Transmission dynamics of an SIS model with age structure on heterogeneous networks

# Transmission dynamics of an SIS model with age structure on heterogeneous networks

## Abstract

Age at infection is often an important factor in epidemic dynamics. In this paper a disease transmission model of SIS type with age dependent infection on a heterogeneous network is discussed. The model allows the infectious rate and the recovery rate to vary and depend on the age of the infected individual at the time of infection. We address the threshold property of the basic reproduction number and present the global dynamical properties of the disease-free and endemic equilibria in the model. Finally, some numerical simulations are carried out to illustrate the main results. The combined effects of the network structure and the age dependent factor on the disease dynamics are displayed.

Key words:  Basic reproduction number, age-structure, scale-free network, global stability.

## 1 Introduction

Infectious diseases remain a major challenge for human society. Epidemic diseases (cholera, tuberculosis, SARS, influenza, Ebola virus, etc.) continue to have both a major impact on human beings and an economic cost to society. Any gain we make in understanding the dynamics and control of epidemic transmission therefore has potential for significant impact — and hence has been the focus of scientific research and attracted much attention [1, 2].

Epidemic dynamic models provide a theoretical method for quantitative studies of infectious diseases. Since Kermack and Mckendrick proposed two fundamental epidemic models, the SIS and SIR compartmental models, to study disease transmission [3, 4], epidemic models for the transmission of infectious diseases have been studied extensively. These classical compartmental models are important tools in analyzing the spread and control of infectious diseases, but usually neglect the population structure or assume that all the individuals have the same possibility to contact the others — they are most effective for well-mixed homogeneous populations with a substantial penetration of infection. However, the spreading of infectious diseases is primarily via specific contacts between individuals, emerging diseases start with a relative small number of infectives, and the possibility to contact others is heterogeneous. Therefore, depicting the spread of disease processes on a contact network can be more realistic [5, 6, 7, 8]. Currently, the most popular transmission models on networks are based on mean-field approximations and follow the framework initially proposed by Pastor-Satorras and Vespignani [9, 10, 11]. They were the first to study SIS and SIR epidemic models on a scale-free network and showed that the epidemic threshold is infinitesimal in the limit of a large number of links and nodes. Since then, a great deal of epidemiological research work followed on scale-free (and other) networks [12, 13, 14, 15, 16, 17].

For some epidemic diseases, such as scarlet fever, poliomyelitis and HFMD (hand-foot-and-mouth), the process of their transmission is related to age and some public health and preventative policies for those diseases depend on the age structure of host population. Hence, in order to reflect the effect of demographic behavior of individuals, researchers have begun to examine age-structured epidemic models. The pioneering work in age-structured epidemic models was that of Hoppensteadt [18, 19], since then, the importance of age structure in epidemic models has been recently stressed by many authors[20, 21, 22, 23]. Although age-dependent epidemic models have been studied extensively, all these models were established on homogeneous networks — in essence a convenient approximation to the homogeneous well-mixed population. There are still few significant results concerning age-structured epidemic models on complex (in this case, scale-free) networks.

The main purpose of this paper is to obtain threshold results for an age-structured epidemic model on scale-free networks. A scale-free network is characterized by a power-law degree distribution  [24], where is the probability that a randomly chosen node has degree , and is a characteristic exponent whose value is usually in the range . We know that for many infectious diseases, transmission can be studied by using the SIS model with S and I representing the susceptible and infected individuals, respectively. Based on the SIS model with age structure, our work provides new insight into epidemic spreading dynamics.

The organization of this paper is as follows. In Section 2, we present our age-of-infection model and give some description and assumptions. In Section 3, we analyze the existence of equilibria and obtain the basic reproduction number. We then present some preliminaries for the analysis of stability, which includes asymptotic smoothness of the semi-flow generated by the system and the uniform persistence of the system. The main results of this paper are given in Section 4, which include the local stability and global stability of the disease-free and endemic equilibria. Some numerical analysis are performed in Section 5. Finally, in Section 6, we give conclusions and discussions.

