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Dynamics of Directly Transmitted Pathogens

Carl

MMED 2025

Mathematical Modelling in Medicine and Public Health

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Motivation

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From: Earn et al. 2000 Science

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From: Earn et al. 2000 Science

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From: Earn et al. 2000 Science

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Goals

Illustrate how simple infectious disease models can give insights into real-world dynamics for acute immunizing infections

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Transmission

Infectious diseases

Mode of transmission

Direct transmission

Direct contact

Droplet spread

Indirect transmission

Airborne

Vehicle-borne (fomites)

Vector-borne (mechanical or biological)

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Transmission

Infectious diseases

Mode of transmission

Direct transmission

Direct contact

Droplet spread

Indirect transmission

Airborne

Vehicle-borne (fomites)

Vector-borne (mechanical or biological)

Sexual contact

(STD’s)

Casual contact

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Directly-transmitted pathogens

“Typical” natural history

Infection

Onset of symptoms

Onset of shedding

Incubation period

Clinical disease

Infectious period

Latent period

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Directly-transmitted pathogens

“Typical” natural history

Acute

Time course of infection� << �normal lifespan of host

Immunizing

infection stimulates antibody production

preventing future infection

Infection

Onset of symptoms

Onset of shedding

Incubation period

Clinical disease

Infectious period

Latent period

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Directly-transmitted pathogens

Examples

Measles

Smallpox

Chicken pox

Whooping cough

Foodbourne

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A simple view of the world

Directly-transmitted pathogens

Infected=Infectious

Infection

Onset of shedding

Infectivity = 1

Don’t worry about symptoms and disease!

Assume immediate infectiousness after exposure…

^

very!

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A simple view of the world

Directly-transmitted pathogens

Infectivity = 1�(everyone exposed becomes infected)

Infected

(not infectious)

Infected

(infectious =

diseased)

Infection

Onset of symptoms

Onset of shedding

Incubation period

Clinical disease

Infectious period

Latent period

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A realistic view of the world

Directly-transmitted pathogens

Infectivity < 1

Infected

(infectious =

diseased)

^

more

Infection

Onset of symptoms

Onset of shedding

Incubation period

Clinical disease

Infectious period

Latent period

Infected

(not infectious)

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A realistic view of the world

Directly-transmitted pathogens

Infectivity < 1

Infected

(infectious =

diseased)

^

more

Infection

Onset of symptoms

Onset of shedding

Incubation period

Clinical disease

Infectious period

Latent period

β = infectivity x per capita contact rate

Infected

(not infectious)

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A realistic view of the world

Directly-transmitted pathogens

Infected

(infectious =

diseased)

Infected

(not infectious)

^

more

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A realistic view of the world

Directly-transmitted pathogens

Susceptible

Recovered

^

more

Infected

(infectious =

diseased)

Infected

(not infectious)

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A realistic view of the world

Directly-transmitted pathogens

^

more

S

R

I

E

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A realistic view of the world

Directly-transmitted pathogens

^

more

S

R

I

E

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A realistic view of the world

Directly-transmitted pathogens

S

R

E

^

more

I

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A useful view of the world

Directly-transmitted pathogens

S

R

E

I

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A useful view of the world

Directly-transmitted pathogens

birth rate

mortality rate

1 / latent period

1 / infectious period

transmission coefficient

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A useful view of the world

Directly-transmitted pathogens

Assume constant population size

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A useful view of the world

Directly-transmitted pathogens

Rate at which an infected individual produces new infections in a naïve population

X

Proportion of new infections that become infectious

X

Average duration of infectiousness

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A useful view of the world

Directly-transmitted pathogens

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A useful view of the world

Directly-transmitted pathogens

Equilibria…

Disease free equilibrium

Endemic equilibrium

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A useful view of the world

Directly-transmitted pathogens

Endemic equilibrium

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A useful view of measles

Directly-transmitted pathogens

individuals/year

years^-1

days^-1

(unknown)

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Some measles data

Grenfell and Harwood 1997

Directly-transmitted pathogens

fadeouts per year

vs

population size

“critical community size”

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Some measles data

Grenfell and Harwood 1997

~ 7,200

~ 300,000

~ 500,000

Directly-transmitted pathogens

“critical community size”

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Directly-transmitted pathogens

From: Anderson & May 1982 Science

If we know R₀, we can calculate endemic prevalence:

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A useful view of measles

Directly-transmitted pathogens

N = 7,200

N = 300,000

N = 500,000

Equilibrium # infected for different N

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More measles data

From: Earn et al. 2000 Science

Directly-transmitted pathogens

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Seasonal SEIR Model

Directly-transmitted pathogens

individuals/year

years^-1

days^-1

seasonal

(school terms)

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Seasonal SEIR Model

Directly-transmitted pathogens

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Seasonal SEIR Model

Directly-transmitted pathogens

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Susceptible Replenishment & Periodicity

Directly-transmitted pathogens

Life expectancy of 40 years

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Susceptible Replenishment & Periodicity

Directly-transmitted pathogens

Life expectancy of 40 years

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Susceptible Replenishment & Periodicity

Directly-transmitted pathogens

Life expectancy of 40 years

Relatively rapid replenishment of the susceptible population

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Susceptible Replenishment & Periodicity

Directly-transmitted pathogens

Life expectancy of 50 years

Intermediate replenishment of the susceptible population

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Susceptible Replenishment & Periodicity

Directly-transmitted pathogens

Life expectancy of 60 years

Slow replenishment of the susceptible population

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More measles data

Directly-transmitted pathogens

From: Earn et al. 2000 Science

Effectively, vaccination reduces the rate of susceptible replenishment

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More measles data

Directly-transmitted pathogens

From: Earn et al. 2000 Science

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Summary

Simple infectious disease models can help us understand

Why diseases persist in some populations but not others
The processes that determine the frequency and size of recurrent epidemics

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“Dynamics of directly-transmitted pathogens (aka Introduction to Infectious Disease Dynamics III)

Prof Juliet Pulliam (SACEMA, Stellenbosch University), Dr Steve Bellan, and Dr Rebecca Borchering

Presented as part of the “Mathematical Modelling in Medicine and Public Health” module of the AIMS/SU Biomathematics Honours Programme, 2023.

For further information or modifiable slides please contact medph@ici3d.org.

This presentation is made available through a Creative Commons Attribution license. Details of the license and permitted uses are available at�http://creativecommons.org/licenses/by/4.0/.

Mathematical Modelling in Medicine and Public Health

19 JUNE – 8 JULY 2023 ~ VIRTUAL EDITION