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Populations:
S = Susceptible
I1 = Infected but not yet infectious (incubation period)
Standard SIR
I1(0) = Number that are infected in the attack
I2 = Infectious
Q = Quarantine (both susceptible and infectious people)
Q = Qs + Qi
Qs = Susceptible people that get quarantined
Qi = Infected people that get quarantined (incubation and infectious)
R1 = Fully recovered (no residual damage)
R2 = Recovered, but sustained some residual damage
D = Dead
S = S+Qs
I = I1+I2+Qi+D
R = R1+R2
Input Parameters:
d = mortality rate for the disease
add in treated vs. untreated later
κ = recovery rate
κ = (1/μ2)*(1-d)
d1 = mortality rate for the disease (if untreated)
d2 = mortality rate for the disease (if untreated)
κ1 = recovery rate (for untreated infectious people)
κ2 = recovery rate (for treated infectious people)
φ = fraction of people treated per day
λ = fraction of infected people that sustain long term residual damage
μ1 = average duration of non-infectious stage (incubation)
μ2 = average duration of infectious stage
β = transmission rate (fraction of people infected per day per infectious person)
αi = fraction of infected people quarantined per day
αs = fraction of susceptible people quarantined per day (contacts of infected
people)
green = known
blue = unknown
(parameterize and/or
optimize to determine)
S+Q+I1+I2+R1+R2+D = N
Track 7 different populations
Susceptible
S
s(t)* i2(t)*β
Infected (not yet infectious)
I1
Assumption: don’t know that
I1 people are infected
s(t)*αs
i1(t)*1/μ1
Infectious
I2
i2(t)*αi
i1(t)*αs
i2(t)*κ(1-λ)
Quarantine
Q = Qs + Qi
qi(t)*κ(1-λ)
Fully Recovered
R1
(insusceptible)
qi(t)*κ*λ
Standard SIR
S = S+Qs
I = I1+I2+Qi+D
R = R1+R2
i2(t)*κ*λ
Recovered
R2
(but disabled)
i2(t)*d
qi(t)*d
Dead
D
Initial Assumptions
• Constant population
– No immigration/emigration, births, or deaths (not related to the
disease)
• One recovery rate
– Don’t separate treated vs. non-treated people (add in later later)
• People in the incubation stage (non-contagious) are
treated as susceptible in terms of quarantine/treatment
since they are not yet known to be infected
• Treatment has no significant side effects (compared to
mortality rate of disease and likelihood of long term
damage from the disease)
– Add this in later (when separate the treated vs. non-treated
people)
Basic SIR Model
st = susceptible fraction of the population at time t
it = infected fraction of the population at time t
rt = recovered fraction of the population at time t
β = transmission rate
κ = recovery rate
st+1= st – β*st*it
rt+1 = rt + κ*it
it+1 = it * (1 + β*st – κ)
Smallpox
Incubation Period
(Duration: 7 to 17 days)
Not contagious
Incubation Period  7-17 days (avg.12-14 days)
(not contagious)
Initial Symptoms (Prodrome)
(Duration: 2 to 4 days)
Sometimes contagious
Prodrome Period  2-4 days
(sometimes contagious)
Early Rash
(Duration: about 4 days)
Most contagious
Pustular Rash
(Duration: about 5 days)
Contagious
Pustules and Scabs
(Duration: about 5 days)
Contagious
Symptomatic Period  20 days
(contagious)
Resolving Scabs
(Duration: about 6 days)
Contagious
Scabs resolved
Not contagious
Recovered
(not contagious)
http://www.bt.cdc.gov/agent/smallpox/overview/disease-facts.asp