1. Modelling infectious diseases
Jean-François Boivin
25 October 2010

4. This decline prompted the U. S
This decline prompted the U.S. Surgeon General to declare in 1967 that "the time has come to close the book on infectious diseases."

6. Dear Sir: In your excellent Science paper (28 January 2000), you quote the US Surgeon General ('the time has come to close the book on infectious diseases'). You did not provide a reference for that quote, and I would like very much to know exactly where it comes from for a lecture I am preparing on infectious diseases. Can you help identify this reference? Thank you very much.
Jean-François Boivin, MD, ScD
Professor
Faculty of Medicine
McGill University
Montreal
Canada

8. I regret that Bob May has been over generous in its attribution
I regret that Bob May has been over generous in its attribution. The only reference I have is that the statement was made in 1967 but I have no formal source.
Best wishes.
George Poste

9. Two objectives:
Understanding population dynamics of the transmission of infectious agents
Understanding potential impact of interventions

10. Rothman, Greenland (1998)

11. Chickenpox
Epidemic spread due to children who do not appear to be sick

12. Parasite becomes infectious for mosquitoes
14 days
Day 10
Parasite becomes infectious for mosquitoes
Malaria (Plasmodium falciparum)
early treatment may affect transmission

13. A public health nightmare: HIV
days
weeks
median > 10 years

14. SiR model S i R 3 population densities (persons per mile2)
X Y Z = N
Reference: Chapter 6 in: Nelson et al. (2001)

15. Population is closed (no entry, no exit)
Process begins here with 1 infectious subject
Infectious subject enters in contact with susceptible and then the movement of subjects begins
S  i  R
Assumptions
Direct transmission
Life-long immunity
(prototype: measles)
Population is closed (no entry, no exit)

16. βXY = incidence of infection (modelling assumptions)
Nelson (2001)
βXY = incidence of infection (modelling assumptions)
γY = incidence of removals (cured, immune, dead)
direct observations from clinical epidemiology

17. Oxford Textbook of Public Health. Volume 2. Second edition. 1991

19. Imagine susceptible and infectious individuals behaving as ideal gas particles within a closed system
X = number of particles of one gas (susceptibles)
Y = number of particles of a second gas (infectious people)
β = collision coefficient for the formation of molecules of a new gas from one molecule each of the original gases (i.e. new cases of infection)

20. Gas particles (individuals) are mixing in a homogeneous manner such that collisions (contacts) occur at random. The law of mass action states that the net rate of production of new molecules (i.e. cases), I, is: I = βXY

21. The coefficient β is a measure of (i) the rate at which collisions (contacts) occur (ii) the probability that the repellent forces of the gas particles can be overcome to produce new molecules, or, in the case of infection, the likelihood that a contact between a susceptible and an infectious person results in the transmission of infection

22. X: number of black molecules
Y: number of white molecules
β: rate of collisions and probability that collision will lead to creation of a new molecule

23. Under these assumptions, the incidence of infection will be increased by larger numbers of infectious and susceptible persons and/or high probabilities (β) of transmission

24. Area of movement = 0.001 mile2 per person per day
Example 1
Area = 1 mile2
S = 8,699 persons
i = 1 person
R = 0 person
random movement
homogeneous distribution of subjects
Area of movement = mile2 per person per day
Probability of infection per contact = 40%
data?
Average duration of a case = 2 days
Incidence of recoveries = 0.5 case/day
Initial rate of infection :
(area of movement) (probability of infection) (i x S) =
0.001 mile2 x x 1 person x 8,699 persons = cases
person·day mile mile day·mile2
Infection rate > recovery rate; infection will spread
The initial case lasted 2 days, generating 3.48 x 2 = 6.96 secondary cases
= basic reproductive rate

25. (Nelson 2001)

26. Example 2
Area = 1 mile2
S = 1,249 persons
i = 1 person
R = 0 person
Area of movement = mile2 per person per day
Probability of infection per contact = 40%
Average duration of a case = 2 days
Incidence of recoveries = 0.5 case/day
Initial rate of infection :
(area of movement) (probability of infection) (i x S) =
0.001 mile2 x x 1 person x 1,249 persons = cases
person·day mile mile day·mile2
Infection rate = recovery rate
Infection will not spread
The initial case lasted 2 days, generating 0.5 x 2 = 1 secondary case

27. Basic reproduction ratio
= basic reproductive rate (R of R0)
= the number of secondary cases generated from a single infective case introduced into a susceptible population
= (initial infection rate) x (duration of infection)
or : Rate of infection
Rate of recovery

28. (Nelson 2001)

30. Although the population biology of measles depends on many factors, such as seasonality of transmission and the social, spatial, and age structure of the population, the fate of an epidemic can be predicted by a single parameter: the reproductive number R, defined as the mean number of secondary infections per infection

31. If the reproductive number is smaller than one, the disease will not persist but will manifest itself in outbreaks of varying size triggered by importations of the disease.

32. If the reproductive number approaches one, large outbreaks become increasingly likely, and, if it exceeds one, the disease can become endemic.

33. If the reproductive number equals one, the situation is said to be at criticality.

34. A decline in vaccine uptake will lead to increasingly large outbreaks of measles and, finally, the reappearance of measles as an endemic disease.

35. The effect of mass immunization is to reduce the basic reproduction ratio ...
Defining R’ to be the basic reproduction ratio after immunization and v to be the proportion vaccinated and effectively immunized,
R’ = R(1 – v)
(Nelson 2001, page 161)

36. NEJM 2003; 349:

37. Science 2003; 300:

40. Nelson KE, William CM, Graham NMH. Infectious disease epidemiology
Nelson KE, William CM, Graham NMH. Infectious disease epidemiology. Theory and practice. Aspen Publishers. Gaithersburg, Maryland. 2001