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APLICACION DEL MODELAMIENTO
MATEMATICO EN LA PREDICCION DE
ESCENARIOS DE RIESGO EN EPIDEMIOLOGIA
Carlos Castillo-Chavez1
Regents Professor, Director
Joaquin Bustoz Professor of Mathematical Biology
Simon A. Levin
Mathematical, Computational
and Modeling Sciences Center
Tempe, AZ 85287-1904, USA Universidad de Lima
Lima, Peru
Miercoles, November 25, 2015
1
[email protected];: https://twitter.com/mcmsc01
Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious d
Dr
aft
Steven Strogatz.
The Real Scientific Hero of 1953.
New York Times (1923-Current file), page 1, 2003.
Three ways of doing science brought by James
D. Watson and Francis Crick & the inventors
of the computer experiment: Enrico Fermi,
John Pasta and Stanislaw Ulam.
The computer experiment offered a third way
of doing science.
Data Science (Big Data) is the fourth way of
doing science
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aft
Enrico Fermi, John Pasta and Stanislaw Ulam – in 1953 invented the concept
of a "computer experiment.
"... the most important lesson ...is how feeble
even the best minds are at grasping the
dynamics of large, nonlinear systems. Faced
with a thicket of interlocking feedback loops,
where everything affects everything else, our
familiar ways of thinking fall apart. To solve
the most important problems of our time,
we’re going to have to change the way we do
science." NYT 2003, S. Strogatz
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Simon A. Levin Modeling Center
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aft
Training of Mathematical Scientists for the 21st Century
“... As Fermi and his colleagues taught us, a complex system like cancer can’t be
understood merely by cataloging its parts and the rules governing their interactions. The
nonlinear logic of cancer will be fathomed only through the collaborative efforts of
molecular biologists.” (Strogatz, 2003).
The world’s ability to train 21st century mathematical scientists must rely on models of
learning and thinking embedded within interdisciplinary educational
research/mentorship models.
Mathematical scientists must become proficient on multiple models of doing science
including the systematic use of computer experiments and in data science.
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Simon A. Levin Modeling Center
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aft
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Source: hlp://www.datasciencecentral.com/forum/topics/the-­‐3vs-­‐that-­‐define-­‐big-­‐data APLICACION DEL MODELAMIENTO
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Simon A. Levin Modeling Center
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Dr
aft
OakRidgeNaTonalLaboratory-June6,2013
Cray-madeTitan–thefastestcomputerintheworld
Chinaannouncesfastercomputer
Milky–Way2onJune17,2013
hl p://www.voanews.com/content/china-boasts-worldsfastest-computer/1683465.html
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Simon A. Levin Modeling Center
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A Brief History Block of Mathematical Epidemiology
Daniel Bernoulli
(1700–1782)
Dr
aft
The field was the result of the work of medical doctors - mathematical
scientists
Sir Ronald Ross
(1857–1932)
Anderson G. McKendrick
(1876–1943)
William O. Kermack
(1898–1970)
www.fameimages.com/daniel-bernoulli
www.nobelprize.org/nobel_prizes/medicine/laureates/1902/ross-bio.html
www.york.ac.uk/depts/maths/histstat/people/
Photograph courtesy of Godfrey Argent Studios
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Simon A. Levin Modeling Center
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The mathematical theory of infectious diseases started by medical doctors
Dr
aft
Sir Ronald Ross (1857–1932)
www.nobelprize.org/nobel_prizes/medicine/laureates/1902/ross-bio.html
Nobel Laureate 1902
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Simon A. Levin Modeling Center
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Basic Malaria Model
The Life–cycle of malaria parasites
Ross-Macdonald Model
Dr
aft
x : Proportion of infected
humans
y : Proportion of infected
mosquitoes
Parameter
M
N
a
b
r
µ
dx
dt
= ab M
N y (1 − x) − rx
dy
dt
= ax (1 − y) − µy
Definition
Number of female mosquitoes per human host
Biting rate on a human per mosquito
Infected mosquito to human transmission efficiency
Per capita human recovery rate
Per capita mortality rate of mosquitos
APLICACION DEL MODELAMIENTO
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Simon A. Levin Modeling Center
Units
day−1
day−1
day−1
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Dr
aft
Basic Reproductive Number for Malaria
R20 =
ma2 b
µr
Biological interpretation
R20 is the number of secondary cases
of infection on hosts or vectors
generated by a single infective host
or infective vector.
a – number of bites per unit time
b – infected bites that produce an infection
m=
1
r
1
µ
M
N
– number of female mosquitoes per human host
– duration of infection in human
– lifetime of a mosquito
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Simon A. Levin Modeling Center
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Holistic Perspective on Malaria
Dr
aft
How can we use what we learned from person to vector to person or vector to
host to vector transmission at higher levels of organization?
