Modeling latent infection transmissions through biosocial stochastic dynamics.

The events of the recent SARS-CoV-2 epidemics have shown the importance of social factors, especially given the large number of asymptomatic cases that effectively spread the virus, which can cause a medical emergency to very susceptible individuals. Besides, the SARS-CoV-2 virus survives for several hours on different surfaces, where a new host can contract it with a delay. These passive modes of infection transmission remain an unexplored area for traditional mean-field epidemic models. Here, we design an agent-based model for simulations of infection transmission in an open system driven by the dynamics of social activity; the model takes into account the personal characteristics of individuals, as well as the survival time of the virus and its potential mutations. A growing bipartite graph embodies this biosocial process, consisting of active carriers (host) nodes that produce viral nodes during their infectious period. With its directed edges passing through viral nodes between two successive hosts, this graph contains complete information about the routes leading to each infected individual. We determine temporal fluctuations of the number of exposed and the number of infected individuals, the number of active carriers and active viruses at hourly resolution. The simulated processes underpin the latent infection transmissions, contributing significantly to the spread of the virus within a large time window. More precisely, being brought by social dynamics and exposed to the currently existing infection, an individual passes through the infectious state until eventually spontaneously recovers or otherwise is moves to a controlled hospital environment. Our results reveal complex feedback mechanisms that shape the dependence of the infection curve on the intensity of social dynamics and other sociobiological factors. In particular, the results show how the lockdown effectively reduces the spread of infection and how it increases again after the lockdown is removed. Furthermore, a reduced level of social activity but prolonged exposure of susceptible individuals have adverse effects. On the other hand, virus mutations that can gradually reduce the transmission rate by hopping to each new host along the infection path can significantly reduce the extent of the infection, but can not stop the spreading without additional social strategies. Our stochastic processes, based on graphs at the interface of biology and social dynamics, provide a new mathematical framework for simulations of various epidemic control strategies with high temporal resolution and virus traceability.

Introduction

Stochastic processes of epidemic spreading in human society comprise a specific type of critical phenomena where the microscopic-scale interactions give raise to collective dynamics. Thus, mathematical modelling approaches are necessary to understand the nature of the process and control parameters that govern the transition from the micro- to global scale behaviours [1-3]. Recently, the critical phenomena in social systems have been researched using the concepts developed in physics of complex systems and networks [4]. Some prominent examples are the emotion spreading in online social networks [5-8], opinion dynamics [9], and constructive engagement for the collective knowledge creation [10]. A detailed analysis of empirical data of human activity online and related theoretical modelling [6, 11, 12] provided evidence that the prominent dynamical mechanisms enabling these collective phenomena lie in the self-organised criticality [13-15]. The appropriate agent-based modeling of these social phenomena [16] includes the individual emotional [5, 6] and cognitive properties [10] of the interacting agents.

In addition to social dynamics, the epidemic spreading processes involve some essential biological factors, such as the biology of pathogens, and certain health factors of individuals and groups. Recent COVID-19 data [17] on the outbreak with the new SARS-CoV-2 virus are an excellent example. In the absence of pharmaceutical interventions against the virus, social and sanitary measures remain of primary importance for controlling the epidemics. In this context, there are new traits to deal with, high latency times, rapid transmission, and the potential of the virus to trigger SARS (severe acute respiratory syndrome), which is a medical emergency. For a summary of COVID-19’s unique properties on pathogenic, epidemiological and clinical issues, see [18, 19] and references therein. Statistically, about 20% of the infected individuals need hospitalisation, some of whom (making about 13% of all infected) have mild and moderate symtoms. However, 7% exhibit severe symptoms and need intensive care; they can further differentiate such that 4.7% have clinical stage of illnesses, ending up with a high fatality rate. Data from around the world confirm this overall picture with some regional variations [20-23]. On the other hand, about 80% of those infected show very mild clinical manifestations or remain completely asymptomatic, so that they recover spontaneously after about two weeks; this group of infected individuals often remains undetected and outside formal health management procedures. Therefore, mortality statistics can serve as an indicator of the actual number of infected individuals. The mortality rate ranges from 1.7% to 9% of registered cases, depending on the country or region. In addition to organisational health issues, it is hypothesized that such variations in clinical manifestations may be associated with the existence of different strains and potential genetic mutations of the virus [20, 24]. Theoretically, a virus that jumps on a new host can trigger its evolutionary development in the direction of better adaptation to human cells, which can make the pathogen progressively less aggressive towards the host [25-27]. Such mutations of the virus would be highly desirable to mitigate current epidemics. Currently, various possible mutations in the SARS-CoV-2 virus are being observed, giving rise to an open hot topic of discussion among researchers [28-30].

