SIR and SIRS ModelsCindy Wu, 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
Dr. Petrillo
Amanda Clemm
Scripps College
Dr. Ou
Cindy Wu Why Math?
Friends Coolest thing you
learned Number
Theory Why SPWM?
DC>Spokane Otherwise,
unproductive
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
Why math?◦ Interesting◦ Challenging
Coolest Thing you Learned◦ RSA Cryptosystem
Why SPWM?◦ Grad school◦ Learn something new
Michelle Zajac
Why Math?◦ Applications◦ Challenge
Coolest Thing you Learned◦ Infinitude of the
primes Why SPWM?
◦ Life after college◦ DC
Amanda Clemm
Study of disease occurrence Actual experiments vs Models Prevention and control procedures
Epidemiology
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
Total population=N ( a constant) Population fractions
◦ S(t)=susceptible pop. fraction◦ I(t)=infected pop. fraction◦ R(t)=removed pop. fraction
Populations
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
PART ONE: SIR Models using ODES
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
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
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
MATLAB Epidemic
PART TWO: SIRS Models using DDES
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
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
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
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
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
When does the Hopf bifurcation occur?
In terms of the original variables…
r=0.005
r=0.005 (Zoomed in)
r=0.02
r=0.02 (Zoomed in)
r=0.03
r=0.03 (Zoomed in)
r=0.9
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
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