
(49)dxada=−λ+μxa−βa∂fs¯a,I¯∂s¯axa−βa∂fs¯a,I¯∂I¯y0,x0=0,0=∫0∞βa∂fs¯a,I¯∂s¯axada+∫0∞βa∂fs¯a,I¯∂I¯day0−λ+μ+δy0.
Solving the first equation yields(50)xa=−y0∫0aβτ∂fs¯τ,I¯∂I¯e−∫τaλ+μ+βξ∂fs¯ξ,I¯/∂s¯dξdτ.
Then substituting (50) into the equation z≔∫0∞βa∂fs¯a,I¯/∂s¯axada, we have
(51)z=−y0∫0∞βa∂fs¯a,I¯∂s¯∫0∞βτ∂fs¯τ,I¯∂I¯e−∫τaλ+μ∣+βξ∂fs¯ξ,I¯/∂s¯dξdτda.
After variable change and integration by parts, we obtain(52)z=−y0∫0∞βτ∂fs¯τ,I¯∂I¯∫0∞βa∂fs¯a,I¯∂s¯e−∫τaλ+μ∣+βξ∂fs¯ξ,I¯/∂s¯dξdadτ=y0∫0∞βτ∂fs¯τ,I¯∂I¯e−∫τaλ+μ∣+βξ∂fs¯ξ,I¯/∂s¯dξ∣τ∞+λ+μ∫t∞e−∫τaλ+μ∣+βξ∂fs¯ξ,I¯/∂s¯dξdadτ=y0∫0∞βτ∂fs¯τ,I¯∂I¯−1+λ+μ∫0∞e−∫τaλ+μ∣+βξ∂fs¯ξ,I¯/∂s¯dξdadτ.
Plugging (53) into the second equation of (49), we admit
(53)y0λ+μ∫0∞βτ∂fs¯τ,I¯∂I¯∫τ∞e−∫τaλ+μ+βξ∂fs¯ξ,I¯/∂s¯dξdadτ=y0λ+μ+δ.
Equation (53) is the characteristic equation of (5). Then, the steady state E¯ is locally (asymptotically) stable if all eigenvalues of the characteristic equation have negative real parts and it is unstable if at least one eigenvalue has a positive real part.
Firstly, we note that the characteristic equation atE0 is
(54)λ+μ∫0∞βτ∂fs0τ,0∂I¯∫τ∞e−∫τaλ+μdξdadτ=λ+μ+δ.
Integrating the inner formula yields(55)λ+μ+δ∫0∞βa∂fΛπa,0∂Ie−∫0aλ+μdξda.
We claim that all roots of (55) have negative real parts. Otherwise, let λ0 be a root of (55) with Reλ0≥0. Then, the module of the left hand side of (55) is larger than μ+δ, while the module of the right hand side of (55) is ∫0∞βa∂fΛπa,0/∂Ie−∫0aλ+μdξda≤ℛ0μ+δ. If ℛ0<1, this leads to a contradiction. Therefore, all the eigenvalues of the characteristic (54) have negative real part and then disease-free steady state E0 is locally asymptotically stable if ℛ0<1.
Secondly, by the virtue off, we readily obtain
(56)λ+μ∫0∞βτ∂fs¯τ,I¯∂I¯∫τ∞e−∫τaλ+μ+βξ∂fs¯ξ,I¯/∂s¯dξdadτ≤λ+μ∫0∞βτfs¯τ,I¯I¯∫τ∞e−∫τaλ+μdξdadτ=∫0∞βτfs¯τ,I¯I¯e−∫0τλ+μdξdadτ.
Next, we show that the characteristic equation (53) has no eigenvalues with nonnegative real parts. Arguing directly, assume that there is one eigenvalue λ1 with Reλ1≥0. Then
(57)∫0∞βτfs¯τ,I¯I¯e−∫0τλ+μdξdadτ≤∫0∞βτfs¯τ,I¯I¯πτdτ<μ+δ.
On the other hand, by the right hand side of (53), we have
(58)λ+µ+δ>µ+δ.
