Effects of heterogeneous self-protection awareness on resource-epidemic coevolution dynamics

For the epidemic spreading model, each node has two possible states: the infected (I) and the susceptible (S) state [33]. At any time step $t$, each I-state node $i$ can transmit the disease to its susceptible neighbors, at the same time it recoveries to S-state with the rate $\mu_{i}(t)$, which depends on the resource quantity $\omega_{i}(t)$ received from its healthy neighbors. We consider that the resource, such as funds and medical, can promote the recovery of the I-state individuals [34, 35]. Thus, the recovery rate $\mu_{i}(t)$ is assumed to be proportional to resource quantity $\omega_{i}(t)$ in this paper and is defined as:

where $\mu_{0}$ is the spontaneous recovery rate that is independent of resource. Since the value of $\mu_{0}$ does not qualitatively affect the results [36], without loss of generality, it is set at a small value $\mu_{0}=0.1$ in this paper. Besides, resource wastes widely exist in the real scene of the medical system during the treatment process [37]. To mimic the phenomenon of resource wasting, a parameter $\varepsilon$ is introduced to represent the resource utilization rate, which has been proved to not qualitatively affect the dynamical properties [18, 36]. Thus, without loss of generality, it is set to be $\varepsilon=0.6$ in this paper. In the spreading process of the epidemic, every S-state node has a probability of being infected by its I-state neighbors. We consider that the healthy (S-state) nodes are the source of resources, they can both generate new resources and donate them to the I-state neighbors. If an S-state node chooses to donate its resources, it will have fewer resources for self-protection, which leads to a greater risk of been infected. Otherwise, if it refuses to donate resources for self-protection, the infection rate will reduce by a factor $c$. As there is a heterogeneous distribution of self-awareness in populations, the infection rate varies from node to node. To reflect the relationship between the donation behavior and infection probability, we denote the basic infection rate as $\beta$, and define the actual infection of node $i$ as:

According to the individual-based mean-field theory [38, 39], the dynamical process of the r-SIS model can be expressed as:

where $a_{ij}$ is the element of adjacency matrix $A$. If there is an edge between nodes $i$ and $j$, $a_{ij}=1$, otherwise, $a_{ij}=0$. Besides, to calculate the spreading size of the epidemic, a parameter $\rho_{i}(t)$ is introduced to represent the probability that node $i$ is in I-state. Thus we can calculate the fraction of infected nodes in a network of size $N$ at time $t$ by averaging overall $N$ nodes:

At last, the prevalence of the epidemic in the stationary state is defined as $\rho\equiv\rho(\infty)$.

As the statements above, the healthy individuals can generate and donate resources to support the recovery of their I-state neighbors during the outbreak of an epidemic. For the sake of simplicity, we assume that each susceptible node will generate one unit resource at each time step. Besides, the S-state nodes can perceive the severity of an outbreak from the state of infection in the neighborhood. The parameter $m_{i}$ is defined to represent the amount of the I-state neighbors of node $i$. Generally, the larger the value of $m_{i}$, the lower probability of resource donation of node $i$ [40, 41]. Besides, the parameter $\alpha_{i}$ is defined to represent the awareness of self-protection for each node $i$. A larger value of $\alpha_{i}$ indicates more sensitive of node $i$ to the disease, and a lower intention to donate resource. We consider that $\alpha_{i}$ obeys the heterogeneous distribution $g(\alpha)\sim{\alpha}^{-\gamma_{\alpha}}$, where $\gamma_{\alpha}$ is the self-awareness exponent. Thus, the resource donation probability of a healthy node $i$ is related to its intrinsic self-awareness and a total of $m_{i}$ infected neighbors, which is defined as:

where $q_{0}$ is a basic donation probability. We consider that the resource of an S-state node will be distributed equally among neighbors.

