Vaccination and population dynamics of an epidemic

Vaccination and population dynamics of an epidemic

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I'm trying to figure out how should a vaccination model be built to correlate with population density, and I'm having problems to understand meanings of the results I receive when I apply theory on specific data I'm provided with.


The initial phase of an outburst of a disease can be described by an exponential growth model. The relevant equation is:

$(1)frac{dI}{dt}=eta n(1-q)I-mu I$ where:

$n$ = the population density. Let us measure it in units of $km^{-2}$.

$I$ = the density of already infected individuals in the population; measured in the same units as $n$.

$q$ = the fraction of the population that is immune to the disease, either naturally of due to vaccination. Consequently, $1-q$ is the fraction of the population that is susceptible, i.e., at risk of getting infected. $q$ is a pure number between $0$ and $1$, and has no units.

$eta$ = is the transmission rate of the disease. It measures how easily and quickly the disease can be transmitted from an infected individual to an non-infected susceptible individual. $eta$ includes within it both the rate at which encounter between infected and non-infected individuals occur, and the probability that such an encounter would result in actual transmission of the disease. $eta$ has dimensions of $frac{1}{time imes density^{2}}$, so let us measure it in units of $week^{-1}km^{4}$.

$mu$ = the rate at which infected individuals are eliminated from the group of infected individual, either because they recover, or because they die. $frac{1}{mu}$ is the average duration of the infection, i.e., the average time that an individual remains infected before it either recovers or dies. Let us measure $mu$ in units of $week^{-1}$.

This equation derives from the differential equation $(2) frac{dN}{dt}=rN$ where $r$ is called instantaneous rate of increase. It is easy to see that $I$ from equation $(1)$ is equivalent to $N$ from equation $(2)$ and therefore, $r$ for equation $(1)$ will be $(3) r=eta n(1-q)-mu$. When we look at equation $(3)$, we see two factors:

$eta n(1-q)$ - A positive factor(ii) $mu$ - A negative factor

Minding the above, when $r=0$, there is no increase in population(iii). From this, we can compute $q_{0}$ which is the minimum fraction of vaccinated/immune individuals in the population that is required in order to prevent the disease from spreading. From equation $(3)$ we can figure out that $q_{0}=1-frac{mu}{eta n}$. Just as $q$, $q_{0}$ is a pure number between $0$ and $1$.

Welcome to the desert of the real (my question):

Suppose we compare two countries with the following data:

  1. Israel: $n=347km^{-2}$, $eta=0.0015week^{-1}km^{4}$, $mu=0.25week^{-1}$
  2. Finland: $n=16km^{-2}$, $eta=0.0015week^{-1}km^{4}$, $mu=0.25week^{-1}$

When we look for $q_{0}$ for Israel we see that $q_{0}(Israel)=1-frac{0.25}{0.0015 imes347}=0.52=52$% while for Finland we see that $q_{0}(Finland)=1-frac{0.25}{0.0015 imes16}=-9.42=-942$%. Assuming that we've got correct data in the first place, $q_{0}$ is a negative pure numbers which is not between $0$ and $1$.

  1. Do such, and similar results make any sense at all? Especially when they are not between the defined boundaries of the variable.

  2. If they do make sense, what does it mean getting a negative results? How should it affect my vaccination policy?


(i) Taken from my Populations Ecology lecture slides

(ii) Positive when looking at it from the epidemic point of view

(iii) Of infected individuals

I think it does make sense - with a population density for finland that is so low, the disease with such a low beta cannot communicate to enough people to propagate.

The number of people who have this disease will be fewer each week. I think this makes sense because at 16 / km^2, you can expect that practically nobody will ever see each other.

This is a flawed model though because it assumes that the mean density is uniform. In a city like Helsinki (2,800 / km^2) you would expect the disease to get caught by nearly everyone in just a week.

Helsinki: n = 94.5%

In Lapland (which has a population density of less than 2/km^2) , the transmission rate (beta) of 0.0015 translates to 0.003 incidents per week. This is not a terribly catchy disease, you probably have to kiss someone, wear their clothes, or eat off their plate to get it. With only 2 people per km^2 the chances of this happening appear to be poor, though even here families tend to get the disease and the model breaks down.

So to sum up, the model is consistent within itself, BUT it is a baby model and makes some broad assumptions that do not help it describe the dynamics of the disease in a national scope or in a highly detailed scope. It probably describes the chances balls will collide in a box as well as disease spreading.

