The Analysis of Epidemic Network Model with Infectious Force in Latent and Infected Period

Abstract

We discuss the epidemic network model with infectious force in latent and infected period. We obtain the basic reproduction number and analyze the globally dynamic behaviors of the disease-free equilibrium when the basic reproduction number is less than one. The effects of various immunization schemes are studied. Finally, the final sizes relation is derived for the network epidemic model. The derivation depends on an explicit formula for the basic reproduction number of network of disease transmission models.

1. Introduction

Disease spreading has been the subject of intense research since long time ago. With the advent of modern society, fast transportation systems have changed human habits, and some diseases that just a few years ago would have produced local outbreaks are nowadays a global threat for public health systems. A recent example is given by influenza A(H1N1). In order to understand the mechanism of diseases spreading and other similar processes, such as rumors spreading, networks of movie actor collaboration and science collaboration, WWW, and the Internet, it is of great significance to inspect the effect of complex networks features on disease spreading. Therefore, it is of utmost importance to carefully take into account as much details as possible of the structural properties of the network on which the infection dynamics occurs. And in the general case, the epidemic system can be represented as a network where nodes stand for individuals and an edge connecting two nodes denotes the interaction between individuals. The degree 𝑘 of a node is the number of its neighbors, that is, the number of links adjacent to the node.

In the past, researchers mainly focused the disease transmission study on the conventional networks [1, 2] such as lattices, regular tree, and ER random graph. Since late 1990s, scientists have presented a series of statistical complex topological characteristics [3–6] such as the small-world (SW) phenomenon [7] and scale-free (SF) property [8] by investigating many real networks. On scale-free networks, it was assumed that the larger the node degree, the greater the infectivity of the node, and the infectivity is just equal to the node degree. Under such an assumption, for instance, Pastor-Satorras et al. concluded that the epidemic threshold 𝜆_𝑐=0 for heterogenous networks with sufficiently large size [9]. Subsequently, the studies of dynamical processes on complex networks also have attracted lots of interests with various subjects [10–15], and as one of the typical dynamical processes built on complex networks, epidemic spreading has been investigating intensively once more. The susceptible-infected-susceptible (SIS) [10, 11], susceptible-infected-recovered (SIR) [16, 17], and susceptible-infected (SI) [18–20] models on complex networks have been extensively studied recently.

In this paper, we will establish the susceptible-exposed-asymptomatically infected, symptomatically infected-recovered(SEAIR) epidemic model on a network. We provide a detailed analytical of the SEAIR model on complex networks. The remainder of this paper presents our model and results. In the next section, we describe the epidemic model on networks with infectious force in latent and infected period. The subsequent section is devoted to determine the stability of the disease-free equilibrium. In Section 4, we consider two models of immunization. In Section 5, we derive the final sizes relation of the network epidemic models.

2. Model and Parameters

We propose a SEIAR model by classifying the population as susceptible (𝑆), exposed (𝐸), asymptomatically infected (𝐴), symptomatically infected (𝐼), and removed/immune (𝑅), for example, the spreading process of H1N1. The asymptomatic infected compartment contains those who fail to show noticeable symptoms or with light flu-like symptoms but are not identified, and are able to spread the H1N1 infection. We also assume that a susceptible individual becomes infected if in contact with an exposed, asymptomatically or symptomatically infective individual. Then the susceptible enters the exposed class 𝐸 of those in the latent period. After the latent period, the individual enters the class 𝐼 or 𝐴 of infectives, who are infectious in the sense that they are capable of transmitting the infection. When the infectious period ends, the individual enters the recovered class 𝑅. We assume that a removed individual will never become susceptible or infected again. In our model, new births, natural deaths and migrations are ignored. The flow diagram of the individuals is depicted in Figure 1.

Figure 1 

Flow diagram.