In the traditional modelling approaches based on mean-field equations, the standard SIR model [1] has been extended to take into account the above-mentioned features of the epidemic manifestation by distinguishing between four SIRU [31] or six SEIHR [32] infection stages and groups involved. Acronymes indicate the initial letter of “Suspected”, “Infected”, “Recovered”, extended by the groups of “Exposed”, “Undetected”, “Hospitalised”, that have own co-evolutionary dynamics. Recently, due to the characteristics of COVID-19 disease, a model has been extended to take into account eight different groups SIDARTHE [33]. These models with a large number of phenomenological parameters that are adapted to the actual data, were able to describe the infection curve (increase in the cumulative number of infected individuals) as well assess the effects of social isolation on the flattening of the curve [34]. A new and promising line of research is opening up through the microscopic agent-based modeling of the epidemic processes [35]. These type of models are increasingly used for describing several specific issues of COVID-19 like epidemics [36-41].

Respiratory droplets and contacts are considered to be the primary routes of transmission of SARS-CoV-2 virus [42]. However, transmission via passive objects (fomits) as well as several other mechanisms (aerosol and fecal-oral transmission) are also reported as highly possible [18, 43–45]. In addition to hospital equipments, various passive objects can get contaminated, e.g., by contacts or respiratory droplets of an infected individual and the infection can be transmitted to a new host. This indirect transmission mechanism is becoming increasingly intriging given the reported long survival time of the SARS CoV-2 virus on different surfaces [44, 45]. Indirect exposure to the virus and the large number of undiagnosed cases in current COVID-19 epidemics underscore the importance of latent infection transmission as a new face of epidemic spreading. So far, this problem has remained outside the radar for standard modeling approaches.

In this paper, we develop an agent-based model that adequately describes these latent transmissions of the infection at microscopic scale and the emergence of global patterns. The model takes into account survival time of the virus and key personal characteristics of individuals, such as susceptibility to the virus and exposure time, which are crucial for the process. We simulate an open system, where the agents are generated over time through social activity fluctuations. As a proxy for social dynamics we use an empirical time series of fluctuations in activity collected from online social networks, precisely a segment of MySpace [46]. Previous studies show [8, 46] that these time series are reflecting compact off-line communities, and they closely represent correlations of fluctuations in community-related activities. Moreover, these time series have essential features of social dynamics, in particular persistent fluctuations with a typical daily periodicity (circadian cycle) [8, 46]. Hence, the underlying social structure is implicitly represented through temporal correlations and a variable intensity of time series. Using long time series of the social activity with an hourly resolution, which determines the time unit of the simulation step, our simulations span several weeks of real-time processes. In the time window, which lasts up to fourteen days after its first appearance in the process, an agent changes its state from “Susceptible” to “Exposed” and possibly “Infected”, followed by either “Hospitalised” or “spontaneously-Recovered”, after which it is removed from the process. During this period, an infected agent generates a number of viruses (contagious spots), which remain infective for other agents within the virus survival time. The most susceptible agents are likely to have severe symptoms; they are hospitalised and thus moved to a controlled environment. Among the remaining agents, most are asymptomatically infected carriers of viruses; thus, they contribute to the latent spread of infection within ongoing social dynamics for a long period before their spontaneous recovery occurs.

The process is presented as a growing directed bipartite graph composed of infected agents and viruses that they spread. This graph-based presentation allows us to identify the pathways of infection that lead to each infected individual and the number of hops the virus has carried from its origin to the current host. Note that chain interactions “host-virus-host” do not necessarily imply a particular social relationship between the implicated hosts. This mathematical framework enables us to simulate possible scenarios with the virus mutation. The development of the network framework based on the stochastic process is essential also due to multifaceted possible side effects and damages that the virus can have in a long run [19]. The simulated high-resolution process revealed the features of dynamic feedback in different scenarios that lead to altered course of the infection propagation. We determine the impact of the level of social activity with/out lockdown and exposure of each individual and the mutation of the virus on the shape of the infection curves.

1 The model

An external input drives the system—a time series of social activity s, which introduces new agents in the process and sets the time t; the resolution is one hour and the total length is t. Thus the total number of agents is . The process is visualised as an evolving bipartite graph with a growing number of nodes and edges. Two types of nodes are the agent nodes representing humans (Hnode), and virus nodes representing contagious spots (Vnode). They are connected via directed edges from Hnode → Vnode → Hnode oriented in the direction of infection transmission. Besides its ID and the creation time, each Hnode possesses several other properties that can influence the process: the agent’s state (“Uninfected”, “Infected”, “Hospitalised” and “spontaneously-Recovered”); the individual susceptibility to infections, h-factor, which is fixed by the creation of the node as a random number h ∈ [0, 1], as well as its exposition time , where T is the maximum exposition time in hours. Besides, we define the virus-host g variable, which keeps track of the number of hops of the virus from its origin (the first infected node in the system) up to the current Hnode. A new Vnode can be produced at every time step by an active carrier (infected agent) proportionally to the severity of its infection, which is measured by its susceptibility; the virus node remains active for a fixed number of hours T. Over time, an agent can experience the transition from the “Uninfected” to “Infected” state, followed by one of the possible scenarios, depending on the agent’s susceptibility level, h. Precisely, the highly susceptible agent (whose h > 0.8) is likely to have a severe illness, and its state will change to “Hospitalised” after a random period between two and seven days. Meanwhile, the less susceptible agents (h ≤ 0.8) represent the asymptomatic cases that will stay mildly infected and unreported until eventually reaching the state “spontaneously-Recovered” after fourteen days. After changing its status to “spontaneously-Recovered”, an agent is removed from the dynamics. Similarly, we remove the agent when its status has changed from “Infected” to “Hospitalised”; even though the hospitalised individuals can transfer infection, e.g., when the so-called nosocomial transmission (via health-care objects) occurs, it takes part in a controlled hospital environment. Meanwhile, the free active virus carriers who take part in the social dynamics primarily contribute to the latent infection transmission. See schematic flow in Fig 1.

