WXML 2020 covidmodeling learning
guide
This page provides a survey of some influential
mathematical models being used to track and forecast COVID19 in
the United States. It was produced by Jarod Alper, an
Associate Professor of Mathematics at the University of Washington
whose research lies in algebraic geometry.
He is not an expert in mathematical epidemiology and has no
prior experience with infectious disease modeling.
This page was put together as a learning guide for a Spring 2020
Washington Experimental Mathematics Lab (WXML) project.
Last updated: May 6, 2020
Influential modeling groups (in no particular order)
 Imperial College London (ICL)

reports: from WHO Collaborating Centre
for Infectious Disease Modeling, MRC Centre for Global Infectious
Disease Analysis, Abdul Latif Jameel Institute for Disease and
Emergency Analytics, Imperial College London, Vaccine Centre,
London School of Hygiene and Tropical Medicine, UK
 ICL March 16 model
 ICL March 26 model
 ICL March 30 model
 Institute for Health Metrics and Evaluation (IHME)
 The IHME model has 4 components:
 identifying and processing covid data,
 a statistical model where for each location,
a curve fit is applied to population death
rates as a function of time since the death rate
exceeds a given threshold in the location
 predicting the time a given location reaches
this given threshold if it hasn't already, and
 an individual agestructured microsimulation
to model hospital usage.
 For the curve fit in (ii),
two different Sshaped (or sigmoid) curves were considered each depending on 3 parameters.
You can see the curves and change the parameter values in this
desmos graph.
 The model is regularlyupdated and provides predictions on
deaths and hospital usage in the US and Europe.
 projections,
paper,
faq,
update notes,
CurveFit documentation.
 Institute for Disease Modeling (IDM)
 IDM
research reports modeling transmission dynamics in
various locations including
King & Snohomish counties, Oregon and subSaharan Africa.
 King & Snohomish counties March 10 report:
Working paper – modelbased estimates of COVID19 burden in
King and Snohomish counties through April 7, 2020 by Klein, et al.
 Provides projections on infections and deaths through April 7
using a stochastic SEIR model with models parameters taken
from the scientific literature. To monitor hospital usage,
each model uses a discrete compartmentalized event care usage model.
 Not agestructured or spatially located but simulations
use a negative binomial transmission model with
heterogeneous parameter k = 0.54.
 Considers affects various social distancing measures
given as a percentage in the reduction of transmission rates
(usual, 25%, 50%, 75%)
 King county March 29 reports: Understanding the Impact of
COVID19 Policy Change in the Greater Seattle Area using Mobility
Data by Burstein, et al and Social distancing and mobility
reductions have reduced COVID19 transmission in King County, WA
by Thakkar, et al.
 Uses mobility data from Facebook Data For Good to determine
effect of policy changes. Namely, mobility data is used
determine (1) population flux between day & night,
(2) perecent difference in daytime residential occupancy,
(3) perecent difference in commuting, and
(4) percent difference in daytime nonresidential occupancy.
 Uses a stochastic SEIR model to estimate the effective
reproductive rate R_e through March 24
from case data. Unknown parameters (assumed to depend on time)
such as the transmission rate and reporting rate
(% of infections detected) are fit to case data.
 King county April 10 report: Physical distancing is working and still needed to prevent
COVID19 resurgence in King, Snohomish, and Pierce
counties by Thakkar, et al.
 Uses similar method as the March 29 reports with updated data
to estimate the effective reproductive rate through March 30.
 King county April 21 report: Short report: Updated analysis of
COVID19 transmission in
King County, WA by Thakkar, et al.
 Uses similar method as the March 29 reports with updated data
to estimate the effective reproductive rate through April 9.
 King county April 29 report: Sustained reductions in transmission
have led to declining COVID 19 prevalence in King County, WA
by Thakkar, et al.
 Estimates the effective reproductive rate through April 15.
 Uses a stochastic SEIR model as in the March 29 report but
incorporates a case detection rate p_t (the probability an infected case is
reported) with independent values before and after March 10.
It also assumes that the transmission rate varies over time
with the structure inferred by the positive vs negative testing data.
Assumes that an unknown quantity z_t of infections were imported
to King County on January 15. The model determines the unknown
parameters p_t and z_t by fitting them to the observed data.
 Global Epidemic and Mobility Project (GLEAM) with contributions from Northeastern

The effect of travel restrictions on the spread of the 2019 novel coronavirus
(COVID19) outbreak (preprint March 6, published April 24), Science,
by Chinazzi M., Davis J.T. Additional details can be found in the
supplement

Model description: individualbased, stochastic, and spatialbased model
based on a metapopulation network approach which subdivides
the world's roughly 200 countries into 3200 geographic subpopulations.
 Uses airline transportation data from Official Aviation Guide
(OAG) and International Air Transport Association (IATA), and
ground mobility from statistics offices for 30 countries on 5 continents.