In contrast to classical compartment models, we consider the whole population and their contacts in networks, each person in a community can be regarded as a vertex in the network, and each contact between two individuals is represented as an edge (line) connecting their vertices. The number of edges emanating from a vertex, that is, the number of contacts a person has, is called the degree of the vertex. We classify the population into groups based on the number of contacts the individual can make per unit of time. The densities of susceptible, exposed, asymptomatically infected, symptomatically infected,vaccinated and recovered nodes of degree 𝑘 at time 𝑡, denoted by 𝑆_𝑘(𝑡), 𝐸_𝑘(𝑡), 𝐴_𝑘(𝑡), 𝐼_𝑘(𝑡), and 𝑅_𝑘(𝑡) respectively. Clearly, these variables obey the normalization condition: 𝑆_𝑘(𝑡)+𝐸_𝑘(𝑡)+𝐴_𝑘(𝑡)+𝐼_𝑘(𝑡)+𝑅_𝑘(𝑡)=1, (2.1)𝑘=1,2,…,𝑛, 𝑛 is the maximum number of contact each individual can make. Therefore the network model with infectious force in latent and infected period follow 𝑑𝑆_𝑘(𝑡)𝑑𝑡=−𝜆₁𝑘𝑆_𝑘(𝑡)Θ^𝐸(𝑡)−𝜆₂𝑘𝑆_𝑘(𝑡)Θ^𝐴(𝑡)−𝜆₃𝑘𝑆_𝑘(𝑡)Θ^𝐼(𝑡),𝑑𝐸_𝑘(𝑡)𝑑𝑡=𝜆₁𝑘𝑆_𝑘(𝑡)Θ^𝐸(𝑡)+𝜆₂𝑘𝑆_𝑘(𝑡)Θ^𝐴(𝑡)+𝜆₃𝑘𝑆_𝑘(𝑡)Θ^𝐼(𝑡)−𝛿𝐸_𝑘(𝑡),𝑑𝐴_𝑘(𝑡)𝑑𝑡=𝑞𝛿𝐸_𝑘(𝑡)−𝛼₁𝐴_𝑘(𝑡),𝑑𝐼_𝑘(𝑡)=𝑑𝑡1−𝑞𝛿𝐸_𝑘(𝑡)−𝛼₂𝐼_𝑘(𝑡),𝑑𝑅_𝑘(𝑡)𝑑𝑡=𝛼₁𝐴_𝑘(𝑡)+𝛼₂𝐼_𝑘(𝑡), (2.2) where Θ^𝑓(𝑡)=(1/⟨𝑘⟩)∑_𝑘𝑘𝑃(𝑘)𝑓_𝑘(𝑡), 𝑓=𝐸,𝐴,𝐼, ⟨𝑘⟩=∑_𝑘𝑘𝑃(𝑘). The factor Θ(𝑡) represents the probability that any given link points to an infected node. It is assumed that all parameters are positive constants and summarized in the following list:𝜆₁, 𝜆₂, 𝜆₃: transmission coefficient between communities 𝑆_𝑘 and 𝐸_𝑖, 𝐴_𝑖, 𝐼_𝑖, 𝑖=1,2,…,𝑛;𝛿: the transfer rate between the exposed and the infectious;𝑞, 1−𝑞: rate of becoming asymptomatically infected and symptomatically infected;𝛼₁, 𝛼₂: recovery rate of asymptomatically infected and symptomatically infected.

The mathematical formulation of the epidemic problem is completed given initial conditions such as 𝑆_𝑘(0)≃1, 𝐼_𝑘(0)≃0, 𝐸_𝑘(0)=𝐴_𝑘(0)=𝑅_𝑘(0)=0.