Holistic view - Public good: Can malaria be controlled at higher levels of
organization?
Problem across scales: how do we use knowledge at the individual level
to understand phenomena at the population level?
Validation of proposed control policies via the existence of a threshold:
the prestige of mathematics and mathematical modeling
Power of abstraction, can we use this framework elsewhere: STDs
APLICACION DEL MODELAMIENTO
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Simon A. Levin Modeling Center
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Gonorrhea: Transmission Dynamics and Control
University of Iowa
Kenneth Cooke
James A. Yorke
Dr
aft
Herbert Hethcote
Pomona College
University of Maryland, College Park
Herbert Hethcote and Jim Yorke changed health policy with their work
on gonorrhea via their concept of Core Group
Ken Cooke and Jim Yorke expanded significantly the work of Ross in
1970s with their work on Gonorrhea transmission and control
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Simon A. Levin Modeling Center
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Ross’ "students" Kermack and McKendrick
Anderson Gray McKendrick
(1876–1943)
Photograph courtesy of Godfrey Argent Studios
www.york.ac.uk/depts/maths/histstat/people/
Dr
aft
William Ogilvy Kermack
(1898–1970)
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Simon A. Levin Modeling Center
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Index case
Dr
aft
The basic reproduction number for the SIR model without vital dynamics
Infected
T(4)
T(3)
T(2)
T(1)
Susceptible
T(0)
Infected
Susceptible
Index case
R0=2
No Transmission
Transmission
The basic reproduction number, R0 , defined as the number of secondary cases
generated by a typical infectious individual during its period of infectiousness in an
entirely susceptible population.
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Simon A. Levin Modeling Center
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aft
System of equations without vital dynamics - single outbreak
S
dS
dt
dI
dt
dR
dt
SI
,
N
SI
= β − γI,
N
= −β
= γI,
N = S + I + R.
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β IS
I
States
S
I
R
Parameters
β
γ
Carlos Castillo-Chavez
γ
R
Meaning
# of Susceptible
# of Infectecd
# of Recovered
Meaning
Transmission coefficient
Per-capita recovery rate
Simon A. Levin Modeling Center
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Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious d
Dr
aft
On the basic reproduction number R0
R0 =
β
γ
1
R0 depends on number of contacts and probability of transmission (both
quantities captured by β) and the infectious period (1/γ).
2
If R0 < 1 then the infection dies out.
3
If R0 > 1 then an epidemic ensues
4
Accurate estimation of the value of the reproductive number are central
in the planning of control of intervention efforts.
→ The goal of public health interventions can often be reduced to that of
bringing R0 below 1.
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Simon A. Levin Modeling Center
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Model fitting and predictions using Influenza outbreak data
Dr
aft
Estimate parameters from data using the SIR model
Prevalence of an influenza
outbreak in a boys boarding
school, in the UK, 1978.
300
Total population size: N = 763,
150
1978 UK Boarding School
Data
Best fit
Parameter estimates and standard
250
200
error:
β̂ = 1.6682 ± 0.0294 days−1
γ̂ = 0.4417 ± 0.0177 days1
100
Initial # of susceptible: S0 = 762,
50
Initial # of infectives: I0 = 1.
0
0
2
4
6
8
10
12
14
SIR epidemic model simulated with estimated parameters
Infective Fraction(I)
0.2
0
0
S+
5
Population Size
0.2
0
0
0.2
0.4
0.6
0.8
Susceptible Fraction (S)
Carlos Castillo-Chavez
10
15
20
25
Time (Days)
1
I
0.4
Infective
0.1
=
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0.6
0.3
N
Initial conditions:
S(0) = 762, I(0) = 1,
R(0) = 0.
