### SIR and SIRS Models

```Cindy Wu, Hyesu Kim, Michelle Zajac, Amanda Clemm
SPWM 2011
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Cindy Wu
Gonzaga
University
Dr. Burke
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Hyesu Kim
Manhattan
College
Dr. Tyler
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Michelle
Zajac
Alfred
University
Dr. Petrillo
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Amanda
Clemm
Scripps
College
Dr. Ou
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Why Math?
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Study of disease occurrence
Actual experiments vs Models
Prevention and control procedures
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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
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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
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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
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λ=daily contact rate
◦ Homogeneously mixing
◦ Does not change seasonally
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γ =daily recovery removal rate
σ=λ/ γ
◦ The contact number
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Model for infection that
confers permanent
immunity
Compartmental diagram
λSNI
NS
Susceptibles
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ϒNI
NI Infectives
(NS(t))’=-λSNI
(NI(t))’= λSNI- γNI
(NR(t))’= γNI
NR Removeds
S’(t)=-λSI
I’(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)
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CASE TWO: σS₀>1
◦ I(t) increases up to a maximum of:
1-R₀-1/σ-ln(σS₀)/σ
Then it decreases to 0 as t goes to infinity (epidemic)
σS₀=(S₀λ)/ϒ
Initial Susceptible
population fraction
Daily contact rate
Daily recovery
removal rate
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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
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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
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dx/dt=-y-εx(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 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
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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
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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
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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
```