3. Global Stability of the Model and Basic Reproduction Number

In this section, we will show the basic reproduction number 𝑅₀ and the global stability of the disease-free equilibrium. Following van den Driessche and Watmough [21], we note that only compartments 𝐸_𝑘, 𝐴_𝑘, and 𝐼_𝑘 are involved in the calculation of 𝑅₀. At the disease-free equilibrium 𝑃⁰(1,0,0,0,0,…,1,0,0,0,0), the rate of appearance of new infections 𝐹 and the rate of transfer of individuals out of the two compartments 𝑉 are given by 𝐹=𝐹^𝑛×𝑛_𝑖𝑗3×3, (3.1) where 𝐹^𝑛×𝑛₂₁, 𝐹^𝑛×𝑛₂₂, 𝐹^𝑛×𝑛₂₃, 𝐹^𝑛×𝑛₃₁, 𝐹^𝑛×𝑛₃₂, and 𝐹^𝑛×𝑛₃₃ are zero matrices, 𝐹^𝑛×𝑛₁₁=(𝜆₁/⟨𝑘⟩)𝑇^𝑛×𝑛, 𝐹^𝑛×𝑛₁₂=(𝜆₂/⟨𝑘⟩)𝑇^𝑛×𝑛, and 𝐹^𝑛×𝑛₁₃=(𝜆₃/⟨𝑘⟩)𝑇^𝑛×𝑛, where 𝑇^𝑛×𝑛=𝑃(1)(2)(𝑛)(1)22𝑃⋯𝑛𝑃2𝑃²𝑃(2)(𝑛)(1)(2)⋯2𝑛𝑃⋮⋮⋮𝑛𝑃2𝑛𝑃⋯𝑛²𝑃(𝑛), (3.2) and 𝑉=(𝑉^𝑛×𝑛_𝑖𝑗)_3×3, where 𝑉^𝑛×𝑛₁₂, 𝑉^𝑛×𝑛₁₃, 𝑉^𝑛×𝑛₂₃, and 𝑉^𝑛×𝑛₃₂ are zero matrices, 𝑉^𝑛×𝑛₁₁=𝛿0⋯00𝛿⋯0⋮⋮⋱⋮00⋯𝛿,𝑉^𝑛×𝑛₂₁=−𝑞𝛿0⋯00−𝑞𝛿⋯0⋮⋮⋱⋮00⋯−𝑞𝛿,𝑉^𝑛×𝑛₂₂=𝛼₁0⋯00𝛼₁⋯0⋮⋮⋱⋮00⋯𝛼₁,𝑉^𝑛×𝑛₃₁=−𝛿1−𝑞𝛿0⋯00−1−𝑞𝛿⋯0⋮⋮⋱⋮00⋯−1−𝑞,𝑉^𝑛×𝑛₃₃=𝛼₂0⋯00𝛼₂⋯0⋮⋮⋱⋮00⋯𝛼₂. (3.3) According to the concepts of next generation matrix and the basic reproduction number presented in [21], then, the reproduction number 𝑅₀=𝜌(𝐹𝑉⁻¹) for (2.2), where 𝜌 represents the spectral radius of the matrix.

To determine the spectral radius of 𝐹𝑉⁻¹, we first represent the inverse of 𝑉 by the following matrix 𝑉⁻¹=(𝑉^{−1𝑛×𝑛}_𝑖𝑗)_3×3, where 𝑉^{−1𝑛×𝑛}₁₂, 𝑉^{−1𝑛×𝑛}₁₃, 𝑉^{−1𝑛×𝑛}₂₃, 𝑉^{−1𝑛×𝑛}₃₂ are zero matrices, 𝑉^{−1𝑛×𝑛}₁₁=1𝛿010⋯0𝛿1⋯0⋮⋮⋱⋮00⋯𝛿,𝑉^{−1𝑛×𝑛}₂₁=𝑞𝛼₁0𝑞0⋯0𝛼₁𝑞⋯0⋮⋮⋱⋮00⋯𝛼₁,𝑉^{−1𝑛×𝑛}₂₂=1𝛼₁010⋯0𝛼₁1⋯0⋮⋮⋱⋮00⋯𝛼₁,𝑉^{−1𝑛×𝑛}₃₁=1−𝑞𝛼₂00⋯01−𝑞𝛼₂⋯0⋮⋮⋱⋮00⋯1−𝑞𝛼₂,𝑉^{−1𝑛×𝑛}₃₃=1𝛼₂010⋯0𝛼₂1⋯0⋮⋮⋱⋮00⋯𝛼₂. (3.4)

Let 𝐶=𝐹𝑉⁻¹, we have 𝐶=(𝐶^𝑛×𝑛_𝑖𝑗)_3×3, where 𝐶^𝑛×𝑛₂₁, 𝐶^𝑛×𝑛₂₂, 𝐶^𝑛×𝑛₂₃, 𝐶^𝑛×𝑛₃₁, 𝐶^𝑛×𝑛₃₂, and 𝐶^𝑛×𝑛₃₃ are zero matrices, 𝐶^𝑛×𝑛₁₁=(1/⟨𝑘⟩)[𝜆₁/𝛿+𝜆₂𝑞/𝛼₁+𝜆₃(1−𝑞)/𝛼₂]𝑇^𝑛×𝑛, 𝐶^𝑛×𝑛₁₂=(𝜆₁/𝛼₁⟨𝑘⟩)𝑇^𝑛×𝑛, 𝐶^𝑛×𝑛₁₃=(𝜆₂/𝛼₂⟨𝑘⟩)𝑇^𝑛×𝑛.