0.8
Infective Fraction (I)
Parameter values:
β = 1.6682, γ = 0.4417
0.4
S − I Phase Plane Por tr ait
1
1
0.5
0
0
5
10
15
Time (Days)
Simon A. Levin Modeling Center
Susceptible
Recovered
S (25) = 0. 02
20
25
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Sara del Valle
Dr
aft
Computational Epidemiology: The case of EpiSims
MTBI alumni – now a leading scientist
Research Interests: Develop mathematical and computational models to
help mitigate he spread of infectious diseases.
(EpiSims slides courtesy of Sara del Valle et al.)
APLICACION DEL MODELAMIENTO
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Simon A. Levin Modeling Center
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Dr
aft
Typical Family Day- Families using Same Locations
P. Stroud, S. Del Valle, S. Sydoriak, J. Riese, S. Mniszewski,
Spatial dynamics of pandemic influenza in a massive artificial society,
Journal of Artificial Societies and Social Simulation, 10, (4) (2007) 9.
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Simon A. Levin Modeling Center
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Mobility in a Network Simulation
Dr
aft
G. Chowell, J. M. Hyman, S. Eubank, and C. Castillo-Chavez.
Scaling laws for the movement of people between locations in a large city.
Phys. Rev. E, 68, 066102 – Published 15 December 2003
APLICACION DEL MODELAMIENTO
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Simon A. Levin Modeling Center
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Dr
aft
Portland-Simulated Flu Pandemic-75% stay at home
S. Eubank, H. Guclu, V. S. Anil Kumar, M. V. Marathe, A. Srinivasan, Z. Toroczkai, N. Wang,
Modelling disease outbreaks in realistic urban social networks,
Nature 429 (6988) (2004) 180–184.
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Simon A. Levin Modeling Center
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aft
Heterogeneous Spread
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Simon A. Levin Modeling Center
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Characteristics of the 2014 Ebola epidemic
Dr
aft
G Chowell, N W Hengartner, C Castillo-Chavez, P W Fenimore, and J M Hyman.
The basic reproductive number of Ebola and the effects of public health measures:
the cases of Congo and Uganda
J Theor Biol, 229(1):119–126, Jul 2004.
The causative Ebola strain in west Africa is closely related to a strain
associated with past outbreaks in Central Africa
Likely common reservoir: Fruit bats
Epidemiological characteristics include:
R0 ∼ 2.0 (substantial uncertainty)
Incubation period 11 days
Serial interval 15 days
Case fatality ratio 70.8% (95% IC 68.6 − 72.8%)
Photograph courtesy of AP Photo/Abbas Dulleh
High mortality rate (50-90% in previous outbreaks, 70% currently)
Bodily fluids are highly infectious, as are the unburied dead
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Simon A. Levin Modeling Center
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Dr
aft
2014 West African Outbreak
Index case Dec 2013, 2
year old girl in
mountainous region of
Guinea
Porous borders and
trafficking have likely
aided the spread of the
disease
(distribution as of Oct 6th., 2014
Image source: bbc.com)
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Simon A. Levin Modeling Center
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Dr
aft
What is unique about this epidemic?
Affected region lacked experience with EVD outbreaks
Substantial delays in detection and implementation of control
interventions in a region characterized by porous borders and high
population movement
Limited public health infrastructure in affected region including
epidemiological surveillance systems and diagnostic testing
Unsafe burials and health-care settings contributed to seeding the
epidemic in multiple districts and across borders
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Simon A. Levin Modeling Center
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aft
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http://www.bbc.com/news/world-africa-28755033
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Simon A. Levin Modeling Center
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Dr
aft
Example 1: Exponential rise in new cases
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Carlos Castillo-Chavez
Dr. Sherry Towers
(Faculty, ASU)
Dr. Oscar Patterson
(Postdoc, Harvard Univ)
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Dr
aft
Example 1: Exponential rise in new cases
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Simon A. Levin Modeling Center
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aft
Exponential rise in new cases
www.newfluwiki2.com
Early in an epidemic, when the population is predominantly naive, initial rise in
cases is exponential,
When everybody is susceptible the reproduction number is called R0 ,
The Effective Reproduction Number measures the changes in exponential
growth as a function of time (Reff (t)),
Reff (t) is the average number of new cases one case will cause over the course
of their infection at time t.
