Methodological Borrowing and Blending for Health Technology Assessments

By Anuj Mubayi and Carolyn Harley

COVID-19 has impacted the healthcare sector in numerous ways, both directly and indirectly. The pandemic has forced researchers to quickly evaluate several pressing questions despite significant levels of uncertainty. Estimating COVID-19’s demand on the healthcare system—especially in terms of intensive care unit beds—is paramount, particularly since demand dynamics change with each variant and overall vaccine availability. Furthermore, understanding the spillover impacts of COVID-19 on the demand for non-COVID services is critical to the estimation of staffing needs, service availability, and revenue. An additional, less observed outcome from the pandemic is the disruption of the biopharma industry’s ability to carry out clinical trials in the normal healthcare setting, which can affect future drug development for non-COVID conditions.

As a result, health economists and other healthcare researchers have been working to produce models that assist stakeholders in planning responses to COVID-19-related pressures. Healthcare researchers have generated strong methods to address demand-related questions for care, costs of care, and “value” assessment; Figure 1 shows stakeholders, value assessment methods, and time courses of health technology. However, the ongoing pandemic has evidenced that traditional techniques may be inadequate. Because this continually growing field has a significant impact on decision-makers in regard to investment in new technologies and coverage of new treatments, scientists should consider additional modeling techniques.

Here we highlight the potential role of mathematical and computational techniques when addressing challenges in Health Economics and Outcomes Research (HEOR). We postulate that model choice is important and assume that other disciplines’ methods may be applicable to the healthcare setting, including in health technology assessment. COVID-19 modeling efforts have demonstrated the challenges and relevance of analytical and modeling decisions in healthcare evaluation; they have also highlighted the predictive value of methods from scientific disciplines that may be unfamiliar to the healthcare community. 

 Mathematician George Box maintained that the primary purpose of modeling is to understand the useful impacts of a system’s selected drivers, even though they fall short of capturing the full complexity of reality. He said that “All models are wrong, but some are useful,” thus cleverly affirming that no model can perfectly replicate the real world. Yet despite significant limitations in data, structure, and uncertainty, model results can nonetheless offer insight into real-world questions. Features such as complex social-behavioral aspects are typically less sophisticated in models than in actuality, and available data for precise estimation of behavioral parameters are often limited. Furthermore, challenges in scaling—like the use of current information to predict future trends and the use of individual-level data to estimate population-level characteristics—can complicate matters. By considering modeling approaches outside of social science and health economic fields, we may be able to address some of these limitations. Here we focus on three specific modeling approaches that might offer deeper insight into healthcare system mechanisms (see Figure 2). 

Collective Behavior Models in Population Dynamics 

Collective behaviors refer to social processes that do not reflect existing social structures (like laws, conventions, and institutions) but rather emerge in a “spontaneous” way. Scientists have developed several theories to explain group behavior, including (i) contagion theory (which assumes that crowds exert a hypnotic influence over their members), (ii) convergence theory or learning theory (which considers how people who want to act in a certain way come together to form crowds), and (iii) emergent norm theory (which examines crowds that begin as collectivities and are composed of people with mixed interests and motives, wherein new norms “emerge” on the spot). Social force models (SFMs), which describe pedestrian interactions in terms of physical and social forces, can capture such behaviors. Social forces are not directly exerted by the pedestrians’ personal environment; instead, they serve as a measure for the internal motivations of the individuals to perform certain actions/movements. The following set of equations captures the SFM:

\[\frac{dr_\alpha(t)}{dt}=v_\alpha(t),\]

\[\frac{dv_\alpha(t)}{dt}=f_\alpha(t)+\xi_\alpha(t),\]

\[f_\alpha(t)=\frac{1}{\tau_\alpha}(v^0_\alpha e^0_\alpha-v_\alpha)+\Sigma_{\beta(\ne\alpha)}f_{\alpha\beta}(t)+\Sigma_if_{\alpha i}(t).\]

Here, \(r_\alpha(t)\) is the change in location of pedestrian \(\alpha\), \(v_\alpha(t)\) is their change in velocity or acceleration, \(f_\alpha(t)\) is the sum of social forces that influence pedestrian \(\alpha\), \(\xi_\alpha(t)\) is the individual fluctuations that reflect unsystematic behavioral variations, and \(f_{\alpha\beta}(t)\) and \(f_{\alpha i}(t)\) are the repulsive forces that maintain a certain safe distance from other pedestrians \(\beta\) and obstacles \(i\). Each individual \(\alpha\) attempts to move in a desired direction \(e^0_\alpha\) with a desired speed \(v^0_\alpha\) and adapts the actual velocity \(v_\alpha\) to the desired velocity within a certain time \(\tau_\alpha\). This SFM can describe the self-organization of several observed collective effects of crowd behavior. Such models are valuable tools for the design and planning of common pedestrian areas. In healthcare settings, the methods help practitioners manage patient movement—including for residents in long-term care facilities—and track the spread of infections that result from air travel. An individual’s motion is governed by the behaviors of other people in close vicinity, which leads to a varied number of contacts that successfully transmit infection. 

