Modeling Ebola Propagation:Understanding, Prediction, Control
Peter [email protected]
Henry R. Luce ProfessorCenter for Complex Systems Studies
Kalamazoo Collegehttp://people.kzoo.edu/ perdi/
andWigner Research Centre, Hungarian Academy of Sciences, Budapest
http://cneuro.rmki.kfki.hu/
with the assistance of Kamalaldin M. Kamalaldin (K’17)
Content
• The Rationalist Perspective: Thinking with Models
• Interpreting Temporal Data: Growth, Saturation, Boom-and-Bust
• The Actual Data Set
• From Data to Understanding and . . . Action
• The Basic Epidemic Model
• A More Advanced Model
• Predictability: Scope and Limits
• Controlling Epidemics
• Propagation of viruses and fear: combined biological and social aspects
• Take home message
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The Rationalist Perspective:Thinking with Models
”In defense of rationality”: To manage extreme events
• Things not just ”happen”
• If we don’t understand it - we cannot control !
• Search for CAUSAL explanations
• Reality and models
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.
Figure 1: Mathematical models are our good friends!
So, you should survive the next slide!
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Interpreting Temporal Data:
Growth, Saturation, Boom-and-Bust, Oscillation
t
x(t)
t
x(t)
t
x(t)
linear exponential super- exponential
boundless growth
v=x=c v=ax v=axn
a>0 a>0,n>1
external! positive feedback
unstable: a<0 → extinction
t
x(t)
t
x(t)
pull-back dynamic
equilibrium
x=c-dx
x=axY - dx2
logistic
gain loss term
boom -bust Y: resources
t
x(t)
Y= - ax
x=ax-bx2
non-renewable
x=axY - dx2
Y= - ax + by renewable
cycle t
x(t)
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The Actual Data Set
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The Actual Data Set
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From Data to Understanding and. . . Action
• How to characterize thechanges?
• Difficulties of predictions
• Initial stage of an exponentialcurve looks linear
• Control strategies
– to reduce increase to reach in-flexion point
– to reduce increase to get sat-uration
INFLEXION point: the signature ofchanging dynamics
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The Basic Epidemic Model
• What is happening?”Susceptible” will be convertedto ”infected”
• What is the velocity of the infec-tion?β determines the ”effectivity”of the encounter
• Infection would be ever-increasing: this is not the wholestory
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The Basic Epidemic Model
Figure 2: SIR model and its varia-tions
• S(t): represents the number ofindividuals not yet infected withthe disease at time t, or thosesusceptible to the disease.
• I(t): the number of individu-als who have been infected withthe disease and are capable ofspreading the disease to those inthe susceptible category.
• R(t): the number of those indi-viduals who have been infectedand THEN removed from thedisease, due to immunization,quarantine or due to death. .
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Do you know what Public Health Authorities Do?
They try to influence the β and γ parameters of the differential equations ofepidemic propagation. (stay tuned: two slides later!)
So, please, please, don’t close your eyes!
ASSUMPTION: rate of contact between two groups in a population isproportional to the size of each of the groups concerned s
• velocity of change in the number of susceptible (S): proportional with the numbers ofS and I
• velocity of change in the number of infected (I): two terms: new infected - removed
• velocity of change in the number of removed: removal rate: proportional with thenumber of infected.
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First law of lectures
DON’T Show Any EQUATION!
(I will violate the law from pedagogical reasons)
x2 = 4 (1)
What do equations have?solution(s).yes: x = 2, more precisely: x = ±2
the equations behind CHANGES: differential equations (I leave it for Dr. Barth).
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The Basic Epidemic Model
• β: how effectives are the
infected
• γ: how effective the
removal of the infected
(good and bad)
• larger β increases the ve-
locity of the epidemics
• larger γ decreases the ve-
locity of the epidemics
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The SIR Model: a typical ”solution”
How the quantities of S,I and R are CHANGING in time?
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The SIR Model and the BasicReproduction Number
Epidemic is a THRESHOLD PHENOMENON:
There is a threshold quantity which determines whether an epidemic occurs or thedisease simply dies out. The threshold is called as the basic reproduction number, oftendenoted by R0, (N is the total population, initially S(0).