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Simon A. Levin Modeling Center
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Dr
aft
Reproduction number
In closed populations, Reff (t) usually declines as an epidemic progresses
(fewer and fewer naïve people available to infect)
As an epidemic progresses social distancing due to fear can also cause
additional reductions in Reff (t)
But poorly designed control strategies can, unfortunately, potentially do
more harm than good.
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Simon A. Levin Modeling Center
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Dr
aft
Temporal variations in Reff (t)
Temporal variations in the effective reproduction number of the 2014 West
Africa Ebola outbreak
Rate of exponential rise, in conjunction with a mathematical model of
the spread of the disease, can be used to determine Reff (t).
For instance, SEIRD model for Ebola.
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Simon A. Levin Modeling Center
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Temporal variations in Reff (t) contd...
Dr
aft
Developed a simple,
model-independent way to
determine if the relative
transmission rate of a disease
in a closed population
appears to increase or
decrease in time;
Method applies piece-wise
exponential fits to the time
series of cases in outbreak to
determine if the rate of
exponential rise in cases
increases over time (evidence
that effective Reff (t) is
increasing).
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Simon A. Levin Modeling Center
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aft
Example 2: Research question
Volume 15, Issue 2, February 2015, Pages 148–149
Modelling the effect of early detection of Ebola
Diego Chowella, b, Carlos Castillo-Chaveza, Sri Krishnab, Xiangguo Qiuc, Karen S Andersonb,
a
Simon A Levin Mathematical, Computational and Modeling Sciences Center, Biodesign Institute, Arizona State University, Tempe, AZ, USA
b
Center for Personalized Diagnostics, Biodesign Institute, Arizona State University, Tempe, AZ, USA
c
National Laboratory for Zoonotic Diseases and Special Pathogens, Public Health Agency of Canada, Winnipeg, MB, Canada
Copyright © 2015 Elsevier Ltd. All rights reserved
What is the effect of pre-symptomatic stage Ebola Virus detection on its
transmission dynamics?
Diego Chowell (PhD student)
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Simon A. Levin Modeling Center
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aft
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APLICACION DEL MODELAMIENTO
Carlos Castillo-Chavez
• 
Twenty'four,“asymptoma2c”,individuals,
exposed,to,Ebola,were,tested,using,P.C.R.,,
• 
Eleven,of,the,exposed,pa2ents,eventually,
developed,the,infec2on.,,
• 
Seven,of,the,11,tested,posi2ve,for,the,
P.C.R.,assay;,none,of,the,other,13,did.,
Simon A. Levin Modeling Center
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Overview Big Data Risk factors and recent outbreaks Modeling disease dynamics using a Lagrangian perspective Spatial spread of infectious d
Dr
aft
Modeling the effect of early detection of Ebola
The polymerase chain reaction (PCR) can
detect Ebola virus in both human beings and
non-human primates in the pre-symptomatic
stage.
We evaluated the potential effect of early
diagnosis of pre-symptomatic individuals in
west Africa.
We used a simple mathematical model
calibrated to the transmission dynamics of
Ebola virus in west Africa. The baseline
model includes the effects of contact tracing
and effective isolation of infectious
individuals in health-care settings.
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Carlos Castillo-Chavez
Members of the European Mobile Laboratory
Project use PCR tests in Guéckédou, Guinea.
G. Vogel, Testing new Ebola tests, Science
(2014).
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S
Class
S
E1
E2
I
J
P
Dr
aft
Model of transmission dynamics of Ebola infection incorporating diagnosis of
infected and pre-symptomatic individuals
E1
E2
Description
Susceptible
Latent undetectable
Latent detectable
Infectious and symptomatic
Isolated
Recovered and Dead
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Carlos Castillo-Chavez
I
J
Parameter
1/k2
1/k2
1/α
1/γ
1/γr
P
Value
4 days
3 days
3 days
6 days
7 days
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Dr
aft
Predictions on the effect of diagnosing pre-symptomatic individuals on the
Ebola epidemic attack rate
100
Effectiveness of isolation 50%
Effectiveness of isolation 60%
Effectiveness of isolation 65%
90
Attack rate (%)
80
70
60
50
40
30
20
10
0
0
5
10
15
20
25
30
35
40
45
Proportion of latent individuals diagnosed before onset of symptoms (%)
50
We can make now two observations:
The effect of early Ebola detection is a function of existing public health
measures and resources.