In the context of air travel, social behaviors—such as taking time to stow luggage and passing by an aisle seat passenger to reach the window seat—may influence overall contact patterns between individuals [3] (see Figure 3). 

Catastrophic Theory in Climate Modeling 

Catastrophe models can simultaneously handle complex linear and nonlinear relationships as well as sudden discontinuous changes. Many health conditions—including seizures, cardiac arrest, strokes, depression, and bipolar disorders—experience key catastrophe flags: bimodality (a two-mode process, normal and abnormal), sudden jumps from one mode to the other, and divergence (the ability of small changes in environmental factors to trigger dramatic changes in health conditions). The cusp catastrophe model, which is captured via the following stochastic differential equations, is a classic example of a catastrophe model: 

\[dz= \frac{dV(z;x,y)}{dz}dt+dW(t), \;\;\;\; \textrm{where} \;\; V(z;x,y)=\frac{1}{4}z^4-\frac{1}{2}z^2y-zx.\]

Here, \(z\) is the outcome variable, \(x\) is the asymmetry control factor where outcome \(z\) changes asymmetrically from one mode to the other as \(x\) increases, \(y\) is the splitting control factor that causes the outcome surface to split and bifurcate from smooth changes to sudden jumps as \(y\) increases, and \(V\) is the potential function. The probability density function of the corresponding limiting stationary stochastic processes is given by

\[f(z)=\frac{\Psi}{\sigma^2}\exp\Big[\frac{x(z-\lambda)+0.5y(z-\lambda)^2-0.25(z-\lambda)^4}{\sigma^2}\Big].\]

To study the impact of predictor variables \(v_1\) and \(v_2\) on response variable \(w\), we should proceed as follows: (i) Define \(x\) and \(y\) as a linear combination of the predictor variables (\(v_1\) and \(v_2\)) and \(z\) as a linear combination of \(w\), (ii) substitute these linear relationships into the above cusp catastrophe model, and (iii) use data on predictor and response variables to identify relationships’ linear coefficients. 

For example, we might want to explain patients’ differences in grip strength (GS), which is impacted by their levels of interleukin-6 (IL-6) and executive function (EF) [2]. While inflammation (captured by IL-6) has degrading effects on bone and muscle mass, muscle strength also depends in part on brain control; this means that the cognitive operation of EF may interact with levels of inflammatory processes. Catastrophe theory can tackle this type of problem because the problem exhibits key catastrophe flags: bimodality (“sarcopenia” or “impaired” versus “normal” form of GS), sudden jump (due to slight changes in individuals’ physiological factors), and divergence (ability to reach the two states by increasing or decreasing certain physiological factors). Here, \(x\) and \(y\) are a linear combination of EF’s predictor variables, and IL-6 and \(z\) are a linear combination of GS. 

Substituting these linear relationships into the cusp catastrophe model and estimating coefficients of linear models with appropriate data can yield an explanation of individual differences in GS. When EF is high, GS is stronger for patients with less severe inflammation (lower levels of IL-6) and vice versa; that is, the gradual continuous linear path of GS is evident in the cusp model (see Figure 4). However, two possibilities exist when EF is low: (i) A very low degree of IL-6 maintains strong GS (upper stable region), and (ii) a very high degree of IL-6 results in comprised GS (lower stable region). Moreover, as the IL-6 level wavers from low to high, it induces a sudden deterioration of GS (see paths B and C in Figure 4). 

Multiscale Analysis in Infectious Diseases 

Real problems rarely stay within one scale and instead cross multiple scales in terms of space, time, or size. Multiscale modeling methods can address computational difficulties that occur when the solution varies rapidly and often abruptly between scales. These methods integrate physics-based knowledge that bridges the scales and efficiently passes information across them. They involve the reduction of a hard multiscale problem into a sequence of relatively easy problems and solutions, which one can later combine with a defined strategy. Researchers use singular perturbation theory—which handles systems whose solutions evolve on different time scales and whose ratio is characterized by a small parameter \(\varepsilon>0\)—for such problems. The system is a fast-slow system when \(\varepsilon<1\). 