Consequently, the endemic steady stateE∗ is locally asymptotically stable.


## 4. Global Stability Analysis

In order to consider the compactness of the orbitΦt,s0·,I0∈Γ, we use the invariance principle which is referred in Theorem 4.2 of Chapter IV [21]. Therefore, we can define ρ:Γ→ℝ+ as
(59)ρst,·,It=It,forst,·,It∈Γ.Let(60)Γ0=s0·,I0∈Γ:thereexistsat0∈ℝ+suchthatρΦt0,s0·,I0>0.Obviously, ifs0·,I0∈Γ/Γ0, then st,·,It→E0 as t→∞.Definition 4.1
([21], page 61).
If there exists anϵ>0, independent of the initial conditions, such that
(61)limsupt→∞ρΦt,s0⋅,I0>ϵresp.,liminft→∞ρΦt,s0·,I0>ϵ,
then (5) is called to be uniformly weakly ρ-persistent (resp., uniformly strongly ρ-persistent), for s0·,I0∈Γ0.To achieve the persistence of the system (5), we firstly prove the weekly ρ-persistence and then give the global compactness of the orbit.Proposition 4.2.
Ifℛ0>1, then (5) is uniformly weakly ρ-persistent.Proof 6.
Supposeℛ0>1, there exists an ϵ0>0 such that
(62)1μ+δ∫0∞βa∂fΛπa−∫0aβsfΛπs,ϵ0πsds,ϵ0∂Ie−λada>1.
Arguing directly, there existss0·,I0∈Γ0 with
(63)limsupt→∞ρΦt,s0⋅,I0≤ϵ02.
Then, there existst0∈ℝ+ such that
(64)ρΦt,s0·,I0≤ϵ0,fort≥t0.
Without loss of generality, we can assume thatt0=0 since we can achieve it by replacing the initial condition with Φt0,s0·,I0. Then, for t≥t0=0 with a notice st,a≤Λπa and (A3) of Assumption 1.1 for the function fst,a,It with respect to variables s and I, we have
(65)∂st,a∂a+∂st,a∂a≥−μst,a−βafΛπa,ϵ0,and this implies that
(66)st,a=Λπa−∫0aβa−ηfΛπa−η,ϵ0πηdη,t≥a,s0a−tπt−∫0tβa−ηfΛπa−η,ϵ0πηdη,t<a.
Therefore,(67)Jt≔∫0∞βafst,a,Itda≥∫0tβafΛπa−∫0aβsfΛπs,ϵ0πsds,Itda.
Trying to consider the third equation of (5), we have
(68)İt=Jt−μ+δIt≥∫0tβafΛπa−∫0aβa−ηfΛπa−η,ϵ0πηdη,Itda−μ+δIt≥∫0tβa∂fΛπa−∫0aβa−ηfΛπa−η,ϵ0πηdη,∈0∂IdaIt−μ+δIt,where we follow the mean value theorem that
(69)Jt≥∫0tβa∂fΛπa−∫0aβa−ηfΛπa−η,ϵ0πηdη,ϵ0∂IdaIt,for 0≤It≤ϵ0. Taking Laplace transform on both side of (68) yields
(70)Iλ^≥1λ+μ+δ∫0∞βa∂fΛπa−∫0aβa−ηfΛπa−η,ϵ0πηdη,ϵ0∂Ie−λadaIλ^+I0λ+μ+δ≥1λ+μ+δ∫0∞βa∂fΛπa−∫0aβa−ηfΛπa−η,ϵ0πηdη,ϵ0∂Ie−λadaIλ^.
By the arbitrary ofϵ0 and λ, we take ϵ0 and λ which are small enough. Then, (62) contradicts ℛ0>1.Atotal trajectory of the semiflow Φ is a function g:ℝ→ℝ+×L+1×ℝ+ such that Φs,gt=gt+s for all t∈ℝ and all s∈ℝ+. For the definition of a total trajectory, the solution st,a,It satisfies the following property for t∈ℝ and a∈ℝ+.Proposition 4.3.