Now we are ready to compute the eigenvalues of the matrix 𝐶=𝐹𝑉⁻¹. Obviously, the 𝐶 and 𝐶^𝑛×𝑛₁₁ have same spectral radius. Since matrix 𝐶^𝑛×𝑛₁₁ has rank 1, the spectral radius 𝜌(𝐶^𝑛×𝑛₁₁) is equal to the trace of matrix 𝐶^𝑛×𝑛₁₁.

Therefore, the basic reproduction number 𝑅₀ is 𝑅₀=𝜆₁𝛿+𝜆₂𝑞𝛼₁+𝜆₃1−𝑞𝛼₂𝑘².⟨𝑘⟩ (3.5)

In summary, we have the following theorem.

Theorem 3.1. If 𝑅₀<1, the infection-free equilibrium 𝑃⁰(1,0,0,0,0,…,1,0,0,0,0) is locally asymptotically stable, and if 𝑅₀>1, the infection-free equilibrium 𝑃⁰ is unstable, an epidemic ensues.

Next, we will prove the global asymptotic stability of the infection-free equilibrium.

Theorem 3.2. If 𝑅₀<1, the infection-free equilibrium 𝑃⁰(1,0,0,0,0,…,1,0,0,0,0) is global asymptotic stability.

Proof. Let us consider the Lyapunov function 𝐿(𝑡)=_𝑘𝑎_𝑘𝑆_𝑘(𝑡)+𝑏_𝑘𝐸_𝑘(𝑡)+𝑐_𝑘𝐴_𝑘(𝑡)+𝑑_𝑘𝐼_𝑘(𝑡), (3.6) where 𝑎_𝑘=2𝑘𝑃(𝑘), 𝑏_𝑘=3𝑘𝑃(𝑘), 𝑐_𝑘=(𝑘/𝑞)𝑃(𝑘), and 𝑑_𝑘=(𝑘/(1−𝑞))𝑃(𝑘). We now compute the time derivative of 𝐿(𝑡) along the solutions of (2.2). It is seen that 𝐿^′(𝑡)=−2_𝑘𝑘𝑃(𝑘)𝜆₁𝑘𝑆_𝑘Θ^𝐸+𝜆₂𝑘𝑆_𝑘Θ^𝐴+𝜆₃𝑘𝑆_𝑘Θ^𝐼+3_𝑘𝑘𝑃(𝑘)𝜆₁𝑘𝑆_𝑘Θ^𝐸+𝜆₂𝑘𝑆_𝑘Θ^𝐴+𝜆₃𝑘𝑆_𝑘Θ^𝐼−𝛿𝐸_𝑘+_𝑘𝑘𝑃(𝑘)𝛿𝐸_𝑘−𝛼₁𝑞𝐴_𝑘+_𝑘𝑘𝑃(𝑘)𝛿𝐸_𝑘−𝛼₂𝐼1−𝑞^𝑘=_𝑘𝑘𝑃(𝑘)𝑆_𝑘𝜆₁𝑘Θ^𝐸+𝜆₂𝑘Θ^𝐴+𝜆₃𝑘Θ^𝐼−_𝑘𝛿𝑘𝑃(𝑘)𝐸_𝑘−_𝑘𝛼₁𝑞𝑘𝑃(𝑘)𝐴_𝑘−_𝑘𝛼₂(𝑘)𝐼1−𝑞𝑘𝑃_𝑘≤_𝑘𝑘𝑃(𝑘)𝜆₁𝑘Θ^𝐸+𝜆₂𝑘Θ^𝐴+𝜆₃𝑘Θ^𝐼−_𝑘𝛿𝑘𝑃(𝑘)𝐸_𝑘−_𝑘𝛼₁𝑞𝑘𝑃(𝑘)𝐴_𝑘−_𝑘𝛼₂(𝑘)𝐼1−𝑞𝑘𝑃_𝑘=𝜆₁Θ^𝐸+𝜆₂Θ^𝐴+𝜆₃Θ^𝐼⟨𝑘²⟩−𝛿Θ^𝐸+𝛼₁𝑞Θ^𝐴+𝛼₂Θ1−𝑞^𝐼⟨𝑘⟩=𝛿Θ^𝐸+𝛼₁𝑞Θ^𝐴+𝛼₂Θ1−𝑞^𝐼⟨𝑘𝜆⟩¹Θ^𝐸+𝜆₂Θ^𝐴+𝜆₃Θ^𝐼𝛿Θ^𝐸+𝛼₁/𝑞Θ^𝐴+𝛼₂/1−𝑞Θ^𝐼⟨𝑘²⟩≤⟨𝑘⟩−1𝛿Θ^𝐸+𝛼₁𝑞Θ^𝐴+𝛼₂Θ1−𝑞^𝐼⟨𝑘⟩𝑅₀−1≤0. (3.7) Furthermore, 𝐿^′(𝑡)=0 if and only if 𝐸_𝑘=𝐴_𝑘=𝐼_𝑘=0. Therefore, by the LaSalle Invariance Principle [22], 𝑃⁰ is the global asymptotic stability if 𝑅₀<1.