## 2 Formulation of the model

Consider a population with connectivity modelled as a complex network N, where each node of N is either vacant or occupied by one individual. In an epidemic spreading process, every node has three optional states: vacant state, susceptible state, and, infected state [25, 26]. In order to consider the heterogeneity of contacts, we divide the population into  groups. Let , , denote the densities of susceptible and infected nodes (individuals) with connectivity (degree) at time , respectively, and let denote the density of infected individuals with respect to the age of infection at time . It is obvious that

 Ik(t)=∫∞0Ik(t,τ)dτ

Noting that , which describes the total density of the individuals with degree at time , then, the density of the vacant nodes with degree is .

In addition, as a disease spreads, a birth event occurs at a vacant node next to a non-vacant node at rate , that is to say, the empty nodes will give birth to new individuals once one of their neighbours is occupied. Thus, the birth process depends on the number of neighboring individuals. All individuals die at rate , causing the occupied node becomes vacant. Let , represent infectious function and removal function with respect to age of infection respectively. Therefore, the SIS epidemic model with the age-of-infection structure on a heterogeneous network is formulated as follows:

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩dSk(t)dt=bk[1−Nk(t)]Ψk−μSk(t)−kSk(t)∫∞0β(τ)Θ(t,τ)dτ+∫∞0γ(τ)Ik(t,τ)dτ∂Ik(t,τ)∂t+∂Ik(t,τ)∂τ=−(μ+γ(τ))Ik(t,τ),                          0<τ<∞Ik(t,0)=kSk(t)∫∞0β(τ)Θk(t,τ)dτ,                                0≤t<∞ (2.1)

where

 Θk(t,τ)=n∑i=1p(i|k)φ(i)iIi(t,τ),  Ψk(t)=n∑i=1p(i|k)AiNi(t)

under the following initial conditions:

 {Sk(0)=Sk0≥0 ,Ik(0,τ)=Ik0(τ)∈L1+(0,∞),(k=1,2,...n);Sk0+∫∞0Ik0(τ)dτ=Nk0(k=1,2,...n) (2.2)

where  is the space of functions on that are nonnegative and Lebesgue integrable.

The meaning of the parameters and variables of the above model are as follows:

• Let  and   be positive constants denoting the birth and natural death rates of all individuals. The additional death rate induced by the infectious disease is not considered.

• is the average degree of the network, i.e., . For a general function , this is defined as . Let be the probability that a node of degree is connected to a node of degree . In present paper, we primarily study epidemic transmissions on uncorrelated networks, the probability is considered independent of the connectivity of the node from which the link is emanating. Therefore, .

•  describes the probability of a link pointing to an infected individual of age . We note that is the infectivity of nodes with degree , i.e., it denotes the average number of edges from which a node with degree can transmit the disease. Thus, represents newly infected individuals per unit time.

• is the probability of fertility contacts between nodes with degree and its neighbours with degree . The factor accounts for the probability that one of the neighboring individual of a vacant node with degree , will activate this vacant node at the present time step. It is assumed that, at each time step, every individual generates the same birth contacts , here  [26]. Therefore, represents density of new born individuals per unit time.

Next we make the following assumptions on parameters, which are thought to be biologically relevant.

###### Assumption 2.1

Consider the system (2.1), we assume that,

1. ;

2. , with respective essential upper bounds and . Furthermore, there exists a constant such that for all ;

3. are Lipschitz continuous on with Lipschitz coefficients and , respectively;

4. For all and any k, . Furthermore, .

Let us define a functional space for system (2.1),

 X=X1×X2×…×Xn,  Xi=R1+×L1+, i=1,2,⋯n.

Note that is a closed subset of a Banach space, and hence is a complete metric space. The norm on is taken to be

 ∥Xk∥=|Sk(t)|+∫∞0|Ik(t,τ)|dτ

By applying tools from [27, 28] and following from Assumption 2.1, it can be verified that the solution of system (2.1) exists and is nonnegative for any initial conditions. Thus, for define a continuous flow : of system (2.1) such that

 Ξ(t,X0)=ϱ(t,X0)=(S1(t),I1(t,τ),S2(t),I2(t,τ),…,Sn(t),In(t,τ))

where is the solution of the model (2.1) with initial condition .

