Resilience of epidemics for SIS model on networks

Complex systems¹ in various fields, ranging from natural to engineering systems, such as ecosystems, financial markets, and electric grids, can be viewed as complex networks. Such complex networks^2–4 are frequently subject to environmental changes or internal fluctuations. The dynamics on networks⁵ may possess the capacity to retain the original state essentially after perturbations. Such an adaptive capability is defined as resilience.^6–10 For example, it is shown that the food chain¹¹ in a biological network can withstand the shocks from a dramatic fall of one species and reorganize into a connected ecological web.

The resilience of epidemics here means that the spreading of epidemics recovers after various perturbations. Most studies on epidemic spreading mainly focus on the phase of the epidemic outbreak on networks.^12–19 Correspondingly, the parameters characterizing the processes of epidemic transmission on networks are nearly invariant in most mathematical epidemic models.^20–22 In some cases,^23–25 the epidemic spreading may be influenced or reduced by external control of self-repairing mechanism.^26,27 However, epidemic may continue to spread by absorbing the perturbations and recover to a stable trajectory, thereby presenting the above-mentioned resilient behaviors.

Resilient behaviors in the context of epidemic propagation have rarely been studied. In this paper, we study the resilience of epidemics on networks based on the classical epidemic model. Three well-known epidemic mathematical models are usually used in the study of epidemic transmission: SI (susceptible-infected) model, SIS (susceptible-infected-susceptible) model, and SIR (susceptible-infected-recovered) model.^12,28,29 The classical SIS epidemic model is one of the most general way to model the epidemic dynamical behaviors on networks such as Erdős-Rényi (ER) networks³⁰ and scale-free (SF) networks.³¹ It is well known that there exists a nonzero epidemic threshold λ_c^12,32 for ER networks in the dynamics of epidemic outbreak. For a given infection rate λ  ≥  λ_c, the epidemic will spread out, and the system will reach a stationary state with a finite stable density ρ of the infected population. However, if the infection rate λ is below the epidemic threshold λ_c, the epidemic will ultimately die out, with no infected individuals (i.e., ρ  =  0). In contrast, it has been demonstrated that the epidemic threshold does not exist for SF networks with 2 < γ  ≤  3.¹⁴ This has prompted the propagation of viruses in such networks. When the parameter γ  >  3, the epidemic will spread on these SF networks with an epidemic threshold λ_c.^33,34

In this paper, we perform studies on the resilience of epidemic transmission with λ varying over time in the classical SIS epidemic model, to model the perturbations (i.e., “control”). When the epidemic propagation is controlled since a certain time instant (i.e., ct) for a duration cd, the infection rate (λ₂) is smaller than λ₁ in the uncontrolled state. Based on different durations of control (i.e., cd), the transmission level of the epidemic would reflect distinct resilient behaviors after the control. The issue being tackled in this paper is whether the epidemic propagation can present resilient behaviors and the corresponding critical condition.

According to the above-mentioned model, we analyze epidemic spreading on three types of real networks (Facebook network, Internet, and social network) and two types of network models (ER networks and SF networks). The simulation results have demonstrated that under certain conditions, the epidemic can bounce back to the initial steady state in the finite network scale. It is shown on ER networks that even for λ₂ < λ_c, the epidemic may eventually bounce back when control time is below a threshold. This is verified by our theoretical analysis of the recovery probability for epidemics, P, calculated by the equation $ρ(ct+cd)→1N$⁠. The critical control time of the resilience (i.e., the critical time for the extinction of epidemics, cd_max) seems to be predicted by the diameter (d) of networks, with the quantitative relation cd_max ∼ d^α.

The contents of this paper are arranged as follows. Section II is devoted to introducing the model used to study the resilience of epidemics on networks based on the classic SIS epidemic model. In Sec. III, we perform simulations on different types of networks. Theoretical analysis is also performed on the obtained results. In Sec. IV, we draw the conclusions and present the discussions.