Fig 1

Different groups of agents and transition between them.

In the simulation time window between the addition of agents through social dynamics time series and their removal by either spontaneous recovery or hospitalisation, are schematically indicated: “Susceptible” (S)→ “Exposed” (E) →”spontaneously- Recovered” (sR) or → “Hospitalised” (H). For completeness, further differentiation of the “Hospitalised” group to “Intensive-Care” (IC), “Recovered” (R) and “Deceased” (D) groups are shown in the darker square on the lower right corner; these latter subgroups do not contribute to the dynamics studied in this work.

The simulations start with one infected agent, placed on the active carriers list Hactive. At each step, an agent in the Hactive list can produce a new Vnode with a probability proportional to its susceptibility factor h. In this way, we take into account different infectiousness of symptomatic and asymptomatic individuals, which is in agreement with the average viral loads measured in the upper respiratory tract for mild and severe courses of the disease [47, 48]. Thus, more susceptible agents, who are likely to have severe symptoms, can spread viruses more often than those who are barely ill. The fluctuating time series introduces s new uninfected agents at each time step t. By creating an agent, we identify its creation time t as current time t and fix its individual susceptibility factor h and the exposition time . The agent is placed on Hexposed list where it remains for hours exposed and can get infected by currently active viruses with the infection rate λ. Apart from the agent’s susceptibility factor, the infection transmission rate λ depends on several factors and fluctuates in time, as explained below, see Eq (1). Once infected, the agent is removed from the list of exposed agents and placed to the list of active carriers Hactive. We keep the identity of the virus that infected the agent and update the number of hops that the virus passed till that infection event occurs (the virus generation g). The time step ends up with revising the contents of each list. The difference between the current time and the node’s time t is computed for each node on a given list. Then the node is removed according to the criteria described above. The detailed program flow is described in the S1 File.

The transmission rate λ varies in time and depending on the actual contact between the agent and the virus; apart from a constant (empirical value) λ0 it is given by The impact of the current fluctuation of the number V(t) of active viruses, i.e., ϕ = dV(t)/dt/H(t) is normalised by the active number of carriers H(t), which represents the upper limit of the new viruses at that instant of time. In addition, the infection rate is proprotional to the individual susceptibility h of the agent in question. As stated above, our network framework allows us to follow the sequence of the virus transmission along the chain of infection events, see Fig 2, the virus generation g. Thus, we can consider the impact of hypothetical virus mutation along the chain. For this scenario, we note that by passing through a new agent node, the virus g increases by one starting from g = 1 at the first infected individual. We assume that its sufficient weakening can be described as g(g) = 2/(1 + g). Alternatively, we simulate the case without the mutation, i.e., by fixing the factor g = 1. The developed methodology readily allows the analysis of other scenarios as well.

Fig 2

Zoom-in the directed bipartite network.

Hnodes (blue) and created by them Vnodes (pale) within the first 72 hours (λ0 = 0.23, with mutations). Infection transmission occurred through virus nodes, which are indicated by red colour, in the direction of the edge. The amount of Vnodes emitted by one Hnode is proportional to its susceptibility factor and the duration of its stay among active carriers. Among white Vnodes are recent ones that still can infect a new exposed agent; otherwise, each Vnode is non-infective four hours after its appearance.

Results

In the simulations, we fix the parameters λ0 = 0.23, T = 4 hours, the hospitalisation (2-7 days) and spontaneous recovery time (14 days) as well as the threshold susceptibility h = 0.8 according to the reported empirical data from SARS-CoV-2 epidemics, described in Introduction. By differentiating between the mutation (“gen”) and non-mutation (“g1”) scenarios, we change the maximum exposition time T and control the intensity of the social dynamics (average number of new uninfected agents) by choosing the corresponding time series. We compute the time-evolving number of created agents and virus nodes, the edges between the infected agents and viruses, the effective transmission rate, the number of active carriers and the number of active virus spots, as well as the number of exposed and the number of infected individuals per time step. We also determine the cumulative number of infected, hospitalised and spontaneously recovered individuals for t = 1364 hours (56 days). Furthermore, we simulate the lockdown scenario with two types of social dynamics for the total period of 1680 hours, and a lockdown–recovery scenarios for t = 3044 hours. These results are presented in the following three sections.

