By Erin C.S. Acquesta, Walt Beyeler, Pat Finley, Katherine Klise, Monear Makvandi, and Emma Stanislawski
As the world desperately attempts to control the spread of COVID-19, the need for a model that accounts for realistic trade-offs between time, resources, and corresponding epidemiological implications is apparent. Some early mathematical models of the outbreak compared trade-offs for non-pharmaceutical interventions [3], while others derived the necessary level of test coverage for case-based interventions [4] and demonstrated the value of prioritized testing for close contacts [7].
Isolated analyses provide valuable insights, but real-world intervention strategies are interconnected. Contact tracing is the lynchpin of infection control [6] and forms the basis of prioritized testing. Therefore, quantifying the effectiveness of contact tracing is crucial to understanding the real-life implications of disease control strategies.
Contact Tracing Demands
Contact tracers are skilled, culturally competent interviewers who apply their knowledge of disease and risk factors when notifying people who have come into contact with COVID-19-infected individuals. They also continue to monitor the situation after case investigations [1].
Case investigation consists of four steps:
- Identify and notify cases
- Interview cases
- Locate and notify contacts
- Monitor contacts.
Most health departments are implementing case investigation, contact identification, and quarantine to disrupt COVID-19 transmission. The timeliness of contact tracing is constrained by the length of the infectious period, the turn-around time for testing and result reporting, and the ability to successfully reach and interview patients and their contacts. The European Centre for Disease Prevention and Control approximates that contact tracers spend one to two hours conducting an interview [2]. Estimates regarding the timelines of other steps are limited to subject matter expert elicitation and can vary based on cases’ access to phone service or willingness to participate in interviews.
Bounded Exponential
The fundamental structure of our model follows traditional susceptible-exposed-infected-recovered (SEIR) compartmental modeling [5]. We add an asymptomatic population \(A\), a hospitalized population \(H\), and disease-related deaths \(D\), as well as corresponding quarantine states. We define the states \(\{S_i, E_i, A_i, I_i,\) \(H, R, D\}_{i=0,1}\) for our compartments, such that \(i=0\) and \(i=1\) correspond to unquarantined and quarantined respectively. Rather than focus on the dynamics that are associated with the state transition diagram in Figure 1, we introduce a formulation for the real-time demands on contact tracers’ time as a function of infection prevalence, while also respecting constraints on resources.
Figure 1. Disease state diagram for the compartmental infectious disease model. Figure courtesy of the authors.
When the work that is required to investigate new cases and monitor existing contacts exceeds available resources, a backlog develops. To simulate this backlog, we introduce a new compartment \(C\) for tracking the dynamic states of cases:
\[\frac{dC}{dt}=[flow_{in}]-[flow_{out}].\]
Flow into the backlog compartment, represented by \([flow_{in}]\), reflects case identification that is associated with the following transitions in the model:
- The rate of random testing: \(q_{rA}(t)A_0(t)\rightarrow A_1(t)\) and \(q_{rI}(t)I_0(t)\rightarrow I_1(t)\)
- Testing triggered by contact tracing: \(q_{tA}(t)A_0(t)\rightarrow A_1(t)\), \(q_{tI}(t)I_0(t)\rightarrow I_1(t)\), and \(q_{tE}(t)E_1(t)\rightarrow \{A_1(t), I_1(t)\}\)
- The population that was missed by the non-pharmaceutical interventions that require hospitalization: \(\tau_{IH}(t)I_0(t)\rightarrow H(t)\).
Here, \(q_{r*}(t)\) defines the time-dependent rate of random testing, \(q_{t*}(t)\) signifies the time-dependent rate of testing that is triggered by contact tracing, and \(\tau_{IH}\) is the inverse of the expected amount of time for which an infected individual is symptomatic before hospitalization. These terms collectively provide the simulated number of newly-identified positive COVID-19 cases. However, we also need the average number of contacts per case. We thus define function \(\mathcal{K}(\kappa, T_S, \phi_\kappa)\) that depends on the average number of contacts a day \((\kappa)\), the average number of days for which an individual is infectious before going into isolation \((T_S)\), and the likelihood that the individual will recall his/her contacts \((\phi_\kappa)\). This yields the following expression:
\[[flow_{in}] := \mathcal{K}(\kappa, T_S, \phi_\kappa)[q_{tE}(t)E_1(t)+(q_{rA}(t)+q_{tA}(t))A_0(t)+(q_{rI}(t)+q_{tI}(t)+\tau_{IH})I_0(t)].\]
The formulation models the rate of increase for the contact tracers’ backlog. This leaves us to estimate the rate that is needed to emerge from the backlog, which reflects the competing demands for contact tracers’ time.
