ESPM-EEP%202010%20Lecture%2016

# ESPM-EEP%202010%20Lecture%2016 - 3 Pure SIVS(commonly...

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ESPM 104/EEP 115 Fall 2010: Lecture 16 1. SIR Models in Epidemiology: see Hastings 1996, Chapt 10 and EpiMaterial.pdf . S : susceptible E : exposed but note infectious I : infectious V : vaccinated D : dead (disease and other) R = D + V (‘removed’ individuals) N = S + E + I + V (living individuals) τ : transmission γ : latent to active disease α X : transition rate of I to X ρ X : transition rate of X back to susceptible b : births (two types) μ : non-disease mortality

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2. Pure SIS model with mass-action transmission τ = β SI dS dt = IS + rI = I ( r S ) dI dt = IS rI = I ( S r ) dN dt = dS dt + dI dt = 0 N ( t ) = N 0 S = N 0 I dI dt = I ( N 0 r I ) dI dt = ( N 0 r ) I 1 ( N 0 r ) I This is logistic growth with intrinsic growth rate N 0 r and carrying capacity K = N 0 r = N 0 r . The epidemic only happens if N 0 > r = N T where N T is called the threshold density.
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Unformatted text preview: 3. Pure SIVS (commonly called SIRS) epidemic dS dt = − IS + ρ V dI dt = IS − α I = I ( S − ) dV dt = I − V dN dt = dS dt + dI dt + dV dt = ⇒ N ( t ) = N V = N − ( S + I ) ⇒ dS dt = − IS + N − ( S + I ) ( ) Thus we have a dynamic system in variables S and I (2 nd and last eqs.) Susceptible nullcline: dS dt = − β IS + ρ N − ( S + I ) ( ) = ⇒ IS + I = N − S ( ) ⇒ I = N − S ( ) S + Infectious nullcline: dI dt = I =0 or S = α / . S I ! / " N...
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ESPM-EEP%202010%20Lecture%2016 - 3 Pure SIVS(commonly...

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