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Transcript
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