Global behavior of a two-stage contact process on complex networks

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Abstract

Masuda and Konno [14] first formulated a two-stage contact process on complex networks with heterogeneous degree distribution, and they derived a critical birth or infection rate βc, above which there exists a unique positive equilibrium. The global behavior of this model is not well understood, and the authors have not given a rigorous mathematical analysis of their model. In this paper, we investigate the global behavior in detail and show that the global behavior is completely determined by a threshold R0. In particular, by comparison arguments, we establish the global asymptotic stability of the trivial equilibrium E0 for R0 < 1; by constructing a bounded function, we show that the system is uniformly persistent for R0 > 1. Furthermore, by means of a monotone iterative approach, we obtain a sufficient condition for the global asymptotic stability of the positive equilibrium E*.

Introduction

The contact process that describes stochastic growing of a population on a spatial lattice was first proposed by Harris in 1974 [1]. In its simplest form, every site of a lattice can be in two states, i.e., occupied or vacant, and much effort has been devoted to this model (see, for example, [2], [3] and references therein). In recent years, there are many model variants on this contact process, such as two or more stages of development [4], or multitype of individuals [5]. Using a multitype framework, Sugimine et al. [6] studied an infection spread model between households spatially located on homogenous trees. Deshayes [7] introduced a contact process with aging that affected a particle’s ability to give birth and proved a sharp theorem for this process conditioned to survive. Foxall [8] studied the contact process with a latent stage and gave upper and lower bounds on the critical values. Moreover, he proved convergence of critical values in the limit of small and large latent time, respectively.

The so-called two-stage contact process was first proposed by Krone [4], and an intermediate state ‘juvenile’ was forced to mature before it can give birth to any descendant. Specifically, it is a growth model of population on Zd (a spatial lattice of d dimension) given by the following transition rules

0 → 1 at rate βn2(x),

1 → 2 at rate r,

2 → 0 at rate μ,

1 → 0 at rate δ,

where n2(x) is the number of neighbors belonging to state 2 of site x on a lattice of d dimension. Basically, it is a continuous-time Markov process whose state space is {0,1,2}Zd; that is, each site is either vacant, occupied by a young individual or occupied by an adult. Only adults can give birth at other neighboring vacant sites at a rate β. Each new descendant is young and the maturation rate is r. The death rates for adults and young individuals are μ and δ, respectively. If r=+, one retrieves the classical contact process with state space {0,2}Zd in [1].

In [4], Krone constructed a multitype dual process to analyze the model and obtained numerous basic properties of the two-stage contact process. For example, he discussed the monotonicity with respect to parameters (increasing in β and r and decreasing in δ) and estimated the bounds on survival region. Later, Foxall [9] continued the work started by Krone on the two-stage contact process, provided a brief proof of the duality relation and solved some of the open problems proposed in Krone [4]. Recently, Xue [10] considered the two-stage contact process on a lattice of high dimension and proved a limit theorem on the critical infection rate in the limit of sufficiently large dimension d.

Lots of population or epidemic systems can be described by a graph (network) where vertices (nodes) represent individuals and an edge linking a pair of nodes signifies the interaction between individuals. Most previous results mainly concern contact process on regular lattices, however, at the end of twentieth century, real networks underlying individual interaction have been revealed to be complex, which can not be well approximated by conventional graphs such as regular lattices, or classical random graphs. In fact, many networks show the phenomena of small-world and scale-free [11], [12]. Specifically, the scale-free phenomenon shows that a large number of nodes have a small number of contacts with other individuals (nodes) and a small number of nodes have a large number of contacts; that is, the number of contacts k follows a power law distribution, P(k)kγ (typically, 2 < γ ≤ 3).

