Download Modeling Infectious Diseases from a Real World Perspective

Modeling Infectious Diseases
from a Real World Perspective Wayne Getz
Department of Environmental Science, Policy and Management
What is disease? Disease is an abnormal condition that impairs
bodily functions
Infectious Disease is transmitted from one
individual to another (airborne, waterborne,
sexually transmitted, contact transmission)
Vectored Disease requires an agent to be involved
in the transfer
Zoonotic Disease has a non human source
Pathogens cause Disease
microparasites: virus, bacteria, protozoans, fungi
macroparasites: cestodes, nematodes, ticks, fleas
Disease is an ecological process Basic Elements
define species: single pop, vectored system,
ecological system
disease categories: infected vs infectious,
latent vs active, normal vs superspreader
demographic categories: gender, age, other
interventions: vaccination, quarantine, drug
regimens, circumcision,
time: fast diseases (e.g. pneumonia, influenza)
vs. slow diseases (e.g. TB, HIV, leprosy).
Emerging Infectious Diseases: What?, Where? How? and Why?
Cover: Vol 6(6), 2000
Emerging Infectious
Disease (CDC Journal)
Japanese color
woodcut print
advertising the
effectiveness of
cowpox vaccine
(circa 1850 A.D.)
WHAT?
(Definition from MedicineNet.com)
Emerging infectious disease: An infectious disease that has
newly appeared in a population or that has been known for some
time but is rapidly increasing in incidence or geographic range.
Examples of emerging infectious diseases include:
* Ebola virus (first outbreaks in 1976)
* HIV/AIDS (virus first isolated in 1983)
* Hepatitis C (first identified in 1989)
* Influenza A(H5N1) (bird ‘flu first isolated from humans in 1997)
* Legionella pneumophila (first outbreak in 1976)
* E. coli O157:H7 (first detected in 1982)
* Borrelia burgdorferi (first detected case of Lyme disease in 1982)
* Mad Cow disease (variant Creutzfeldt-Jakob: first described 1996)
More WHAT!
CDC National Center for Infectious Disease information list
for emerging and re-emerging infectious diseases
drug-resistant infections, bovine spongiform encephalopathy (Mad cow
disease) and variant Creutzfeldt-Jakob disease (vCJD), campylobacteriosis,
Chagas disease, cholera, cryptococcosis, cryptosporidiosis (Crypto),
cyclosporiasis, cysticercosis, dengue fever, diphtheria, Ebola hemorrhagic
fever, Escherichia coli infection, group B streptococcal infection, hantavirus
pulmonary syndrome, hepatitis C, hendra virus infection, histoplasmosis,
HIV/AIDS, influenza, Lassa fever, legionnaires' disease (legionellosis) and
Pontiac fever, leptospirosis, listeriosis, Lyme disease, malaria, Marburg
hemorrhagic fever, measles, meningitis, monkeypox, MRSA (Methicillin
Resistant Staphylococcus aureus), Nipah virus infection, norovirus (formerly
Norwalk virus) infection, pertussis, plague, polio (poliomyelitis), rabies, Rift
Valley fever, rotavirus infection, salmonellosis, SARS (Severe acute
respiratory syndrome), shigellosis, smallpox, sleeping Sickness
(Trypanosomiasis), tuberculosis, tularemia, valley fever (coccidioidomycosis),
VISA/VRSA - Vancomycin-Intermediate/Resistant Staphylococcus aureus,
West Nile virus infection, yellow fever
More WHAT!
CDC National Center for Infectious Disease information list
for emerging and re-emerging infectious diseases
drug-resistant infections, bovine spongiform encephalopathy (Mad cow
disease) and variant Creutzfeldt-Jakob disease (vCJD), campylobacteriosis,
Chagas disease, cholera, cryptococcosis, cryptosporidiosis (Crypto),
cyclosporiasis, cysticercosis, dengue fever, diphtheria, Ebola hemorrhagic
fever, Escherichia coli infection, group B streptococcal infection, hantavirus
pulmonary syndrome, hepatitis C, hendra virus infection, histoplasmosis,
HIV/AIDS, influenza, Lassa fever, legionnaires' disease (legionellosis) and
Pontiac fever, leptospirosis, listeriosis, Lyme disease, malaria, Marburg
hemorrhagic fever, measles, meningitis, monkeypox, MRSA (Methicillin
Resistant Staphylococcus aureus), Nipah virus infection, norovirus (formerly
Norwalk virus) infection, pertussis, plague, polio (poliomyelitis), rabies, Rift
Valley fever, rotavirus infection, salmonellosis, SARS (Severe acute
respiratory syndrome), shigellosis, smallpox, sleeping Sickness
(Trypanosomiasis), tuberculosis, tularemia, valley fever (coccidioidomycosis),
VISA/VRSA - Vancomycin-Intermediate/Resistant Staphylococcus aureus,
West Nile virus infection, yellow fever
=: first recognized ’93, rodent excretions, rare but deadly
More WHAT!
CDC National Center for Infectious Disease information list
for emerging and re-emerging infectious diseases
