Epidemic spreading dynamics on complex networks with adaptive social-support

Epidemic spreading dynamics on complex networks with adaptive social-support

Accepted Manuscript Epidemic spreading dynamics on complex networks with adaptive social-support Rong Zhou, Qingchu Wu PII: DOI: Reference: S0378-43...

571KB Sizes 1 Downloads 122 Views

Accepted Manuscript Epidemic spreading dynamics on complex networks with adaptive social-support Rong Zhou, Qingchu Wu

PII: DOI: Reference:

S0378-4371(19)30342-5 https://doi.org/10.1016/j.physa.2019.03.107 PHYSA 20742

To appear in:

Physica A

Received date : 11 January 2019 Revised date : 6 March 2019 Please cite this article as: R. Zhou and Q. Wu, Epidemic spreading dynamics on complex networks with adaptive social-support, Physica A (2019), https://doi.org/10.1016/j.physa.2019.03.107 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Epidemic spreading dynamics on complex networks with adaptive social-support Rong Zhou, Qingchu Wu∗ College of Mathematics and Information Science, Jiangxi Normal University, Nanchang, Jiangxi 330022, P. R. China

Abstract Using conditional quenched mean-field (cQMF) method, we develop a continuous-time susceptibleinfected-susceptible epidemic model on quenched complex networks with combined self-recovery and social support from healthy network nodes. We determined the lower bound of the epidemic threshold which was found to depend directly on social support α. Our analytical results of the generalized cQMF model were in good agreement with numerical simulations on Erd˝os-R´enyi (ER) random graphs and different scale-free (SF) network configurations. We observed an interaction effect between network type (ER vs. SF) and social support level (α), finding a significant difference between the final epidemic sizes observed on ER versus SF networks when α was large, but not when α level was small or zero. Our findings suggest that social support from non-infected individuals contributes substantially towards the inhibition of an epidemic outbreak, and that the type of network structure plays a role in determining the final epidemic size but only when the social support level is sufficiently high. Keywords: Complex networks, Epidemic spreading, Susceptible-infected-susceptible epidemics, Conditional quenched mean-field approach, Social-support, Adaptive networks.

1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

1. Introduction Infectious diseases compromise not only health and survival of individuals, but they can also be a cause of a great economic burden to the whole society [1]. In order to prevent and control disease outbreaks, governments need to impose intervention strategies [2], such as isolation, news distribution about the emerging disease, and vaccination [3]. On the other hand, individuals can locally protect themselves by employing simple mitigation strategies such as crowd avoidance or face-mask application [4], and generally by reacting adaptively to the circulating pathogens [5, 6, 7, 8]. Additionally, the dynamics of disease transmission can be modified considerably by the level of social support that emerges in a given population, such as support from the available medical resources and hospitals, the support from family and friends, or any help provided by organizations such as the Red Cross. Hence, the epidemic risk of infectious disease transmission and the associated economic burden of mass vaccination and isolation, can raise the awareness about the importance of increasing the investment in social resources [9]. Indeed, allocation of resources in epidemic outbreaks has recently been identified as an important component of efficient recovery policies [10], but also as a crucial factor for the early-stage epidemic mitigation, especially in the case of some highly lethal pathogens such as the Ebola virus [11]. ∗

Corresponding author. Tel.: +86 15270881015; fax: +86 791 88120360. Email address: [email protected] (Qingchu Wu )

Preprint submitted to Physica A

March 28, 2019

17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

In spite of many theoretical and computational studies of epidemic dynamics on complex networks [12, 13], our conclusive understanding of the role of resources in disease prevention and control is still lacking. Recently, mathematical models have provided some insights into development of optimal control and sensible resource allocation strategies [14]. B¨otcher et al. [15] studied the impact of resource constraints on epidemic spreading finding a discontinuous transition in the disease system. Chen et al. [16] investigated how the level of social support affects spreading dynamics in social-contact multiplex networks. The social support means here that the recovery of infected nodes depends on resources from healthy neighbors in the social layer of the multiplex network. Their results suggest the existence of an explosive transition, which appears to be closely related to both degree heterogeneity and edge overlap. When all edges overlap in a multiplex network, there is no explosive transition of the infected density in the steady state. In Ref. [17], Chen and colleagues further examined the effects of hybrid resources comprising both local and global social support, observing that the explosive transition and hybrid phase depend on the initial values of the infected nodes. We note here that related previous work largely assumed that each healthy node can only generate a single-unit new resource at each time step, regardless of the number of its infected neighbors. However, it is reasonable to consider a more general resource-generation or allocation mechanism in which the number of resources produced by a healthy node changes adaptively with the number of its infected neighbors. In addition, the process of self-recovery has not been considered in earlier models. Since the treatment and recovery of an infected individual are also related to her or his own particular situation (e.g., economic status, physical fitness, status of the immune system of an individual, etc.), various aspects of the self-recovery process cannot be ignored. Therefore, we would like to analyze the dynamical behavior of a susceptible-infected-susceptible (SIS) model when an infected node recovers via the combination of self-recovery and adaptive social-support. More recently, various studies of complex social dynamics [18, 19, 20, 21, 22] including those of spreading phenomena [23, 24, 25] have shown that many factors cannot unfold their full potential if they act completely individually and in isolation, but that instead specific combinations of various factor levels may actually be required to produce the desired effects. The presence of such nontrivial interaction effects between different mechanisms has largely been neglected in the earlier epidemic modeling literature, and studies have mostly considered the isolated effects of individual variables on epidemic size, threshold, and other relevant dependent measures. In order to address this problem, and to better understand the effects of social support and network structure on epidemic dynamics, we developed an individual-based analytical model with social support on complex networks by using the conditional quenched mean-field (cQMF) method developed recently by Wu et al. [26]. In our study design, we considered network type (Erd˝os-R´enyi and scale-free networks) and social support α (0, 0.1, 0.2, ..., 1) as the two independent variables, and we measured the density of infected individuals and the associated epidemic thresholds as dependent variables. We found that the generalized cQMF model agrees well with stochastic simulations in both network types, and that the epidemic threshold is strongly affected by the social-support level. Our results suggest that the social support contributes substantially toward the inhibition of epidemic transmission via increasing the epidemic threshold and decreasing the epidemic prevalence. Additionally, we observed that the type of network structure plays a role in determining the epidemic size only when the social support level α is sufficiently high, but not when α is low or zero. Thus, relative to a less supportive structure, a typically more favorable network structure will not significantly reduce the final epidemic size in the absence of a sufficient social-support level α. Moreover, our generalized model confirms that cQMF-based approaches [26] are generally better in predicting the final epidemic size than the previously introduced recurrent dynamic message passing method [27], and its generalized variant which we consider in the present paper. The remaining parts of our paper are structured as follows. In Section 2 we first give some prelim2