Infection transmission network and sampled quantities

In Fig 2, we show a part of the bipartite network that embodies the infection transmissions during the first three days. Agents (blue nodes) are enumerated by order of appearance (addition to the network), starting from the first infected agent. Along with the outgoing links from each agent node, we have Vnodes (pale colour) that were emitted by the agent during its active infectious period. Some of the Vnodes, shown in red colour, appear to infect another agent along the red edge. Meanwhile, the majority of other Vnodes are no longer infectious, excluding recently posted ones, which are still infectious and can change the colour to red by connecting with a new uninfected agent. Given that an agent can become infected only once during the period of interest here, the actual network has a tree structure; thus, the uninfected Hnodes remain disconnected from this graph (not shown).

Following the creation of new agents from a given time series s, we sample several time-dependent quantities, as shown in Fig 3 and in the following figures. We determine the time fluctuations in the number of infected and exposed agents, the number of active carriers and active viruses (see a sample data in S1 Text), as well as the cumulative infectious curve and the number of spontaneously recovered and hospitalised agents. Notably, in Fig 3, we describe the differences between the cases with and without virus mutations, meanwhile, the input time series and the other parameters (λ0 = 0.23, T = 24 and T = 4 hours) are kept the same. In Fig 4, we show how the effective transmission rates vary in these two cases. Even though the number of exposed individuals is practically the same, the number of infected ones per hour is smaller in the case with virus mutations than without mutations, leading to the two different cumulative infection curves in the panel (B). A similar difference then occurs in the fractions of hospitalised and spontaneously recovered cases, as shown in the panel (C). Computing the temporal variations in the number of active carriers (infected agents) and the number of active viruses nodes completes the picture. As shown in panel (D), these quantities are significantly higher in the case without mutations. It should be stressed that, given Eq (1), the fluctuations in these quantities have dynamical feedback to the effective transmission rate. We note that the obtained infection curves can be fitted by logistic function with different parameters, see more in the following section.

Fig 3

Comparative simulations for virus mutations (“gen”) and without mutations (“g1”).

Results are for the same social dynamics–time series s depicted in panel (A). Time fluctuations of the number of exposed agents and the number of infected agents per hour, panel (A), and the corresponding infectious curves–the cumulative number of infected agents, panel (B). Fits according to the logistic function (parameters shown in the legend). The cumulative number of spontaneously recovered and hospitalised agents are shown in panel (C) and the respective fluctuation of the number of active carriers and active viruses, in panel (D). In panels (C) and (D), the infection curves from panel (A) are shown by dotted lines, for comparison.

Fig 4

Effective transmissibility.

Top: Transmission rate λ vs time t for the case with the virus mutations (full, red curve) and without mutations (dashed, black curve) for λ0 = 0.23 corresponding to the course of the process in Fig 3. Bottom: Related to these transmission rates, the number of infected followers k of a given Hnode i indicated along the horisontal axis. For clarity, only the first 2000 nodes are shown.

As the infection network in Fig 2 shows, the infection path can be followed forwards along with the directed links via red virus nodes. It appears that the number of agents that get infected from a given previously infected one varies from agent to agent, as it is shown in the bottom panel of Fig 4. Averaging over the infected agents in a given period, we get the values < k > that are larger than one, see the legend. Note that this is a quantity similar to what is in the epidemiology literature known as “R-factor” [49, 50]. Our network presentation of the process clearly shows that the R-factor is given by the relative ratio between two successive layers of the Hnods. Recently, studies have shown that the reproduction rate is a local measure with a limited predictive value, see the discussion and the empirical data analysed in [49-51].

Influence of social dynamics and exposure times on the course of the infection curve

To assess the impact of the intensity of social dynamics to the infection curve, here, we simulate the scenarios with the social lockdown, which is modelled by another time series; a representative example is shown in the top panel in Fig 5. Specifically, starting with a moderately high social activity (we use the same time series as in Fig 3), the process lasts for six weeks, then the input time series is changed. Here we chose another empirical time series (corresponding to the negative-emotion activity in MySpace data set [46]), which exhibits about four times smaller intensity but also almost absent correlations (the Hurst exponent is close to 0.5). Some simulation results are shown in the main panel of Fig 5 for the case with the mutations. These results reveal how the reduced social activity leads to an effective flattening of the infection curve, in qualitative agreement with the overall empirical data. (For easier comparison, the parameters are such that the initial part of the bottom curve corresponds to the lower curve in panel (B) of Fig 3). However, the plateau level results from the course of the entire curve from the beginning of the infection. In this way, the impact of social lockdown depends on other factors that are built into the infection process before the intervention. In this figure, we further demonstrate how the increased exposure time of the agents leads to the increased level of the plateau, with other parameters fixed. In the following Fig 6, we show comparative results with/out mutations and two different lockdown scenarios, while keeping the exposure fixed.

Fig 5

Moderate social dynamics with a lockdown.

The fluctuations of social activity s (top panel) and corresponding infection curves I vs time t for different range of exposure times, indicated in the legend (bottom), for the case with virus mutations.

Fig 6

Impact of the social dynamics preceding a lockdown.

