I am a post doc at the Network Dynamics and Simulation Science Laboratory, part of the Virginia Bioinformatics Institute at Virginia Tech. My research interests primarily concern the mathematical and computational modeling of infectious diseases, particularly healthcare-associated infections. I am particularly interested in the intersection of computational models and observational data as well as ways to visualize and quantify uncertainty in large-scale epidemiological simulations.

Computational Epidemiologist with research focused on providing public health decision making support and policy guidance.

West Africa is currently experiencing an unprecedented outbreak of Ebola, a viral hemorrhagic fever. On March 23, 2014 the World Health Organization announced through the Global Alert and Response Network that an outbreak of Ebola virus disease in Guinea was unfolding^{,}^{,}

As of October 5, 2014, the World Health Organization has reported 8,033 cases of Ebola virus disease in Sierra Leone, Liberia, Guinea, Nigeria and Senegal, with sporadic cases occurring outside West Africa

Mathematical models of disease outbreaks can be helpful under these conditions by providing forecasts for the development of the epidemic that account for the complex and non-linear dynamics of infectious diseases and by projecting the likely impact of proposed interventions before they are implemented. This in turn provides policy makers, the media, healthcare personnel and the public health community with timely, quantifiable guidance and support^{,}^{,}^{,}

We use a mathematical model to describe the development of the Ebola outbreak to date, provide short term projections for its future development, and examine the potential impact of several interventions, namely increased contact tracing, improved access to PPE for healthcare personnel, and the use of a pharmaceutical intervention to improve survival in hospitalized patients.

A time series of reported Ebola cases was collected from public data released by the World Health Organization, as well as the Ministries of Health of the afflicted countries. These data sets do not include patient-level information, but rather laboratory confirmed, suspected or probable cases of the disease, which is thought to represent the best available estimate of the current state of the epidemic. A curated version of this data is available at https://github.com/cmrivers/ebola.

A compartmental model was used to describe the natural history and epidemiology of Ebola, adapted from Legrand

The population is divided into six compartments: Susceptible (S), Exposed (E), Infectious (I), Hospitalized (F), Funeral (F) – indicating transmission from handling a diseased patient’s body, and Recovered/Removed (R). Arrows indicate the possible transitions, and the parameters that govern them. Note that λ is a composite of all β transmission terms described in Table 1.

A deterministic version of the model was fit and validated to the current outbreak data using least-squares optimization, with seed values from the Uganda outbreak described in Legrand

This validated model provides a mathematical description of the epidemic up to the present. In order to forecast into the future, a stochastic version of the model was implemented using Gillespie’s algorithm with a tau-leaping approximation^{,}

Based on interventions that are technically, but not necessarily socially, feasible in the foreseeable future, we model five scenarios to examine their likely impact on the development of the epidemic. First, we model improved contact tracing by increasing the proportion of infected cases that are diagnosed and hospitalized from the baseline scenario of 51% in Liberia and 58% in Sierra Leone to 80%, 90% and 100%, and a concordant decrease in the time it takes for an infected individual to be hospitalized by 25%. This scenario could also be considered to represent improved access to healthcare, or improved public support for the hospitalization of sick individuals. Second, we explore the impact of simultaneously (1) decreasing the contact rate for hospitalized cases (β_{H}) to represent the increased use of PPE as supplies and awareness of the outbreak increase as well as (2) eliminating the possibility of post-mortem infection from hospitalized patients due to inappropriate funereal practices. Third, we model both a simultaneous (1) decrease in β_{H} (lack of post-mortem infection from hospitalized cases) and (2) increase in the proportion of hospitalized cases. This models the effect of a joint, intensified campaign to identify and isolate patients – the conventional means of containing an Ebola outbreak – with the necessary supplies and infrastructure to treat these patients using appropriate infection control practices. Finally, we model a pharmaceutical intervention that increases the survival rate of hospitalized patients by 25%, 50% and 75%, with a moderately high level of contact tracing (80%).

Parameter
Liberia Fitted Values
Sierra Leone Fitted Values
Contact Rate, Community (β
_{I})0.160
0.128
Contact Rate, Hospital (β
_{H})0.062
0.080
Contact Rate, Funeral (β
_{F})0.489
0.111
Incubation Period (1/α)
12 days
10 days
Time until Hospitalization (1/γ
_{H})3.24 days
4.12 days
Time from Hospitalization to Death (1/γ
_{DH})10.07 days
6.26 days
Duration of Traditional Funeral (1/γ
_{F})2.01 days
4.50 days
Duration of Infection (1/γ
_{I})15.00 days
20.00 days
Time from Infection to Death (1/γ
_{D})13.31 days
10.38 days
Time from Hospitalization to Recovery (1/γ
_{IH})15.88 days
15.88 days
Probability a Case is Hospitalized (ι)
0.197
0.197
Case Fatality Rate, Unhospitalized (δ
_{1})0.500
0.750
Case Fatality Rate, Hospitalized (δ
_{2})0.500
0.750

As this study uses publicly available data without personal identifiers, it was determined not to require IRB approval.