69

inaries necessary for the further model development. Section 3 employs the cQMF approach to study how self-recovery and social-support affect the spreading dynamics. Continuous-time stochastic simulations are performed to compare the simulation results with our theoretical model predictions. Finally, in Section 4 we discuss our main findings, suggest some further research directions, and conclude with some final remarks.

70

2. Methods and preliminaries

65 66 67 68

71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102

In the SIS-type model analysis, the microscopic Markov-chain (MMC) method is frequently employed to analyze the spreading dynamics on quenched networks [28, 29]. Van Mieghem et al. [30] developed a continuous-time MMC approach and called it as the N-intertwined method [31], which is also referred to as the quenched mean-field (QMF) method [32]. The QMF method uses the adjacency matrix to predict epidemic sizes and thresholds, and neglects the dynamical correlations among the states of neighbors, which induces deviations between the theoretical predictions and the corresponding numerical simulations [33, 34]. Mata and Ferreira [32] developed a pair-type formulation of quenched meanfield method, also called the pair QMF method [35]. It employs the two-order neighbor information and provides a more precise prediction [36], also on multiplex networks, as has been recently verified by Wu and Hadzibeganovic [37]. In Ref. [26], a conditional quenched mean-field (cQMF) method was introduced to analyze the recurrent-state spreading processes such as the SIS model. It was observed that the cQMF method has a comparable performance with the pQMF method but a simpler mathematical form in terms of the modeling derivation and structure. Moreover, the cQMF method clearly outperformed the recurrent dynamic message passing and quenched mean-field models [26]. In the present work, we will use the cQMF method to analyze the spreading dynamics with social-support. Hence, we first give some necessary notations and a brief illustration of the cQMF method for the standard SIS epidemic model. We assume that an infectious disease follows SIS recurrent dynamics on a given network denoted by an unweighted and undirected graph G = (V, E) with N = |V | and M = |E|. This graph can also be fully determined by its adjacency matrix A = (aij ): if the nodes i and j are connected links in G, then aij = aji = 1; otherwise aij = aji = 0. In addition, each node may stay in either susceptible (S) state or in infected (I) state. In the course of an infinitesimal time interval (t, t + ∆t], an infected node carries the infectious disease to its susceptible neighbors with probability β∆t and afterwards, it recovers and becomes susceptible again with probability µ∆t. Similar to the microscopic Markov-chain approach [28, 29], we focus on the dynamic of the probability of each individual node i to be infected at time t, denoted by ρi . Let Xi (t) ∈ {0, 1} denote the state of node i at time t [30]. If Xi (t) = 0, node i is susceptible; if Xi (t) = 1, node i is infectious. Then ρi = P[Xi (t) = 1] and 1 − ρi = P[Xi (t) = 0], where P[A] represents the probability of event A occurring. Moreover, we define the conditional probability θj|i = P[Xj (t) = 1|Xi (t) = 0], which holds for the case P[Xi (t) = 0] > 0. In order to apply it to all possible scenarios, it suffices to add that θj|i = 0 when P[Xi (t) = 0] = 0. Using these notations, in Ref. [26] we built a cQMF model describing the standard SIS epidemic dynamics comprising N + 2M equations X d ρi = −µρi + β(1 − ρi ) θi|j dt j∈N (i)

