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
dS



b
S

I

R


SI

dS
dt
dI




SI

D


I
dt
dR   I  dR
dt
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)
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
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
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?
8
Virulence: Process
 Pathogen Takes Resources from Host
Energy, Nutrients
 “Virulence Factors”
Pathogen Growth Releases
Substance Toxic to Host
Increase Host Mortality
Decrease Host Fecundity
9
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
10
Virulence: Trade-Off
Greater Transmission Rate 
Requires Greater Virulence 
11
Natural Selection and Virulence?
Evolutionarily Stable Virulence 𝛼 ∗
Maximizes 𝑅0 =
Number of Infections/Infection When Rare
Equivalently, 𝛼 ∗
Minimizes Density of Susceptible Hosts at
Endemic Equilibrium
12
Evolutionarily Stable Virulence
13
ES Virulence Maximizes 𝑹𝟎
14
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
15
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?
16
Viral Load: Heterogeneity
17
Transmission Rate-Duration Trade-Off
Transmission Rate
Duration of Infectious Period
18
E[Infections per Case]
19
𝑹𝟎 and E[Growth Rate]
20