drug-resistant infections, bovine spongiform encephalopathy (Mad cow
disease) and variant Creutzfeldt-Jakob disease (vCJD), campylobacteriosis,
Chagas disease, cholera, cryptococcosis, cryptosporidiosis (Crypto),
cyclosporiasis, cysticercosis, dengue fever, diphtheria, Ebola hemorrhagic
fever, Escherichia coli infection, group B streptococcal infection, hantavirus
pulmonary syndrome, hepatitis C, hendra virus infection, histoplasmosis,
HIV/AIDS, influenza, Lassa fever, legionnaires' disease (legionellosis) and
Pontiac fever, leptospirosis, listeriosis, Lyme disease, malaria, Marburg
hemorrhagic fever, measles, meningitis, monkeypox, MRSA (Methicillin
Resistant Staphylococcus aureus), Nipah virus infection, norovirus (formerly
Norwalk virus) infection, pertussis, plague, polio (poliomyelitis), rabies, Rift
Valley fever, rotavirus infection, salmonellosis, SARS (Severe acute
respiratory syndrome), shigellosis, smallpox, sleeping Sickness
(Trypanosomiasis), tuberculosis, tularemia, valley fever (coccidioidomycosis),
VISA/VRSA - Vancomycin-Intermediate/Resistant Staphylococcus aureus,
West Nile virus infection, yellow fever
=: identified ’72, stomach flu on cruise ships, schools, hotels
More WHAT!
CDC National Center for Infectious Disease information list
for emerging and re-emerging infectious diseases
drug-resistant infections, bovine spongiform encephalopathy (Mad cow
disease) and variant Creutzfeldt-Jakob disease (vCJD), campylobacteriosis,
Chagas disease, cholera, cryptococcosis, cryptosporidiosis (Crypto),
cyclosporiasis, cysticercosis, dengue fever, diphtheria, Ebola hemorrhagic
fever, Escherichia coli infection, group B streptococcal infection, hantavirus
pulmonary syndrome, hepatitis C, hendra virus infection, histoplasmosis,
HIV/AIDS, influenza, Lassa fever, legionnaires' disease (legionellosis) and
Pontiac fever, leptospirosis, listeriosis, Lyme disease, malaria, Marburg
hemorrhagic fever, measles, meningitis, monkeypox, MRSA (Methicillin
Resistant Staphylococcus aureus), Nipah virus infection, norovirus (formerly
Norwalk virus) infection, pertussis, plague, polio (poliomyelitis), rabies, Rift
Valley fever, rotavirus infection, salmonellosis, SARS (Severe acute
respiratory syndrome), shigellosis, smallpox, sleeping Sickness
(Trypanosomiasis), tuberculosis, tularemia, valley fever (coccidioidomycosis),
VISA/VRSA - Vancomycin-Intermediate/Resistant Staphylococcus aureus,
West Nile virus infection, yellow fever
=: mosquito vector, 1st case N.Am. ’99 now ≈ 15000 cases 500 deaths
WHERE?
Global trends in emerging infectious diseases
Jones et al. Nature 451, 990-993(21 February 2008)
335 events
all pathogen types: 1940-2004
WHAT?
by decade
Jones et al.
Nature 451,
990-993(21
February 2008)
WHAT?
by decade
Jones et al.
Nature 451,
990-993(21
February 2008)
WHAT?
by decade
Jones et al.
Nature 451,
990-993(21
February 2008)
•
HOW?
Contacts with
wildlife
•
Vulnerability to
infection (elderly,
HIV+)
•
Strains evolving
to resist
treatments
•
Contact networks
particularly global
travel
•
new diagnostic
tools
SARS
Outbreak
Current risk of an EID zoonotic pathogen from
wildlife Jones et al. Nature 451, 990-993(21 February 2008)
Disease Categories and Transmission in
Kermack-Mckendrick Models
W. O. Kermack and A. G. McKendrick: A Contribution to the
Mathematical Theory of Epidemics, I, II (endemicity), and III (endemicity cont.)
I. Proc. R. Soc. Lond. A, 1927, 115, 700-721 (doi: 10.1098/rspa.1927.0118)
II. Proc. R. Soc. Lond. A, 1932, 138, 55-83 (doi: 10.1098/rspa.1932.0171)
III. Proc. R. Soc. Lond. A, 1933, 141, 94-122 (doi: 10.1098/rspa.1933.0106)
Hethcote, H. W. 2000. The mathematics of infectious disease.
SIAM Rev. 42, 599–653. (doi:10.1137/S0036144500371907)
Disease Categories and Transmission
SIR Models
S: susceptible,
I: infected & infectious
R: “recovered & immune” (V) or “removed” (D)
N: Does N=S+I+V change with time?
Units: numbers vs. densities. vs proportions.
Transmission: mass action (densities of SxI)
frequency dependent (proportion of SxI)
Be Warned!: transmission = bSI holds for both
frequency or mass action if N is constant or for
variable N(t) if units are density (mass action) or
proportions (frequency)
Epidemics with “lumped” demography
S: susceptible
E: exposed (infected)
I: infectious
V: recovered immune
D: dead
N: S+E+I+V
b0 bV: birth rate
!
!
! I !V
µ
! D !V
transmission rate
refraction rate (latent period)
reversion rate
natural mortility
disease induce mortality
Outline of remaining material
Preliminaries:
Discrete versus continuous models in biology
Discrete versus continuous models in epidemiology
Discrete multi-compartment formulations based on
probabilities
Case studies:
Bovine TB and Vaccination