There is a tipping point, where early diagnosis of high risk individuals,
combined with adequate isolation, can lead to rapid reduction in Ebola
transmission.
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Simon A. Levin Modeling Center
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Example 3: A disease model framework under virtual dispersal and varying
residence times
Dr
aft
Difficulties in defining a contact in the
context of communicable diseases.
Notion of contact is well-defined in
STDs and vector-born diseases.
Estimate the average risk of acquiring
TB or flu to individuals spending 3
hours on the average per day, in public
transportation.
http://pilgrimagetoindia.com/gallery/26.html
Contacts or variable environmental risks?
From differential susceptibility to local
environmental risk infectivity.
http://www.livetradingnews.com
Overall Question: How does environmental pathogen risk defined by risk/transmission vector B and patch
residence time distribution P = (pij ) impact disease transmission dynamics and control.
Formulation of epidemic models (host parasite dynamics) where risk of infection (parasitism) is a function of local residence times.
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Simon A. Levin Modeling Center
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Dr
aft
Two approaches to incorporate heterogeneity
Eulerian Approach:
Patch 1
Focus on the whole population stratified
into different patches.
Individuals take the identity of the host
patch after leaving their residence patch.
More realistic for long term dispersal.
Lagrangian Approach:
12
p
21
Patch 1
Keep track of individuals for population
in each patch through time and space.
Focus on the patch level of population
interacting between different patches .
More realistic for short scale
movements.
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Patch 2
p
Carlos Castillo-Chavez
Patch 2
p12
p21
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aft
Model formulation
p12 is the proportion of times residents of Patch 1 spend in Patch 2.
p21 is the proportion of times residents of Patch 2 spend in Patch 1.
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Simon A. Levin Modeling Center
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Dr
aft
Patch-specific dynamics
We define a Patch-specific basic reproduction number:
Ri0 (P) = Ri0 ×
n X
βj
j=1
βi

pij dbi
i
.
pij  Pn
bk
k=1 pkj d

k
If Ri0 (P) > 1 then the disease persists in Patch i.
If pkj = 0 for all k = 1, .., n, and k 6= i, whenever pij > 0 and Ri0 (P) < 1, then the disease dies
out in Patch i.
Remarks:
• This results also include the non-strongly connected patches case.
• The connectivity of Patch i to other patches may promote or suppress endemicity:
β
Via the presence of high risk patches. For instance, if there exists a patch j such that βj is large
i
enough. In this last case Patch j is actually a source and Patch i a sink.
Whenever individuals spend more time in high risk patches than in low risk patches.
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Simon A. Levin Modeling Center
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Findings
Dr
aft
We have used residence times in heterogeneous environments as a proxy for “effective” contacts
over a certain window in time;
We have formulated general multi-patch SIS epidemic models with residence times that provide
conditions for extinction or persistence of the disease at global level and patch-specific level.
Derdei Bichara, Yun Kang, Carlos Castillo-Chavez, Richard Horan, and Charles Perrings.
SIS and SIR Epidemic Models Under Virtual Dispersal.
pages 1–31, 2015.
Joint work with:
Derdei Bichara
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Yun Kang
Carlos Castillo-Chavez
Charles Perrings
Richard Horan
Simon A. Levin Modeling Center
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Heterogeneity challenges
Dr
aft
A cordon sanitaire is a medieval control strategy that involves creating boundaries around an area
experiencing an epidemic of disease.
This strategy was used in 1821 in the border of France and Spain to avert the spread of a deadly
fever.
Prior 2014, it was last used in 1918 at the Polish-Russian border in an attempt to stop the spread of
typhus.
cordons sanitaire were used in 2014 in some
of the Ebola-stricken countries.
The effectiveness of cordons sanitaire is
controversial and debatable.
Soldiers enforcing the cordon securitaire in Kanema, Sierra Leone (via
the NYT)
Research question: How does the movement of individuals between a low and high risk area promote,
mitigate or suppress disease dynamics?
A Lagrangian conceptual framework that models the movement of individuals across different areas
keeping track of residence times is used on the Ebola outbreak.
