An SEIRD epidemiological model with communication

Among all possible infectious diseases, we decided to use the Ebola virus disease (EVD) in this study for several reasons. Even though we are in the middle of a COVID-19 pandemic, and considering the worldwide context it would have made more sense to work in those lines, our understanding about EVD is fare more mature than that of COVID-19. Moreover, a vast literature regarding the dynamics of EVD has been published since 2014 [33–40], arriving to a certain degree of consensus about fundamental characteristics of this disease such as its spreading, lethality and mortality, basic and time-dependant reproductive numbers, R₀ and R_t, respectively, having an R₀ for Ebola virus within the interval 1.51–2.53 [33, 37, 40]. The latter is extremely relevant for us, as our intention is to explore cases where, due to the action of exchanging information, the result is a feasible reduction of the disease’s negative impact in the population. In other words, this can be interpreted as the possibility of modulating R_t by means of a strategy of information delivery, reaching a R_t < 1 at some point along the simulation.

Furthermore, the extreme symptoms of EVD and the sociological impact of its outbreaks can lead to undesired social behaviors from both authorities and the population such as fear, discrimination, overreaction in the implementation of public policies, and the rejection of scientific evidence. Examples of these anomalous behaviors are extensively discussed in [34, 41–44], showing why considering human behavior, and not only epidemiological factors, is essential to understand the dynamics of the spreading of infectious diseases. Of note, previous EVD outbreaks were successfully controlled by implementing public policies that helped to prevent contagion [45–49]. Thankfully, the situation can potentially change now, as there are two EVD vaccines that were approved between 2019 and 2020 for the Zaire strain: rVSV-ZEBOV approved by the FDA [50, 51] and ZEBOV/MVA-BN-Filo which was approved by the European Union [52, 53]. However, considering the case of the current COVID-19 pandemic, and moreover future epidemics, producing models to study how non-pharmaceutical interventions based on communication strategies may impact on the spreading of infectious diseases, is crucial.

When it comes to modelling the dynamics of an epidemic, the compartmental models, first proposed by Kermack and McKendrick in 1927 [54], have shown their success and flexibility. Under this scheme, individuals are assigned to specific compartments depending on their current epidemiological state, and they can transit from one compartment to another along time, depending on the specific disease and dynamics. Common compartments are Susceptible (S), Infected (I), and Removed (R), among others. Using these elements as building blocks we may assemble a variety of epidemiological models such as the SIS (susceptible-infected-susceptible) or the SIR (susceptible-infected-removed) models. The former has been used to describe diseases such as the common cold or influenza [55, 56], whilst the latter to describe, for instance, measles [57, 58]. When modeling the EVD dynamics, the SEIRD model has shown to have good agreement with real data [36, 40]. Here, the E and D compartments stand for Exposed and Dead, respectively. Including these compartments explicitly defines an incubation period—i.e. defined as exposed state E—and, unlike other diseases, that susceptible individuals can still get infected with EVD through the contact with dead individuals, if no proper precautions are taken into consideration. Notably, despite controversy, this seems to be also the case of the COVID-19 pandemic [59–62].

Finally, to implement a model in which we may study the influence of communication strategies on the spread of EVD, we followed the approach proposed by Funk et al. [32], assuming that only trustworthy information is present in the system. Despite that the existence and importance of unreliable information, particularly in the form of fake-news, has been widely discussed [63, 64], we decided to focus only in reliable information as a first approach. Thus, the influence of fake-news during the spread of epidemic diseases, will be explored elsewhere. As a consequence, considering only truthful information implies that individuals having information are less likely to get infected than that of the ones without information. In our model, information can spread by three mechanisms: i) we consider that individuals can acquire information from a central entity corresponding to a global source of information; ii) through individual-individual interaction, which is a local source; and iii) when agents get infected and hence, they become informed. Of note, under this approach, individuals can affect each other not only from the epidemiological point of view –i.e. transmitting the disease–, but also because they disseminate information in the system. In the following section, we extend these ideas into a more technical and mathematical description of our model and simulations.

