1. Populations: Input Parameters: S = S+Qs I = I1+I2+Qi+D R = R1+R2
S = Susceptible
I1 = Infected but not yet infectious (incubation period)
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
Standard SIR
Input Parameters:
green = known
blue = unknown
(parameterize and/or optimize to determine)
add in treated vs. untreated later
d = mortality rate for the disease
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)

2. Track 7 different populations S+Q+I1+I2+R1+R2+D = N
s(t)* i2(t)*β
i1(t)*1/μ1
Infectious I2
Susceptible S
Infected (not yet infectious) I1
Assumption: don’t know that I1 people are infected
i2(t)*αi
s(t)*αs
i1(t)*αs
i2(t)*κ(1-λ)
Fully Recovered R1
(insusceptible)
qi(t)*κ(1-λ)
Quarantine
Q = Qs + Qi
i2(t)*κ*λ
qi(t)*κ*λ
Recovered R2
(but disabled)
i2(t)*d
S = S+Qs
I = I1+I2+Qi+D
R = R1+R2
Standard SIR
qi(t)*d
Dead D

3. Initial Assumptions Constant population One recovery rate
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)

4. 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)

5. Smallpox
Incubation Period (Duration: 7 to 17 days) Not contagious
Incubation Period  7-17 days (avg 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
Symptomatic Period  20 days
(contagious)
Pustules and Scabs (Duration: about 5 days) Contagious
Resolving Scabs (Duration: about 6 days) Contagious
Recovered
(not contagious)
Scabs resolved Not contagious