A new three-layer metapopulation networks model coupling online-offline social interactions with epidemic spread

Abstract

The dissemination of epidemic-related information can effectively enhance public awareness of preventive measures and decrease infection risks. However, information dissemination through multiple channels, including both online and offline sources, and the dynamics of information dissemination across these channels are influenced by individual mobility. To address this gap, this study develops a three-layer metapopulation network coupled model that integrates online information dissemination, epidemic spreading, and offline information dissemination. The proposed model systematically investigates the patterns of information dissemination through both online and offline channels, as well as the epidemic reaction–diffusion process driven by individual migration. Furthermore, the social interactions dynamics of both information channels are influenced by migration, manifested as migrated individuals acquire offline information in destination patches and establish online connections with individuals residing there. Theoretical analysis using the Microscopic Markov Chain Approach (MMCA) derives epidemic thresholds, while Monte Carlo (MC) simulations validate the model and examine factors influencing propagation dynamics. Results indicate that migration enhances individuals’ offline information acquisition, and accelerates cross-patch online information dissemination. Both online and offline information interactions exert inhibitory effects on epidemic spread, effectively increasing the infection threshold and reducing the outbreak scale. Furthermore, increasing the connectivity and dissemination rate of online information, particularly the information dissemination via weak relationships between patches, can effectively inhibit epidemic outbreaks. Notably, when the suppression effect of information on the epidemic is weak, population migration facilitates epidemic spread. In contrast, when the suppression effect is strong, migration-driven information dissemination significantly reduces the probability of infection, thereby further suppressing the outbreak.

1. Introduction

The epidemics have long posed a substantial threat to global public health and socioeconomic stability, such as the global spread of COVID-19, which has had an enormous impact on the safety of human life and the world economy since its outbreak (Morens et al., 2004; Tonnelier et al., 2025; Xue et al., 2026). Accelerated urbanization and intensified human activities have significantly altered epidemic transmission patterns, propelling complex network theory as a paradigm-shifting tool for epidemiological research. A large number of scholars have extensively studied the mechanisms of epidemic transmission through complex network theory, modeling and exploring the important factors influencing the spread of epidemics (Al-Amery et al., 2025; Iacopini et al., 2019; Xu et al., 2019).

The spread of epidemic is not only influenced by biological factors (e.g., viral infectivity, host immunity), but is also closely related to human behavior and social factors (Elsonbaty et al., 2026; Liu, Geng, et al., 2026; Wu & Huo, 2025). Studies have shown that information dissemination can significantly affect the dynamics of epidemic transmission by altering the individuals’ sense of protection (Liu, Peng, et al., 2026; Masoomy et al., 2023; Yu & Huo, 2025). It is well known that the spread of an epidemic may lead to the diffusion of epidemic-related information (Hong et al., 2022). In turn, individuals who are aware of epidemic-related information will take self-protective measures to avoid contracting the epidemic (Fan et al., 2022; Li et al., 2024). Therefore, the multi-layer network of information and epidemic coupling becomes an important framework for modeling the spread of epidemics (Chen et al., 2023; Granell et al., 2013; Liu et al., 2025; Zhang, Zhu, et al., 2025). For example, Ma et al. (Ma & Wang, 2025) proposed a UAU-SIS model based on a two-layer community structure and analyzed the effect of community structure on the spread of epidemic in a multi-layer network. Zhang et al. (Zhang et al., 2023) proposed an activity-driven multiplex framework with partial node mapping between layers and simplicial-complex interactions, showing that higher-order information interactions and interlayer mapping patterns can substantially affect epidemic spreading. Sagar et al. (Sagar et al., 2018) investigated the effect of time-varying transmission rates on the co-evolution of epidemics on a two-layer multiplexed network, where the infection rate in the epidemic layer depends on its perception probability in the awareness layer, and the information transmission rate in the awareness layer depends on its infection probability in the epidemic layer.

In the real world, information related to epidemics is diverse (Gallotti et al., 2020). For example, diversified information access channels, including offline information dissemination (Liu et al., 2022), online information dissemination (Saggu & Sinha, 2020), and mass media information dissemination (Granell et al., 2014), among others. Different information dissemination mechanisms have different impacts on epidemic transmission. Du et al. (Du & Fan, 2024) introduced the SHI₁I₂R information dissemination model for investigating the interaction dynamics of information in two-layer networks, online and offline, in emergency situations, and showed that information spreads faster and more widely in two-layer networks than in single-layer networks. Sun et al. (Sun et al., 2024) examined the SIRS-UAU disease-awareness transmission model incorporating the influence of mass media, noting that the implementation of mass media broadcasts contributed to an increase in the incidence of awareness and a decrease in the proportion of recovered individuals. In addition, during the epidemic spread, relevant information spreads not only positive information that is useful for epidemic prevention and control, but also negative information (Chen et al., 2022). Bao et al. (Bao & He, 2025) developed an epidemic model with an awareness cascade of positive and negative information on a time-delay multiplex networks, and investigated the effect of the awareness cascade of positive and negative information on the model. There may also be competition (Fang et al., 2023) or cooperation (Zhang et al., 2022) between the various types of information. Xiao et al. (Xiao et al., 2019) proposed the MM-SIS (Multiple information and Multiplex network-SIS) model to explore the propagation mechanism of multiple information in multiplexed networks. Meanwhile, the concept of influence factor is introduced to describe the complex interaction between different information.

Complex network theory provides an analytical tool for the spatial heterogeneity of epidemic transmission. Traditional contact models assume that epidemic transmission is characterized by significant spatial clustering through interactions between individuals, and in recent years, research has been further extended to metapopulation networks (Vikram et al., 2025; Wang et al., 2022). In metapopulation network models, where nodes represent different patches (e.g., regions and cities) and links represent migration routes between these patches, metapopulation models have been widely used to explore epidemiologic trends that depend on spatial structure (Tanaka et al., 2014; Valgañón et al., 2022). Colizza et al. (Colizza et al., 2007) first proposed a reaction-diffusion (RD) model in metapopulation networks, analyzing the effects of individual mobility and network heterogeneity on transmission. Gómez et al. (Gómez-Gardeñes et al., 2018) developed a Migration-Interaction-Return (MIR) framework based on reaction-diffusion model that takes into account the recurrent mobility patterns of human movement in response to underlying epidemiological processes, spatial distribution of populations, and commuting mobility patterns. Recent studies have further considered the spread of epidemics in multiple metapopulation networks (Han et al., 2025; Soriano-Paños et al., 2018; Wang et al., 2025). Zhang et al. (Zhang, Hong, et al., 2025) in order to explore the co-effects of herd awareness and resource allocation on the spatial spread of epidemics, a three-layer metapopulation network model was proposed to characterize the complex interactions between information dissemination, resource allocation, and epidemic spreading. From the perspective of considering individual behavior (e.g., individual mobility), Zhou et al. (Zhou et al., 2024) proposed a two-layer network framework, with an individual contact network at the information layer and a reaction-diffusion metapopulation network at the disease layer, and explored the coupling mechanism between information dissemination and epidemic spreading.