We begin by emphasizing that the time for contact monitoring \((w_m)\) is independent of the time that is necessary to investigate new cases and their contacts \((w_c)\). Again, we leverage the state variables to derive the contact tracers’ total amount of work:
- Total work required to monitor known cases: \(w_m(A_1(t)+I_1(t))\)
- Total work required to investigate new cases: \(w_cC(t)\).
The amount of work that contact tracers accomplish per day will be capped by the available resources and time. Let \(N_{trace}\) represent the number of contact tracers who are available to execute the tasks, and \(q_w\) be the fraction of a day that consists of the work hours of each tracer.
When enough resources are available to execute the tasks, we expect \(work_{applied} \approx work_{demand}\) until we hit our carrying capacity, at which point \(work_{applied} \approx q_wN_{trace}\). The bounded exponential function provides a smooth approximation for the relationship between \(work_{applied}\) and \(work_{demand}\) (see Figure 2). When the demand on work is less than the available resources, the bounded exponential simulates \(work_{applied} > work_{demand}\). Though performing more work than demanded depletes the backlog at a higher-than-nominal rate, it does not ultimately overshoot the actual case queue. Recall that the goal is to determine the rate at which contact tracers can investigate new cases, so we further constrain the formulation to only reflect the proportion of work that tracers conduct for new cases:
\[\frac{w_cC(t)}{[w_m(A_1(t)+I_1(t))+w_cC(t)]}[q_wN_{trace}(1-\exp(-0.05(w_m(A_1(t)+I_1(t))+w_cC(t))))].\]
To obtain the rate at which tracers remove individuals from the backlog of contacts, we divide by the average workload per contact. Therefore,
\[[flow_{out}]:=\frac{C(t)}{[w_m(A_1(t)+I_1(t))+w_cC(t)]}[q_wN_{trace}(1-\exp(-0.05(w_m(A_1(t)+I_1(t))+w_cC(t))))].\]
This notation provides a smooth approximation for the rate at which contact tracers can reach and isolate newly-identified COVID-19 cases and quarantine their contacts. It is constrained by the available resources and informed by the current prevalence of infections.
Figure 2. Bounded exponential as a smooth approximation to the linear piecewise continuous relationship between \(work_{demand}\) and \(work_{applied}\). Figure courtesy of the authors.
Application to Vaccine Distribution
The bounded exponential formulations also have practical implications in the distribution of vaccines within compartmental model dynamics. The initial vaccine supply will be limited when vaccines first become available, with an increasing rate of availability in the future. By using a monotonically increasing vaccine distribution function \(v(t)\), we can formulate the continuously increasing nature of vaccine availability with linear approximations. Since the function is independent of population size, it will be difficult to bound the amount of distributed vaccines when the available vaccines begin to exceed the size of the eligible population.
If we utilize the bounded exponential function, we can approximate the fraction of distributed vaccines, \(f_v(t)\), as a function of the current population that is eligible for vaccination. We assume that \(S_0(t)\) is the model’s only compartment to be vaccinated. If \(S_0(t)<v(t)\), we reduce the vaccine distribution to reflect the fraction of the available population with respect to the total amount of available vaccines, \(\frac{S_0(t)}{v(t)}\). Doing so results in an expression for the fraction of distributed vaccines:
\[f_v(t):=1-\exp\bigg(-\alpha\frac{S_0(t)}{v(t)}\bigg).\]
For an appropriate \(\alpha>0\), the product \(f_v(t)v(t)\) must have the following properties:
- If the population is greater than the number of available vaccines, we will simulate distribution for all available vaccines.
- If the amount of vaccines exceeds the total population to which they can be distributed, the simulated distribution will only reflect a fraction of the available vaccines.
