Original articles
Stability, bifurcation and chaos of a discrete-time pair approximation epidemic model on adaptive networks

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Abstract

Adaptive behavior on networks leads to bifurcation and other special phenomenon, which has been proved for differential systems. In this paper, one will discuss the stability, bifurcation and chaos of the discrete pair epidemic model on adaptive networks, which will bring some new challenges. The stability of the disease free equilibrium and endemic equilibrium with respect to basic reproduction number R0 is studied. Under certain conditions, as the time step-size increases, flip bifurcation that occurs at the endemic equilibrium is discussed, the period-doubling bifurcation and the chaos phenomenon of the system are exhibited, some figures are provided for illustration. According to the results in this paper, the dynamical behaviors of the discrete model are multitudinous and quite different from the continuous model.

Introduction

For a long time, mathematical models have played an important role in studying the spread of disease, and it can help people better understand the spread of disease [2], [18]. The classic mathematical infectious diseases model focuses on the nature of the disease itself [9], [12], [26]. It assumes that the population is uniformly mixed, and that there is no difference between individuals in contact with each other. In the past 20 years, the theory of complex networks has developed rapidly, and it has been found that the structure of the network has an important influence on the function of the network and the information transmission on it [1], [23], [28]. Social networks consist of many different people and their relationships, and can be abstracted as nodes and edges. Defining a single contact between people as a connection, the diversity of contact forms (one-way or two-way) and the number of different times per unit of time makes social networks complex. These factors may affect the spread of diseases and viruses in social networks [17]. As a result, the spread of disease on complex networks has attracted a lot of attention [19], [25].

In [20], applying mean-field theory, the authors emphasized the important role of topology in epidemic modeling and proposed an epidemic model on scale-free networks, it indicated the threshold and critical behavior may disappear in the scale-free networks, which changes the standard conclusions in epidemic spreading. In order to reflect the dynamical interaction amongst neighboring individuals on networks, a pair approximation theory of ecological models was introduced in [16]. Applying the pair approximation method, Ref. [11] considered the local spatial heterogeneities in an epidemic model by studying the pairs of connected individuals, and more recommendations about the basic reproductive ratio and the final size of the pair SIR epidemic model was discussed in [10].

Adaptive behavior often exists in complex networks, namely, the ability to adapt the network topology dynamically in response to the dynamic state [6]. For example, people tend to avoid contact with people infected to deal with the spread of the disease. The authors in [6] proposed a new SIS epidemic model on adaptive network, it indicates that the adaptive nature of the system produces complex behaviors, such as bistability and cycles. A further study gives an epidemic model on an adaptive networks, it investigated the fluctuations in the infectives near the bifurcation point and the effects on the topological structure of networks [22]. Based on the earlier researches, the adaptive behavior has attracted much attention and achieved many interested results [7], [24], [30]. In [29], adaptive defense on SIS computer virus model with time varying parameters was investigated, the authors provide adaptive control strategies to ensure the extinction of virus. An SIS epidemic model with limited treatment capacity on adaptive networks was presented in [15], the backward and forward bifurcations of the disease free equilibrium were derived. Recently, detailed analysis of pair SIS epidemic on adaptive networks was discussed in [31], the authors analyzed the backward bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and limit-cycles of the system. For pair epidemic models, there are many problems to be solved, especially the dynamics on adaptive networks.

For a long time, the difference equation models have been widely considered in the ecological field because its advantages in statistics and data analysis [4]. For many biological systems, discrete model played an important role, such as in population forecast model and insect population problems. Disconnection and reconnection in the network are more like discrete processes, in which the network changes from one state to another and the model in discrete form is more reasonable. On the other hand, simple difference equations may exhibit complex properties, such as period-doubling bifurcation and chaos, which is more challenging in the study of mathematical theory [3], [8]. A discrete-time closure model was established in [5], and the authors found that the pair-based models are more precise in prediction of the disease evolution than the individual-based. In [27], a discrete-time SIS epidemic model was studied, the properties of the solutions are discussed and the stability of the equilibriums is considered. Compared with the continuous system in [19], more complex phenomenon was shown in the discrete system of [27]. As can be seen, the adaptive behavior of the network and the discretized model make the dynamic behavior of the system richer, which brings more mathematical challenges. Therefore, as an important part of mathematical epidemic models, discrete pair models on adaptive networks involve great value to explore and study.

Based on the previous considerations, this paper will focus on the dynamic analysis of discrete pair epidemic models on adaptive networks, and the main contributions is given as follows. Firstly, stability of the equilibriums is discussed under the conditions of R0<1, R0=1 and R0>1. Secondly, the flip bifurcation of the system is studied. Thirdly, period-doubling bifurcation and chaos for discrete system with respect to time step-size are illustrated.

The organizational structure of this paper is as follows: Section 2 introduces the formulation of the discrete pair epidemic model on adaptive networks. The stability conditions of the equilibriums are provided in Section 3. In Section 4, bifurcation and chaos of the system are discussed, and the conclusion is summarized in Section 5.

Section snippets

Model formulation

In order to describe the topological structure of the network, we here introduce some notation, more details can be found in [16]. Assuming each individuals belongs to a set of types A,B,C,, the connections over the whole network can be counted as Table 1.

[AB] means the direct connections from A to B, and [ABC] denotes the occurrences of an A neighboring a B which neighbors a C(distinct from A). If the network is bidirectional connected, then the triples in the form of [ABA] and the pairs in

Stability of equilibrium points

In this section, the stability of the disease free equilibrium and the endemic equilibrium will be studied.

Bifurcation and chaos

In this section, the bifurcation and chaos of system (3) will be discussed.

Conclusion

In this paper, a discrete-time pair epidemic model on adaptive networks is established, and the discrete model shows complex dynamical behaviors under certain conditions. According to the results in [6], [22], [31], adaptive characteristics of complex network will lead to the generation of complex dynamic behaviors, and it has been verified by the analysis of differential models. For discrete system, simple system may exhibit complex dynamical properties, and it is worth noting that the

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    This work was supported by the National Natural Science Foundation of China (Nos. 61973199, 61973200, 61573008) and the Shandong University of Science and Technology Research Fund of China (2018 TDJH101).

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