Existing studies have investigated the coupled dynamics of online information dissemination and physical epidemic spread. However, offline social interaction remains a crucial channel for information dissemination, as individuals continue face-to-face communication or receive broadcasts within their residential patches. Moreover, within metapopulation networks, individual mobility redistributes populations across regions through reaction–diffusion processes, thereby changing individuals’ physical locations. Consequently, migrating individuals shifted their information sources from their residential patches to their destination patches in the offline information dissemination channel, while simultaneously adjusting their connections in online information channels—for instance, by forming new virtual connections with individuals in their destination patches. The physical movement of individuals introduces time-varying characteristics to information environments and connections, thereby facilitating cross-patch information exchange—features that existing network frameworks fail to capture.

To bridge these research gaps, we propose a novel three-layer network framework that integrates an epidemic reaction-diffusion process in metapopulation network, an offline information propagation network, and an online information contact network. Considering the interplay between individual migration and dual-channel information dissemination, the model systematically investigates how the offline information environment in residential and destination patches, together with migration-driven time-varying online information connections, co-evolve with epidemic transmission. The main innovations of this study are as follows:

• (1)We propose a three-layer coupled metapopulation network model consisting of an epidemic spread layer, an offline information layer, and an online information layer. The epidemic spread layer represents the reaction-diffusion process of epidemic spread across patches; the offline information layer represents the offline information dissemination process in the metapopulation network; while the online information layer represents the online information dissemination process in the individual’s virtual social network.
• (2)The model examines the effect of individual migration on offline information dissemination. During movement from their residential patch to a destination patch, individuals exclusively access and adapt to the offline information environment of their new location. Specifically, the individuals’ source of offline information switches from the offline information dissemination in their residential patch to those in the destination patch.
• (3)The model further captures the effect of individual migration on online information dissemination. Upon arriving at the destination patch, increased offline interaction with local residents motivates migrants to establish new online informational weak relationships with a certain probability, thereby promoting cross-patch information exchange.

The rest of this paper is organized as follows. In Section 2, we introduce our model and theoretically deduce the epidemic spreading threshold by MMCA. Subsequently, we apply MC simulations to validate the accuracy of MMCA, and explore the effects of various factors on both information dissemination and epidemic spreading in Section 3. Finally, we discuss our findings and present a summary of our work in Section 4.

2. A coupling three-layer metapopulation network model

2.1. Model description

During epidemics, individuals acquire epidemic-related information through multiple channels. Traditionally, they are exposed to offline dissemination channels in their physical environment, such as community bulletins and broadcast announcements. However, with the rapid development of the internet, they engage in online information exchange with friends and family through digital devices and internet platforms. Furthermore, convenient transportation systems enable frequent population mobility across cities and regions, including commuting and travel. When an individual moves to a new area, they encounter local offline epidemic prevention publicity, which reshapes their immediate information environment. Simultaneously, migration prompts individuals to establish new online connections with local residents through social media and other digital platforms, facilitating additional information exchange (Zhou et al., 2024). Such online connections established between migrating individuals and local residents at their destinations are temporary and unstable, and are defined as online informational weak relationships (such as nodding acquaintances or temporary partners) (Feng et al., 2024). Consequently, human mobility facilitates cross-regional information flow and integration.

In this paper, we propose a three-layer network model to characterize the coupled interactions among epidemic spreading, offline information dissemination, and online information dissemination, as shown in Fig. 1. The epidemic spread layer ($L$) represents the epidemic reaction-diffusion metapopulation network, where nodes correspond to patches, and a homogeneous mix of individuals within the patches. Within each patch, individuals are homogeneously mixed, with epidemic transmission governed by intra-patch reaction processes and inter-patch diffusion through individual migration. The offline information layer ($V_1$) represents the offline information dissemination network, with nodes as patches (one-to-one correspondence with the patches in the epidemic spread layer), and individuals within patches accessing information through their residential patches or neighboring patches. The online information layer ($V_2$) represents the online information dissemination network, where the nodes are individuals (one-to-one correspondence with the individuals in the epidemic spread layer), and information diffuses through connections between individuals. The specific epidemic and information interaction processes in each layer of the network are shown in Fig. 2.

Fig. 1.

A coupling three-layer network framework. The middle epidemic spread layer ($L$) network defines the propagation of the epidemic, where the large circles as nodes are patches, the links between nodes represent individual migration paths, and the particles within each circle are resident individuals in two possible states: susceptible state ($S$) and infected state ($I$). The middle and upper layers describe the information dissemination process, where the lower layer ($V_1$) is the offline information dissemination network, the nodes are patches containing certain individuals, and the links between nodes and nodes correspond to the $L$ layer. Individuals have two possible states: unaware state ($U$) and aware state ($A$). The upper layer ($V_2$) is the online information dissemination network, the nodes are individuals, which correspond one-to-one with the individuals in the $L$ layer, and the blue solid lines between the nodes indicate strong relationships for online information and the blue dashed lines indicate weak relationships for online information.

Fig. 2.

Specific interaction process in the three-layer network. (a) The Migration-Interaction-Return (MIR) process of the epidemic within a time step. Node $i$ denotes the migrating individual. (b) The disseminating process of offline information. Individuals that do not migrate during time step $t$ can only access information at the residence patch; while migrating individuals access offline information about their destination patch. (c) The disseminating process of online information. After migration, migrating individuals have a probability $\delta$ of establishing an online information weak relationship with destination patch individuals. After return, this connection has a probability $\tau$ of disconnecting in subsequent time steps.

Table 1 lists the main parameters in this paper.

Table 1.

Definition of parameters.

