Introduction: The 2014 Ebola outbreak in West Africa raised many questions about the control of infectious disease in an increasingly connected global society. Limited availability of contact information made contact tracing diffcult or impractical in combating the outbreak.

Methods: We consider the development of multi-scale public health strategies that act on individual and community levels. We simulate policies for community-level response aimed at early screening all members of a community, as well as travel restrictions to prevent inter-community transmission.

Results: Our analysis shows the policies to be effective even at a relatively low level of compliance and for a variety of local and long range contact transmission networks. In our simulations, 40% of individuals conforming to these policies is enough to stop the outbreak. Simulations with a 50% compliance rate are consistent with the case counts in Liberia during the period of rapid decline after mid September, 2014. We also find the travel restriction to be effective at reducing the risks associated with compliance substantially below the 40% level, shortening the outbreak and enabling efforts to be focused on affected areas.

Discussion: Our results suggest that the multi-scale approach can be used to further evolve public health strategy for defeating emerging epidemics.

The initial medical response to Ebola in 2014 focused on caring for individuals in hospital settings and using contact tracing as the primary preventative measure^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}

In general, early detection of Ebola-like symptoms is necessary for early care of patients with Ebola and limiting new infections. This is due to the extended infectious period and tendency of the disease to become more contagious as it progresses^{,}^{,}

If all new infections could be perfectly isolated through the complete monitoring of the population, these policies would evidently drastically limit new Ebola cases and result in an abrupt halt to the epidemic. However, an analysis must account for how effective the implementation of screening will be. It is infeasible and undesirable to constrain the population using high levels of force, so the level of compliance that is achieved is a key variable in efficacy. Here we model compliance as a probability that individuals will adhere to the community-level policies. This captures both the possibility of defiance as well as other sources of performance failure such as accidents or lack of awareness or information. We analyze the level of compliance necessary for the policies to work effectively. As shown in Fig. 1, we find that even with 40% compliance the community level policies curtail the epidemic and a 60% compliance rapidly ends the outbreak. We used a combination of simulations and mean field analysis, which are complementary. The importance of simulations is twofold: first it demonstrates that our results can be generalized to a wide range of geographic contact network properties, and second it allows for simulations of travel restrictions and community interventions. The mean field analysis provides intuition about the reason that community monitoring is robust compared to contact tracing and for the threshold of compliance that is needed for successful intervention. Both mean field analysis and simulations of different population contact network structures imply that these results are robust to the simulation assumptions, as well as variations in real world network transmission properties.

Screening begins at the vertical dotted line, with a level of compliance indicated by label and color (green 0 to blue 1.0). A. Number of cases with or without symptoms. Note that even 40% compliance (0.4) results in decrease in cases. B. Cumulative cases. C. _{t}_{t}_{t}_{t}

Simulations of Ebola and other infectious diseases have been performed on complex networks^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}^{,}

Our model is a Susceptible, Exposed, Infectious, Removed (SEIR) model on a spatial lattice of individuals (Fig. 2) with periodic boundary conditions. Individuals can be in one of four states: disease-free and never previously infected (susceptible), infected in the latent period without symptoms (exposed), infected with symptoms (infectious), and recovered or dead (removed). Newly infected individuals progress through a latent period for Δ days where they are asymptomatic and not contagious. They then become contagious for a period of Γ days, at the end of which they have either died or have recovered and have acquired immunity from further reinfection.

We simulate several transmission networks. In our baseline model, each individual interacts with all four of its nearest neighbors on the lattice once per day, and an infectious individual infects a susceptible neighbor with probability τ during a given interaction. Each individual also interacts with another randomly chosen individual from the population. If one of them is infectious, they have a probability η of infecting the other by this long-range interaction. A schematic of these interactions is shown in Fig. 2. This mix of local and long-range disease transmission was chosen for our model to reflect the tendency for Ebola to spread both within households and through non-local interactions in shared taxis, hospitals, or through other travel

Black squares indicate individuals of a spatially structured population, blue lines denote partitions between communities. A. Neighbor infection within a neighborhood. B. Cross-partition neighbor infection to another community. C. Long-range transmission within a community. D. Long-range transmission across a partition.