4. Immunization Strategy

Immunization is very helpful in controlling diseases. In this section, we will discuss various immunization schemes [23–25].

4.1. Uniform Immunization Strategy

Uniform immunization strategy is simplest immunization schemes [23–25], and vaccinates is also a fraction of the population randomly. Immune nodes cannot become infected and, thus, do not transmit the infection to their neighbors. In this case, for a fixed spreading rate, the relevant control parameter is the immunity 𝑝(0<𝑝<1), defined as the fraction of immune nodes present in the network. At the mean-field level, the presence of uniform immunity will effectively reduce the spreading rate by a factor (1−𝑝) [23], that is, the probability of finding and infecting a susceptible. By substituting 𝜆₁→(1−𝑝)𝜆₁, 𝜆₂→(1−𝑝)𝜆₂, and 𝜆₃→(1−𝑝)𝜆₃ in the system (2.2), then the system (2.2) becomes 𝑑𝑆_𝑘(𝑡)𝑑𝑡=−𝜆₁𝑘1−𝑝𝑆_𝑘(𝑡)Θ^𝐸(𝑡)−𝜆₂𝑘1−𝑝𝑆_𝑘(𝑡)Θ^𝐴(𝑡)−𝜆₃𝑘1−𝑝𝑆_𝑘(𝑡)Θ^𝐼(𝑡),𝑑𝐸_𝑘(𝑡)𝑑𝑡=𝜆₁𝑘1−𝑝𝑆_𝑘(𝑡)Θ^𝐸(𝑡)+𝜆₂𝑘1−𝑝𝑆_𝑘(𝑡)Θ^𝐴(𝑡)+𝜆₃𝑘1−𝑝𝑆_𝑘(𝑡)Θ^𝐼(𝑡)−𝛿𝐸_𝑘(𝑡),𝑑𝐴_𝑘(𝑡)𝑑𝑡=𝑞𝛿𝐸_𝑘(𝑡)−𝛼₁𝐴_𝑘(𝑡),𝑑𝐼_𝑘(𝑡)=𝑑𝑡1−𝑞𝛿𝐸_𝑘(𝑡)−𝛼₂𝐼_𝑘(𝑡),𝑑𝑅_𝑘(𝑡)𝑑𝑡=𝛼₁𝐴_𝑘(𝑡)+𝛼₂𝐼_𝑘(𝑡). (4.1) We obtain that the basic reproduction number is 𝑅₀=1−𝑝𝑅₀. (4.2) If no immunization were done, then 𝑅₀=𝑅₀; when 0<𝑝<1, 𝑅₀<𝑅₀, that is, the immunization scheme is effective; while as 𝑝→1, 𝑅₀→0, that is, in the case of a full immunization, it would be impossible for the epidemic to spread in the network.