In closing, we offer a method for incorporating smooth mathematical formulations into the traditional SEIR compartmental modeling structure to simulate constraints on resource allocation. These formulations will provide a framework for the application of methods of dynamically constrained optimization problems in support of optimal resource allocation for a COVID-19 vaccine.
The Adaptive Recovery Modeling Team at Sandia National Laboratories brought together multidisciplinary researchers to provide their unique and experienced perspectives to the challenges of modeling the novel COVID-19 pandemic. In discussions with the New Mexico Department of Health, the practical aspects that public health professionals employ in response to intervention strategy policies have been used to assess baseline model formulations.
This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
Acknowledgments: Research was supported by the DOE Office of Science through the National Virtual Biotechnology Laboratory, a consortium of DOE national laboratories focused on response to COVID-19, with funding provided by the Coronavirus CARES Act. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. User acknowledges that neither the Government nor operating contractors of the above national laboratories makes any warranty, express or implied, of either the accuracy or completeness of this information, or assumes any liability or responsibility for the use of this information.
References
[1] Centers for Disease Control and Prevention. National Center for Immunization and Respiratory Diseases, Division of Viral Diseases. (2020, April 29). Case Investigation and Contact Tracing: Part of a Multipronged Approach to Fight the COVID-19 Pandemic. Retrieved from: https://www.cdc.gov/coronavirus/2019-ncov/php/principles-contact-tracing.html.
[2] European Centre for Disease Prevention and Control (2020, March 2). Resource estimation for contact tracing, quarantine and monitoring activities for COVID-19 cases in the EU/EEA. Stockholm: ECDC.
[3] Flaxman, S., Mishra, S., Gandy, A., Unwin, H.J.T., Mellan, T.A., Coupland, H., ..., Monod, M. (2020). Estimating the effects of non-pharmaceutical interventions on COVID-19 in Europe. Nature, 584(7820), 257-261.
[4] Gottlieb, S., Rivers, C., McClellan, M.B., Silvis, L., & Watson, C. (2020, March 28). National coronavirus response: a road map to reopening. American Enterprise Institute.
[5] Hethcote, H.W. (2000). The mathematics of infectious diseases. SIAM Rev., 42(4), 599-653.
[6] Heymann, D.L. (Ed.). (2015). Response to an Outbreak. Control of Communicable Diseases Manual, 20th Edition (pp. A8-A13). Washington, D.C.: APHA Press.
[7] Siddarth, D., & Weyl, E.G. (2020). Why We Must Test Millions a Day. COVID-19 Rapid Response Impact Initiative (White Paper 6). Cambridge, MA: Harvard University.
Erin C.S. Acquesta of Sandia National Laboratories (SNL) has experience with analyzing the mathematical properties and controllability of dynamical systems that are designed to simulate the spread of infectious diseases. Her broader area of research focuses on the application of methods of formal mathematical analyses to provide evidence of model credibility. Walt Beyeler of SNL has experience developing models of complex systems for the simulation of disease transmission, control, and resource utilization. Application contexts include evaluating strategies for pandemic influenza, detecting and countering epidemics in herd animals in Afghanistan, improving medical laboratory systems to speed the diagnosis of Ebola patients in West Africa, and using medical logistics modeling to anticipate and resolve resource shortfalls that arise from the COVID-19 pandemic. Pat Finley leads biosurveillence and disease modeling efforts at SNL, focusing on rapid-response operational models for developing outbreaks. His current research involves machine learning approaches to predict zoonotic transitions of emergent pathogens in developing countries. Katherine Klise of SNL applies her research experience in infrastructure resilience, sensor placement optimization, and data analytics in the context of water distribution systems, electricity grids, fossil energy, and renewable energy systems to the needs of modeling complex architectures for the simulation of disease transmission. Monear Makvandi of SNL is an infectious disease epidemiologist with experience in emerging/re-emerging infectious disease surveillance and outbreak response. At SNL, she leads the development and implementation of strategic and sustainable biological risk reduction through enhanced laboratory security, improved diagnostic methods, infectious disease control, and health system strengthening. Emma Stanislawski is an epidemiologist with the New Mexico Department of Health, where she specializes in respiratory infectious diseases. She has previously worked as a vaccine-preventable disease epidemiologist and with an immunization advocacy coalition.