Impact of cross-protective vaccines on epidemiological and evolutionary dynamics of influenza

Large-scale immunization has profoundly impacted control of many infectious diseases such as measles and smallpox because of the ability of vaccination campaigns to maintain long-term herd immunity and, hence, indirect protection of the unvaccinated. In the case of human influenza, such potential benefits of mass vaccination have so far proved elusive. The central difficulty is a considerable viral capacity for immune escape new pandemic variants, as well as viral escape mutants in seasonal influenza, compromise the buildup of herd immunity from natural infection or deployment of current vaccines. Consequently, most current influenza vaccination programs focus mainly on protection of specific risk groups, rather than mass prophylactic protection. Here, we use epidemiological models to show that emerging vaccine technologies, aimed at broad-spectrum protection, could qualitatively alter this picture. We demonstrate that sustained immunization with such vaccines could—through potentially lowering transmission rates and improving herd immunity—significantly moderate both influenza pandemic and seasonal epidemics. More subtly, phylodynamic models indicate that widespread cross-protective immunization could slow the antigenic evolution of seasonal influenza these effects have profound implications for a transition to mass vaccination strategies against human influenza, and for the management of antigenically variable viruses in general.

Influenza is a major disease of humans and animals (1). Current influenza vaccines induce immunity primarily against the variable viral surface antigen hemagglutinin (HA). Owing to ongoing evolution of HA—manifested on the population level as antigenic “drift” (2)—current vaccines must be reviewed semiannually, in anticipation of the upcoming winter influenza season in the Northern and Southern hemispheres (3). There is evidence that current vaccines can elicit a degree of herd immunity (4 ⇓ ⇓ –7), even when imperfectly matched (8) however, ongoing antigenic drift renders this effect too short-lived for any lasting epidemiological impact. Similarly, vaccination for pandemic response is limited to a largely reactive function that can only be fully initiated once a pandemic virus has emerged (9).

There is therefore increasing interest in developing vaccines targeting viral proteins more conserved than currently targeted HA epitopes (10 ⇓ ⇓ ⇓ –14). By inducing immunity against different subtypes of influenza, and different strains of the same subtype, such vaccines do not require knowledge of what strain is emerging they could thus provide better pandemic and epidemic mitigation than antiviral drugs or social distancing (15), by permitting long-term suppression of transmission. In doing so, these vaccines may afford qualitatively unique opportunities concerning the epidemiology and evolution of influenza, which we explore here.

There are many different cross-protective vaccine candidates under study (10 ⇓ ⇓ ⇓ –14). Such vaccines do not necessarily prevent infection, but can be effective in reducing viral shedding, and can greatly reduce morbidity and mortality in animal models (10, 11). Fig. 1A shows recent experimental results (11) of a vaccine based on two conserved viral components, the matrix protein M2 and the nucleoprotein NP derived from an H1N1 strain. Vaccinated ferrets challenged with H5N1 exhibited a striking (order-of-magnitude) drop in nasal viral shedding over the course of infection. Further studies reporting comparable results are shown in Fig. S1 in SI Materials and Methods. If reducing viral shedding also reduces infectiousness (Table S1), then studies such as these suggest a potential impact of cross-protective vaccines on limiting onward transmission.

Reduction in viral shedding and potential effects on transmission and epidemiology. (A) Results of cross-protective vaccination and lethal H5N1 challenge in ferrets. Adult ferrets, 6 per group, were immunized with three doses of DNA vaccine encoding influenza A matrix 2 and nucleoprotein (NP+M2) or influenza B nucleoprotein (B/NP) given intramuscularly at 2-wk intervals, followed by intranasal boosting with recombinant adenovirus vectors (rAd) expressing the same antigen(s) 1 mo later. Animals were challenged with 5 LD50 of A/Vietnam/1203/04 (H5N1) 6 wk after boosting, and virus titers in nasal wash samples from days 1, 3, 5, 7, and 9 after challenge determined by 50% egg infectious dose (EID50) assay. Mean virus titers are shown ±SEM. The NP+M2 group differs significantly from the B/NP group at all times, (P < 0.05 by one-way ANOVA). Note that the ferret model shows susceptibility to influenza infection, similar symptoms to those in humans, and is transmission-competent. For additional details, see ref. 11. (Reprinted from ref. 11, Copyright 2009, with permission from Elsevier.). (B) Simulated outcome of pandemic emergence in a vaccinated population, taking R0 = 2, consistent with previous pandemics, and assuming that vaccinated individuals have transmission potential reduced by a proportion c.