The deterministic model fit well for both Liberia and Sierra Leone, with the predicted curve of cumulative cases following the reported number of cases in both countries. The end-of-year forecast shows a range of uncertainty for each country of several thousand cases between the most optimistic and pessimistic scenarios. However, the number of cumulative cases is forecast to continue rising extremely rapidly, with the bulk of the epidemic yet to come (Figure 2). This suggests an extremely poor outlook for the course of the epidemic without intensive interventions.

In the baseline end-of-year forecasts for both Sierra Leone and Liberia, person-to-person transmission within the community made up the bulk of transmission events, with a median (IQR: Interquartile Range) of 117,877 (115,100– 120,585) cases arising from the community in the Liberia forecast and 30,611 (29,667 – 31,857) in the forecast for Sierra Leone. Both had fewer hospital transmissions – 21,533 (21,025 – 21,534) in Liberia and 5,474 (5,306 – 5,710) in Sierra Leone, than transmissions arising from funerals – 35,993 (35,163 – 36,789) in Liberia and 9,768 (9,470 – 10,137) in Sierra Leone. For brevity, only the results of the Liberia model are reported below, with the results from Sierra Leone in the electronic supplement. The epidemic trajectories for all modeled interventions may also be found in the

Red dots depict the reported number of cumulative cases of Ebola in each country, with the black line indicating the deterministic model fit. Each blue line indicates one of two hundred and fifty stochastic simulated forecasts of the epidemic, with areas of denser color indicating larger numbers of forecasts.

The basic reproduction number (R_{0}) for the baseline scenario was calculated in the same manner as in Legrand _{0} is broken into three components, representing the respective contributions of community, hospital and funereal transmissions, as well as an overall R_{0} reflecting the epidemic potential for the disease. In the baseline scenario, we estimate an overall R_{0} of 2.22, made up of an R_{0 }of 1.35 from the community, 0.35 from hospitals and 0.53 from funerals for Liberia. Sierra Leone's R_{0} was estimated to be 1.78, made up of an R_{0} of 1.11 from the community, 0.24 from hospitals and 0.43 from funerals. These estimates are similar to estimates for the current outbreak reported elsewhere. For brevity, only the overall R_{0} estimates will be reported here. The breakdown of R_{0 }by source can be found in the

The forecasted distribution of cases under intensified contact tracing is shown in Figure 3. There is a shift from community transmission toward hospital transmission, though at extremely high levels of contact tracing and hospitalization, the impact of the intervention on the course of the outbreak also results in fewer hospitalized cases. There is a less pronounced but still substantial downward shift in funeral cases, and a decrease in total cases in Liberia. This decrease is not sufficient to shift the cumulative case curve off its steep upward trajectory, but only lessens its magnitude. The 80%, 90% and 100% of patients traced and hospitalized scenarios resulted in an overall reduction of R_{0} to 2.11, 2.01 and 1.89 respectively.

Box plots depict the median, interquartile range and 1.5 times the interquartile range for each scenario. Each individual simulated forecast is shown as a single dot, jittered so as to depict the complete distribution of the data.

The improved infection control scenario decreased β_{H }to represent decreased risk of hospital transmission due to increased PPE, increased number of healthcare workers or greater awareness of the epidemic resulting in greater care while treating patients with undiagnosed febrile illness. Additionally, it eliminated the potential for post-mortem transmission during the funereal process. This combination of interventions resulted in a marked decrease in the overall number of cases (Figure 4), and a reduction of R_{0} to 2.13, 2.05 and 1.96 for 25%, 50% and 75% reductions in the hospital transmission contact rates and improvements in the disposal of the remains of Ebola victims.

Box plots depict the median, interquartile range and 1.5 times the interquartile range for each scenario. Each individual simulated forecast is shown as a single dot, jittered so as to depict the complete distribution of the data.

Major reductions in all sources of cases were seen, with the most dramatic drop in the relative number of cases arising from the reduction of within-hospital transmissions. However, even with substantially reduced transmission and a decrease in the burden of mortality from the outbreak, improved infection control was also insufficient to push the epidemic off its steep upward trajectory.