Top: Infection curves for high-intensity dynamics and lockdown after week 4, and low-intensity dynamics with the lockdown after week 6; two lines in each case are the scenarios with/without virus mutations. Fits according to the logistic function (see text). Bottom: Temporal fluctuations of the number of exposed agents and the number of carriers and the number of active virus spots corresponding to the infection curves in the top panel.

To predict the extent of the infection [33, 49, 52], the course of the infection curve is standardly fitted by the sigmoid (logistic) function The parameter K is the so-called “carrying capacity”, and r is the rate by which the curve reaches it, while the parameters m and n relate to the beginning and position of the inflexion point. The rationale behind the occurrence of this functional form (and the related derivative, the bell-like function of the infection rate curve) lies in the very nature of the infectious spreading in a given population size, a fraction of susceptible individuals that will be infected. In the beginning, the infection spreads to ever-larger number having practically unlimited resources. When the accelerated growth reaches the maximum infection rate (the inflexion point of the infectious curve), the process starts experiencing the limited space: the number of potential susceptible individuals that are not yet infected is reducing. Consequently, the infection rate starts decreasing while the cumulative number of infected cases asymptotically approaches the final capacity K.

Moreover, all simulations demonstrate that lowering the intensity of social dynamics, even if for a small factor such as natural day-night fluctuations, will affect the process, but with a delay. A particularly considerable lockdown comes in effect only with a ten days delay, cf. Fig 5. A closer investigation reveals, see Fig 6 that, after the lockdown, the number of exposed agents starts decreasing, reaching the corresponding lower level after the period comparable to T. Then the number of active carriers takes about a week to ten days longer to reduce and adjust to the lockdown dynamics. The number of active viruses follows this curve with a small delay (4 hours).

Predicting the course of events after lockdown is lifted

In the literature, the impact of the imposed social lockdown during COVID-19 pandemics has been investigated from several different angles. For example, apart from economic issues, diverse social and psychological factors have been reported [53], which manifest in altering the social dynamics after a particular lockdown is lifted. In this context, our model allows us to simulate different scenarios. Notably, Fig 7 presents the results where we simulate the impact of the social activity after the 4-weeks lockdown, meanwhile keeping all other parameters at the same level. In one case, a moderate social activity takes part, which also does not have prominent dynamical correlations (here, we amplified the time series that characterises the lockdown). In the other scenario (“back-to-normal”), we use the same time series as in the period before the lockdown and continue the process for another eight weeks. Note that this time series contains a reasonable level of dynamic correlations, as explained above. For comparison, we also simulate the situation without lockdown keeping the original level of the social activity for the period corresponding to 16 weeks of real-time. The results showing the course of the infection curve are given in the central panel of Fig 7. The driving social dynamics time series and the corresponding number of exposed agents for the scenario without lockdown are shown in the top panel and for two scenarios with the lockdown, in the bottom panel of Fig 7. These results reveal that the infection curve increases after the lockdown in all cases, but the increase rate is low for the case of moderate social activity. In the “back-to-normal” case, the increase is much faster even if compared to the corresponding segment of the curve “1”, but slower than at the beginning of infection process before the lockdown. Fitting the segment III of the curve gives the overall capacity which is still lower than the projection from the fit of the curve “1”. It is also essential to notice that the curve “1” has a different course from the possible extension driven from fitting its first part (see black dotted line). We can conclude that even small fluctuations that lower the social dynamics s (see top panel) can, through the above-described feedback mechanisms, affect the course of the infection curve, thus reducing the overall projection of the infection.

Fig 7

Simulations of exiting scenarios.

Central panel: Infection curves I vs time for the extended period with the naturally fluctuating social dynamics, curve (1), and for two scenarios with the lockdown intervention followed by moderate-intensity dynamics (2) and “back-to-normal” scenario (3). Periods before, during, and after lockdown are indicated as I, II, III. The fluctuating intensity of the susceptible s and exposed e agents corresponding to the case (1) are shown in the top panel, and to the curves (2) and (3) in the bottom panel. The last part of the signal s in the case “back-to-normal” corresponds to the second half of the signal in the top panel.

Discussion and conclusions

We have designed an agent-based model with the high-resolution dynamics of infection process that explicitly observes the survival time of viruses and personal properties of individuals who produce them. We simulate an open system which is driven by social dynamics of the involved agents, and represented by a growing bipartite graph with the agents and viruses nodes. Considering the actual SARS-CoV-2 epidemics, the model adequately takes into account latent infection transmissions by asymptomatic virus carriers as well as indirect transmissions that can occur, for example, through contaminated surfaces during the virus survival period. These processes comprise a significant part of infection transmissions by each agent occurring in the time window which can last up to two weeks since the agent enters the system through social dynamics until it spontaneously recovers or otherwise moves to a controlled hospital environment.

