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SIR and SIRS Models

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.

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SIR and SIRS Models

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  1. SIR and SIRS Models Cindy Wu, Hyesu Kim, Michelle Zajac, Amanda Clemm SPWM 2011

  2. 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

  3. Cindy Wu • Why Math? • Friends • Coolest thing you learned • Number Theory • Why SPWM? • DC>Spokane • Otherwise, unproductive

  4. Hyesu Kim • Why math? • Common language • Challenging • Coolest thing you learned • Math is everywhere • Anything is possible • Why SPWM? • Work or grad school? • Possible careers

  5. Michelle Zajac • Why math? • Interesting • Challenging • Coolest Thing you Learned • RSA Cryptosystem • Why SPWM? • Grad school • Learn something new

  6. Amanda Clemm • Why Math? • Applications • Challenge • Coolest Thing you Learned • Infinitude of the primes • Why SPWM? • Life after college • DC

  7. Epidemiology • Study of disease occurrence • Actual experiments vs Models • Prevention and control procedures

  8. Epidemic vs Endemic • 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

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

  10. SIR vs SIRS Model • 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

  11. PART ONE: SIR Models using ODES

  12. Variables and Values of Importance • λ=daily contact rate • Homogeneously mixing • Does not change seasonally • γ =daily recovery removal rate • σ=λ/ γ • The contact number

  13. The SIR Model without Vital Dynamics • Model for infection that confers permanent immunity • Compartmental diagram • (NS(t))’=-λSNI • (NI(t))’= λSNI- γNI • (NR(t))’= γNI λSNI ϒNI NR Removeds NI Infectives NS Susceptibles S’(t)=-λSI I’(t)=λSI-ϒI

  14. Theorem • 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 recovery removal rate Daily contact rate

  15. MATLAB Epidemic

  16. PART TWO: SIRS Models using DDES

  17. Equations and Variables • 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

  18. Equilibrium Solutions • 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

  19. Rescaled Equations • 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

  20. More about r • 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

  21. Characteristic Equation • 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

  22. When does the Hopf bifurcation occur?

  23. In terms of the original variables…

  24. r=0.005

  25. r=0.005 (Zoomed in)

  26. r=0.02

  27. r=0.02 (Zoomed in)

  28. r=0.03

  29. r=0.03 (Zoomed in)

  30. r=0.9

  31. Conclusion • 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

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