A novel epidemic model considering demographics and intercity commuting on complex dynamical networks
Introduction
Various epidemics are one of the important factors that threaten the human survival, such as smallpox [1], malaria [2], Ebola [3], and have caused serious problems for human beings. In particular, commuting is one of the significant means that promote the epidemic propagation from one region to another, which can make the spread of infectious diseases between regions easier. For instance, after the first case of SARS was found in Guangdong province in China, SARS quickly spread to other provinces in China due to the infected commuters [4]. H1N1 quickly spread globally after the outbreak in Mexico in 2009, which induced a large number of people to be infected [5]. With the rapid development of transportation, e.g., airplanes and high-speed trains, the number of migrant workers and commuters has increased significantly, which leads to the spread of infectious diseases within different cities or countries [6]. Hence, understanding the spreading mechanism of infectious diseases among multiple cities and take some effective measures to control them has become a crucial issue, which has attracted a variety of researchers in applied mathematics [7], [8], public health management [9], [10], and population dynamics [11], [12].
In recent years, epidemic models considering the mobility of population have been widely explored. As an example, considering the influence of travelers on the epidemic spread, Arino et al. [13] established one mathematical model containing 2n2 equations to characterize the epidemic spreading among n cities. Cui et al. [14] considered the impact of population spread and traffic infection on epidemic propagation between two cities. Liu and Zhou [15] constructed a model describing the diffusion of SIRS-type epidemics, showing that transportation can promote the epidemic propagation. Liu and Stechlinski [16] analyzed a multi-city SIR model of pulse control strategy. All of the above-mentioned contributions, however, are based on randomly mixed homogeneous population; the heterogeneity of contacts among individuals is often exhibited in the real-world population, but neglected in the aforementioned works.
The theory of complex networks, over the past two decades, has achieved some important achievements [17], [18], [19], [20], one of the more important research is used to model and analyze the epidemic spreading behavior [21], [22], [23], [24], [25], [26], [27], since the topological structure of complex networks can accurately depict the contact between different individuals. Nevertheless, most of current studies about epidemic propagation on complex networks focus on static networks [28], [29], [30], [31], which are effective tools to investigate the infectious diseases of short duration, such as influenza [32]. But for the infectious diseases of long duration, such as HIV [33], many factors (e.g., birth, death, immigration or emigration) may change the topology of the population network during the epidemic spreading [34], [35], [36]. Thereby, it is of great importance to investigate the epidemic propagation in dynamic networks. Jin et al. [37] designed an epidemiological model that incorporates demographics into complex networks theory to study the influence of demographics on population distribution. Pan et al. [38] established an SIS epidemic model community structure and discussed the impact of demographics and short-time travel on disease transmission. In addition, Yao and Zhang [39] proposed an epidemic model considering demographic characteristics on complex networks to study the dynamics of simultaneous transmission of two different epidemics, which pointed out that the demographics can create a great impact on the propagation characteristics of epidemics.
Motivated by the aforementioned contributions, this paper extends the epidemic spread model of two cities on dynamic networks. Considering the fact that some infectious diseases can hold the characteristic of lifelong immunity after recovery, such as chickenpox and measles [40], [41], the SIR model is introduced to characterize the epidemic propagation. Moreover, the effect of demographics (birth, natural death) on infectious disease spreading is also discussed at the same time. What is more, demographics will change the number and internal relationships of individuals in cities, forming a complex dynamical networks. Therefore, we analyze the dynamic change of population degree distribution over time. Furthermore, the epidemic could spread from one city to the other through intercity commuters, and hence we consider the impact of intercity commuting rate and exposed individuals during commuting on epidemic propagation in the proposed model. In general, the results of this work mainly include the following three aspects. First, we propose a novel two-city SIR propagation model considering demographics and intercity commuting on dynamic networks. Second, we calculate the basic reproduction number R0 of the proposed model by using spectral analysis method and show that two equilibria of the proposed epidemic model are both globally asymptotic stable under the corresponding conditions (R0 < 1 and R0 > 1). Finally, we investigate the evolution of population degree distribution over time and explore the influences of demographics, intercity commuting rates and the number of individuals who are exposed to others during commuting on the epidemic propagation between two cities.
The remainder of this paper is divided into four sections as follows. First, the proposed model is described in Section 2. In Section 3, by use of the spectral analysis, the basic reproduction number R0 is obtained, then the stability of two equilibria are analytically proved. After that, in Section 4, the theoretical prediction is confirmed by a large plethora of numerical simulations, and the sensitivity analysis of the parameters in the model is performed. At last, Section 5 summarizes the main contributions of this paper and points out some potential directions in the future.
Section snippets
The SIR model with demographics and intercity commuting
In this section, we use the theory of complex networks to describe the proposed two-city epidemic model in Fig. 1. The propagation process in the model is depicted by use of susceptible-infected-recovered model, which classifies the individuals into three compartments within each city: S(Susceptible), I(Infected) and R(Recovered). Moreover, some susceptible individuals are added into these two cities according to a certain probability at each time step, and some individuals are also removed
Dynamical analysis of the proposed model
In the cause of deeply characterizing the epidemic process, R0 is often used as an important indicator, to determine whether infectious diseases can outbreak. If R0 < 1, the disease will die out naturally; otherwise, epidemics will exist and evolve into the endemic state. In what follows, R0 is firstly calculated; and then the stability of two equilibria for the current model is analyzed in detail.
Numerical simulations
Here, we will further verify the aforementioned analytical predictions through a large number of numerical simulations. In the simulation experiments, we select two cities with different population sizes, where the initial sizes in the network are set to and . The initial number of individuals in each city are assumed to be and (i ∈ {1, 2}, ), that is, the degree distribution of the original two networks obeys
Conclusions
To sum up, we put forward a novel two-city SIR epidemic model on complex dynamical networks and discuss the impact of demographics and individual intercity commuting on the epidemic propagation behavior. By resorting to the spectral analysis approach based on the next generation matrix, we obtained R0 of the proposed model and analyzed the stability of two equilibria including the endemic equilibrium and disease-free one. Through theoretical analysis, we proved that it is globally asymptotic
Acknowledgments
This work is partially supported by the National Natural Science Foundation of China (NSFC) under Grant nos. 61773286 and 61873154. Matthias Dehmer thanks the Austrian Science Fund for financial support (P 30031).
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