The analysis of an epidemic model on networks

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Abstract

The paper consider an epidemic model with birth and death on networks. We derive the epidemic threshold R0 dependent on birth rate b, death rate d (natural death) and μ from the infectious disease and natural death, and cure rate γ. And the stability of the equilibriums (the disease-free equilibrium and endemic equilibrium) are analysed. Finally, the effects of various immunization schemes are studied and compared. We show that both targeted, and acquaintance immunization strategies compare favorably to a proportional scheme in terms of effectiveness. For active immunization, the threshold is easier to apply practically. To illustrate our theoretical analysis, some numerical simulations are also included.

Introduction

Epidemic, one of the most important issues related to our real lives, such as computer virus on Internet and venereal disease on sexual contact networks, attracts a lot of attention. The dynamical behaviors of epidemic diseases have been studied for a long time, SIS and SIR are the two important and fundamental epidemic models [1]. When disease transmission is modeled over networks [2], [3], it is usual to model the infectivity, that is, the rate of transmission between infected and susceptible nodes, by assuming that transmission is equally likely over all links. For an idealized model this is the natural way to consider infectivity. However, when the underlying complex network is scale-free, the situation becomes unrealistic in the extreme tail of the distribution.

In 1999, Barabási and Albert addressed a new model of complex networks: scale-free networks (BA) [2]. In a scale-free network the probability P(k) for any node with k links to other nodes is distributed according to the power law P(k) = Cf(k)kr, r  (2, 3], where f(k) is the function of k. While it has frequently been observed that real human social and disease transmission networks exhibit scale-free properties over several orders of magnitude, the tail of the distribution observed from data is always bounded.

The spread of epidemic diseases on networks has been studied by many researchers [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25]. For the mechanism of the spreading of epidemic on complex networks, different researcher gave different explanations [7], [14], [20]. 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 λc = 0 for heterogenous networks with sufficiently large size [7]. Zhou et al. [20] suggested that this hypothesis is not always appropriate. For example, in epidemic contact networks, the hub node has many acquaintances, however, he/she cannot contact all his/her acquaintances in one time step. So they assumed that the infectivity is identical for every node regardless of their different degrees, the threshold λc is a constant value, regardless of the size of networks and the degree distribution. Joo and Lebowitz [21] examined cases where the transmission of infection between nodes depends on their connectivity, and a saturation function C(k), which reduces the infection transmission rate across an edge going from a node with high connectivity k, was introduced. For some infectious disease with the longer spreading time, we can consider birth, death etc. population dynamics in the model. Liu [10] consider the spread of epidemic diseases with birth and death on networks. They obtain the epidemic threshold.

Based on the above results, in this paper we take a more realistic model on networks and discuss the stability of the equilibrium. We further discuss the model with respect to the effects of various immunization schemes. In the next section, we will describe the epidemic model on networks with birth and death. The subsequent section is devoted to determine the stability of equilibriums. In Section 4 we consider several models of immunization. Numerical simulations supporting the theoretical analysis are given in Section 5. The paper ends with a conclusion and discussion in Section 6.

Section snippets

Epidemic model on networks

On the network N, each site of N is empty or occupied by only one individual. We give each site a number: 0, 1 or 2. Alternatively, we can interpret the three states as 0: vacant, 1: a healthy individual occupation, 2: an infected individual occupation. The state of the system at time t can be described by a set of numbers 0, 1, 2. That means if the system are in state A and the site x  N, then At(x)  {0, 1, 2}. Each site can change its state with a certain rate. An empty site can give birth to a

The stability of the equilibrium

In this section, we consider the stability of the equilibrium E0 and E.

Firstly, we consider that the Jacobin matrix at disease-free equilibrium E0 isJ(E0)=-b-d-b+γ0-λS1g(j)0-λS1g(K)0-(γ+μ)0λS1g(j)0λS1g(K)00-b-d-b-λjSjg(j)+γ0-λjSjg(K)000-(γ+μ)+λjSjg(j)0λjSjg(K)000-λKSKg(j)+γ-b-d-b-λKSKg(K)+γ000λKSKg(j)+γ0-(γ+μ)+λKSKg(K)2K×2K,where g(j)=(j-1)P(j)k,j=1,2,,K. The matrix J(E0) has K eigenvalues equal to −(b + d). The (K + 1)th eigenvalue is −(γ + μ). The rest K  1 eigenvalues

Immunization strategy

Vaccination is very helpful in controlling vaccine preventable diseases. We give a parallel comparison of the effect of different immunization schemes [14] in the case of the Barabási and Albert network.

Numerical results

For disease spreading with birth and death on networks, we know that the stability of the disease-free equilibrium and the endemic equilibrium. Here we present numerical simulations to support the results obtained in previous sections. Our simulations are based on the BA network with P(k)=(γ1-1)mγ1-1k-γ1, and N = 1000.

In Fig. 2, we show the outcome of the system (2.1) when the basic reproductive number R0 < 1. The parameters are chosen as b = 0.45, d = 0.02, λ = 0.08, γ = 0.5, μ = 0.09, and γ1 = 3.68. Fig. 2

Conclusion and discussion

In previous sections we have discussed epidemic dynamics on networks with birth and death. We derive the epidemic threshold R0 dependent on birth rate b, death rate d (natural death) and μ from the infectious disease and nature death, and cure rate γ. We analyse the stability of the disease-free equilibrium and the endemic equilibrium. If R0 < 1, the infection-free periodic solution is global asymptotic stability which implies that the disease will extinct. If R0 > 1 and γ > b, the endemic

Acknowledgments

This work is supported by the National Natural Sciences Foundation of China (60771026), Graduate Students’ Excellent Innovative Item of Shanxi Province (20093018), Science Foundation of Shanxi Province (2010011007), Science Foundation of Shanxi Province (2009011005-1), National Natural Science Foundation of China (10901145), and National Natural Science Foundation of China (70871072).

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