4.2. Targeted Immunization

We can use a targeted immunization scheme [25]. We introduce the lower and upper threshold 𝜅₁ and 𝜅₂, such that if 𝑘>𝜅₂, all nodes with connectivity 𝑘 are immunized, if 𝜅₁<𝑘<𝜅₂, 𝑝_𝑘 is defined as the fraction of immune individuals, that is, we define the immunization rate 𝜎_𝑘 by 𝜎_𝑘=1,𝑘>𝜅₂,𝑝_𝑘,𝜅₁<𝑘≤𝜅₂,0,𝑘≤𝜅₁, (4.3) where 0<𝑝_𝑘≤1, and ∑_𝑘𝜎_𝑘𝑃(𝑘)=𝜎. 𝜎 is the average immunization rate. The epidemic model (2.2) becomes 𝑑𝑆_𝑘(𝑡)𝑑𝑡=−𝜆₁𝑘1−𝜎_𝑘𝑆_𝑘(𝑡)Θ^𝐸(𝑡)−𝜆₂𝑘1−𝜎_𝑘𝑆_𝑘(𝑡)Θ^𝐴(𝑡)−𝜆₃𝑘1−𝜎_𝑘𝑆_𝑘(𝑡)Θ^𝐼(𝑡),𝑑𝐸_𝑘(𝑡)𝑑𝑡=𝜆₁𝑘1−𝜎_𝑘𝑆_𝑘(𝑡)Θ^𝐸(𝑡)+𝜆₂𝑘1−𝜎_𝑘𝑆_𝑘(𝑡)Θ^𝐴(𝑡)+𝜆₃𝑘1−𝜎_𝑘𝑆_𝑘(𝑡)Θ^𝐼(𝑡)−𝛿𝐸_𝑘(𝑡),𝑑𝐴_𝑘(𝑡)𝑑𝑡=𝑞𝛿𝐸_𝑘(𝑡)−𝛼₁𝐴_𝑘(𝑡),𝑑𝐼_𝑘(𝑡)=𝑑𝑡1−𝑞𝛿𝐸_𝑘(𝑡)−𝛼₂𝐼_𝑘(𝑡),𝑑𝑅_𝑘(𝑡)𝑑𝑡=𝛼₁𝐴_𝑘(𝑡)+𝛼₂𝐼_𝑘(𝑡). (4.4) We obtain that the basic reproduction number is 𝑅₀=𝜆₁𝛿+𝜆₂𝑞𝛼₁+𝜆₃1−𝑞𝛼₂𝑘²1−𝜎_𝑘,⟨𝑘⟩ (4.5) or 𝑅₀=𝑅₀−𝜆₁𝛿+𝜆₂𝑞𝛼₁+𝜆₃1−𝑞𝛼₂𝑘²𝜎_𝑘.⟨𝑘⟩ (4.6)

5. The Final Sizes Relation

First, we show that the disease will eventually die out, that is, 𝐸_𝑘(∞)=0,𝐴_𝑘(∞)=0, and 𝐼_𝑘(∞)=0.

Note that the positive quadrant is invariant, so all solutions of (2.2) lie in the nonnegative, bounded set defined by 𝑆_𝑘(𝑡),𝐸_𝑘(𝑡),𝐴_𝑘(𝑡),𝐼_𝑘(𝑡),𝑅_𝑘(𝑡)≥0 and 𝑆_𝑘(𝑡)+𝐸_𝑘(𝑡)+𝐴_𝑘(𝑡)+𝐼_𝑘(𝑡)+𝑅_𝑘(𝑡)=1. Observing that 𝑑𝑆𝑑𝑡^𝑘(𝑡)+𝐸_𝑘(𝑡)=−𝛿𝐸_𝑘(𝑡), (5.1) we see that 𝑆_𝑘(𝑡)+𝐸_𝑘(𝑡) is decreasing whenever 𝐸_𝑘(𝑡)>0. However, 𝑆_𝑘(𝑡)+𝐸_𝑘(𝑡) is bounded below by 0, hence it has a limit as 𝑡→∞. Because 𝑆_𝑘(𝑡) is bounded and decreasing, it has a limit as 𝑡→∞. Therefore, 𝐸_𝑘(𝑡) has a limit as 𝑡→∞.

In analyzing the system (2.2) we adopt the conventions that for an arbitrary continuous function 𝑤(𝑡) with nonnegative components, 𝑤_∞=lim_𝑡→∞𝑤(𝑡),𝑤=^∞₀𝑤(𝑡)𝑑𝑡. (5.2)

Integrating (5.1) from 0 to 𝑡, we have 𝐸_𝑘(𝑡)−𝐸_𝑘(0)+𝑆_𝑘(𝑡)−𝑆_𝑘(0)=−𝛿^𝑡₀𝐸_𝑘(𝑡)𝑑𝑡. (5.3) Thus, 𝐸_𝑘(0)−𝐸_𝑘(∞)+𝑆_𝑘(0)−𝑆_𝑘(∞)=𝛿𝐸_𝑘, (5.4) as 𝑡→∞.