In the SIS epidemic model, nodes in the network are divided into two compartments: susceptible individuals (S) and infected individuals (I). Initially, a fraction of nodes, which are randomly selected in the network, are infected. At each time step, each susceptible node is infected by each of its infected neighbors in the network with probability β. Each infected node is cured and becomes susceptible again with probability δ simultaneously. The effective infection rate is defined as λ  =  β/δ.

In this paper, we study the resilience of epidemics under perturbations, by changing the infection rate λ to model the perturbations (i.e., “control”). The model applied in networks describing the whole propagation process can be classified into three sub-phases by adding “control,” where the starting time and the duration of the “control” are defined as ct and cd, respectively:

• Phase 1: The epidemic spreads with an infection rate λ₁ = β/δ₁ lasting ct time steps since the beginning t = 0. The probability β is set to a constant value in the entire process of epidemic transmission.

• Phase 2: When the “control” is introduced in the second phase since time ct, the recovery rate δ₂ in the controlled network is given larger than δ₁, i.e., λ₂ is smaller than λ₁. This process will last for cd time steps.

• Phase 3: The control is removed at time instant ct + cd and the epidemic propagates with the infection rate λ₃, equal to λ₁ in phase 1.

In addition, the infection densities that we mainly observe are defined as ρ₁, ρ_c, and ρ_r for each stage. The model can be explained by the following example. In the Internet, some computers may become infected by a certain virus. Accordingly, anti-virus efforts (“control”), i.e., operations including certain immunization process and restoration with anti-virus software,³⁵ will be implemented to stop or lower the epidemic spreading, thus leading to a dramatic decline of the infection rate. When the control is removed due to the limited budgets or adaptation of virus, for different control durations cd, the epidemic may lose its resilience or resume to spread, therein presenting different resilient behaviors. We study the critical condition by both simulation and theoretical results that the epidemic can continue propagating after the control phase.

We show the results of epidemic resilience on three types of real networks (Facebook network, Internet, and social network) and two types of network models (ER networks and scale-free networks). The Facebook network originates from the database on Stanford Large Network Dataset Collection site.³⁶ It contains 4039 nodes and 88 234 edges, with the average degree of $〈k〉≈43.691$⁠. The data of Internet come from the UCI Network Data Repository,⁴¹ whose average degree is $〈k〉≈4.219$⁠, including 22 963 nodes and 48 436 edges. The dataset of social network (Google+) with $〈k〉≈3.322$ was collected by the Koblenz Network Collection,^36,42 which contains 23 628 nodes and 39 242 edges. In addition, these real networks have degree distribution, following a power-law distribution with the exponent γ  ≈  1.3, 2.5, and 2.6, respectively. For an ER network with N nodes, each node pair is independently connected with a probability p. Then, we create an ER graph with a Poisson degree distribution³⁷ described as follows:

We create scale-free networks via the configuration model,³ following a scale-free distribution P(k) ∼ k^–γ, where γ is the degree exponent.

In this paper, we first focus on the key quantity about the epidemics, i.e., the infection density ρ to study the resilient behaviors of epidemic propagation. Meanwhile, the critical condition for the emergence of the resilient behaviors will be studied. We also explore the probability of recovery for epidemics, P, as a function of infection rate λ₂ in the second phase and control duration cd. In order to find the critical recovery time of resilience for the epidemic to “bounce back,” we calculate the infection density ρ_r in the third stage, as a function of control duration (cd). In addition, understanding the relation between the resilience of epidemics and network structure is essential to design the resilience strategy. The relation between the critical control time of resilience cd_max and diameters (d) seems to be identified on ER and SF networks.