Details of our ABM

We implemented our model using an ABM scheme having 10,000 agents in Netlogo 6.1.1 [65]. Netlogo is a free software designed to run multi-agent simulations, with a large set of included functionalities, having an extensive user community. It supports agent dynamics embedded in space, which makes it ideal for simulations in epidemiology when the dependency for the spreading dynamics on the spatial proximity between agents, is a desired characteristic [66–68].

In our simulations, an agent is characterized by the parameters z_i, Q_i, and r_i, where z_i denotes the agent’s position in space, Q_i denotes its epidemiological state, and r_i denotes its information state. The suffix i denotes the ith agent. For simplicity, we assumed that each agent moves in a 2-torus (a 2D space with periodic boundary conditions) following a 2-D random walk (with unitary steps following random directions) and without affecting or being affected by other agents, at least in terms of its spatial dynamics. In other words, they diffuse all over the space in such a way that the probability of finding any agent in any position of space for a sufficiently long simulation time converges to a uniform distribution. It is worth noting that, even though this is true theoretically speaking, in the limit of t → ∞, we are far away from that limit given that we explore at most 1000 days of simulation time.

Agents were uniformly distributed in the 2-torus space at the start of the simulation. To define proximity between agents, we have divided the 2-torus into 103 × 103 square patches of length size equal to 1. This leads to a grid-like space in which both epidemiological interactions and information exchange occur only between agents within the same patch. These interactions happen at rates which are modulated by the spatial density of agents. Thus, the number of total patches was selected to define a rate of interactions so to adjust our ABM to the results of the SEIRD model based on ODEs proposed to explain the EVD dynamics by Weitz and Dushoff [36]. For a deeper dive into the SEIRD model, see S1 Eq. The comparison between the results of our ABM model against the ODE model is shown in S1 Fig.

The epidemiological dynamics of the simulation has two parts: an infection dynamics and a transition dynamics. In general, agents can be in any of the five epidemiological states, i.e. Q_i ∈ {S, E, I, R, D}, however, depending on their particular state their dynamics will be of one type or the other. It is worth noting that, S and S represent the susceptible epidemiological state and the susceptible compartment, respectively. The same occurs for the other states and compartments. Importantly, the infection dynamic occurs when a pair of agents susceptible-infected or susceptible-dead is simultaneously found at the same patch. Then, following a probabilistic process, guided by a Montecarlo algorithm, the susceptible agent may suffer a transition from S to E, which we will denote from now and on by S → E, or it may remain in the S state. In practice, when one susceptible agent or multiples susceptible agents encounter either an infected or a death agent in the same patch at the same time, we resolve the Montecarlo step by sampling a random number ν from a uniform distribution . Then, we compare ν with β_I,D (1 − r_i) < 1 for each susceptible agent in the patch to resolve the infection: if ν < β_I,D (1 − r_i), then the susceptible agent gets exposed. β_I and β_D are the infection rates for transitions S → E due to the action of I and D, respectively, when there is absence of information in the system. These parameters were extracted from reference [36] and they are specific for the 2014–2016 EVD outbreak (these are provided in S1 Table). Of note, the term β_I,D (1 − r_i) contains the dependency of the information state r_i. This is actually the modification to the infection rates suggested by Funk el. al [32], and the one we adopt in this article, to couple the communication dynamics to the infection dynamics. Furthermore, since the mean density of agents per patch is approximately 0.94, then the interactions along the simulation are basically pair-wise, this will be also true for the communication dynamics between agents.

On the other hand, the transition dynamic occurs when agents are in any of the epidemiological states but susceptible. In those cases the transitions are E → I, I → R, I → D, and D → ∅. The former and the latter occur in a totally deterministic fashion, after T_E and T_D days respectively. However, I → R and I → D occur both after T_I days but at rate 1 − f and f, respectively. The empty state is represented by ∅, and D → ∅ symbolizes that the agents are being removed from the simulation, which in most of the cases, accounts for a person being buried. All these parameters were also obtained from [36] and are provided in S1 Table.