Group structure and containment of SARS
TB and drug therapies, TB-HIV dynamics
General theory of heterogeneous transmission
Goals:
Provide a flavor of how to incorporate complexity
Illustrate how output used to understand complexities
Lead you into some literature for you to explore further!
Continuous versus discrete models in biology
Simplest model: constant pop N = S + I;
S → I, transmission β NS I:
! "
!
"
dI
S
I
= βI
= βI 1 −
, I(0) = I0.
dt
N
N
Logistic model with solution:
I0N Text
I(t) =
I0 + (N − I0)e−βt
Discretized system ODE:
!
"
I(t)
I(t + ∆t) ≈ I(t) + ∆tβI(t) 1 −
.
N
Discretized Solution:
I(t)N
I(t + ∆t) =
I(t) + (N − I(t))e−β∆t
Which is more appropriate?
Continuous versus discrete models in biology
Simplest model: constant pop N = S + I;
S → I, transmission β NS I:
! "
!
"
dI
S
I
= βI
= βI 1 −
, I(0) = I0.
dt
N
N
Logistic model with solution:
I0N Text
I(t) =
I0 + (N − I0)e−βt
Discretized system ODE:
!
"
I(t)
I(t + ∆t) ≈ I(t) + ∆tβI(t) 1 −
.
N
Discretized Solution:
I(t)N
I(t + ∆t) =
I(t) + (N − I(t))e−β∆t
Which is more appropriate?
Continuous versus discrete models in biology
Simplest model: constant pop N = S + I;
S → I, transmission β NS I:
! "
!
"
dI
S
I
= βI
= βI 1 −
, I(0) = I0.
dt
N
N
Logistic model with solution:
I0N Text
I(t) =
I0 + (N − I0)e−βt
Discretized system ODE:
!
"
I(t)
I(t + ∆t) ≈ I(t) + ∆tβI(t) 1 −
.
N
Discretized Solution:
I(t)N
I(t + ∆t) =
I(t) + (N − I(t))e−β∆t
Which is more appropriate?
Continuous versus discrete models in biology
Simplest model: constant pop N = S + I;
S → I, transmission β NS I:
! "
!
"
dI
S
I
= βI
= βI 1 −
, I(0) = I0.
dt
N
N
Logistic model with solution:
I0N Text
I(t) =
I0 + (N − I0)e−βt
Discretized system ODE:
!
"
I(t)
I(t + ∆t) ≈ I(t) + ∆tβI(t) 1 −
.
N
Discretized Solution:
I(t)N
I(t + ∆t) =
I(t) + (N − I(t))e−β∆t
Which is more appropriate?
Continuous versus discrete models in biology
Simplest model: constant pop N = S + I;
S → I, transmission β NS I:
! "
!
"
dI
S
I
= βI
= βI 1 −
, I(0) = I0.
dt
N
N
Logistic model with solution:
I0N Text
I(t) =
I0 + (N − I0)e−βt
Discretized system ODE:
!
"
I(t)
I(t + ∆t) ≈ I(t) + ∆tβI(t) 1 −
.
N
Discretized Solution:
I(t)N
I(t + ∆t) =
I(t) + (N − I(t))e−β∆t
Which is more appropriate?
Which is the
better
discretization
scheme?
Continuous versus discrete models in biology
Time (Δt=0.25)
Time (Δt=0.05)
Solid line: Iteration using solution
Circles: Iteration using discretized equations
Continuous
Models
with
Demography
dS
= f recruitment (S, I, R) − f transmission (S, I, R)S − µS
dt
dI
= f transmission (S, I, R)S − (α + µ ) I
dt
dR
= α I − µR
dt
f recruitment : recruits and/or births
µ: natural mortality rate
Elaborations:
α : infectious → removed/recovered
1. exposed class E
2. constant rate “exponential” transfers: → Weibull distribution
OR
→ “box car” staging: gamma distribution
breeders and long-lived species). Data reporting the proportion pµ of individuals
step.
Fortunately,
a good
approximation
can be
obtained by
that die in a unit of
time can
be converted
to a mortality
rate parameter
µ appearing
linear modeling
approach,
as= follows.
−µN by noting that the solution
in a differential equation
model of the
form dN
dt
we write
down
time
to this equation over First
any time
interval
[k, kthe
+ 1]continuous
is N (k + 1)
= Nmodel
(k)e−µof
. interest
This
Proportion thatample,
die or equations
make
transitions:
e.g.
mortalitytotal
rate recruitment rate λ
(4.1) dying
with
ais constant
implies that the proportion
of individuals
Some basics on discrete epi models
(6.1)
class
and a constant per-capita
natural mortality rate µ. Then,
N (k) − N (k + 1)
N (k)(1 − e−µ )
pµτ=
= (see equation=(3.1)),
1 − e−µthese equations be
(I, N ) := pT C(N )I/N
N (k)
N (k)
"
!
1
dS
or,
equivalently,
µ
=
ln
.
1−pµ
= λ − µS − τ (I, N )S
S(0) = S0
Continuous model SEI:
It is often advantageous to formulate dt
epidemiological and demographic models
dE
in discrete time. The
primary
advantage
of differential
equation
(6.2)
= τ (I, N )S
− (δ +models
µ)E disappears
E(0) = E0
dt systems rather than trying to obtain
once we resort to numerical simulation of
MODELING
THE INVASION
AND
SPREAD
OF CONTAGIOUS
dIimpossible
analytical
results, which
are difficult
if not
to obtain forDISEASES
most detailed 1
= δE − (α + µ)I
I(0) = I0 .
nonlinear models. Indeed, discrete time models
can
be
implemented
very
naturally
dt
and
easily
indiscrete
computer
simulations,
whilewenumerical
solutions
heEquivalent
time
interval
[k, k +SEI:
1].
In
this
case
obtain on
the
note
transmission
depends
k: modelof differential equaNow assume
over
a small interval
t ∈ [k, kAlso
+ 1]parameters
that the propor
tions requires algorithms
that use
discretizing
approximations.
in
6.5)

