RPI (2009) To: The Rensselaer Community From: Leslie Lawrence, M.D. Medical Director, Student Health Center Date: November 23, 2009 Re: H1N1 Update As of Friday, Nov. 20, we have experienced 232 cases of influenza among students on the Troy campus. Some 18 students have active cases of the illness. Of those, none are currently in isolation rooms and 18 are recuperating at home with their families. The remaining 214 students are fully recovered. In addition, we continue to receive reports of small numbers of faculty and staff with influenza-like illnesses.

SIR model for epidemics (compartmental model) S  I with rate  (infection rate) I  R with rate  (recovery rate) N: number of individuals in the population S: number of Susceptible individuals I: number of Infective individuals R: number of Removed (recovered/dead) individuals homogeneous mixing:

SIR model for epidemics s=S/N: density of Susceptible individuals i=I/N: density of Infective individuals r=R/N: density of Removed (recovered/dead) individuals s  i  (infection rate) i  r  (recovery rate) R o : basic reproduction number (the # of individuals a sick person will infect)

SIR model for epidemics s  i  (infection rate) i  r  (recovery rate) s: susceptible i: infected r: recovered

SIR model for epidemics: numerical integration condition for outbreak:

epidemic is spreading How do you control epidemics? make smaller, or

Epidemic controls:  Reduce s(t): vaccination  Reduce  : wash hands, isolate sick persons, shut down public events, close schools  Increase  : better/faster acting medicine, antivirals How do you control epidemics? make smaller, or fraction of people got the disease (cumulative) epidemic is spreading fraction of people sick at a given time t

Mass Vaccination i.e., at any time (preferably before the outbreak), if we can sufficiently reduce the density of susceptible individuals (by vaccinating), the epidemics will die out for example, for R o = 1.5  s c = 0.66, i.e., roughly 33% percent of the population should be vaccinated for R o = 3.0  s c = 0.33, i.e., roughly 66% percent of the population should be vaccinated i.e., for a successful vaccination campaign, the fraction of the population that should be vaccinated: (within the limitations of the simple SIR model) density of vaccinated individuals

 “For a population with sufficiently high vaccine coverage, a disease can be eradicated without vaccinating everyone.  Therefore, as coverage increases, there is a greater individual incentive not to vaccinate, since non-vaccinators can gain the benefits of herd immunity (population-level immunity) without the risk of vaccine complications” Zhifeng Sun (2009) See also: “Vaccination and the theory of games”, C.T. Bauch and D.J.D. Earn, PNAS 101, (2004).C.T. Bauch and D.J.D. Earn, PNAS 101, (2004). Game Theory and Vaccination

RPI (2009) To: The Rensselaer Community From: Leslie Lawrence, M.D. Medical Director, Student Health Center Date: November 23, 2009 Re: H1N1 Update As of Friday, Nov. 20, we have experienced 232 cases of influenza among students on the Troy campus. Some 18 students have active cases of the illness. Of those, none are currently in isolation rooms and 18 are recuperating at home with their families. The remaining 214 students are fully recovered. In addition, we continue to receive reports of small numbers of faculty and staff with influenza-like illnesses.

RPI (2009) fraction of people got the disease (cumulative ) fraction of people sick at a given time t vaccination

The SIR Model for Spread of Disease:  National Science Digital Library

The effect of the airline transportation network Colizza et al. (2007) PLoS Medicine 4, 0095 Main modeling features (SARS, H1N1, etc.):  SIR model with empirical population and airline traffic/network data  homogeneous mixing within cites  network connections and stochastic transport between cities

Stochastic SIR on global networks Colizza (2006) Bulletin of Math. Biol. 68, Gaussian noisestochastic travel operator Main modeling features:  SIR model with empirical population and airline traffic/network data  homogeneous mixing within cites  network connections and stochastic transport between cities

H1N1 Flu Mobility networks in Europe (network-coupled SIR model): Left: airport network; Right: commuting network.

H1N1 Flu

H1N1 Flu P Bajardi, C Poletto, D Balcan, H Hu, B Goncalves, JJ Ramasco, D Paolotti, N Perra, M Tizzoni, W Van den Broeck, V Colizza, A Vespignani, Emerging Health Threats Journal 2009, 2:e11.

DiseaseTransmissionR0R0 MeaslesAirborne12-18 PertussisAirborne droplet12-17 DiphtheriaSaliva6-7 SmallpoxSocial contact5-7 PolioFecal-oral route5-7 RubellaAirborne droplet5-7 MumpsAirborne droplet4-7 HIVHIV/AIDSAIDSSexual contact2-5 [2] [2] SARSAirborne droplet2-5 [3] [3] Influenza (1918 pandemic strain) Airborne droplet2-3 [4] [4]

Model fit to the daily number of hospital notifications during the first two waves of the 1918 influenza pandemic in the Canton of Geneva, Switzerland. Chowell et al., Vaccine 24, Issues 44-46, (2006)