We simulated several alternative transmission networks to test the robustness of our analysis. This also tests the robustness of the intervention policies to changes in the societal transmission network. We included networks in which each person had different immediate neighbors (Moore neighborhood), and ones for which the contact structure itself changes to a small world network model (Kleinberg network). For the Moore neighborhood network, each individual is connected to its closest eight rather than four nearest neighbors. The Kleinberg network was implemented by allowing individuals to have a varying number of long-range contacts with their probability of having a contact decreasing inversely with the distance squared. In addition, we tested the effect of changes in transmission by allowing the lengths of periods of the disease to vary for individuals across values found for Ebola

The community-level public health interventions are modeled based on a proposed policy draft_{0} days, after which the intervention policies are put into place. The time until the start of the intervention is important in measuring the impact of early response on the control of an emerging outbreak. We assume that, even after the start of the intervention, individuals cannot be isolated and are fully capable of infecting others on the first day of their infectious period. This captures the idea that an individual can become infectious and infect someone else between symptom checks on consecutive days. With probability κ, infectious individuals are chosen to be compliant, which means that, after the first day of their infectious period, they will be perfectly isolated and incapable of interacting with others for the final Γ - 1 days of their infectious period. The lack of compliance, occurring with probability 1 - κ, is assumed to be complete in the sense that noncompliant individuals continue to infect others for the duration of their infectious period. The compliance analysis in the model is a robustness analysis of the community based response strategy.

The travel restrictions are implemented based on cordons outlined in the policy draft

We used data from the World Health Organization

It is useful to consider the value of the basic reproduction number, _{0}_{0}_{0}_{0}_{0}

where r _{0}_{0}

First, we considered a latency period of Δ = 5 days and an infectious period of Γ = 6 days to reflect the modeling assumptions of Althaus_{0 }^{,}

Second, we used the parameter values Δ = 10 and Γ = 7 to compare our model with that of Chowell and Nishiura, who estimated that _{0}_{0}^{,}

For each set of parameters, we simulated compliances ranging from 0.0 to 1.0 in steps of 0.05, and intervention delay times of 50, 70, 90, and 110. Results are averaged over 1,000 simulations of a population of 10,000 individuals, 100×100 square lattice, with neighborhoods of size 100, 10×10 sublattices, initialized with 0.02% of the population infectious and 0.02% latent.

Fig. 1 shows the number of current and cumulative cases for various levels of compliance with community level interventions implemented at _{0 }

The impact of intervention policies can be readily seen by plotting the effective reproduction number _{t}_{t}_{0}_{t}_{t}_{t}

To observe the effects of the travel restrictions, we show the outbreak length and the cumulative number of infections for interventions implemented at time _{0 }= 70

A and B: Blue shows the case with travel restrictions, and red shows the case without such restrictions. Differentiation between the two occurs because the travel restrictions compensate for low levels of compliance. This decreases the length of the epidemic A and reduces the cumulative number of infections B in cases of low compliance. C. The cumulative number of infections over the entire epidemic, as a function of compliance levels and intervention times. Colors from brown to yellow signify intervention times (50, 70, 90, 110). Without enforced travel restrictions (dotted lines), a low compliance results in little differentiation between early and late policy implementation. The travel restrictions (solid lines) dramatically reduce infection number for earlier interventions at low compliance.

Comparing the cumulative cases as a function of compliance for different delay times (Fig. 3C), interventions with an earlier start time _{0}

We visualized the evolution of neighborhood types over time with the cordon to demonstrate the spatial constriction of the disease with this policy (Fig. 4). The first panel (top left) shows the geographical distribution of neighborhood types shortly after the policies come into effect. The infected area shrinks and the ratio of type A (red) to type B (green) neighborhoods decreases over time.

A simulated epidemic run on a 300×300 lattice with neighborhoods of size 10×10, with 70% compliance (0.7) and a delay of _{0 }

In Fig. 5 we plot both the reported number of cases in Liberia_{0 }_{t}

Normalized, linear-log plot of Liberia empirical values (red) compared with simulation data (blue) with T0 = 50 and 50% compliance (0.5).

