Slide 1 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Infectious disease epidemiology
and mathematical models
in the context of Public Health
Hans-Peter Duerr
Kurs Infektionsepidemiologie, Studiengang Public Health, Master of Science, Universitätsklinikum Düsseldorf
Slide 2 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Program Slide Lesson 1 - Infectious diseases - how they emerge and disappear
29 Lesson 2 - SARS 2002/2003: why modeling, and what is a mathematical model?
36 Exercise 1 - Design a mathematical model yourself: the bacterial growth curve
42 Exercise 2 - Homemade solution: solve the model numerically (bacterial growth curve with Excel)
45 Lesson 3 - Deterministic models: SIR-model, basic reproduction number R0
55 Exercise 3 - Be professional: simulate an epidemic with professional algorithms
63 Exercise 4 - Harvest: proportion of susceptibles after an epidemic infection
66 Exercise 5 - Think longterm: the influence of time and demography
69 Lesson 4 - Vaccination: final size of an epidemic, critical vaccination coverage
81 Exercise 6 - Predict: how many newborns to vaccinate? (critical vaccination coverage)
93 Lesson 5 - SIR, add-ons: extending SIR to SEIR, SEIRS, etc
99 Exercise 7 - Design: specific models for specific diseases (Extending the SIR to SEIR, SEIRS, etc.)
101 Lesson 6 - Interventions: pandemic influenza preparedness planning using InfluSim
103 Exercise 8 - Be an intervention planner: control an influenza pandemic (using InfluSim)
109 Lesson 7 - Stochastic Models: from theory to reality, the epidemic as a random event
157 Lesson 8 - "Super-"reality: The role of superspreaders
Slide 3 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Literature selection
• Infectious Disease Epidemiology: Theory and Practice. Neil Graham. Jones and Bartlett Publishers, Inc.
• Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation (Wiley
Series in Mathematical and Computational Biology), O. Diekmann and J. A. P. Heesterbeek.
• Epidemic Models: Their Structure and Relation to Data (Publications of the Newton Institute). Denis Mollison
Slide 4 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
National online ressources
www.promedmail.org www.destatis.de
www.gbe-bund.de
• CDC: Traveler’s Health:
http://www.cdc.gov/travel/
• Emerging Health Threats Forum
(EHTF)
http://www.eht-forum.org/
• GIDEON: Global Infectious
Disease & Epidemiology Network:
http://www.GIDEONonline.com
Slide 5 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
WHO ressources
• The Weekly Epidemiological Record
(WER): www.who.int/wer/en/
• WHO Outbreak News:
http://www.who.int/csr/don/en/
• WHO Weekly Epidemiological
Record - WER
http://www.who.int/wer/
• WHO Report on Infectious Diseases
2000: Overcoming Antimicrobial
Resistance
http://www.who.int/infectious-
disease-report/2000/
http://gamapserver.who.int/GlobalAtlas/home.asp
http://www.who.int/csr/outbreaknetwork/en/
Slide 6 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Infectious diseases in ICD-10
Schlüssel Text
A00-B99 Bestimmte infektiöse und parasitäre Krankheiten
C00-D48 Neubildungen
D50-D89 Krankheiten des Blutes und der blutbildenden Organe sowie
bestimmte Störungen mit Beteiligung des Immunsystems
E00-E90 Endokrine, Ernährungs- und Stoffwechselkrankheiten
F00-F99 Psychische und Verhaltensstörungen
G00-G99 Krankheiten des Nervensystems
H00-H59 Krankheiten des Auges und der Augenanhangsgebilde
H60-H95 Krankheiten des Ohres und des Warzenfortsatzes
I00-I99 Krankheiten des Kreislaufsystems
J00-J99 Krankheiten des Atmungssystems
K00-K93 Krankheiten des Verdauungssystems
L00-L99 Krankheiten der Haut und der Unterhaut
M00-M99 Krankheiten des Muskel-Skelett-Systems und Bindegewebes
N00-N99 Krankheiten des Urogenitalsystems
O00-O99 Schwangerschaft, Geburt und Wochenbett
P00-P96 Zustände, die ihren Ursprung in der Perinatalperiode haben
Q00-Q99 Angeborene Fehlbildungen, Deformitäten und
Chromosomenanomalien
R00-R99 Symptome und abnorme klinische und Laborbefunde, die
anderenorts nicht klassifiziert sind
S00-T98 Verletzungen, Vergiftungen und bestimmte andere Folgen
äußerer Ursachen
V01-Y98 Äußere Ursachen von Morbidität und Mortalität
Z00-Z99 Faktoren, die den Gesundheitszustand beeinflussen und zur
Inanspruchnahme des Gesundheitswesens führen
Schlüssel Text
A00-A09 Infektiöse Darmkrankheiten
A15-A19 Tuberkulose
A20-A28 Bestimmte bakterielle Zoonosen
A30-A49 Sonstige bakterielle Krankheiten
A50-A64 Infektionen, die vorwiegend durch
Geschlechtsverkehr übertragen werden
A65-A69 Sonstige Spirochätenkrankheiten
A70-A74 Sonstige Krankheiten durch Chlamydien
A75-A79 Rickettsiosen
A80-A89 Virusinfektionen des Zentralnervensystems
A90-A99 Durch Arthropoden übertragene Viruskrankheiten
und virale hämorrhagische Fieber
B00-B09 Virusinfektionen, die durch Haut- und
Schleimhautläsionen gekennzeichnet sind
B15-B19 Virushepatitis
B20-B24 HIV-Krankheit [Humane Immundefizienz-
Viruskrankheit]
B25-B34 Sonstige Viruskrankheiten
B35-B49 Mykosen
B50-B64 Protozoenkrankheiten
B65-B83 Helminthosen
B85-B89 Pedikulose [Läusebefall], Akarinose [Milbenbefall]
und sonstiger Parasitenbefall der Haut
B90-B94 Folgezustände von infektiösen und parasitären
Krankheiten
B95-B97 Bakterien, Viren und sonstige Infektionserreger als
Ursache von Krankheiten
B99 Sonstige Infektionskrankheiten
ICD-10 Infectious diseases
Slide 7 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Blank page for documentation
Slide 8 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Blank page for documentation
Slide 9 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Slides Topic Lesson
Introduction: Infectious diseases - how they emerge and disappear
1
SARS 2002/2003: why modeling, and what is a mathematical model? (the example of bacterial growth)
2
Deterministic models: SIR-model, theory, basic reproduction number R0, epidemic vs. endemic case
3
Vaccination: final size of an epidemic, critical vaccination coverage
4
SIR, add-ons: extending SIR to SEIR, SEIRS, etc. 5
Intervention planning: pandemic influenza preparedness planning using InfluSim (can interventions do harm to a population?)
6
Stochastic Models: from theory to reality, the epidemic as a random event, the role of chance: be lucky or not
7
The role of superspreaders 8
Program
Le
sso
n 1
Slide 10 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Infectious diseases:
how they emerge and disappear
Slide 11 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Leishmaniasis
Climate change
Measles
Population growth
SARS
Mobility
Globalisation
Animals
Slide 12 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
http://vcolizza.googlepages.com
SARS 2002 / 2003 N
um
be
r in
fecte
d
0
1000
2000
3000
4000
20. March 4. April 19. April 4. May 19. May 3. June 18. June 3. July
China
Hong Kong Taiwan Remaining countries
2003
16:17
Slide 13 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Where did SARS come from?
Literatur: Li W, et al: Bats Are Natural Reservoirs of SARS-Like Coronaviruses. Science 2005.
Hotel Metropole,
Hong Kong
http://www.msnbc.msn.com/id/12371160/
June
2003
March
2003
2005
Slide 14 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Infectious diseases: a matter of opportunities
http://www.msnbc.msn.com/id/12371160/
Slide 15 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Opportunities for emerging infectious diseases
Mobility
Slide 16 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
How did we get rid of SARS?
Am
oy G
ard
ens B
lock E
Culling of Zibet-cats
Isolation of diseased people, quarantine
Travel restrictions, control
"stay-at-home"-policy,
face masks, etc.
Slide 17 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Diseases come and go
How they come How they go
SARS Bat → Cat → Human Control
AIDS Monkey → Human -
Plague Wild animals → Rodents
→ Fleas → Human
Humans die
(animals stay)
Ebola Bat → Ape, Gorilla → Human Humans die
Smallpox Cattle, Monkey, Cat, Mouse...
→ Human
Eradication by
vaccination
Measles ? Eradication not
yet achieved
Slide 18 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Why are infectious disease there at all?
Measles: Children who have been infected attain life-long immunity.
0 10 20 30 40 50 60 70 Years
susceptible infected immune - - Natural history of disease
another 10 days later Look at this family 10 days later
Slide 19 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Why are infectious disease there?
A kindergarden
10 days later
another 10
days later
some weeks
later
Measles always eliminate themselves
Slide 20 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Computer simulations make us understand
Slide 21 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
When did measles attack humans?
Small populations Large populations
…probably in ancient Mesopotamian civilizations 2000–
4000 years BC, when the critical population size for
measeles (about 300 000) has been exceeded. Lit.: Black FL 1966. Measles endemicity in insular populations: critical
community size and its evolutionary implication. J. Theor. Biol. 11, 207–211.
