Infectious Diseases of Humans
Seminar
The basic model
Zhuobin Li
Outline
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Framework of the basic model
Basic reproduction number, R0
Average age of infection
Transmission parameter,
Frame work of the basic model
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Assumption
• Previous model
Age-specific host death rate, per capita
Age-specific recovery rate
Age-specific disease-induced death rate
‘force of infection’ at time t
Frame work of the basic model
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Assumption
• X(a),Y(a),and Z(a) are time independent
• Birth rate and death rate are exactly balanced
• Ignore mortality associated with the infection
is assumed to be zero
Frame work of the basic model
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Model
Frame work of the basic model
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Boundary condition
Frame work of the basic model
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Solve the model
where
Survivorship function
Frame work of the basic model
Frame work of the basic model
Basic reproductive number R
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0
R0 for a microparasite
• The basic reproductive number R0, is essentially the
average number of successful offspring that a parasite is
intrinsically capable of producing.
• For a microparasite, R0 is defined as the average number of
secondary infections produced when one infected
individual is introduced into a host population where
everyone is suscetible.
• R 0 >1 is the condition of being capable of invading and
establishing itself within a host population
Basic reproductive rate R
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0
R0 for a microparasite
• At the equilibrium, the rate of the susceptible being
infected is balanced against a rate of newly susceptible
individuals appearing.
• At the equilibrium, each infection will be on average
produce exactly one secondary infection;
R=R0 x*=1
where x* is the fraction of the host population that is
susceptible at equilibrium
Basic reproductive rate R
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0
R0 for a this model
• It seems to more plausible that the net rate of acquisition of
new infectious is proportional to than to .
“doubling the number of susceptibles in a school is arguably more
likely to double the incidence of infection than is doubling the
number of infectious individual”
• Basic reproductive number
Basic reproductive number R
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Two types of mortality
Type I:
Type II:
where L is life expectancy.
0
Basic reproductive number R
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Two types of mortality
Type I:
Type II:
Survivor ship function:
0
Basic reproductive number R
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Two types of mortality
Type I:
Type II:
0
Basic reproductive number R
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0
Two types of mortality
Type I:
Type II:
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Approximation of L
“The difference between L and G can be significant, especially in some
developing country. For an example, life expectancy L of India is
around 40 years, while births are around 40 per 1000 per annum
which corresponds to G~25 years”
Average age at infection
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The average age at which individuals acquire infection, A.
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Type I
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Type II
Average age at infection
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The average age at which individuals acquire infection based
on the proportions susceptible.
If λ is age-dependent, then
Average age at infection
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Come back to basic reproductive number
Type I:
Transmission parameter,
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Transmission parameter 
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Combines a multitude of epidemiolagical, environmental, and social
factors that affect transmission rates
•  is “ force of infection”, the per capita rate of acquisition of infection.
If  is time dependent, (t)t represents the probability that a given
susceptible host will become infected in the small time interval t.
Transmission parameter,
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Type I:
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Type II: