1. Mathematical models of disease transmission can inform public health policy responses to pandemics like COVID-19. 2. Early modeling of COVID-19 in Michigan showed that early and sustained social distancing interventions would be necessary to flatten the curve and prevent overwhelming the healthcare system. 3. These models also demonstrated that relaxing social distancing restrictions too early could lead to a second wave of infections and that some level of continued social distancing may be required until herd immunity is achieved through vaccination or widespread infections.

1. Joseph Eisenberg, PhD, MPH
Department of Epidemiology
School of Public Health
University of Michigan
Epidemiology of
Pandemics: How
Mathematical Modeling
Informs Public Health
Policy
COVID -19

2. Overview  What do we know about COVID-19?
 Transmission modeling: The basics
 Examples of models that have
informed policy
 COVID-19: Lessons from models

3. COVID-19
Disease
Trajectory
Exposure
18 d
Symptoms
Death
25 d Hospital
Dischar
ge
Incubation
5 d (2-14
d)
When do people become
infectious?

4. The continuum of infection in a
population
Deaths
Severe Cases
Symptomatic Cases (fever)
Mild or Asymptomatic
Cases
Cases
Detected
DIAGNOSED INFECTIONS: TIP OF THE ICEBERG
Spectrum of Coronavirus Cases, Diagnosed and Undiagnosed

5. Case fataily rate varies
across countries:
● Timing of epidemic
● Who /how many
tested
● Demography (age,
comorbidties)
● Healthcare capacity
COVID-19
Case-Fatality
Rate by Country
(Among Those with
500+ Confirmed Cases
and 1+ Deaths)

6. Greater
Mortality in
Older
Individuals
with
Conditions

7. How Many
People in the
US Are
Infected?
How do we
estimate
total cases?
Reported cases and
deaths (as of
4/4/2020)

8. Thinking through ‘How many people are
infected?”

9. One death today
means...~4 weeks ago there were 100 infections (assuming case
fatality rate 1/100)
or 1000 infections (assuming case fatality rate 1/1000)
1 Death

10. Assuming a doubling time of 1 week
1 death today means 1600-16,000
infections today.
(Assume 100-1000 infections per death 4 weeks
ago and a doubling time of 1 week)
1 Death

11. Assuming a doubling time of 1 week
Example (Michigan): 500 deaths today
indicates 50,000 to 500,000 estimated
cases
(Assuming 100 infections per death 4 weeks ago and a doubling
time of 1week)
50,000
500
deaths
100,000
200,000
400,000
800,000

12. Probability of
infection given
contact
Duration of
infectious
period (12 d)
Social distancing
can diminish
contact
Opportunity
to intervene
Contact rate
among
individuals
Pathogen R0
Measles 12-18
SARS 2-5
SARS-COV-2 2-3
Seasonal
Influenza
1-2
MERS <1
What Drives
Transmission?
Reproductive number, a
measure of spread, is a
product of three terms
Reproductive number
(R0)

13. How Does
Spread
Happen in
the Real
World?
Idealized situation Data from SARS
pandemicWuhan: Ro = 2.2
Epidemic doubles every 7 days
“A” infects 33
people, most
of whom
don’t infect
others

14. What Are We
Up Against?
▪ 80% of cases are mild or
asymptomatic
▪ Pre-symptomatic transmission is
likely
(maybe 2 days prior to symptoms)
▪ Dose matters
▪ 60% of population will get sick with
no interventions (based on R0=2.5)
Therefore, finding and quarantining cases will
be difficult.
Social distancing is our best option.

15. Goal:
Flatten the
Curve

16. Three Levels of Preventive
Measures
▪ Infection prevention and
control in the community
▪ Social distancing measures
▪ Travel-related measures
▪ Screening (questionnaires, testing)
▪ Contact tracing and case finding,
isolation and quarantine
▪ Surveillance
▪ (Vaccine 12-18 months away)
▪ Supportive/intensive care (Medication in
clinical trials)

17. But, Does
Social
Distancing
Work?
SCALE MATTERS
 Measles transmission
driven by school
 Mixed evidence of the
success in school
closings for Influenza
A
Concerts,
Sporting
Events
Schools,
Daycare
Centers
Restaurants,
Home
Gatherings

18. Modeling
transmission
helps us
understand
details about
how disease
spreads
Germ theory leads to the theory of mass
action
Sir Ronald
Ross
Contact with an
infectious
individual
Recover and become
susceptible again
Susceptibl
e
Infected