The left hand side of (5.4) is finite because the components of 𝑆_𝑘(0), 𝑆_𝑘(∞), 𝐸_𝑘(0) and 𝐸_𝑘(∞) are bounded by the initial total population size. Therefore the right hand side (5.4) is also finite and because 𝛿 is positive, 𝐸_𝑘<∞. Since 𝐸_𝑘(𝑡) is a smooth nonnegative function, 𝐸_𝑘(∞)=0 (Similarly 𝐴_𝑘(∞)=0,𝐼_𝑘(∞)=0), and 𝐸_𝑘=𝛿⁻¹𝑆_𝑘(0)−𝑆_𝑘(∞)+𝛿⁻¹𝐸_𝑘(0). (5.5) Similarly, we can obtain 𝐴_𝑘(𝑡)−𝐴_𝑘(0)=𝑞𝛿^𝑡₀𝐸_𝑘(𝑡)𝑑𝑡−𝛼₁^𝑡₀𝐴_𝑘(𝑡)𝑑𝑡, (5.6) when 𝑡→∞, we have 𝐴_𝑘(∞)−𝐴_𝑘(0)=𝑞𝛿𝐸_𝑘−𝛼₁𝐴_𝑘. (5.7) Because 𝐴_𝑘(∞)=0 and 𝐸_𝑘 are bound, we have 𝐴_𝑘=𝑞𝛿𝐸_𝑘+𝐴_𝑘(0)𝛼₁. (5.8) For the same reason, we obtain 𝐼_𝑘(𝑡)−𝐼_𝑘(0)=1−𝑞𝛿^𝑡₀𝐸_𝑘(𝑡)𝑑𝑡−𝛼₂^𝑡₀𝐼_𝑘(𝑡)𝑑𝑡, (5.9) when 𝑡→∞, we have 𝐼_𝑘(∞)−𝐼_𝑘(0)=1−𝑞𝛿𝐸_𝑘−𝛼₂𝐼_𝑘. (5.10) Because 𝐼_𝑘(∞)=0 and 𝐸_𝑘 are bound, we have 𝐼_𝑘=1−𝑞𝛿𝐸_𝑘+𝐼_𝑘(0)𝛼₂. (5.11)