Letgt be a total trajectory in Γ. Then, st,a is positive and It is positive or identically zero for all t∈ℝ.Proof 7.
First, we show thatst,a is strictly positive for all t∈ℝ and a∈ℝ+. From s equation of system (5), we obtain that
(71)∂st,a∂t+∂st,a∂a≥−μst,a−βafs′Λπa,Λμst,a.
Integrating this equality with the boundary equationst,0=Λ yields
(72)st,a≥Λe−μae−∫0aβsfs′Λπs,Λμds>0.
Second, we claim thatIt is identically zero if there exists some t0 such that It0=0. From the third equation of (5), we follow that
(73)I′t≤∫0∞βafI′st,a,0daIt−μ+δIt.
Solving the above inequality, we have(74)It≤It0e∫t0t∫0∞βafI′sξ,a,0dadξe−μ+δt−t0=0.
By the nonnegativity ofIt, we conclude that It is identically zero for all t≥t0. Similarly, we repeat this process and conclude that It=0 for all t<t0. Consequently, our claim is true.
Finally, we show thatIt is strictly positive if there does not exist some t0∈ℝ such that It0=0. By way of contradiction, we redo the second process and arrive at the strict positivity of It for all t∈ℝ.By Propositions4.2, 2.5, and 4.3, together with Theorem 3.2 in [25], the following theorem holds.Theorem 4.4.
System (5) is uniformly strongly ρ-persistent if ℛ0>1.Corollary 4.1.
Supposeℛ0>1 and I0>0. Let st,a,It be a total trajectory in A. Then, there exists an ε0>0 such that st,a>ε0πa and It>ε0 for all t∈ℝ.Proof 8.
First of all, we claim thatIt>0 for all t∈ℝ. Otherwise, suppose that there exists
(75)t0=supt∈−∞,0t∈ℝ∣It=0.
By the assumption ofI0>0, we conclude that for all t∈0,t0, It>0, and It0=0. Then by the virtue of the function f with respect to I, we have
(76)I′t0=∫0∞βafst0,a,It0da−μ+δIt0=0,which is a contradiction with I′t0>0. We repeat this process and obtain It>0 for all t∈ℝ. In review of Theorem 4.4, we immediately have that there exists some positive constant ϵ1>0 such that It>ϵ1. On the other hand, for such ϵ1>0, we recall the process of 4.3 to obtain st,a≥ϵ2πa for t∈ℝ and a∈R+. Hence, ϵ0=minϵ1,ϵ2 is the value that we are looking for and
(77)st,a≥ϵ0πa,It≥ϵ0.From Theorem 5.7 in [21], we have the following result.Theorem 4.5.
Suppose Assumptions1.1 and 1.2 hold. If the basic reproduction number ℛ0>1, then there exists a compact attractor A that attracts all solutions with initial condition belonging to Γ\Γ0.For constructing the Lyapunov functionals to get the global stability of the steady states, we defineφ:0,∞→ℝ as
(78)φx=x−1−lnx.It is well known thatφ attains a global minimum only at 1 with φ1=0 and φx>0 for x≠1. In order to guarantee a well definition of φ, we give the following proposition.Proposition 4.6.
For alls0·,I0T∈A, the following estimation holds.
(79)st,as∗a≥ϵ0Λ,ItI∗t≥ϵ0I∗,for all t∈ℝ and a∈ℝ+.Proof 9.
This is a direct result from Corollary4.1. We omit the details.Theorem 4.7.
Supposeℛ0>1 and (A3) of Assumption 1.1 hold, then the disease-free steady state E0=s0a,0=Λπa,0 is globally asymptotically stable.Proof 10.
Define a Lyapunov function as(80)Vt=Vst+VIt,where Vst=∫0∞s0aφst,a/s0ada, and VI=It. Deviating Vst along the solution of system (5) yields
(81)dVStdt=∫0∞s0a1s0a−1st,a∂st,s∂tda=∫0∞s0a1s0a−1st,a−∂st,a∂a−μst,a−βafst,a,Itda.
