Containing Epidemic Spreading on Networks with Neighbor Resource Supporting

Previous studies revealed that the susceptibility, contacting preference, and recovery probability markedly alter the epidemic outbreak size and threshold. The recovery probability of an infected node is closely related to its obtained resources. How to allocate limited resources to infected neighbors is extremely important for containing the epidemic spreading on complex networks. In this paper, we proposed an epidemic spreading model on complex networks, in which we assume that the node has heterogeneous susceptibility and contacting preference, and susceptible nodes are willing to share their resources to neighbors. Through a developed heterogeneous mean-ﬁeld theory and a large number of numerical simulations, we ﬁnd that the recovered nodes provide resources uniformly to their infected neighbor nodes, and the epidemic spreading can be suppressed optimally on homogeneous and heterogeneous networks. Besides, altering the susceptibility and contacting preference does not qualitatively change the results. The susceptibility of the node decreases, which makes the outbreak threshold of epidemic spreading increase, and the outbreak size decreases. Our theory agrees well with the numerical simulations.


Introduction
Extensive phenomena, such as infectious disease spreading, information diffusion, and rumor propagation, in real-world systems, can be described as "epidemic" spreading on complex networks [1][2][3][4][5]. In such a description framework, nodes represent elements and edges stand for the relationship among nodes. Previous studies have revealed that the evolution mechanisms, network topology, and social factors are markedly affecting the epidemic outbreak size and threshold point [6][7][8][9][10][11][12]. Researchers found that strong degree heterogeneity makes the outbreak threshold vanish [6]. When epidemic spreading dynamics on multilayer networks, the system may exist a hybrid phase transition; that is to say, the epidemic outbreaks globally in one network and spreads locally in the other network [13][14][15][16][17][18][19][20]. Davis et al. proposed a Maki-ompson rumor-spreading model on multipopulations and found that the dynamical behaviors and threshold are closely correlated to the interactions between spreading and subpopulations [21].
Designing effective strategies for fast recovery infected nodes is extremely important for epidemic spreading. On the one hand, researchers investigated the effects of recovery probability on epidemic spreading, including the epidemic outbreak size and threshold. Shu et al. [22] investigated the recovery probability on the classical susceptible-infectedrecovered (SIR) model, by using an edge-based compartmental theory and synchronous updating numerical simulations, and they found that the outbreak threshold increases with the recovery probability for a given effective infection probability. For epidemic spreading dynamics with general recovery probability, the dynamic always exhibits a non-Markovian character, which makes the theoretical predictions deviate from the numerical simulations [23][24][25][26][27]. Karrer and Newman [28] proposed a general messagepassing approach for epidemic spreading with a general recovery probability distribution and markedly agree with numerical simulations. de Arruda and his colleagues [29] studied the roles of heterogeneous recovery probability and revealed that the critical infectivity is smaller than the values predicted by the quenched mean-field approaches.
On the other hand, researchers revealed that resources (e.g., masks and specific drugs) are an essential factor for epidemic recovery [30]. How to allocate the resources to individuals is important. Chen et al. [31] assumed that the recovery probability of infected nodes depends on the amount of received resources. rough a mean-field theory, they found that the system exhibits a discontinuous phase transition when the resources are insufficient. ey further consider the resource allocation correlates to a social network, and they investigated the effects of different resource allocation strategies, network topologies, interlayer degreedegree correlations on the epidemic spreading, and phase transition [32][33][34][35]. By using extensive numerical Monte-Carlo simulations and the message-passing approach, they revealed that the phase transition closely correlated to those essential factors. Recently, Zhang et al. [36] revealed an optimal resource allocation strategy, at which the epidemic will be significantly suppressed.
In reality, distinct individuals always have different susceptibility, inelastic resources (e.g., time), and medical resources (e.g., masks). When a pandemic epidemic is spreading, such as SARS and COVID-19, the above three factors are essential. Neglecting any one may lead to a misunderstanding about the spreading dynamics. To our best knowledge, there is still a lack of a mathematical model, which includes the three aspects simultaneously. Specifically, we will propose a generalized SIR epidemic spreading dynamics considering the following three aspects. (i) Each node with an inherent infection threshold is used to reflect the susceptibility of nodes. e larger the values of infection threshold, the smaller the susceptibility. Miller [37] revealed that an epidemic is a more likely outbreak for homogeneous infectivity. Watts [38] revealed that the phase transition is discontinuous when the infection threshold is included. (ii) We preferred contact with neighbors due to the limited energy and time. Previous studies found that preferring contacting nodes with small degrees promotes the spreading dynamics [39,40]. (iii) e resource supporting is determined by the degree of nodes. We propose an epidemic spreading dynamic on complex networks, including the above three aspects simultaneously, and discuss the effects of the above three aspects of epidemic spreading dynamics. e organization of this paper as follows: in Section 2, we describe our model. In Section 3, we develop a heterogeneous mean-field theory to describe the spreading dynamics. e numerical simulations are performed in Section 4. Finally, we draw the conclusions in Section 5.