 over
 this interval due to a change in I(t) is sufficiently smallth
discrete
(1 −bepµmore
)(1 −easily
pτk ) related to0data that have been
0 collated
S(k +time
1) models can
overE(k
discrete
rates, N
etc.).
well
approximated
by τk(1=−τp(I(k),
(k)) (e.g. over

=
 (e.g.
i
+ 1) intervals
(1vital
− pµand
)pτktransmission
0 the time
µ )(1 − pδ )
Discrete timein
equations,
cannot properly
account for
the interactions
τ (I(t), Nhowever,
(t))
percent,
our
I(k + 1)
0 may be a few
(1 −
pµ )pδ but (1
− estimates
pµ )(1 − pαof) the
of simultaneously pnonlinear
processes,
as individuals
simultaneously
subject
to un
N
as defined
byexpression
(3.2), may
have
τ (I(t),
(t)),such
T in
S(k)
(1 Obviously
−time
pµ )λ
the processes of infection
death:
in each
interval
weassumption
either first account
for
several and
times
as large).
this
will influence
 +orvice versa.
infection and thenstep
natural
mortality
It 
does
a difference
how we
E(k)
0 time
×
, make
duration,
with shorter
steps
required
for
faster-growin
28
schedule things [54].
Alternatively
we
can
treat
the
two
processes
simultaneously,
I(k)
0
replacing τ (I(t), N (t)) with the constant τ for t ∈ [k, k + 1], e
k
dI1
= δ(En − I1 )
Ex: Use analytical/ numerical methods
dtto
dIj
Characterize the distribution of R(t) in
the=SE
n Ij−1
m R−model
δ(I
Ij ), j with
= 2, .S(0)
. . , m=
tribution
R(t)
the
with
S(0) ==0 in terms of β,
S0 , Ei (0) of
= 0,
i = in
1, ...,
n, SE
Ij (0)
=R0, model
j dt
= 1, ...,
m, R(0)
n Im
dR
δ, µ,
discrete
and compare
, ...,
n,mIjand
(0) n=for
0, the
j =continuous
1, ..., m, and
R(0)
= 0 =informulations
terms
of β,
δIm − µR
(start with µ =and
δ = 1 discrete
and m = 1formulations
and investigatedt
inand
the discrete
model δ < 1)
e continuous
compare
!model
#δ < 1)
dm
=
1
and
investigate
in
the
discrete
m
"
Continuous
Discrete
dS
= −β
Ij S
! m #
! m
#
dt
"
"
j=1
dS
! m S(t
# + 1) = S(t) − β
= −β
Ij S
Ij (t) S(t)
"
dt
dE1
j=1
j=1
Ij S − δE1
! m # dt = β
! m #
j=1
"
"
dE1
dEδE
= β
Ij S −
i 1
E1E(t),+ i1)= 2,=. . . ,βn
Ij S + (1 − δ)E1
= δ(Ei−1 −
dt
i
j=1
dt
j=1
dEi
dt
dI1
dt
dIj
dt
dR
dt
dI1
= . .δ(E
I1i)(t + 1) = δEi−1 (t) + (1 − δ)Ei (t)
= δ(Ei−1 − Ei ), dt
i = 2,
. , nn − E
i = 2, . . . , n
dIj
= δ(Ij−1 − Ij ), j = 2, . . . , m
dt
= δ(En − I1 )
I1 (t + 1) = δEn (t) + (1 − δ)I1 (t)
dR
= δIm − µR
Ij (t + 1) = δIj−1 (t) + (1 − δ)Ij (t)
dt
= δ(Ij−1 − Ij ), j = 2, . . . , m
= δIm − µR
j = 2, . . . , m
R(t!+"
1) =# δIm (t) − µR(t)
m
S(t + 1) = S(t) − β
Ij (t) S(t)
j=1
29
First Case Study:
Bovine TB in African Buffalo
Cross & Getz (2006) Ecological Modelling 196: 494-504.
Important elements:
Includes demography
Herd structure: focus on one herd embedded in background
prevalence assuming balanced movement into and out of
herd
SVEID structure (Susc, Vaccinated, Exposed, Infected,
Dead)
BTB model with demography & ecology
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Second Case Study: SARS
Lloyd-Smith, Galvani, Getz (2003) Proc. Royal Soc. B 270: 1979-1989.
Important elements:
No demography but group structure for disease classes
Group structure relates to intervention and control strategies
Time iteration is daily: relates to reporting and data structure
Global emergence
of SARS, 2003
Hong Kong
>195 cases
Guangdong
province,
A
China
Canada
F,G
29 cases
F,G
A
A
H,J
H,J
K
Hotel M,
Hong Kong
B
K
Ireland
0 cases
C,D,E
I,L,M
C,D,E
B
Vietnam
58 cases
I,L,M
United
States
Singapore
71 cases
1 case
Adapted from Dr. J. Gerberding, Centers for Disease Control
SARS transmission chain, Singapore 2003
Morbidity & Mortality Weekly Report (2003)
Group-level heterogeneity for SARS
Health care workers (HCWs) comprised 18-63% of cases in
different locales
• Main control measures were hospitalization and quarantine.
• Strict infection control implemented in hospitals, and contacts
with visitors were reduced.
β
ρβ
Community
ηβ
HCWs
ρβ
SARS Patients
κηβ
Detailed
structure of
SARS: results
from daily
iterated
stochastic
simulations
ij
(c) symptomatic
modifying transmission for different settings:
ε, η, γ, substructure
κ, and ρ (all on the interval
21
r 0.15
1– individuals
1– r (indaybaseline
b
0.08–0.26
2 1transmission
[0,) 1]). In
particular,
the
reduced
rate
of
exposed
(E)
transmission rate (day )
b
I
I
I
x1
x2
x3
x4
cluded because the extent of pre-symptomatic transmission
of SARS was
= 3)
(R0 = 1.5–5)
(R0Iunknown
((IR
R
h:when
health
workers;
c: general
managed
r
r
1 m: within
1
thecare
model
was created)
is εβ. community;
All transmission
the patients
hospital
setting
occursowing
at a reduced
rate ηβ to reflect contact precautions adopted by all hospital
ionfactors
rate,
to: transmission
d ) transmission
substructure
modifying
rate, (owing
to:
and patients, such as the use of
and gowns. Additionally,
« masks, gloves
ssionpersonnel
0–0.1
0.1
hbk
«
pre-symptomatic transmission
I
l ha factor γ, yieldm
quarantine of exposed individuals reduces their contact
rates by
cautions
h
0–1
0.5
hospital-wide
contact rate
precautions
hidentified
ing
a total transmission
of γεβ, while specific
isolation
measures
for
hb
Ih + e Eh 0–1
ty mixing
r
0.5
reduced
HCW–community
mixing
r factor
SARS patients (Im ) in the hospital reduces their transmission rby ba further
Ic + e Ec
k
0–1
1
isolation
rb transmissionk rates beκ.case
Finally,
we considered the impact of measures to reduce
0–1 these assumptions,
0.5
quarantine
g e Em
tween
HCWs and community members gby a factor ρ. Under
lc
bg
the transmission hazards are:
daily probability of:
β(Ic + εEc ) + ρβ(Ih + εEh ) + γβεEm
τ
=
c
quarantining
of community
incubating
individuals
g individuals
in the
0–1 ‘m’), representing
0q
q Ninc the community
case management
(subscript
qu
or case isolation. We categorized disease classes a
and(Ec)
ηβ(Iin
εEhcommunity
+ κIm )0–1(Ec , Eh , Em 0.3
isolation
individuals
hcsymptomat
h +the
individuals
in of
thesymptomatic
community
),
τh = ρτc +tible h(cSc , Sh ), incubating
,
(Ic)
Nh
owing to recovery or death (Rc ,
Im ) and removed
where
Ii , i = c, h, represent
over all in
sub-compartments
in
isolation
of symptomatic
HCWsmodel
(sums
Ih)hhupdates
hhthe incu- t
1-day
HCWs
(IhE
) i and
0.9timesteps, representing
0.9
bating and symptomatic classes for est
pool
j, andfor which people’s activities can be th
interval
beEequivalent.
Substructures associated with dai
Nh = S h +
h + Ih + Vh + Im
ments are summarized in figure 1, with details
and
the caption.
Stochastic
transitions
of individuals
Hospital-wide
contact
precautions,
such
as
the
use
at
all
possible.
In
cautions, such as N
the
use
at
all
possible.
In
the
case
of
SARS,
non-sp
+ Vc + ρ(S
Ehthe
+ Iprobabilities
c = Sc + Ec + Ic classes,
h +on
h + Vh ).
based
shown
in figur
times of sterile gowns, filtration masks and gloves, modify
ures may
ha
Equations: transmission hazard
ation masks and gloves, modify
ures may have helped or hindered earl
and
Epi Equations:
Nc = Sc + Ec + Ic + Vc + ρ(Sh + Eh + Ih + Vh ).
The detailed form of the SID equations that were formulated are:
Community and HCW equations:












