Hyesu Kim Michelle Zajac Amanda Clemm SPWM 2011 Cindy Wu Gonzaga University Dr Burke Our group Hyesu Kim Manhattan College Dr Tyler Michelle Zajac Alfred University ID: 697454
Download Presentation The PPT/PDF document "SIR and SIRS Models Cindy Wu," is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Slide1
SIR and SIRS Models
Cindy Wu,
Hyesu
Kim, Michelle
Zajac
, Amanda
Clemm
SPWM 2011Slide2
Cindy Wu
Gonzaga University
Dr. Burke
Our group!
Hyesu KimManhattan College Dr. Tyler
Michelle ZajacAlfred UniversityDr. Petrillo
Amanda ClemmScripps CollegeDr. OuSlide3
Cindy Wu
Why Math?
Friends
Coolest thing you learned
Number TheoryWhy SPWM?DC>SpokaneOtherwise, unproductiveSlide4
Why math?
Common language
Challenging
Coolest thing you learnedMath is everywhereAnything is possible
Why SPWM?Work or grad school?Possible careersHyesu KimSlide5
Why math?
Interesting
Challenging
Coolest Thing you LearnedRSA CryptosystemWhy SPWM?
Grad schoolLearn something newMichelle ZajacSlide6
Why Math?
Applications
Challenge
Coolest Thing you LearnedInfinitude of the primesWhy SPWM?
Life after collegeDCAmanda ClemmSlide7
Study of disease occurrence
Actual experiments
vs ModelsPrevention and control procedures
EpidemiologySlide8
Epidemic: Unusually large,
short term
outbreak of a diseaseEndemic
: The disease persistsVital Dynamics
: Births and natural deaths accounted forVital Dynamics play a bigger part in an endemicEpidemic vs EndemicSlide9
Total population=N ( a constant)
Population fractions
S(t)=susceptible pop. fraction
I(t)=infected pop. fractionR(t)=removed pop. fraction
PopulationsSlide10
Both are epidemiological models that compute the number of people infected with a contagious illness in a population over time
SIR: Those infected that recover gain
permanent
immunity (ODE)SIRS: Those infected that recover gain temporary immunity (DDE)
NOTE: Person to person contact onlySIR vs SIRS ModelSlide11
PART ONE: SIR Models using ODESSlide12
λ=daily contact rate
Homogeneously mixing
Does not change seasonally
γ =daily recovery removal rateσ=
λ/ γThe contact numberVariables and Values of ImportanceSlide13
Model for infection that confers permanent immunity
Compartmental diagram
(NS(t))’=-
λ
SNI(NI(t))’= λSNI- γNI(NR(t))’= γNIThe SIR Model without Vital Dynamics
NS SusceptiblesNI InfectivesNR RemovedsλSNI
ϒNIS’(t)=-λSII’(t)=λSI-ϒISlide14
S’(t)=-
λ
SI
I’(t)=λSI-ϒI
Let S(t) and I(t) be solutions of this system.CASE ONE: σS₀≤1I(t) decreases to 0 as t goes to infinity (no epidemic)CASE TWO: σS₀>1I(t) increases up to a maximum of: 1-R₀-1/σ-ln(σS₀)/σThen it decreases to 0 as t goes to infinity (epidemic)TheoremσS₀=(S₀λ)/
ϒInitial Susceptible population fractionDaily contact rateDaily recovery removal rateSlide15
MATLAB Epidemic Slide16
PART TWO: SIRS Models using DDESSlide17
dS
/
dt
=μ[1-S(t)]-Β
I(t)S(t)+r γ γ e-μτI(t-τ)dI/dt=ΒI(t)S(t)-(μ+γ)I(t)dR/dt=γI(t)-μR(t)-rγγe-μτI(t-τ)μ=death rateΒ
=transmission coefficientγ=recovery rateτ=amount of time before re-susceptibility e-μτ=fraction who recover at time t-τ who survive to time trγ=fraction of pop. that become re-susceptibleEquations and VariablesSlide18
Focus on the endemic steady state (
R
0S=1)
Reproductive number: R
0=Β/(μ+γ)Sc=1/R0Ic=[(μ/Β)(ℛ0-1)]/[1-(rγγ)(e-μτ )/(μ+γ)] Our goal is now to determine stability
Equilibrium SolutionsSlide19
dx/
dt
=-y-ε
x(a+by)+ry
(t-τ)dy/dt=x(1+y) where ε=√(μΒ)/γ2<<1 and r=(e-μτ rγγ)/(μ+γ) and a, b are really close to 1Rescaled equation for r is a primary control parameterr is the fraction of those in S who return to S after being infectedRescaled EquationsSlide20
r=(e-
μτ
r
γγ)/(μ+
γ)What does rγ=1 mean?Thus, r max=γ e-μτ /(μ+γ)So we have: 0≤r≤ r max<1More about rSlide21
λ2
+
εa
λ+1-re-λτ
=0 Note: When r=0, the delay term is removed leaving a scaled SIR model such that the endemic steady state is stable for R0>1Characteristic EquationSlide22
When does the
Hopf
bifurcation occur?Slide23Slide24Slide25
Slide26
In terms of the original variables…
Slide27
r=0.005Slide28
r=0.005 (Zoomed in)Slide29
r=0.02Slide30
r=0.02 (Zoomed in)Slide31
r=0.03Slide32
r=0.03 (Zoomed in)Slide33
r=0.9Slide34Slide35
In our ODE we represented an epidemic
DDE case more accurately represents longer term population behavior
Changing the delay and resusceptible
value changes the models behaviorBetter prevention and control strategies
Conclusion