Model Descriptions
In this section, we describe the epidemic spreading dynamics model on complex networks.

Network Description.
To describe the complex networks, we adopt two types of artificial networks: ER network and uncorrelated configuration networks. We denoted the network size, degree distribution, and average degree as N, P(k), and 〈k〉 � k kP(k), respectively. To build the ER network, two nodes are connected with probability p. erefore, we have 〈k〉 � pN. For the uncorrelated configuration networks, we build as follows: (i) assign the values of N and P(k). Specifically, we assume the degree distribution follows a power-law distribution P(k) � ζk − c D , where ζ � 1/ k k − c D , c D represents the degree exponent. e larger of c D , the homogeneous of the degree distribution. (ii) Assign degree for each node according to P(k). (iii) Assign stubs for each node, and the number of stubs equals to its degree. (iv) Randomly connect two stubs to build an edge. Note that we disallow multiple edges and self-loops. (v) Repeat step (iv) until there are no stubs left. e pseudocode of building the uncorrelated configuration networks is shown in Algorithm 1 [41].

Epidemic Spreading Model.
To describe the epidemic spreading dynamics on complex networks, we adopt a generalized susceptible-infected-recovered (SIR) model. A susceptible node represents that the epidemic does not infect it. Infected node stands for that it is infected by the epidemic and can transmit the epidemic to susceptible neighbors. Recovered nodes means that it is ever infected by the epidemic and recovered. Note that the recovered nodes will not participate in the remaining spreading dynamics. e immune system response, such as allergic response, is dependent on multiple exposures [42]. For the case of social contagions, an individual should eliminate the adoption risk before adopting a product, innovation, or behavior. erefore, multiple exposures are needed [43,44]. To include this essential factor, we assume that every node has an inherent infectious threshold f to reflect the fraction of exposures that a susceptible node should have before become an infected node. e epidemic spreading dynamics evolves as follows. Initially, randomly select a fraction of s nodes as the seeds (i.e., in the infected state), and the remaining nodes are set to be in the susceptible state. At each time interval t + δt, every infected node i tries to transmit the infection to a neighbor according to some strategies. Since every individual has limited time and energy, he always preferentially contact some neighbors to transmit the infection. We here assume that node i selects node j to transmit the infection with probability: where Ω(i) represents the neighbor set of node i. e parameter β is used to reflect the preference of contacting. For the case of β < 0, the infected node prefers to select neighbors with a small degree. e opposite situation happens when β > 0. For the case of β � 0, the infected node contacts neighbors randomly. If node j is in the infected or recovered state, nothing happens. If node j is in the susceptible state, 2 Complexity node i transmits the infection to node j with probability λδt. e number of infections y j is that node j plus one, i.e., y j � y j + 1. If y j ≥ fk j , node j becomes infected, where k j is the degree of node j and ⌈x⌉ means ceil of x.
e recovery of infected nodes always needs some resources from public resources and friends' support. In this paper, we focused on the latter one and assumed that every susceptible node is generous and willing to share resources with neighbors. e number of resources of nodes is always related to their inherent characteristics (e.g., age, profession and education), which can be reflected by their degrees to some extent. e simplest way to describe the individuals' characteristics is by using their degrees. erefore, we assume that a susceptible node u with degree k u provides k α u pieces of resources. For the case of α > 0, nodes with a large degree can provide more resources. e opposite situation happens when α < 0. For the case of α � 0, all nodes provide the pieces of the resource. To normalize the resources, we set a susceptible node u can provide k α u /max k α min , k α max resources, where k min and k max represent the minimum and maximum degrees, respectively. erefore, the infected node i receives r i pieces of resources and can be expressed as With r i pieces of resources, Γ(i) is the set of susceptible neighbors of node i, and the infected node i becomes recovered with rate: where μ 0 is the basic recovery probability with one piece of resource. w is the resource utilization, and μ 1 is endogenous probability. e spreading dynamics terminates when there are no nodes in the infected state. In the final state, the epidemic spreading size R is the fraction of nodes in the recovered state. e pseudocode of the spreading dynamics is presented in Algorithm 2. e model proposed in this section includes three important aspects: (i) we use an inherent infection threshold to describe the heterogeneous susceptibility of nodes, which is similar to the threshold model, (ii) we preferentially contact neighbors according to equation (1), and (iii) the recovery probability is determined by equation (3).