Process	Parameter	Definition
Information	$U$	The unaware state
	$A$	The aware state
	${\lambda }_1$	The offline information dissemination rate
	${\lambda }_2$	The online information dissemination rate
	${\lambda }_2^{\prime }$	The online information weak relationships dissemination rate
	$\alpha$	The decay rate of online weak relationships
	$\delta$	The online information weak relationships connection rate
	$\tau$	The online information weak relationships disconnection rate
	$\gamma$	The information forgetting rate
Epidemic	$S$	The susceptible state
	$I$	The infected state
	$\beta$	The epidemic infection rate
	$\mu$	The epidemic recovery rate
	$\theta$	The inhibitory effects of aware individuals on epidemic
Migration	$p$	The migration rate
	${\omega }_{N{N}^{\prime }}$	The weight of the edges between patch $N$ and patch $N^{\prime }$
Network	$L$	The epidemic reaction-diffusion metapopulation network
	$V_1$	The offline information dissemination network
	$V_2$	The online information dissemination network
	$N$	The number of patches
	$n$	The total number of individuals
	$A,a_{ij}\in A$	The online information strong relationships adjacency matrix
	$B\left(t\right),b_{ij}\left(t\right)\in B\left(t\right)$	The online information weak relationships adjacency matrix
	$t$	The time step

The $L$ layer describes the epidemic spread process, employing the Susceptible-Infected-Susceptible ($SIS$) model. There are two possible states of an individual: susceptible ($S$) or infected ($I$). Simultaneously, the reaction-diffusion process is used to characterize the spatial spread of the epidemic between patches. Following the reaction-diffusion framework introduced by (Gómez-Gardeñes et al., 2018) in the context of epidemic transmission dynamics, which describes periodic recurrent migration patterns, the proposed model consists of three steps at each time step: migration, interaction and return. As shown in Fig. 2(a).

• (1)Migration (M): first, any individual $i$ selects to migrate out of their residential patch $N$ with migration rate $p$, and determines the destination patch $N^{\prime }$ through the inter-patch transition matrix $W$, $W_{N{N}^{\prime }}={\omega }_{N{N}^{\prime }}/{\sum }_{N^{″}}{\omega }_{N{N}^{″}}$, where ${\omega }_{N{N}^{\prime }}$ denotes the weight of the edges between patch $N$ and patch $N^{\prime }$.
• (2)Interaction (I): after migration, epidemic interactions occur in each patch. Susceptible individuals ($S$) acquire infection through contact with infected counterparts ($I$) at a transmission rate $\beta$, while infected individuals ($I$) spontaneously revert to the susceptible state with a recovery rate $\mu$.
• (3)Return (R): finally, migrating individuals return to the residential patches after updating their status and the next time step begins.

The information dissemination layers capture the propagation dynamics of epidemic-related information, employing the unaware-aware- unaware ($UAU$) model. Each individual may receive information and become aware of the epidemic, so the individual may be in one of two states: unaware ($U$) or aware ($A$).

The $V_1$ layer describes the offline dissemination dynamics of information. Typically, unaware individuals become more alert to the epidemic and gradually become aware of the risk by interacting with aware individuals or information campaigns within the patch. It is reasonable to assume that the offline information dissemination rate exhibits a positive correlation with the local awareness density. Therefore, the probability that an individual at time $t$ receives information offline within patch $N$ is defined as ${\lambda }_N^1\left(t\right)$,

$${\lambda }_N^1\left(t\right)={\rho }_N^A\left(t\right){\lambda }_1$$ (1)

where ${\lambda }_1$ is the offline information dissemination rate and ${\rho }_N^A\left(t\right)$ is the proportion of aware individuals in patch $N$.

Furthermore, for an individual migrating at time $t$, the recurrent migration pattern leads to movement from the residence patch to the destination patch. As a consequence, individuals that do not migrate during time step $t$ can only access information at the residence patch $N$; while migrating individuals access offline information about their destination patch $N^{\prime }$, as shown in Fig. 2(b). Consequently, we define $r_N^1\left(t\right)$ as the probability that an unaware individual in patch $N$ has not been disseminated information through offline information interaction at time $t$ as below:

$$r_N^1\left(t\right)=p\sum _{N^{\prime }}{\omega }_{N{N}^{\prime }}\left(1-{\lambda }_{N^{\prime }}^1\left(t\right)\right)+\left(1-p\right)\left(1-{\lambda }_N^1\left(t\right)\right)$$ (2)

where the first term on the right of Eq. (2) represents the probability that a migrated individual did not obtain information in the destination patch, and the second term represents the probability that a non-migrated individual did not obtain information in the residential patch.

The $V_2$ layer describes the online dissemination dynamics of information. Individuals also interact with each other online through virtual socialization, and it is assumed that individuals have stable online friendships with their relatives and friends, i.e., online information strong relationships (Friedkin, 1982). These strong relationships persist over time and are not disrupted by individual migration. The online information strong relationship adjacency matrix between individuals is defined as $A,a_{ij}\in A$, and $a_{ij}=1$ indicates that there is an online information strong relationship between individual $i$ and individual $j$; $a_{ij}=0$ indicates that there is no online information strong relationship between individual $i$ and individual $j$.

For migrated individuals, their offline interactions with destination patches become closer, thus affecting online socialization. Individuals have the probability to establish online informational weak relationships (Larson, 2017) (connecting edges with low frequency of interactions or weak relationships, such as nodding acquaintances or temporary partners) with individuals in the patch where they are located after migration is completed, as shown in Fig. 2(c). Specifically, after an individual migrates offline to a destination patch, the individual’s online information social network also has a probability of generating an information weak connection with an individual within that destination patch, with a connection rate $\delta$. Thus, the probability that individual $i$ in patch $N$ and individual $j$ in patch $N^{\prime }$, who are not already connected to each other, establish an online informational weak relationship at time $t$ can be expressed as follows:

$$g_{Ni,N^{\prime }j}\left(t\right)=\delta \left[p\left(1-p\right){\omega }_{N{N}^{\prime }}+p\left(1-p\right){\omega }_{N^{\prime }N}+p^2\sum _{N^{″}}{\omega }_{N{N}^{″}}{\omega }_{N^{\prime }{N}^{″}}\right]$$ (3)

where the three parts of Eq. (3) denote the probability that individual $i$ migrates to the patch where individual $j$ is located and establishes a connection with individual $j$ who stays in residence; the probability that individual $j$ migrates to the patch where individual $i$ is located and establishes a connection with individual $j$ who stays in residence; and the probability that both individual $i$ and individual $j$ migrate to another patch and establish a connection at the same time.

