Global behavior of a two-stage contact process on complex networks
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 (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 ; 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 one retrieves the classical contact process with state space 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, (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 the two-stage contact process could be reduced to the network-based SIS epidemic model proposed by Pastor–Satorras and Vespignani [17], however, if 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
References (34)
- et al.
On global and local critical points of extended contact process on homogeneous trees
Math. Biosci.
(2008) The critical infection rate of the high-dimensional two-stage contact process
Stat. Probab. Lett.
(2018)Stochastic epidemic models: a survey
Math. Biosci.
(2010)- et al.
Multi-state epidemic processes on complex networks
J. Theor. Biol.
(2006) A note on the global behaviour of the network-based SIS epidemic model
Nonlinear Anal. Real World Appl.
(2008)- et al.
Global analysis of an SIS model with an infective vector on complex networks
Nonlinear Anal. Real World Appl.
(2012) - et al.
Analysis of epidemic spreading of an SIRS model in complex heterogeneous networks
Commun. Nonlinear Sci. Numer. Simul.
(2014) - et al.
Complex dynamics of epidemic models on adaptive networks
J. Differ. Equ.
(2019) - et al.
Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission
Math. Biosci.
(2002) - et al.
Stability of a delayed SIRS epidemic model with a nonlinear incidence rate
Chaos Solitons Fractals
(2009)
Monotone iterative techniques to SIRS epidemic models with nonlinear incidence rates and distributed delays
Nonlinear Anal. Real World Appl.
Contact interactions on a lattice
Ann. Probab.
Interacting Particle Systems
Stochastic Interacting Systems: Contact, Voter and Exclusion Processes
The two-stage contact process
Ann. Appl. Prob.
Two-scale multitype contact process: coexistence in spatially explicit metapopulations
Markov Process. Relat. Fields
The contact process with aging
ALEA Lat. Am. J. Probab. Math. Stat.
Cited by (12)
Bifurcation and propagation dynamics of a discrete pair SIS epidemic model on networks with correlation coefficient
2022, Applied Mathematics and ComputationCitation Excerpt :Inspired by the influential work [17], epidemic models on heterogeneous networks were widely investigated and attracted much attention [19,20]. Indeed, the mean-field theory provides an easy way to establish epidemic models on networks, and a plenty of publications are proven efficient to formulate the spread of disease on networks [21,22]. However, there are many difficulties in studying infectious diseases on networks with correlation coefficients, especially for mathematical modeling [23].
Global dynamics of a network-based SIQS epidemic model with nonmonotone incidence rate
2021, Chaos, Solitons and FractalsCitation Excerpt :As a matter of fact, transmission of infectious diseases at the group level occurs mainly through social contact networks, consequently, it is more realistic to use complex networks for investigating the spread of epidemics in populations. In recent years, an increasing number of scholars have begun to focus on the dynamics of infectious diseases on complex networks [1–10]. Pastor-Satorras and Vespignani [1,2] first established the SIS model in scale-free networks using a mean-field approximation.
Transmission dynamics, global stability and control strategies of a modified SIS epidemic model on complex networks with an infective medium
2021, Mathematics and Computers in SimulationCitation Excerpt :For this reason, the epidemic model on complex networks has been widely concerned [6,11,12,28]. Since Pastor-Satorras and Vespignani built the first epidemic model on complex networks [13], more and more interesting results about the spread of diseases on complex networks have been obtained [9,10,15,16,18,20,22,26,29,30]. Today, studying the epidemic model on complex networks is still a hot issue.
Dynamics of a competing two-strain SIS epidemic model with general infection force on complex networks
2021, Nonlinear Analysis: Real World ApplicationsStability, bifurcation and chaos of a discrete-time pair approximation epidemic model on adaptive networks
2021, Mathematics and Computers in SimulationCitation Excerpt :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.
Global dynamics of an epidemic model with incomplete recovery in a complex network
2020, Journal of the Franklin Institute