Stochasticity of disease spreading derived from the microscopic simulation approach for various physical contact networks

Abstract

COVID-19 has emphasized that a precise prediction of a disease spreading is one of the most pressing and crucial issues from a social standpoint. Although an ordinary differential equation (ODE) approach has been well established, stochastic spreading features might be hard to capture accurately. Perhaps, the most important factors adding such stochasticity are the effect of the underlying networks indicating physical contacts among individuals. The multi-agent simulation (MAS) approach works effectively to quantify the stochasticity. We systematically investigate the stochastic features of epidemic spreading on homogeneous and heterogeneous networks. The study quantitatively elucidates that a strong microscopic locality observed in one- and two-dimensional regular graphs, such as ring and lattice, leads to wide stochastic deviations in the final epidemic size (FES). The ensemble average of FES observed in this case shows substantial discrepancies with the results of ODE based mean-field approach. Unlike the regular graphs, results on heterogeneous networks, such as Erdős–Rényi random or scale-free, show less stochastic variations in FES. Also, the ensemble average of FES in heterogeneous networks seems closer to that of the mean-field result. Although the use of spatial structure is common in epidemic modeling, such fundamental results have not been well-recognized in literature. The stochastic outcomes brought by our MAS approach may lead to some implications when the authority designs social provisions to mitigate a pandemic of un-experienced infectious disease like COVID-19.

1. Introduction

The wake of COVID-19 strongly sheds light on the importance of mathematical epidemiology. The mile-stone work on epidemic modeling presented by Kermack–McKendrick [1] in 1927, called the SIR (susceptible–infected–recovered (or removed)) model, has revolutionized the field of mathematical epidemiology. Since then, there has been rich stock of modeling works in line with the so-called ordinary differential equation (ODE) approach, where a set of ODEs depicting the dynamics of respective compartments is introduced [2,3]. The ODE approach presumes a deterministic process, although some stochastic situations could still be analytically treated even in ODE frameworks [4], [5], [6]. From the viewpoint of the application to real-world problems, one of the most important issues related to stochastic disease spreading process is the spatial structure, i.e., the underlying network connecting individuals where each agent is physically linked by a complex social network. Underling network structure decides what nodes, stochastically appeared as initial infectious individuals, substantially affect the final epidemic size (FES) and its time-evolution. Thus, a stochastic multi-agent simulation (MAS) framework based on a certain appropriate algorithm, such as Gillespie algorithm [7], plays an essential role in drawing a precise prediction. However, there have been several works in literature which allow an ODE framework to approximately quantify the basic stochastic features [8], [9], [10], [11]. The MAS approach accounting underlying network effect, which can also capture various realistic scenarios inevitably brought by stochastic components, could be extended to more practical applications than simple epidemiological analysis. For instance, the so-called vaccination game [12], [13], [14], where disease spreading dynamics and human decision dynamics on whether or not committing a vaccination could be mutually dovetailed. The agent-based simulation on networks delivers stochastic outcomes using a robust statistical procedure which may not be tractable in analytic procedures.

A recent study by Okabe and Sudo [15] has reported stochastic deviations in FES by a microscopic SIR simulations on networks. The authors mainly focused on Erdős–Rényi random network [16] (called ER-random) and scale-free (SF) network [17]. Although the study successfully delivered results highlighting its stochasticity, which was compared to the result from the ODE prediction with mean-field approximation (MFA), there was none of the holistic discussion when underlying topology would be different from ER-random and SF graphs. Therefore, a more comprehensive and profound picture of the focal point is highly needed. This work intends to draw a quantitative answer to this question.

The remainder of this paper is organized as follows. In Section 2, we establish a MAS model. In Section 3, we present the results and discussion. Finally, Section 4 presents the conclusion.