For offline information dissemination, individuals are affected by migration only in relation to the migration status of the individual at the current time step, and are not affected by previous time steps, i.e., migrating individuals will no longer receive offline information transfers from the migrating patch once they return to the residential patch.

However, for online information dissemination, it is assumed that individual $i$ establishes an online informational weak relationship with individual $j$ of the migration destination patch due to migration, and the connection will not be disconnected immediately with the return of the individual $i$, i.e., the online social relationship due to migration is time-delayed, and the weak connection formed by an individual at time $t$ is disconnected at subsequent moments with a probability $\tau$. Let the online information weak relationship adjacency matrix between individuals be $B\left(t\right),b_{ij}\left(t\right)\in B\left(t\right)$, and $b_{ij}\left(t\right)=1$ denotes that individual $i$ and individual $j$ have an online information weak relationship at time $t$, and $b_{ij}\left(0\right)=0$ means that there is no online information weak relationship. Thus, the evolution equation for the probability of the existence of an online information weak relationship between individual $i$ and individual $j$ at time $t$ is expressed as:

$$b_{ij}\left(t+1\right)=b_{ij}\left(t\right)\left(1-\tau \right)+\left(1-b_{ij}\left(t\right)\right)\left(1-a_{ij}\right)g_{Ni,N^{\prime }j}\left(t\right)$$ (4)

where the first term on the right of Eq. (4) represents the probability that the online information weak relationship is not broken, and the second term represents the probability that the online weak relationship is established between individuals without an online information strong relationship. ($a_{ij}=0$).

Therefore, we define $r_{N,i}^2\left(t\right)$ as the probability that an unaware individual $i$ in patch $N$ has not been disseminated information through online information interaction at time $t$ as below:

$$r_{N,i}^2\left(t\right)={\prod }_j\left(1-a_{ij}{\lambda }_2{\rho }_j^A\left(t\right)\right)\left(1-b_{ij}\left(t\right){\lambda }_2^{\prime }{\rho }_j^A\left(t\right)\right)$$ (5)

where ${\rho }_j^A\left(t\right)$ denotes the probability that individual $j$ is in the aware state. ${\lambda }_2$ is the online strong relationship information dissemination rate and ${\lambda }_2^{\prime }$ is the online weak relationship information dissemination rate, ${\lambda }_2^{\prime }=\left(1-\alpha \right){\lambda }_2$, $\alpha$ represents the decay rate of online weak relationship. Accordingly, $1-a_{ij}{\lambda }_2{\rho }_j^A\left(t\right)$ represents the probability that individual $i$ does not receive information from aware individual $j$ through their online strong relationship, and $1-b_{ij}\left(t\right){\lambda }_2^{\prime }{\rho }_j^A\left(t\right)$ represents the corresponding probability for their online weak relationship.

Combining both offline and online information interactions, we define $r_{N,i}\left(t\right)$ as the comprehensive probability that an unaware individual $i$ in patch $N$ transitions to an aware state at time $t$, which is computed as follows:

$$r_{N,i}\left(t\right)=1-r_N^1\left(t\right)r_{N,i}^2\left(t\right)$$ (6)

In addition, an aware individual cannot maintain awareness of the epidemic information and forgets the information at a forgetting rate $\gamma$.

By integrating individual states in the three-layer network, each individual may be in one of four coupled states: unaware susceptible ($US$), unaware infected ($UI$), aware susceptible ($AS$), or aware infected ($AI$). The state transfer probability tree is shown in Fig. 3. We define the probability that a $US$ state individual in a plaque is infected at time $t$ as ${\beta }^U=\beta$. On the contrary, AS state individuals will take protective measures, such as wearing masks and getting vaccinated, to avoid infection due to the knowledge of epidemic-related information, so the infection rate can be modified as ${\beta }^A=\theta \beta$, where $\theta \in \left[0,1\right]$ is a parameter characterizing the information inhibitory intensity.

Fig. 3.

The individual states transition trees. (a)-(d) show the state transfer probability trees for $US,AS,UI,AI$ state individuals, respectively, where nodes are states, branches are transition processes, and weights on branches are probabilities of moving from one state to another, with the first level of branches denoting the individual information state transfer probability, and the second level of branches denoting the individual epidemic state transfer probability.

2.2. Theoretical analysis

The quantities ${\rho }_N^{US}\left(t\right)$, ${\rho }_N^{UI}\left(t\right)$, ${\rho }_N^{AS}\left(t\right)$, ${\rho }_N^{AI}\left(t\right)$ denote the respective percentages of individuals occupying the states $US$, $UI$, $AS$ and $AI$ at time step $t$ of patch $N$.The Markovian evolution (Zhang et al., 2025c, 2026) governing the dynamics of these four states unfold as follows:

$$\{\begin{array}{l}{\rho }_N^{US}\left(t+1\right)=\frac{{\sum }_i{\rho }_{N,i}^S\left(t\right)U_{N,i}\left(t+1\right)}{n_N}\left(1-{\pi }_N^U\left(t\right)\right)+\frac{{\sum }_i{\rho }_{N,i}^I\left(t\right)U_{N,i}\left(t+1\right)}{n_N}\mu \\ {\rho }_N^{UI}\left(t+1\right)=\frac{{\sum }_i{\rho }_{N,i}^I\left(t\right)A_{N,i}\left(t+1\right)}{n_N}{\pi }_N^U\left(t\right)+\frac{{\sum }_i{\rho }_{N,i}^I\left(t\right)U_{N,i}\left(t+1\right)}{n_N}\left(1-\mu \right)\\ {\rho }_N^{AS}\left(t+1\right)=\frac{{\sum }_i{\rho }_{N,i}^S\left(t\right)A_{N,i}\left(t+1\right)}{n_N}\left(1-{\pi }_N^A\left(t\right)\right)+\frac{{\sum }_i{\rho }_{N,i}^I\left(t\right)A_{N,i}\left(t+1\right)}{n_N}\mu \\ {\rho }_N^{AI}\left(t+1\right)=\frac{{\sum }_i{\rho }_{N,i}^S\left(t\right)A_{N,i}\left(t+1\right)}{n_N}{\pi }_N^A\left(t\right)+\frac{{\sum }_i{\rho }_{N,i}^I\left(t\right)A_{N,i}\left(t+1\right)}{n_N}\left(1-\mu \right)\end{array}$$ (7)

where ${\rho }_{N,i}^S\left(t\right)$, ${\rho }_{N,i}^I\left(t\right)$ denotes the probability that individual $i$ in the patch $N$ at time $t$, is in a different epidemic state. The shifts in the state of awareness are described as follows:

$$\{\begin{array}{l}{U}_{N,i}\left(t+1\right)=U_{N,i}\left(t\right)\left(1-r_N\left(t\right)\right)+A_{N,i}\left(t\right)\gamma \\ A_{N,i}\left(t+1\right)=A_{N,i}\left(t\right)\left(1-\gamma \right)+U_{N,i}\left(t\right)r_N\left(t\right)\end{array}$$ (8)