Slide 22 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Causality: Population density
Weltbevölkerung
0
1000
2000
3000
4000
5000
6000
7000
-1000 -500 0 500 1000 1500 2000
Jahr
Mio
. M
en
sch
en
Slide 23 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Opportunities for emerging infectious diseases
Mobility
0
1000
2000
3000
4000
5000
6000
7000
-1000 -500 0 500 1000 1500 2000
Jahr
Mio
. M
enschen
Population growth
Slide 24 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Culex
Phlebotomus
(Sandfly)
Emerging diseases: example Leishmaniasis
Cutaneous Leishmaniasis
Sandflies
Slide 25 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Causality: Climate change D
evia
nce
fro
m a
ver
age
1961
-19
90
Slide 26 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Opportunities for emerging infectious diseases
Mobility
Climate change
0
1000
2000
3000
4000
5000
6000
7000
-1000 -500 0 500 1000 1500 2000
Jahr
Mio
. M
enschen
Population growth
Industrialisation
→ CO2 →
Global Warming
Poverty
Mig
ration
Limited
Resources,
Water, Food,
Hygiene
Ecological system
Globalisation
Opportunities
Slide 27 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Blank page for documentation
Slide 28 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Blank page for documentation
Slide 29 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Slides Topic Lesson
Introduction: Infectious diseases - how they emerge and disappear
1
SARS 2002/2003: why modeling, and what is a mathematical model? (the example of bacterial growth)
2
Deterministic models: SIR-model, theory, basic reproduction number R0, epidemic vs. endemic case
3
Vaccination: final size of an epidemic, critical vaccination coverage
4
SIR, add-ons: extending SIR to SEIR, SEIRS, etc. 5
Intervention planning: pandemic influenza preparedness planning using InfluSim (can interventions do harm to a population?)
6
Stochastic Models: from theory to reality, the epidemic as a random event, the role of chance: be lucky or not
7
The role of superspreaders 8
Program
Le
sso
n 2
Slide 30 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Am
oy G
ard
en
s B
lock E
Country Cases Deaths Case fatality [%]
China 5327 349 7
Hong Kong 1755 296 17
Taiwan 665 180 27
Kanada 251 41 16
Singapur 238 33 14
Vietnam 63 5 8
USA 33 0 0
Philippinen 14 2 14
Deutschland 9 0 0
8355 906 10.8%
For comparison Malaria: ~300 Mio. Cases / Year,
~1 Mio Deaths / Year (predominantly children)
Example SARS 2002/2003
Slide 31 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Am
oy G
ard
ens B
lock E
SARS, initial spread
CDC, taken from http://en.wikipedia.org H
ote
l M
etr
opole
Slide 32 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Spread from Hotel Metropole
Ke
y: B
asic
re
pro
du
ctio
n n
um
be
r
Slide 33 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Problem Global Networks
e.g. between
Chicago and New
York 25.000
Passengers per day
Passengers per day
L. Hufnagel et al. 2004, PNAS 101: 15124-9 Ke
y w
ord
s:
Mo
de
l, N
etw
ork
s
Slide 34 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Prediction
Fig. 2. Global spread of SARS. (A) Geographical
representation of the global spreading of probable
SARS cases on May 30, 2003, as reported by the WHO
and Centers for Disease Control and Prevention. The
first cases of SARS emerged in mid-November 2002 in
Guangdong Province, China (17). The disease was
then carried to Hong Kong on the February 21, 2003,
and began spreading around the world along
international air travel routes, because tourists and the
medical doctors who treated the early cases traveled
internationally. As the disease moved out of southern
China, the first hot zones of SARS were Hong Kong,
Singapore, Hanoi (Vietnam), and Toronto (Canada), but
soon cases in Taiwan, Thailand, the U.S., Europe, and
elsewhere were reported. (B) Geographical
representation of the results of our simulations 90 days
after an initial infection in Hong Kong, The simulation
corresponds to the real SARS infection at the end of
May 2003. Because our simulations cannot describe the
infection in China, where the disease started in
November 2002, we used the WHO data for China.
Ke
y w
ord
: M
od
el p
red
ictio
n
L. Hufnagel et al. 2004, PNAS 101: 15124-9
Slide 35 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
What is a mathematical Model?
Model: tBtB 20
Growth from one generation to the next: 12 ii BB
Logari
thm
ic
Lin
ear
Problems of this approach: - only valid for initial growth
- too simple for describing complex processes
Example bacterial groth: bacteria divide 2 times per hour.
Duration between divisions D = 0.5 hours.
Rate of division = 2 per hour D/1
Slide 36 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 1:
preliminary considerations
E
xerc
ise 1
Draw into each graph the bacterial growth curve you would expect if
• … the before-
mentioned changes
occur simultaneously
• … the generation
time of the bacterium
was not 0.5h but 1h
• … the culture was
started with 10000
bacteria, rather than
1 bacteria
Target: intuitive prediction
of quantitative relations
Slide 37 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 1: iterative solution
of the model in Excel
Exerc
ise
1 (
File
"0
0_
ba
cte
ria
lGro
wth
.xls
", s
heet "g
en
era
tio
n t
ime")
=A2*tGen 1. Parameter:
"Bakt0"
2. Parameter: "tGen"
3. Parameter "multFaktor"
=Bakt0
=A2*tGen =C2*multFaktor
Complete cells B2 to C32
in file "00_Bakterienwachstum.xls", spreadsheet "generation time"
Slide 38 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 1
results V
erify
the p
relim
inary
consid
era
tions
of th
e p
revio
us s
lide
Exerc
ise
1 (
File
"0
0_
ba
cte
ria
lGro
wth
.xls
", s
heet "g
en
era
tio
n t
ime")
Slide 39 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Design models with differential equations
tB
dt
tdB~
"The speed by which the
total number of bacteria
B changes over time t…"
"…is proportional
to the individual
rate of division
…"
"…and proportional
to the number of
bacteria reproducing
at time t "
=
teconsttB ~
Total number of bacteria at time t
Example bacterial growth: bacteria divide 2 times per hour.
Duration between divisions D = 0.5 hours.
Rate of division = 2 per hour D/1
Integration Derivative
Slide 40 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Differential equations offer more…
Lin
ear
K
tB1
Previously: the bacterial culture grows indefinitely (unrealistic in a finite world)
Now: the culture cannot exceed a certain capacity K (realistic: test tube)
"The growth rate
approaches zero when
the bacterial culture
approaches the value of
the capacity (B(t)=K)."
teBKB
KBtB
~
00
0
)(
Slide 41 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Summary
• A mathematical Model is just a mathematical way to describe a process.
• Simple processes may be intuitively described "by hand"
• Differential equations are a useful tool to describe complex processes.
• Differential equations allow for describing dynamic processes by means of
interpretable parameters.
Other ways to
model
processes: (Galileos planet model.
In: Universal Dictionary
of Arts, Sciences, and
Literature, Plates, Vol.
IV (London 1820) )
Slide 42 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 2:
preliminary considerations
Assume A) a bacterium which, under optimal conditions, reproduces 2 times per
hour (per capita-reproduction rate=2/h) and
B) a volume which can harbour at maximum 1,000,000 bacteria.
• Fill in the numbers
for the upper and
lower bounds of
each axis into the
white boxes
• Draw a qualitative
curve for the per
capita-reproduction
rate and the num-
ber of bacteria over
time, B(t). Where is
the inflection point
of B(t)?
E
xerc
ise 2
Nu
mbe
r of bacte
ria
B(t
)
Per
ca
pita-r
ep
rod
uctio
n r
ate
Aim: intuitive prediction
of a dynamic process
Time
Slide 43 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 2:
spreadsheet solution
Exerc
ise
2 (
"00_ba
cte
rialG
row
th.x
ls",
sheet "d
iffe
ren
ce
equ
ation
s")
1. Parameter: "deltaT"
2. Parameter: "Bak0"
3. Parameter "lambda"
=lambda
=C2+(A3-A2)*B3
=lambda*C2*(1-C2/kapazitaet)
Complete cells A3 to E1000
in file "00_bacterialGrowth.xls", spreadsheet "difference equations")
4. Parameter "kapazitaet"
=0
=A2+deltaT/60
=Bak0
=Bak0*kapazitaet/
(Bak0+(kapazitaet-Bak0)
*EXP(-lambda*A2))
=B3/C2
Aim: programming an
iterative solution
Slide 44 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 2: results
Ve
rify
the p
relim
inary
consid
era
tions
of th
e p
revio
us s
lide
Exerc
ise
2 (
"00_ba
cte
rialG
row
th.x
ls",
sheet "d
iffe
ren
ce
equ
ation
s")
Slide 45 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Slides Topic Lesson
Introduction: Infectious diseases - how they emerge and disappear
1
SARS 2002/2003: why modeling, and what is a mathematical model? (the example of bacterial growth)
2
Deterministic models: SIR-model, theory, basic reproduction number R0, epidemic vs. endemic case
3
Vaccination: final size of an epidemic, critical vaccination coverage
4
SIR, add-ons: extending SIR to SEIR, SEIRS, etc. 5
Intervention planning: pandemic influenza preparedness planning using InfluSim (can interventions do harm to a population?)