X d θj|i = −µθj|i + β(1 − θj|i ) θj|l dt l∈N (j) l6=i

(1) 103

where N (i) (N (j)) denotes the set of the first neighbors of node i (j). 3

104

105 106 107 108 109 110 111 112 113 114

3. The social support model 3.1. The model In this paper, we consider a generalized SIS epidemic model on a given network, where the recovery of infected nodes depends on resources obtained from healthy neighbors. In Ref. [16], Chen et al. used a multiplex-network framework to study the impact of social support on the epidemic spreading. They made the following basic assumptions about the role of resources: (i) Each healthy node allocates its resources to each infected neighbor uniformly; (ii) The total resources are not cumulative in the system, i.e., if healthy nodes have no infected neighbors to which they can allocate their resources, they consume these resources by themselves; (iii) Infected nodes consume all of the received resources at the current time step; (iv) Each healthy individual generates a new single-unit resource at the next time step. Herein, we revise assumption (iv) and assume that a healthy individual j with nj,inf infected neighbors can adaptively generate ζj = cj nj,inf unit resources at the next time step. Other above assumptions remain unchanged. In this case, the resource that the infected node j gives to node i is Rj→i = cj . Let Ri (t) denote the total resource that node i receives from healthy neighbors. Then, we have X Ri (t) = cj [1 − Xj (t)]. j∈N (i)

115 116 117

118 119 120 121

Without any resource support, an infected node can spontaneously recover at a rate µi0 . For an infected node i with ki total neighbors and ni,sus susceptible neighbors, the recovery rate of i at time t takes a general form µi (t) = f (µi0 , ki , Ri (t)). (2) In [16], f ∼ Ri (t)/ki . In [17], f ∼ Ri (t). In [38], the authors introduced a biased recovery strategy to cure the hubs in a population, and studied the SIS epidemic model with f ∼ kiν . Here, we consider both the self-recovery and social-support by which an infected node recovers. We assume that the transition rate from the infected state to the susceptible state is given by [17] f = 1 − (1 − µi0 )(1 − µi0 )α1 Ri (t)/ki .

122 123

124 125

For simplicity, we set µi0 = µ0 and cj = c, i.e., taking a homogeneous self-recovery and resource generating rates. Hence, in the continuous-time process, the recovery rate is given by   ni,sus µi (t) = µ0 1 + α . (4) ki Here, α = α1 c. In what follows, we will use the cQMF method [26] to derive the dynamical model. Using the law of total probability, one finds P[Xi (t + ∆t) = 1] = P[Xi (t + ∆t) = 1, Xi (t) = 1] + P[Xi (t + ∆t) = 1, Xi (t) = 0].

126

(3)

Then, the first term in the right-hand side (RHS) of Eq. (5) can be written as P[Xi (t + ∆t) = 1, Xi (t) = 1] X P[Xi (t + ∆t) = 1, ni,sus = x, Xi (t) = 1] = x

=

X

P[Xi (t + ∆t) = 1|ni,sus = x, Xi (t) = 1]P[ni,sus = x, Xi (t) = 1]

x

   X x = 1 − µ0 1 + α ∆t P[ni,sus = x|Xi (t) = 1]P[Xi (t) = 1] ki x 4

(5)

Following Lemma 1 in Ref. [35], we have X X xP[ni,sus = x|Xi (t) = 1] = P[Xj (t) = 0|Xi (t) = 1]. x

127

j∈N (i)

Hence, P[X (t + ∆t) = 1, Xi (t) = 1]  i    X 1 − µ0 ∆t − µ0 αki−1 ∆t P[Xj (t) = 0|Xi (t) = 1] P[Xi (t) = 1] =   j∈N (i)

128

(6)

While, the second term in the RHS of Eq. (5) is given by

P[Xi (t + ∆t) = 1, Xi (t) = 0] ki X = P[Xi (t + ∆t) = 1, Xi (t) = 0, ni,inf = x] x=0

= =

ki X

x=0 ki X x=0

129 130

P[Xi (t + ∆t) = 1|Xi (t) = 0, ni,inf = x]P[Xi (t) = 0, ni,inf = x] [1 − (1 − β∆t)x ]P[ni,inf = x|Xi (t) = 0]P[Xi (t) = 0] (7)

Thus, subtracting P[Xi (t) = 1] and dividing by ∆t, as well as letting ∆t → 0 in Eq.(5) and using Lemma 1 in Ref. [35], together with Eqs. (6) and (7), we can get X X d ρi = −µ0 ρi − µ0 αki−1 θ¯j|i ρi + β(1 − ρi ) θj|i dt j∈N (i)

131 132

j∈N (i)

Here, θ¯j|i = P[Xj (t) = 0|Xi (t) = 1] and θj|i = P[Xj (t) = 1|Xi (t) = 0]. The conditional probability θ¯j|i is computed as follows θ¯j|i ρi = P[Xi (t) = 1, Xj (t) = 0] = P[Xi (t) = 1|Xj (t) = 0]P[Xj (t) = 0] = θi|j (1 − ρj ).

133

Using this expression, Eq.(8) becomes X X d ρi = −µ0 ρi − µ0 αki−1 θi|j (1 − ρj ) + β(1 − ρi ) θj|i dt j∈N (i)

134

j∈N (i)

Next, it is needed to derive the dynamic of θj|i . By the total probability formula, we have P[Xj (t + ∆t) = 1|Xi (t) = 0] X = P[Xj (t + ∆t) = 1, Xj (t) = x|Xi (t) = 0] x

= P[Xj (t + ∆t) = 1, Xj (t) = 1|Xi (t) = 0] +P[Xj (t) = 0|Xi (t) = 0]P[Xj (t + ∆t) = 1|Xj (t) = 0, Xi (t) = 0] = P[Xj (t + ∆t) = 1, Xj (t) = 1|Xi (t) = 0] + P[Xj (t) = 0|Xi (t) = 0] × kj −1

