Cindy Wu, Hyesu Kim, Michelle Zajac, Amanda Clemm SPWM 2011

   Cindy Wu Gonzaga University Dr. Burke    Hyesu Kim Manhattan College Dr. Tyler    Michelle Zajac Alfred University Dr. Petrillo    Amanda Clemm Scripps College Dr. Ou

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

  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

  ◦ ◦ ◦ Total population=N ( a constant) Population fractions S(t)=susceptible pop. fraction I(t)=infected pop. fraction R(t)=removed pop. fraction

    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

   λ=daily contact rate ◦ ◦ Homogeneously mixing Does not change seasonally γ =daily recovery removal rate ◦ σ=λ/ γ The contact number

  Model for infection that confers permanent immunity Compartmental diagram λSNI NS Susceptibles NI Infectives ϒNI    (NS(t))’=-λSNI (NI(t))’= λSNI- γNI (NR(t))’= γNI NR Removeds

S’(t)=-λSI I’(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)/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

   dS/dt=μ[1-S(t)]-ΒI(t)S(t)+r dI/dt=ΒI(t)S(t)-(μ+γ)I(t) γ γ e -μτ I(t-τ) dR/dt=γI(t)-μR(t)-r γ γe -μτ I(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 r γ =fraction of pop. that become re-susceptible

 Focus on the endemic steady state (R 0 S=1) Reproductive number: R 0 =Β/(μ+γ)   S c =1/R 0 I c =[(μ/Β)(ℛ 0 -1)]/[1-(r γ γ)(e -μτ )/(μ+γ)] Our goal is now to determine stability

    dx/dt=-y-εx(a+by)+ry(t-τ) dy/dt=x(1+y) where ε=√(μΒ)/γ 2 <<1 and r=(e -μτ r γ γ)/(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

   r=(e -μτ r γ γ)/(μ+γ) What does r γ =1 mean?

Thus, r max =γ e -μτ /(μ+γ)  So we have: 0≤r≤ r max <1

 λ 2 +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 R 0 >1

    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