Visualization dashboard for number of infections,
deaths, hospital and ICU beds needed in the US.
 GLEAM also offers:
 EpiRisk:
"EpiRisk is a computational platform designed to allow a
quick estimate of the probability of exporting infected
individuals from sites affected by a disease outbreak to
other areas in the world through the airline transportation
network and the daily commuting patters. It also lets the user to
explore the effects of potential restrictions applied to airline
traffic and commuting flows."
 GLEAMviz: a desktop application
allowing you to configure and run and analyze your own simulations.
 Centre for the Mathematical Modeling of
Infectious Diseases (CMMID) and
the London School of Hygiene & Tropical Medicine (LSHTM)
 COVIDProjections by Youyang Gu (YYG)

model details
 Uses a basic SEIR/SEIS model to simulate the covid epidemic in each
country/state/region. The model is neither agestructure
nor models hospital usage.
The parameters of the model are learned using
machine learning techniques attempting to minimize the error between
the actual data on the number of deaths (as reported by John Hopkins CSSE)
and their projections.

model summary with comparison to IHME

projections of their model

tracker of the basic reproduction number R_0 and number of infections
 Univisty of Texas COVID19 Modeling Consortium

Report: UT COVID19 Mortality Forecasting Model (April 16)
 Uses a statisticalcurve fitting approach to a fit a sigmoid curve
(in fact, the same curve as the IHME model depending on 3 parameters)
and a probabalistic error model to observed death rates. In comparison
to the IHME model, they "reformulated the approach in a generalized
linear model framework to correct a statistical flaw that leads
to the underestimation of uncertainty in the IHME forecasts."
 Uses local mobilephone GPS data from SafeGraph from each state to
quantify the effects of socialdistancing measures
 projections

reports and publications
 Los Alamos National Laboratory (LANL)
 model description and forecasets
 Uses a basic SIR model combined with a statistical process
where for each state, the model
attempts to determine/learn (1) the "growth parameter"
(i.e. transmissibility rate) as a function
of time (assumed to decrease over time) based on trends in the number of
new cases reported, (2) the case fatality rate determined
by the number of new deaths and reported
cases, and (3) the discrepancy between the actual and reported number
of cases/deaths as a function of time.
 In forecasting, the model assumes the case fatality rate to be constant
and that deaths are synchronous with a positive test.
 Columbia University

Severe COVID19 Risk Mapping :
This tool shows projections for hospital demand for each US county
and the expected date of peak capacity.

paper summarizing results: Flattening the curve before it flattens us:
hospital critical care capacity limits and mortality from novel
coronavirus (SARSCoV2) cases in US counties
 3 and 6 week projections from April 2, 2020

paper with modeling details
(contains model details)
Simulation of SARSCoV2 Spread and Intervention Effects in the
Continental US with Variable Contact Rates, March 24, 2020
 Uses a metapopulation SEIR model (similar to this
Science paper) with S, E, I, R components for each county incorporating
commuting and random movements of individuals using data from the US Census
Bureau on countytocounty commuting patterns. Transmissions are separated
into daytime (8 hours) and nighttime (16 hours).
 University of Minnesota

Model with links to video briefing,
slides, faq, infographic and technical documentation
 Estimates number of daily cases & deaths and ICU usage in Minnesota
and predicts peak usage. Examines impact of social distancing
(assumed to reduce transmission rates by 50%) and shelter in place
(assumed to reduce transmission rates by 80%).
 Uses an age and comorbidity structured compartamentalized SEIR
model.
 Geneva

COVID19 Epidemic Forecasting Dashboard
 "We calculate the growth rate of cumulative cases
(resp. deaths) between two days ago and today. If it's greater
than 5%, we use an exponential model to forecast the cumulative
number of cases (resp. deaths), and then derive the daily number of
cases (resp. deaths).
If it's less than 5%, we use an linear model instead."
 No other descriptive model documentation is provided but there
is a public GitHub repository.
Modeling papers (in no particular order)
 Kissler (Harvard), et al. Science paper

Projecting the transmission dynamics of SARSCoV2 through
the postpandemic period (preprint March 6, published in Science April 14)

Supplementary materials
 Explores the role of seasonality, length of immunity, crossimmunity
(with two other coronavirus HCoVOC43 and HCoVHKU1),
social distancing and adding critical care capacity to
model the transmission dynamics through 2025.
 Uses a compartmentalized deterministic SEIRS model with
gammadistributed waiting times.
 To estimate the incidence of each coronavirus,
used the percentage of positive tests multiplied by the populationweighted
proportion of influenzalike illnesses.
 homogeneous model: no agestructure, not spaciallylocated,
did not include school closings.
 Dandekar (MIT)Barbastathis (MIT) paper in PNAS