The information state r_i ∈ [0, 1] accounts for the information that an agent i has about the epidemic at a given time. Whilst r_i = 0 indicates that agent i is completely unaware of the epidemic, when r_i = 1, agent i becomes completely aware of the epidemic and the state of its environment, knowing exactly how to prevent the infection. We consider r_i as a function of time given by , where ρ_i ∈ [0, 1] is the awareness decay constant of agent i. On the other hand, is the information quality constant of agent i, which is also a function of time, being q(t) = 0 the maximum information quality. Importantly, each agent has its own awareness decay constant that is sampled from a certain distribution p(ρ). This sampling procedure introduces heterogeneity into the system in the sense that every agent may follow a different trajectory along the simulation not only due to the stochastic nature of the Montecarlo simulation, but also influenced by their own awareness. Let us say, we want to apply this model to describe the evolution of an epidemic in a certain society, we state that p(ρ) should be somehow assembled by considering sociological, economic, political, and cultural elements, among others, of that specific society. Nevertheless, to the best of our knowledge, there is no clear theory that could help us to infer the proper p(ρ) based on the actual relationship among all those factors. Still, for simplicity, we can assume that p(ρ) exists, expecting that societies with higher trust, either between each other, the institutions, or the communications media, will have a tendency to have a higher degree of awareness and for longer times, than that of societies with lower trust, particularly when they are exposed to valuable information on how to deal with the spreading of an infectious disease.

The information quality constant q_i(t) has a very interesting dynamics. Counter-intuitively, as mentioned before, maximum information quality is reached when q_i(t) = 0, which makes β_I,D (1 − r_i) to vanish resulting in a form of Behavioral Immune System for that agent, at least during that time step. As stated before, new information can be gathered by agents through three different sources: i) through the action of a global source that periodically feeds information into the system, setting q_i(t) = 0 for all the informed agents; ii) from direct contact with agents in the same patch having information of higher quality, setting q_i(t) = q_j(t) + 1; or by iii) acquiring the disease, in which case the exposed agent gets informed when receiving the virus, setting q_i(t) = 0. Furthermore, information quality degrades in time, one unit per time iteration. Thus, for the case of t + 1, i.e. when a time unit goes by in the simulation, and according to the previous paragraph, the information quality constant of an agent i can turn into: (1) Of note, when agents i and j interact, information transfer from an agent with higher quality to an agent with lower quality occurs, and simultaneously, a time unit goes by in the simulation. Hence, we add one unit because of communication and one unit because time goes by, resulting in q_i(t + 1) = q_j(t) + 2. As expected, communication between agents only occurs if both agents are alive. A schematic representation of our ABM is shown in Fig 1, where we show graphically most of the aforementioned concepts.

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Fig 1. Graphical scheme of our agent-based model.

The dynamics of the spreading of the infectious disease is depicted in the left side, where β_I and β_D represent the infection rate of infected agents and dead agents, respectively. Black lines represent deterministic transitions, in days, between states. Blue lines represent infections and, in orange, agents that get buried or removed from the system. The dynamics of information transfer between agents (local communication) is depicted in the right side using blue dashed lines. Information coming from a central entity to the population (global communication), is represented by red dashed lines. Each agent i in the middle zoomed circle is defined by a vector (z_i, Q_i, r_i) where the set of parameters represent its position in the space, its epidemiological state and its information state, respectively. The physical space representation, where each square represent a patch and the shaded agents represent a sample of agents distributed in space, is represented by gray discontinued lines in the background.

https://doi.org/10.1371/journal.pone.0257995.g001

To account for the heterogeneity of space in the simulation, agents were located randomly in the grid following a uniform distribution. The initial information quality constant was defined as q_i(0) = 100 for all agents, except for the ones that were initially infected, whose initial information quality constant was defined as q_i(0) = 0. Despite unbounded, we did set q_i(0) = 100 because it is a number representing low information quality such, when applied to , it gives a result close to zero. In other words, when q_i(0) = 100, for ρ ∈ [0.1, 0.9] the information state r_i ∈ [10⁻¹⁰⁰, 2 × 10⁻⁵], meaning that agents have a negligible awareness of the pandemic situation. To ensure both information and epidemiological dynamics, we choose to start the simulation with 100 agents having Q_i = I and the remaining agents starting in the susceptible compartment, i.e. with Q_i = S.