To study the resilient behaviors of epidemic propagation, we first perform simulations to obtain the infection density ρ as a function of time on different types of networks. To observe the processes of epidemic transmission on real networks, Facebook network, Internet, and social network are examined with the SIS epidemic model, respectively. As shown in Fig. 1(a), the epidemic spreads rapidly in the uncontrolled stage, since there exits hub nodes in the topological structure of Facebook network. The infection density is decreased when the epidemic is controlled with a lower infection rate since time instant ct. When the control stage is finished and infection rate is recovered, the epidemic can soon restore to the same state before the control. For the Internet shown in Fig. 1(b), the epidemic can bounce back to the original steady state when the “control” is removed. As shown in Fig. 1(c), the simulations on social network show that the infection density can return to the initial stationary state after the “control,” when λ₂ is below that of uncontrolled state. The infected individuals are decreased to zero with a quite small λ₂ and the epidemic propagation comes to an end without exhibiting resilience. The findings on the real networks illustrate that under certain conditions, the epidemic can bounce back to the original steady state in the finite network scale.

Then, we carry out simulations of the epidemic propagation on ER and SF networks. In the case of ER networks, we acquire the results obtained from simulations shown in Fig. 2(a). It shows that the infection density ρ is decreased quickly once the“ control” is added. When the “control” is removed, the epidemic can also continue to spread and recover to a steady state. It is known that the epidemic threshold $λc=1/〈k〉$ through the theoretical calculation,³⁸ when there is no control in ER networks. Here, the infection rate for control phase is smaller than λ_c. Therefore, the transmission capability of epidemics decreases with “control” added. Afterwards, it recovers in the uncontrolled stage showing resilient behaviors. For SF networks with γ  =  2.5 illustrated in Fig. 2(b), epidemics can always bounce back to its original state in our simulations at different control durations. This may be due to the fact that the network has no epidemic threshold for 2 < γ  ≤  3. The findings explicitly demonstrate the existence of the resilient behaviors for the epidemic propagation on SF networks. In addition, in the inset of Fig. 2(b), the changes of infected density ρ with various given cd on SF networks with γ  =  3.5 show that the epidemic could not exhibit resilience with a quite large cd. Meanwhile, the dynamical mean-field rate equation is applied to describe the processes analytically. For comparison, the classical mean-filed rate equation is shown below:

The first quantity we study here in ER networks is the density of infected nodes ρ(t). The governing equation, which is mainly used for the analysis of this work, follows:

when t ≤ ct or t > ct + cd.

when ct < t ≤ ct + cd.

It is found that the simulation results on ER networks can be well matched by the theoretical analysis obtained from Eqs. (3) and (4). However, we found that the classical mean-field rate equation can generate different results compared with our simulations. Through the analysis of each part of two equations, we found that the problem mainly comes from the infection process of classical mean-field theory, which is not consistent with that of governing equation in this work. Because, in the process of infection, we assume that a node being infected by each of its infectious neighbors is not independent. And the further explanation is that item $β〈k〉ρ(t)$ in the classical mean-field theory of SIS model has different probability consideration from item $1−(1−β)〈k〉ρ(t)$ in Eqs. (3) and (4).

In order to find the critical condition for the emergence of the above-mentioned resilient behaviors, we perform the theoretical and simulation analysis for the probability of epidemics returning to the steady state in ER networks, P, as a function of infection rate λ₂ in the controlled stage. For theoretical analysis, the probability of recovery for epidemics P can be calculated by the mean-field equation written as

In Fig. 3(a), the probability of epidemic restoration, P, grows to 1.0 eventually with increasing λ₂ for a given cd. There exists a critical infection rate $λ2c$ in control stage for epidemic resilience. Meanwhile, even for λ₂ < λ_c, where λ_c is the epidemic threshold (⁠$λc=1/〈k〉$⁠) shown by dash line, the epidemic may still bounce back, which is due to the fact that the infected density did not drop below the necessary threshold during the time cd. When the equation satisfies $ρ(ct+cd)≤1N$ in theory, the epidemic is considered to lose the resilience. The theoretical results, based on the above-mentioned calculation method, can predict the critical $λ2c$⁠, verified by the simulation results. It illustrates that the epidemics will bounce back with $λ2≥λ2c$ (λ₂ < λ_c) for a given control duration cd. This threshold for epidemic resilience depends on the combined effect of epidemic spread and control processes.