Theoretical Analysis
To describe the epidemic spreading dynamics on complex networks, we here use a heterogeneous mean-field theory inspired by [6]. In this theory, we make the following three assumptions: (i) nodes with the same values of degrees have the same statistical properties, and (ii) the infected neighbors, e.g., node u and v, of a susceptible node i transmit the infection to node i independently. at is to say, we neglect the dynamical correlations among the states of neighbors. (iii) In theory, we assume the epidemic spreading on largescale systems.

Evolution Dynamics.
Denoting S k (t), I k (t), and R k (t) as the probability that a randomly selected node with degree k at time t is in the susceptible, infected, and recovered state, respectively. Since at very time step, a node should be in one of the three states, we have In what follows, we will derive the evolutions of S k (t), I k (t), and R k (t).
For a susceptible node u with degree k, the probability that u is in the susceptible state decreases when u is infected by neighbors. From the descriptions in Section 2, we know that node u infected by neighbors at time t should fulfill a situation; that is to say, node u at least received fk infections from neighbors. We denote θ k (t) as the probability that a susceptible node with degree k receives one piece of infection from a neighbor. erefore, the probability that node u receives m pieces of infections from neighbors is We further get that probability that node u becomes infected at time t with probability: (1) Input: N and P(k) (2) Output: Network G (3) Assigning degree k i for each node i according to P(k) (4) Assigning stubs ℓ i for each node with same value of degree k i , i.e., (6) Randomly select two stubs m and n ≠ m to build an edge e mn (7) h ⟵ h − 2 (8) end while ALGORITHM 1: Building uncorrelated configuration networks.
By using equations (5) and (6), we can describe the infection threshold in our model. e evolution of S k (t) is Until now, we do not know the value of θ k (t).
To obtain θ k (t), we should compute the following three aspects: (i) e probability of node u connects to an infected neighbor node v with degree k ′ . In uncorrelated network, we know the probability that an edge of node u connects to node v is k ′ P(k ′ )/〈k〉, where 〈k〉 is the average degree of the network. (ii) e infected node v selects node u to contact according to equation (1). e weight of node v selecting node u is k β . e weight of remaining k ′ − 1 neighbors is H(k ′ − 1), where H is the average weight of node v selecting one node to contact except for node u, and it is expressed as where 〈k β+1 〉 is the β + 1-th moment of the degree distribution. erefore, the preferred contact probability that node v selects node u to contact is (1) Input: Network G and dynamical parameters λ, μ 1 , μ 0 , w, α, β, f, ρ 0 ; (2) Output: Epidemic spreading size R; (3) Randomly selecting ρ 0 seeds, and put them into queue Q 1 ; (4) t ⟵ 0; (5) Initialize y j for every node j; (6) while Q 1 is not empty do (7) Initialize Q 2 to be empty; (8) δt ⟵ 1/length(Q 1 ); (9) for i � 1 to length(Q 1 ) do (10) Node i of Q 1 selects one neighbor j to contact according to equation (1); (11) Node i transmits the infection to node j with probability λδt if j is in the susceptible state; (12) if Node j receives the infection from i then y j ←y j + 1 (13) end if (14) if y j ≥ fk j then (15) Node j becomes infected, and put it into queue Q 2 ; (16) end if (17) end for (18) for i � 1 to length(Q 1 ) do (19) Recovering node i according to equation (3); (20) if Node i recovers then (21) Delete node i from queue Q 1 ; (22) end if (23) end for (24) for i � 1 to length(Q 2 ) do (25) Adding node i to queue Q 1 ; (26) Deleting node i from queue Q 2 ; (27) end for (28) t←t + δt (29) end while ALGORITHM 2: Epidemic spreading dynamics. 4 Complexity (iii) Node v successfully transmits the infection to node u with probability λ. Combining the above three aspects, we obtain Inserting equation (10) into equation (7), we get the evolution of S k (t).
For the evolution of I k (t), we should consider two aspects. On the one hand, the susceptible node u with degree k becomes infected with the probability Ψ k (t), which induces the increase of I k (t). On the other hand, the infected nodes with degree k become recovered μ k , which causes the decrease of I k (t). e expression of μ k is where ϕ k is the pieces of resources that node v receives from neighbors. According to equation (2), we know On the right hand of equation (12), we times k − 1 since an infected node v must connect to an infected neighbor. By using equations (11) and (12), the recovery probability can be computed. erefore, we get the evolution of I k (t) as Combining equations (7), (13), and (14), we can get the fraction of nodes in the susceptible, infected, and recovered state as where X ∈ S, I, R { }. When t ⟶ ∞, the epidemic outbreak size is

