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

 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

 Why math? ◦ Interesting ◦ Challenging  Coolest Thing you Learned ◦ RSA Cryptosystem  Why SPWM? ◦ Grad school ◦ Learn something new

 Why Math? ◦ Applications ◦ Challenge  Coolest Thing you Learned ◦ Infinitude of the primes  Why SPWM? ◦ Life after college ◦ DC

 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  (NS(t))’=-λSNI  (NI(t))’= λSNI- γNI  (NR(t))’= γNI NS Susceptibles NI Infectives NR Removeds λSNIϒNI 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 γ γ e -μτ I(t-τ)  dI/dt=ΒI(t)S(t)-(μ+γ)I(t)  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