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4.5
×10 4
Infected class patch 1
3
10000
Dr
aft
3.5
Infected class patch 2
12000
p12=0
p12=0.15
p12=0.3
p12=0.45
4
p12=0
p12=0.15
p12=0.3
p12=.0.45
8000
2.5
6000
2
1.5
4000
1
2000
0.5
0
0
0
500
1000
1500
Time
Figure:
0
500
1000
1500
Time
Dynamics of incidence in each patch for p21 = 0 and varying p12 . Parameter values: εD = 1, β1 = 0.35,
β2 = 0.1, fdeath = 0.8, k = 1/24, α = 0, ν = 1/2 and γ = 1/5.6. The blue graph is the case where the patches are isolated. The
disease reaches its highest pick in the high risk Patch 1 whereas it dies off without a major outbreak in the low risk patch 2.
Besides the isolated case, we note that the disease prevalence decreases
in both patches as p12 increases.
While the results in Patch 1 are expected, the results are counterintuitive
in Patch 2, as we expect more infections as the flow of individuals from
the high risk area increases.
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aft
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Figure: Dynamics of the cumulative incidence in each patch for p21 = 0 and varying p12 .
The predicted dynamics of the overall infected individuals is similar to those in Patch
1 since it is assumed that the high risk Patch 1 is overpopulated when compared to
the low risk Patch 2.
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aft
Conclusion
A conceptual framework to test the effectiveness of control strategies
aimed at limiting the movements of individuals across different risk areas
has been introduced.
It was observed, for example, that increases in the time that residents of
high risk areas spend in low risk areas do not necessarily generates
notable prevalence increases in low risk areas.
Joint work with:
Baltazar
Derdei
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Victor
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Mass Media and contagion of fear
Dr
aft
Sherry Towers, Shehzad Afzal, Gilbert Bernal, Nadya Bliss, Shala Brown, Baltazar Espinoza, , et al.
Mass media and the contagion of fear: The case of ebola in america.
PLoS ONE, 10(6):e0129179 EP –, 06 2015.
Ebola related searches and tweets originating in the U. S.
during the outbreak tied in to public interest or panic.
Of interest, how would public interest, curiosity, or panic
on certain topic affects social media and Internet search
dynamics?
A mathematical model has been employed to simulate
the potential influence of Ebola-related news videos on
peoples’ tendency to perform Ebola-related Internet
searches or tweets.
Fits of the news media contagion model, and a simple linear regres
model, to the sources of data used in this study.
Overall Goal: Determine if news coverage was a significant factor on the
temporal patterns in Ebola-related Internet and Twitter data.
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aft
News on Ebola
Significant evidence of contagion was found, with each Ebola-related
news video inspiring tens of thousands of Ebola-related tweets and
Internet searches.
There is no significant evidence of contagion due to effects other than
news videos.
There is no statistically significant evidence that people return to the
susceptible class after recovery.
In all cases, the contagion model had a better predictive power than the
linear regression model.
There is no statistically significant evidence that Ebola-related Internet
searches and tweets Granger-cause temporal patterns in Ebola-related
news videos, but there is evidence in several cases that the reverse is true.
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SIR with diffusion
Dr
aft
SIR no diffusion
Ṡ = −βSI
İ = βSI − αI
Standard model with diffusion
St = −βSI + Ds Sxx
It = βSI − αI + Di Ixx
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aft
Necessary and sufficient conditions for Aggregation Turin’s Method
Superposition and Instability
ci (x, t) = αi cos(qx)eσt
∂c1
∂t
∂c2
∂t
values of q s.t. <(σ) > 0
α1 (σ − a11 + D1 q2 ) − α2 a12 = 0
∂ 2 c1
∂2x
∂ 2 c2
= a21 c1 + a22 c2 + D2 2
∂ x
= a11 c1 + a12 c2 + D1
−α1 a21 + α2 (σ − a22 + D2 q2 ) = 0
Necessary and Sufficient Conditions
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Carlos Castillo-Chavez
1
a11 + a22 < 0
2
a11 a22 − a12 a21 > 0
3
a11
pD2 + a22 D1 >
2 D1 D2 (a11 a22 − a12 a21 )
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aft
RESULT
K. E. Yong, E. Díaz Herrera, and C. Castillo-Chavez.
From bee species aggregation to models of disease avoidance: The Ben − Hur effect.
ArXiv e-prints, October 2015.