6
Stochastic Models: from theory to reality, the epidemic as a random event, the role of chance: be lucky or not
7
The role of superspreaders 8
Program
Le
sso
n 3
Slide 46 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Polio virus type 1
Extensions of
the SIR-model:
SIRS
SEIR
SEIRS
R S I
Susceptible Immune Infectious Common way of
representing a model:
Compartiments
SIR-Model
& Transititions
Slide 47 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Information needed
• Durations: latent and infectious period
• Rates: contact rate(s)
• Probabilities: P(infection | transmission)
• Demography: birth and death rate,
age structure of the population
• Disease: Proportion of inapparent infections
• ....
Slide 48 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Birth of (susceptible) individuals
dS(t) / dt = m
Dynamic description: Birth
S Proportion susceptibles m per capita birth rate
R
I
S
=1
[ S(t)+I(t)+R(t) ]
Slide 49 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
new Infections
dS(t) / dt = m - bc I(t) S(t)
dI(t) / dt = bc I(t) S(t)
Dynamic description: Infection
S Proportion susceptible
I Proportion infectious
R Proportion immune
c P (infection | contact) m Per capita birth rate
b Contact rate
R
I
S
Slide 50 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Mass action law
The probability of encounterings between
susceptible and infectious individuals
depends on:
• the contact rate b ("Temperature")
• the ratio Susceptible : Infectious
Susceptible Infectious
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
Proportion susceptible
P
S*S
2(S*
I)I*I
Sum
Slide 51 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
new infections
dS(t) / dt = m - bc I(t) S(t)
dI(t) / dt = bc I(t) S(t)
Dynamic description: Infection
R
I
S
S Proportion susceptible
I Proportion infectious
R Proportion immune
c P (infection | contact) m Per capita birth rate
b Contact rate
Slide 52 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Loss of infectiousity
dS(t) / dt = m - bc I(t) S(t)
dI(t) / dt = bc I(t) S(t) - g I(t)
dR(t) / dt = g I(t)
Dynamic description: Loss of infection
R
I
S
S Proportion susceptible
I Proportion infectious
R Proportion immune
c P (infection | contact) m Per capita birth rate
b Contact rate
g rate of loss of infectiousity
Slide 53 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Mortality
dS(t) / dt = m - bc I(t) S(t) - m S(t)
dI(t) / dt = bc I(t) S(t) - g I(t) - m I(t)
dR(t) / dt = g I(t) - m R(t)
Dynamic description: Mortality
R
I
S
S Proportion susceptible
I Proportion infectious
R Proportion immune
c P (infection | contact) m Per capita birth rate
b Contact rate
g rate of loss of infectiousity
Slide 54 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
• Initialisation – choose parameter values for b, c, g and m
– choose initial values for S(0), I(0), and R(0)
• First iteration (time = 0) – compute for a short time step D the changes
dS(0)/dt, dI(0)/dt and dR(0)/dt
– extrapolate changes to S(0+D), I(0+D) and R(0+D)
• Following iterations (time = t) 1. compute for a short time step D the changes
dS(t)/dt, dI(t)/dt and dR(t)/dt
2. extrapolate changes to S(t+D), I(t+D) and R(t+D)
t=t+D, goto 1
Numeric solution of the model
A r
ead
y-t
o-u
se
so
ftw
are
of
the S
IR m
ode
l is
ava
ilable
fro
m
ww
w.u
ni-
tue
bin
gen.d
e/m
odelin
g/M
od_P
ub
_S
oft
ware
_S
IR_e
n.h
tml
Slide 55 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 3:
preliminary considerations
The SIR model, defined as, … produces qualitatively a graph like
dS(t) / dt = m - bc I(t) S(t) - m S(t)
dI(t) / dt = bc I(t) S(t) - g I(t) - m I(t)
dR(t) / dt = g I(t) - m R(t) Zeit
Zahl der
Infizie
rten
the contact rate between
people increases? (b ↑)
Time
No.
of
infe
cto
us p
eople
Draw your qualitative prediction into the graph on how the course of the
epidemic would change (higher, faster, slower, etc) if
patients recover more
rapidly? (g ↑)
Time
No. of
infe
cto
us p
eople
b and g increase at the
same time
Time
No.
of
infe
cto
us p
eople
Aim: understanding the role
of the parameters in the SIR
Exerc
ise 3
Slide 56 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 3: solving differen-
tial equations numerically
Exerc
ise 3
(F
ile "10_SIR.txt")
• Aim: quantitative
epidemiology of
infectious diseases –
learning by doing
• Complete file "10_SIR.txt" with the
equations of the SIR-
model (save your work),
and specify the initial
values (INIT S, I, R).
• Copy the text into the
program editor of
Berkeley-Madonna
• Click "Run"
Program here, using
the parameters listed
beyond
{--- Parameters ---}
Slide 57 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 3: M
ake y
our
slid
ers
in M
enu
Parameters|Define Sliders...
Aim: Performing a
sensitivity analysis
… and verify your preliminary considerations of the previous slide
Exerc
ise 3
(F
ile "10_SIR.txt")
Slide 58 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
R0: Basic reproduction number
• Average number of secondary infections
which one infectious individual would cause
in a fully susceptible population
Definition: R0 = b c D
• R0>1: Infection can persist;
an endemic state ist possible
• R0<1: Infection cannot persist; goes extinct
D = 1 / (gm average duration of the infectious period
bc Number of (sufficiently close) contacts per unit of time
Wichtig! Wichtig!
Wichtig!
Slide 59 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
R0 for some infectious diseases
Disease
Average
age at
infection
[years]
R0
Critical
vaccination
coverage
pcrit [%]
Measles 5.0 15.6 94
Pertussis 4.5 17.5 94
Mumps 7.0 11.5 91
Rubella 10.2 7.2 86
Polio 10.4 6.1 84
Diphteria 10.4 6.1 84
Slide 60 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
bc = 0,5/day, g = 0,1/day, m = 0/day R0 = 5
SIR Model; without births and deaths
Epidemic
Slide 61 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
bc = 0,2/Tag, g = 0,1/Tag, m = 0/Tag R0 = 2
SIR Model; without births and deaths
Epidemic
S
Slide 62 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
At the end of an Epidemic...
Infectious
... susceptible individuals may remain
Susceptible Resistent
Slide 63 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 4: reparameteri-
zation of the SIR model
Exerc
ise 4
(F
ile "11_SIRreparameterized.txt")
• Complete file "11_SIRrepara-meterized.txt" with the parameters at the bottom, and
define how b is derived from R0 and g (save your work as *.txt)
• Copy the text into the program editor of Berkeley-Madonna
• Define sliders as before, in menu Parameters|Define Sliders...
• Click "Run", and evaluate the role of "R0", in particular investigate the the proportion of susceptibles after an epidemic
dependent on R0
Aim: Understanding the
basic reproduction number
Slide 64 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 4: the proportion of
susceptibles after an epidemic
Exerc
ise 4
(F
ile "11_SIRreparameterized.txt")
Aim: Automation of a
sensitivity analysis
Step 1: check position and upper limits of sliders
(in particular: lifeExpectYears=10000 and
STOPTIME=1000)
Step 2: open menu Parameters|Parameter Plot…
and (try to) tell the software: "please show me on Y the final value of S
for different values of R0 on X"
Step 3: click Run and view results
Slide 65 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
0
0.2
0.4
0.6
0.8
1
1 2 3 4 5
Basic reproduction number R0
Pro
port
ion s
usceptib
le
- log (S) = R0 (1 - S)
Proportion S susceptible at the end of the epidemic
Slide 66 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 5:
endemic infection
Exerc
ise 5
(F
ile "11_SIRreparameterized.txt")
• For R0=15 (Measles-like) simulate an ex-tended period of time by increasing STOP-
TIME from 100 to 5000 days (=13.7 years)
• Simulate a population with a lower life ex-
pectancy (developing countries) by decrea-sing lifeExpectYears from 50 to 30 years.
Aim: Understanding the
influence of demography
Parameter Minimum Use Maximum
STOPTIME 0 100 5000
DT 0 0.1 1
DTOUT 0 1 10
iniInfected 0 0.0001 1
lifeExpectYears 0 50 100
durationInfected 0 10 20
R0 0 15 20
• Make sure that slider settings in file "11_SIRreparameterized.mmd"
are as follows
• What is the reason for recurrent epidemics?
• Why reduces the time between epidemics?
Slide 67 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
0
0.2
0.4
0.6
0.8
1
0 1000 2000 3000
Time [days]
Pro
po
rtio
ns
suszeptible
infizierte
immune
SIR Modell with demography
bc = 0,5/day, g = 0,1day, m = 0,0005/day R0 = 5
Endemic case
Slide 68 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Summary
• Neglecting births and deaths, – we model an epidemic scenario;
– after the epidemic, a proportion of susceptibles,
which depends on R0, remains
• Considering births and deaths, – we model an endemic scenario
– the model variables (S,I,R, ...) approach the
endemic state (if R0 > 1);
– the equilibirum prevalence depends on R0.
logKurve
Slide 69 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Slides Topic Lesson
Introduction: Infectious diseases - how they emerge and disappear
1
SARS 2002/2003: why modeling, and what is a mathematical model? (the example of bacterial growth)
2
Deterministic models: SIR-model, theory, basic reproduction number R0, epidemic vs. endemic case
3
Vaccination: complications, contra-intuitive effects, final size of an epidemic, critical vaccination coverage
4
SIR, add-ons: extending SIR to SEIR, SEIRS, etc. 5
Intervention planning: pandemic influenza preparedness planning using InfluSim (can interventions do harm to a population?)