Accordingly we refer to “cross-protection” as broadly including: (i) protection of vaccinated individuals by reducing viral shedding and/or morbidity and mortality, but not necessarily preventing infection, and (ii) offering this protection against different subtypes (e.g., H3N2 and H5N1), and against divergent strains of the same subtype (e.g., drift variants of H3N2). Most important in our context is the reduction in transmission that could result from vaccination.

The simple math of herd immunity

When a new infection enters a fully susceptible population, as SARS-CoV-2 infection did in early 2020, each infectious case on average infects R0 other cases. As the disease spreads, it leaves in its wake immunity to infection in some or all of those who have recovered. Moreover, once society recognizes the threat, measures can be put in place (masks, distancing, handwashing, movement restrictions, etc.) that may further hamper spread. At some time t when the disease has been spreading for a while, the average case in a population will infect R(t) others in that population, where usually R(t) < R0, thanks to the combined effect of immunity to infection that has accumulated, and control measures. As has become common knowledge thanks to the teachable moments of this pandemic, when each individual infects on average more than one other, the number of cases will rise, and when each individual infects on average less than one other, the number of cases will fall.

Vaccines can help reduce R(t). We often refer to a critical vaccine coverage f*, which is the proportion of randomly chosen individuals in the population that must be vaccinated to achieve R(t) < 1. If we have a vaccine that reduces transmission by a factor x, then we want to create the situation where Rvac(t) = (1 – xf*) Runvac(t) = 1 — that is, where the average case in a vaccinated population infects less than one additional person (we use an equals sign because we are trying to get the value of f* that just barely achieves this threshold any value of f > f* will create Rvac(t) < 1. If we rearrange that expression, we get />: the proportion that needs to be vaccinated is greater for large values of Runvac(t) and for smaller values of x. Textbooks often write that />, which makes sense because as we noted in the previous paragraph, R(t) < R0 in most cases, because of immunity and control measures. Therefore if we can control spread without any other immunity or control measures, we can control it even better with the help of those things. When we think about “getting back to normal” we are thinking of controlling the disease with vaccination alone, and not the control measures we have imposed.


Electronic supplementary material is available online at

Published by the Royal Society. All rights reserved.


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Biological Sciences and the COVID-19 Pandemic

New webinar from The Scientist: SARS-CoV-2, the coronavirus behind the COVID-19 pandemic, has infected hundreds of thousands of people. In response, the scientific community sprang into action to uncover its sequence, structure, and potential biological mechanisms. This information helps guide public health policies as well as drug, vaccine, and diagnostic test development. In this webinar, Biological Sciences Professor Emily Troemel and David Wang of the Washington University School of Medicine in St Louis discuss the emergence of SARS-CoV-2, diagnostic tests and possibilities for treatment.

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Throughout this paper, we showed the possibilities to investigate practical and intriguing questions using a within-host viral dynamic model and an age-structured network model. The advantages of using explicitly within-host dynamics are the availability of experimental data, the possibility of conducting experiments to characterize transmission parameters, and the ability to provide high-resolution subject-specific responses to infection. The advantages of using an age-structured network model are its simple implementation, its representativeness for disease transmission, and the availability of the age-structured data. Therefore, immunological studies of infectious agents could be seamlessly integrated into studies of between hosts transmission, promoting evidence-based public health practices.

Long-term COVID-19 containment will be shaped by strength and duration of natural, vaccine-induced immunity

New research suggests that the impact of natural and vaccine-induced immunity will be key factors in shaping the future trajectory of the global coronavirus pandemic, known as COVID-19. In particular, a vaccine capable of eliciting a strong immune response could substantially reduce the future burden of infection, according to a study by Princeton researchers published in the journal Science Sept. 21.

A new study led by Princeton researchers suggests that the impact of natural and vaccine-induced immunity will be key factors in shaping the future trajectory of the global coronavirus pandemic, known as COVID-19. In particular, a vaccine capable of eliciting a strong immune response could substantially reduce the future burden of infection.

“Much of the discussion so far related to the future trajectory of COVID-19 has rightly been focused on the effects of seasonality and non-pharmaceutical interventions [NPIs], such as mask-wearing and physical distancing,” said co-first author Chadi Saad-Roy, a Ph.D. candidate in Princeton’s Lewis-Sigler Institute for Integrative Genomics . “ In the short term, and during the pandemic phase, NPIs are the key determinant of case burdens. However, the role of immunity will become increasingly important as we look into the future. ”

“Ultimately, we don’t know what the strength or duration of natural immunity to SARS-CoV-2 — or a potential vaccine — will look like,” explained co-first author Caroline Wagner, an assistant professor of bioengineering at McGill University who worked on the study as a postdoctoral research associate in the Princeton Environmental Institute (PEI).