2. Model depiction

We tailored an SIR process using multi-agent simulation (MAS), where N agents are placed at nodes (vertices) on an underlying network. Agents are connected via links. Throughout this study, according to our previous MAS studies [13,14], we presumed N = 10⁴ (except for three-dimensional (3D) regular graph of which population is presumed N = 9, 261) as acceptably enough to avoid the effect of finite domain size. The underlying network demonstrates physical contacts among individuals. An epidemic triggered by some initial infected and infectious individuals, which is denoted by I ₀, spreads on the network. We varied I ₀ either 5 or 50 to explore how an initial condition affects the stochasticity of disease spreading.

2.1. Disease spreading process

We presumed the SIR process, which divides a population into three groups: susceptible (S), infected and infectious (I), and recovered (R). Susceptible individuals become infected at a rate provided by parameter β—the disease transmission rate per day per person. Here the physical unit of β [person⁻¹ day⁻¹] has “person⁻¹,” since the transmission “rate” inherently means how many susceptible people a single infectious “person” can infect per day. Individuals recover from the disease with a probability determined by parameter γ—the recovery rate per day (i.e., the inverse of the mean number of days required to recover from the disease). Referring to the previous works [18], we assumed γ = 1/3.

In a MAS procedure, introducing the equivalent transmission rate (i.e., the transmission probability, as mentioned afterward), β_eq[day⁻¹ person⁻¹ link⁻¹], as defined in Section 2.2, we can formulate the state transfer process as follows:

$$\mathrm{I}+\underset{\_}{\mathrm{S}}\stackrel{{\beta }_{\text{eq}}}{\to }\mathrm{I}+\underset{\_}{\mathrm{I}}$$ (1-1)

$$\underset{\_}{\mathrm{I}}\stackrel{\gamma }{\to }\underset{\_}{\mathrm{R}}$$ (1-2)

where underline implies the change of state of a focal individual. Since our MAS model fully obeys a stochastic process, we applied the Gillespie algorithm [7] to numerically simulate the SIR dynamics on a social network. Here, a discrete-time increment allowing a single state transition event varies according to the prevalence of infected individuals at a certain time-step, which clearly reflects the stochasticity of the dynamical system. This process starts by assuming randomly selected infected individuals among susceptible. At each time step, the state of susceptible and infected person changes according to each transition probabilities determined by the Gillespie algorithm. Each simulation episode ends when there are no infected individuals in the system, that is, all infected individuals recover. At each simulation episode, we quantitatively evaluate the final epidemic size (FES), which can be observed when all infected agents recover. Section 3 presents how each FES under the same condition can stochastically deviate. It should be noted that each outcome, e.g., FES, is estimated by taking an average over 10⁴ independent realizations.

2.2. Physical network

We investigated the stochastic effect of the underlying network that depicts the physical contact among agents. We varied topology and its average degree. Results generated from MAS have been compared with that of an ODE based mean-field approximation.

In a series of MAS experiments, we presumed ring (i.e., 1D regular graph), lattice (i.e., 2D regular graph), and 3D regular graph, which exhibit relatively larger average path lengths than those of general heterogeneous graphs. These three topologies inherit a strong locality effect. We also investigated a random regular graph (RRG). As for heterogeneous networks, we explored Erdős–Rényi or ER-random [16] and Barabasi–Albert scale-free graph (BA-SF) [17].

If an ODE-based SIR model is concerned, the transmission rate is unequivocally defined by β = R ₀ · γ [person⁻¹ day⁻¹], where R ₀ is the basic reproduction number. However, we must be careful when taking a MAS approach on networks because an appropriate equivalent transmission rate must be defined. More precisely, such an equivalent transmission rate should be called the transmission probability because it physically means with what probability a focal infected agent spreads infection to susceptible neighbors through each of its link. Thus, this physical property, called the equivalent transmission probability β_eq, should have [link⁻¹ day⁻¹] as the physical unit. Note that the physical meaning (physical unit) is different from that of β [person⁻¹ day⁻¹].

In this study, irrespective of network topology, we thoroughly presumed β_eq as follows:

$${\beta }_{\text{eq}}=\frac{R_0·\gamma }{〈k〉}$$ (2)

Note that β_eq< 1 because β_eq [link⁻¹ day⁻¹] is defined as the transmission probability in an interaction event between S and I agents.