q:
Si (t + 1) = exp (−τi (t)) Si (t)
Ei1 (t + 1) = [1 − exp (−τi (t))] Si (t)
Eij (t + 1) = (1 − pj−1 )(1 − qij−1 )Eij−1 (t) j = 2, . . . , 10
!10
Ii1 (t + 1) = j=1 pj (1 − qij )Eij (t)
i = c, h,

Ii2 (t + 1) = (1 − hi1 )Ii1 (t)




Ii3 (t + 1) = (1 − hi2 )Ii2 (t) + (1 − r)(1 − hi3 )Ii3 (t)



Iij (t + 1) = r(1 − hi j−1 )Ii j−1 (t) + (1 − r)(1 − hij )Iij (t) j = 4, 5 


i
Vi (t + 1) = Vi (t) + rIi5 (t) + rIm5 (t)
& i
'

i
Em,j (t + 1) = (1 − pc j−1 ) Em,j−1 (t) + qj−1 Ec j−1 (t)
j = 2, . . . , 10 

&
'
!

10
i
i

Im1 (t + 1) = j=1 pj Emj (t) + qij Eij (t)



i
i
Im2 (t + 1) = hi1 Ii1 (t) + Im1 (t)
(
)
i = c, h.
i
i
i

Im3 (t + 1) = hi2
Ii2 (t) + Im2 (t) + (1 − r) )hi3 Ii3 (t) + Im1 (t)

(

i
i


Imj (t + 1) = r hi j−1 Ii j−1 (t) + Im( j−1 (t)

)