In Eq. (7), ${\pi }_N^U\left(t\right)$, ${\pi }_N^A\left(t\right)$ can be expressed as:

$$\{\begin{array}{l}{\pi }_N^U\left(t\right)=\left(1-p\right)P_N^U\left(t\right)+p\sum _{N^{\prime }}{W}_{N{N}^{\prime }}{P}_{N^{\prime }}^U\left(t\right)\\ {\pi }_N^A\left(t\right)=\left(1-p\right)P_N^A\left(t\right)+p\sum _{N^{\prime }}{W}_{N{N}^{\prime }}{P}_{N^{\prime }}^A\left(t\right)\end{array}$$ (9)

where the first term to the right of ${\pi }_N^U\left(t\right)/{\pi }_N^A\left(t\right)$ defines the probability that a $US/AS$ status individual in the patch $N$ is infected while remaining in the patch $N$ at time $t$, and the second term denotes the probability that a $US/AS$ status individual is infected while migrating to a neighboring patch $N^{\prime }$. In addition, $P_N^U\left(t\right)/P_N^A\left(t\right)$ denotes the probability that a susceptible individual ($US/AS$) located in the patch $N$ (but not necessarily belonging to the patch) is infected by an infected individual in the patch $N$ at time $t$. Under the mixed approximation, the meaning is:

$$\{\begin{array}{l}{P}_N^U=1-{\prod }_{N^{\prime }}{\left(1-{\beta }^U{\rho }_{N^{\prime }}^I\left(t\right)\right)}^{n_{N^{\prime }N}\left(t\right)}\\ P_N^A=1-{\prod }_{N^{\prime }}{\left(1-{\beta }^A{\rho }_{N^{\prime }}^I\left(t\right)\right)}^{n_{N^{\prime }N}\left(t\right)}\end{array}$$ (10)

where $n_{N^{\prime }N}\left(t\right)$ is the total number of individuals moving from patch $N^{\prime }$ to patch $N$. $n_{N{N}^{\prime }}\left(t\right)$ is given by the equation:

$$n_{N^{\prime }N}\left(t\right)=\left(1-p\right){\varphi }_{N{N}^{\prime }}{n}_{N^{\prime }}+p{W}_{N^{\prime }N}{n}_{N^{\prime }}$$ (11)

where ${\varphi }_{N{N}^{\prime }}$ is the Kronecker delta function, if $N=N^{\prime }$, then ${\varphi }_{N{N}^{\prime }}=1$; otherwise ${\varphi }_{N{N}^{\prime }}=0$.

When $t\to \infty$, the information dissemination and epidemic spread in the coupled model will reach a steady state, and ${\rho }_N^x\left(t+1\right)\approx {\rho }_N^x\left(t\right)={\rho }_N^x$ holds, $x\in \left\{US,AS,UI,AI\right\}$. When $\beta \to {\beta }_c$, the number of infected individuals in the patch $N$ is negligible, i.e., ${\rho }_{N,i}^I\left(t\right)\ll 1$, ${\rho }_{N,i}^S\left(t\right)\approx 1$, ${\rho }_N^I={\epsilon }_N^{\ast }\ll 1$. The equations for ${\rho }_N^U$ and ${\rho }_N^A$ can be obtained from Eq. (8):

$${\rho }_N^U=\frac{{\sum }_i{U}_{N,i}}{n_N},{\rho }_N^A=\frac{{\sum }_i{A}_{N,i}}{n_N}$$ (12)

Substituting Eq. (12) into Eq. (7) gives:

$$\mu {\epsilon }_N^{\ast }={\rho }_N^U{\pi }_N^U+{\rho }_N^A{\pi }_N^A$$ (13)

Moreover, when $\beta <{\beta }_c$, ${\rho }_N^I\ll 1$ is small enough for each patch. Eq. (10) can be approximated as:

$$\{\begin{array}{l}{P}_N^U=1-{{\prod }_{N^{\prime }}\left(1-{\beta }^U{\rho }_{N^{\prime }}^I\left(t\right)\right)}^{n_{N^{\prime }N}}\approx \sum _{N^{\prime }}{n}_{N^{\prime }N}{\beta }^U{\rho }_{N^{\prime }}^I\left(t\right)\\ P_N^A=1-{{\prod }_{N^{\prime }}\left(1-{\beta }^A{\rho }_{N^{\prime }}^I\left(t\right)\right)}^{n_{N^{\prime }N}}\approx \sum _{N^{\prime }}{n}_{N^{\prime }N}{\beta }^A{\rho }_{N^{\prime }}^I\left(t\right)\end{array}$$ (14)

Substituting Eq. (14) into Eq. (9) and combining it with Eq. (13), the epidemic threshold can be obtained as:

$$\frac{\mu }{\beta }{\epsilon }_N^{\ast }=\sum _{N^{\prime }}{M}_{N{N}^{\prime }}{\epsilon }_{N^{\prime }}^{\ast }$$ (15)

where $M_{N{N}^{\prime }}$ is denoted as:

$$M_{N{N}^{\prime }}=\left({\rho }_N^U+\theta {\rho }_N^A\right)\left[{\left(1-p\right)}^2{\delta }_{N{N}^{\prime }}+p\left(1-p\right)W_{N{N}^{\prime }}+p\left(1-p\right)W_{N{N}^{\prime }}{\delta }_{N{N}^{\prime }}+p^2{\left(W{W}^T\right)}_{N{N}^{\prime }}\right]n_{N^{\prime }}$$ (16)

Epidemic threshold can then be obtained by deriving the maximum eigenvalue of matrix M:

$${\beta }_c=\frac{\mu }{{\mathrm{\Lambda }}_{\max }\left(M\right)}$$ (17)