“For instance, if reinfection is possible, what does a person’s immune response to their previous infection do?” Wagner asked. “Is that immune response capable of stopping you from transmitting the infection to others? These will all impact the dynamics of future outbreaks.”

The current study builds on Princeton research published in Science May 18 that reported that local variations in climate are not likely to dominate the first wave of the COVID-19 pandemic and included many of the same authors, who are all affiliated with the Climate Change and Infectious Disease initiative funded by PEI and the Princeton Institute for International and Regional Studies (PIIRS).

In the most recent paper, the researchers used a simple model to project the future incidence of COVID-19 cases — and the degree of immunity in the human population — under a range of assumptions related to how likely individuals are to transmit the virus in different contexts. For example, the model allows for different durations of immunity after infection, as well as different extents of protection from reinfection. The researchers posted online an interactive version of model’s predictions under these different sets of assumptions .

As expected, the model found that the initial pandemic peak is largely independent of immunity because most people are susceptible. However, a substantial range of epidemic patterns are possible as SARS-CoV-2 infection — and thus immunity — increases in the population.

“If immune responses are only weak, or transiently protective against reinfection, for example, then larger and more frequent outbreaks can be expected in the medium term,” said co-author Andrea Graham, professor of ecology and evolutionary biology at Princeton and an associated faculty member in PEI .

The nature of the immune responses also can affect clinical outcomes and the burden of severe cases requiring hospitalization, the researchers found. The key question is the severity of subsequent infections in comparison to primary ones.

Importantly, the study found that in all scenarios a vaccine capable of eliciting a strong immune response could substantially reduce future caseloads. Even a vaccine that only offers partial protection against secondary transmission could generate major benefits if widely deployed, the researchers reported.

F actors such as age and superspreading events are known to influence the spread of SARS-CoV-2 by causing individuals within a population to experience different immune responses or transmit the virus at different rates. “Our models show that these factors do not affect our qualitative projections about future epidemic dynamics,” said Bryan Grenfell , the Kathryn Briger and Sarah Fenton Professor of Ecology and Evolutionary Biology and Public Affairs and an associated faculty member in PEI. Grenfell is a co-senior author on the paper with C. Jessica Metcalf , associate professor of ecology and evolutionary biology and public affairs and also a PEI associated faculty member.

“As vaccine candidates emerge, and more detailed predictions of future caseloads with vaccination are needed, these additional details will need to be incorporated into more complex models,” Grenfell said.

The researchers used a simple model to project the future incidence of COVID-19 cases — and the degree of immunity in the human population — under a range of assumptions on host immune responses following natural infection or vaccination. The middle flowchart (above) corresponds to the simplest model used by the researchers and allows for the incorporation of these different immune assumptions. The model found that, after the pandemic peak, a substantial range of epidemic patterns are possible as SARS-CoV-2 infection — and thus immunity — increases in the population. In all scenarios, a vaccine capable of eliciting a strong immune response could substantially reduce future caseloads.

The study authors also explored the effect of “vaccine hesitancy” on future infection dynamics. Their model found that people who decline to partake in pharmaceutical and non-pharmaceutical measures to contain the coronavirus could nonetheless slow containment of the virus even if a vaccine is available.

“Our model indicates that if vaccine refusal is high and correlated with increased transmission and riskier behavior such as refusing to wear a mask, then the necessary vaccination rate needed to reach herd immunity could be much higher,” said co-author Simon Levin , the James S. McDonnell Distinguished University Professor in Ecology and Evolutionary Biology and an associated faculty member in PEI. “In this case, the nature of the immune response after infection or vaccination would be very important factors in determining how effective a vaccine would be.”

“When so much uncertainty in the underlying processes exists, it can be challenging to make accurate projections about the future,” Grenfell said. “We argue in this study that ultimately, a family of both simple and more complex models is the best way to proceed under these circumstances. Comparing the predictions of these models carefully and then coming up with a carefully averaged picture of the future — as with weather prediction — can be very helpful.”

One of the main takeaways of the study is that monitoring population-level immunity to SARS-CoV-2, in addition to active infections, will be critical for accurately predicting future incidence.