If a network, with degree distribution P(k), is not regular and not randomly connected, i.e., if the network does not belong to either 1D regular graph (ring), 2D regular graph (lattice), or 3D regular graph, the basic reproduction number representing such a networked SIR dynamical system can be evaluated as follows [9]:

$$R_o^{\prime }=\sum _{k=1}^{k_{max}}\frac{k·P\left(k\right)}{k}·\left(k-1\right)·\frac{{\beta }_{eq}}{{\beta }_{eq}+\gamma }$$ (3)

where k_max and 〈k〉 represent the maximum degree and average degree of the underlying network, respectively. This formulation comes from the following situation. Suppose that at a certain time, a focal agent (node) becomes infected by getting infection from one of its k neighbors. The term $\frac{k·P\left(k\right)}{〈k〉}$ indicates the probability that this focal agent has k links. One of the neighbors is either I or R because this neighbor lets the focal agent become infected. The remaining k − 1 neighbors are susceptible now. Considering this focal agent, it exhibits a chance to those k − 1 neighbors to become infected. The probability that the focal agent driving one of the k − 1 neighbors to become infected before its recovery can be evaluated by $\frac{{\beta }_{eq}}{{\beta }_{eq}+\gamma }$. Thus, the right side of Eq. (3) quantifies the expected number of individuals in the network being infected by a single infectious agent, which literally depicts the definition of basic reproduction number. Therefore, $R_o^{\prime }<1$ brings a time-evolution that must end with a disease-free equilibrium (DFE). Note that in the case of ODE-based SIR process, $R_0=\frac{\beta }{\gamma },$ which also suggests that R ₀ < 1 leads to a DFE. By solving Eq. (3), we explicitly obtain β_eq, and then using Eq. (2), we can obtain the relationship between R ₀ and $R_o^{\prime }$ as follows:

$$R_o^{\prime }=R_o\frac{〈k^2〉-〈k〉}{<k{>}^2+R_o〈k〉}$$ (4)

2.3. Stochastic deviation observed in MAS trial

Generally speaking, any result based on MAS could not avoid the influence of stochastic noise because each simulation trial randomly places initial infected individuals in the domain which leads to a different final result. Note that this stochastic noise results from two reasons. One is the locality effect each topology incurs, which is what we intend to highlight in the present study. Another reason results from the so-called domain size effect, where a finite domain size cannot escape from the error when compared to the result with an infinite domain size, which is impossible to obtain as long as taking a MAS approach. Compromising those, we took, as explained above, an appropriately large domain size (10⁴) and observe a healthy statistic ensured by sufficient number of independent realizations (10⁴). Averaging outcomes of these realizations, we can get the estimation for FES.

3. Result and discussion

Before discussing the results, we first demonstrate the graphical representations of all the network topologies that we considered in our work. Fig. 1 provides the illustration of each topology in which panels (a)-(d) correspond to regular graphs (i.e., every node has the same degree), such as ring, 2D and 3D lattice, random regular graph, whereas panels (e) and (f) layout the instances of heterogenous networks (degrees of agents follow a probability distribution) such as ER random, BA scale-free network.

Fig. 1.

A pictorial representation of all the network topologies that we considered. Panels (a)-(d) represent the instances of regular networks (every node has the same degree), whereas panels (e)-(f) provide the illustrations of heterogeneous networks such as Erdős–Rényi (ER) random and Barabási–Albert (BA) scale-free.