i
+(1 − r) hij Iij (t) + Imj (t) j = 4, 5
In the analysis presented here, the probabilities qij and hij vary between 0
quarantine
h: hospitalization
rates;forr:delays
recovery/death
and a fixed rates;
value less
than 1 and account
in contact tracing or case
identification. In addition, we did not analyze scenarios where health care workers
are quarantined so that qhj = 0 for all j. Deterministic solutions to this SARS
Parameter values used in
simulations
1980 J. O. Lloyd-Smith and others SARS transmission in a community and hospital
Table 1. Summary of transmission and case-management parameters, including the range of values used throughout the study
and the three control strategies depicted in figure 3.
parameter
symbol
range
examined
figure 3
(1)
figure 3
(2)
figure 3
(3)
0.15
(R0 = 3)
0.15
(R0 = 3)
0.15
(R0 = 3)
baseline transmission rate (day2 1)
b
0.08–0.26
(R0 = 1.5–5)
factors modifying transmission rate, owing to:
pre-symptomatic transmission
hospital-wide contact precautions
reduced HCW–community mixing
case isolation
quarantine
«
h
r
k
g
0–0.1
0–1
0–1
0–1
0–1
q
daily probability of:
quarantining of incubating individuals in the community
(Ec)
isolation of symptomatic individuals in the community
(Ic)
isolation of symptomatic HCWs (Ih)
0.1
0.5
0.5
1
0.5
0.1
0.9
1
0.5
0.5
0.1
0.5
0.5
0.5
0.5
0–1
0
0.5
0.5
hc
0–1
0.3
0.9
0.9
hh
0.9
0.9
0.9
0.9
1984 J. O. Lloyd-Smith and others SARS transmission in a
Individual runs: Cumulative cases for different R (effective
reproduction numbers--i.e.
R0 when some control is applied)
(a)
100
R = 2.0
R = 1.6
R = 1.2
cumulative cases
80
60
40
20
0
0
50
100
150
days since initial case
200
Probability of epidemic containment for different effective
(b)
R’s
200
1.0
Pr(containment)
probability of containment
1.0
0.8
0.6
0.9
0.8
0.7
0.6
0.8
0.4
1.0
R
1.2
0.2
0
0
2
4
6
reproductive number, R
8
10
R=1 contours (right side of curves guarentees control of
epidemic) for the effects of isolation leves hc and
transmission curtailment (1-κ) for epidemics with
different R0
η: hospital precautions
reduce transmission by
1/2
3 lines right to left:
increasing delays
in isolation of patients
20
0
50
0
probability
probabil
cumula
40
0.4
0.8
0.4
1.0
0.6
R
0.8
1.2
1.0
0.2
Combinations of policies
that lead to
containment: plots of
R=1 contours
0
100
150
200
50
100
150
days since initial case
days since initial case
0.2
20
0
1.2
R
4
6
8
10
0 reproductive
2
4
6
8
number, R
reproductive number, R
(three lines represent increasing delays in isolating patients)
(d )
h=1
0.8
0.6
0.4
0.2
0
0.2 0 0.4 0.2 0.6 0.4 0.8 0.6
1.00.8
transmission intransmission
isolation, kin isolation, k
(e)
(d
R 0) = 5.0
1.0
h=1
0.8
0.6
0.4
0.2
0
1.0
(f)
1.0
daily probability of isolation, hc
R 0 = 1.5
daily probability of isolation, hc
daily probability of isolation, hc
R 0 = 1.5
(c) 3.0 2.5 2.0
0 4.0
5.0 4.0 3.0 2.5 2.0
1.0
200
= 0.5
0.8
B
B
0.6
0.4
0.2
0
0.2 0
0
4.0
3.0
2.5
2.0
3.0
2.5
R 0 = 5.0 4.0
h = 0.5
h
A
1.0
0.8
transmissiontransmission
in isolation,ink isolation, k
(f)
0.40.2
A
0.6
0.4
0.8
0.6
proba
cumulativ
control
ect case isolation. Here, h = 0.5, r = 1,
0.9, so the case isolation strategy is
h h = 100
to commence after the first day of
degree50
to which transmission is reduced
es. From right to left, two solid lines of
t 1-day 0and 5-day delays in contact
0
ntining begins.
Solid50lines show100
the case 150
(days)
mission can occur during thetime
incubation
ines show the case « = 0 (please note
=) 2.44, 2.92 and 3.90, rather than 2.5, 3
(c)
ty of 1.0
effective reproductivecontrol
number
R to
precautions
n-control parameters. In allquarantine and isolation
, q = 0.5, h c = 0.3 and h h = 0.9;
all measures
combined
h were0.8
set to 0.5, then varied one at a
0.2
Probability of containment
in terms of
implementation of control
after
epi
onset
2000 0
50
100
0.6
1.0
probability of containment
probability of containment
in implementation
Left: 3 strategies; Right: combineddelay
measure
for 3(days)
R0
0.8
150
R0 = 2
R0 = 3
R0 = 4
0.6
egies and delays in
on 0.4
0.4
the importance of various control meairs, we now consider the effects of intetegies0.2
on SARS outbreaks. We treat a
0.2
3, similar to outbreaks in Hong Kong
e median and 50% confidence intervals
0
0
75th percentile
values)
of cumulative
0
50
100
150 0
50
100
hat such an epidemic
is
likely
to
spread
delay in implementation (days)
delay in implementation (days)
population if uncontrolled (figure 3a,
Figure 3. (Caption overleaf.)
)ol strategies emphasizing contact pre1.0 lines) or quarantine and isoa, green
150
number
10
Importance of HCW mixing
restrictions
ρ
in
5
preventing epidemics (control after 14 days):
0
20 and
4050060runs
80
100
988 J.histograms
O. Lloyd-Smith and others
transmission
community
-- 1SARSrun;
piein 0acharts
--hospital
time (days)
c=community pool,
h=hospital
pool
Third, the hospital
pool is considered to in
(b)
25
c–to– c
13%
number of infections
20
c– to –h
15%
15
h –to– h
57%
h– to –c
15%
10
5
0
0
20
40
60
time (days)
and SARS cases, but other patients are
25
explicitly. Infection
of other patients
has pla
r = 0.1
c– to –c
c –to– h
r=1
cant part
4% in some
4% outbreaks, though it
important
in hospitals that eliminate nonh –to–
c
20
cedures
while SARS remains a significant ri
2%
al. 2003; Maunder et al. 2003), and for regi
h– to–hospitals
h
opened dedicated SARS
or wards.
15
90%
on hospital-community SARS outbreaks co
ate patient dynamics, and could also evaluat
staff reductions (Maunder et al. 2003) or of
10
tine of hospital staff following diagnosis of
(as reported by Dwosh et al. 2003).
5
Some caution is required in identifying R0
with that obtained from incidence data for p
breaks. As discussed already, reproduct
0
derived
incorporate
80
100 0
20 from40data inevitably
60
80
100so
control owing to routine healthcare practice
(days)
late R0 from itstime
formal
definition, however, as
number of infections
a)
Third Case Study: TB in Humans
Salomon, Lloyd-Smith, Getz, Resch, Sanchez, Porco, & Borgdorff,
2006. PLoS Medicine. 3(8), e273.
Sánchez M. S., J. O. Lloyd-Smith, T. C. Porco,B. G. Williams, M. W.
Borgdorff, J. Mansoer, J. A. Salomon, W. M. Getz, 2008. Impact of
HIV on novel therapies for tuberculosis control. AIDS 22:963-972.
Important elements:
Includes important disease classes relating to latent vs.
active, sputum smear positive vs. negative TB, DOTS vs
Non-DOTS treatment, detectable vs. non-detectable
Follows a competing rates formulation
Time iteration is monthly: relates well to treatment regimen
TB in and HIV background
Core model of TB – elaborated SEIR framework
Active TB
Susceptible Latent
Slo
ss
Fas
ss
Sus
Re
Recovered
Not shown:
all classes suffer natural mortality
active cases suffer additional mortality
TB treatment model