X x=0

[1 − (1 − β∆t)x ]P[nj,inf = x|Xj (t) = 0, Xi (t) = 0] 5

(8)

135

By using the total probability formula and Lemma 1 in Ref. [35], one can obtain P[Xj (t + ∆t) = 1, Xj (t) = 1|Xi (t) = 0] X = P[Xj (t + ∆t) = 1, Xj (t) = 1, nj,sus = x|Xi (t) = 0] x

=

X x

P[Xj (t + ∆t) = 1|Xj (t) = 1, nj,sus = x, Xi (t) = 0] × P[nj,sus = x|Xj (t) = 1, Xi (t) = 0]P[Xj (t) = 1|Xi (t) = 0]

136

=

X

[1 − µ0 ∆t − µ0 αkj−1 x∆t]P[nj,sus = x|Xj (t) = 1, Xi (t) = 0]θj|i

 x   X = 1 − µ0 ∆t − µ0 αkj−1 ∆t P[Xl (t) = 0|Xj (t) = 1, Xi (t) = 0] θj|i   l∈N (j)       X −1 −1 P[Xl (t) = 0|Xj (t) = 1] θj|i = 1 − µ0 (1 + αkj )∆t − µ0 αkj ∆t     l∈N (j) l6=i

137 138 139 140

For the last equality, we assume that the topological connectivity is unclustered, which means that the network has no cycle of length 3. In that case, we have P[Xl (t) = 0|Xj (t) = 1, Xi (t) = 0] = P[Xl (t) = 0|Xj (t) = 1] = θ¯l|j when l 6= i. Similar to the computation of θ¯j|i , we have θ¯l|j ρj = θj|l (1 − ρl ). Therefore, the dynamic of θj|i is governed by X θj|l (1 − ρl ) X d θj|i = −µ0 (1 + αkj−1 )θj|i − µ0 αkj−1 θj|i + β(1 − θj|i ) θl|j . dt ρj l∈N (j) l∈N (j) l6=i

141 142 143 144 145 146 147 148 149 150 151 152 153 154 155

Remark 1: Let us denote the feasible region for the model as  ∆ = ρi , θj|i , i ∈ V, (i, j) ∈ E | 0 ≤ ρi ≤ 1, 0 ≤ θj|i ≤ 1 .

(9)

l6=i

By using the “outer normals” method in [39], it is easy to show that all solutions of (8) and (9) are bounded and positively invariant in the set ∆. Remark 2: When α = 0, i.e., there is no social support, and the dynamical model composed of (8) and (9) is just the cQMF model for the standard SIS epidemic spreading in networks [26]. In Ref. [26], the dynamic of θj|i is uncoupled with the first equations of ρi . However, this is not the case for the social support model. One can see that the second equations are coupled with the first ones. Remark 3: To investigate cyclic recurrent-state epidemic dynamics like SIS and SIRS, Shrestha et al. [27] introduced a recurrent dynamic message-passing (rDMP) method. In this approach, ρi is the probability that i is infected and θj→i denotes the probability that j was infected by one of its neighbors other than i. So, this probability (or say “message”) is defined on the directed edges of a network to carry information of the contagion and prevent the messages from immediately backtracking to the initial source node. Although the authors in [27] proposed the rDMP models for the SIS, SIRS, and SEIS recurrent-state models, a general derivation of the model structure was not included. Noting the mathematical similarity between the cQMF and rDMP models for the SIS model [26], we replace θj|i by θj→i in all equations but (1 − θj|i ) by (1 − ρj ) in the last term of the second equations, and we then get

6

156

the following equations and call it the generalized rDMP (G-rDMP) model. X X d ρi = −µ0 ρi − µ0 αki−1 θi→j (1 − ρj ) + β(1 − ρi ) θj→i dt j∈N (i)

j∈N (i)

X θj→l (1 − ρl ) X d θj→i = −µ0 (1 + αkj−1 )θj→i − µ0 αkj−1 θj→i + β(1 − ρj ) θl→j . dt ρj l∈N (j) l∈N (j) l6=i

l6=i

(10) 157 158 159 160 161 162 163

3.2. Epidemic thresholds In epidemiological studies, it is always useful to determine the epidemic threshold [40], to estimate whether the infectious disease will spread in a population. Note that ρj lies in the denominator and hence the system is not smooth at the disease-free equilibrium (i.e., ρi = θj|i = 0), that is similar to the effective degree model for local awareness [41, 42]. So, we cannot apply the linear stability approach to calculate the condition for epidemic outbreak. However, as an alternative, we can get its lower evaluation by considering the following system X d ρi ≤ −µ0 ρi + β(1 − ρi ) θj|i dt

(11)

j∈N (i)

164

and

X d θj|i ≤ −µ0 (1 + αkj−1 )θj|i + β(1 − θj|i ) θl|j dt l∈N (j)

(12)

l6=i

165 166

Apparently, the stability of the above system (here, the inequality is changed to the equal sign) is completely determined by the stability of the system (12). The Jacobian matrix for Eq.(12) then reads Lij,hl = −µ0 (1 + αkj−1 )δlj δih + βBij,hl.

167

Here, δij is a Dirac delta function and B is a 2M × 2M non-backtracking matrix [43, 44] Bij,hl = δjh ajh (1 − δil ail ).