∂S
∂t
∂I1
∂t
∂I2
∂t
∂2S
β
SI1 + αI2 + DS 2
1 + I2
∂x
2
β
∂ I1
=
SI1 − δI1 + DI1 2
1 + I2
∂x
2
∂ I2
= δI1 − αI2 + DI2 2
∂ x
= −
(1)
Theorem (Diffusive Instability in Epidemics)
The linearization of the system (1) satisfies the necessary and sufficient
conditions for instability if and only if βδ > 1 and αβ > 1
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Different diffusion rates: linear model
time = 0
time = 6
0
20
20
40
40
60
60
80
80
0
20
40
60
80
0
time = 19
0
20
20
40
40
60
60
80
80
20
40
60
80
time = 38
20
20
40
40
60
60
80
80
0
20
40
20
40
60
20
40
20
20
20
40
40
40
60
60
60
60
80
80
80
80
0
20
60
80
40
60
80
20
40
20
0
20
40
80
60
80
0
20
20
20
20
40
40
40
40
60
60
60
80
20
40
60
80
20
40
60
80
0
20
time = 38
40
80
0
0
20
20
20
20
40
40
40
40
60
60
60
80
60
80
20
40
60
80
0
20
40
60
80
60
80
60
80
0
40
time = 50
0
40
20
time = 44
0
20
80
80
60
0
0
60
60
80
0
40
time = 31
0
0
20
time = 25
0
D2 D1
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80
0
80
80
60
0
time = 50
60
40
time = 19
80
0
0
time = 31
60
time = 13
0
40
time = 44
0
time = 6
0
20
80
0
0
time = 0
0
time = 25
0
0
time = 13
0
Dr
aft
0
80
60
80
0
20
40
D2 > D1
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aft
Different diffusion rates: nonlinear model
u
1
0.5
0
150
2
0
−2
140
120
140
100
100
150
120
80
100
60
80
100
50
60
40
40
20
50
20
0
0
0
Slow aggregation
Fast aggregation
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Beyond Ebola: lessons to mitigate future pandemics
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Beyond Ebola
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,
Carlos Castillo Chavez
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Poste
:
lessons to mitigate future pandemics
,
Roy Curtiss
Peter Daszak
,
,
Simon A Levin
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,
Oscar Patterson Lomba
,
Charles Perrings
George
Sherry Towers
Open Access
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DOI
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/10.1016/ 2214-109 (15)00068-6
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Article Info
Summary
Full Text
References
It is now just more than a year since the official confirmation of an outbreak of Ebola haemorrhagic
fever in west Africa
.1
With new cases occurring at their lowest rate for
,
outbreak in sight for all three countries predominantly affected
,
strategies to prevent future outbreaks of this
like many other emerging diseases
,
,
and other
2015,2
and the end of the
now is the time to consider
.
zoonotic pathogens
.
,
The Ebola outbreak
,
illustrates the crucial role of the ecological
,
social
,
political
and
economic context within which diseases emerge
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So what have we learned from influenza pandemics, SARS and Ebola?
The cost of management of infectious disease outbreaks is almost always greater than
the cost of avoiding them. “For severe acute respiratory syndrome (SARS), the global
cost of a single outbreak was estimated to be between US 13 billion and US 50 billion at
the currency values of the 2003 outbreak. For Ebola, the cost might be higher? both in
the direct, short-term cost of control, patient care, and hospital admission, and in the
indirect, longer-term dislocation of the regional economies in west Africa." Lancet
Global Health, Volume 3, No. 7, e354-355, 2015)
The economic costs of disease emergence are projected to continue to rise in line with
increasing frequency of outbreaks driven by expanding socioeconomic and
environmental changes that cause diseases. Lancet Global Health, Volume 3, No. 7,
e354-355, 2015)
Mitigation of future pandemic threats such as Ebola is therefore more cost-effective than
the current approach of responding to outbreaks after they have begun to spread rapidly
in the human population. Lancet Global Health, Volume 3, No. 7, e354-355, 2015)
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Is it possible to design a strategy for an as-yet unknown pathogen?
This task seems daunting, but it has already begun, partly though reduction of the size of
the problem and allocation of resources in an objective way to the locations most at risk.