Susc.
Latent
Slo
Active TB
ss
ss−
det
ss
ss+
det
“Detectable”
cases
Sus
Fas
Re
Recovered
“Nondetectable”
cases
TB treatment model
Susc.
Latent
Slo
Active TB
ss
ss−
det
ss
ss+
det
Sus
Fas
Under treatment
ss+ non-DOTS
full
rec
Recovered
partial
rec.
ss+
Partially
recovered
Defaulters
Tx
Completers
TB treatment model
Susc.
Latent
Slo
Active TB
ss
ss−
det
ss
ss+
det
Sus
Fas
Under treatment
ss+ non-DOTS
full
rec
partial
rec.
ss+
partial
rec.
ss-
TB treatment model
Slo
ss− DOTS
ss
ss−
det
ss− non-DOTS
ss
ss+
det
ss+ DOTS
Sus
Fas
ss+ non-DOTS
full
rec
partial
rec.
ss+
TB/HIV treatment model
TB
HIV−
TB
HIV+ stage I
TB
HIV+ stage II
TB
HIV+ stage III
TB
HIV+ stage IV
HIV
incidence
(external input)
HIV
progression
HIV
mortality
partial
rec.
ss-
TB treatment model
Slo
ss− DOTS
ss
ss−
det
ss− non-DOTS
ss
ss+
det
ss+ DOTS
Sus
Fas
ss+ non-DOTS
full
rec
partial
rec.
ss+
TB
REINFECTION
HIV
no HIV
progression progression
Slow
Fas
HIV+ stage II
full
rec
Slow
HIV+ stage III
Fas
TB-HIV CO-DYNAMICS
IN KENYA:
Monitoring Interacting Epidemics
Sánchez M. S., J. O. Lloyd-Smith, B. G. Williams, T. C. Porco,
S. J. Ryan, M. W. Borgdorff, J. Mansoer, C, Dye, W. M. Getz,
2009. Incongruent HIV and Tuberculosis Co-dynamics in
Kenya: Interacting Epidemics Monitor Each Other. Epidemics
1:14-20.
Tuberculosis notification rate, 2004
Notified TB cases (new
and relapse) per 100 000
population
0 - 24
25 - 49
50 - 99
100 or more
No report
The boundaries and names shown and the designations used on this map do not imply the expression of any opinion whatsoever on the part of the World Health Organization concerning the legal status of any
country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries.
Dotted lines on maps represent approximate border lines for which there may not yet be full agreement.
 WHO 2005. All rights reserved
HIV prevalence in adults, 2005
38.6 million people [33.4-46.0 million] living with HIV, 2005
Estimated HIV prevalence in new adult TB cases,
2004
HIV prevalence in TB
cases, 15-49 years (%)
0-4
5 - 19
20 - 49
50 or more
No estimate
The boundaries and names shown and the designations used on this map do not imply the expression of any opinion whatsoever on the part of the World Health Organization concerning the legal status of any
country, territory, city or area or of its authorities, or concerning the delimitation of its frontiers or boundaries.
Dotted lines on maps represent approximate border lines for which there may not yet be full agreement.
 WHO 2005. All rights reserved
350
TB case notification rate
300
Misfit of TB Data in Kenya in HIV-TB model
Data
Calibration to data up to 2004
Calibration to data up to 1997
Uncertainty bounds to pre-1997
250
200
150
100
50
0
1980
1985
1990
Year
1995
2000
2005
Case Study: Circumcision & HIV
Williams, B.G., Lloyd-Smith, J.O., Gouws, E., Hankins, C., Getz,
W.M., Dye, C.,1, Hargrove, J., de Zoysa, I., Auvert, B, 2006.
The potential impact of male circumcision on HIV incidence, HIV
prevalence and AIDS deaths in Africa. PLoS Medicine 3(7):e262.
Important elements:
two sex model
circumcised versus uncircumcised male categories
Weibull
Circumcision reduces female to male transmission of HIV by 70%
Circumcision reduces female to male transmission of HIV by 70%
green: S. Af.; red. E. Af.; orange: cent. Af.; blue, W. Af.
Circumcision%
reduces
female to male transmission
of HIV by 70%
% prevalence
circumcised
Currently
MC equivalent to a vaccine with 37% efficacy
%prevalence
reduction
impact
numbers
1000s p.y.
Stochastic models in
homogeneous populations
Discrete Markov Chain Binomial Models
Reed-Frost (class room lectures late 1920s at Johns Hopkins)
E.g. Daley and Gani’s book: Epidemic Modelling, 1999
Graph theory interpretations of Reed-Frost models
unidirected graph on N nodes, probability p of connections
Giant component iff R0=pN>1 ⇒ z = 1 - exp(R0z)
where z is expected value for (1-S∞)
Stochastic models in
homogeneous populations
Continuous time stochastic jump process models
SIR + demography
E.g Ingemar Nasell, Math. Biosci. 179:1-19, 2002.
Stochastic simulation of discrete time equivalents of
SIR models with demography (including age structure)
(e.g. HIV models, TB models, SARS models, bovine TB models)
Problem with homogeneity!
1. Variation in host behavior: contact rates
2. Variation in host susceptibility: probability of infection
3. Variation in intensity of host infectivity: probability of
infection
4. Variation in period of infectiousness: number of contacts
and probability if infection
5. Several host strains with varying transmissibility and
virulence.
6. Lots of others!
Superspreaders: the effect of
heterogeneity on disease emergence
Lloyd-Smith, J. O., S, J. Schreiber, P. E. Kopp, and W. M. Getz, 2006. Superpreading
and the impact of individual variation on disease emergence. Nature 438:335-359.
Heterogeneity and epidemiology
We have discussed disease models that
assume homogeneous
What about populations with heterogeneity?
Common approach: break population into many
sub-groups, each of which is homogeneous.
What about continuous variability among
individuals within well-mixed groups?
Homogeneous models of disease: Individual Level
Galton-Watson branching process theory:
A probability generating function approach
∞
1. Probability that I infects k individuals is qk : q = {qk }k = 0
∞
2. Probability generating function gq (z) = ∑ qk z k , 0 ≤ z ≤ 1
k =1
3. zn is probability I(t) = 0 at generation n : zn = gq (zn −1 ), z1 = q0
4.
gq (0) = q0 ,
gq (1) = 1,
gq′ (1)=R0
5. Each individual expects to infect ν : Poisson process: gq (z) = eν (z −1)
Invasion condition (infinite pop size assumption, fixed generation time):
Determistic: R0>1
Stochastic (homogeneous): R0>1 ⇒ prob{invasion}=1-1/R0
Heterogeneous models of disease:
Individual Level
5. Each individual expects to infect ν (homogenous ⇒ Poisson process)
6. If ν is itself distributed (e.g. gamma) then process
is not Poisson (e.g negative binomial)
Parent distribution:
Individual reproductive
number ν
Offspring distribution:
Distribution of cases
caused by particular
individuals
Standard Model I
Completely homogeneous population, all ν = R0
Constant
Parent distribution
ν
Poisson
Offspring distribution
Z
fν (x) = δ ( x − R0 )
∞
g(z) = ∫ e
− x (1− z )
0
= e − R0 (1− z )
fν (x)dx
Standard Model II (SIR)
Homogeneous transmission, constant recovery
Exponential
Parent distribution
ν
Geometric
1 − x R0
fν (x) =
e
R0
∞
Offspring distribution
g(z) = ∫ e − x (1− z ) fν (x)dx
Z
= 1 + R0 (1 − z )
0
New Model
Heterogeneous force of infection
(superspreaders in right-hand tail)
Gamma
Parent distribution
1  kx 
fν (x) =
Γ ( k )  R0 
k −1
 k  − kx R0
 R  e
0
ν
∞
Negative Binomial
Offspring distribution
Z
g(z) = ∫ e − x (1− z ) fν (x)dx
0
R0


