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

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

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

4. Dynamics of General Epidemic

5. 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?

6. Vaccination Control (Ricklefs & Miller, 2000)

7. 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

8. 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?

9. 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

10. 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

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

12. 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

13. Evolutionarily Stable Virulence

14. ES Virulence Maximizes 𝑹 𝟎

15. 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

16. 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?

17. Viral Load: Heterogeneity

18. Transmission Rate-Duration Trade-Off
Duration of Infectious Period

19. E[Infections per Case]

20. 𝑹 𝟎 and E[Growth Rate]