1. © 2014 by Princeton University Press
4.3 Spread of SARS
Angela B. Shiflet and George W. Shiflet
Wofford College
© 2014 by Princeton University Press

2. Spread of Disease The first case of SARS (several acute respiratory
syndrome) occurred in November, 2002 in southern China.
SARS is spread by close personal contact and perhaps
by airborne transmission. It was contained by July, 2003,
but resulted in 812 deaths.
Many models of the spread of disease (including SARS
model) are extensions of SIR Model (three populations are
considered: Susceptible, Infected, Recovered).
In SIR Model, assume the rate of change of the number
of recovered is proportional to the number of infected.

3. Spread of Disease
Also, assuming no births, deaths, immigration, or emigration in SIR Model.
recovery rate a = 1/(number of days infected) = 1/d.
Thus, dR/dt = a*I = (1/d)*I
The rate of change of the number of susceptible is proportional to the product of the number of infected and
the number of susceptible (due to the interactions
between S and I): dS/dt = -r*S*I where r is called the
transmission constant and usually is a small constant.

4. Spread of Disease
Another explanation is that the rate of change of S is minus the product of the mean number of contacts per day an infected has (k), the probability such a contact is with a susceptible (S/N where N is the total population size), the probability that the disease is spread during such a contact (b), and the number of infected:
dS/dt = -k*(S/N)*b*I = -(k*b/N)*S*I = -r*S*I
The rate of change of the infected: dI/dt = - dS/dt – dR/dt
I gains from what S has lost; and what I loses, R
acquires.

5. Spread of Disease Diagram of model? Differential equations?
dS / dt = -rSI
dI / dt = rSI - aI
dR / dt = aI

6. Spread of Disease

7. Spread of SARS and SEIR Model
Efforts including quarantine the exposed individuals to
separate them from the susceptible population, and isolate
those who had SARS.
SEIR Model (susceptible, exposed, infected, recovered)
has an intermediate exposed population of individuals
who have the disease but are not yet infectious.
The assumptions are

8. Spread of SARS and SEIR Model

10. Spread of SARS and SEIR Model

13. As in the SIR Model, the product Iu*S gives the total
number of possible interactions. Thus, (k/N0)*b*Iu*S
= k*b*Iu*S/N0 is the number of new cases of SARS
each day. Of these new cases, a fraction of (q) go into
category exposed_quarantined (EQ), while the fraction
(1 – q) go into exposed (E).
For those transferring from susceptible (S) to
susceptible_quanrantined (SQ), although they have
been exposed, the disease was not transmitted. The total
Number of non-transmission contact is (k/N0)*(1-b)*IU*S =
k*(1-b)*IU*S/N0. The rate of change of those going from
Susceptible (S) to SQ is q*k*(1-b)*IU*S/N0.

15. Reproduction Number
Due to exponential growth, it’s important that R < 1.
If R < 1, there is no epidemic. For R > 1, there is an
epidemic.
Three paths from IU to recovered_immune at rate of v, to SARS_death
at a rate of m, to ID at a rate of w. The total rate of change leaving IU
is (v + m + w)/day. Thus, 1/(v + m + w) is the average duration of
infectiousness. R0 = k*b*(1-q)/(v + m + w).