6
Stochastic Models: from theory to reality, the epidemic as a random event, the role of chance: be lucky or not
7
The role of superspreaders 8
Program
Le
sso
n 4
Slide 70 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Can vaccinations do harm to the population? (1)
• Typical childhood disease, relatively trivial infection, disease
is often mild or proceeds asymptomatically. Duration of
disease: 1-3 days, fever rarely >38°C. R0: 5-7.
• Epidemiology: worldwide. Humans are the only host, no
animal reservoir. Before vaccination was introduced in 1969,
widespread outbreaks usually occurred every 4-8 years,
mostly affecting children in the 5-9 year old age group.
• Congenital Rubella Syndrome (CRS) if the mother is
infected within the first 20 weeks of pregnancy.
• Vaccination. Two live attenuated virus vaccines can prevent
adult disease. Use in prepubertile females not reliable.
Vaccine is given as part of the MMR vaccine. Schedule: 1st
dose at 12 to 18 months, 2nd dose at 36 months. Pregnant
women are usually tested for immunity to rubella early on.
Women found to be susceptible are not vaccinated until after
the baby is born because the vaccine contains live virus.
• Problem: not eradicated, reintroduction from endemic
countries always possible.
Example & clinical background: Rubelly ("German measles", "Röteln")
© 2
009 N
ucle
us M
edic
al M
edia
, In
c
Countries using Rubella vaccine in National
Immunization Schedule, 2009 (WHO)
Source: NIH
Microcephaly
PDA
Cataracts
Question: Does vaccination always or proportionally decrease CRS?
CRS
Slide 71 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Example Rubella vaccination
Simplifications:
Lit.: K. Dietz & D. Schenzle, 1985: Epidemiologische Auswirkungen von Schutzimpfungen gegen Masern, Mumps und Röteln. In: Schutzimpfungen:
Notwendigkeit, Wirkung/Nebenwirkungen, Impfpolitik. Bericht von der Tagung des Deutschen Grünen Kreuzes, in Verbindung mit der Deutschen
Vereinigung zur Bekämpfung der Viruskrankheiten e.V. Herausgegeben von H. Spiess, Medizinische Verlagsgesellschaft mbH,Marburg/Lahn 1985.
• All mothers give birth to a child at the age of A = 25 years
• Annual Risk of infection with Rubella: R= 0.1 = 10% per year
Incidence of CRS )1()1( pRA
CRS epRI
Vaccination can increase the
risk for higher age groups
0.1 per year ∙ 0.75 ∙ 0.115
= 0.0086 per year
0.1 per year ∙ 0.082
= 0.0082 per year
Incidence of CRS
(ICRS)
e(-25 years∙0.1 per year ∙0.75)
≈ 0.153 (among 75% not vaccinated)
≙0.153 ∙ 0.75 = 0.115 (11.5%) (among all mothers)
e(-25 years∙0.1 per year)
≈ 0.082 (8.2%)
Proportion of
susceptible
mothers at the age
of 25 years
With 25% of newborns
vaccinated (p=0.25)
Without
vaccination
R is reduced
because vacci-
nation has
reduced the
overall rate of
transmission
0
0.5
1
0 10 20 30 40 Age
agee 1.0
Slide 72 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Example Rubella vaccination
Relative Incidence of CRS
)01(
)1(
)01(
)1(
RA
pRA
CRSeR
epRI
0
0.5
1
1.5
0 0.2 0.4 0.6 0.8 1
Proportion p vaccinated at birth
Rela
tive I
ncid
ence
ICR
S
0
0.02
0.04
0.06
0.08
0.1
0 1 2 3 4 5 6 7 8 9 10
Age [years]
Pro
po
rtio
n a
t risk
Novaccination
25% ofnewborns arevaccinated
Reason for an increased incidence
under (low) vaccination coverage:
Hazard rate is lower
→ hazard increases at higher age.
Oversimplifications so far: R0, Age- and contact-
structure not considered, assumption of constant risk
over time (implies exponential distribtution),
immunization is complete and homgenious, no
immigration, maternal immunity after birth, etc.
Vaccination can increase the
risk for higher age groups
with A =25 years, R =0.1 per year
Slide 73 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Example Rubella vaccination Seperate and
average effects
1) Unvaccinated women
have an increased risk,
(except shortly before
elimination) because the
overall transmission is
reduced and hence, the
average age of infection
will increase.
2) Vaccinated women have
always a low risk,
dependent on vaccine
efficacy
3) The overall effect for the
population is a weighted
average
(Risk∙Proportion)unvaccinated
+ (Risk∙Proportion)vaccinated
Elim
ination
From: Dietz K & Eichner M. The effect of heterogeneous interventions on the spread of infectious diseases. Proceedings 21. Tagung Internationalen Biometrischen Gesellschaft, 2002.
Slide 74 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Rubella vaccination in Greece: BMJ 1999
Slide 75 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Can vaccinations do harm to the population? (2)
Example & clinical background: Measles
Age-dependent risk factors
after measles infection
Otitis
Pneumonia
Orchitis
Embryopathy
Lit.: K. Dietz & D. Schenzle, 1985: Epidemiologische Auswirkungen von Schutzimpfungen gegen Masern, Mumps und Röteln. In: Schutzimpfungen:
Notwendigkeit, Wirkung/Nebenwirkungen, Impfpolitik. Bericht von der Tagung des Deutschen Grünen Kreuzes, in Verbindung mit der Deutschen
Vereinigung zur Bekämpfung der Viruskrankheiten e.V. Herausgegeben von H. Spiess, Medizinische Verlagsgesellschaft mbH,Marburg/Lahn 1985.
Average age of measles infection
dependent on vaccination coverage
Vaccination reduces the overall risk of infection and hence, increases the
average age of infection. More children may suffer from a "late" infection
if complications after measles infection occur more likely at higher age.
Slide 76 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Example Measles vaccination
Example & clinical background: Measles
Age-dependent risk factors
after measles infection
Otitis
Pneumonia
Orchitis
Embryopathy
Lit.: K. Dietz & D. Schenzle, 1985: Epidemiologische Auswirkungen von Schutzimpfungen gegen Masern, Mumps und Röteln. In: Schutzimpfungen:
Notwendigkeit, Wirkung/Nebenwirkungen, Impfpolitik. Bericht von der Tagung des Deutschen Grünen Kreuzes, in Verbindung mit der Deutschen
Vereinigung zur Bekämpfung der Viruskrankheiten e.V. Herausgegeben von H. Spiess, Medizinische Verlagsgesellschaft mbH,Marburg/Lahn 1985.
Vaccination reduces the overall risk of infection and hence, increases the
average age of infection. More children could suffer from a "late" infection
if complications after measles infection occur more likely at higher age.
Expected changes in the relative
incidence of complications
dependent on vaccination coverage
Orchitis
Slide 77 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Examples Mumps & Rubella vaccination
Mumps Rubella
Congenital
Rubella
Syndrome
(CRS)
Current status of CRS:
Morbidity and Mortality Weekly Report (MMWR)
October 15, 2010 / 59(40);1307-1310. Progress
Toward Control of Rubella and Prevention of
Congenital Rubella Syndrome --- Worldwide, 2009
"...concern that the risk for CRS
might increase if high vacci-
nation coverage could not be
achieved. Low coverage might
result in decreased virus
circulation, which could increase
the average age of rubella
infection for females from child-
hood to the childbearing years."
"...Globally, a total of 165 CRS
cases were reported from 123
member states during 2009,
compared with 157 CRS cases
reported from 75 member states
during 2000" ... "CRS, which
affects an estimated 110,000
infants each year in
developing countries"
Slide 78 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Summary
• Vaccination not necessarily yield benefits
proportional to vaccination coverage. At least
two complications need to be adressed:
• Vaccination decreases the overall virus
circulation and thus can increase the average
age of infection.
• Age-dependently increasing complications of
infection can bring about situations where low
vaccination coverages produce a higher number
of cases (clinical, fatal, etc.).