From the model (2.1), we get that  satisfies the following differential equation,

 dNk(t)dt=bk[1−Nk(t)]Ψ−μNk(t) (2.3)

Let ,we get , which corresponds to the equilibrium solution of extinction, and another solution satisfies

 Nk=bkΨμ+bkΨ (2.4)

Putting the above equation  (2.4) to , we obtain

 Ψ=1⟨k⟩∑iip(i)bΨμ+biΨ

Noting that

 f(Ψ)=1−1⟨k⟩∑ibip(i)μ+biΨ

it is clear that , . Thus, has a unique positive solution if and only if . That is ,when , the equation (2.3) has a unique positive solution , which satisfies

 N∗k=bkΨ∗μ+bkΨ∗ , Ψ∗=1⟨k⟩n∑i=1p(i)N∗i(t)

Therefore, from  (2.3) and [38], when , there is , the population becomes extinct and there is no other dynamic behaviors any more. While, when , . Therefore, we only consider the condition of in the following sections.

Since there are the same long-playing behaviors between the original system and the limiting system. To study the stability of system (2.1), we consider the limiting system under which as follows,

 ⎧⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪⎩dSk(t)dt=bk[1−N∗k]Ψ∗k−μSk(t)−kSk(t)∫∞0β(τ)Θ(t,τ)dτ+∫∞0γ(τ)Ik(t,τ)dτ∂Ik(t,τ)∂t+∂Ik(t,τ)∂τ=−(μ+γ(τ))Ik(t,τ)Ik(t,0)=kSk(t)∫∞0β(τ)Θ(t,τ)dτ (2.5)

Finally, we define the state space for system (2.5) as

 Γ={(S1(t),I1(t,⋅),⋯,Sn(t),In(t,⋅))∈X:0 ≤ Sk(t)+∫∞0Ik(t,τ)dτ≤ 1,k=1,⋯,n}

The following proposition shows that is positively invariant with respect to system (2.5) for .

###### Proposition 2.1

is positively invariant of system (2.5) for . Moreover, is point dissipative (there exists a bounded set which attracts all points in ). That is , , for any the solution of system (2.5), noting as ,with the initial condition  . Then  .

Proof    Define the arbitrary initial condition , which satisfies (2.2). Thus, . From (2.3), we get

 dΨ(t)dt=(b−μ)Ψ(t)−bΨ(t)⟨k⟩Σiip(i)Ni(t)

It is obvious that . It follows from (2.4) that , and

 0≤ Sk(t)+∫∞0Ik(t,τ)dτ=Nk=bkΨμ+bkΨ ≤ 1

Therefore, is point dissipative and attracts all points in X . This completes the proof.

According to Assumption 2.1 and the above results, we have the following proposition.

###### Proposition 2.2

There exists a constant satisfied , then the following statements hold true for and k(k=1,2,…n):
(1) ;
(2) , and
(3) The function and are Lipschitz continuous with coefficient , on .

## 3 Preliminaries

### 3.1 Equilibria and the basic reproduction number

Firstly, with the above boundary conditions and initial conditions (2.2) , we obtain by integrating the second differential equation in  (2.5) along the characteristic line  [29],

 Ik(t,τ)=⎧⎨⎩Ik(t−τ,0)H(τ),    t≥τ;Ik(0,τ−t)H(τ)H(τ−t)  0

Where .
Next, we consider the steady states of system (2.5). For simplicity, we note . System (2.5) always has a disease-free equilibrium .

Then, let us investigate the positive equilibrium of system (2.5). Any positive equilibrium should satisfy the following equations,

 ⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩bk[1−N∗k]Ψ∗−μS∗k−kS∗k∫∞0β(τ)Θ∗(τ)dτ+∫∞0γ(τ)I∗k(τ)dτ=0dI∗k(τ)dτ=−(μ+γ)I∗k(τ)Z∗k=kS∗k∫∞0β(τ)Θ∗(τ)dτΘ∗(τ)=1⟨k⟩∑ni=1φ(i)p(i)I∗i(τ) (3.2)

For ease of notation, let

 K1(λ)=∫∞0β(τ)e−λτH(τ)dτ,   K2(λ)=∫∞0γ(τ)e−λτH(τ)dτ

We will get , which satisfies

 I∗k(τ)=I∗k(0)H(τ)=Z∗kH(τ)
 S∗k=⟨k⟩Z∗kkK1(0)∑φ(i)p(i)Z∗i,  Z∗k=bk[1−N∗k]Ψ∗−μS∗k1−K2(0)

To make sure that , and if and only if . It is clear that

 K2(0)=∫∞0γ(τ)e−μτ−∫τ0γ(ξ)dξdτ≤∫∞0γ(τ)e−∫τ0γ(ξ)dξdτ=1−e−∫∞0γ(ξ)dξ<1

Hence, we obtain .