On the influence of homogeneity and heterogeneity of agents

One of the main critiques to traditional ODE-based compartmental models is the restriction imposed by the well-mixed assumption where mass-action dynamics occur. Its equivalence in ABM is when we assume that all agents have the same characteristics, being uniformly distributed in the space, so they are chosen randomly at the moment of updating their epidemiological states. Several authors have discussed this issue suggesting that including heterogeneity is desirable when modeling the dynamics of epidemic outbreaks [32, 69–71]. Heterogeneous and quenched mean-field theories using representations of the space-based on complex networks [72], can also be used to represent heterogeneity in dynamical systems. However, ABM models represent a straightforward way to deal with heterogeneity, since specific features that are unique to each agent can be individually associated. Furthermore, different types of spatial environments with heterogeneous features can be added into the system, for example transit of pedestrians or public gatherings in specific geometries, such as cities or neighbors. Another approach that has shown to be quite successful in this line is the use of complex networks, where nodes can represent either an individual or a group of them. In this context, it has being shown, for instance, that large fluctuations in connectivity between population networks, may strengthen the incidence of epidemic outbreaks [73].

We have in our simulation certain degree of heterogeneity given by the spatial dynamics: agents which start randomly distributed in the space and become spatially closer as the simulation goes by, are more likely to interact than those being distant apart. More importantly, we explored the effect of having homogeneous and heterogeneous societies, by considering both the information quality q_i that each agent has in the system, and different distributions of the awareness decay constant p(ρ). Of note, as q_i changes along simulation time following the interaction dynamics between agents, heterogeneity of information quality is expressed by the different distributions of q_i that may arise from the simulation. On the other hand, as the awareness decay constant ρ is defined as a parameter of the simulation, to enforce heterogeneity we sampled a distribution of ρ considering that ρ_i ∈ [0, 1]. Hence, two options arise: i) the first one representing a homogeneous society, where all agents have the same awareness decay constant ρ_i = ρ_m, where ρ_m represents the mode of the distribution. Therefore, the decay constant for such a society could be formally described as sampling ρ_i from a delta distribution, given by (2) where δ is the Dirac delta and ρ_m represents its center. The second case corresponds to a ii) heterogeneous society, for which the decay constant can be obtained by sampling ρ_i from a truncated Gaussian distribution, given by (3) In this case, σ is the standard deviation of the associated untruncated Gaussian distribution (which was chosen as σ = 0.2), ρ_m is the mode of the distribution, H is the Heaviside step function, and erf is the Gauss error function. Examples of truncated Gaussian distributions p₁ and p₂ using different values of ρ_m, are shown in panels (a) and (b) from Fig 2, respectively. Importantly, we decided to use the mode of the truncated Gaussian distribution so to produce a distribution of ρ surrounding a central representative value ρ_m, which can be understood as a delta distribution with lateral non-symmetrical diffusion, in both directions. As noted in Fig 2b, when producing a truncated Gaussian distribution of ρ using as center the ρ_m obtained from the Dirac delta (Fig 2a), three heterogeneous distributions of ρ, denoted A, B and C, were produced. Whereas distribution A resembles a skewed right distribution, distribution C resembles a skewed left distribution.

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Fig 2. Accessing homogeneity and heterogeneity effects on the ABM simulation.

Different probability distributions of the awareness decay constant ρ were used to evaluate the effect of heterogeneity by considering a sampling process from a (a) Dirac delta and (b) a truncated Gaussian distribution. In both plots, bars are the sampled distribution and solid lines are the analytic curves, where the arrows in (a) indicate the delta function. (c) Evolution along time of susceptible agents for different values of the mode of decay constant ρ_m vs the ratio of informed agents α. Continuous lines represent simulations where ρ_m is sampled from a Gaussian distribution and dotted lines, the sampling from a Dirac delta distribution. Shaded areas represent the standard deviation obtained from 100 independent simulations. (d) Density plot of the final ratio of susceptible agents at the end of an epidemic when there is a heterogeneous distribution of ρ, for different values of ρ_m and α. Contours lines show the boundary for high and low impact epidemic. (e) Density plot of the difference of final susceptible agents between homogeneous and heterogeneous systems for different ρ_m and α. Contour lines delimit areas of low and high difference between systems. In panel (a) and (b), points A, B and C represent different values of ρ_m, 0.8, 0.6 and 0.1, respectively. In figures (c), (d) and (e) those points represent the pair (ρ_m, α), being A′ = (0.8, 0.7), B′ = (0.6, 0.5) and C′ = (0.1, 0.4). Points are set in interest regions, high (A′), middle (B′) and low (C′) impact of the information on the epidemic dynamics.

