SIR Epidemic and Vaccination

Infection TransmissionPlus

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

b Transmission Rate (Mass Action)

d Non-Disease Mortality (All)

D Infective Mortality

(D – d): Virulence > 0 Recovery from Infection

Dynamics of General Epidemic

dSSIRISbdtdS

IDSIdtdI

dRIdtdR

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 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

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

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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