Download Epidemic (Compartment) Models

Epidemic (Compartment) Models
St
Susceptible Hosts, Time t
It
Infectious Hosts; Initially Rare
Rt
Removed Hosts
May Fix Population Size: 𝑆𝑡 + 𝐼𝑡 + 𝑅𝑡 = 𝑛
May Allow Host Birth, Non-Disease Mortality
Epidemic without Removal
SI Process
Only Transition: Infection Transmission
SIS Process
Two Transitions: Infection and
Recovery without Immunity
Epidemic with Removal
SIR Epidemic
Two Transitions: Infection and Removal
SIRS Epidemic
Three Transitions: Infection, Removal
with Immunity, Loss of Immunity
Infection Dynamics
Epidemic: Pathogen Advances from Rarity;
Infects Detectable Number of Hosts
Endemic: Dynamic Equilibrium; Constant Positive
Density of Infectious Hosts
Disease-Free Equilibrium: No Infected Hosts
AND No Free-Living Pathogens
Infection Dynamics
R0 : Number of New Infections per Infection when
Pathogen (Infection) Is Rare
𝑅0 > 1
Implies Epidemic Advance
Virulence: Reduction in Infected Host’s Fitness
Usually, Increase in Host Mortality
SIR Epidemic; Fixed Population Size
𝑑𝑆𝑡
𝑑𝐼𝑡
𝑑𝑅𝑡
𝑑𝑡
= − 𝛽 𝑆𝑡 𝐼𝑡
𝑑𝑡
= 𝛽 𝑆𝑡 𝐼𝑡 − 𝛾 𝐼𝑡
𝑑𝑡
= 𝛾 𝐼𝑡
Infection Transmission
Infection - Removal
Removal
Infection Rare: Will Epidemic Occur?
SIR Epidemic: Pathogen Invasion
Growth/Decline per Infected Host at t = 0
𝑑𝐼 𝑑𝑡
𝐼
= 𝛽 𝑆0 − 𝛾
New Infections – Removal
Calculate R0 from preceding expression.
SIR Epidemic: Pathogen Invasion
Rate of New Infection
 S0
Infections/Time
Duration of Infectious Period
1/
Time
(Infections/Time) Time = New Infections
SIR Epidemic: Pathogen Invasion
𝑅0 =
𝛽 𝑆0
𝑆𝑐𝑟𝑖𝑡 =
𝛾
𝛾
𝑆0 > 𝑆𝑐𝑟𝑖𝑡
𝛽 ⇒ 𝑅0 = 1
Epidemic