https://doi.org/10.1371/journal.pone.0257995.g002

In order to characterize both the similarities and differences between the homogeneous and the heterogeneous cases, and considering the stochastic nature of the ABM, we executed a large number of simulations, evaluating the effect of different parameters of the system. We used a scheme where information is delivered from a central entity (global information) to a certain portion α, randomly selected, of the total population in every step of the simulation. Thus, we have a 2-D parameter space with (ρ_m, α) ∈ [0, 1] × [0, 1] and we systematically explore it by sampling (ρ_m, α) with steps dρ_m = 0.02 and dα = 0.1. For each point in the parameter space, we run 100 simulations lasting for 1000 simulation days each one, which adds up to a total of 56, 100 independent simulations for each case. We observe that after this extended simulation time, the system has virtually reached its equilibrium state, as can be noted by tracking the evolution of the number of susceptible agents along time. Specifically, we compute 〈S(t)〉, which is the ratio between the amount of susceptible agents and 10⁴, which is the total number of agents at the beginning of the simulation, at time t and averaged over 100 simulations. A comparison of 〈S(t) > as a function of time for three homogeneous and heterogeneous cases, marked with dotted and continuous lines, respectively, is presented in Fig 2c. Selected points in the parameter space are labeled as A′ = (0.8, 0.7), B′ = (0.6, 0.5), and C′ = (0.1, 0.4). Whilst in A′, the disease is stopped right away from a very early stage of the propagation, on the other hand in both B′ and C′, the disease spreads over the population. We quantify the difference between the outcome of the homogeneous and the heterogeneous cases by computing, (4) where the superscript (1) and (2) denotes the difference between the homogeneous and heterogeneous cases. For simplicity, from now on we will adopt the notation , or simply S_f, with suffix f and without the brackets 〈⋅〉, to indicates the value of 〈S(t)〉 at the end of the simulation, i.e. at t = 1000 [days]. The use of notation with superscript will be only in sections where multiple cases are discussed at once. In general, S_f is an interesting metric because it allows us to analyze the system at the equilibrium state. Furthermore, Δ is bounded by 0 and 1 and it represents a difference in terms of fraction of agents relative to 10⁴ (initial total population). Its limiting values indicate that both cases are identical, when Δ = 0, or both cases are totally disparate, when Δ = 1. Remarkably, the outcomes of the simulations in cases A′ and C′ are quite similar, we have Δ < 0.02, whilst in case B′ the difference increases and we have Δ ≈ 0.07. Now, we thoroughly explore Δ over the entire parameter space. In Fig 2d we show how looks as a function of the parameters α and ρ_m in the heterogeneous case. From here we can analyze the effect of the parameters over the outcome of the system. The dark blue zone indicates the region where the disease spread is barely affected by the available information, contrary to what happens in the bright yellow region where the disease is under control. Extreme examples of societies for these two opposite scenarios are those with ρ_m = 0 and ρ_m = 1, respectively. Additionally, in stark contrast to what our initial intuition could have told us, homogeneous and heterogeneous simulations are very similar for most of the parameter space, as shown in Fig 2e. Here, we can see that the major difference between the simulations is reached when there is very little information being delivered by the central entity, i.e. α → 0, but only in societies with a tendency towards ρ_m closer to 1, but not exactly 1. As α increases, Δ tends to decrease, however, this does not happen in a monotonic way for all values of ρ_m. Instead, for some values of ρ_m we observe first a subtle increase for small values of α and then a descending behavior.