Slide 79 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Model without vaccination
dS(t) / dt = m - bc I(t) S(t) - m S(t)
dI(t) / dt = bc I(t) S(t) - g I(t) - m I(t)
dR(t) / dt = g I(t) - m R(t)
Dynamic description: SIR without vaccination
R
I
S
S Proportion susceptible
I Proportion infectious
R Proportion immune
c P (infection | contact) m Per capita birth rate
b Contact rate
g rate of loss of infectiousity
Slide 80 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
A proportion of newborns will be vaccinated
dS(t) / dt = m ........ - bc I(t) S(t) - m S(t)
dI(t) / dt = bc I(t) S(t) - g I(t) - m I(t)
dR(t) / dt = ....... + g I(t) - m R(t)
Dynamic description: SIR with vaccination
R
I
S 1-p
m p
S Proportion susceptible
I Proportion infectious
R Proportion immune
c P (infection | contact)
p Proportion vaccinated
m Per capita birth rate
b Contact rate
g rate of loss of infectiousity
Slide 81 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 6: Vaccination E
xerc
ise 6
(F
ile "11_SIRreparameterized.txt")
Aim: Understanding the
concept of thresholds
• Define slider for "p" in
Menu
Parameters|Define
Sliders... and choose
slider settings as shown in
the screenshot to the right
• Implement parameter "p" for the proportion of vaccinated newborns (see
previous slide) in the equations of file "11_SIRreparameterized.txt"
file and save it as "12_SIRvaccination.txt"
Slide 82 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 6: Vaccination E
xerc
ise 6
(F
ile "12_SIRvaccination.txt")
Aim: Understanding the
concept of thresholds
• Technical remark:
1.
2.
For purposes of better inspectation, we change in the
output window axis settings for compartment I to "Auto" –
see below and ask the lecturer.
Slide 83 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 6: Vaccination E
xerc
ise 6
(F
ile "12_SIRvaccination.txt")
Aim: Understanding the
concept of thresholds
• For R0=15, increase p up to a value when
there is no epidemic
anymore. This is the
ciritical vaccination
coverage p*. Repeat
the procedure for
R0=10, 5 and 2, and
plot your results in
the graph to the right.
p*
0 5 10 15
0
0.2
0.4
0.6
0.8
1
R0
What is the critical vaccination coverage when the basic repro-duction number tends to values of R0→1
Slide 84 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 6: Vaccination E
xerc
ise 6
(F
ile "12_SIRvaccination.txt")
Aim: Understanding the
concept of thresholds
• For R0=15, increase p up to a value when
there is no epidemic
anymore. This is the
ciritical vaccination
coverage p*. Repeat
the procedure for
R0=10, 5 and 2, and
plot your results in
the graph to the right.
p*
0 5 10 15
0
0.2
0.4
0.6
0.8
1
R0
What is the critical vaccination coverage when the basic repro-duction number tends to values of R0→1
Slide 85 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Endemic equilibrium
no change of model variables in the endemic equilibrium
0 = m 1-p - bc I(t) S(t) - m S(t)
0 = bc I(t) S(t) - g I(t) - m I(t)
0 = m p + g I(t) - m R(t)
R
I
S
S Proportion susceptible
I Proportion infectious
R Proportion immune
c P (infection | contact)
p Proportion vaccinated
m Per capita birth rate
b Contact rate
g rate of loss of infectiousity
Slide 86 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Endemic equilibrium
no change of model variables in the endemic equilibrium
0 = m 1-p - bc I S - m S
0 = bc I S - g I - m I
0 = m p + g I - m R
I=...
S=...
R=... R
I
S
S Proportion susceptible
I Proportion infectious
R Proportion immune
c P (infection | contact)
p Proportion vaccinated
m Per capita birth rate
b Contact rate
g rate of loss of infectiousity
Slide 87 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Endemic equilibrium
S = (g + m) / (bc)
I = (1 - 1/R0 - p) m / (g + m)
R = 1 - S - I
R0 = bc / (g+m)
= 1 / R0
Estimate R0 from the
proportion of
susceptibles in the
endemic equilibrium
R
I
S
S Proportion susceptible
I Proportion infectious
R Proportion immune
c P (infection | contact)
p Proportion vaccinated
m Per capita birth rate
b Contact rate
g rate of loss of infectiousity
Slide 88 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Critical vaccination coverage
Parameters of the right hand side are known, except p
I = (1 - 1/R0 - p) m / (g + m)
R
I
S
I Proportion infectious
Basic reproduction number:
R0 = bc / (g+m)
p Proportion vaccinated m Per capita birth rate = death rate
g rate of loss of infectiousity
Slide 89 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Critical vaccination coverage
Parameters of the right hand side are known, except p
0 = (1 - 1/R0 - pcrit) m / (g + m)
pcrit = 1 - 1 / R0
To eliminate a disease, it is not necessary to
vaccinate the whole population
R
I
S
Slide 90 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
0
0.2
0.4
0.6
0.8
1
0 2 4 6 8 10 12 14 16 18 20
Basic reproduction number R0
Pro
port
ion v
acc
inate
d
Elimination
Persistence
Critical vaccination coverage
Slide 91 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Summary (2)
• The proportion of susceptibles in the endemic
equilibrium does not depend on the proportion p of
vaccinated children
• Transmission stops if p pcrit
• The critical vaccination coverage is
• The model can be used for sensitivity analyses into the
effects of different vaccination strategies:
- What is the critical vaccination coverage?
- How does vaccination impact on the prevalence and incidence of the
infection?
- what is the best vaccination strategy (e. g. ring vaccination vs. mass
vaccination)?
pcrit = 1 - 1/R0
Slide 92 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
m
g
R0
pcrit
bc
Basic reproduction number 1 / R0 is the endemic prevalence of susceptibles
Per capita birth rate = death rate 1 / m is the life expectancy
Loss-of-infection rate 1 / (gm) is the average duration of the infectious period
Critical vaccination coverage pcrit = 1 - 1 / R0
Effective contact rate bc = R0 (gm)
Estimation of model parameters
Slide 93 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Slides Topic Lesson
Introduction: Infectious diseases - how they emerge and disappear
1
SARS 2002/2003: why modeling, and what is a mathematical model? (the example of bacterial growth)
2
Deterministic models: SIR-model, theory, basic reproduction number R0, epidemic vs. endemic case
3
Vaccination: final size of an epidemic, critical vaccination coverage
4
SIR, add-ons: extending SIR to SEIR, SEIRS, etc. 5
Intervention planning: pandemic influenza preparedness planning using InfluSim (can interventions do harm to a population?)
6
Stochastic Models: from theory to reality, the epidemic as a random event, the role of chance: be lucky or not
7
The role of superspreaders 8
Program
Le
sso
n 5
Slide 94 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Extensions: from SIR to SEIR
SIR
dS(t) / dt = m 1-p - bc I(t) S(t) - m S(t)
dI(t) / dt = bc I(t) S(t) - g I(t) - m I(t)
dR(t) / dt = m p + g I(t) - m R(t)
R
I
S
S Proportion susceptible
I Proportion infectious
R Proportion immune
c P (infection | contact)
p Proportion vaccinated
m Per capita birth rate
b Contact rate
g rate of loss of infectiousity
Slide 95 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Infections with latent stage
dS(t) / dt = m 1-p - bc I(t) S(t) - m S(t)
Incorporate latent stage
dI(t) / dt = bc I(t) S(t) - g I(t) - m I(t)
dR(t) / dt = m p + g I(t) - m R(t)
R
I
S
S Proportion susceptible
I Proportion infectious
R Proportion immune
c P (infection | contact)
p Proportion vaccinated
m Per capita birth rate
b Contact rate
g rate of loss of infectiousity
Slide 96 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
SEIR
dS(t) / dt = m 1-p - bc I(t) S(t) - m S(t)
dE(t) / dt = . . . . .
dI(t) / dt = bc I(t) S(t) - g I(t) - m I(t)
dR(t) / dt = m p + g I(t) - m R(t)
R
I
S
E
S Proportion susceptible
E Proportion latent
I Proportion infectious
R Proportion immune
c P (infection | contact)
p Proportion vaccinated
m Per capita birth rate
b Contact rate
g rate of loss of infectiousity
Slide 97 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
SEIR
dS(t) / dt = m 1-p - bc I(t) S(t) - m S(t)
dE(t) / dt = bc I(t) S(t) - . . . . .
dI(t) / dt = . . . . . - g I(t) - m I(t)
dR(t) / dt = m p + g I(t) - m R(t)
R
I
S
E
S Proportion susceptible
E Proportion latent
I Proportion infectious
R Proportion immune
c P (infection | contact)
p Proportion vaccinated
m Per capita birth rate
b Contact rate
g rate of loss of infectiousity
Slide 98 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
SEIR
dS(t) / dt = m 1-p - bc I(t) S(t) - m S(t)
dE(t) / dt = bc I(t) S(t) - d E(t) - m E(t)
dI(t) / dt = d E(t) - g I(t) - m I(t)
dR(t) / dt = m p + g I(t) - m R(t)
R
I
S
E
S Proportion susceptible
E Proportion latent
I Proportion infectious
R Proportion immune
c P (infection | contact)
p Proportion vaccinated
m Per capita birth rate
b Contact rate
g rate of loss of infectiousity
Slide 99 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 7: SEIR model E
xerc
ise 7
(F
ile "13_SEIRvaccination.txt")
Aim: customize an SIR
model yourself
• Complete file "13_SEIRvacci-nation.txt" with the parameters for the latent stage.
• Copy the text into the program editor of Berkeley-Madonna
• Define sliders as before, in menu Parameters|Define Sliders...
•Run…
Technical skills: What is the value for d that reduces an SEIR model back to an SIR model?
