Research paperCooperative epidemic spreading in simplicial complex
Introduction
Epidemic spreading is an essential branch of network science, and its ideas can be applied to describe many phenomena in ecological, biological, and social systems [1], [2], [3], [4], [5], [6]. Extensive researches are devoted to predicting and controlling the spread of the epidemic [7], [8]. For single epidemic spreading, scholars found that the network topologies (e.g., degree distribution, weight distribution, cliques, and community) markedly affect the spreading dynamics [9], [10], [11], [12]. In reality, the spreading of epidemics, behaviours, and information is rarely independent of each other, but they are coevolving with strong interactions [13], [14], [15], [16], [17]. For example, the hosts may be killed or be provided permanent immunity by the first epidemic. Thus the second epidemic cannot infect the hosts, which means that the two epidemics compete with each other [18], [19], [20], [21], [22], [23]. Another different situation is that if the host is already infected by one epidemic, the infection probability of infecting the second epidemic is enlarged. In other words, the two cooperate epidemics with each other [24], [25], [26], [27], [28]. In real life, viruses or messages are usually rarely independent. Their spread often co-evolves with each other. HIV, for example, dramatically reduces the body’s immunity while making it easier for other viruses to destroy the organism. Coevolution refers to how one epidemic interacts with another epidemic in an evolutionary process [29]. In the coevolution of epidemic and information spreading, there exists an asymmetric interaction, i.e., the epidemic spreading promotes the information diffusion in social networks, while the information spreading suppresses the epidemic spreading [30], [31], [32], [33], [34].
In this paper, we mainly focus on the cooperate epidemics, while the researches about other interacting mechanisms can be found in the recent review [13]. Cai et al. [35] proposed a model of cooperative multiple SIR epidemic, and the system experienced a mixed discontinuous transition. The influence of network topology on dynamics is also the focus of the attention of scholars. Hébert-Dufresne et al. pointed out that the spread of disease in clustered networks is faster than in non-clustered networks [36]. Grassberger et al. [37] Further discussed the influence of network topology on spreading dynamics of cooperative disease and revealed the importance of loops on discontinuous phase transition in the network. Soriano-Pnos et al. [38] proposed a unified theoretical method to describe the transmission dynamics of multiple epidemics competing and promoting each other, the coupling partner of the two processes is indicated by a scaling parameter , if , diseases promote each other, otherwise diseases compete with each other. In addition, resource control is also a matter of concern. Li et al. [39] study resource control strategies in the scenario of mutually promoting disease transmission, and they found that when available resources are limited, it may be more effective to inhibit diseases with low infection rates preferentially.
Most of the above researches are based on a simple graph (i.e., networks only exist pairwise interactions), which neglected the impacts of higher-order interactions (e.g., school attendance, meetings, and parties) in the systems [40], [41], [42], [43], [44], [45], [46]. In real systems, there exists the pairwise and higher-order interactions among individuals simultaneously and can be described as higher-order networks (or hypergraph) [47], [48], [49], [50], [51]. Researchers have revealed that higher-order networks markedly affect the dynamical characteristics and phase transition of synchronization, game, and single epidemic spreading dynamics. The hypergraph provides an unconstrained higher-order structure [52], [53], [54], [55], and nodes interact together in some ways to form a hyperedge. However, this extreme flexibility also brings high complexity. Jhun et al. [56] considered the SIS infection problem on scale-free uniform hypergraphs and found that a continuous or hybrid epidemic transition occurs when the hub effect is dominant or weak, respectively. Theoretically, simplicial complex [57] is a widely higher-order network model since its solvability. A simplicial complex is a higher-order structure composed of simplices, and the nodes are connected through simplex types. If a simplex belongs to a simplex complex, all subsets in the simplex belong to the simplicial complex. Iacopini et al. [58] found that there are discontinuous transition, hysteresis loops, and bistable phenomena in the simplicial epidemic model. Following this work, researchers investigated the effects of network heterogeneous [59] and temporality [40].
