SIR Epidemic and Vaccination Infection Transmission Plus Host Birth, Death Purpose: What fraction of hosts must be vaccinated in order to eradicate disease?

Compartments & Virulence Susceptible, Infective, Recovered Hosts Background Mortality: All classes Virulence: Extra mortality, Infected hosts only Reproduction: All classes; Hosts Born S

Parameters b Per-capitum Birth Transmission Rate (Mass Action) d Non-Disease Mortality (All) D Infective Mortality (D – d): Virulence > 0  Recovery from Infection

Dynamics of General Epidemic

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?

Vaccination Control (Ricklefs & Miller, 2000)

Disease Locale R0 pV Smallpox Developing World 3 - 5 0.7 – 0.8 Measles May, 1983; Includes corrections for vaccine efficiencies Disease Locale R0 pV Smallpox Developing World 3 - 5 0.7 – 0.8 Measles England 13 0.92 Whooping cough 17 0.94 Rubella 6 0.83 Chicken Pox US 9 - 10 0.9 Diphtheria 4 - 6 0.8 Scarlet fever 5 - 7 Mumps 4 - 7 Polio Netherlands Malaria Nigeria 80 0.99 16

Why is virulence so diverse? Define Virulence Reduction in Host Fitness Due to Pathogen’s Reproduction Increased Host Mortality Rate “Sub-lethal” Diminished Host Fecundity Cost of Immune Response Decline in Competitive Ability Why is virulence so diverse?

Increase Host Mortality Decrease Host Fecundity Virulence: Process Pathogen Takes Resources from Host Energy, Nutrients “Virulence Factors” Pathogen Growth Releases Substance Toxic to Host Increase Host Mortality Decrease Host Fecundity

Virulence: Trade-Off Pathogen Evolves Faster than Host Benefit of Increased Virulence Faster Pathogen Growth  Increased Transmission Rate Cost of Increased Virulence Duration of Infectious Period Reduced Via Greater Host Mortality

Virulence: Trade-Off Greater Transmission Rate  Requires Greater Virulence 

Natural Selection and Virulence? Evolutionarily Stable Virulence 𝛼 ∗ Maximizes 𝑅 0 = Number of Infections/Infection When Rare Equivalently, 𝛼 ∗ Minimizes Density of Susceptible Hosts at Endemic Equilibrium

Evolutionarily Stable Virulence

ES Virulence Maximizes 𝑹 𝟎

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

Fraser et al. 2007. PNAS 104:17441-17446 Set-Point Viral Load of HIV-1 Peripheral 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?

Viral Load: Heterogeneity

Transmission Rate-Duration Trade-Off Duration of Infectious Period

E[Infections per Case]

𝑹 𝟎 and E[Growth Rate]