The developed mathematical formalism with the infection network makes it possible to trace a path along which the virus hops over different hosts before infecting a particular individual. It thus allows us to model potential mutations of the virus along its evolution path. The results revealed the key components of this biosocial stochastic process that significantly influence the course of infection spread and the predictions of outcomes. Specifically, we have found that:

The intensity of social dynamics in conjunction with the individual susceptibility of each agent is crucial for the latent infection transmission. Hence, a lockdown measure becomes effective, but with a typical delay. The simulated process with a high temporal resolution uncovers the underlying mechanisms at work. Lowering the social activity level gradually reduces the number of exposed individuals until it reaches the level of new social activity; with a delay, it causes a decrease in the number of active viruses carries and the number of active viruses that they produce. The opposite trends occur by lifting the lockdown measures. Depending on the renewed social activity, the outcome can be lower than in the case of the process without lockdown. In particular, in the “back-to-normal” situation, the final projection of the number of infected is still smaller or comparable with the original one. However, it can be reached after a much more extended period. It is interesting to point out that much smaller, natural fluctuations in social dynamics (such as day-night or workdays vs weekends) that appear periodically, as in our time series, can have profound effects on lowering the slope of the infection curves.

The exposure time of each individual, is another factor that can considerably increase the course of the infection curve even with a low or moderate social activity level. Thus, modifying the exposure time of individuals or groups is an additional essential characteristic to be considered in conjunction with social dynamics measures.

Virus mutations scenarios towards gradually reducing the transmission rate can slow down the growth of the infection curve. Even though these virus mutations are favourable, they have no potential to stop the spread of infection without additional social strategies.

A new mathematical framework, developed in this work, provides a robust tool for the analysis of biosocial processes in SARS-CoV-2 like epidemics. Its essential new ingredients are high-resolution dynamics, open system, and the network-assisted process presentation. Meanwhile, the traditional mean-field equations and most agent-based models in the literature have limiting factors due to the fixed size of the system. On the other hand, given the infection-network representation in our model, we do not consider direct physical distance and mobility patterns of the agents. This fact is a limiting factor for some applications of the model, for example, to describe specific geographical locations and spaces with different social groups involved (see a different study in [35, 41, 54]). Instead, the social dynamics that drive the system in our model can be varied. It represents cumulative activity participation, originating from different communities and social events. Considering the biology aspects, we have neglected possible temporal variations of the viral load of a host (besides its constant susceptibility factor), which might affect the impact that a particular host has in the process. With sufficient empirical data (current studies [47, 48, 55] are for symptomatic individuals), a modification of the model can take into account the appropriate profile of the viral load for each infected agent. Moreover, possible extensions of the model are to include a specific set of connections of each agent, a kind of ego-network, and modified transmission rates inside it. Consequently, a more heterogeneous pattern of the exposure times can emerge. However, such non-random distribution of the exposure time per agent could not significantly influence its impact to the global features of the epidemics found in this work. To reveal the nature of the underlying stochastic process, here we have simulated the infection spreading from a single source. In the meantime, the spreading from different sources or different times, which may increase the slope of the infection curve, can also be considered. Lastly, as we already mentioned, the issue of SARS-CoV-2 virus mutations is an open problem that has received increasing attention of researchers in different fields [28-30]. The infection network presentation allows us to model different ways of how particular virus strains (virus-host interaction) change over time. These aspects of stochastic biosocial processes, among others highlighted here, could represent a compelling direction for further research.

Program flow.

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Sample output data.

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22 Sep 2020

PONE-D-20-23763

Modeling latent infection transmissions through biosocial stochastic dynamics

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Reviewer #1: Yes

Reviewer #2: Partly

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2. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: Yes

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3. Have the authors made all data underlying the findings in their manuscript fully available?

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Reviewer #1: Yes

Reviewer #2: Yes

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4. Is the manuscript presented in an intelligible fashion and written in standard English?

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Reviewer #1: Yes

Reviewer #2: Yes

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5. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: This work develop an agent-based model for simulations of infection transmission in an open system driven by

the dynamics of social activity; the model takes into account the personal characteristics of individuals, as well as the survival time of the virus and its potential mutations. The paper is well organized. It can be seen that the proposed model reveal complex feedback mechanisms that shape the dependence of the infection curve on the intensity

of social dynamics and other sociobiological factors.

Reviewer #2: Thank you for the opportunity to read this interesting manuscript. The manuscript is very easy to read. Authors clearly describe their aims, provide clear operationalization of all variables, and the reasoning of the procedure. In general, I consider this type of simulation to have a contribution over the mean-field theory whose results are biased by the well-mixed assumption and missing the temporal dynamics in real-world epidemic spreading.

The authors propose an agent-based model to address the latent infection transmissions by introducing temporal dynamics in simulation. The most impressive and practical thing is the authors treat the human and virus as separate elements in their model. The introduction of exposure time for the human and survival time for the virus can also reflect the latent characteristic of infection transmissions. Besides, the application of parameter gv helps capture the weakening process of the virus due to evolution.