Slide 100 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Blank page for documentation
Slide 101 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Slides Topic Lesson
Introduction: Infectious diseases - how they emerge and disappear
1
SARS 2002/2003: why modeling, and what is a mathematical model? (the example of bacterial growth)
2
Deterministic models: SIR-model, theory, basic reproduction number R0, epidemic vs. endemic case
3
Vaccination: final size of an epidemic, critical vaccination coverage
4
SIR, add-ons: extending SIR to SEIR, SEIRS, etc. 5
Intervention planning: pandemic influenza preparedness planning using InfluSim (can interventions do harm to a population?)
6
Stochastic Models: from theory to reality, the epidemic as a random event, the role of chance: be lucky or not
7
The role of superspreaders 8
Program
Le
sso
n 6
Slide 102 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
download: http://www.uni-tuebingen.de/modeling/Mod_Pub_Software_InfluSim_en.html
Lesson 6: InfluSim (Version 1.3)
Parameter panel Output panel
Output
table
Output
graphs
Output for
population with
N=100 000
individuals
Slide 103 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper
Output tab Output vari-
able (column)
At which stage of
the epidemic?*
R0 =2 R0 1.5
Infection Dead at end* 166 114
Infection Immune at end* 76855 55572
Ressource use Hospital beds at peak* 191 79
Cumulative Work loss [wk] at end* 29059 21647
Costs Total at end* 36130 26708
Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 8: Influsim E
xerc
ise 8
(F
old
er
"Influsim1.3")
Aim: use a model to evaluate
intervention strategies
• Start Influsim from folder "InfluSim1.3" (screenshot see previous slide).
• Get familiar with using sliders, customizing panel arrangement and switching between tabs
for parameter values and output. First exploration:
Keep the default parameter values (or restart the program to dismiss your changes).
This will produce output based on R0 =2 ("severe" influenza epidemic).
Lookup values in the output table as listed in the table below and transfer the values
predicted by InfluSim into this table under column "R0 =2" (values refer to N=100 000).
Change R0 in parameter tab 'Contagiousness' from the default value of R0 =2 to R0
=1.5 and denote the new output values under column "R0 =1.5" ("moderate" influenza).
* click into the graphics window and/or scroll to the relevant row in the output table.
Slide 104 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper
Parameter section Slider Value Number
of deaths
Baseline 166
Treatment of severe cases Treatment fraction [%] increase to 100% 117
Treatment of extremely sick cases Treatment fraction [%] increase to 100% 65
Ressources Antiviral availability [%] decrease to 10% 113
Treatment of severe cases Treatment fraction [%] decrease to 0% 93
Treatment of extremely sick cases Begin on day increase to 60 days 161
Treatment of extremely sick cases Begin on day decrease to 30 days 94
Treatment of extremely sick cases Begin on day decrease to 0 days 93
What is your optimal antiviral intervention with 10% antivirals for the population?
Exercise 8: Influsim E
xerc
ise 8
(F
old
er
"Influsim1.3")
Aim: explore the effects of
antiviral treatment
• Restart influSim to restore default parameter values (assumes R0 =2).
• Choose parameter tab ''Treatment'
• Change parameter sucessively as listed in the first three columns of the table below and
• denote the predicted number of deaths at the end of the epidemic (output tab 'Infection',
column 'Dead' in output table).
* go to output tab 'Cumulative' and scroll through table column 'Antiviral': this column lists the
number of individuals to whom antivirals have been given. How many?
** why less than before?
*
**
Slide 105 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper
Exercise 8: Influsim E
xerc
ise 8
(F
old
er
"Influsim1.3")
Aim: explore the effects of
social distancing measures
• Restart influSim to restore default parameter values.
• Choose parameter tab ''Social distancing '
• We ignore initially 'Closing of day care centers and schools' and restrict on the two parameter sections
'General reduction of contacts' and
'Cancelling of mass gathering events'
Questions:
• What do you think: A) to which extent can we reduce the general contact rate in the population such
that the country's infrastructure will not be substantially impaired? Denote here: I guess the contact in our population can be reduced by ____________ %. B) guess how many deaths this contact reduction should safe: My guesstimate: this should reduce the number of deaths from 166 per 100 000 to _____________ per 100 000.
Move slider 'Contact reduction by [%]' in section 'General reduction of contacts' to the value you
have assumed under A). Note: For effects to be simulated you must specify how long this
contact reduction will be performed → next point …
Specify with slider 'End on day' in the same section 60 days (produces a scenario with contact
reduction from day 0 to 60 - quite a long time from a perspective of a population): Denote the
result of this simulation: My scenario reduced the number of deaths to _____________ per 100 000.
Move slider 'Contact reduction by [%]' (still in this section 'General reduction of contacts' ) to the unrealistically high value of 50% - why does this increase the number of deaths ??
• Evaluate the same considerations under section 'Cancelling of mass gathering events' - Question:
Which intervention is more efficient: 'General reduction of contacts' or 'Cancelling of mass
gathering events' ?
Slide 106 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Antivirals & social distancing
Sta
y a
t h
om
e
Slide 107 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Exercise 8: Influsim Aim: explore effects of delayed inter-
ventions (here: isolation measures)
Exerc
ise 8
(F
old
er
"Influsim1.3")
• Restart influSim to restore default parameter values (assumes R0 =2).
• Choose parameter tab 'Contagiousness' and specify with slider "Begin on day" a delay of
30 days as baseline delay for isolation measures: "Begin on day" =30 days.
• Vary parameters under 'Partial Isolation': we assume that 50% of Moderately sick cases,
Severe cases (home), or Severe cases (hospital) can be isolated with a delay of 30 days
(Row A1, A2, A3) or with a delay of 60 days (Row B1, B2, B3).
* Columns "Dead" and "Immune" in the ouput table, tab "Infection"
INPUT (tab 'Contagiousness', 'Partial Isolation') OUTPUT Infection tab
Moderately
sick cases
Severe
cases
(home)
Severe
cases
(hospital)
Begin
on day
Number of
deaths*
Number
infected*
No isolation (default) 0 0 0 30 166 76855
A1) Only 'moderate' 50 0 0 30 139 65697
A2) Only 'home' 0 50 0 30 118 58008
A3) Only 'hospital' 0 0 50 30 165 76665
B1) Only 'moderate' + delay 50 0 0 60 165 76278
B2) Only 'home' + delay 0 50 0 60 163 75925
B3) Only 'hospital' + delay 0 0 50 60 166 76837
Slide 108 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper
Exercise 8: Influsim - Duration of measures E
xerc
ise 8
Don't stop when it seems to be over,
but later.
Slide 109 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Slides Topic Lesson
Introduction: Infectious diseases - how they emerge and disappear
1
SARS 2002/2003: why modeling, and what is a mathematical model? (the example of bacterial growth)
2
Deterministic models: SIR-model, theory, basic reproduction number R0, epidemic vs. endemic case
3
Vaccination: final size of an epidemic, critical vaccination coverage
4
SIR, add-ons: extending SIR to SEIR, SEIRS, etc. 5
Intervention planning: pandemic influenza preparedness planning using InfluSim (can interventions do harm to a population?)
6
Stochastic Models: from theory to reality, the epidemic as a random event, the role of chance: be lucky or not
7
The role of superspreaders 8
Program
Le
sso
n 7
Slide 110 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Vergleich:
deterministische vs. stochastische Modelle
Deterministische Modelle stochastische Modelle
• werden i.d.R. durch explizite
Formeln (Differenzialgleichungen)
erstellt
• liefern bei gleichen Anfangs-
bedingungen stets identische
Ergebnisse
• ihre Ergebnisse sind meist besser
verallgemeinerbar Zur Planung
von Interventionsmaßnahmen sind
deterministische Modelle oft besser
geeignet
• werden i.d.R. durch (individuen-
basierte) Simulationsprogramme
erstellt
• liefern bei gleichen Anfangs-
bedingungen zufallsbedingt
unterschiedliche Ergebnisse
• ihre Ergebnisse sind meist
realitätsnäher da sie zufällige
Effekte wiedergeben können
(Stochastizität)
• Für die Untersuchung von Effekten
in kleinen Populationen besser
Wichtig! Wichtig!
Wichtig!
Slide 111 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Lesson 7:
• General modeling, SEIR, diseases
• Influenza Vaccination, Antivirals
• SARS Isolation, Contact tracing
Behaviour
• Contact structures Networks
• Conclusions
• ... if time is left Playmulation (Simulator as a playstation)
Interventions in epidemics of influenza-like diseases -
insights from individual-based computer simulations
Slide 112 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
The simulator: interface
www.uni-tuebingen.de/modeling
Slide 113 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
The simulator: network
www.uni-tuebingen.de/modeling
Contacts of person 1446: Contacts of person 8969:
Slide 114 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza-like SEIR(S)
S susceptible
I infectious
R immune
E latent
S susceptible
• genetic drift in
inluenza
• negligible for short-
term investigations
Slide 115 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Simulation algorithm
Build up network of contacts
Introduce infection
Sample for an infected individual
and each of his/her contacts:
Number of transmitted
infections
Time of infection
Duration
• of latency period • of prodromal phase
• of infectivity D • of immunity
Place events in a priority queue
and execute them consecutively
Scalefree network
8 close contacts
8 remote contacts
10 index cases
R0, close =1
R0, remote=1
Exponential distribution
with mean 1/b = D/R0
Gamma distribution Gamma distribution
Gamma distribution
Binary heap algorithm
R0=2
Slide 116 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Events: dependence & triggering
Demography
(birth of
susceptibles)
Intervention
Contact
structures
Surveillance
Case detection
Contact tracing
Observation
Symptoms Infection process
(Proportions)
Slide 117 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
A 'standard' epidemic
20% have not
been infected
and remain
susceptible
80% have been
infected
and become
immune 10 simulations
latent
infectious
80
29 c
ases
10 index cases initiate at t=0
an epidemic in a fully
susceptible population of
10000 people.