168 169 170 171

172 173 174

175 176 177 178 179

(13)

(14)

Let ΛL denote the the largest eigenvalue of matrix L. Then the epidemic must go to extinction if ΛL (β) < 0. This shows that the critical value of epidemic outbreaks βc satisfies ΛL (βc ) ≥ 0. As for a random regular network (RRN), all N nodes have the same connectivity m, P (k) = δkm . So, the largest eigenvalue of Lij,hl is given by α ΛL = −µ0 (1 + ) + β(m − 1) (15) m where the largest eigenvalue of the Aij is computed by ΛA = m and the largest eigenvalue of the Bij,hl is m − 1. By the Chaplygin comparison theorem, the epidemic threshold βc of system (8) and (9) is then obtained as µ0  α βc ≥ 1+ (16) m−1 m

3.3. Simulations In this section, we compare stochastic simulations with the numerical predictions of theoretical models including the cQMF model (8)(9) and the G-rDMP model (10). To test the above argument, and also to enable its comparability with our earlier results [26], we perform stochastic simulations over two typical random network types: Erd¨os-R´enyi (ER) random network [45] and scale-free (SF) networks with 7

0.50

0.35

(a)

0.45

(b) 0.30

0.40

0.25

(t)

(t)

0.35

0.30

0.20

0.25

0.20 0.15 0.15

G-rDMP

cQMF

simulation

G-rDMP

0.10

cQMF

simulation

0.10 0

50

100

150

200

250

300

0

t

50

100

150

200

250

300

t

Figure 1: The density of infected individuals ρ(t) as a function of time t for the two different models and simulations in an ER random graph with mean degree hki = 5 and the size of the network N = 103 . In the stochastic simulations, β = 0.05, µ0 = 0.1, α = 0.5 (a), α = 1 (b) and the initial condition comprised 100 infectious nodes. The simulation results are the averages taken over a total of 100 independent simulation runs.

180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209

degree distribution P (k) ∼ k −γ for k0 ≤ k ≤ kc . The ER random network is fixed with mean degree z = 5 and√the SF network is built from the standard configuration model by taking γ = 2.7, k0 = 3 and kc = N [46]. Both network kinds have a small clustering coefficient. We use the same initial conditional of the infected nodes on two types of networks (ER and SF networks), and investigate the effect of different values of α. To simulate an SIS epidemic process on a contact network, we set the network size N = 103 and perform the Gillespie algorithm (GA) continuous-time stochastic simulations [47, 48]. At each time step t → t + ∆t, there is no more than one single event (infection or recovery) to occur. On each type of networks, we performed 100 independent simulation runs of epidemic dynamics, and we compare the ensemble average of simulation results with the corresponding numerical solutions of conditional QMF (cQMF) models and generalized rDMP (G-rDMP) models using the same contact network and disease parameters. In Figure 1, we compare the stochastic simulations with the model predictions for the temporal evolution of infectious density in an ER random graph with α = 0.5 (a) and α = 1 (b). In Figure 2, we use the same disease parameters and initial condition as that in Figure 1 to compare them in an SF network. From Figures 1 and 2, we can see that our model is significantly more accurate than the GrDMP model nearly during the complete evolution of the epidemic process. The cQMF predictions also agree well with the ensemble averages of the stochastic simulations. Thus, one can conclude from this the effectiveness of the conditional QMF model in analyzing the social support model. We further check their performances on the infection density at the metastable state in Figure 3. Since the self-recovery rate µ0 only affects the time scale of the system evolution from the theoretical model, we fix its value µ0 = 1.0 and only vary β values. It is easy to see that the conditional QMF method provides a good prediction about the final epidemic size and epidemic threshold when compared to the stochastic simulations, regardless of whether one uses ER graphs or SF networks. In Figure 3, the black arrows represent the lower bounds of epidemic threshold according to Eqs. (11) and (12). From this, one can observe a large discrepancy between the epidemic threshold and its lower bound, and hence the epidemic threshold can not be predicted by such a simple system. Also, our simulations indicate that the system under the present study has no discontinuous phase transition, which is similar to the result for the fully-overlapping case [16]. It is interesting to investigate the impact of social support on the epidemic threshold, since it is not 8

0.55

0.45

(b)

(a)

0.50

0.40

0.45 0.35 0.40 0.30

(t)

(t)

0.35

0.30

0.25

0.25 0.20 0.20 0.15

0.15

G-rDMP

cQMF

simulation

G-rDMP

0.10

cQMF

simulation

0.10 0

50

100

150

200

250

300

0

t

50

100

150

200

250

300

t

Figure 2: The density of infected individuals ρ(t) as a function of time t for the two different models and simulations in a scale-free network with γ = 2.7, k0 = 3 and N = 103 . In the stochastic simulations, β = 0.05, µ0 = 0.1, α = 0.5 (a), α = 1 (b) and the initial condition consists of 100 infectious nodes. The simulation results are the averages taken over a total of 100 independent simulation runs.