Overall, these results suggests that the heterogeneous component added by the distribution p₂(ρ) into the simulation is not that relevant, at least in a large domain of the parameter space where its effect is marginal compared with choosing p₁(ρ). It is worth noting that if we could certainly test many distributions to investigate the equivalence of the system, a truncated Gaussian distribution is enough to prove that the system behaves similarly with a slight degree of heterogeneity. With a slight degree of heterogeneity, we refer to a distribution where not heavy tails are present. For this reason, in what follows on this article, we work only in homogeneous systems with p(ρ) = p₁(ρ) as described by Eq (2).

Non-pharmaceutical strategies based on communications

We now examine the impact of different communication strategies on the epidemic spreading. To do so, we rely on different patterns of information delivery from a central entity to a certain fraction α of the population. Hence, we study three different strategies, namely strategy I, II, and III. By following strategy I, we inform on a daily basis to a fraction α of the agents in the simulation, that is chosen randomly among the living agents, following a uniform distribution. In strategy II, we inform to the entire population, but contrary to the previous strategy, this happens every τ days. Finally, in strategy III, we inform on a daily basis to the entire population, only after δ days have passed, since the beginning of the epidemic.

As we previously proceeded, in all three strategies we are interested in the outcome of the epidemic by evaluating the number of susceptible agents at the end of the simulation, i.e. at t = 1000 [days]. However, at t = 600 [days] the system has virtually reached the equilibrium state. In this section, we use a slightly different notation compared to that of the one used in the previous section. The superscript (x) in can take values in {α, τ, δ} and it now indicates which strategy is being analyzed. Again, depending on the context we use either or simply S_f.

Strategy I: Daily information to a fraction α of the population.

The first strategy, discussed in general terms in the previous section, will be analyzed in further details. As seen in Fig 3a, the value of is depicted for 11 simulations executed considering different values of α, ranging from 0 to 1. We also present a control case, indicated by a dashed line at the bottom and a double asterisk symbol (**), for which no information at all is present in the system. In this case, we obtain a value of S_f ≈ 0.16. We may recall that there are three mechanisms for the agents to acquire information: i) from agent to agent; ii) when agents acquire the disease and get informed, and the last one, iii) by agents acquiring information from a central entity. In the case of Strategy I, the first two mechanisms are always on, therefore, even when α = 0, agents may still have access to new information. Of note, the latter case may act as a control situation in which α = 1. We termed this case as the ideal situation considering the complexity involved in implementing such a strategy during a real epidemic—accounting for logistic and other factors. As expected, the ideal case has the best results in terms of reducing the impact of the epidemic for all values of ρ_m. This case is highlighted with a single asterisk symbol (*) and appears on top of all the other more realistic cases. It is worth noting that, both controls, i.e. cases with single asterisk and double asterisk, appear inherently for the three strategies, as shown in panels (b) and (c) of Fig 3.

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Fig 3. Accessing the effect of different communication strategies on the epidemic outcome.

In all panels the final ratio of susceptible agents after 1000 days of simulation for different values of ρ_m. (a) Central information delivered to the population while the ratio of informed agents changes. Different curves represent different ratios of informed agents α, ranging from 0 (olive line) to 1 (blue line) with 0.1 interval. (b) Periodicity of information delivery. Different curves represent different periodicity of information τ, delivered to the population. We have tested periodicity daily from 1 day (blue line) to 7 days (pink line), and then weekly, at 14, 21 and 28 days (green line). (c) Delay in starting information delivery. Different curves represent different delays δ, considering the elapsed time to deliver the message from the beginning of the epidemic. We have tested delay in the first message from 0 (blue line) to 90 days (light blue line) with 10 days interval. After the first message arrives, subsequent information is delivered daily. In all panels, shaded areas represent the standard deviation for 100 simulations. A * above blue line indicates the ideal strategy, and ** above orange dashed line indicates the worst strategy, i.e., when there is no central information delivered to the population neither information delivered to infected agents. Curves with higher values of where x ∈ {α, τ, δ} implied a better strategy result, i.e., less infected agents.