From the above analysis, we get the following theorem.

###### Theorem 3.1

Define the basic reproduction number as follows,

 R0=K1(0)n∑i=1iφ(i)p(i)S0i

If , the system  (2.5) has a unique disease-free equilibrium ; if , there exist two equilibria and , which satisfy .

### 3.2 Asymptotic smoothness

In order to prove the global stability of model (2.5), we need to make the following preparations. First, we establish asymptotic smoothness of the semigroup . The semigroup is asymptotically smooth, if, for any nonempty, closed and bounded set for which (t,B) B, there is a compact set such that attracts . In order to obtain it, we will need the following lemmas and proposition.

###### Lemma 3.1 ([30])

For each , suppose has the property that is completely continuous and there is a continuous function :: such that as  and if . Then , is asymptotically smooth.

###### Lemma 3.2 ([31])

Let be closed and bounded where . Then is compact if and only if the following conditions hold:
(i) uniformly for . ( if ).
(ii) uniformly for .

###### Proposition 3.1

Let . For , suppose that is a bounded Lipschitz continuous function with bound and Lipschitz coefficient . Then the product function is Lipschitz continuous with coefficient .

From the above two lemmas, we have the following theorem.

###### Theorem 3.2

The semigroup is asymptotically smooth.

Proof    To apply Lemma 3.1, we define the projection of about any bounded set of by decomposing into the following two operators,

 Ξ(t)=U(t)+C(t)

where,

 U(t)=(0,U1(t),0,U2(t),…,0,Un(t)),  C(t)=(S1(t),C1(t),S2(t),C2(t),…,Sn(t),Cn(t))
 Ui(t)=⎧⎨⎩Ii(0,τ−t)H(τ)H(τ−t)   τ≥t0                               τ

From equation (3.1), it is easy to get . Then,

 ∥Ui(t)∥ = ∫∞0|Ui(t)|dτ=∫∞tIi(0,τ−t)H(τ)H(τ−t)dτ = ∫∞tIi(0,τ−t)e−μt−∫ττ−tγ(ξ)dτdτ ≤ e−(μ+α)t∫∞tIi(0,τ−t)dτ ≤ e−(μ+α)t∥Ii(0,τ)∥

If , we note , then, as and for any .

Next, we verify that is completely continuous. We need to pay more attention to the state space, since is a component of our state space . Hence a notion of compactness in is necessary. In an infinite dimensional Banach space, boundedness does not necessarily imply precompactness. Hence, we need to prove it by applying Lemma3.2.

Suppose that is bounded for any initial condition . From Proposition 2.1, it is easy to see that remains in the compact set . Thus, we only need to verify that the following conditions valid for remaining in a precompact subset of .

To check condition , according to (3.3) ,

 Ci(t,τ) = iSi(t−τ)∫∞0β(τ)1⟨k⟩n∑k=1φ(k)p(k)Ik(t−τ,τ)dτH(τ) = iSi(t−τ)1⟨k⟩n∑i=1φ(k)p(k)∫∞0β(τ)Ik(t−τ,τ)dτH(τ) ≤ i⟨φ(k)⟩⟨k⟩¯¯¯βΛ2e−(μ+α)τ

Note that for all , . Therefore, is satisfied for the set .
To check condition (i), for sufficiently small , we observe

 ∫∞0|Ci(t,τ)−Ci(t,τ+h)|X0dτ = ∫t−h0|Zi(t−τ−h)H(τ+h)−Zi(t−τ)H(τ)|dτ+∫tt−h|Zi(t−τ)H(τ)|dτ ≤ ∫t−h0Zi(t−τ−h)|H(τ+h)−H(τ)|dτ+∫t−h0|Zi(t−τ−h)−Zi(t−τ)|H(τ)dτ + ∫tt−hH(τ)iSi(t−τ)∫∞0β(τ)1⟨k⟩∑kφ(k)p(k)Ik(t−τ,τ)|dτ

It is clear that , and is a decreasing function.