228

fully illustrated in the theoretical analysis. To this end, the epidemic threshold βc is obtained in the following way. Let β increase systematically by 0.01 in the interval [0, 1] and we compute the final epidemic size ρ for each β. When ρ > 0.0003 at β1 , we set βc = β1 − 0.01. In Figure 4(a), we show the impact of α on the epidemic threshold βc in the ER and SF networks used in Figure 2. Note that α reflects the strength of social support. Our results show that the epidemic threshold increases with α, which can also be seen in the expression of its lower bound. When β = 0.5 and µ0 = 1.0, we also investigate the impact of α on the epidemic prevalence (i.e., the final epidemic size) in Figure 4(b). Comparing the different values of α in the same network, we see that the larger the value of α, the smaller the epidemic prevalence. Moreover, we see an interesting interaction effect between the network structure type and the social support α. Specifically, a significant difference is observed in the final epidemic sizes between ER and SF networks when α is sufficiently large, but not when α is small or when α = 0, i.e. when there is no social support at all. Thus, the network structure plays a role in determining the epidemic size but only when the level of α is sufficiently high. For smaller values of α, the levels of infected individuals ρ will be roughly comparable, irrespective of the underlying network structure. Our additional analyses revealed that the infection density in our model increases with the degree k but decreases with α, up to a certain degree k after which the infection density largely becomes comparable irrespective of the level of α (not shown). Together with the simulation results shown in Figures 1 and 2, we can conclude that the social support indeed remarkably suppresses the spreading process in our model.

229

4. Discussion, conclusions, and future research directions

210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227

230 231 232 233 234 235 236 237

In summary, we used the susceptible-infected-susceptible model in a complex network to study how resource allocation affects the spreading dynamics. Infected nodes can be supported via medical resources provided by healthy neighbors and they can subsequently transform into a susceptible state. By using the conditional quenched mean-field method [26], we built a dynamical system described by a set of ordinary differential equations. Moreover, we conducted extensive stochastic simulations using the continuous-time GA method. The results showed that our model predictions agree well with stochastic simulations, regardless of the underlying network topology. It is also found that the social support has a strong inhibition effect on the epidemic spreading, and that the effect of the network structure depends 9

0.8

0.7

0.8

(a): ER

0.7

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.2

(b): SF

0.3

0.2

G-rDMP

lower bound

lower bound

G-rDMP

cQMF 0.1

0.0 0.0

cQMF

0.1

simulation

simulation

0.0 0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Figure 3: The density of infected agents ρ at the metastable state vs. β for the two different models. Simulations were performed in an ER random graph with mean degree hki = 5 (a) and in a scale-free network with γ = 2.7 and k0 = 3 (b). In the stochastic simulations, N = 103 , µ0 = 1.0 and the initial configuration consisted of 100 infectious nodes. Error bars show the standard deviation induced by 100 epidemic runs. The arrow indicates the lower bound of epidemic threshold obtained by Eqs. (11) and (12).

238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254

on the level of social support α. Individual behavioral responses based on local information have been identified as an important factor in the modeling of epidemic dynamics in complex networks [8], where the infection rate decreases with the number of infected neighbors [49, 50]. On the other hand, our social-support model considers the recovery rate, which increases in the present model with the number of susceptible neighbors. In epidemiology, both the recovery rate and infection rate are basic dynamical parameters. Hence, it is naturally interesting to consider and model the coupled effects of human behavioral responses and socialsupport. Mathematical models can clearly contribute to the advancement of our understanding of infectious disease dynamics and help us develop preventive measures to control infection spread efficiently. In future studies, the theoretical work reported in this paper could be extended to analyze spreading dynamics with local resources in multiplex networks [16, 51, 52] and hybrid (local and global) resources in complex networked systems [17]. Additionally, since epidemic modeling approaches are frequently used in other areas such as social contagion spreading [53, 54] and information diffusion [55], next generalizations of our present work could also be applied to such non-epidemic domains. Moreover, in the present study, the epidemic threshold was not analytically obtained from the mathematical model, and further research should therefore explicitly establish the condition for the epidemic outbreak [56].

257

Acknowledgments This work was jointly supported by the NSFC Grant (No. 61663015) and the Natural Science Foundation of Jiangxi Province of China (No. 20161BAB202051).

258

Reference

255 256

259 260

261 262

[1] J.M. Kirigia, L.G. Sambo, A. Yokouide, et al., Economic burden of cholera in the WHO African region, BMC Int. Health Human Rights 9 (2009) 8. [2] H.F. Zhang, P.P. Shu, Z. Wang, M. Tang and Michael Small, Preferential imitation can invalidate targeted subsidy policies on seasonal-influenza diseases, Appl. Math. Comput. 294 (2017) 332-342.

10

0.60

0.40

0.36

(b)

(a) 0.55

0.32

0.50

0.28

c

0.45 0.24 0.40 0.20 0.35

0.16

ER

ER 0.12

0.30

SF

0.0

SF

0.25

0.08 0.2

0.4

0.6

0.8

0.0

1.0

0.2

0.4

0.6

0.8

1.0

Figure 4: Plot of the effect of α on the epidemic spreading when µ0 = 1.0. Panel (a) illustrates the change of the epidemic threshold βc with α, and panel (b) shows the change of the final epidemic size ρ for β = 0.5 as a function of α. All the results are obtained by numerically solving Eqs. (8) and (9) on the same quenched networks as used in Figure 3.