3.1. Regular graph

Fig. 2 shows the result when we presume ring. In all panels, X-axis indicates R ₀, and Y-axis indicates FES. The left and right panels show respective results varying the number of initially infected individuals; I ₀ is randomly placed in the network, either 5 (panels (a-*)) or 50 (panels (b-*)). Panels (*-ii), (*-iii), and (*-iv) vary the degree of 8, 16, and 64, respectively. Because of stochasticity, each plot of FES is widely scattered from which we took the ensemble average. The blue (k = 8), yellow (k = 16), and red (k = 64) lines correspond to panels (*-ii), (*-iii), and (*-iv), respectively. The violet curve indicates the frequency of DFE. In both panels (*-i), the ODE result is added by a green broken line as a reference to the respective MAS results. The visual format mentioned in this paragraph will be followed in the subsequent figures.

Fig. 2.

Representations of FES as a function of R₀ on a ring graph. The left panels (a-*) and right panels (b-*), panels show the results of I₀ = 5 and 50, respectively. Panels (*-ii), (*-iii), and (*-iv) correspond the results for degrees 64, 16, and 8, respectively. The ensemble average of scattered plots in (*-ii), (*-iii), and (*-iv) is supplemented in panels (a-i) and (b-i) by red (k = 64), yellow (k = 16), and blue (k = 8) lines, respectively. The violet curve suggests the frequency of disease-free equilibrium (DFE). In panels (*-i), several average values of MAS results are compared with the mean-field outcomes (green broken lines). Panels (*-i) show larger difference between the results of MAS and ODE based mean-field approximation. Interestingly, the case k = 8 leads to a disease-free situation or a smaller outbreak (panels (*-iv)).

Figs. 3 and 4 show the results when lattice and 3D regular graphs are presumed. In these cases, we investigate the outcomes for two scenarios; k = 6 and 18.

Fig. 3.

Illustrations of FESs as a function of R₀ on a 2D lattice. Results are shown for two different initial conditions. The visual explanation is similar to that in Fig. 2. Contrasting panels (*-i) with panels (*-i) in Fig. 2 (ring), we perceive relatively less deviations in MAS results (on lattice) compared to mean-field approximation.

Fig. 4.

The FESs are shown as a function of R₀ on a 3D regular graph. The explanations for several panels are similar to that in Fig. 2 and 3. Unlike Figs. 2 and 3, in this case, we demonstrate results for degree k = 6 and 18. Results corresponding to higher degree are closer to the mean-field outcome which is conceivable.

It is worth stressing that the extent of FES plots of each episode being scattered is most significant for the case of the ring with presumption of I ₀ = 5 and k = 8 (Fig. 1 (a-ii)). This is because the specific case presumes the following: (i) a most strong locality effect brought by the 1D regular structure than 2D and 3D ones, (ii) the smallest degree that makes far away from the well-mixed situation, postulated by mean-field approximation (MFA), and (iii) a smaller initially infected individuals that mechanically allows a larger stochastic deviation than that of a large number of initially infected individuals.

Note that the largest degree (k = 64) of the ring (Fig. 2) and lattice (Fig. 3) presuming sufficiently large R_o (approximately R_o> 4 for ring and R_o> 3 for lattice) exhibits the assembled-averaged FES (red line in panels (*-i)) being almost consistent with the result from MFA (denoted by a green broken line in these panels). This indicates that the network effect for these settings becomes negligible. Other settings and situations more or less deviated from those predicted by the deterministic MFA, resulting from more or less scattered plots of MAS episodes. Such deviations arise from the stochastic effect brought by the underlying network and initial conditions.

3.2. Random regular graph

Fig. 5 shows the result on a random regular graph (RRG), which is presented in the same format as in Fig. 2. Because of random connections, unlike 1D (Fig. 2), 2D (Fig. 3), and 3D (Fig. 4) regular graphs, the extent of scattering is much less than previous results. Thus, the assemble-averaged FES of any degrees is close to the deterministic prediction by MFA in which the tendency is more in the case of I ₀ = 5 (Fig. 5 (a-i)) than I ₀ = 50 (Fig. 5 (b-i)). Although no degree distribution is found like other regular graphs, RRG shows a quite different tendency from that of other regular graphs, as described above. Therefore, random connection realizes fewer scattering plots, i.e., entails less extent of stochasticity brought by the underlying network.