“This is not an easy thing to do accurately, particularly when the nature of this immune response is not well understood,” said co-author Michael Mina, an assistant professor at the Harvard School of Public Health and Harvard Medical School. “Even if we can measure a clinical quantity like an antibody titer against this virus, we don’t necessarily know what that means in terms of protection.”

“Studying the effects of T-cell immunity and cross-protection from other coronaviruses are important avenues for future work,” Metcalf said.

Additional authors on the paper include Rachel Baker , a PEI postdoctoral research associate Sinead Morris, a postdoctoral research scientist at Columbia University who received her Ph.D. in ecology and evolutionary biology from Princeton and Jeremy Farrar, director of the Wellcome Trust.

The paper, “Immune life-history, vaccination, and the dynamics of SARS-CoV-2 over the next five years,” was published online by Science Sept. 21. This work was supported by funds from the Natural Sciences and Engineering Research Council of Canada, the Life Sciences Research Foundation, the Cooperative Institute for Modelling the Earth System (CIMES) at Princeton University, the James S. McDonnell Foundation, the Digital Transformation Institute, the National Science Foundation, the US Centers for Disease Control and Prevention, and Flu Lab.

Study Data And Methods

Study Design

We used a simple mathematical model to estimate the population benefits of a vaccine against COVID-19. We considered vaccines with varying degrees of preventive benefit (transmission effect) and disease-modifying benefit (progression and mortality effect). We considered different assumptions regarding the speed of manufacturing/distribution (pace) and the extent of vaccine delivery (coverage)—two implementation parameters that are independent of vaccine clinical trial results. We also considered different background epidemic severities, as measured by the reproduction number ( R t ). Outcomes of interest—including total infections, deaths, and peak hospital or intensive care unit (ICU) use—were reported both on an absolute basis and as a percentage reduction from a “no vaccination” scenario during a six-month planning horizon. We initialized the simulation with a population size of 100,000 people, of whom 100 (0.1 percent) were exposed and 9,000 (9 percent) were recovered cases. 15 The model was implemented as a spreadsheet and parameterized and validated using population-average data inputs (see online appendix exhibit 1). 16

Compartmental Model

The SEIR (susceptible-exposed-infectious-recovered) model is one of the simplest deterministic, mathematical frameworks for portraying the trajectory of an infectious disease through an at-risk population. Briefly stated, the SEIR framework treats the process of viral transmission and disease progression as a sequence of transitions among a finite number of health states (or “compartments”). Transitions are governed by mathematical equations that capture both the transmission dynamics of the virus and what is known about the natural history of disease.

We adapted the classic SEIR framework in two important ways (appendix exhibit 4). 16 First, we divided the “infected” compartment into four distinct subcompartments to capture the increasing severity and resource use associated with more advanced COVID-19 disease: “asymptomatic,” “mild” (outpatient), “severe” (hospitalized), and “critical” (hospitalized in an ICU). Second, we introduced the possibility of vaccination by creating a parallel set of compartments to the ones described above. People receiving the vaccine moved from the “susceptible unvaccinated” state to the “susceptible vaccinated” state. From there, their progress to exposure, infection, recovery, and death was adjusted to reflect the transmission and disease-modifying benefits of the vaccine. This modeling device also permitted us to adjust the infectiousness of people who received an imperfect vaccine but who nevertheless became infected (that is, breakthrough infections).

Vaccine Efficacy

To capture the broad definition of vaccine efficacy in the FDA’s June 2020 guidance, we considered three different vaccine types (appendix exhibit 2) 16 : a preventive vaccine that decreases susceptibility to infection in uninfected people a disease-modifying vaccine that improves the course of disease in infected people, slowing progression, speeding recovery, reducing mortality, and decreasing infectiousness and finally, a composite vaccine that combines the attributes of both the preventive and disease-modifying vaccines. We set the efficacy for each of these attributes at 50 percent in the base case and examined ranges of 25–75 percent in sensitivity analysis. (For the recovery rate increase, the base-case value was 100 percent [that is, cutting recovery time in half] with a range of 75–150 percent.) We considered lag times between vaccine administration and when effects take hold ranging from fourteen days (representing a fast-acting, single-dose vaccine) to thirty days in the base case (representing a two-dose vaccine with administration thirty days apart and partial efficacy after the first dose) and forty-two days (representing a two-dose vaccine with no efficacy after the first dose). 17 , 18

Implementation Effectiveness

The challenges of vaccine development do not end once an effective vaccine is identified.