263 264

265 266

267 268

269 270

271 272

273 274

275 276

277 278 279

280 281

282 283

284 285

[3] Z. Wang, C.T. Bauch, S. Bhattacharyya, et al., Statistical physics of vaccination, Phys. Rep. 664 (2016) 1-113. [4] Z.K. Zhang, C. Liu, X.X. Zhan, X. Lu, C.X. Zhang, Y.C. Zhang, Dynamics of information diffusion and its applications on complex networks, Phys. Rep. 651 (2016) 134. [5] T. Gross, C.J. D’Lima, B. Blasius, Epidemic dynamics on an adaptive network, Phys Rev Lett 96 (2006) 208701. [6] E.P. Fenichel, C. Castillo-Chavez, M.G. Ceddia, G. Chowell, et al., Adaptive human behavior in epidemiological models, Proc. Natl Acad. Sci. USA 108 (2011) 6306-6311. [7] H. Sayama, I. Pestov, J. Schmidt, et al., Modeling complex systems with adaptive networks, Comput. & Math. Appl. 65 (2013) 1645-1664. [8] P. Hu, L. Ding, T. Hadzibeganovic, Individual-based optimal weight adaptation for heterogeneous epidemic spreading networks, Commun. Nonlinear Sci. Numer. Simulat. 63 (2018) 339-355. [9] S. Funk, E. Gilad, V.A. Jansen, Endemic disease, awareness, and local behavioural response, J. Theor. Biol. 264(2) (2010) 501-509. [10] A. Dadlani, M. S. Kumar, K. Kim, F.D. Sahneh, Transient analysis of a resource-limited recovery policy for epidemics: A retrial queueing approach, 2016 IEEE 37th Sarnoff Symposium. doi: 10.1109/SARNOF.2016.7846752 [11] N.M. Shahtori, T. Ferdousi, C. Scoglio, F.D. Sahneh, Quantifying the impact of early-stage contact tracing on controlling Ebola diffusion. Math. Biosci. Eng. 15 (2018) 1165-1180. [12] R. Pastor-Satorras, C. Castellano, P. Van Mieghem, et al, Epidemic processes in complex networks, Rev. Mod. Phys. 87 (2015) 925. [13] C. Nowzari, V. M. Preciado, G. J. Pappas, Analysis and control of epidemics: A survey of spreading processes on complex networks, IEEE Control system 36(1) (2016) 26-46.

11

286 287

288 289

290 291

292 293

294 295

[14] X.L. Chen, W.J. Wang, S.M. Cai, et al., Optimal resource diffusion for suppressing disease spreading in multiplex networks, J. Stat. Mech. 2018 (2018) 053501. [15] L. B¨otcher, O. Woolley-Meza, N.A.M. Ara´ ujo, et al., Disease-induced resource constraints can trigger explosive epidemics, Sci. Rep. 5 (2015) 16571. [16] X.L. Chen, R.J. Wang, M. Tang, et al., Suppressing epidemic spreading in multiplex networks with social-support, New J. Phys. 20(1) (2018) 013007. [17] X.L. Chen, R.J. Wang, C. Yang, et al., Hybrid resource allocation and its impact on the dynamics of disease spreading, Physica A 513 (2019) 156-165. [18] T. Hadzibeganovic, D. Stauffer, X.P. Han, Randomness in the evolution of cooperation, Behav. Process. 113 (2015) 86-93.

297

[19] J. Garcia, M. van Veelen, A. Traulsen, Evil green beards: Tag recognition can also be used to withhold cooperation in structured populations, J. Theor. Biol. 360 (2014) 181-186.

298

[20] T. Clutton-Brock, Cooperation between non-kin in animal societies, Nature 462 (2009) 51-57.

296

299 300 301

302 303

304 305

306 307

308 309

310 311

312 313

314 315 316

317 318

319 320

[21] K. Lichtenegger, T. Hadzibeganovic, The interplay of self-reflection, social interaction and random events in the dynamics of opinion flow in two-party democracies, Int. J. Mod. Phys. C 27 (2016) 1650065. [22] T. Hadzibeganovic, D. Stauffer, X.P. Han, Interplay between cooperation-enhancing mechanisms in evolutionary games with tag-mediated interactions, Physica A 496 (2018) 676-690. [23] P. Manfredi, A. d’Onofrio, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases. New York, Springer, 2013. [24] Z. Zhang, Z.Q. Zhang, An interplay model for rumour spreading and emergency development, Physica A 388 (2009) 4159-4166. [25] C. Kamp, Untangling the interplay between epidemic spread and transmission network dynamics, PLoS Comput. Biol. 6(11) (2010) e1000984. [26] Q.C. Wu, R. Zhou, T. Hadzibeganovic, Conditional quenched mean-field approach for recurrentstate epidemic dynamics in complex networks, Physica A 518 (2019) 71-79. [27] M. Shrestha, S.V. Scarpino, C. Moore, Message-passing approach for recurrent-state epidemic models on networks, Phys. Rev. E 92 (2015) 022821. [28] Y. Wang, D. Chakrabarti, C.X. Wang, et al., Epidemic spreading in real networks: An eigenvalue viewpoint, In: Proceedings of 22nd International Symposium on Reliable Distributed Systems. Pittsburgh, PA: Carnegie Mellon University, 2003. [29] S. G´omez, A. Arenas, J. Borge-Holthoefer, S. Meloni, Y. Moreno, Discrete-time Markov chain approach to contact-based disease spreading in complex networks, Europhys. Lett. 89 (2010) 38009. [30] P. Van Mieghem, J. Omic, R.E. Kooij, Virus Spread in Networks, IEEE ACM T. Network 17 (2009) 1.