 ∫t−h0|H(τ+h)−H(τ)|dτ = ∫t−h0H(τ)dτ−∫t−h0H(τ+h)dτ = ∫t−h0H(τ)dτ−∫thH(τ)dτ = ∫h0H(τ)dτ−∫tt−hH(τ)dτ≤h

Then, we note

 Δ=∫t−h0H(τ)|Zi(t−τ−h)−Zi(t−τ)|dτ

From Proposition 2.2 and Proposition 3.1, we have

 Δ = ∫t−h0H(τ)|Zi(t−τ−h)−Zi(t−τ)|dτ = i∑kφ(k)p(k)⟨k⟩∫t−h0H(τ)|Si(t−τ−h)∫∞0β(τ)Ik(t−τ−h,τ)dτ − Si(t−τ)∫∞0β(τ)Ik(t−τ,τ)dτ|dτ ≤ i⟨φ(k)⟩⟨k⟩[MβΛ+¯¯¯βΛMs]hμ+α

is the coefficient of Lipschitz continuous function . It is easy to see that is bounded, and therefore, is Lipschitz on with coefficient .

 ∫∞0|Ci(t,τ)−Ci(t,τ+h)|X0dτ≤Λh+i⟨φ(k)⟩⟨k⟩[MβΛ+¯¯¯βΛMs]hμ+α+i⟨φ(k)⟩⟨k⟩¯¯¯βΛ2h

This converges uniformly to 0 as . Therefore, the condition (i) is verified for . From Lemma 3.2, we have that is completely continuous. Finally, according to Lemma 3.1, we conclude that is asymptotically smooth. This completes the proof.

Next, we show that the solution semigroup has a global compact attractor in .

We first give the following definition of global attractors.

###### Definition 1 ([30])

A set in is defined to be an attractor if is non-empty, compact and invariant, and there exists some open neighborhood of in such that attracts . A global attractor is defined to be an attractor which attracts every point in .

From the results above, we can get the existence of a global attractor by applying the following Lemma.

###### Lemma 3.3 ([30])

If is asymptotically smooth and point dissipative in , and orbits of bounded sets are bounded in , then there is a global attractor in .

Propositions 2.1 and Theorem 3.2 show that the semigroup of system  (2.5) is asymptotically smooth and point dissipative on the state space . The proof of Proposition 2.1 can verify that every forward orbit of bounded sets is bounded in . Therefore, by Lemma 3.3, we have the following theorem.

###### Theorem 3.3

The semigroup generated by the system (2.5) on the state space has a global attractor in .

### 3.3 Uniform Persistence

In this section we study the uniform persistence of system (2.5). Let us define a function that as

 ρk(X)=Zk(t),  k=1,2,…n

Before introducing the result of persistence, we introduce the following important lemmas.

###### Lemma 3.4 (Fatou’s Lemma)

Let be a non-negative measurable function sequence, then it satisfies

 ∫limt→+∞inffn≤limt→+∞inf∫fn≤limt→+∞sup∫fn≤∫limt→+∞supfn (3.4)
###### Lemma 3.5 (Fluctuation Lemma)

Let

 limt→+∞supΦ(t)=Φ∞, limt→+∞infΦ(t)=Φ∞

and  be a bounded and continuously differentiable function. Then there exist sequences and such that , . , as .

###### Lemma 3.6

If , then there exists a positive constant , such that for any ,

 limsupt→∞Zk(t)>ε (3.5)

Proof    If , there exists a sufficiently small such that

 1n∑i=1iφ(i)p(i)bi[1−N∗i]Ψ∗−εμ∫∞0β(τ)H(τ)dτ>1 (3.6)

We now show that this small is the in (3.5). We will do this by contradiction. Assume that there exists a constant which is sufficiently large such that

 Zk(t)≤ε          for t≥ T

Together with (2.5), we have

 dSk(t)dt = bk[1−N∗k]Ψ∗−μSk(t)−Zk(t)+∫∞0γ(τ)Ik(t,τ)dτ ≥ bk[1−N∗k(t)]Ψ∗−μSk−ε

Then , according to the comparison principle,

 Sk(t)≥bk[1−N∗k(t)]Ψ∗−εμ         for t≥ T (3.7)

Furthermore, it follows from  (3.1) together with (3.7) and the Fluctuation Lemma, if