R0 =2.
Latency: 1.9 days.
Infectivity: 4 days.
Lifelong immunity.
SR
eS10
... N index cases
Slide 118 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
A 'standard' epidemic Variation: number of index cases
10 simulations
100 index cases initiate at t=0
an epidemic in a fully
susceptible population of
10000 people.
R0 =2.
Latency: 1.9 days.
Infectivity: 4 days.
Lifelong immunity. 80
58 c
ases
Epidemic develops
more rapid
... latency
Slide 119 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
A 'standard' epidemic Variation: duration of latency period
10 simulations
10 index cases initiate at t=0
an epidemic in a fully
susceptible population of
10000 people.
R0 =2.
Latency: 3.8 days.
Infectivity: 4 days.
Lifelong immunity. 80
61 c
ases
Epidemic delayed
... infectious period
Slide 120 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
A 'standard' epidemic Variation: duration of infectious period
10 simulations
10 index cases initiate at t=0
an epidemic in a fully
susceptible population of
10000 people.
R0 =2.
Latency: 1.9 days.
Infectivity: 8 days.
Lifelong immunity. 80
41 c
ases
... but the number
of cases hardly
increases
Epidemic delayed,
appears more severe...
Slide 121 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Lesson 7:
• General modeling, SEIR, diseases
• Influenza Vaccination, Antivirals
• SARS Isolation, Contact tracing
Behaviour
• Contact structures Networks
• Conclusions
• ... if time is left Playmulation (Simulator as a playstation)
Interventions in epidemics of influenza-like diseases -
insights from individual-based computer simulations
Slide 122 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Estimates: Infection
Influenza SARS R0 1.8
close contacts: 1
casual contacts: 0.8
2.6
close contacts: 1
casual contacts: 1.6
Latency ~2 days (1 day)
Range: 1-3 days
4.5 days (1.5 day)
Range: 3-6 days
Infectivity 4 days (2 days)
Range: 2-6 days
14 days (5 days)
Range: 9-19 days
Immunity forever forever
Symptoms 68% 100%
Case fatality rate 5% 10%
Slide 123 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Vaccination
Mass-vaccination Traced vaccination
Ring vaccination
Problems: • Vaccine stock limited,
Production takes time
• Who get's vaccinated ?
• Individual decision
• Which strategy
after stock is used up
• Contact tracing: who was a
contact?
• Where is he/she?
• Individual decision
• Who is a close contact?
• Fraction of contacts traced
Example: • Capacity: how many
persons can be vaccinated
per day?
• Delay between onset of
outbreak and begin of
vaccination
Slide 124 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Standard epidemics: influenza
General Infection
10 index cases
R0 = 1.9
Latency: 1.9 days.
Infectivity: 8 days.
Immunity: lifelong
... the ultimate vaccination campaign
Slide 125 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza: Mass-Vaccination (1)
General Infection Vaccination
10 index cases
R0 = 1.9
Latency: 1.9 days.
Infectivity: 8 days.
Immunity: lifelong
Overoptimistic: All people are vaccinated. Vaccine
efficacy: 100%. All vaccinations can be performed
when the first case has been detected.
... Compliance: 50%, vaccine efficacy: 80%
Slide 126 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza: Mass-Vaccination (2)
General Infection Vaccination
10 index cases
R0 = 1.9
Latency: 1.9 days.
Infectivity: 8 days.
Immunity: lifelong
50% of the population are eligible. Vaccine efficacy:
80%. Overoptimistic: All vaccinations can be
performed when the first case has been detected.
... increase compliance, consider limitations
Slide 127 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza: Mass-Vaccination (3)
General Infection Vaccination
10 index cases
R0 = 1.9
Latency: 1.9 days.
Infectivity: 8 days.
Immunity: lifelong
80% of the population are eligible. Vaccine efficacy:
80%. Vaccination starts without delay (Overoptimistic),
but only 2000 vaccinations per day are feasible.
... vaccination cannot start on day "zero"
Slide 128 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza: Mass-Vaccination (4)
General Infection Vaccination
10 index cases
R0 = 1.9
Latency: 1.9 days.
Infectivity: 8 days.
Immunity: lifelong
80% of the population are eligible. Vaccine efficacy:
80%. Vaccination starts 2 weeks after the first case
has been detected, 2000 vaccinations per day.
... reduce delays
Slide 129 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza: Mass-Vaccination (5)
General Infection Vaccination
10 index cases
R0 = 1.9
Latency: 1.9 days.
Infectivity: 8 days.
Immunity: lifelong
80% of the population are eligible. Vaccine efficacy:
80%. Vaccination starts 1 week after the first case
has been detected, 2000 vaccinations per day.
?
Slide 130 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza: No intervention vs. Mass vaccination
5331 cases prevented
-4949 vaccinations
382
only 382 cases of
transmission prevented
mass vaccination
no intervention
Slide 131 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Conclusions: Vaccination (Influenza)
• Compliance
• Public Health capacities
• Intervention delays
Factors determining the prospects of success of (voluntary) vaccination campaigns:
Given proper compliance and capacities,
reducing intervention delays should be the main goal
For realistic parameter values,
the concomitant profit of vaccination is low
Summary: not very efficient, expensive,
makes voluntary vaccination sense?
Slide 132 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Blank page for documentation
Slide 133 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Lesson 7:
• General modeling, SEIR, diseases
• Influenza Vaccination, Antivirals
• SARS Isolation, Contact tracing
Behaviour
• Contact structures Networks
• Conclusions
• ... if time is left Playmulation (Simulator as a playstation)
Interventions in epidemics of influenza-like diseases -
insights from individual-based computer simulations
Slide 134 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Treatment / Prophylaxis with antivirals
Prophylaxis Treatment
Questions: • When shall we start taking
prophylaxis and for how long?
• How many people will take
prophylaxis?
• Efficacy: reduction of
susceptibility in non-infected
• How quickly can infection be
diagnosed?
• Efficacy: reduction of
infectiousness in infected
Examples: • Compliance
• Duration
• Delay
• Compliance
• Duration
• Delay
• Shall close contacts take
antivirals prophylactically?
Slide 135 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza: without Intervention
General Infection Prophylaxis Treatment
10 index
cases
R0 = 1.9
Latency: 1.9 d
Infectivity: 4 d
Immunity:
- -
... Population prophylaxis
Slide 136 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza: Prophylaxis with Antivirals (1)
Useless intervention
General Infection Prophylaxis Treatment
10 index
cases
R0 = 1.9
Latency: 1.9 d
Infectivity: 4 d
Immunity:
- 50% of the population take
antiviral prophylaxis
- (which reduces the
susceptibility to 30%)
- for a period of 2 weeks
-
... prolonge prophylaxis
Slide 137 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza: Prophylaxis with Antivirals (2)
"Sometimes useful" intervention
General Infection Prophylaxis Treatment
10 index
cases
R0 = 1.9
Latency: 1.9 d
Infectivity: 4 d
Immunity:
- 50% of the population take
antiviral prophylaxis
- (which reduces the
susceptibility to 30%)
- for a period of 4 weeks
-
epidemics in
4 of 10 simulations
... consider delay
Slide 138 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza: Prophylaxis with Antivirals (3)
Delays
General Infection Prophylaxis Treatment
10 index
cases
R0 = 1.9
Latency: 1.9 d
Infectivity: 4 d
Immunity:
- 50% of the population take
antiviral prophylaxis
- (which reduces the
susceptibility to 30%)
- for a period of 4 weeks
- but delayed by 10 days
-
epidemics in
all 10 simulations
... Treatment
Slide 139 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza: without Intervention
General Infection Prophylaxis Treatment
10 index
cases
R0 = 1.9
Latency: 1.9 d
Infectivity: 4 d
Immunity:
- -
... Treatment
Slide 140 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza: Treatment with Antivirals
General Infection Prophylaxis Treatment
10 index
cases
R0 = 1.9
Latency: 1.9 d
Infectivity: 4 d
Immunity:
- - Cases receive treatment for 7 days
- 1 day after having been detected
- Compliance: 80%
- Efficacy: 80% (reduction infectiousness)
epidemics
in 8 of 10 simulations
... Treatment + Prophylaxis
Slide 141 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Influenza: Treatment & Prophylaxis
General Infection Prophylaxis Treatment
10 index
cases
R0 = 1.9
Latency: 1.9 d
Infectivity: 4 d
Immunity:
- Close contacts take antiviral
prophylaxis (Compliance:
80%), reducing their
susceptibility to 30%
- for a period of 2 weeks
- delayed by 1 day
- Cases receive treatment for 7 days
- 1 day after having been detected
- Compliance: 80%
- Efficacy: 80% (reduction infectiousness)
C
Slide 142 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Conclusions: Antivirals (influenza)
xx
Prophylaxis with of antivirals in the population appears not very efficient
given realistic values for compliance and delays
Major effect: epidemic slows down, but last for much longer
Treating patients is more efficient,
but should be supplemented with
prophylaxis for close contacts (e.g. family)
The delay in prophylaxis for close contacts
is not that critical
A combination
of therapeutic and prophylactic use
seems to be
the most promising approach
Germany: "antivirals for 9% of the population"
Slide 143 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Blank page for documentation
Slide 144 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Blank page for documentation
Slide 145 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Lesson 7:
• General modeling, SEIR, diseases
• Influenza Vaccination, Antivirals
• SARS Isolation, Contact tracing
Behaviour
• Contact structures Networks
• Conclusions
• ... if time is left Playmulation (Simulator as a playstation)
Military hospital, flu in Kansas, 1918
Interventions in epidemics of influenza-like diseases -
insights from individual-based computer simulations
Slide 146 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Estimates: Infection
Influenza SARS R0 1.8
close contacts: 1
casual contacts: 0.8
2.6
close contacts: 1
casual contacts: 1.6
Latency ~2 days (1 day)
Range: 1-3 days
4.5 days (1.5 day)
Range: 3-6 days
Infectivity 4 days (2 days)
Range: 2-6 days
14 days (5 days)
Range: 9-19 days
Immunity forever forever
Symptoms 68% 100%
Case fatality rate 5% 10%
Slide 147 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Isolation and Contact tracing (SARS)
Problems: • Contact tracing: who is / was a contact?