Fig. 5.

The FESs are shown as a function of R₀ on a random regular graph (RRG). The assemble averaged FESs illustrated in panels (*-i) are pretty close to the mean-field approach (MFA). The main horizontal axes in panels (*-i) correspond to the scaling for MFA. The three horizontal axes (panels (*-i)) beneath the main axis correspond to the cases for k = 64,  16, and 8. The purpose of showing these three lines is to demonstrate the epidemic threshold $R_0^{\prime }=1$ for each case.

For RRG, we compare our MAS result with the theoretical prediction using the following quantification [19]:

$$FES=R\left(\infty \right)=1-{\left(1-\frac{R_0^{\prime }}{k-2+R_0^{\prime }}+\theta ·\frac{R_0^{\prime }}{k-2+R_0^{\prime }}\right)}^k$$ (5-1)

$$\theta ={\left(1-\frac{R_0^{\prime }}{k-2+R_0^{\prime }}+\theta ·\frac{R_0^{\prime }}{k-2+R_0^{\prime }}\right)}^{k-1}$$ (5-2)

Note that Eqs. (5-2) forms a transcendental equation with regard to θ. Fig. 6 shows the result, where R ₀ is introduced as the main X-coordinate, and $R_o^{\prime }$ is put as its supplement, which visually indicates $R_o^{\prime }=1$, demonstrating the so-called epidemic threshold.

Fig. 6.

The theoretical estimation of FES along R₀ on a random regular graph (RRG). Supplementary x-axes beneath the main horizontal axis provide $R_o^{\prime }$ corresponding to the degree k = 64, 16, and 8 (upper to lower). Each horizontal axis is attributed to a different scaling in which we show the epidemic threshold $R_0^{\prime }=1$ (black vertical dotted lines).

Fig. 6 should be compared with respective blue, yellow, and red lines in Fig. 5 (a-i) and (b-i). Note that the discrepancy between the theoretical prediction and assemble-averaged FES from MAS is small. Even though a theoretical approach may predict “averaged” FES, a deviation could hardly be predicted. To cope with that, one must establish much higher order moment equations, which is impossible in practice.

3.3. Heterogeneous graph

Fig. 7 shows the result when the ER-random graph is presumed, which is presented in the same format as in Fig. 2. For heterogeneous graphs, such as BA scale-free in Fig. 8 , $R_o^{\prime }$ is supplemented as another X-coordinate, confirming that $R_o^{\prime }=1$ points out the epidemic threshold as discussed in Fig. 6.

Fig. 7.

The FESs are shown as a function of R₀ on an ER-random graph. The assemble averaged FESs illustrated in panels (*-i) seems pretty close to the mean-field approach (MFA). The main horizontal axes in panels (*-i) correspond to the scaling for MFA. The three horizontal axes (panels (*-i)) beneath the main axis correspond to the cases for the average degree 〈k〉 = 64,  16, and 8. Like the random regular graph, we again perceive a moderate agreement between the results of MAS and ODE based mean-field approach. We further show the corresponding epidemic threshold $R_0^{\prime }=1$ for each horizontal axis.

Fig. 8.

The FESs are shown as a function of R₀ on a BA-SF (scale-free) network. Unlike the other heterogeneous network, e.g., ER random network, MAS results in BA-SF show significant discrepancies compared to the mean-field result (panels (*-i)). The main horizontal axes in panels (*-i) correspond to the scaling for MFA. The three horizontal axes (panels (*-i)) beneath the main axis correspond to the cases for the average degree 〈k〉 = 64,  16, and 8. We also show the epidemic threshold point $R_0^{\prime }=1$ on each line.

Qualitatively, the plot scattering tendency resembles the outcomes of RRG (Fig. 5). However, ER-random shows slightly more scattered than RRG, resulting from its heterogeneous degree distribution (Poison distribution). Like RRG, the plot scattering range and feature do not much differ among different average degrees and different I ₀, which would be paraphrased by the statement that the randomness brought by randomly connected links dilutes the extent of mutual discrepancy resulting from the average degree and the number of initially infected individuals.