12

321 322 323

324 325

326 327

328 329

330 331

332 333

[31] C. Li, R. van de Bovenkamp, P. van Mieghem, Susceptible-infected-susceptible model: A comparison of N-intertwined and heterogeneous mean-field approximations, Phys. Rev. E 86(2) (2012) 026116. [32] A.S. Mata, S.C. Ferreira, Pair quenched mean-field theory for the susceptible-infected-susceptible model on complex networks, Europhys. Lett. 103 (2013) 48003. [33] W. Wang, M. Tang, H.E. Stanley, et al., Unification of theoretical approaches for epidemic spreading on complex networks, Rep. Prog. Phys. 80(3) (2017) 036603. [34] W. Wang, Q.H. Liu, J. Liang, et al., Coevolution spreading in complex networks, arXiv preprint arXiv:1901.02125, 2019. [35] Q.C. Wu, C.F. Chen, L.L. Zha, Epidemic spreading over quenched networks with local behavioral response, Chaos Solit. Fract. 96 (2017) 17-22. [36] T. House, M. J. Keeling, Insights from unifying modern approximations to infections on networks, J. R. Soc. Interface 8(54) (2011) 67-73.

335

[37] Q.C. Wu, T. Hadzibeganovic, Pair quenched mean-field approach to epidemic spreading in multiplex networks, Appl. Math. Model. 60 (2018) 244-254.

336

[38] Z. Dezso, A.L. Barabasi, Halting viruses in scale-free networks, Phys. Rev. E 65 (2002) 055103(R).

334

337 338

339 340

341 342

343 344

345 346

347 348

349 350

351 352

353 354

355 356

[39] A. Lajmanovich, J.A. Yorke, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Bios. 28(3-4) (1976) 221-236. [40] R. Pastor-Satorras, A. Vespignani, Epidemic spreading in scale-free networks, Phys. Rev. Lett. 86 (2001) 3200. [41] Q.C. Wu, W.F. Zhu, Toward a generalized theory of epidemic awareness in social networks, Int. J. of Mod. Phys. C 28(05) (2017) 1750070. [42] Q.C. Wu, S.F. Chen, Susceptible-infected-recovered epidemics in random networks with population awareness, Chaos 27(10) (2017) 103107. [43] F. Krzakala, C. Moore, E. Mossel, et al., Spectral redemption in clustering sparse networks, Proc. Natl. Acad. Sci. U. S. A. 110(52) (2013) 20935-20940. [44] X.H. Chen, S.M. Cai, W. Wang, et al., Predicting epidemic threshold of correlated networks: A comparison of methods, Physica A 505 (2018) 500-511. [45] P. Erd¨os, A. R´enyi, On the evolution of random graphs, Publ. Math. Inst. Hung. Acad. Sci 5 (1960) 17-61 . [46] M.E.J. Newman, S.H. Strogatz, D.J. Watts, Random graphs with arbitrary degree distributions and their applications, Phys. Rev. E 64 (2001) 026118. [47] J. Lindquist, J.L. Ma, P. van den Driessche, et al., Effective degree network disease models, J. Math. Biol. 62 (2011) 143–164. [48] P.G. Fennell, S. Melnik, J.P. Gleeson, Limitations of discrete-time approaches to continuous-time contagion dynamics, Phys. Rev. E 94(5) (2016) 052125. 13

357 358

359 360

361 362

363 364 365

366 367

368 369

370 371

372 373

[49] Q.C. Wu, X. Fu, M. Small, X.J. Xu, The impact of awareness on epidemic spreading in networks, Chaos 22(1) (2012) 013101. [50] H.F. Zhang, J.R. Xie, M. Tang, et al., Suppression of epidemic spreading in complex networks by local information based behavioral responses, Chaos 24(4) (2014) 043106. [51] G.F. de Arruda, F.A. Rodrigues, Y. Moreno, Fundamentals of spreading processes in single and multilayer complex networks, Phys. Rep. 756 (2018) 1-59. [52] J.Q. Kan, H.F. Zhang, Effects of awareness diffusion and self-initiated awareness behavior on epidemic spreading - An approach based on multiplex networks, Commun. Nolinear Sci. Numer. Simulat. 44 (2107) 193. [53] W. Wang, M. Cai, M. Zheng, Social contagions on correlated multiplex networks, Physica A 499 (2018) 121-128. [54] W. Wang, M. Tang, H.E. Stanley, et al., Social contagions with communication channel alternation on multiplex networks, Phys. Rev. E 98(6) (2018) 062320. [55] W. Wang, Z.X. Wang, S.M. Cai, Critical phenomena of information spreading dynamics on networks with cliques, Phys. Rev. E 98(5) (2018) 052312. [56] W. Wang, Q.H. Liu, L.F. Zhong, et al., Predicting the epidemic threshold of the susceptibleinfected-recovered model, Sci. Rep. 6 (2016) 24676.

14

Title: Epidemic spreading dynamics on complex networks with adaptive social-support

Highlights

● We study continuous-time epidemic model with self-recovery and adaptive social-support. ● An individual-based analytic model is derived by using the conditional quenched mean-field method. ● The epidemic threshold increases with the provided social-support. ● There is an interaction effect between the selected network type and social-support.