We simulated different types of networks to test the robustness of the simulations to variations in the transmission network structure. We compared the original von Neumann neighborhood with long-range contacts, a Moore neighborhood with long-range contacts, and a Kleinberg network where the probability of making a connection decreases with distance. We also allowed the length of the of latent and infectious periods to vary for individuals across values found by Chowell_{0} in agreement with the observed results for Ebola. The results show the intervention effectiveness remains high and a compliance above 50% is sufficient to rapidly stop the epidemic. This provides additional evidence for the relevance of the results to real world conditions despite approximations used in modeling the transmission network. It also indicates that the community level interventions are highly robust.

Screening at the dotted lines. Results are normalized to the number of infected individuals at the time of the intervention. The intervention is robust against variation in network structure. A. Von Neumann neighborhood of four nearest neighbors. B. Moore neighborhood of eight nearest neighbors. C. Kleinberg small world network, with four nearest neighbors and longrange neighbors with probability of connection decreasing as inverse distance squared.

We note that recent research on contagious processes on networks that are not geographically local have considered heterogeneous connectivities across nodes, and specifically power law node connectivity. Under these conditions highly connected nodes enable the disease to spread across the entire network at arbitrarily low contagion rates

We analyzed the effect of community monitoring using a mean field, spatially averaged approach. The analytic equations are constructed by considering the dynamics of infection across the entire population. The susceptible population becomes infected by short range and long range transmission from noncompliant infectious individuals. The parameters are the compliance κ, the short range infectious rate τ, the long range infection rate η, the latent period Δ, the infectious period Γ, the average number of neighbors per individual

where S represents susceptible, S exposed, and S infectious groups. The first equation can be written as

where _{0}_{t}

For a similar analysis of contact tracing, a mean field treatment should separate exposed and infected populations based on whether or not they were traced_{c}_{c}_{0}_{c}_{c}

We found that a policy of community response can be effective at combating disease outbreaks without relying on information about infected individual's contact networks. This highlights the possibility of alternate methods to contact tracing for combating outbreaks. We have shown the policies require a surprisingly low compliance to end the outbreak. Notably, we see that for estimated Ebola parameters, 40% compliance is sufficient, and the cumulative number of infections in an outbreak is not substantially decreased by compliance higher than 60%. We also found that travel restrictions can be used to reduce the risks associated with compliance below 40%, and that the pairing of community-level interventions and travel restrictions can result in saving a substantial fraction of individuals from infection at any level of compliance. Public health interventions implementing variants of these policies have helped the number of active Ebola cases to reach zero in Liberia in March 2015^{,}

We model Ebola using a Susceptible, Exposed, Infectious, Removed (SEIR) model with a finite, spatially structured population with periodic boundary conditions. In an SEIR model, each individual can be in one of four states of health:

Individuals transition from state to state in the order

The standard non-spatial SEIR model is governed by a nonlinear system of differential equations. Let

Equation 1

where δ represents the rate at which exposed individuals become infectious and γ represents the rate at which the disease removes infectious individuals (either via death or survival with acquired immunity)

This system of differential equations provides a mean-field representation of the dynamics of an epidemic with an SEIR structure. We consider potential policy recommendations that involve explicit change of the contact network structure of the population. For our purpose, a model based upon this mass-action system of differential equations is insufficient.

We model a population on a square lattice where each individual has three properties: their current state of health (

Using the definitions from the SEIR model, we consider the rate of transition from state

We initialized the population by randomly setting 0.02% of the individuals to be in each of states

The transmission of the disease involves both local and long-range spreading mechanisms. Each infectious individual has probability τ of infecting each of its susceptible neighbors during a given day. Additionally, each susceptible individual chooses at random an individual on the grid with whom they will interact on a given day and, if infectious, the susceptible will become infected with probability η. The formal value of the network connectivity is N as all nodes are connected by the long range transmission probability.

The basic reproduction number, _{0}_{0}

Equation 2

The number of neighbors that are infected is given by the second term, and the number of non-neighbors that are infected is given by the first term. The number of neighbors that become infected is complicated by the reduction in number of susceptible neighbors as they become infected from day to day during the infectious period. For the long range interactions, the effect is small due to the large number of possible sites so that a few new infections do not affect the number of susceptible individuals the long range interactions can affect.

