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  • Richard M. Adler

Flattening the Curve and Modeling Viral Pandemics

The phrase “flattening the curve” dominates government briefings and media coverage of the COVID-19 coronavirus pandemic. The curve refers to a graphic plot of the number of reported active viral infections over time. In countries that fail to contain their initial viral outbreaks, curves rise at stunning rates, triggering disastrous social and economic consequences. The steepness of these curves is often quantified as the number of days it will take to double the number of cases. During a runaway contagion, doubling can occur in just a few days but slows to a few weeks for states or countries that manage to get their outbreaks under control. If left unattended, the curve grows unabated until it sweeps through the population—much like a wildfire only burns out after it decimates all combustible material in its path.

Cumulative reported cases in the United States (Source: CSC)


Imposing public health measures that reduce direct personal contacts can flatten this curve, slowing the explosive increase of new infections. Asian countries, where COVID-19 surfaced first and that imposed social isolation measures aggressively, seem to have throttled their outbreaks. European countries that applied these measures more belatedly are also reporting promising trends: steep infection curves have peaked and are now dropping.

Flattening the curve (adapted from CDC)


Unfortunately, the measures required to arrest a full-blown pandemic wave shut down all non-essential travel, business operations, and public services. Communities must also curtail most of the routine face-to-face and group interactions for work, education, and recreation that sustain us as social beings. The economic, psychological, and cultural costs of such clampdowns are punitive and unsustainable in the long run.

These dismal facts raise urgent questions about when and how to dial back draconian public health policies to allow economic and social activities to resume without triggering new waves of contagion. Public health experts look to answer these questions with epidemiological models that simulate the progression of viral outbreaks with and without policy interventions. They analyze projected futures to compare the outcomes of alternative strategies, make cost-benefit trade offs, and provide informed recommendations to government officials. In short, simulation models play a crucial role in driving government policies, investments, and other actions that will determine our safety and welfare—making it even more important to understand how these models work, their dependencies, and shortcomings.

The dominant approach to modeling the progression of infectious diseases projects the flow of individuals among three disjoint populations or compartments. The diagram below displays a basic three-compartment model developed by William O. Kermack and Anderson G. McKendrick in 1927. It is called SIR: S stands for susceptible individuals, I for infected, and R for recovered. The basic SIR model assumes that nearly the entire population starts off in the S compartment, equally vulnerable to the novel virus in question. When an individual becomes infected, they move from the S to the I compartment. If that person recovers from the infection, they shift from I to R; otherwise, they die, dropping out of I (and decreasing the total population).

SIR model of the progression of viral outbreaks

A set of ordinary differential equations dictate how S, I, and R change over time as the viral contagion plays out, determining the shape of the curve that represents the projected increase in the size of the I compartment. These equations feature two parameters, the infection rate (β) and the recovery rate (γ), which dictate the rate of flow of individuals from S to I and from I to R respectively and must be estimated from the best available diagnostic test and clinical patient data.

Parameter β depends on three factors: the infectiousness of the disease (COVID-19); the probability of a susceptible person being exposed to the pathogen given contact with an infected individual; and the level of mixing, meaning how frequently individuals (from S and I) come into contact and interact with one another. The first two factors depend on the nature of the viral pathogen (SARS-Cov-2) and its modes of transmission. The third factor is driven by social density and patterns of movement in the area of interest. Parameter γ depends on how the disease progresses in humans and its lethality, which, in turn, depend on environmental factors such as population density, demographics, life expectancy, health, nutrition, and socio-economic conditions.

Epidemiologists define a constant called the reproductive number (R0) for comparing viral outbreaks. It estimates the average number of people that will be infected by contact with a single infected individual (before they recover or die). For the SIR model, R0 is proportional to β / γ. Intuitively, if R0 is greater than 1, each infected individual infects more than one other person , setting off a chain reaction of spreading contagion, whereas if R0 is less than 1, the scourge will die out. Current (early) estimates of R0 range from 1.5 to 3.5. By contrast, R0 for measles ranges from 12 to 18, influenza ranges from 2 to 3, polio 6, Ebola 1.5, and SARS (3.5). R0 represents the maximal susceptibility of a population (with zero immunity) to a novel viral infection. A second metric, the effective reproductive number R’ (not related to the “R” for Recovered in SIR), reflects policy interventions to flatten the curve.

Public health measures focus primarily on lowering β. Mixing is reduced by restricting travel, closing non-essential businesses and other social institutions, halting public gatherings, sheltering (and working) at home, and imposing quarantines on people who are infected (or exposed). Further hygienic measures aim to lower the probability of exposure by preventing indirect transfers of viral residues left by infected individuals: maintaining physical separation when people do have to interact, wearing masks and gloves around others when distancing isn’t feasible, washing hands frequently, and sterilizing shared surfaces and objects. You can think of β as a valve controlling flow from S into I.

Medical researchers and pharmaceutical companies are racing to develop effective antiviral treatments and antibody therapies specific to COVID-19. Antiviral drugs can lessen the severity of disease symptoms, reduce the risk of complications such as respiratory failure, and potentially shorten the duration of the infection. Antibody therapies bolster patients’ immune systems to better defend against the virus. Thus, medical treatments and therapies act primarily to increase γ by facilitating recovery and reducing deaths. You can think of γ as a valve that regulates the opening of the drain from I into R.

