R0 is the basic reproduction number that measures the average number of people infected by one infectious person in a totally susceptible population. It is important because if R0 is less than 1, the disease will eventually die out, but if greater than 1 it may persist in the population. R0 can be estimated from the intrinsic growth rate of new cases over time by fitting an exponential curve, though this assumes exponential growth which may not always occur. Modified estimators have been developed that do not require this assumption. Estimates of R0 have been obtained for different attack rates of diseases, but estimating R0 is challenging and improvements are needed to speed up calculations and account for non-exponential growth over longer time periods.

1. Estimating R0

2. What is R0
?
• R0
is defined to be the mean number of infections caused by
a (typical) infective in a totally susceptible population.
• This quantity is also known as the basic reproduction
number, the basic reproductive ratio, and the basic
reproductive rate.
• R0
is a measure of the growth of an epidemic on a
generation basis.
• The effective R0
is the mean number of infections caused by
a (typical) infective in a population.
• Factors that may affect R0
are generally ignored.

3. Why is R0
important?
• If a disease is introduced into a totally susceptible (naive)
population and R0
≤ 1, then it will eventually die out with
probability 1.
• If R0
> 1, then it will die out with probability less than 1.
• If R0
> 1 and 100(1 - 1/ R0
)% of a naive population is
vaccinated, then the effective R0
will be less than 1.
• R0
can be used to determine the final attack rate.
• Control strategies can be evaluated based on what they
make the effective R0
.

4. Estimating R0
from the Intrinsic Growth
Rate
• Incidence on day t: the number of new cases on day t.

5. Estimating R0
from the Intrinsic Growth
Rate
• Incidence on day t: the number of new cases on day t.

6. Estimating R0
from the Intrinsic Growth
Rate
• Suppose .
• The intrinsic growth rate is defined to be r.
• r can be estimated in various ways.
o It can be estimated using least-squares.
o It can be estimated by making a certain technical
assumption about the spread of the disease.
• Generation time: the time between infection of a primary
case and infection of a secondary case caused by the
primary case.
• the density of the generation time.

7. Estimating R0
from the Intrinsic Growth
Rate
• We have
• If we estimate using the technical assumption, then we
can get a confidence interval for R0
.
• But we need to assume that the incidence initially grows
exponentially.
• There's a delay before exponential growth begins, if it
begins at all.

8. Estimating R0
without exponential
growth
• We can modify the previous estimator so that the
assumption of exponential growth is unnecessary.
• Let be some day. An estimator of R0
is
• This estimator just requires incidence and generation time
data up to some day t. Also, a confidence interval can be
constructed for R0
.
• We have to choose t though.

9. Estimates of R0
for various attack rates
Attack Rate R0
Mean, Variance
6.3% 1.0623, 0.0001
19.4% 1.2116, 0.0002
25.3% 1.2795, 0.0002
29.3% 1.3259, 0.0005

10. Issues to deal with in the future
• My code using the first estimator is very slow.
o The estimation takes upwards of a day on my laptop.
o It takes several hours on unicron.
• The data on who infected whom and when each infective
was infected is stored in a matrix; is there a better way to
store it?
• The exponential curve fit to the incidence data fits well for
the first few weeks, but diverges afterwards.

11. Issues to deal with in the future