SIR and SIRS Models

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SIR and SIRS Models ndy Wu, Hyesu Kim, Michelle Zajac, Amanda Clemm SPWM 2011

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SIR and SIRS Models. Cindy Wu, Hyesu Kim, Michelle Zajac , Amanda Clemm SPWM 2011. Our group!. Cindy Wu Gonzaga University Dr. Burke. Amanda Clemm Scripps College Dr. Ou. Hyesu Kim Manhattan College Dr. Tyler. Michelle Zajac Alfred University Dr. Petrillo. Cindy Wu. - PowerPoint PPT Presentation

Transcript of SIR and SIRS Models

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SIR and SIRS ModelsCindy Wu, Hyesu Kim, Michelle Zajac, Amanda Clemm

SPWM 2011

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Cindy Wu Gonzaga

University

Dr. Burke

Our group!

Hyesu Kim Manhattan

College Dr. Tyler

Michelle Zajac

Alfred University

Dr. Petrillo

Amanda Clemm

Scripps College

Dr. Ou

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Cindy Wu Why Math?

Friends Coolest thing you

learned Number

Theory Why SPWM?

DC>Spokane Otherwise,

unproductive

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Why math?◦ Common language◦ Challenging

Coolest thing you learned◦ Math is everywhere◦ Anything is possible

Why SPWM?◦ Work or grad school?◦ Possible careers

Hyesu Kim

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Why math?◦ Interesting◦ Challenging

Coolest Thing you Learned◦ RSA Cryptosystem

Why SPWM?◦ Grad school◦ Learn something new

Michelle Zajac

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Why Math?◦ Applications◦ Challenge

Coolest Thing you Learned◦ Infinitude of the

primes Why SPWM?

◦ Life after college◦ DC

Amanda Clemm

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Study of disease occurrence Actual experiments vs Models Prevention and control procedures

Epidemiology

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Epidemic: Unusually large, short term outbreak of a disease

Endemic: The disease persists

Vital Dynamics: Births and natural deaths accounted for

Vital Dynamics play a bigger part in an endemic

Epidemic vs Endemic

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Total population=N ( a constant) Population fractions

◦ S(t)=susceptible pop. fraction◦ I(t)=infected pop. fraction◦ R(t)=removed pop. fraction

Populations

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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 only

SIR vs SIRS Model

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PART ONE: SIR Models using ODES

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λ=daily contact rate◦ Homogeneously mixing◦ Does not change seasonally

γ =daily recovery removal rate σ=λ/ γ

◦ The contact number

Variables and Values of Importance

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Model for infection that confers permanent immunity

Compartmental diagram

(NS(t))’=-λSNI (NI(t))’= λSNI- γNI (NR(t))’= γNI

The SIR Model without Vital Dynamics

NS Susceptibles

NI Infectives

NR Removeds

λSNI ϒNI

S’(t)=-λSII’(t)=λSI-ϒI

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S’(t)=-λSI I’(t)=λSI-ϒI Let S(t) and I(t) be solutions of this system. CASE ONE: σS₀≤1

◦ I(t) decreases to 0 as t goes to infinity (no epidemic)

CASE TWO: σS₀>1◦ I(t) increases up to a maximum of: 1-R₀-1/σ-ln(σS₀)/σThen it decreases to 0 as t goes to infinity

(epidemic)

Theorem

σS₀=(S₀λ)/ϒInitial Susceptible population fraction

Daily contact rate

Daily recovery removal rate

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MATLAB Epidemic

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PART TWO: SIRS Models using DDES

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dS/dt=μ[1-S(t)]-ΒI(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 rate Β=transmission coefficient γ=recovery rate τ=amount of time before re-susceptibility e-μτ=fraction who recover at time t-τ who

survive to time t rγ=fraction of pop. that become re-susceptible

Equations and Variables

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Focus on the endemic steady state (R0S=1)Reproductive number:R0=Β/(μ+γ)

Sc=1/R0 Ic=[(μ/Β)(ℛ0-1)]/[1-(rγγ)(e-μτ )/(μ+γ)]

Our goal is now to determine stability

Equilibrium Solutions

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dx/dt=-y-εx(a+by)+ry(t-τ) dy/dt=x(1+y)

where ε=√(μΒ)/γ2<<1and r=(e-μτ rγγ)/(μ+γ)and a, b are really close to 1

Rescaled equation for r is a primary control parameter

r is the fraction of those in S who return to S after being infected

Rescaled Equations

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r=(e-μτ rγγ)/(μ+γ) What does rγ=1 mean? Thus,

r max=γ e-μτ /(μ+γ)

So we have:0≤r≤ r max<1

More about r

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λ2+εaλ+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>1

Characteristic Equation

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When does the Hopf bifurcation occur?

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In terms of the original variables… 

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r=0.005

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r=0.005 (Zoomed in)

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r=0.02

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r=0.02 (Zoomed in)

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r=0.03

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r=0.03 (Zoomed in)

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r=0.9

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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 behavior Better prevention and control strategies

Conclusion