To obtain Eq. 3, consider a neighbor that can be infected by the local infection process with probability τ and independently, by the long-range infection mechanism with probability

Equation 3

For ^{2}

Equation 4

For large ^{Γ}).

Individuals outside of the infected individual's neighborhood can only be infected by long-range interaction. Neglecting corrections of O(1/^{2}), the infected individual chooses one such individual to visit and infects that individual according to the probability η. Since it is assumed that the concentration of infected individuals is small, the average number of individuals infected is η. The additional infection is only 1 in G^{2}, which doesn't affect the calculation in the next period, so the total number over the infectious period is Γη.

We note that the characterization of _{0}_{0}_{0}

The dynamics of the epidemic can be more completely described by the effective reproduction number, _{t}_{t}

Equation 5.

where _{t}_{t}_{t }_{t }= z

In Fig. 7, we compare the simulated time-series of values for _{t }_{t}_{t}_{t}_{t}_{t }

The value of _{t}_{t}_{t}_{t}

In order to make relevant comparisons between the results of our simulations and the actual 2014 Ebola outbreak, it is useful to choose model parameters so that the spread of our simulated epidemic matches the real-world spread of Ebola. We chose parameters so that the simulated epidemic matches the growth of cumulative case numbers observed in the Liberia outbreak during the period of exponential growth. Values obtained by comparison with event data and other methods do not change the conclusions. In addition, we consider values of Δ and Γ consistent with the actual mean latent and infectious periods for Ebola

Given a time series of cumulative cases,

We consider two sets of Δ and Γ that have been used to describe the Ebola outbreak. First, we use Δ = 5, Γ = 6, approximating values of Althaus

We also considered the values Δ = 10, Γ = 7, which are consistent with the values used by Chowell and Nishiura^{,}

It is common to characterize simulated and real-world outbreaks using _{0}_{0}

The cumulative number of cases ^{,}^{,}^{,}_{0}

Equation 6.

Using Eq. 6 and the exponential growth rates generated from our simulations, we see that _{0}_{0 }

We simulated our epidemic with the parameters values Δ = 10, Γ = 7, τ = 0.18, and η = 0.015. As shown in Figs. 8 and 9, the qualitative behavior of the epidemic and the impact of intervention policies were similar to the Δ = 5, Γ = 6 case. For compliances of 0.6, 0.8, and 1.0, the outbreak ended quickly after the implementation of community-level isolation policies (Fig. 1A), and the value of _{t}

From Fig. 9C, we see that a compliance of 0.6 or higher limits the number of cumulative infections, and that travel restrictions still substantially help to limit the loss of life in the case of low compliance. Early implementation of the intervention (lower values of _{0}_{0}

Screening begins at the vertical dotted line, with a level of compliance indicated by label and color (green 0 to blue 1.0). A. Number of cases with or without symptoms. Note that, compared to the simulations in the main paper, 40% compliance (0.4) is no longer sufficient to end this more virulent outbreak. B. Cumulative cases. C. For greater than 40% compliance (0.4), _{t}

A,B. Simulations with (blue) and without (red) travel restrictions. The travel restrictions compensate for low levels of compliance, and their differences are comparable to Fig. 3 in the main paper. C. The cumulative number of infections over the entire epidemic, as a function of compliance levels and intervention times. Colors from brown to yellow signify intervention times (70, 90, 110, 130). Without enforced travel restrictions (dotted lines), a low compliance results in minimal differences between early and late policy implementation. Travel restrictions (solid lines) dramatically reduce infection numbers for earlier interventions at low compliance. We chose a slightly later set of intervention times _{0}^{,}

All data are publicly available and cited in the paper.

The authors have declared that no competing interests exist.

We thank Irving Epstein, Joseph Norman, Francisco Prieto-Castrillo, Alfredo Morales, and Matthew Hardcastle for comments on the manuscript and Joa Ja'keno Okech-Ojony, Lorenzo Dorr, Stephen Paul Ayella Ataro, Subarna Mukherjee, and Katherine Collins for help in obtaining or providing information about the Ebola response in Liberia.