Figure 5 shows the median decrease in cases as compared to baseline for simulations combining increased contact tracing and a reduction in the risk of hospital transmission from those who are isolated and treated. The most optimistic of these scenarios, with complete contact tracing and a 75% reduction in hospital transmission results in more than 165,000 fewer total cases over the course of the forecasted period, as compared to the baseline scenario (Figure 6). The overall R_{0} in this scenario is reduced to 1.72. This represents a ten-fold reduction in the number of cases, and is a major improvement epidemic trajectory. However even under this scenario, the epidemic is slowed and mitigated, rather than fully stopped, with transmission still ocurring after the end of the year.

Each box represents the median result of 250 forecasted epidemics, each with a % of contacts traced and a % decrease in hospital transmission. Areas of deeper blue indicate progressively greater reductions of the median number of cases.

The solid black line represents the deterministic model fit of the epidemic to present, with each grey line representing a single simulated forecast with no interventions in place, and each blue line representing a single simulated forecast of the epidemic with 100% of contacts traced, a 75% reduction in hospital transmission (βH) and no post-mortem infections from hospitalized patients. Areas of darker color indicate more forecasts with that result.

The introduction of a pharmaceutical intervention that dramatically improves the survival rate of hospitalized patients also leads to a less severe outbreak, shown in Figure 7. Compared to contact tracing alone, there is a small reduction in the number of hospitalized cases (as the scenario implies no change in infection control practices), but a stronger decrease in the number of community, funeral and overall cases depending on the efficacy of the hypothetical pharmaceutical. An efficacy that reduces the case fatality rate of hospitalized patients by 25%, 50% or 75% results in a corresponding reduction of R_{0} to 2.03, 1.94 and 1.85 respectively. As with the other forecasts above, this intervention also fails to halt the progress of the epidemic, though it does considerably reduce the burden of disease.

Box plots depict the median, interquartile range and 1.5 times the interquartile range for each scenario. Each individual simulated forecast is shown as a single dot, jittered so as to depict the complete distribution of the data.

The control of Ebola outbreaks in the past has been a straightforward, albeit difficult application of infection control and quarantine policies. In principle, these types of interventions should be applicable to this outbreak as well. However, it remains unclear whether they can be implemented at the unprecedented scale of the current outbreak. This study attempts to address whether or not aggressive interventions could arrest, or at least mitigate, the epidemic.

Our findings suggest that, for at least in the near term, some form of coordinated intervention is imperative. The forecasts for both Liberia and Sierra Leone in the absence of any major effort to contain the epidemic paint a bleak picture of its future progress, which suggests that we are in the opening phase of the epidemic, rather than near its peak. These findings are in line with predictions from other models which, despite using different methods and different data sources, have all estimated similar basic reproductive numbers, and forecast that the epidemic is currently beyond the point where it can be easily controlled^{,}^{,}

Of the modeled interventions applied to the epidemic, the most effective by far is a combined strategy of intensifying contact tracing to remove infected individuals from the general population and placing them in a setting that can provide both isolation and dedicated care. This intervention requires that clinics have the necessary supplies, training and personnel to follow infection control practices. Although both of these interventions in isolation also have an impact on the epidemic, they are much more effective in parallel. In particular, the slight increase in cases in Sierra Leone at the lower end of the modeled contact tracing range, when unaccompanied by a concordant increase in infection control, highlights the necessity of these two interventions being implemented side-by-side.

The hypothetical mass application of a novel pharmaceutical like the one administered to two American aid workers^{,}

Despite the considerable impact the proposed interventions have on the burden of disease, none of them are forecast, at least in the short term, to halt the epidemic entirely. It is possible these interventions will have a longer-term impact on the epidemic, however we have avoided projecting out further than the end of the year due to the inherent uncertainty in an emerging epidemic. This in turn suggests another communication challenge for public health planners. While all of the proposed interventions are worth pursuing, and will have an impact on the epidemic and public health, the attention of the international community must be sustained in the long term in order to ensure the necessary supplies and expertise remain present in the affected areas. Additionally, in light of public resistance to the limited types of these interventions already in place, public awareness and acceptance of intensified interventions must be built, as there will not be an immediate cessation to the epidemic, or even necessarily a clear sign that the situation is improving.

This study is not without limitations. As with all mathematical models, the results of the study depend on the assumptions about the natural history of Ebola, its epidemiology, and the values of the parameters used – as well as the quality of the data used to fit the model. Particularly, this model is validated against data that records cases on their time of reporting, rather than their time of onset, so the model time series may be shifted by several days. Because of the relatively small number of historical Ebola outbreaks, the unusual size of this outbreak, and the difficulty collecting data during an emerging epidemic, the accuracy of this model and its parameters is difficult to ascertain. It does however represent our best understanding of the epidemic using available data, and the model has proven capable of predicting the ongoing development of the epidemic, as well as having been used to model previous Ebola outbreak. The uncertainty inherent in model prediction has been addressed with the short forecasting window, and with the use of stochastic simulation to aid in quantifying uncertainty inherent within the model system.

