Modelling infectious diseases

Jean-François Boivin

25 October 2010

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

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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, ScDProfessorFaculty of MedicineMcGill UniversityMontrealCanada

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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

Two objectives:

• Understanding population dynamics of the transmission of infectious agents

• Understanding potential impact of interventions

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Rothman, Greenland (1998)

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Chickenpox

Epidemic spread due to children who do not appear to be sick

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Malaria (Plasmodium falciparum)

14 days

Parasite becomes infectious for mosquitoes

early treatment may affect transmission

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Day 10

A public health nightmare: HIV

daysweeks

median > 10 years

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SiR model

S i R

3 population densities (persons per mile2)

X + Y + Z = N

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Reference: Chapter 6 in: Nelson et al. (2001)

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)

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Nelson (2001)

βXY = incidence of infection (modelling assumptions)

γY = incidence of removals (cured, immune, dead)

direct observations from clinical epidemiology

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

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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)

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

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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

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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

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

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Example 1

Area = 1 mile2

S = 8,699 persons

i = 1 person

R = 0 person

Area of movement = 0.001 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) =

Infection rate > recovery rate; infection will spread

The initial case lasted 2 days, generating 3.48 x 2 = 6.96 secondary cases

• random movement• homogeneous

distribution of subjects

data?

0.001 mile2 x 0.4 x 1 person x 8,699 persons = 3.48 casesperson·day mile2 mile2 day·mile2

= basic reproductive rate24

(Nelson 2001)

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Example 2

Area = 1 mile2

S = 1,249 persons

i = 1 person

R = 0 person

Area of movement = 0.001 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) =

Infection rate = recovery rate

Infection will not spread

The initial case lasted 2 days, generating 0.5 x 2 = 1 secondary case

0.001 mile2 x 0.4 x 1 person x 1,249 persons = 0.5 casesperson·day mile2 mile2 day·mile2

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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 infectionRate of recovery

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(Nelson 2001)

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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

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.

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

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

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

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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)

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NEJM 2003; 349: 2431-244136

Science 2003; 300: 1966-197037

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

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