• What is a close, what is a casual contact?
• Number of traced contact persons per patient
• Limited resources to isolate patients
• Which strategy after resources have been exhausted
• Delay in case detection (Infectivity before showing symptoms?)
Example: • Number of isolation units available
• Detection delay
• Effect of seclusion
• Fraction of asymptomatic (but infectious) infections
Slide 148 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
SARS: without intervention
... Isolation
Slide 149 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
SARS: Isolation
only 7 epidemics
in 10 simulations
Isolation: 100 cases (per 10000)
can be isolated at the same time.
Case detection:
• Cases can be detected
within 4 days (2 days).
if occurring,
the size of the epidemic
(number of cases)
is almost the same
... + seclusion
2727 cases prevented
-2951 cases isolated
-224 excess effort
Slide 150 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
SARS: Isolation + Seclusion (1)
if occurring,
the size of the epidemic
(number of cases)
is not much smaller
Isolation: 100 cases (per 10000)
can be isolated at the same time.
Case detection:
• Cases can be detected
within 4 days (2 days).
Seclusion: 100 cases (per 10000) can be
additionally observed at the same time.
only 3 epidemics
in 10 simulations
... 90% symptoms
6528 cases prevented
-1835 cases isolated
- 925 cases secluded
3768 cases
of transmission
prevented
Slide 151 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
SARS: Isolation + Seclusion (2)
Isolation: 100 cases (per 10000)
can be isolated at the same time.
Case detection:
• Cases can be detected
within 4 days (2 days).
Seclusion: 100 cases (per 10000) can be
additionally observed at the same time.
Symptoms: 90% of infected subjects
develop detectable symptoms
... improving diagnosis
900 cases prevented
-3914 cases isolated
-1538 cases secluded
-4552 excess effort
Slide 152 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
SARS: Isolation + Seclusion (3)
Isolation: 100 cases (per 10000)
can be isolated at the same time.
Case detection:
• Cases can be detected
within 4 days (2 days).
• Case detection improves in course of the
epidemic by 5% per day and, finally, cases
will be detected within 2 days (1 day).
Seclusion: 100 cases (per 10000) can be
additionally observed at the same time.
Symptoms: 90% of infected subjects
develop detectable symptoms
... + tracing
8185 cases prevented
- 519 cases isolated
- 140 cases secluded
7526 cases
of transmission
prevented
Slide 153 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
SARS: Isolation + Seclusion + Tracing
Isolation: 100 cases (per 10000)
can be isolated at the same time.
Case detection:
• Cases can be detected
within 4 days (2 days).
• Case detection improves in course of the
epidemic by 5% per day and, finally, cases
will be detected within 2 days (1 day).
Seclusion: 100 cases (per 10000) can be
additionally observed at the same time.
Symptoms: 90% of infected subjects
develop detectable symptoms
Contact tracing:
• Contact persons can be traced
within 2.0 days (0.5 day).
• 50% of the contacts of a case can be
traced, at maximum 20 contacts per
case.
• 200 contacts (per 10000) can be
observed at the same time for a period
of 7 days.
C
9015 cases prevented
- 110 cases isolated
- 57 cases secluded
-1052 contacts traced
-1003 contacts observed
6739 cases
of transmission
prevented
Slide 154 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Conclusions: Isolation, contact tracing
Isolation of cases is an effective intervention (again: targeted),
but its effectiveness strongly depends on the capacities
of the public health system
Seclusion of cases is an effective complement
As these interventions depend on the detection of cases,
rapid diagnostics becomes more and more important
Improving procedures (e.g. case detection occurs more quickly)
can substantially aid controlling
the spread of infection (again: time)
Contact tracing: very effective ( contact structures...)
Slide 155 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Summary Lesson 7
Successful interventions are immediate & targeted:
• Diagnose quickly
• Act quickly
• target Intervention !
Slide 156 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Blank page for documentation
Slide 157 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Program
Slides Topic Lesson
Introduction: Infectious diseases - how they emerge and disappear
1
SARS 2002/2003: why modeling, and what is a mathematical model? (the example of bacterial growth)
2
Deterministic models: SIR-model, theory, basic reproduction number R0, epidemic vs. endemic case
3
Vaccination: final size of an epidemic, critical vaccination coverage
4
SIR, add-ons: extending SIR to SEIR, SEIRS, etc. 5
Intervention planning: pandemic influenza preparedness planning using InfluSim (can interventions do harm to a population?)
6
Stochastic Models: from theory to reality, the epidemic as a random event, the role of chance: be lucky or not
7
The role of superspreaders 8
Le
sso
n 8
Slide 158 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Networks: Random vs. Scale-free
Barabasi AL, Bonabeau E. Scale-free networks. Sci Am 2003;288(5):60-9
Random network Scale-free network
Slide 159 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
The contact network
log(H
äufigkeit)
log(Kontakte)
Scale-free network H
äufig
ke
it
Kontakte
log(H
äufigkeit)
log(Kontakte)
Random network
Häufig
ke
it
Kontakte
log(H
äufigkeit)
log(Kontakte)
Scale-free network
mit Familienstruktur
Häufig
ke
it
Kontakte
"will
mei R
uah"
Fam
ilien
Slide 160 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
The structure of the contact network & corresponding epidemics
0
100
200
300
400
500
600
0 30 60 90 120 150 180 210 240 270 300
Day
Num
ber
infe
cted
.
IAR: 10%
IAR: 13%
IAR: 21%
1
10
100
1000
10000
1 10 100 1000
Degree
Ab
solu
te f
req
uen
cy
Without
superspreaders
Epidemic: slow
low infection attack
rate (IAR)
0
100
200
300
400
500
600
0 30 60 90 120 150 180 210 240 270 300
Day
Nu
mb
er
infe
cte
d
.
IAR: 26%
IAR: 30%
IAR: 33%
1
10
100
1000
10000
1 10 100 1000
Degree
Ab
solu
te f
req
uen
cy
With
superspreaders
Dmax=479 Dmax=33
Epidemic: fulminant
high infection attack
rate (IAR)
Slide 161 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Attacking a scale-free network
The accidental failure of a
number of nodes in a
random network can
fracture the system into
noncommunicating islands.
Scale-free networks are
robust in the face of such
failures.
...but they are highly
vulnerable to a coordinated
attack against their hubs.
Barabasi AL, Bonabeau E. Scale-free networks. Sci Am 2003;288(5):60-9
Slide 162 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Social networks
Sci Am 2005;292(3):42-49
Slide 163 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
What is a contact?
Kis
sin
g: "h
ello
"
Kis
sin
g: w
ith
in
tention
s
Ha
vin
g s
ex
Ha
ndsh
akin
g
In c
ine
ma
, p
ub
...
Ta
lkin
g to
each
oth
ers
Slide 164 / 164
1: In
tro
2: S
AR
S
3: S
IR, R
0
4: V
accin
atio
n
5: S
EIR
S…
6: In
fluenza
7: S
tochastic
8: S
uper Hans-Peter Duerr, www.uni-tuebingen.de/modeling
Summary Lesson 7
• The structure of the contact network determines the mode of
transmission of infection – it represents, however, often the most
unknown variable.
• The efficacy of interventions depends on the contact network if
specific groups or individuals are targeted, as is the case for contact
tracing or isolation.
• Stochasticity can change the actual course of an epidemic
decisively.
• Superspreaders can cause profound stochastic fluctuations in the
course of an epidemic, depending on, for instance, when they come
into play or how long they are infectious.