The art of modeling

Discussion

Discussion

• Estimation
• Inference
• Prediction
• Scenario analysis
• Coherence (“sanity check”)
• Synthesis

1. The model is only as good as the data it's based on

2. A model is just an idea

Exercise: Critique these perspectives

Note: Models can't make up for what we lack in empirical information

Infectious disease management requires a multi-disciplinary approach

• Medicine
• Genetics/Genomics
• Microbiology
• Immunology
• Vaccines & Drugs

Although each of these approaches are important for understanding individual response and clinical care they don't address important questions at the population level

• Will this disease invade the population or go extinct?
• How fast will it grow?
• When will the epidemic peak?
• When will it disappear?
• Is it evolving? If so, why and how fast?
• Many, many more

• How to prevent spatial spread?
• How to prevent reintroduction?
• When is it best to implement interventions?
• How drastic of an intervention is required for containment?
• Should interventions be general or targeted?
• What interventions will be most effective?
• Many, many more

Models are appropriate for both emerging and endemic diseases, but depending on the epidemiology may emphasize different aspects of transmission and control.

Emerging pathogens arise in a wholly susceptible population and may exhibit transient trajectories (epidemics) that are far from equilibrium

Endemic pathogens exhibit quasi-stationary behavior

• A model is an abstract representation of a system
• A model is the mathematical or computational description of an idea
• A model is described in terms of its state and processes that result in a change of state
• In compartmental models, the primary focus of this course, these are often represented by two kinds of variables, referred to as the state variables and the parameters

Developments in recent years have blurred these categories significantly

• Statistical (including AI) vs mechanistic
• Compartmental models
• Deterministic vs stochastic
• Agent-based models
• Network models

(Generalized) linear models, nonlinear regression, anova, additive models, multivariate adaptive regression splines, boosted regression trees, convolutional neural nets

\( y = f(x) + \epsilon \)

All model terms refer to parts or processes

Each individual in the population belong to one and only one compartment.

This is the extent to which individuals have attributes. Individuals flow from compartment to compartment.

States at one time, together with parameters, uniquely determine states at all other times (past and future). Flows are expressed by ordinary differential equations (continuous time) or maps (discrete time).

States at one time, together with parameters, probabilistically determine states at all other times (past and future).

We distinguish among the concepts of stochasticity, variability, and uncertainty

Computer simulations use the paradigm of object-oriented programming in which attributes (e.g. infection status, location, etc.) are attached to objects (individuals) that belong to declared classes (e.g. hosts).

• Time lagged models (renewal equation)
• McKendrick-von Foerster type models (partial differential equation)
• Reaction-diffusion models (partial differential equation)

Choice of model depends on

1. The purpose for which the model is constructed (estimation, inference, prediction…)
2. The information (or data) available to inform the model

There is no validation without an objective.

“The multiplicity of models is imposed by the contradictory demands of a complex, heterogeneous nature and a mind that can only cope with few variables at a time; by the contradictory desiderata of generality, realism, and precision; by the need to understand and also to control; even by the opposing esthetic standards which emphasize the stark simplicity and power of a general theorem as against the richness and diversity of living nature. These conflicts are irreconcilable. Therefore, the alternative approaches even of contending schools are part of a larger mixed strategy. But the conflict is about method, not nature, for theindividual models, while they are essential for understanding reality, should not be confused with that reality itself.”

Levins, R. 1966. The strategy of model-building in population biology. American Scientist 54:421-431

• Precision/Accuracy

Model complexity (number of state variables, nonlinearity)

Ability to account for observations

• Intelligibility

Intelligible to human consumers/decision-makers

• Flexibility

Adaptability to new scenarios

Brett T, et al. (2020) Detecting critical slowing down in high-dimensional epidemiological systems. PLoS Comput Biol 16(3): e1007679.

Brett T, et al. (2020) Detecting critical slowing down in high-dimensional epidemiological systems. PLoS Comput Biol 16(3): e1007679.

• Express concept mathematically (equations, rates of change, etc.)
• Solve analytically (the general solution) or numerically (specific solutions) – Note: only the simplest models are analytically tractable
• Ready made software (e.g. ModelMaker, R packages, Berkeley Madonna)
• “Big” simulators (GLEAM, MOBS, etc.)
• Note on terminology: simulation vs solution

• Anderson & May (1991)
• Otto & Day (2007)
• Keeling & Rohani (2008)
• Vynnycky & White (2010)
• Diekmann et al. (2012)