Besides the heterogeneous number of contacts, most early models are stochastic, and the type of questions that were addressed are for example: What is the probability of population or disease survival? How long is the population or disease likely to persist (with or without intervention)? One generalization of the initial simple stochastic contact process was to study deterministic contact processes, which is of course preferable when investigating a community of large size. Without a doubt, both stochastic and deterministic models have their benefits and drawbacks [13], respectively; however, the attention of the present paper is paid to deterministic contact processes. For deterministic contact processes, Masuda and Konno [14] first extended a stochastic two-stage contact process in [4] to allow for non-uniform/heterogeneous mixing between individuals (i.e., individuals do not contact all individuals equally likely), and they established a threshold condition for the existence of positive equilibrium corresponding to the coexistence of young individuals and adults. Li and Han [15] studied a two-stage contact process on scale-free networks as a model for the spread of disease and estimated the metastable density.

In this paper, we consider the two-stage contact process on complex networks introduced in [14]. A fundamental problem in population or epidemic models is to investigate the global dynamics of population persistence or disease transmissions, that is, to investigate the long-term behavior of the persistence of a population or an infection. In particular, the focus is on the analysis of the local/global stability of the trivial equilibrium (disease-free equilibrium) and the positive equilibrium (endemic equilibrium) of the population (epidemic) models. Adapting the methods used for multi-group SIS models, d’Onofrio [16] first rigorously showed the global behavior of the network-based SIS epidemic model proposed by Pastor–Satorras and Vespignani [17]. By constructing monotone sequences, Wang and Dai [18] gave another strict mathematical proof of the global attractivity of the endemic equilibrium for the same network-based SIS epidemic model. Wang et al. [19] derived a threshold condition to determine the global dynamics of a network SIS model with an infective vector. By constructing suitable Lyapunov functions, Li et al. [20] established the global asymptotical stability of the disease-free equilibrium and endemic equilibrium of a network SIRS epidemic model. Zhang et al. [21] investigated complex dynamics of epidemic in an adaptive network, including degenerate Hopf bifurcation, homoclinic bifurcation and Bogdanov–Takens bifurcation.

Our aim is to give a strict mathematical analysis of the two-stage deterministic contact process on complex networks introduced in [14]. If r=+, the two-stage contact process could be reduced to the network-based SIS epidemic model proposed by Pastor–Satorras and Vespignani [17], however, if 0<r<+, the methods used to proving the global attractivity of the endemic equilibrium in [16] and [18] are invalid. To this end, we apply a new monotone technique to obtain a sufficient condition that ensure the global asymptotic stability of the positive equilibrium of the system. This paper is organized as follows. In Section 2, we first introduce the deterministic two-stage contact process and derive some basic properties of it. We study the local and global stability of the equilibria in Section 3, respectively. In particular, we analyze the existence and uniqueness of the equilibria and the global stability of the trivial equilibrium, as well as the uniform persistence of the system and the global stability of the positive equilibrium, using a bounded function method and a novel monotone technique, respectively. Finally, further discussion is given to conclude this work in Section 4.

Section snippets

The model

The two-stage contact process in a homogeneous mixing population is extended to a population with heterogeneous mixing patterns by Masuda and Konno [14]. The flow charts of the two-stage contact process and the SIRS (susceptible-infectious-recovered-susceptible) epidemic model are depicted in Fig. 1(A) and (B), respectively. They differ in the aspect that the change rate from 0 to 1 is proportional to the density of different neighbors of a state-0 node; in particular, in the two-stage contact

The analysis

In this section, we demonstrate a condition determining the existence and uniqueness of the equilibria, and the local/global stability of the equilibria.

Concluding remarks

The goal of this paper was to investigate the global behavior of a two-stage contact process on complex networks with arbitrary degree distribution formulated by Masuda and Konno [14]. We discussed the global asymptotic stability of the trivial equilibrium and the positive equilibrium of system (1), respectively. For system (1), the global behavior are shown to be completely determined by a threshold parameter R0: (i) if R0 < 1, by comparison arguments, we established that the trivial

Acknowledgments

This project was supported by (i) the National Natural Science Foundation of China under Grants 11801532, 11747142, 11501338 and 61833005, (ii) China Postdoctoral Science Foundation under Grant 2018M630490, (iii) Hubei Provincial Natural Science Foundation of China under Grant 2018CFB260, (iv) Fundamental Research Funds for the Central Universities, China University of Geosciences (Wuhan) under Grant CUG170622, (v) Jiangsu Provincial Key Laboratory of Networked Collective Intelligence under

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