Fig. 8 shows the result for BA-scale-free network, which is presented in the same format as in Fig. 2. Plot scattering extent resembles that of the ER-random observed in Fig. 7. However, significant differences compared to ER-random network are found as follows:

First, the epidemic threshold displayed in R_o as the X-coordinate appears less than R_o = 1 regardless of the average degree and the number of initially infected individuals. Note that this observation comes when one is concerned about the assemble-averaged FES. Observing each evolutionary destiny of scattered plots, one can realize that still quite few episodes are absorbed by DFE in the range of R_o < 1, suggesting that it is entailed with the characteristic of bifurcation. Such a stochastic characteristic results from the specific topological feature of the SF graph, i.e., whether relatively higher degree nodes, called hub agents, are infected in an early stage of a time-evolution triggers an outbreak of disease spreading or not (i.e., leading to eradication).

Second, under a highly infective situation, i.e., higher R_o, the assemble-averaged FES deviates from the ODE result. Even with the highest average degree, k = 64, and with a larger number of initially infected individuals (I ₀ = 50), a non-negligible gap exists between the red and green broken lines in Fig. 8 (b-i) at R_o = 5.0. This phenomenon is also attributed to the specific topological feature of the scale-free network. In other words, the heterogeneous degree distribution of scale-free network (entailing more heterogeneity than Poison distribution) avoids full-scale infection, i.e., pandemic situation, even exposed to a quite higher transmission rate.

These two points imply that the scale-free network can be featured with fragileness and robustness to the disease spreading.

4. Conclusive remarks

This study systematically investigates the stochastic outcomes of agent-based simulations for networked SIR process by varying topology, average degree, and initial number of infected individuals. Each result displays scattered FES. As displayed for the case of the RRG graph, a theoretical framework fairly accounts for the effect of spatial structure and can reproduce assemble-averaged FES of agent-based simulations. However, the theoretical framework does not show the extent of stochasticity present in the system, i.e., how the results are stochastically scattered. In this sense, the MAS approach is still meaningful.

We visually elucidated that a topology with a strong locality effect, such as the 1D regular graph (ring) or 2D regular graph (lattice), exhibits a wider range of scattered FES. This indicates a more significant stochasticity of the topology when we randomly place initial infected individuals in the network. On the other hand, outcomes on random regular graph and heterogenous networks, like ER random or scale-free, show less stochastic variations (or less scatteredness) compared to regular graphs such as ring or lattice. As a result, we perceived a better agreement between MAS and mean-field results (Figs. 5, Fig. 7, Fig. 8).

This finding brings the following implication. When the government issues the so-called lock-down to control a new emerging epidemic like COVID-19, the physical contact represented by a static network would be likened by a 2D lattice or a specific small-world network [20] with a smaller short-cut probability, where people are only allowed to take 2D and short-distance migration. It would be unlikely to correspond a movement control provision to randomly connected networks such as usual small-world network [21] or scale-free networks. Under such a circumstance, the stochastic effect resulting from the underlying network significantly affects the real destiny of whether a locally started epidemic spread could grow to a global scale or go to eradication. The authority therefore should pay careful attention to controlling such a situation. Eqs. (1-1), (1-2), 2-4, (5-1) (5-2)

PACS numbers

Theory and modeling; computer simulation, 87.15.Aa; Dynamics of evolution, 87.23.Kg.

Funding

Funding was received for this work.

Intellectual property

We confirm that we have given due consideration to the protection of intellectual property associated with this work and that there are no impediments to publication, including the timing of publication, with respect to intellectual property. In so doing we confirm that we have followed the regulations of our institutions concerning intellectual property.

Research ethics

We further confirm that any aspect of the work covered in this manuscript that has involved human patients has been conducted with the ethical approval of all relevant bodies and that such approvals are acknowledged within the manuscript.

Data Availability

• Data will be made available on request.