Download Modern Applications of the SIR Epidemic Model

Nik Addleman and Jen Fox
Traditional SIR Model

 Susceptible, Infected and Recovered
S' = - ßSI
I' = ßSI - γI
R' = γI
 Assumptions
 S and I contact leads to infection
 Infection is a disease, allows for recovery (or death…)
 Fixed population
Jacobian Analysis

S' = - ßSI = 0
I' = ßSI – γI = 0
R' = γI = 0
æ -b I, b S, 0 ö
ç
÷
J = ç b I, b S - g , 0 ÷
ç
÷
è 0, g , 0 ø
bS
C0 =
g
Equilibrium points: I = S = 0, R = R*
Example of SIR Model

 Infectious contact rate
S
 β = # daily contacts *
transmission
probability given a
contact
 Infectious Period
 γ= time until recovered
and no longer
infectious
t
Extensions

 Vaccinations
 Vaccinated members of susceptible pop.
are not as likely to contract disease
 Temporary infective/immunity periods
Modeling Influenza

 Modeling Seasonal Influenza Outbreak in a Closed
College Campus. (K. L. Nichol et al.)
 Compartmentalized, fixed-population ODE model
 Modification of the SIR model
 Minimize Total Attack Rate
 Experimentally determine
parameters
Compartments

 Students and Faculty
 Vaccinated versus Unvaccinated
 Symptomatic and Asymptomatic infections
 Different β and γ values for various populations
 Categories (following slide)
 Four susceptible categories
 Eight infected
 One recovered

Constructing Equations

 Determining parameters
 β varies between students/faculty and
symptomatic/asymptomatic
 γ has different values for symptomatic/asymptomatic
and vaccinated/unvaccinated populations
 Vaccine 80% effective
 Apply to all compartments
Susceptible

dSvs
= -(.2)Svs (bvws I vws + bvos I vos + bnws I nws + bnos I nos + bvwf I vwf + bvof I vof + bnwf Inwf + bnof I nof )
dt
dSns
= -Sns (bvws I vws + bvos I vos + bnws I nws + bnos I nos + bvwf I vwf + bvof I vof + bnwf I nwf + bnof I nof
dt
dSvf
dt
dSnf
dt
= -(.2)Svf (bvws I vws + bvos I vos + bnws I nws + bnos I nos + bvwf I vwf + bvof I vof + bnwf I nwf + bnof I nof )
= -Snf (bvws I vws + bvos I vos + bnws I nws + bnos I nos + bvwf I vwf + bvof I vof + bnwf I nwf + bnof I nof )
Infectious

dI vws
= j (.2)Svs (bvws I vws + bvos I vos + bnws I nws + bnos I nos + bvwf I vwf + bvof I vof + bnwf I nwf + bnof I nof ) - g vws I vws
dt
dI vos
= (1- j )(.2)Svs (bvws I vws + bvos I vos + bnws I nws + bnos I nos + bvwf I vwf + bvof I vof + bnwf I nwf + bnof I nof ) - g vos I vos
dt
dI nws
= j (.2)Sns (bvws I vws + bvos I vos + bnws I nws + bnos I nos + bvwf I vwf + bvof I vof + bnwf I nwf + bnof I nof ) - g nws I nws
dt
dI nos
= (1- j )(.2)Sns (bvws I vws + bvos I vos + bnws I nws + bnos I nos + bvwf I vwf + bvof I vof + bnwf I nwf + bnof I nof ) - g nos I nos
dt
… etc

Conclusions

 Can use SIR model to determine best way to cut
down on infections
 Stay home when you are sick because you are
infectious. Gross.
 Get vaccinated!
 Even late vaccinations are effective
 Vaccine helps you and those around you
 60% vaccination means none of us gets sick