The challenges of vaccine development do not end once an effective vaccine is identified. The model includes two implementation measures: pace and coverage. Pace, the percentage of the population that could be vaccinated on a given day, is a measure of manufacturing and logistical preparedness. We assumed a base-case value of 0.5 percent for the pace parameter to approximate the daily rate of influenza vaccination in the US during the peak period of vaccination efforts each fall. 19 This reflects our assumption that although a COVID-19 vaccine may need to be administered in two doses, the urgency of the pandemic may prompt sponsors to bring production and distribution to scale at twice the rate of the influenza vaccine. Given the uncertainty surrounding these assumptions, we considered alternative values ranging from 0.1 percent to 2 percent in sensitivity analysis. We defined coverage as the percentage of the population ultimately vaccinated—a measure of public acceptance and the success of public health efforts to make vaccines available to all who desire them. We used a base-case value of 50 percent (range, 25–75 percent), reflecting recent US polling data on vaccine acceptability. 20 At a daily pace of 0.5 percent, it would take 0.5/0.005 = 100 days to achieve a 50 percent coverage goal.

Epidemiology And Natural History

We defined three epidemic severity scenarios: a base case with a reproduction number ( R t ) of 1.8, a best case ( R t = 1.5) representing strict adherence to social distancing and other preventive best practices, and a worst case ( R t = 2.1) reflecting the higher risks associated with winter weather and greater indoor activity. We also report results for R t = 1.2 in the appendix. 16

Input data on the development and natural history of COVID-19 (including incubation times likelihood of symptoms and rates of progression, recovery, and fatality) were obtained primarily from modeling guidance issued by the Centers for Disease Control and Prevention (CDC) and the Office of the Assistant Secretary for Preparedness and Response in the Department of Health and Human Services, supplemented by published literature. 21 We attempted to use the most current input data available. However, as clinical care and outcomes improve, as testing services magnify, and as the COVID-19 pandemic expands its demographic reach, our analysis will require adjustment and updating. Specifically, hospitalization and mortality rates are improving as the pandemic is controlled among the elderly and extends its reach to younger populations. Recognizing how quickly these statistics are evolving, we deliberately focused our attention in this analysis on infections, not deaths.

Appendix exhibit 1 documents all inputs and sources. 16


Similar to any model-based analysis, our evaluation has important methodological limitations. First, we assumed a model of homogenous mixing. Although this simplified the underlying mathematics, recent evidence suggests that spikes in local positivity—and the resultant protective immunity—may be attributable to spatially correlated, small group gatherings. Vaccine hesitancy may also vary by setting and other demographics. Future refinements might consider more complex geospatial or age-based mixing assumptions.

Second, we did not stratify vaccine deployment or coverage scenarios across different at-risk and vulnerable populations, as suggested recently by the National Academies of Sciences, Engineering, and Medicine’s Framework for Equitable Allocation of COVID-19 Vaccine. 22 To some extent, sensitivity analyses on R t might serve as a surrogate for a stratified assessment of outcomes across communities with different epidemic severities. Furthermore, our framework did not allow for differential prioritization or uptake among groups at higher risk for hospitalization and death. Published data used to populate the model were necessarily taken from early in the pandemic course. Additional evidence (for example, age-adjusted outcomes, new strategies for COVID-19 clinical care, geographic case clustering, and patterns of vaccine hesitancy and acceptance among the public) may permit the model to be stratified by age or other dimensions and updated for risk for complications and death at the individual level. 23

Third, we assumed constant rates of transition from one model compartment to the next. This produced exponentially distributed residence times—time spent in a given state can be quite long, even if the mean duration is short—and could have biased the analysis against prevention and in favor of rapid implementation. As better data on the natural history of disease emerge, it may be possible to address the problem using multiple sequential compartments.

Finally, our base-case analysis restricted attention to a six-month horizon. Although we also report projections over the course of twelve months, this should be interpreted with caution, as waning immunity after disease and vaccine durability remain ongoing concerns. 24

A comprehensive description of the model, its parameters, governing equations, and input data values is in the appendix. 16


In circumstances in which vaccine production is delayed due to technological or logistical barriers, as seen with the pH1N1 vaccine, it is critical to have a good estimate of the timing of the epidemic peak before making policy decisions on vaccination strategies. Careful modeling may provide decision makers with estimates of these effects before the epidemic peak to motivate production efficiencies and inform policy decisions. Integration of real-time surveillance data with mathematical models is paramount to detect early upswings in illness activity heralding an epidemic peak and to enable public health to optimize the community benefits from proposed interventions before that occurs.