Based on the above resource allocation scheme, the amount of resource that node $i$ donates to node $j$ at a time can be expressed as:

With the resource allocation scheme defined in Eq. (6), we can further define the resource quantity $\omega_{i}(t)$, which is expressed as:

where $\mathcal{N}_{i}$ represents the neighbor set of node $i$, and the expression $[1-\rho_{j}(t)]$ represents the probability that a neighbor $j$ is in S-state. Combining Eq.( 6), we can get the expression of infection rate of any node $i$ at time $t$ as:

In order to investigate the effects of heterogeneous distributions of self-awareness and node degree on the epidemic dynamics, we first generate networks with degree distribution $P(k)\sim{k^{-\gamma_{D}}}$ using the uncorrelated configuration model (UCM) [47, 48], where $\gamma_{D}$ is the degree exponent. The maximum and minimum degree of the network are set to be $k_{\rm max}\sim\sqrt{N}$ and $k_{\rm min}=3$ respectively, which assures no degree correlation of the network when $N$ is sufficient large [49, 50], and the mean degree is set to be $\langle{k}\rangle=8$. Then we generate a self-awareness sequence $\{\alpha_{i}\}_{i=1}^{N}$ according to the distribution $g(\alpha)\sim{\alpha}^{-\gamma_{\alpha}}$. The maximum and minimum values are set to be $\alpha_{\rm max}\sim{N}^{1/2}$, $\alpha_{\rm min}=5$ respectively. To ensure $\alpha\in[0,1]$, we rescale each value as $\alpha/\alpha_{\rm max}$. At last, each node is assigned an value of self-awareness randomly.

In addition, we employ the susceptibility measure  [51] $\chi$ to numerically determine the epidemic threshold:

where $\langle\cdots\rangle$ is the ensemble averaging. To obtain a reliable value of $\chi$, we perform at least $2\times 10^{3}$ independent realizations on a specific network with fixed self-awareness distribution for each basic infection rate $\beta$. At the threshold $\beta_{c}$, the value of $\chi$ exhibits a maximum value. And then, by performing the simulations on 100 different networks, we can obtain the average value of $\beta_{c}$.

Fig. 1 (a) displays the prevalence $\rho$ in the stationary state as a function of basic infection rate $\beta$ at different degree and awareness exponents. We can observe that the dynamic process converges to two possible stationary states: the completely healthy state when $\beta<\beta_{c}$, and nearly all infected state $\beta>\beta_{c}$, which indicates a first-order transition at $\beta_{c}$. As shown in Fig. 1 (a), the epidemic threshold $\beta_{c}$ decreases with the heterogeneity of degree distribution, which is consistent with the epidemic outbreak on networks without self-awareness [52]. The phenomenon is induced by the hub nodes that exist on strong heterogeneous networks. When a given heterogeneity of degree distribution, the value of $\beta_{c}$ increases with the heterogeneity of self-awareness distribution, which is in contrast to the effects of degree heterogeneity. For instance, when $\gamma_{D}=2.1$, the $\beta_{c}$ for $\gamma_{\alpha}=2.1$ is larger than that for $\gamma_{\alpha}=4.0$. Fig 1 (b) exhibits the value of relative threshold $\beta_{c}/\beta_{c}^{0}$ as a function of $\gamma_{\alpha}$ for three degree exponents $\gamma_{D}=2.1$, $\gamma_{D}=2.5$ and $\gamma_{D}=4.0$ respectively. We can observe that the epidemic threshold $\beta_{c}$ decreases gradually with the increases of $\gamma_{\alpha}$, which reveals that the self-awareness heterogeneity restrains the epidemic spreading.

