Basic Diagram
B(t)
E (exposed)
S (susceptible)
QS(t)
QE(t)
C(t)
Q=Q1+Q2
Q1 (quarantined
non-symptomatic)
I (infectious)
QI(t)
DI(t)
RI(t)
QQ(t)
Q2 (quarantined
symptomatic)
DQ(t)
D (dead/disabled)
* Transitions are the # of people moving between states
(NOT the fraction of people who transition)
RQ2(t)
R (recovered)
RQ1(t)
States (sum up to the whole population)
S(t) = # of susceptible people at time t
E(t) = # of exposed people at time t (incubation state)
I(t) = # of infectious people at time t
Q1(t) = # of non-symptomatic people in quarantine at time t
Q2(t) = # of symptomatic (infectious) people in quarantine at time t
Q(t) = # of people in quarantine at time t
D(t) = # of people dead/disabled at time t
R(t) = # of people recovered at time t
Transitions (# of people transitioning between states during the time interval)
B(t) = # of susceptible people who become infected (exposed) during the time interval
C(t) = # of exposed people who develop symptoms during the time interval
QS(t) = # of susceptible people who are quarantined during the time interval
QE(t) = # of exposed people who are quarantined during the time interval
QI(t) = # of infectious people who are quarantined during the time interval
QQ(t) = # of quarantined exposed people who develop symptoms during the time interval
(currently do not include the QQ transition in the model)
DI(t) = # of infectious people who die or become disabled during the time interval
DQ(t) = # of quarantined people who die during the time interval
RI(t) = # of infectious people who recover during the time interval
RQ1(t) = # of quarantined non-symptomatic people who recover during the time interval
RQ2(t) = # of quarantined infectious people who recover during the time interval
B(t)
E (exposed)
β=transmission rate (# new people
exposed per infectious person)
β*I(t-1)
α=close contacts of newly
quarantined people
S (susceptible)
α*ΔQ*S(t-1)/(E(t-1)+S(t-1))
QS(t)
QE(t) α*ΔQ*(E(t-1)/(E(t-1)+S(t-1))
C(t)
[E(t-1)-QE(t)]*(1/μ1)
ΔQ=QI(t-1)
Q1 (quarantined
non-symptomatic)
q=quarantine rate
q*I(t-1) QI(t)
I (infectious)
φ=treatment rate
Q2 (quarantined
symptomatic)
RI(t)
(1-d)*[I(t-1)-QI(t)]*(1/μ2)
DI(t)
RQ2(t) (1-d)*Q2(t-1)*(1/μ2)
d*[I(t-1)-QI(t)]*(1/μ2)
DQ(t) d*Q2(t-1)*(1/μ2)
d=mortality rate of the disease
D (dead/disabled)
RQ1(t)
φ*Q1(t-1)
+ Q1(t-1)(1-φ)*(1/μ1)
R (recovered)
• subtract off those who go to quarantine before determining how many people go to the other states
• use 1/μ2 to account for time delay before a person dies or recovers (if they were infectious)
• use 1/μ1 to account for time delay before a quarantined non-symptomatic person is considered recovered
(unless they receive treatment … then they go immediately to recovered)
β=transmission rate (# new people
exposed per infectious person)
States (sum up to the whole population)
S(t) = # of susceptible people at time t
E(t) = # of exposed people at time t (incubation state)
I(t) = # of infectious people at time t
Q(t) = # of people in quarantine at time t
D(t) = # of people dead/disabled at time t
R(t) = # of people recovered at time t
Discretized model
S(t+h) = S(t) – QS(t) – B(t)
E(t+h) = E(t) + B(t) – C(t) – QE(t)
I(t+h) = I(t) + C(t) – QI(t) – RI(t) – DI(t)
Q1(t+h) = Q1(t) + QS(t) + QE(t) – RQ1(t)
Q2(t+h) = Q2(t) + QI(t) – DQ(t) – RQ2(t)
D(t+h) = D(t) + DI(t) + DQ(t)
R(t+h) = R(t) + RI(t) + RQ1(t) + RQ2(t)
• constraints are included in the Excel
implementation of this model
• limits on the number of people being
sent to quarantine (setting to a large
number effectively removes the
constraint)
• limits on the B(t) transition (since it
doesn’t depend on S(t) limits are
needed to prevent more people from
transitioning than are available)
State Transitions
(# of people moving
between states)
α=close contacts of newly
quarantined people
q=quarantine rate
d=mortality rate of the disease
φ=treatment rate
• subtract off those who go to quarantine first
• use 1/μ2 to account for time delay before a person dies
or recovers (if they were infectious)
• use 1/μ1 to account for time delay before a
quarantined non-symptomatic person is considered
recovered (unless they receive treatment … then they
go immediately to recovered)
ΔQ=QI(t-1)
B(t) = β*I(t-1)
C(t) = [E(t-1)-QE(t)]*(1/μ1)
QS(t) = α*ΔQ*S(t-1)/(S(t-1)+E(t-1))
QE(t) = α*ΔQ*E(t-1)/(S(t-1)+E(t-1))
QI(t) = q*I(t-1)
DI(t) = d*[I(t-1)-QI(t)]*(1/μ2)
DQ(t) = d*Q2(t-1)*(1/μ2)
RI(t) = (1-d)*[I(t-1)-QI(t)]*(1/μ2)
RQ1(t) = φ*Q1(t-1) + Q1(t-1)*(1-φ)*(1/μ1)
RQ2(t) = (1-d)*Q2(t-1)*(1/μ2)