Though efforts to reduce β for COVID-19 aim to squelch runaway contagion, they only delay the flow from S to I. Returning to the SIR diagram, the S compartment is at a higher position than I to highlight the fact that flow from S to I is as inexorable as the constant force of gravity—a steady pressure exerted by contacts with infected individuals on the susceptible population. The primary goal of lowering β is to reduce the peak of the curve. Given that 20% of infected persons require hospitalization and 5% intensive care, a rising curve signals an impending surge of serious cases into hospitals. Flattening the curve serves to spread the influx of new admissions at any given time, preventing surges that would overwhelm limited treatment capacity. This avoids painful decisions regarding rationing scarce resources and gives providers breathing room and options to offer the best possible care to all patients as they need it.

A more recent approach to simulating viral outbreaks relies on a software technology called agent-based models (ABMs). An ABM consists of a collection of software objects called agents that represent individuals rather than populations. Agents have properties (location, age, occupation, health status), behaviors (commuting, shopping, recreation), and decision rules that trigger changes in behavior (switching to working at home, adopting social distancing and hand-washing practices). Agents operate in environments such as cities or rural communities. ABMs have also been used to model traffic jams, ecosystems, and financial crises (think of investor panic and forced selling as a type of contagion).

The Washington Post recently published a simple ABM model of viral contagion that consists of a bunch of dots (agents) moving about randomly (a behavior) in a box (a community). One agent becomes infected, a property signified by the color brown. As this agent comes into contact with an uninfected but susceptible blue agent, the second becomes infected (turns to brown), and infects further agents, and so on. The entire population succumbs rapidly and graphically to the viral infection. To illustrate the effects of public health policies (decision rules), a subset of uninfected agents shelter in place and stop moving. This slows the rate of contacts with other agents, which visibly slows the time it takes for the infection to spread universally.

You can add properties, behaviors, and decision rules to ABMs to make them more realistic. Assigning different values to groups of agents produces interesting population segments (young vs. elderly and sick, professional vs. service workers). They can be assigned different behaviors (commute by car vs. public transportation), placed in different environments (poor urban neighborhoods vs. affluent suburbs) and assumed to adopt different combinations of social distancing and sanitation decisions, or none at all.

Anatomy of an Agent-Based Model to support pandemic policy decisions

The two approaches to modeling the progression of viruses are complementary. SIR models look at viral spread in terms of entire populations, whereas ABMs view contagion from the perspective of the individuals that make up a population. Physicists use a similar two-pronged strategy to study gases; top-down using system-level parameters such as pressure, temperature, volume and entropy vs. bottom-up, looking at individual atoms or molecules and the effects of collisions on their positions and momenta. Each approach highlights different facets of the dynamic behavior of the gas.

Similarly, SIR models offer valuable “big picture” insights into the proportions of S, I, and R individuals and their rates of change. But SIR models are also coarse, relying on just two flows to model the dynamics between three compartments. This forces analysts to combine and reduce everything they know (or assume) about rates of adoption for public health measures and their likely efficacy into one adjustment to a single model parameter, the rate of infection β. By contrast, ABMs contain large numbers of “moving parts”the forest (population) is harder to see for the trees (individual agents). However, this increased granularity provides superior insight into how public health measures such as social distancing and enhanced hygiene combine to alter viral spread. Recall that ABMs model personal behaviors explicitly (e.g., travel, hygiene, social interactions). ABMs also model how decisions to adopt new defensive measures modify personal behaviors. These detailed assumptions are preserved rather than collapsed into a single number. Instead, ABMs project rather than assume the effects of interventions on viral spread; curve flattening emerges from simulating the accumulating consequences of individuals making independent decisions about adopting behavioral changes to protect themselves and others. In short, analysts and policy-makers get the best of both worlds by using flow and agent-based models in tandem.

Government leaders around the world were almost universally stunned by the explosive spread of COVID-19 infection within their countries. Political considerations aside, their tardy recognition stems from limitations in most peoples’ ability to grasp changing rates of flow. Our inability to anticipate outcomes correctly is particularly acute in situations when flows accelerate (or decelerate) non-linearly, as they do in the spread of viral infections. Cognitive psychologists have demonstrated this problem in experiments that require people to predict the net effects of water pouring into and out of sinks at different rates, businesses trying to balance orders and shipments to respond to changes in demand across supply chains, and heat accumulating globally to drive climate change. Pandemic models compensate for the deficiencies in our intuitions about viral dynamics. Most leaders did not heed the projections from these models. They acted too slowly to contain their countries’ outbreaks, leading to a much more difficult and costly struggle to arrest runaway contagion. Now that infection curves are flattening, leaders must play closer attention to pandemic models and their projections to guide their strategies for relaxing painful public health measures and restarting their economies. Of course, pandemic models are not silver bullets; they have dependencies and shortcomings of which decision-makers should be aware. My next article reviews these caveats, or the “fine print” for pandemic models.

©2019 by Richard M. Adler

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