# Reproductive number

Reproductive number.

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__Definition: __

The basic reproductive ratio, R_{0} (also called the basic reproductive “rate” or “number”), is the average number of persons directly infected by an infectious case during their entire infectious period when the infectious case enters a totally susceptible population.

Notice: In your reference book, Nelson et al. use “R” to refer to R_{0}. For most other authors, R is a slightly different concept (see below).

__Question 1: __

In a sense, R_{0} is an abstraction though it can in fact be estimated from simple observations. Calculate R_{0} based on the following schematic spread of an infectious disease.

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__Question 2: __

Severe Acute Respiratory Syndrome (SARS) is a recently described viral respiratory illness caused by a coronavirus, called SARS-associated coronavirus (SARS-CoV). The main way that SARS seems to spread is by close person- to-person contact (droplets). At present, there is no direct evidence of SARS transmission from asymptomatic persons. Epidemics of SARS were reported in 29 countries between November 2002 and July 2003. A total of 8,098 probable SARS cases were reported to the World Health Organization (WHO), including 29 cases from the United States; 774 SARS-related deaths (case-fatality rate: 9.6 percent) were reported (none of which occurred in the United States).

In Singapore, 205 probable cases of SARS were reported in a population of 4.6 million inhabitants. The mean time from the onset of SARS symptoms in an index case to the onset of SARS symptoms in a subsequent case infected by the index case in this Asian country was 8.4 days. What term would you use to describe this parameter?

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__Question 3: __

The number of secondary SARS cases per index case fell from a mean of 7 for index

cases with symptom outset in the first week of the Singapore outbreak to a mean of 1.6 in the second week, and to a mean of below 1 in most weeks thereafter. Based on these data, what is the best estimate of the basic reproductive ratio, R_{0}, for the SARS epidemic in Singapore?

** Information for question 4:** Note that R

_{0}refers to the situation that prevails

*early*during the course of an outbreak in an

*entirely susceptible*population. R is generally used to refer to the actual (or effective) reproductive ratio– the reproductive rate

*actually*observed at

*any specific time*during the course of an epidemic in a population that may include

*any*proportion of

*non- susceptible*individuals. R

_{0}can be estimated from the value of R at a given time during the epidemic and the proportion of susceptibles in the population at that time.

Where R_{t} = value of R at time *t *and S_{t} = number of susceptible in population of size N at time *t. *

Equivalently,

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__Question 4: __

Based on the above formula and data from question 3, calculate estimates of the proportion of persons susceptible to SARS in Singapore during the second week ** and** the following weeks of the outbreak.

__Question 5: __

Based on the data from question 3 and the calculations from question 4, is the decline with time in number of secondary cases per index case (R) likely to be explained by the simple depletion of the number of susceptibles over the natural course of the epidemic? (Provide a rationale for your answer.)

__Question 6: __

As a continuation from question 5, besides the simple depletion of the number of susceptibles over time, can you think of another reason why the value of R dropped so markedly during the course of the epidemic?

__Question 7: __

Using information from the table below, how would you characterize the epidemic potential of SARS compared to that of measles and smallpox?

Disease |
Latent Period (days) |
Incubation Period (days) |
T (days) |
R |

SARS |
~7 |
~5 |
~10 |
~3 |

Measles |
~7 |
~12 |
~11 |
~17 |

Smallpox |
~15 |
~12 |
~21 |
~6 |

__Question 8: __

Since neither effective antiviral therapy nor vaccine was available, control of SARS outbreaks was achieved by implementing aggressive measures of isolation, segregation, and quarantine. Considering that the best estimate of R_{0} for SARS is ≈ 3 and that control measures were successful, what minimal reduction in overall SARS infectiousness was achieved with these methods?

__Question 9: __

Fortunately, extensive contact tracing in Hong Kong identified a known symptomatic

SARS contact for 91.4% of reported cases. Would your conclusion regarding potential for success in SARS control be changed if new evidence were shown that infected persons who never become symptomatic contributed substantially to transmission? (Provide a rationale for your answer.)

__Question 10: __

Early studies of viral titers in nasopharyngeal aspirates suggest that viral shedding increases over the first 10 days after the onset of SARS symptoms and then drop abruptly. What is the implication of this finding with regard to the potential efficacy of isolation as a measure of outbreak control?

__Question 11: __

In practice, what factors may limit the effectiveness or feasibility of quarantine? (Propose

at least three factors.)

__Question 12:__** **

In practice, what factors may limit the effectiveness or feasibility of isolation? (Propose at least three factors.)

**The figure below show trends in measles rate in New York between 1944 and 1964 (i.e., before measles vaccine became available). Use this figure as a basis for answering questions 13-15.**

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__Question 13: __

Describe the variations in incidence over time for the disease.

__Question 14: __

How do you explain these variations?

__Question 15: __

What do these variations tell you about average age of measles in New York during this period?