To explore the effect of the control duration cd on the probability for epidemics to return to the steady state, we perform the theoretical and simulation analysis for P as a function of cd in ER networks. As shown in Fig. 3(b), the probability P is decreased with the increasing cd for a given infection rate λ₂ (< λ_c = 0.1). When the control duration (cd) is small, the probability maintains 1.0. As cd increases, the recovery probability is decreased and finally reaches to zero. It is shown that there exists a threshold (i.e., cd_max) that makes the epidemic lose the resilience completely. And cd_max can also be regarded as the critical recovery time of resilience. The theoretical results (solid line) are obtained based on the Eq. (5). Our theoretical analysis can also predict the critical control duration, cd_max, above which the epidemic will lose its resilience. It further illustrates that even for λ₂ < λ_c, the epidemic is still possible to bounce back, when the control duration, cd, is not long enough.

As the resilience for the epidemic propagation is affected by the infection rate λ₂ in the second stage of transmission and the control duration (cd), we study the resilience output with the steady-state density in the third stage, ρ_r, as a function of cd. As shown in Fig. 4(a), for a fixed λ₂, the infection density ρ_r is decreased as cd is increased for an ER network. When cd is increased to a certain value, ρ_r is almost reduced to zero. This confirms our finding for cd_max in Fig. 3. It can also be seen that ρ_r falls to zero earlier with a smaller infection rate λ₂ than that with a large λ₂. To study the effect of network size on the resilience of epidemics, we perform simulations to obtain ρ_r as a function of cd in ER networks with different sizes. As shown in Fig. 4(b), in the case of ER networks, ρ_r is reduced to zero gradually with increasing cd time steps for various N. Infection density after control stage, ρ_r, is decreased more quickly with a relatively small N. It is considered that cd_max is larger for a large N.

It is meaningful to understand the relation between epidemic resilience and network structure, which can help to design the resilience strategy. Therefore, we perform simulations for cd_max as a function of diameters (d),^1–3,39,40 which is calculated by averaging the number of hops between each pair of nodes in ER networks and SF networks, respectively. In Fig. 5(a), for a given λ₂, the cd_max seems to be found to scale with network diameters in ER networks, cd_max ∼ d^α. In the case of SF networks with γ  =  3.5, as shown in Fig. 5(b), there exits the critical recovery time of resilience (cd_max), which has a different quantitative relation with diameters (d) from that of ER networks. The findings on ER and SF networks illustrate that the time (distance) that epidemics take to spread over the entire network is critical to estimate the resilience of this process. In addition, α is increasing with increasing λ₂.

By adding a “control” stage in the original SIS model, we model the resilience of epidemic propagation under perturbations. When the network is in the controlled state, the epidemic transmits with a smaller infection rate λ₂ than that in the uncontrolled state. When the control is removed, the epidemic may restore to a steady state exhibiting resilient behaviors.

Based on the above-mentioned model, we performed numerical simulations on three types of real networks (Facebook network, Internet, and social networks) and two types of network models (ER and SF networks). The simulation results indicate that under certain condition, the epidemic can restore to the original steady state in the finite network size. Through the simulations and theoretical analysis on ER networks, it is shown that even for λ₂ < λ_c, the epidemic may eventually bounce back when the control duration, cd, is smaller than a threshold cd_max. It can be verified by the theoretical results of the recovery probability for epidemics, P, computed by the equation, $ρ(ct+cd)→1N$⁠. The critical value cd_max is strongly related to the network structure, where cd_max appears to be predicted by the diameter d of networks with the quantitative relation, written as cd_max ∼ d^α. Note that the maximum time of control for epidemics extinction is increasing with system size and will diverge for an infinite system. The resilience is an intrinsic property for epidemics to adapt to the external perturbations and the changes of internal conditions. It can also issue signals for systems to mitigate the epidemic rapidly and accurately when they have been infected. The discovery of cd_max may provide advanced indicator for the resilience of the epidemic, which can help to design protection strategy keeping systems from a secondary epidemic outbreaks.