On the other hand,(82)∂φst,a/s0a∂a=∂∂ast,as0a−1−lnst,as0a=∂∂ast,as0a−∂∂alnst,as0a=sa′t,as0a−st,asa0as0a2−s0ast,asa′t,as0a−st,asa0′as0a2=1s0a−1st,asa′t,a−st,asa0′as0a1s0a−1st,asa′t,a+1s0a−1st,aμst,a.
We substitute (82) into (81) and obtain
(83)dVStdt=∫0∞s0a−∂φst,a/s0a∂a−1s0a−1st,aβafst,a,Itda=−s0aφst,as0a0∞−μ∫0∞s0agst,as0ada−∫0∞βa1−s0ast,afst,a,Itda.
DeviatingVIt with respect to time t yields
(84)dVItdt=∫0∞βafst,a,Itda−μ+δIt.
Substituting them into the deviation ofV gives
(85)dVtdt=dVstdt+dVItdt=−μ∫0∞s0aφst,as0ada+∫0∞βas0ast,afst,a,Itda−μ+δIt≤−μ∫0∞s0aφst,as0ada+∫0∞βa∂fs0a,0∂Ida−μ+δIt=−μ∫0∞s0aφst,as0ada+μ+δℛ0−1It,where we use s0a/st,a≤fs0a,It/fst,a,It by (A3) of Assumption 1.1. Therefore, if ℛ0>1, then V′t≤0. Since Vt is bounded on g·, the alpha limit set of g· must be contained in ℳ, the largest invariant subset of dVt/dt=0=E0.In order to establish the global stability of the endemic steady state, we need the additional assumption for nonlinear functionf.Assumption 4.1.
Forst,a>0,
(86)xI∗<s∗afst,a,xst,afs∗a,I∗<1,0<x<I∗,1<s∗afst,a,xst,afs∗a,I∗<xI∗,x≥I∗.Now, we are in the position to prove the following result.Theorem 4.8.
Suppose thatℛ0>1 and Assumption 4.1 hold, then the endemic steady state E∗ is globally asymptotically stable in Γ0.Proof 11.
By Theorem3.2 and Theorem 4.5, it suffices to show A=E∗. Let gt=st,·,It be a total trajectory in A. By Corollary 4.1, there exists ϵ0>0, for any t∈ℝ and a∈ℝ+, such that 0≤φx<ϵ0 for x=st,a/s∗a and It/I∗. Define
(87)Vt=Vst+VIt,where Vst=∫0∞s∗aφst,a/s∗ada and VIt=I∗φIt/I∗. Then Vt is bounded and well defined based on Proposition 4.6.
Next, we show that the upper right derivativedVt/dt along the solution is nonpositive. We firstly have
(88)dVstdt=∫0∞s∗a1s∗a−1st,a∂st,a∂tda=−∫0∞s∗a1s∗a−1st,a∂st,a∂t−∫0∞s∗a1s∗a−1st,aμst,a+βafst,a,Itda.
Furthermore, it follows from(89)∂∂agsjt,as∗a=1s∗a−1st,a∂∂ast,a−st,as∗addas∗a.
Then,(90)dVstdt=−∫0∞s∗a∂∂agst,as∗ada+∫0∞βafs∗a,I∗1−s∗ast,ast,as∗a−fst,a,Itfs∗a,I∗da.
DifferentiatingVIwith respect to t, we obtain
(91)dVItdt=∫0∞βafs∗a,I∗1−I∗Itfst,a,Itfs∗a,I∗−ItI∗da.
Summing updVst/dt and dVIt/dt together with the expressions of st,a and s∗a arrives at
(92)dVtdt=−s∗aφst,ast0∞+∫0∞ds∗adaφst,astda+∫0∞βafs∗a,I∗×s∗ast,a+s∗afst,a,Itst,afs∗a,I∗−ItI∗−I∗Itfst,a,Itfs∗a,I∗da=−μ∫0∞s∗aφst,as∗ada−∫0∞βafs∗a,I∗φst,as∗ada+∫0∞βafs∗a,I∗φs∗ast,a+φs∗afst,a,Itst,afs∗a,I∗−φItI∗−φI∗Itfst,a,Itfs∗a,I∗da=−μ∫0∞s∗aφst,as∗ada−∫0∞βafS∗,J∗φI∗Itfst,a,Itfs∗a,I∗da+∫0∞βafs∗a,I∗φs∗afst,a,Ist,afs∗a,I∗−φItI∗da.