The authors used the data collected from online social networks, MySpace, as the input data regarding human dynamical fluctuations over time. However, the disease spreading in the real-world depends on contacts (either face-to-face contacts or indirect face-to-face contacts), which may significantly distinct from online social interactions between people. Rather than data from online social media, I miss a modification of this model using data concerning direct or indirect face-to-face contacts. Or, perhaps, the authors could provide a more solid justification for their using of the online social interactions in the context of infection transmission.

As a more specific remark, I would have expected clarification (or supporting materials) of this assumption: “thus, more susceptible agents, who are likely to have severe symptoms, can spread viruses more often than those who are barely ill.” I miss a brief description of the schematic of how virus nodes infect human nodes, or which virus node infects which human node. I also miss I miss a more robust discussion on the limitations of the model.

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2 Oct 2020

We have found the editor's suggestions and reviewers' comments very encouraging and instructive, and we have revised the manuscript accordingly.

Specifically, we have added a description of the social activity time series, which is selected such that it closely reflects the cumulative activity in off-line social communities. We also stated that it serves as a first proxy, having no real data sets from a specific geographical location.

We have also expanded the Discussion section to highlight the absence of physical distance and mobility of the agents that limits some applications of the model, for example, to describe epidemics at specific geographical locations and social groups involved. We also comment the potential impact that changed (more realistic) distributions of the number of contacts as well as temporal variations of the viral load may have on the process that we are simulating. Several concrete extensions of the model mentioned in this context will remain for future work.

We believe that the presented model captures the essential features of the stochastic processes underlying SARS CoV-02 epidemics, in which the latent infection transmissions have a substantial contribution.

Prompted by your suggestion, we have removed funding information from Acknowledgments. Therefore, we would appreciate it if you could update our Funding Statement by adding the following contents:

B.T. work supported by the Slovenian Research Agency (research code funding number P1-0044).

R.M. research was funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada and Canada Research Chairs (CRC) Program.

All Figures are uploaded as separate .tif files (with required dimension and resolution).

We have also uploaded as Supporting Information S2 Text.dat a sample of the simulation output data for a specified set of parameters (mentioned in the main text). We have also moved the program flow from the Appendix to Supporting information as S1 File.pdf.

A detailed response to the reviewers is attached. All changes are traced in the PDF file.

*

5. Review Comments to the Author

Reviewer #1: This work develop an agent-based model for simulations of infection transmission in an open system driven by the dynamics of social activity; the model takes into account the personal characteristics of individuals, as well as the survival time of the virus and its potential mutations. The paper is well organized. It can be seen that the proposed model reveal complex feedback mechanisms that shape the dependence of the infection curve on the intensity

of social dynamics and other sociobiological factors.

Authors response:

We appreciate the Reviewer’s time and a deep understanding of our work.

Reviewer #2: Thank you for the opportunity to read this interesting manuscript. The manuscript is very easy to read. Authors clearly describe their aims, provide clear operationalization of all variables, and the reasoning of the procedure. In general, I consider this type of simulation to have a contribution over the mean-field theory whose results are biased by the well-mixed assumption and missing the temporal dynamics in real-world epidemic spreading.

Authors response:

We thank the Reviewer for his/her time and for very encouraging statements about our work.

The authors propose an agent-based model to address the latent infection transmissions by introducing temporal dynamics in simulation. The most impressive and practical thing is the authors treat the human and virus as separate elements in their model. The introduction of exposure time for the human and survival time for the virus can also reflect the latent characteristic of infection transmissions. Besides, the application of parameter gv helps capture the weakening process of the virus due to evolution.

The authors used the data collected from online social networks, MySpace, as the input data regarding human dynamical fluctuations over time. However, the disease spreading in the real-world depends on contacts (either face-to-face contacts or indirect face-to-face contacts), which may significantly distinct from online social interactions between people. Rather than data from online social media, I miss a modification of this model using data concerning direct or indirect face-to-face contacts. Or, perhaps, the authors could provide a more solid justification for their using of the online social interactions in the context of infection transmission.

Authors response:

We thank the Reviewer for pointing out these issues, which, indeed, need to be better clarified, as we do in the revised text. Specifically, in the absence of real-life data sets, we use a first proxy of the social dynamics available. Among the online social networks, the MySpace data sets are such that they most closely reflect the off-line community contacts and their mutually triggered activity, as our previous studies in Refs.[8,46] show. Moreover, these data sets contain temporal correlations (persistent fluctuations that precisely match the off-line activity fluctuations) and circadian cycles, both of which define essential features of human activity patterns. We added a brief explanation in favour of this empirical time series, as compared to the alternatives (a computer-generated time series or a random number).

We have expanded the Discussion section to once again stress the absence of geographical location and mobility patterns of the agents, also referring to different approaches in Ref.[35,41,54], where a concrete data of such kind are collected and used within the model, aiming to predict the impact of mobility that provides contacts and transmission among different groups of the population.

Furthermore, we pointed out the limitations of our model as well as possibilities to expand the properties of the agents. For example, the model can take into account the heterogeneity in the number of contacts along which the virus could be transmitted, possibly with a changed rate, as well as temporal variations of the viral load, as some recent empirical data indicate. These issues remain to be studied in future work.

