Mathematical Analysis of Epidemiological Models II Jan Medlock - PowerPoint PPT Presentation

Endemic Model Vaccination Tomorrow Mathematical Analysis of Epidemiological Models II Jan Medlock Clemson University Department of Mathematical Sciences 22 July 2009

Endemic Model Vaccination Tomorrow Endemic Model d t = µ N − β I ( t ) d S N S ( t ) − µ S ( t ) d t = β I ( t ) d I N S ( t ) − γ I ( t ) − µ I ( t ) d R d t = γ I ( t ) − µ R ( t ) Constant population size: d N d t = d S d t + d I d t + d R d t = 0 .

Endemic Model Vaccination Tomorrow Endemic Model So divide by population size s = S i = I N , N � S = 1 d s d t = d � d S d t = µ N N − β I N − µ S S N = µ − β is − µ s d t N N N d i d t = β is − γ i − µ i d r d t = γ i − µ r s + i + r = 1

Endemic Model Vaccination Tomorrow Find equilibria d s d t = d i d t = 0 d s d t = 0 = µ − β is − µ s d i d t = 0 = β is − γ i − µ i Two equilibria: • Disease-free equilibrium: E 0 = ( s = 1 , i = 0 ) • Endemic equilibrium: s = γ + µ , i = µ ( β − γ − µ ) � � E e = β β ( γ + µ )

Endemic Model Vaccination Tomorrow Linearize equations Write as vector differential equation � � � � d s µ − β is − µ s = = f ( s , i ) β is − γ i − µ i d t i By Taylor’s theorem �� � � �� s s 0 f ( s , i ) = f ( s 0 , i 0 ) + J ( s 0 , i 0 ) − + · · · i i 0 At equilibrium, f ( s 0 , i 0 ) = 0 , so the dynamics near ( s 0 , i 0 ) are governed by the linear part J ( s 0 , i 0 )

Endemic Model Vaccination Tomorrow Analysis Jacobian derivative of f � ∂ f 1 � � � ∂ f 1 − β i − µ − β s ∂ s ∂ i J ( s , i ) = = ∂ f 2 ∂ f 2 β s − γ − µ β i ∂ s ∂ i • Disease-free equilibrium � � − µ − β J ( 1 , 0 ) = 0 β − γ − µ Eigenvalues {− µ, β − µ − γ } • λ 1 = − µ < 0 • λ 2 = β − µ − γ β • β − µ − γ < 0 ⇐ γ + µ < 1, stable, No epidemic ⇒ β • β − µ − γ > 0 ⇐ γ + µ > 1, unstable, Epidemic ⇒ β R 0 = γ + µ

Endemic Model Vaccination Tomorrow Analysis • Endemic equilibrium − µβ � � − γ − µ � γ + µ , µ ( β − γ − µ ) � γ + µ = J µ ( β − γ − µ ) β β ( γ + µ ) 0 γ + µ � � � µ 2 β 2 − µβ Eigenvalues µ + γ ± ( µ + γ ) 2 − 4 µ ( β − γ − µ ) β • R 0 = γ + µ > 1, stable β • R 0 = γ + µ < 1, unstable

Endemic Model Vaccination Tomorrow Summary β R 0 = γ + µ E 0 = ( s = 1 , i = 0 ) s = γ + µ = 1 , i = µ ( β − γ − µ ) � = µ � E e = β ( 1 − R 0 ) β ( γ + µ ) β R 0 • R 0 < 1 Disease-free equilibrium is stable Endemic equilibrium is unstable (and nonsense!) • R 0 > 1 Disease-free equilibrium is unstable Endemic equilibrium is stable

Endemic Model Vaccination Tomorrow Vaccination model d s d t = ( 1 − p ) µ − β is − µ s d i d t = β is − γ i − µ i d r d t = γ i − µ r d v d t = p µ − µ v s + i + r + v = 1

Endemic Model Vaccination Tomorrow Analysis Disease-free equilibrium: E 0 = ( s = 1 − p , i = 0 , v = p ) Jacobian:   − β i − µ − β s 0 J ( s , i , v ) = β s − γ − µ 0 β i     0 0 − µ   − µ − β ( 1 − p ) 0 J ( E 0 ) = 0 β ( 1 − p ) − γ − µ 0     0 0 − µ

Endemic Model Vaccination Tomorrow Analysis   − µ − β ( 1 − p ) 0 J ( E 0 ) = 0 β ( 1 − p ) − γ − µ 0     0 0 − µ λ 1 , 2 = − µ < 0 λ 3 = β ( 1 − p ) − γ − µ β λ 3 > 0 ⇐ ⇒ R v = γ + µ ( 1 − p ) = R 0 ( 1 − p ) > 1 λ 3 < 0 ⇐ ⇒ R v < 1 Stability determined by R v

Endemic Model Vaccination Tomorrow Critical vaccination level ⇒ p ∗ = 1 − 1 R v = R 0 ( 1 − p ∗ ) = 1 = R 0 p > p ∗ = ⇒ R v < 1 No epidemic!

Endemic Model Vaccination Tomorrow Tomorrow • R 0 for complex models • Vector-borne disease model • Age-structured model