In the following, we show that(93)Ht,a=φs∗afst,a,Itst,afs∗a,I∗−φItI∗.
This, together with Assumption4.1, implies that Ht,a<0and then dVt/dt≤0. So that Vt is nonincreasing. Since Vt is bounded on g·, the ω-limit set of g· must be contained in ℳ, the largest invariant subset of dVt/dt=0. It follows from dVt/dt=0 that st,a=s∗a and s∗afst,a,It/st,afs∗a,I∗=It/I∗. Immediately, we have
(94)fst,a,Itfs∗a,I∗=ItI∗,for t∈ℝ and a∈ℝ+. This, together with the first equation of system (5), yields fs∗a,It=fs∗a,I∗ for all t∈ℝ and a∈ℝ+. By the virtue of fx,y stated in Assumption (A1), we have that It=I∗ for all t. Therefore, ℳ=E∗.
The above analysis indicates that theω-limit set of g⋅ consists of just the endemic steady state E∗ and hence Vgt≥VE∗ for all t∈ℝ. Thus, A=E∗.


## 5. Numerical Simulation

In this section, we carry out numerical simulations to verify our theoretical results. To show the threshold property for system (1), we pick up the parameters as follows:
(95)Λ=0.8,µ=0.01,γ=0.01,δ=0.01,and
(96)βa=β,t≥τ,0,t<τ,where β is varied as our demands. The incidence rate takes in the form of
(97)fx,y=x1+αy,where α is the maturate rate and we fix α=0.5. The initial condition is chosen as
(98)s0a=Λe−µa,I0=1.First, we pick up the transmission rate value asβ=0.0035. Then, we directly estimate the basic reproduction number R0=0.9149<1. Theorem 4.7 indicates the global stability of disease-free equilibrium E0. Figure 1(a) shows that the disease-free steady state E0 decreasingly converges to zero. Second, we take β=0.005 and calculate R0=1.3016>1. Theorem 4.8 and Figure 1(b) show that the endemic steady state E∗s∗a,I∗ is asymptotically stable.Figure 1
The evolution of the density of the infectedIt with parameters listed in text. (a) β=0.00035 and R0≃0.9149<1. (b) β=0.0005 and ℛ0≈1.3016.
(a)
(b)


## 6. Discussion

In this paper, we proposed an SIR model with age structure in susceptibility and a general incidence rate. We gave the existence, uniqueness, and regularity of the solution of system (5). We calculated the basic reproduction number, which is the spectral radius of the next-generation operator. Based on the assumptions on the incidence function and the parameters, we obtain the asymptotical smoothness of the system and existence, uniqueness, local stability, and global stability of the steady states. The interesting thing is that the global behavior of system (5) is rigorously proved by constructing Lyapunov functional method.Incidence rate plays an important role in investigating the disease transmission pattern. As mentioned in Introduction, bilinear type and nonlinear type had been studied in many literatures. We used a general functionfs,I to extend the two classical types and so it can cover some existing results [8, 9]. Some complex dynamics (Hopf bifurcation [26], backward bifurcation [27], homoclinics [28], and so on) inducing by complex incidence rate had been investigated. While in this paper, we just focus on the global dynamics based on some rigorous limitations for the nonlinear incidence rate fs,I. Oscillation is a nature phenomenon in realistic life, and exploring the intrinsic mechanism has been become a hot topic in epidemic modelling processes. In the future, we will turn our focus on this aspect to discover some substantial mechanisms to result in disease oscillations.



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*Source: 9613807-2018-10-14.xml*