Quantifying the effect of quarantine control in Covid19 infectious
spread using machine learning article
(preprint April 3, 2020, published in PNAS)
 Uses a classical SIR approach with
"a neural network added as a non linear function approximator
(Rackauckas et al. 2020) informs the infected variable I in the SIR model.
This neural network encodes information about the quarantine strength
function in the locale where the model is implemented.
The neural network is trained from publicly available infection and
population data for Covid19 for a specific region under study".
 The neural network is a twolayer fullyconnected neural network
which trains Q(t), the percentage of the infected population I(t) that are in
quarantine at time t, so that Q(t)I(t) is the number of people in the
quarantine compartment. The funnction Q(t) is calculated from the
(S, I, R, T) vector by the formula Q = f o A_2 o f o A_1
where A_2:R^10 > R and A_1:R^4 > R^10 are affinelinear transformations
fitted to data,
and f is the nonlinear activation function ReLu. This is a rather small
neural network as it is training only 61 parameters.
 Compares their SIR + neural network controlling quarantine strength Q to a
classical SEIR approach (which does not incorporate
a compartment for quarantine)
 Giodano (Trento), et al paper in Nature Medicine

Modeling the COVID19 epidemic and
implementation of populationwide interventions in Italy
(published Aprill 22, 2020)
 Uses a compartamentalized SIDARTHE model; this is a variation of the
classical SIR/SEIR epidemiological model but here
S=susceptible, I=infected, D=diagnosed, A=ailing, R=recognized,
T=threatened, H=healed, E=extinct. This distinguishes between infected
individuals on whether they have been diagnosed and the severity.
 They run simulations using data from Italy modeling affects of
socialdistancing, populationwide testing ad contact tracing.
The CDC has has a very informative summary of
COVID19 forecasts and models
Comparison of models

COVID Projections Tracker by David Yu
 Compares the various predictions of
IHME and LANL based on the update date of their projections.
 UMass Amherst Reich Lab

COVID Forecast Hub ensemble :Unifies multiple models to give more accurate predictions.
 Updated every Monday, this
provides an interactive visualization comparing the most recent projections
from various teams including ICL, IHME, UT, GLEAM, LANL
and YYG.

Where The Latest COVID19 Models Think We're Headed — And Why They Disagree
by FiveThirtyEight
 With help from UMass Amherst's Reich Lab,
compares predictions from IHME, Columbia, GLEAM, UT, MIT
and LANL.
Recommended books
 Keeling and Rohani,
Modeling infectious diseases in humans and animals
 This is one of the standard texts in
mathematical epidemiology.
 Kiss, Miller, and Simon, Mathematics of Epidemics on Networks:
From Exact to Approximate Models
Online COVIDrelated lectures and courses

Imperial College London online course
 Recommend watching Weeks 13
and taking a look at the suggested readings
 UW seminar series:
Exploring and understanding the COVID19 pandemic

Network Epidemiology Online Workshop Series
offered by the University of Maryland
 Lectures/podcasts:

Mathematics of the Corona outbreak (March 13, 35 min), Tom Britton (Stockholm)

COVID19: The Exponential Power of Now (March 18, 1 hr), Jewell (Berkeley) hosted by MSRI

COVID Math: trailing indicators, leading indicators,
math models, and stats models (April 17, 1 hr),
Flaxman (IHME) hosted by UW Math
 UW panel:
Emerging from COVID19: A GroundZero Perspective (April 23, 1 hr)
with Keith Jerome (Head of UW Virology), Elizabeth Halloran (UW Biostats) and
Jeffrey Duchin (UW Medicine).

TED podcast
with Adam Kucharski (LSHTM)

NYU seminar on "Introduction to Infectious Disease Modeling"
 Fun youtube videos:
Basic data and visualizations
Some educational and informative visualizations

Modeling COVID19 Spread vs Healthcare Capacity
by Alison Hill (Harvard),
 has links to other recommended covid19 apps in
'about' section
 An
informative model
from NY Times regarding impacts of social distancing
 article by Nicholas Kristof and Stuart A. Thompson
 model created with Gabriel Goh, Steven De Keninck,
Ashleigh Tuite and David N. Fisman

An interactive visualization
by John BurnMurdoch
 tracks the exponential spread of COVID19
by country/state

COVID19 Scenarios
 by Richard Neher, et al
at the Biozentrum, University of Basel
 basic homogeneous SEIR model with compartments
including various hospital stages.
 individual model parameters can be varied.
 Impacts of social distancing measures are considered.

Washington post covid simulator
 article by Harry Stevens
 Most viewed Washington post article ever.
 Covid Trends
 By Aatish Bhatia in collaboration with Minute Physics.
 Visualization of spread by country and US state.

Breaking the wave
 by Jon McClure, a Reuters Graphics editor
 explains how the rate of increase in covid deaths is decreasing

Forecasting scurves is hard by Constance Crozier
 Several interactive notebooks
by YongYeol Ahn