Modelling infectious diseases Jean-François Boivin 25 October 2010

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."

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

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

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

Rothman, Greenland (1998)

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

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

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

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

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)

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

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

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

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

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

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

(Nelson 2001)

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) = 0.001 mile2 x 0.4 x 1 person x 1,249 persons = 0.5 cases person·day mile2 mile2 day·mile2 Infection rate = recovery rate Infection will not spread The initial case lasted 2 days, generating 0.5 x 2 = 1 secondary case

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

(Nelson 2001)

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.

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

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

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

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)

NEJM 2003; 349: 2431-2441

Science 2003; 300: 1966-1970

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