Integrating the first equations of (2.2) from 0 to 𝑡 gives 𝑆ln^𝑘(0)𝑆_𝑘(𝑡)=^𝑡₀𝜆₁𝑘Θ^𝐸+𝜆₂𝑘Θ^𝐴+𝜆₃𝑘Θ^𝐼=𝜆𝑑𝑡₁𝑘⟨𝑘⟩_𝑘𝑘𝑃(𝑘)^𝑡₀𝐸_𝑘(𝑡)𝜆𝑑𝑡+²𝑘⟨𝑘⟩_𝑘𝑘𝑃(𝑘)^𝑡₀𝐴_𝑘(𝑡)𝜆𝑑𝑡+³𝑘⟨𝑘⟩_𝑘𝑘𝑃(𝑘)^𝑡₀𝐼_𝑘(𝑡)𝑑𝑡. (5.12) When 𝑡→∞, we have 𝑆ln^𝑘(0)𝑆_𝑘(∞)=𝜆₁𝑘⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝐸_𝑘+𝜆₂𝑘⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝐴_𝑘+𝜆₃𝑘⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝐼_𝑘=𝜆₁𝑘⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝐸_𝑘+𝜆₂𝑘⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝑞𝛿𝐸_𝑘+𝐴_𝑘(0)𝛼₁+𝜆₃𝑘⟨𝑘⟩_𝑘(𝑘)𝛿𝑘𝑃1−𝑞𝐸_𝑘+𝐼_𝑘(0)𝛼₂=𝜆₁𝑘⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝐸_𝑘+𝜆₂𝑘𝛼₁⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝑞𝛿𝐸_𝑘+𝐴_𝑘(0)+𝜆₃𝑘𝛼₂⟨𝑘⟩_𝑘𝑘𝑃(𝑘)1−𝑞𝛿𝐸_𝑘+𝐼_𝑘(0)=𝜆₁𝑘𝛿⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝑆_𝑘(0)−𝑆_𝑘(∞)+𝐸_𝑘(0)+𝜆₂𝑘𝛼₁⟨𝑘⟩_𝑘(𝑘)×𝛿𝑘𝑃𝑞𝛿⁻¹𝑆_𝑘(0)−𝑆_𝑘(∞)+𝛿⁻¹𝐸_𝑘(0)+𝐴_𝑘(0)+𝜆₃𝑘𝛼₂⟨𝑘⟩_𝑘𝑘𝑃(𝑘)1−𝑞𝛿𝛿⁻¹𝑆_𝑘(0)−𝑆_𝑘(∞)+𝛿⁻¹𝐸_𝑘(0)+𝐼_𝑘(0)=𝜆₁𝑘𝛿⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝑆_𝑘(0)−𝑆_𝑘(∞)+𝐸_𝑘(0)+𝜆₂𝑘𝛼₁⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝑞𝑆_𝑘(0)−𝑆_𝑘(∞)+𝐸_𝑘(0)+𝐴_𝑘(0)+𝜆₃𝑘𝛼₂⟨𝑘⟩_𝑘𝑘𝑃(𝑘)1−𝑞𝑆_𝑘(0)−𝑆_𝑘(∞)+𝐸_𝑘(0)+𝐼_𝑘(0)=𝜆₁𝛿+𝜆₂𝑞𝛼₁+𝜆₃1−𝑞𝛼₂𝑘⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝑆_𝑘(0)−𝑆_𝑘(∞)+𝐸_𝑘(0)+𝜆₂𝑘𝛼₁⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝐴_𝑘(0)+𝜆₃𝑘𝛼₂⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝐼_𝑘(0)=𝑅₀𝑘𝑘²_𝑘𝑘𝑃(𝑘)𝑆_𝑘(0)−𝑆_𝑘(∞)+𝐸_𝑘(0)+𝜆₂𝑘𝛼₁⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝐴_𝑘(0)+𝜆₃𝑘𝛼₂⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝐼_𝑘(0). (5.13)(1) If 𝑆_𝑘(0)=𝑆_𝑘0, 𝐼_𝑘(0)=𝐼_𝑘0, and 𝐸_𝑘(0)=𝐴_𝑘(0)=0, then the final size relation is 𝑆ln^𝑘0𝑆_𝑘∞=𝑅₀𝑘𝑘²_𝑘𝑘𝑃(𝑘)𝑆_𝑘0−𝑆_𝑘∞+𝜆₃𝑘𝛼₂⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝐼_𝑘0. (5.14) If 𝑆_𝑘(0)=𝑆_𝑘0, 𝐼_𝑘(0)=𝐼_𝑘0, 𝐸_𝑘(0)=𝐸_𝑘0, and 𝐴_𝑘(0)=𝐴_𝑘0, then the final size relation is 𝑆ln^𝑘0𝑆_𝑘∞=𝑅₀𝑘𝑘²_𝑘𝑘𝑃(𝑘)𝑆_𝑘0−𝑆_𝑘∞+𝑅₀𝑘𝑘²_𝑘𝑘𝑃(𝑘)𝐸_𝑘0+𝜆₂𝑘𝛼₁⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝐴_𝑘0+𝜆₃𝑘𝛼₂⟨𝑘⟩_𝑘𝑘𝑃(𝑘)𝐼_𝑘0. (5.15) Therefore, we show that the final size of the susceptible 𝑆(∞)=𝑁∑_𝑘𝑃(𝑘)𝑆_𝑘∞, where 𝑁 is the whole population size. Similarly, we can obtain the final sizes under various immunization schemes.

6. Conclusions

In this paper, we describe a network epidemic model and calculate the basic reproduction number 𝑅₀ and the final sizes relation. The basic reproduction number is the spectral radius of the matrix 𝐹𝑉⁻¹. We prove the global asymptotic stability of the disease-free equilibrium when the basic reproduction number is less than 1. The effects of various immunization schemes are studied. Finally, a final size relation is derived for network epidemic models. The derivation depends on an explicit formula for the basic reproduction number of network of disease transmission models.

Acknowledgments

This work is supported by the Graduate Students' Excellent Innovative Item of Shanxi Province (no. 20090318), National Natural Science Foundation of China (no. 60771026), Science Foundation of Shanxi Province (no. 2010011007), Science Foundation of Shanxi Province (no. 2009011005-1), National Natural Science Foundation of China (no. 10901145), and National Natural Science Foundation of China (no. 70871072).