As a more specific remark, I would have expected clarification (or supporting materials) of this assumption: ‚thus, more susceptible agents, who are likely to have severe symptoms, can spread viruses more often than those who are barely ill.‚Äù I miss a brief description of the schematic of how virus nodes infect human nodes, or which virus node infects which human node. I also miss I miss a more robust discussion on the limitations of the model.

Authors response:

We thank the Reviewer for pointing out this issue. Different infectiousness of the agents, especially between asymptomatic and symptomatic individuals, needs to be clarified. Recent studies, for example, in Refs. [47,48,55], found that the viral load varies during the infectious time and that the average viral load in the upper respiratory tract of mild and severe symptomatic individuals is proportionally different. Also, a summary in Ref.[18] points out that “80,9% were considered asymptomatic or mild pneumonia but released large amount of viruses at the early phase of infection”. We do not have enough data to adequately introduce another variable for the temporal variations of the viral load of each agent; therefore, we use the susceptibility h^i that effectively differentiates asymptomatic from symptomatic ones. In the Model description, we added a comment regarding different viral loads that motivates the corresponding model rule. Furthermore, we mentioned the occurrence of the temporal patterns of viral loads in the Discussion by pointing out the limitations and potential extension of our model.

In the graph, we have precisely the link along which one host infects the other one via its virus. Perhaps, the Reviewer is pointing to potential (social) relationships between the hosts, which we do not take into account in this version of the model, as above described. Thus, we randomly select among currently existing active viruses; each of them is posted by a known active carrier. In the expanded Discussion, we pointed out several limitations of the current version of the model, as stated above.

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Submitted filename: response_n.pdf

Click here for additional data file.

12 Oct 2020

Modeling latent infection transmissions through biosocial stochastic dynamics

PONE-D-20-23763R1

Dear Dr. Tadić,

We’re pleased to inform you that your manuscript has been judged scientifically suitable for publication and will be formally accepted for publication once it meets all outstanding technical requirements.

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Ning Cai, Ph.D.

Academic Editor

PLOS ONE

Additional Editor Comments (optional):

Reviewers' comments:

Reviewer's Responses to Questions

Comments to the Author

1. If the authors have adequately addressed your comments raised in a previous round of review and you feel that this manuscript is now acceptable for publication, you may indicate that here to bypass the “Comments to the Author” section, enter your conflict of interest statement in the “Confidential to Editor” section, and submit your "Accept" recommendation.

Reviewer #1: All comments have been addressed

Reviewer #2: All comments have been addressed

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2. Is the manuscript technically sound, and do the data support the conclusions?

The manuscript must describe a technically sound piece of scientific research with data that supports the conclusions. Experiments must have been conducted rigorously, with appropriate controls, replication, and sample sizes. The conclusions must be drawn appropriately based on the data presented.

Reviewer #1: Yes

Reviewer #2: Yes

**********

3. Has the statistical analysis been performed appropriately and rigorously?

Reviewer #1: Yes

Reviewer #2: Yes

**********

4. Have the authors made all data underlying the findings in their manuscript fully available?

The PLOS Data policy requires authors to make all data underlying the findings described in their manuscript fully available without restriction, with rare exception (please refer to the Data Availability Statement in the manuscript PDF file). The data should be provided as part of the manuscript or its supporting information, or deposited to a public repository. For example, in addition to summary statistics, the data points behind means, medians and variance measures should be available. If there are restrictions on publicly sharing data—e.g. participant privacy or use of data from a third party—those must be specified.

Reviewer #1: Yes

Reviewer #2: Yes

**********

5. Is the manuscript presented in an intelligible fashion and written in standard English?

PLOS ONE does not copyedit accepted manuscripts, so the language in submitted articles must be clear, correct, and unambiguous. Any typographical or grammatical errors should be corrected at revision, so please note any specific errors here.

Reviewer #1: Yes

Reviewer #2: Yes

**********

6. Review Comments to the Author

Please use the space provided to explain your answers to the questions above. You may also include additional comments for the author, including concerns about dual publication, research ethics, or publication ethics. (Please upload your review as an attachment if it exceeds 20,000 characters)

Reviewer #1: No comments for the author, including concerns about dual publication, research ethics, or publication ethics.

Reviewer #2: I am satisfied with the way the authors have incorporated my comments. Especially, they did great work in explaining the problem of using online social media data instead of real face-to-face contact data. They also described the limitations of their study.

15 Oct 2020

PONE-D-20-23763R1

Modeling latent infection transmissions through biosocial stochastic dynamics

Dear Dr. Tadić:

I'm pleased to inform you that your manuscript has been deemed suitable for publication in PLOS ONE. Congratulations! Your manuscript is now with our production department.

If your institution or institutions have a press office, please let them know about your upcoming paper now to help maximize its impact. If they'll be preparing press materials, please inform our press team within the next 48 hours. Your manuscript will remain under strict press embargo until 2 pm Eastern Time on the date of publication. For more information please contact onepress@plos.org.

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