R0 :=βN
γ, (2)
The basic reproduction number, R0, is interpreted as the average numberof secondary cases caused by a typical infected individual throughout its entirecourse of infection in a completely susceptible population and in the absence ofcontrol intervention. The targets of any control strategies are the β andγ rate parameters.
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A More Advanced Model
”Rivers et al model(PLOS Currents: Outbreaks:2014 October/November)
The population is divided into sixcompartments:
Susceptible (S), Exposed (E),Infectious (I), Hospitalized (F), Fu-neral (F) - indicating transmissionfrom handling a diseased patient’sbody, and Recovered/Removed(R).
Arrows indicate the possible transi-tions, with the rate parameters.
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A More Advanced Model
• Kamal’s simulations
• ”boom and bust” dynamics: the TIME and MAGNITUDE of the maximal value
• I. the temporal course of the epidemics (the paradoxical meaning of ”zero infected”)
• II. effect of changing the effectiveness of the contact between susceptible and infected
• III. effect of changing the fatal rate
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A More Advanced Model
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A More Advanced Model
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A More Advanced Model
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A More Advanced Model: furtherpossibilities
• More compartments: (say: ”good” and ”bad” hospital)
• More processes: back to the community from ”Temporarily removed”
• Spatial Models
• Interregional Mobility
• Additional ’normal’ birth-and-death rate
• Randomness; the world of stochastic models
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Predictability: Scope and Limits
Is West Africa Approaching a Catastrophic Phase or is the 2014 Ebola EpidemicSlowing Down? Different Models Yield Different Answers for Liberia(Gerardo Chowell, Lone Simonsen, Cecile Viboud, Yang Kuang) November 20, 2014):
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Controlling Epidemics
Control strategies:to decrease infection probabilityto remove and cure infected (and infectious) subpopulation
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BIOLOGICAL and SOCIALEPIDEMICS: PROPAGATION and
CONTROL of INFECTIOUSDISEASES AND IDEAS
Coupled Contagion Dynamics of Fear and Disease: Mathematical
and Computational Explorations
Joshua M. Epstein (2008) PLoS One.
very important topic: beyond the scope of this presentation
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Take home message I
In response to the immediate need for solutions in the field of computational biology against
Ebola, The International Society for Computational Biology (ISCB) announces the ISCB Fight
Against Ebola Award. ISCB will give out the ISCB Fight Against Ebola Award, along with a prize
of $2000, at its July 2016 annual meeting (ISCB ISMB 2016, Orlando, Florida). All computational
approaches should include a significant component of Ebola research. In the development of any
modern drug, computational biology is positioned to contribute through comparative analysis
of the genome sequences of Ebola strains, and 3-D protein modeling. Other computational
approaches to Ebola include large-scale docking studies of Ebola proteins with human proteins and
with small-molecule libraries, computational modeling of the spread of the virus, computational
mining of the Ebola literature, and creation of a curated Ebola database. Taken together, such
computational efforts could significantly accelerate traditional scientific approaches.
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Take home message II
Mathematical models: A key tool for outbreak response.(Lofgren et al: (including Marisa Eisenberg) Proc Natl Acad Sci USA .December 23,2014):
• The 2014 outbreak of Ebola in West Africa is unprecedented in its size and geographicrange, and demands swift, effective action from the international community
• Understanding the dynamics and spread of Ebola is critical for directing interventionsand extinguishing the epidemic
• Mathematical models can clarify how the disease is spreading and provide timely guid-ance to policymaker
• However, the use of models in public health often meets resistance
– Public skepticism (and ignorance)– Models are often portrayed as arcane and largely inaccessible thought experiments– However, the role of models is crucial: they can be used to quantify the effect
of mitigation efforts, provide guidance on the scale of interventions required toachieve containment, and identify factors that fundamentally influence the courseof an outbreak
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if you want to LEARN a little more about modeling Ebola:
Kamalaldin M. Kamalaldin presents a
DEMO
otherwise enjoy your lunch!
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