3. Simulation results

In reality, due to the complexity and uncertainty of epidemic propagation, it is difficult to obtain the detailed propagation form of information dissemination during an epidemic outbreak. Therefore, in order to reveal how information dissemination changes the dynamics of an epidemic, we simulate the co-evolution of offline and online information dissemination-epidemic spreading on a synthetic three-layer network. In particular, the form of information dissemination is complex and usually has a heterogeneous network structure. The online network considers two network structures: the BA network and the ER network, with 4000 individuals and an average degree of 10. There were 20 patches in the offline network with a total number of 4000 individuals. For comparison, we consider two population distributions in the offline layer: homogeneous distribution (HOD) and heterogeneous distribution (HED). Under HOD, each patch contained an average of 200 individuals, reflecting a uniform distribution of the population across patches. Under HED, the number of individuals in each patch is proportional to the sum of its edge weights $W_{N{N}^{\prime }}={\omega }_{N{N}^{\prime }}/{\sum }_{N^{″}}{\omega }_{N{N}^{″}}$, which is determined by the physical network structure.

The MMCA starts with 0.1% infected individuals as seeds in each patch. The same initial condition is applied to the MC simulations for comparison. Epidemic prevalence is a commonly used measure of the extent of disease transmission, defined as ${\rho }^I$, ${\rho }^I=\frac{{\sum }_N{\rho }_N^I{n}_N}{{\sum }_N{n}_N}$, which represents the proportion of infected individuals in the population as a whole. Unless otherwise stated, the parameters $\mu$, $\gamma$, $\alpha$, and $\beta$ were set to 0.2, 0.3, 0.5 and 0.4, respectively. In order to gain a comprehensive understanding of the combined effects of information dissemination and individual mobility on epidemic spread, we next analyzed the effects of key parameters, including offline information dissemination rate ${\lambda }_1$, online information dissemination rate ${\lambda }_2$, information suppression rate $\theta$, and migration rate $p$. In the next sections, we explore the specific impact of these parameters on the spread of the epidemic. For each parameter set, we performed 50 mutually independent MC realizations and calculated the average result. Each realization was run for 200 time steps, and the result was recorded after confirming that the system had reached a stable state.

3.1. Impact of information dissemination on epidemic spread

• (1) Effect of offline information dissemination rate on epidemic spread

First, we explored the effect of information dissemination on the spread of epidemic, as shown in Fig. 4, which reflects the change in infection scale with the offline information dissemination rate ${\lambda }_1$, and it can be seen that as ${\lambda }_1$ increases, the size of epidemic becomes smaller. The change in infection scale is not obvious at a small value of offline information dissemination rate, but when ${\lambda }_1$ exceeds a certain value, the epidemic scale begins to show obvious changes. The above results indicate that the dissemination of offline information can effectively increase the epidemic threshold and reduce the epidemic scale, and this effect is especially obvious when the offline information dissemination rate is large. Furthermore, comparing the results in the homogeneous distribution (HOD) and heterogeneous distribution (HED) metapopulation networks reveals that population heterogeneity exerts negligible influence on the coupled dynamics of information and epidemic spread.

Fig. 4.

The effect of offline information dissemination rate on epidemic size. (a-c) represent the final infection scale ${\rho }^I$ in the panel of $\left(\beta ,{\lambda }_1\right)$ when $p=0.2,0.5,0.8$ in the HOD metapopulation network, respectively. (d-f) represent the final infection scale ${\rho }^I$ in the panel of $\left(\beta ,{\lambda }_1\right)$ when $p=0.2,0.5,0.8$ in the HED metapopulation network, respectively. The white dashed curves denote the epidemic thresholds predicted by the MMCA. The other parameters are set as $\gamma =0.3$, $\mu =0.2$, $\theta =0.2$, ${\lambda }_2=0$ and $\alpha =0.5$.

By comparing the variations in infection scale with the offline information dissemination rate under different migration rate $p$, we found that as $p$ increases, the epidemic threshold increases and the infection scale decreases. This phenomenon differs from previous studies (Colizza et al., 2007; Tanaka et al., 2014; Valgañón et al., 2022; Vikram et al., 2025), and we will analyze it further in Section 3.2.

• (2)Effect of online information dissemination rate on epidemic spread

Then the effect of online information dissemination on the spread of epidemic was explored, as shown in Fig. 5, and the results were similar to those of offline information dissemination. As the online information dissemination rate increases, the epidemic size becomes smaller. When the online information dissemination rate takes a small value, the change in the infection size is not obvious, but when it exceeds a certain value, the epidemic size begins to show obvious changes. Furthermore, comparing the variations in infection scale with the online information dissemination rate under different migration rate $p$, we observe the same conclusion as in Fig. 4: increased migration rates suppress epidemic spread. For the two online information interaction networks, ER and BA, it can be seen that the epidemic scale and infection threshold in the BA network is slightly smaller than that in the ER network, which may be due to the existence of a few highly connected hub nodes in the BA network, which can diffuse information rapidly and thus inhibit the outbreak of epidemic.

Fig. 5.

The effect of online information dissemination rate on epidemic size. (a-c) represent the final infection scale ${\rho }^I$ in the panel of $\left(\beta ,{\lambda }_2\right)$ when $p=0.2,0.5,0.8$ in the ER network, respectively. (d-f) represent the final infection scale ${\rho }^I$ in the panel of $\left(\beta ,{\lambda }_2\right)$ when $p=0.2,0.5,0.8$ in the BA network, respectively. The white dashed curves denote the epidemic thresholds predicted by the MMCA. The other parameters are set as $\gamma =0.3$, $\mu =0.2$, $\theta =0.2$, ${\lambda }_1=0$ and $\alpha =0.5$.

Comparing Fig. 4, Fig. 5 we find that the epidemic size is more sensitive to online information dissemination than offline information. For the same dissemination rate, online information diffusion exerts a stronger epidemic suppression effect than offline information diffusion. When the online information dissemination rate is greater than a smaller value, a significant decrease in epidemic size occurs. When the online information dissemination rate is large, the change in ${\lambda }_2$ almost no longer affects the epidemic size, due to the fact that at this point the probability of an individual obtaining information online is so high that there are almost no unaware individuals. Furthermore, we find that the MMCA-derived theoretical threshold boundaries agree well with the transition boundaries observed in the MC results, thereby validating the effectiveness of the theoretical threshold analysis.