Except for a few examples, little is known about the coevolution of epidemics on higher-order networks. Li et al. [60], used the mean-field theory to study the competing two epidemics on the simplicial complex. When the higher-order interaction is strong, the system phase diagram is divided into nine regions, and the system has a hysteresis loop, discontinuous phase transition, and alternate dominant region. However, the mean-field theory ignores the heterogeneity among individuals, limiting the preservation of contact networks’ structural information and the tracking of individual node evolution. Nie et al. made improvements based on this work, and they use Microscopic Markov Chain Approach (MMCA) to track the state of each node in the simplicial complex, which further improves the accuracy of the competing dynamics on the real-world simplicial complex.
The studies in Refs. [60] consider scenarios where two epidemics compete with each other. In the real world, the cooperative propagation model on higher-order interactions also has research significance. For example, if the host is infected by the Corona Virus 2019 (COVID-19), it will increase the possibility of hepatitis [61]. In this paper, we have focused on the cooperative spreading dynamics on higher-order networks. To describe the circumstances, we propose a mathematical cooperative epidemic model on a simplicial complex. First, we use the Microscopic Markov Chain Approach(MMCA) to analyse the evolution equations in each state. In order to facilitate the study of outbreak threshold, we also introduce the homogeneous mean-field method to linearized the evolution equations. We found that the invasion threshold is always a fixed value, and the persistence threshold decreases non-linearly as the higher-order infection strength increases. Next, according to a large number of simulation results, we find that any large reinforcement effect will lead to a hysteresis loop in the system. At the same time, we also know that the dynamical reinforcement effect makes the system easier to global outbreak than the structural reinforcement effect under the same conditions. In this paper, we introduce our model on simplicial complex in Section 2. In Section 3, we use the microscopic Markov chain approach and mean-field method for theoretical analysis. We analyse the results in Section 4, and finally, summarize in Section 5.
Section snippets
Model descriptions
We propose a cooperative epidemic spreading on simplicial complex, which includes two types of cooperative effect (or reinforcement effect): structural and dynamical reinforcement effect. In the following, we first introduce the simplicial complex and then describe the cooperative epidemic spreading dynamics.
Microscopic Markov Chain Approach
In this paper, we use the Microscopic Markov Chain Approach (MMCA) [63], [64], [65], [66], [67] to track the evolution of every node. On the basis of the model description in the precious section, a node has SS, IS, SI and II four possible states at any time. We use to represent the probability that node is in the state at time . According to the microscopical rules mentioned above, the evolution of is as follows
Numerical simulation results analyses
In this section, we carry out some theoretical and computer simulations on cooperative epidemic spreading dynamics on simplicial complex networks. The basic parameters of the network are , the average degree of 1-simplex and 2-simplex and , respectively, and the recovery probability . In order to express the reinforcement effect more conveniently, we rewrite dynamical reinforcement effect and as and , respectively. In this paper, for
Conclusion
In this paper, we explored the cooperative spreading of two epidemics in a simplicial complex. In order to better find the final spreading size of the two diseases, we use the Microscopic Markov Chain Approach(MMCA) for theoretical calculation. At the same time, we also use the theoretical analysis method of homogeneous mean-field to solve the threshold point of the global outbreak of two viruses. We found that due to the mutual promotion of the structural reinforcement effect and the dynamic
CRediT authorship contribution statement
Xiaoyu Xue: Project administration, Validation, Formal analysis, Writing – original draft. WenYao Li: Project administration, Validation, Formal analysis, Writing – original draft. Yanyi Nie: Project administration, Formal analysis. Xun Lei: Formal analysis. Tao Lin: Formal analysis. Wei Wang: Conceptualization, Project administration, Validation, Formal analysis, Writing – original draft.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
This work was partially supported by the Social Science Foundation of Chongqing, China, No. 2021PY53, Natural Science Foundation of Chongqing, China, No. cstc2021jcyj-msxmX0132, Natural Science Foundation of Yuzhong District, Chongqing, China, No. 20210117, and National Natural Science Foundation of China under Grants No. 61903266.
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These authors have contributed equally to this work.