19. Disease
spreads by
direct contact
with infected
Susceptib
le
(S)
Infected
(I)
Removed
(R)

Recovery and
removal from
transmission
process
Furthered
framework of
transmission (SIR
model)
When infected people
recover, they can become
immune

20. Rate of new
cases depends
on infectivity,
but also
susceptible and
infected
individuals
𝒅𝑰
𝒅𝒕
=SI
Probability of
infection given
contact
Contact rate
among
individuals
Rate of
generatin
g new
cases
Product of
infectious
X
susceptibleTransmission
rate
X

21. What can we learn
from models of other
diseases?

22. Models help
us
understand
herd
immunity
Vaccinated
Person
Direct
protection
(individual
effect of
intervention)
Indirect
protection
(community
effect of
intervention)
Infectious
Person
B
A
Need a threshold of vaccine
coverage to create herd
immunity

23. Models help us
understand
herd protection
from other
interventions
(not vaccine
related)
Indirect
protection
(community
effect of
intervention)
Direct
protection
(individual
effect of
intervention)
Susceptible
Infectiou
s
As opposed to vaccines,
bed nets only have effect
while used.

24. Models help us
think about
policy
implications
Rubella vaccination. High-risk group:
reproductive-age women
Herd immunity (all children) is optimal,
but if not feasible, vaccinate high-risk
group
Water treatment. High-risk group: HIV+
individuals
In-home water treatment of high-risk
group is optimal (cheaper) but if disease
spreads from general population to
high-risk group then improving

25. Models help us
think about
policy
implications
Contact tracing for sexually
transmitted diseases. Small high
risk groups (core) can drive
transmission in low risk groups
(non-core)
Targeted tracing will be more
effective

26. Models help us
think about
policy
implications
Determining coverage threshold for building
latrines. High-risk households create
community-level risks.
High coverage minimizes contamination to
community
Communi
ty
Environm
ent
No Latrine
Low-quality
latrine
High-quality
latrine

27. Building a
COVID-19
transmission
model with the
natural history of
the disease

28. Goals for
COVID-19
transmission
model
1. Forecasting to assess burden on
hospitals
2. Quantifying the delay between onset
of intervention and changes in
reported cases/deaths
3. Estimating the type and length of
social distancing needed to flatten the
curve
a) How that will vary as a function of
when intervention begins
4. Developing an adaptive strategy to
keep cases and mortality minimal

29. Models built
iteratively with
greater
complexity
Initial step is crude characterization of
social distancing:
Minimal, Moderate, Intensive
Add more detail in next iterations:
Characterize close contact by size and
duration of gathering
Larger scale events (concerts, sporting
events)
Medium scale events (subways, dorms,
nursing homes)
Small scale gatherings (restaurants,
households)

30. Initial
projections for
Michigan
Estimating time of
peak cases is
difficult when in
exponential phase
of outbreak
Michigan COVID-19 Modeling Dashboard
https://epimath.github.io/covid-19-modeling/#current-forecasts

31. Early
intervention
will flatten the
curve
65% reduction
in contact
Michigan COVID-19 Modeling Dashboard
https://epimath.github.io/covid-19-modeling/#current-forecasts

32. Late
intervention
will not
change the
curve much
65% reduction
in contact
Michigan COVID-19 Modeling Dashboard
https://epimath.github.io/covid-19-modeling/#current-forecasts

33. Strong social
distancing
relaxed too
soon simply
delays
outbreak
90% reduction
in contact
Michigan COVID-19 Modeling Dashboard
https://epimath.github.io/covid-19-modeling/#current-forecasts

34. Models can help us understand disease
transmission dynamics
Models help us think through policy
implications
Importantly, how much herd immunity is
needed?
• In homogeneous social setting – 60%
of population
• In a more realistic setting with social
structure – less than 60%
Assuming R0 = 2.5
Conclusions

35. Social distancing delays the epidemic
Once social distancing is relaxed people
are again at risk
Second peak possible in fall
Will seasonal changes help us?
Flattening the curves relies on
Continuation of social distancing
Slowly building herd immunity
Conclusions

36. Conclusions Developing an adaptive social
distancing strategy
1. Wait until we observe flattening
with current level of social
distancing
2. Relax social distancing and observe
response
a) Requires good testing and
contact tracing
b) Need to wait to see effect (1 – 4
weeks)

37. Future directions Building individual-based
models (IBM) to better
characterize contact patterns in
different work sectors
Develop guidelines for low-risk
conditions in different work
sectors and public gatherings