reshold Analysis.
In what follows, we study the threshold point λ c of the system. Previous studies indicated that when λ ≤ λ c , the global epidemic spreading is impossible. Otherwise, when λ > λ c , the global epidemic outbreak becomes possible. Initially, we only randomly a vanishingly small fraction of seeds. us, we have I k (0) ≈ 0, S k (0) ≈ 1, and the high order of I k (t) can be neglected when t ⟶ 0. Rewriting equation (13) and neglecting the high orders of O( I k (t) 2 ), when fk � 1, we have where δ kk′ � 1 only when k � k ′ , and ϕ k � (k − 1) where In a similar way, we study the case of fk > 1. We have Complexity 5 where L kk′ ″ � (2k − k 2 )λ[ k′ ((k ′ P(k ′ ))/〈k〉)G(k, k ′ ) − δ kk′ (μ 0 wϕ k + μ 1 )]. Combining the above two cases, we have where According to the stability analysis in [45], we know the global epidemic outbreak only when the largest eigenvalue Λ 1 (L) of L is larger than 0. Otherwise, the global epidemic outbreak is very small. e critical epidemic transmission probability λ c fulfills

Simulation Results and Analysis
We perform a large number of numerical simulations on the Erdös-Rényi (ER) and scale-free (SF) networks with network size N � 10 4 , and average degree 〈k〉 � 10. e basic recovery probability is μ 0 � 0.1, the endogenous recovery is μ 1 � 0.1, the resource utilization is w � 0.1, and the initial seed ratio ρ 0 � 0.0005. By performing extensive numerical simulations, we found that other values of μ 0 , μ 1 , and w do not qualitatively affect the results presented in this paper. rough a large number of theoretical analysis and experimental simulation, we find that other parameters change the conclusion presented in this paper qualitatively. All results presented in this paper are averaged at least 1000 times on 10 different networks.
On the ER network, we first study the variation of the outbreak size R(∞) with the transmission probability λ with different preference selection parameters β as shown in Figure 1. It can be found from Figure 1(a) that as the transmission probability increases, the probability of susceptible nodes gets the infection increased; thus, the outbreak size also increases. e outbreak size decreases first and then increases with the resource provision parameter α. Specifically, when λ is the same, the outbreak size corresponding to α � − 0.5 is the largest, the outbreak size of α � 0.5 is between three α values, and the smallest size is among the three α values when α � 0.0, while the outbreak threshold increases first and then decreases with the resource provision parameter α according to equation (22). Specifically, the outbreak threshold corresponding to α � − 0.5 is the smallest, the outbreak threshold of α � 0.5 is between three α values, and the largest size is among the three α values when α � 0.0. erefore, there is α opt at which outbreak size R(∞) is the smallest and outbreak threshold is the largest (to be detailedly discussed in Figure 2). We can understand the above phenomena as follows. For the case of α � 0.0, a susceptible node j provides a piece of resource to its infected neighbor i. For other values of α, the node j at most provides one piece of resource to node i. In this situation, node i receives the maximum resources r i � |Γ i | when α � 0.0, where |Γ i | is the number of the set Γ i . erefore, node i has the largest recovery probability when α � 0.0.
ree peak values of Δ in Figure 1(d) represent the simulated outbreak threshold for α � − 0.5, α � 0.0, and α � 0.5, respectively. Δ is obtained through variability calculation, and the calculation formula is [46] Δ � ey correspond to the arrows in Figure 1(a) representing the theoretical outbreak threshold. It can be seen that the theoretical prediction of R(∞) and theoretical outbreak threshold well agree with the numerical predictions. e reason for the deviation between the theoretical and simulation results is the strong correlation between the neighbor state and the finite size network. e most important thing is that we found that changing the value of β did not qualitatively change the above results. e numerical simulation results are good agreement with the theoretical predictions.
In Figures 3 and 4, we study the effects of different values of f on the epidemic spreading dynamics. With the increase of f, the epidemic outbreak threshold increases and the epidemic outbreak size decreases. Different from previous studies [38], in which the phase transition is discontinuous when the infection threshold is large enough, we find that R(∞) always increases continuously with λ regardless of f. We explain the result as follows. In our model, an infected node's contact ability is limited; that is, an infected node only contacts one neighbor to transmit the infection. In this situation, the susceptible state node becomes infected gradually and cannot induce a finite fraction of susceptible nodes becoming infected state simultaneously. us, R(∞) increases with λ continuously. In addition, we also observed a phenomenon similar to Figure 1. Specifically, when the propagation probability λ is the same, the outbreak size R(∞) reaches the minimum when α � 0.0 and the outbreak threshold reaches the maximum when α � 0.0. at is, there is also an optimal α value here. Above all, from Figures 1, 3, and 4, we can see that the change in f value has no effect on α opt � 0.0, and α opt always exists, since the recovery probability reaches the maximum value at α opt . Next, it will be systematically discussed and studied.