• (3)The synergistic effect of offline and online information dissemination rate on epidemic spread

To further reveal how the epidemic spreading process is affected by the online and offline information dissemination channels, we demonstrate the final density of infected with different combinations of ${\lambda }_1$ and ${\lambda }_2$ in Fig. 6. As shown in the figure, the change of online information dissemination rate has a more significant effect on the epidemic scale, and increasing online information dissemination is more effective in suppressing the spread of epidemic than increasing offline information dissemination, which verifies the above experimental results. Summarizing the above simulation results, we conclude that both offline and online information dissemination have inhibitory effects on epidemic spreading, which can increase the epidemic threshold and reduce the infection scale. However, online information dissemination is more effective, and compared with offline propaganda, improving the propaganda of online epidemic prevention information can increase the prevention and control efforts.

Fig. 6.

The combined effect of offline and online information dissemination rates on epidemic size. (a-c) represent the final infection scale in the panel of $\left({\lambda }_1,{\lambda }_2\right)$ when $p=0.2,0.5,0.8$, respectively. The other parameters are set as $\gamma =0.3$, $\mu =0.2$, $\theta =0.2$, $\beta =0.3$ and $\alpha =0.5$.

To validate the above conclusions, we analyzed changes in epidemic scale under different values of ${\lambda }_1$ and ${\lambda }_2$. As shown in Fig. 7, increasing both ${\lambda }_1$ and ${\lambda }_2$ reduces the epidemic scale, but increasing ${\lambda }_2$ produces a more pronounced effect. Meanwhile, an increase in ${\lambda }_2$ significantly raises the epidemic threshold, whereas variations in ${\lambda }_1$ have minimal impact on it. Overall, simultaneously increasing ${\lambda }_1$ and ${\lambda }_2$ yields the most effective inhibitory effect on epidemic transmission. The results demonstrate that both online information diffusion and offline information dissemination can effectively suppress epidemic spread. During an epidemic, while vigorously promoting the dissemination of online epidemic prevention information, offline information dissemination should not be neglected. Furthermore, by comparing the theoretical and simulation results in Fig. 7, the good agreement between the two further demonstrates the accuracy of the MMCA theoretical analysis and the reliability of the MC simulations.

Fig. 7.

The change in epidemic size at different values of ${\lambda }_1$ and ${\lambda }_2$. Dashed curves with triangular markers represent the MMCA results, whereas solid curves with circular markers represent the MC simulation results. The other parameters are set as $\gamma =0.3$, $\mu =0.2$, $\theta =0.2$, $\beta =0.3$ and $\alpha =0.5$.

In addition, Fig. 4, Fig. 6 above also compared the results in the case of different migration rates. We found that the size of epidemic decreases as $p$ increases. This is at variance with the findings of previous studies, which may be due to the effect of the coupling between the information dissemination mechanism and the migration mechanism on the spread of epidemic. Since the above experiment $\theta$ took a smaller value of 0.2, we specifically investigated the effect of the migration mechanism in the subsequent experiments under different values of $\theta$.

3.2. Interplay between individual migration and information dissemination

• (1)Effect of migration rate on epidemic spread

In this subsection, we further explore the effect of the level of human migration on the spread of the epidemic. Fig. 8 illustrates the variation of the stationary density of infected individuals with the migration rate $p$ for different information inhibitory intensity $\theta$. It can be seen that when the information inhibitory intensity $\theta$ is small, the final infection density ${\rho }^I$ decreases with the increase of the mobility rate $p$, and the epidemic threshold increases accordingly, as shown in Fig. 8(a). However, when the information inhibitory intensity $\theta$ is large, the final infection density ${\rho }^I$ increases with the increase of the mobility rate $p$ and the epidemic threshold decreases, as shown in Fig. 8(b) and (c). This phenomenon explains that the discrepancy between the simulation results and previous studies is attributed to the value of parameter $\theta$. The results show that the effect of the migration rate on the epidemic receives the influence of the information inhibitory effect, and the control or not of population movement in an epidemic need to depend on the different effects of the information on the epidemic suppression.

• (2)The combined effect of migration rate and information inhibitory intensity on epidemic spread

Fig. 8.

The effect of migration rate on epidemic size. (a-c) represent the final infection scale in the panel of $\left(\beta ,p\right)$ when $\theta =0.2,0.5,0.8$, respectively. The other parameters are set as $\gamma =0.3$, $\mu =0.2$, ${\lambda }_1=0.2$, ${\lambda }_2=0.1$ and $\alpha =0.5$.

To further explore the combined effect of information dissemination and individual migration on the spread of epidemic, we demonstrate the final density of infected with different combinations of $p$ and $\theta$ in Fig. 9(a). As shown in the figure, the size of the infection decreases as $\theta$ decreases, and migration inhibits the spread of the epidemic when the information inhibitory intensity $\theta$ is small. However, as $\theta$ increases, the inhibitory effect of migration is limited. When $\theta$ exceeds a certain value, the inhibitory effect is transformed into a facilitating effect, and the increase in the migration rate promotes the spread of the epidemic. To highlight the study of this phenomenon, the variation of the infection size with the value of $\theta$ was analyzed for different values of $p$, as shown in Fig. 9(b). For example, when $\theta =0.1$ and $p$ = 0.1, 0.5, and 0.9, the final infection densities are 0.32, 0.25, and 0.19, respectively, while when $\theta =0.9$ and $p$ is the same, the final infection densities are 0.46, 0.49, and 0.51, respectively.

Fig. 9.

The combined effect of migration rate and information inhibitory intensity on epidemic size. (a) shows the final infection scale in the panel of $\left(p,\theta \right)$. (b) shows the change in epidemic size at different $p$. The other parameters are set as $\gamma =0.3$, $\mu =0.2$, ${\lambda }_1=0.2$, ${\lambda }_2=0.1$ and $\alpha =0.5$.

Thus, we come to the interesting conclusion that the inhibitory intensity of the information is significant when the individual is aware of the information about the epidemic prevention and control and take effective prevention and control measures can significantly reduce the likelihood of infection, when no intervention on population mobility is needed and it is more favorable to promote population mobility. However, when the inhibitory intensity of the information is not obvious, i.e., it is not useful for the aware individual to take protective measures, then mobility needs to be controlled to reduce the likelihood of infection. Measures that combine information dissemination and population mobility can be more effective in controlling the spread of epidemics.