Next, we explain qualitatively the results above by exploring the time evolutions of the critical parameters when the basic infection rate is set to be $\beta=0.04$. Fig 2 (a) plots the time evolution of the average donation probability $\langle{q}\rangle$ (top pane) and average number of infected neighbors $\langle{m}\rangle$ for varies values of $\gamma_{D}$ and $\gamma_{\alpha}$ (bottom panel). It shows that when $\gamma_{D}=2.1$, the donation probability $\langle{q}\rangle$ first decreases and then increases with time $t$ for $\gamma_{\alpha}=2.1$ (see blue upper-circles). While, for $\gamma_{\alpha}=4.0$ (see red circles), the value of $\langle{q}\rangle$ first decreases, and then increases slightly in the middle time, at last, it declines with time $t$. This phenomenon can be qualitatively explained as follows: When there is a strong heterogeneity of the self-awareness distribution, for instance, $\gamma_{\alpha}=2.1$, the network has a larger number of nodes with very low self-awareness, and more nodes with high self-awareness. Since the more nodes with large self-awareness means a lower willingness to donation resources, and a lower donation probability in the early stage, thus the value of $\langle{q}\rangle$ drops more abruptly than that for $\gamma_{\alpha}=4.0$, which induces a larger number of infected neighbors $\langle{m}\rangle$, as less resource is donated from healthy nodes to their neighbors. This phenomenon can be verified by studying the bottom pane of Fig. 2 (a). With less resource, the I-state nodes will recovery with a relatively lower recovery rate [see blue upper-triangles and red circles in Fig. 2(b)]. With the spread of the disease, more nodes with a very small value of $\alpha$ participate in the behavior of donating resources. The smaller value of $\alpha$ means a larger willingness to donate resource, which leads to a greater growth of $\langle{q}\rangle$ for $\gamma_{\alpha}=2.1$ that $\gamma_{\alpha}=4.0$ [see the blue upper-triangles and red circles in the top pane of Fig 2 (a)]. Consequently, with the increase of donation probability of the entire network, the value of $\langle{m}\rangle$ decreases gradually, which leading to a slower decrease of $\langle{\mu}\rangle$ for $\gamma_{\alpha}=2.1$ that $\gamma_{\alpha}=4.0$ [see the blue upper-triangles and red circles in Fig 2 (b)]. The relative larger recovery rate leads to a lower effective infection rate, which is defined as $\langle{\lambda}\rangle=\langle\beta\rangle/\langle{\mu}\rangle$, as shown in Fig. 2 (c). Thus the prevalence $\rho(t)$ increases slower for $\gamma_{\alpha}=2.1$ than $\gamma_{\alpha}=4.0$, as shown in Fig 2 (d). From the statement above, we can explain the reason why the heterogeneity of self-awareness distribution can suppress the outbreak of an epidemic.

When the degree heterogeneity of the network decreases, for instance, $\gamma_{D}=4.0$, the donation probability for both $\langle{q}\rangle$ for $\gamma_{\alpha}=2.1$ that $\gamma_{\alpha}=4.0$ increases with time $t$ [see the yellow squares and green snowflakes in the top pane of Fig 2 (a)], which leads to an increase of the recovery rate of the whole network, as shown in Fig. 2 (b). Consequently, the effective infection rate of the network $\langle{\lambda}\rangle$ decreases gradually to a very small value with time $t$. Thus we see that the prevalence decreases gradually to zero [see Fig 2 (d)]. From the statement above, we can also explain the phenomenon that the threshold $\beta_{c}$ decreases with an increase of $\lambda_{D}$.

In this section, we focus on how the correlation between the node degree and self-awareness affects the spreading dynamics. A network with a given correlation coefficient is built as follows:

• 1.

  A network with degree distribution $P(k)\sim{k}^{-\gamma_{D}}$ is built by the steps in Section 3.1;

• 2.

  A self-awareness sequence is generated from the distribution $P(\alpha)\sim\alpha^{-\gamma_{\alpha}}$;

• 3.

  Sorting the nodes of the network by the degree in ascending order, and then sorting the self-awareness sequence in ascending or descending order, respectively.

• 4.

  Rearranging the order of each self-awareness value with a given probability $\pi$, and then assigning each self-awareness value to the corresponding node.

According to the steps above, we can get a network with a given awareness-degree correlation. The correlation coefficient is:

If the self-awareness sequence is sorted in ascending order, we can get a positive awareness-degree correlation with coefficient $\sigma$; otherwise, we can get a negative awareness-degree correlation with coefficient $\sigma$.

First of all, we study the case when there is a positive degree-awareness correlation, i.e., $\sigma=0.8$. Figs. 3 (a), (b) and (c) exhibit the prevalence $\rho$ in the stationary as a function of $\beta$ when degree exponent is $\gamma_{D}=2.1$, $\gamma_{D}=2.5$ and $\gamma_{D}=4.0$, respectively. It shows that for a network with a fixed structure, the epidemic threshold $\beta_{c}$ increases with self-awareness heterogeneity. To verify the results obtained in Figs. 3 (a) to (c), we explore the relationship between $\beta_{c}$ and the awareness exponent $\gamma_{\alpha}$ in Fig. 3 (d). Obviously, for each fixed $\gamma_{D}$, the value of $\beta_{c}$ decreases gradually with the $\gamma_{\alpha}$, which suggests that when node degree and self-awareness correlated positively, the heterogeneity of self-awareness inhibits the epidemic spreading.