https://doi.org/10.1371/journal.pone.0257995.g003

As expected, all curves S_f in Fig 3 suffer a transition from a state where communication does not affect the epidemic spreading at all, for values of ρ_m closer to 0, to a state where communication conduces to a complete suppression of the epidemic, stopping its spreading, for values of ρ_m closer to 1. The way how these transitions occur for different values of α, however, is quite interesting and highly nontrivial. In all cases, we observe a type of generalized logistic behavior in the transitions. As a consequence of this transition, the effectiveness of the strategy depends strongly on the characteristic ρ_m of the society and, of course, the parameter α. For example, for societies with ρ_m = 0.8 we have that if no actions are taken in terms of informing agents from any central entity, that means α = 0, then the effect is quite similar compare to the control double asterisk (see Fig 3). On the contrary, if we inform as little as α = 0.1 of the total population, the increment in the amount of susceptible agents at the end of the simulation is considerable, having S_f ≈ 0.59. When α = 0.3 we basically stop the epidemic from the beginning and S_f ≈ 0.92 in this case, which is remarkable considering the small amount of agents being informed. Nonetheless, if we perform the same analysis but this time we consider ρ_m = 0.2, the impact of the strategy is very limited, even if we have α = 1, which is the ideal case. The scenario at the end of the epidemic, in this case, is S_f ≈ 0.28. In a more intermediate case, such as considering a society with ρ_m = 0.5, the results are more sensitive to the variation of α over a larger range of parameters. In this case, for example, we can jump from S_f ≈ 0.34, for α = 0.3, to S_f ≈ 0.51, for α = 0.6, or to S_f ≈ 0.69, for α = 0.9. As noted in Fig 3a, there is an evident non-linear relationship between α and ρ_m. Particularly, we can see that our system behaves similarly for different values of α when ρ_m is close to 0.0 or close to 1.0, but for intermediate values of ρ_m, the high non-linearity of the system arise, denoted by a higher difference for the α curves. When ρ_m is close to 0.0, all the population has a fast decay of awareness, and inversely, when ρ_m is close to 1.0 all the population have a slow decay of awareness, so the percentage of informed people α does not contribute substantially to the final size of the epidemic. Now, for an intermediate case of ρ_m, there is not a clear predominance of the agents with fast decay of awareness or agents with slow decay of awareness, resulting in a substantial difference when informed different fraction of population α.

This high variability in the results depending on the election of ρ_m is not uncommon at all in these type of studies. The inference of the right parameters is one of the most difficult parts when trying to apply these type of models in a real situation. For this reason, the use of empirical data to feed models should be mandatory when a more concrete or applied analysis is required. However, everything is not lost and we can still use these models to learn valuable lessons and generate intuition.

Accessing the similarities between strategies.

An interesting observation from Fig 3 is that several curves S_f, as a function of ρ_m, share a similar shape between them for different strategies. This leads to an interest from our part in quantifying this similarity and try to establish a kind of equivalence relation, between specific pairs obtained from different strategies. For example, by simple visual inspection one could be tempted to say that informing every X days to the entire population is more or less equivalent than informing every day to a Y portion of the total population. However, this equivalence immediately raises various questions. For instance, a fundamental one is how can we say that two curves are equivalent, or similar, in this context? To address this interesting question, we used a measure of the distance between two curves and given by (5) where x and y symbolize the value of the parameters for the different strategies that we are comparing. To illustrate this, let us say that we are interested in comparing strategy II with strategy I for the parameters τ = 2 and α = 0.6. In this case, we have ζ(2, 0.6) = 0.17 where this number indicates the area between the two curves. The closer ζ is to zero, the closer the curves are to each other. We have observed that when ζ is less than 0.35 the two curves are virtually indistinguishable from each other and each of them is within the standard deviation of the other for almost all of the values of ρ_m. We are now interested in solving a minimization problem for ζ, in which given a certain value of the parameter x for one strategy, we are interested in finding the value of the parameter y for another strategy, such that it minimizes ζ, i.e. there is no other value of y that gives a lower value for ζ. Some examples of pair of curves that minimize ζ are shown in S5 Fig for different strategies. Furthermore, we explore this in a systematic way and find a large collection of pairs (τ, α), (τ, δ), and (α, δ), which are summarized in Table 1. Interestingly, when comparing strategy II with strategy I, we can find very similar results for the outcome of the epidemic for a large collection of parameters—for all the values of τ that we explored we were able to find a value of α such that it minimizes ζ and also holds the condition ζ(β, α) < 0.35. Therefore, we conclude that, independently on the value of ρ_m for the society and under the assumptions of our simulations, informing to the entire population every 2 [days] conduces to similar results than informing to just a 0.60 of the population every day. On the same page, informing to the entire population every 3 [days] conduces to similar results than informing to just a 0.38 of the population every day. And so on, following the values in Table 1. We would like to highlight now a point that is extremely valuable, and definitively it is something that should be worth to take into consideration when exploring communication strategies in a more practical scenario: α decays rather quickly as τ increases, as shown in Table 1. This remarks the notorious importance of keeping a well-informed population under a quite frequent scheme during an epidemic.