=  1 + (1 − z )


k
−k
offspring
Z
ν ~ gamma
Z ~ negative binomial
Geometric
k=1
Relatively
few
Poisson
k→∞
parent
ν
0.1
1
10
100
k
greater Dispersion
individualparameter,
heterogeneity
(parameter k)
∞
Empirical distributions
The unprecedented global effort to contain SARS generated extensive
datasets through intensive contact tracing: unique opportunity to study
individual variation in a disease of casual contact.
Beijing: Shen et al. EID (2004)
Singapore: Leo et al. MMWR (2003)
Superspreading events: Definition?
Useful concept?
Currently not useful!
Should measure
variation
Beijing SARS hospital outbreak, 2003
Number of secondary cases: note superspreader
events in tail
What fits best?
1.
ν ~ constant
⇒ Z ~ Poisson
2.
ν ~ exponential
⇒ Z ~ geometric
3.
ν ~ gamma
⇒ Z ~ negative binomial
Singapore SARS outbreak, 2003
Singapore SARS outbreak, 2003
Singapore SARS outbreak, 2003
Singapore SARS outbreak, 2003
ν parent
distribution
Z offspring
distribution
ΔAICc
Akaike weight
ν ~ constant
Poisson
250.4
< 0.0001
ν ~ exponential
Geometric
41.2
< 0.0001
ν ~ gamma
Negative binomial
0
>0.9999
Model selection strongly favours NB distribution with MLE
parameters R0=1.63, k=0.16.
ν
Singapore SARS outbreak, 2003
Parent distribution ν is highly overdispersed:
variance-to-mean ratio = 16.4
Singapore SARS outbreak, 2003
c.f. “20/80 rule”: 20% of cases cause 80% of transmission
Evidence
heterogeneity in
other diseases
SARS, smallpox,
monkeypox, pneumonic
plague, avian influenza,
rubella
All show strong evidence
for individual variation
P = Poisson model for Z
generally rejected
G = geometric model
NB = negative binomial
model
Revisiting the 20/80 rule
Superspreaders
Superspreading Events (SSEs)
How many cases make an SSE?
SARS, 2003:
•
•
•
•
Z ≥ 8, Shen et al. Emerg. Infect. Dis. (2003)
Z > 10 Wallinga & Teunis, Am. J. Epidem. (2004)
Z ≥ 10 Leo et al. MMWR (2003)
“many more than the average number”, Riley et al. Science(2003)
But what about measles (R0~18) or monkeypox (R0~0.8)?
How to account for the influence of stochasticity?
We need a general, scaleable definition of a SSE, based on probabilistic
considerations.
Proposed definition for superspreading events
1.
2.
3.
Set context for transmission by estimating effective R0.
Generate Poisson (R0) representing expected range in Z due to
stochastic effects in absence of individual variation
Define an SSE as any case who infects more than Z(99) others,
where Z(99) is the 99th percentile of Poisson (R0).
Superspreading events (SSEs)
■ R0
+ 99th percentile
of Poisson (R0)
♦ reported SSEs
 SSEs with >1
index case
Superspreading Load
Calculate R0 from data and ZPois-99 using Poisson model
(number of infections demarcating 99 percentile)
Fit negative binomial NegB(R0,k) to data
Construct cummulative distribution ΦNB(ZPois-99 )
Calculate proportion in tail beyond ZPois-99
ΨNB(ZPois-99 ) =1-ΦNB(ZPois-99 )
Superspreader load (SSL) is 1-ΨNB(ZPois-99 ) /0.01
Predicting frequency of SSEs in Negative
Binomial epidemics NegB(R0,k)
NegB(10.3,1): SSL≈18
Implications for disease invasion
Data from 10 diseases of casual contact show that
individual variability in ν is a universal phenomenon.
How does this variability affect:
• Probability of stochastic extinction? (infinite population)
• Timing of extinction?
• Size of minor outbreak? (i.e prior to extinction)
• Rate of growth if major outbreak occurs?
We explored these questions using
branching process models for ν ~
gamma
Various Gamma distributions with R0=1.5
Special cases: k = 1
k=infty
exponential ν: Geometric offspring dist.
constant
ν: Poisson offspring dist.
smaller k greater variance in ν: Neg Biomial offspring dist. more aggregated
Probability of
disease extinction
Greater variation in ν
favors stochastic
extinction, due to
higher Pr(Z=0).
Time to stochastic
extinction
High variability in ν
(small k) ⇒
extinction happens
fast or not at all.
Implications for
detection of emerging
pathogens
Expected size of minor outbreak
(i.e. epidemic in infinite pop goes extinct)
R0 < 1 E(total # cases) = 1/(1-R0)
i.e. independent of k
R0 > 1 E(total # cases)
depends very weakly on k
Rate of growth of major epidemic
Greater variability ⇒ major outbreaks are rare but explosive!
Conclusion
• Data imply considerable heterogeneity in epidemics
• Heterogeneity needed to explain rare explosive
outbreaks, as in SARS
• To estimate level of heterogeneity we need BOTH R0
and p0 (proportion of cases NOT transmitting) or
SSL statistic
• Control measures should target individuals in tails of
parent distribution and hence reduce probability of
explosive outbreaks
How to do this an important area of research?
Thanks!
The End
This research is supported by the US NIH, NSF,
and James S. McDonnell Foundation