The ongoing Ebola epidemic in West Africa demands international action, and the results of this study support that many of the interventions currently being implemented or considered will have a positive impact on reducing the burden of the epidemic. However, these results also suggest that the epidemic has progressed beyond the point wherein it will be readily and swiftly addressed by conventional public health strategies. The halting of this outbreak will require patient, ongoing efforts in the affected areas and the swift control of any further outbreaks in neighboring countries.

The authors would like to thank Katie Dunphy, Jesse Jeter, P. Alexander Telionis, James Schlitt, Jessie Gunter and Meredith Wilson for their assistance and support. The authors would also like to acknowledge the participants in the weekly briefings organized by DTRA, BARDA and NIH, including Dave Myer, Aiguo Wu, Mike Phillips, Ron Merris, Jerry Glashow, Dylan George, Irene Eckstrand, Kathy Alexander and Deena Disraelly for their comments and feedback.

We agree with Richard and Christian. There is an inconsistency in the parameterization and the model equations. In the parameter table iota is the probability that a case is hospitalized. 1/gammaH is the average time until hospitalization. Simplify the model (the ideas carry over): dS/dt = - beta I(t) S(t), dI/dt = beta I(t) S(t) – gammaH iota I(t) – r (iota) I(t), dH/dt = gammaH iota I(t), where r (iota) = (1 – iota) (gammaI (1 – delta1) + gammaD delta1) accounts for all other exits from I (none returning to S or I). First, the rate of exit to hospitalization is gammaH iota. So, by the probabilistic interpretation of linear loss rates, the average time until hospitalization is 1/(gammaH iota), not 1/gammaH. Second, from the rates in the differential equation and the definition of iota, the probability a case is hospitalized is gammaH iota / (gammaH iota + r (iota)) = iota. The parameter values do not satisfy this equation. These issues can be easily fixed by adjusting the parameters and loss rates. In addition to the probabilistic interpretation of iota as a ratio of loss rates in the I(t) differential equation, the probability a case is hospitalized can also be derived from the solution of the nonlinear model. From the model, the long-term fraction of cumulative hospitalized cases is also gammaH iota / (gammaH iota + r (iota)), since (gammaH iota times the integral of I(t) from 0 to infinity)/ (S(0) – lim t-> infinity S(t)) = gammaH iota / (gammaH iota + r(iota)). Glenn Webb and Cameron Browne

I think Richard White is right. There seems to be a problem with the weighting of the transition rates. The same problem appears in the study by Legrand et al. (2007). Let's consider hospitalized individuals H. They can either die or recover. According to Table 1, the parameters for Sierra Leone are: Time from hospitalization to death, 1/gamma_DH = 6.26; time from hospitalization to recovery, 1/gamma_IH = 1/15.88; case fatality rate in hospital, delta_2 = 0.75. One can easily calculate the average time an individual will stay in the hospital as follows: delta_2*1/gamma_DH + (1-delta_2)*1/gamma_IH = 8.7 days. This is the weighted mean of the time to death and time to recovery. According to the differential equations in the study by Rivers et al. (2014), the total rate at which individuals will leave the hospital compartment is gamma_DH*delta_2 + gamma_IH*(1-delta_2). The inverse of this rate gives the average time individuals in the model stay in the hospital: 7.4 days. Hence, lower than what it should be.

Edit (Oct 23): After further consideration we agree with commenters that there may be errors in the equations given by Legrand et al and used in this paper. We are working on investigating further. Original: Transition probabilities for compartmental models are indeed 1/rate, and that is the case in this model. It is customary to omit this notation from transition diagrams for the sake of simplicity. See for example Legrand et al 2006, which is the source of this model structure. Table 1 shows our parameters are 1/rate.

Statistician working at Norwegian Institute of Public Health

Are the weights in Figure 1 incorrect? The authors weight the rates by probability, but they should be weighting the average time spent in each compartment (i.e. 1/rate) and then inverting this again. For a simple R example, let us take a compartment spawning two options, one with rate 1/4 and one with rate 1/1, each with 50% probability: 1/mean(c(rexp(1000000,rate=1/4),rexp(1000000,rate=1/1))) This gives a value of 0.4. Which is equal to 1/(4*0.5+1*0.5). This is NOT equal to 1/4*0.5+1/1*0.5=0.625