Next, we explore the case when there is a negative degree-awareness correlation. Figs. 4 (a) to (c) display the value of $\rho$ versus $\beta$ for $\gamma_{D}=2.1$, $\gamma_{D}=2.5$ and $\gamma_{D}=4.0$ when $\sigma=-0.8$ respectively. It shows that for a fixed network, for instance $\gamma_{D}=2.1$, the epidemic threshold $\beta_{c}$ first increases when $\gamma_{\alpha}$ increase from $\gamma_{\alpha}=2.1$ to $\gamma_{\alpha}=2.5$, and then it decreases when $\gamma_{\alpha}$ increases from $\gamma_{\alpha}=2.5$ to $\gamma_{\alpha}=4.0$. We next study systematically the effects of negative correlation between node degree and self-awareness on the spreading dynamics by exploring the relationship between threshold $\beta_{c}$ and awareness exponent $\gamma_{\alpha}$ in Fig. 4 (d) for $\gamma_{D}=2.1$ (green upper-triangles), $\gamma_{D}=2.5$ (red circles), and $\gamma_{D}=4.0$ (yellow squares) respectively. It shows that the threshold $\beta_{c}$ first increases and then decreases with $\gamma_{\alpha}$, and an optimal value $\gamma_{\alpha}^{\rm opt}$ ($\gamma_{\alpha}^{\rm opt}$ is around $2.6$) exists, at which the value of $\beta_{c}$ reaches maximum. This result can be qualitatively explained as follow: When the self-awareness heterogeneity is very strong, e.g., $\gamma_{\alpha}=2.1$, there is a large number of nodes with very small values of self-awareness $\alpha$ (strong willingness of resource donation), and many nodes with very large values $\alpha$ (weak willingness of resource donation). When the coefficient $\sigma=-0.8$, there is a strong negative correlation between node degree and self-awareness. Under this circumstance, the large degree nodes will have a strong willingness to donate resource, while the small degree nodes (covering most nodes of the network) have a weak willingness to donate resource, which leads to a high infection rate of these hub nodes. Besides, the epidemic spreading dynamics exhibits hierarchical features [32]. That is to say, the hubs with large degrees are more likely to be infected firstly, and then the disease propagates from hubs to the intermediate nodes, and finally to nodes with small degrees. Therefore, the large numbers of small degrees will be infected rapidly by the hub nodes in this situation, and the epidemic will outbreak easily.

When the heterogeneity of self-awareness distribution is weak, i.e., $\gamma_{\alpha}=4.0$, there is a small fraction of nodes with very large or small value of $\alpha$, many nodes have an intermediate $\alpha$ around the mean value $\langle{\alpha}\rangle$. The awareness level of the small degree nodes reduces compared to the case of $\gamma_{\alpha}=2.1$, which leads to a raise of both the donation probability $\langle{q}\rangle$ and effective infection rate $\langle{\lambda}\rangle$ of these nodes. As the strong negative correlation between node degree and self-awareness, the hub nodes still have a very small value of $\alpha$ (large value of donation probability $\langle{q}\rangle$). During the outbreak of an epidemic, these hubs nodes are more likely to be infected in the early stage, and then transmit the disease to those small degree nodes rapidly as they have a highly effective infection rate $\langle{\lambda}\rangle$. Thus the epidemic will break out more easily than the case of $\gamma_{\alpha}=2.1$ in this situation.

According to the above statement, we have qualitatively explained the optimal phenomenon by explaining why diseases are more likely to break out when there is a strong heterogeneity ($\gamma_{\alpha}=2.1$) and weak heterogeneity ($\gamma_{\alpha}=4.0$).

Next, we study the effects of correlation between node degree and self-awareness by exploring the relationship between epidemic threshold $\beta_{c}$ systematically and correlation coefficient. Fig. 5 displays the $\beta_{c}$ as a function of $\sigma$ for $\gamma_{\alpha}=2.1$, $\gamma_{\alpha}=2.5$, $\gamma_{\alpha}=3.1$ respectively. We find that for each fixed heterogeneity of self-awareness, the epidemic threshold $\beta_{c}$ increases monotonously with correlation coefficient $\sigma$. For instance, for $\gamma_{\alpha}=2.1$, the threshold increases from $\beta_{c}\approx 0.027$ to $\beta_{c}\approx 0.044$, which suggests that the more positive of the correlation between node degree and self-awareness, the better the disease can be suppressed. This phenomenon can be qualitatively explained as follows: When there is a larger positive correlation, the hub nodes will have a larger value of $\alpha$, which indicates a smaller probability of resource donation $\langle{q}\rangle$, and consequently a lower effective infection rate $\langle{\lambda}\rangle$ of the hubs. This phenomenon reduces the infection probability of the hub nodes in the early stage. Meanwhile, the small degree nodes have a larger probability of resource donation, which increases the recovery rate of the I-state neighbors, including the hub nodes. Thus the outbreak of the epidemic is effectively delayed. In addition, it also shows that the parameter pane $(\sigma-\beta_{c})$ is separated into two phases: phase I and phase II, by a critical value of $\sigma\approx 0.4$. In phase $I$, the threshold $\beta_{c}$ first increases and then decreases with the increase of $\gamma_{\alpha}$, namely the optimal value $\gamma_{\alpha}^{\rm opt}$ exists in this region, as shown the curves in Fig. 4 (d) for $\sigma=-0.8$. In phase $II$, the threshold $\beta_{c}$ decreases monotonously with $\gamma_{\alpha}$, as shown the curves in Fig. 1 (b) for $\sigma=0$, and the curves Fig. 3 (d) for $\sigma=0.8$ respectively.