In Figure 2, we explore the effects of α and λ on the epidemic spreading on ER networks. From Figures 2(a)-2(c), we observe that when α < 0, the R(∞) decreases with α while α > 0 and increases with α. e variation of the outbreak threshold is opposite to the outbreak size. When α < 0, the outbreak threshold increases with α while α > 0 and decreases with α. erefore, there is an optimal value α opt � 0, i.e., white dotted line, where R(∞) reaches the minimum value, and λ c is the largest. Because when α � 0, the volume of resources received by the infected nodes for recovery is the largest, which makes them become recovered quickly. e value of the parameter β does not qualitatively change the above phenomena. We also mainly realized that the change of the parameter f cannot qualitatively change these phenomena, but the increase of f can reduce the outbreak size and increase the outbreak threshold, which is the same as the conclusion obtained in Figures 1, 3, and 4.
We further research the effects of λ and α on spreading dynamics on SF networks with power-law degree distribution P(k) � ck − c , where c is the degree distribution exponent and c � 1/ k k − c , and the maximum degree is set as . e simulations are performed on a strong heterogeneity network with c � 3. We plot the phase diagram of the theoretical outbreak size R(∞) as a function of α and λ on the SF network with different values of β in Figure 5. We find similar phenomena in Figure 2.
Specifically, in Figures 5(a)-5(c), for α < 0, the R(∞) decreases with α and outbreak threshold increases with α. For α > 0, the R(∞) increases with α and outbreak threshold decreases with α. us, there is an optimal value of α that makes R(∞) is the smallest, and the outbreak threshold is the largest, i.e., α opt � 0. e explanation of this optimal phenomenon is the same as that on the ER network of Figure 2. No matter how the values of β and f change, the optimal value of α will not disappear. As f increases, the final outbreak size decreases, and the outbreak threshold increases. e theoretical and simulation results agree well.

Conclusion
In the process of the real epidemic spreading, first of all, the possibility of each node being susceptible to infection is different; that is, each node itself has an inherent infection threshold to reflect the susceptibility of the node. Next, due to the limited energy and time of epidemic disseminators, the process of epidemic transmission is affected. Finally, the resource supporting is determined by the degree of nodes. e above three aspects have a significant impact on the spread and recovery of the epidemic. Previous works have separately studied the influence of three aspects of epidemic dynamics, but no studies have included the three aspects into the same dynamics of epidemic spreading at the same time.
erefore, we proposed a model, including three aspects to study them simultaneously. Besides, we extended the heterogeneous mean-field theory to describe the model. rough theoretical analysis and a large number of numerical simulations, we find that the recovered nodes provide resources uniformly to their infected neighbor nodes, and the epidemic spreading can be suppressed optimally, i.e., the outbreak threshold is maximum, and the outbreak size is minimum for α � 0.0 on homogeneous and heterogeneous networks. e change of threshold parameter f and preference selection parameter β has no qualitative change to the above conclusion. e increase of f increases the infection threshold of each node; that is, the susceptibility of the node decreases, thus increasing the outbreak threshold of epidemic spreading and decreasing the outbreak size. e conclusion of this paper can be used as a reference for the suppression of epidemic spreading, and the theory proposed can be applied to other network dynamic processes. For example, uninfected individuals may provide uniform resources and help susceptible individuals who can be contacted. e research in this paper can be extended to more real networks, such as sequential networks and multilayer networks. Finally, our research results may shed some light on studying the critical phenomena on complex networks.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.