One example involves the following epidemic model that depicts transmission dynamics of a vector-borne disease like dengue: 

\[\frac{dI}{dt}=\frac{\beta}{M}\left(N-I\right)V-\mu I,\]

\[\frac{dV}{dt}=\frac{\vartheta}{N}\left(M-V\right)I-\upsilon V,\]

where \(I\) and \(V\) are infected human and vector populations, \(N\) and \(M\) are total human and vector populations, and \(\beta\) and \(\vartheta\) are transmission rates between the two populations. Since mosquito-vector dynamics occur at a much faster time scale than human-host dynamics, we can express the above system as follows: 

\[\frac{dI}{dt}=\varepsilon g(V,I,\varepsilon)=\varepsilon\left(\frac{\beta}{M}\left(N-I\right)V-\mu I\right),\]

\[\frac{dV}{dt}=f\left(V,I,\varepsilon\right)=\frac{\vartheta}{N}\left(M-V\right)I-\upsilon V.\]

When \(\varepsilon=0\), we have the fast system 

\[\frac{dV}{dt}=f(V,I(0),0)=\frac{\vartheta}{N}(M-V)I(0)-\upsilon V,\]

\[\frac{dI}{dt}=0.\]

A change of timescale \(\tau=\varepsilon t\) in the original model could lead to slow system for \(\varepsilon<<1\). After substituting \(\varepsilon=0\), 

\[\frac{dI}{dt}=g(V,I,\varepsilon)=\frac{\beta}{M}(N-I)V-\mu I,\]

\[\varepsilon\frac{dV}{dt}=f(V,I,\varepsilon)=\frac{\vartheta}{N}(M-V)I-\upsilon V.\]

One can find the solution of the perturb system by using the Levinson-Tihonov theorem and Hoppensteadt theorem [1].

As another example, if we wish to determine potential improvements in lifetime benefits due to treatment for sickle cell disease (SCD) in children, we can identify different scales that measure the benefits. In this case, the treatment of SCD may influence biomarkers like hemoglobin and blood oxygenation levels, which transpire on different scales than observed changes in lifetime income/earnings. We can link the scales through various factors, including the number of silent infarcts and strokes during childhood, changes in patient IQ level, impact on education and academic achievement, and influence on annual income (see Figure 5). 

The rapid response to COVID-19 resulted in significant contributions to pandemic modeling and related healthcare problems across various fields. While other scientific disciplines have utilized techniques that may be unfamiliar to the healthcare community, certain research questions can undoubtedly benefit from novel approaches that address behavioral factors, extreme events, and multiple scales. Future research will hopefully identify the conditions during which these new methods show improvements over more familiar methods and better prepare us for the next pandemic. We must think critically about our procedures and avoid automatically defaulting to familiar thinking and methods. We must challenge ourselves to discover better methods, expand our thinking, and improve our estimation, even if doing so requires nontraditional datasets. 

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References
[1] Brauer, F. (2019). A singular perturbation approach to epidemics of vector-transmitted diseases. Infect. Disease Model., 4, 115-123.
[2] Chen, D.G, Lin, F., Chen, X., Tang, W., & Kitzman, H. (2014). Cusp catastrophe model: A nonlinear model for health outcomes in nursing research. Nurs. Res., 63(3), 211-220.
[3] Islam, T., Lahijani, M.S., Srinivasan, A., Namilae, S., Mubayi, A., & Scotch, M. (2021). From bad to worse: Airline boarding changes in response to COVID-19. Roy. Soc. Open Sci., 8(4), 201019.

Further Reading
Cutler, D.M., & Summers, L.H. (2020). The COVID-19 pandemic and the $16 trillion virus. Jama, 324(15), 1495-1496.
DebRoy, S., Prosper, O., Mishoe, A., & Mubayi, A. (2017). Challenges in modeling complexity of neglected tropical diseases: A review of dynamics of visceral leishmaniasis in resource limited settings. Emerg. Them. Epidemiol., 14(10), 1-4. 
Helbing, D., & Johansson, A. (2013). Pedestrian, crowd, and evacuation dynamics. Preprint, arXiv:1309.1609.
Zeeman, E.C. (1976). Catastrophe theory. Sci. Amer., 234(4), 65-83.

Carolyn Harley is a Head of Operational Excellence in the Precision Medicine Group. She has more than 25 years of research experience in health clinical solutions, has served as an advisor and expert to a wide variety of healthcare organizations, and is responsible for excellence in organizational leadership and strategic vision.

Anuj Mubayi is a principal scientist (Infectious Disease Forecasting Lead) with The Public Health Company. He is also a Distinguished IBA Fellow in the Center for Collaborative Studies in Mathematical Biology at Illinois State University, a senior fellow at the Kalam Institute of Health Technology in India, and an adjunct faculty member in the Department of Mathematics and Computer Science at the Sri Sathya Sai Institute of Higher Learning in India. Mubayi's expertise is in health decision science as well as infectious disease modeling and dynamics.