3.3. Impact of associated factors on epidemic spread

• (1)Effect of information suppression rate on epidemic spread

Fig. 10 demonstrates the effect of information suppression rate on epidemic size. From the figure, it can be seen that as the information suppression rate increases, i.e., the value of $\theta$ becomes smaller, the infection size decreases and the epidemic threshold increases. This result indicates that information has an inhibitory effect on epidemic infection, validating the above conclusion. It indicates that in the process of epidemic transmission, the more effective epidemic protection measures taken by aware individuals, the more effective the inhibition of infection. In addition, comparing Fig. 10(a–c), it can be found that as the migration rate $p$ increases, the more sensitive the infection size and threshold are to the change of $\theta$, indicating that the population mobility has a significant effect on the diffusion of information.

• (2)Effect of online weak relationships decay rate on epidemic spread

Fig. 10.

The effect of online information suppression rate on epidemic size. (a-c) represent the final infection scale in the panel of $\left(\beta ,\theta \right)$ when $p=0.2,0.5,0.8$, respectively. The other parameters are set as $\gamma =0.3$, $\mu =0.2$, ${\lambda }_1=0.2$, ${\lambda }_2=0.1$ and $\alpha =0.5$.

The effect of online weak relationships decay rate on epidemic spread was further investigated. The variation of the final infection density with the online weak relationships decay rate is illustrated in Fig. 11. As shown, increasing decay rate $\alpha$ lead to larger infection scales and lower epidemic thresholds. This occurs because weakened online interactions between migrating individuals and destination patch residents hinder cross-patch information flow, thereby diminishing the information’s inhibitory effect on the epidemic. Moreover, the effect of this decay rate $\alpha$ becomes more pronounced as migration rates increase. This occurs because higher migration rate lead to a greater number of online weak relationships. The findings indicate that enhancing the online communication intensity between migrating individuals and residents in different patches can promote cross-patch information exchange, thereby more effectively suppressing epidemic spread.

• (3)Effect of online information weak relationships connection rate on epidemic spread

Fig. 11.

The effect of online weak relationships decay rate on epidemic size. (a-c) represent the final infection scale in the panel of $\left(\beta ,\alpha \right)$ when $p=0.2,0.5,0.8$, respectively. The other parameters are set as $\gamma =0.3$, $\mu =0.2$, ${\lambda }_1=0.2$, ${\lambda }_2=0.1$, $\theta =0.2$ and $\delta =0.5$.

Finally, we explored the effect of weak relationships for online information on epidemic transmission. Fig. 12 demonstrates the effect of online information weak relationships connection rate on the spread of epidemic, from which it can be seen that as connection rate $\delta$ increases, the size of the epidemic decreases significantly and the threshold of infection increases. It shows that individuals establishing more online information exchanges with dissemination patch individuals after migration facilitates information dissemination, thereby suppressing epidemic spread. Furthermore, comparing Fig. 12(a–c) reveals that as the migration rate $p$ increases, more migrants establish additional online connections, leading to a reduction in the final infection scale. This trend thereby validates the conclusion above.

• (4)Effect of online information weak relationships disconnected rate on epidemic spread

Fig. 12.

The effect of online information weak relationships connection rate on epidemic size. (a-c) represent the final infection scale in the panel of $\left(\beta ,\delta \right)$ when $p=0.2,0.5,0.8$, respectively. The other parameters are set as $\gamma =0.3$, $\mu =0.2$, ${\lambda }_1=0.2$, ${\lambda }_2=0.1$, $\theta =0.2$ and $\alpha =0.5$.

The effect of online information weak connection disconnected rate on epidemic spread was further explored, as shown in Fig. 13, yielding results that are opposite to the online information weak relationships connection rate, with significantly higher epidemic size and lower infection threshold as the disconnected rate $\tau$ increases. It suggests that maintaining online weak relationships between individuals and other patch individuals after returning to their residential patch enhances cross-patch information exchange, thereby facilitating the suppression of epidemic spread.

Fig. 13.

The effect of online information weak relationships connection rate on epidemic size. (a-c) represent the final infection scale in the panel of $\left(\beta ,\tau \right)$ when $p=0.2,0.5,0.8$, respectively. The other parameters are set as $\gamma =0.3$, $\mu =0.2$, ${\lambda }_1=0.2$, ${\lambda }_2=0.1$, $\theta =0.2$ and $\alpha =0.5$.

4. Conclusion

This study develops a novel three-layer metapopulation network framework to investigate the coupled dynamics of epidemic spreading and dual-channel information dissemination under individual mobility. By integrating epidemic migration-interaction-return processes, offline information dissemination, and online information dissemination into a unified structure, the model reveals how migration-driven shifts in physical environments and adjustments in virtual social connections influence the information-epidemic interplay. Key findings and contributions are summarized as follows:

• (1)In this study, we have examined a dual-channel information dissemination mechanism: offline information dissemination through physical metapopulation networks, where individuals have acquired awareness through local interactions within residence/destination patches; online information dissemination through a virtual social network between individuals. The results have shown that the dual-channel information dissemination is effective in suppressing epidemic, but enhancing online information dissemination has proved a more pronounced impact on reducing the epidemic scale. (2) Furthermore, we have explored the migration-information coupling mechanisms. For offline information dissemination, individual migration expands the scope of information dissemination through cross-patch exposure to offline information environments. For online information dissemination, the weak relationships formed between migrants and residents in destination patches accelerate the dissemination of epidemic-related information across patches. Simultaneously, the strength and stability of these relationships influence the final infection scale. (3) We have also investigated the complex relationship between the inhibitory effects of information on epidemic and individual mobility. The results have indicated that when information has only a weak suppressive effect, migration promotes epidemic transmission; in contrast, when information suppression is strong, migration inhibits epidemic spread. Epidemic control measures targeting population mobility need to take into account the specific strength of information’s suppressive effect on epidemic. This study assumed homogeneous mixing within patches and simplified online connection rules. Future research could incorporate dynamic network rewiring mechanisms, heterogeneous individual responses, and real-world mobility data. Extending the model to competitive information (e.g., misinformation vs. official guidance) and multi-pathogen scenarios would further enhance its applicability.

CRediT authorship contribution statement

Nan Chen: Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Methodology, Formal analysis, Data curation. Liang’an Huo: Writing – review & editing, Supervision, Resources, Project administration, Methodology, Investigation, Funding acquisition.

Data availability

The data used in this study can be obtained from the corresponding author for a reasonable request.

Declaration of interests

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.

Handling Editor: Dr. Raluca Eftimie