(Lancet Global Health, Volume 3, No. 7, e354-355, 2015)
“Analysis of trends in disease emergence provides a strategy to identify the places most
likely to propagate the next pandemic... hotspots for disease emergence tend to be
tropical regions with high wildlife diversity that harbour known or unknown zoonoses,
and high levels of socioeconomic and environmental change." Lancet Global Health,
Volume 3, No. 7, e354-355, 2015)
USAID’s Emerging Pandemic Threats (PREDICT) programme targets these hotspots to
identify known and previously unknown viruses in wildlife species known to be zoonotic
reservoirs, analyses patterns of high-risk human behaviour, tests people for evidence of
these viruses moving across the species barrier, and enables the design of strategies to
reduce the risk of even the first spillover event. Lancet Global Health, Volume 3, No. 7,
e354-355, 2015)
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Is it possible to generate massive beahviour changes?
Targeted programmes for behaviour change, focusing on incentives for bushmeat
hunting, should be part of the mitigation strategy...This approach was trialled in central
Africa, with education programmes designed to reduce the consumption of primates
found dead in forests, and has been shown to off er a cost-eff ective way to mitigate the
risk of an Ebola outbreak (Lancet Global Health, Volume 3, No. 7, e354-355, 2015)
Projects aimed to reduce dependency on bushmeat need to be supported, either through
creative approaches to farming of some wildlife species, or by expansion of livestock
production, with appropriate biosecurity and surveillance to prevent emergence of other
zoonoses Lancet Global Health, Volume 3, No. 7, e354-355, 2015)
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What about the use of technology ... training and preparedness?
The acceleration of vaccine development for Ebola as part of an outbreak control
strategy could also have a crucial role to mitigate future outbreaks. (Lancet Global
Health, Volume 3, No. 7, e354-355, 2015)
Ebola’s propensity for nosocomial spread (noted in west Africa and in many previous
Ebola outbreaks) could be curtailed by preoutbreak vaccination of critical care workers
in Ebola virus hotspots. (Lancet Global Health, Volume 3, No. 7, e354-355, 2015)
Targeted training in infection control, and efforts to maintain surge capacity between
outbreaks, will be crucial for rapid response to the first cases in a future emergence
event. (Lancet Global Health, Volume 3, No. 7, e354-355, 2015)
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Can these approaches be scaled up to mitigate future pandemics on a global
scale?
“Global mitigation of future pandemic risk must focus on the large scale behaviours that
lead to zoonotic spillovers. This approach means engaging with the sectors that drive
disease emergence, including industries involved in land-use change, resource
extraction, livestock production, travel, and trade, among others” (Lancet Global
Health, Volume 3, No. 7, e354-355, 2015)
“Large economic development programmes will need health impact assessments that
deal explicitly with the risk of emergence of novel diseases, and plans to set up new
clinics and surveillance programmes listed as project deliverables.” (Lancet Global
Health, Volume 3, No. 7, e354-355, 2015)
“An improved understanding of the liability for disease emergence will drive this
change; when all are at risk, collective action is needed to strengthen the weakest links
in the chain”, (Lancet Global Health, Volume 3, No. 7, e354-355, 2015)
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Dr
aft
Mathematical and Theoretical Biology Institute, 1996
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aft
Presidential award for excellence in Sciences, Math and Engineering mentoring, 2011
– Given in recognition to the Mathematical and Theoretical Biology Institute – MTBI
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Dr
aft
MCMSC Center Renaming Ceremony in Honor of Simon A. Levin
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Dr
aft
Grad students and researchers involved in the Ebola research at SAL-MCMSC - all
MTBI alumni
Dr. Derdei Bichara: Epidemiology,
Infectious Diseases
Baltazar Espinoza: Epidemiology,
Game theory, Economics
Diego Chowell: Evolutionary
biology, Cancer, Computational
modeling
Maryam Khan: Epidemiology, Social
Sciences, Environment
Kamal Barley: Epidemiology, Health
disparities
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Carlos Castillo-Chavez
Dr. Kamuela Yong: Ecology,
Diffusion
Victor Moreno: Epidemiology,
Graph theory
Claudia Rodriguez: Math education
and policy, Retention of
underrepresented students
Dr. Edgar Herera Diaz:
Epidemiology, Diffusion
Dr. Anuj Mubayi: Mathematical
Epidemiology, Quantitative Social
Sciences
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aft
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