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Table 1. Similarity between different communication strategies.

https://doi.org/10.1371/journal.pone.0257995.t001

When comparing τ with δ, even though we can still formally solve the minimization problem, we find that the curves are quite different, at least for a broad range of the parameter ρ_m. This is shown in Table 1 by the large values obtained for ζ, which are all above 1.3, except for the control cases τ = 1 and δ = 0. Something very similar to this happens when comparing δ with α, however, additionally to the control case, we also have a single nontrivial case with δ = 10 and α = 0.96 that provides ζ(10, 0.96) = 0.33.

On the impact of agents’s communication in the spread of an infectious disease

Now we turn to explore the importance of communication between agents on the impact of the spread of an epidemic disease. To do so, we evaluated the impact of strategies focusing on encouraging agents’s communication to decrease the size of an epidemic. Hence, we consider communication between agents as the act of sharing information about the disease between agents, when they are physically close to each other.

Assessing the impact of agents’s communication without the delivery of central information.

First, we explore the impact on the epidemic when there is only communication between agents, without central information delivery. In other words, agents do not receive information from mass media/central entity, neither infected agents receive information when acquiring the virus. Therefore, the only source of information is the initial information that agents have at the beginning of the simulation. We have tested this strategy to determine if solely the communication between agents is capable to stop the spread of an infectious disease. To do so, we set the system with ratios 1, 0.1, 0.01 and 0.001 of the initial number of informed agents, allowing agents to communicate freely between them. Results in Fig 4a, denote that the impact on the epidemic outcome of this strategy is meaningless, as the four curves appear overlapped. In other words, these results tell us that no matter how many agents are informed at the beginning of the epidemic, when central information is absent, the final size of the epidemic will not be affected.

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Fig 4. Assessing the impact of communication between agents on the outcome of an epidemic.

(a) Only agents to agents communication without central information in the system is allowed, considering different proportions of the initial population informed: 1, 0.1, 0.01 and 0.001. The four curves represent the final ratio of susceptible agents of a 600 [days] simulation when the mode of the awareness decay constant ρ_m, increases. Shaded areas represent the standard deviation for 100 simulations. The inset figure zooms in the interval ρ_m ∈ (0.70, 0.74) to graphically show how close to each other the curves actually are. (b) Evolution in time of susceptible agents for different values of α and ρ_m when central information is available. Continuous lines represent simulations where communication between agents is inactivated and dotted lines where communication is activated. (c) Density plot of the difference between systems where agents communication is inactivated and activated for different values of ρ_m and α. For figures (b) and (c) the points A, B and C represent the pair (ρ_m, α) with values (0.8, 0.7), (0.7, 0.3) and (0.2, 0.2), respectively. (d) Difference between two systems, where in the first one agents communication is inactivated and the second one is activated, for different ratio of informed agents α. Different curves represent different ρ_m. Shadows areas represent the propagation of uncertainty (both systems have a standard deviation resulting from sampling 100 simulations, and because of this, we propagate the uncertainty of the difference between the systems).

https://doi.org/10.1371/journal.pone.0257995.g004

Although is expected that this strategy is not effective, due to the fast decay of information quality in time, given by q_i(t + 1) = q_i(t) + 1, what it is surprising is that it has zero impact even when all population is informed at the beginning of the simulation. This result shows us that the initial information of the system has no impact on the output of an infectious disease when there is no central information being delivered to population. Of note, we may also say that a strategy relying purely on agents’s communication should not be enough to stop the spread of the disease, for most of the values of ρ_m.