In this section, we verify the results obtained on artificial networks by conducting the simulations on the real-world networks. The following four typical real-world networks are chosen in our paper: (i). The OpenFlights network [53]. This network describes the flights between airports in the world. The nodes represent a portion of the world’s airports, and edges represent flights from one airport to another. There are $N=2939$ nodes and $V_{e}=30501$ edges in the network with maximum degree $k_{\rm max}=473$ [54]. (ii). The Euroroad network [55] is a international road network. Most road in the network is located in Europe. The nodes of the network represent cities, and the edge between two nodes denotes that an E-road connects them. There are $N=1174$ nodes and $V_{e}=1417$ edges with $k_{\rm max}=10$. (iii). The face-to-face contact network [56]. Nodes in this network represent the individuals who attended the exhibition INFECTIOUS: STAY AWAY in 2009 at the Science Gallery in Dublin, edges describe the face-to-face contacts that were active for at least 20 seconds among individuals during the exhibition [57]. There are a total numbers of $N=410$ nodes and $V_{e}=17298$ edges (contacts) in the network. The maximum degree is $k_{\rm max}=294$. (iv) The Facebook network [58]. This dataset consists of ’circles’ (or ’friends lists’) from Facebook. Facebook data was collected from survey participants using this Facebook app. The dataset includes node features (profiles), circles, and ego networks. There are $N=4039$ nodes and $V_{e}=88234$ edges in the network.

Figs. 6 (a) to (d) display the prevalence $\rho$ in stationary state as a function of $\beta$ when there is no degree-awareness correlation on OpenFlights network, Euroroad network, face-to-face contact network and the Facebook network respectively for different heterogeneities of self-awareness. We find that on the OpenFlights network, the threshold $\beta_{c}$ decreases with the increase of $\gamma_{\alpha}$, which is consistent with the result on artificial networks [see Fig. 1 (a)], and the prevalence $\rho$ increases with $\gamma_{\alpha}$ for a fixed basic infection rate $\beta$. However, there is a difference when disease propagates on the other three real-world networks, i.e., the awareness heterogeneity does not alter the threshold of $\beta_{c}$. Besides, it shows that the prevalence increases with the decreases of awareness heterogeneity, which is consistent with the result on OpenFlights network. The results above reveal that the awareness heterogeneity can suppress the outbreak of epidemic on real-world networks when there is no correlation between node degree and self-awareness.

At last, we study the effects of degree-awareness correlation on the epidemic spreading on the four real-world networks. The awareness exponent is fixed at $\gamma_{\alpha}=2.5$. It shows that when disease propagates on the OpenFlights and face-to-face contact networks, the threshold $\beta_{c}$ does not be altered, but the prevalence $\rho$ decreases with correlation coefficient $\sigma$. This result indicates that a more positive correlation between node degree and self-awareness can better suppress the disease spreading, which is consistent with the result of artificial networks. When disease propagates on the Euroroad network, the correlation does not alter both threshold $\beta_{c}$ and prevalence $\rho$. At last, when disease propagates on the Facebook network, the threshold $\beta_{c}$ increases with $\sigma$, which is consistent with the result on artificial networks [see Fig. 5],

Based on the above research, we can see that the results on the real-world networks are consistent with those on the artificial network. However, the complex structural features of the real networks, such as clustering, community structure, and small-world characteristics, have an important impact on the results, and needs to be further studied in our future research.