sir epidemic and vaccination infection transmission plus host birth, death purpose: what fraction of...
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SIR Epidemic and Vaccination
Infection TransmissionPlus
Host Birth, Death
Purpose: What fraction of hosts must be vaccinated in order to eradicate disease?
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Compartments & Virulence
Susceptible, Infective, Recovered Hosts
Background Mortality: All classes
Virulence: Extra mortality, Infected hosts only
Reproduction: All classes; Hosts Born S
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Parameters
b Per-capitum Birth
b Transmission Rate (Mass Action)
d Non-Disease Mortality (All)
D Infective Mortality
(D – d): Virulence > 0 Recovery from Infection
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Dynamics of General Epidemic
dSSIRISbdtdS
IDSIdtdI
dRIdtdR
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Assumptions
When Rare, Pathogen Invades Host Pop.
R0 > 1 Invasion Criterion
Equilibrium: Endemic Infection if R0 > 1
Vaccine Available: What level of vaccination (reduction in susceptibility) would prevent disease form advancing when rare?
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Vaccination Control (Ricklefs & Miller, 2000)
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Disease Locale R0 pV
Smallpox Developing World
3 - 5 0.7 – 0.8
Measles England 13 0.92
Whooping cough
England 17 0.94
Rubella England 6 0.83
Chicken Pox US 9 - 10 0.9
Diphtheria US 4 - 6 0.8
Scarlet fever US 5 - 7 0.8
Mumps US 4 - 7 0.8
Polio Netherlands 6 0.83
Malaria Nigeria 80 0.99
Malaria Nigeria 16 0.94
May, 1983; Includes corrections for vaccine efficiencies
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Define VirulenceReduction in Host Fitness
Due to Pathogen’s Reproduction
Increased Host Mortality Rate “Sub-lethal”
Diminished Host FecundityCost of Immune ResponseDecline in Competitive Ability
Why is virulence so diverse?
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Virulence: Process Pathogen Takes Resources from Host
Energy, Nutrients
“Virulence Factors”
Pathogen Growth Releases Substance Toxic to Host
Increase Host MortalityDecrease Host Fecundity
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Virulence: Trade-Off
Pathogen Evolves Faster than Host
Benefit of Increased VirulenceFaster Pathogen Growth Increased Transmission Rate
Cost of Increased VirulenceDuration of Infectious Period Reduced Via Greater Host Mortality
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Virulence: Trade-Off
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Greater Transmission Rate Requires Greater Virulence
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Natural Selection and Virulence?
Evolutionarily Stable Virulence
Maximizes =Number of Infections/Infection When Rare
Equivalently, Minimizes Density of Susceptible Hosts at
Endemic Equilibrium
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Evolutionarily Stable Virulence
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ES Virulence Maximizes
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Maximal Infections per Infection
Assumes:
Fully Mixed Population {0, 1} Host Infections Direct-Contact Transmission
Before Host Death or Recovery Monomorphic Solution!
Example Where Assumptions Fit
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Fraser et al. 2007. PNAS 104:17441-17446
Set-Point Viral Load of HIV-1Peripheral Density, Asymptomatic Period
Dependence of Set-Point on Viral Replication Unclear
Infectiousness Increases with Viral Load Duration of Infectious Period Declines Viral Life-History Trade-Off?
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Viral Load: Heterogeneity
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Transmission Rate-Duration Trade-Off
Duration of Infectious Period
Transmission Rate
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E[Infections per Case]
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and E[Growth Rate]
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