Marieke Jesse and Maite Severins

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Mathematical Epidemiology of infectious
diseases Chapter 1 Section 1.1 until 1.3.3
Marieke Jesse and Maite Severins Theoretical
Epidemiology Faculty of Veterinary Science Utrecht
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• Consider a virgin closed population and assume
  that at least one disease-causing organism is
  introduced in at least one host.
• Then the following questions arise
• Does this cause an epidemic?
• If so, with what rate does the number of
  infected hosts increase during the rise of the
  epidemic?
• What proportion of the population will
  ultimately have experienced infection?

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1.2 Initial growth 1.2.1 Initial growth on a
generation basis R0 (the basis reproduction
ratio) The expected number of secondary cases
per primary case in a virgin population This
ratio is a threshold parameter in the sense of
an epidemic, if R0 lt 1 the number of infected
decreases and the disease dies out R0 gt 1 an
epidemic can occur.
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The probability p that contact between a
susceptible and an infective leads to
transmission depends on the time elapsed since
the infective was itself infected. Assume
the time since infection took place
(age of infection)
Infectious period T2 T1 ?T
where c is the expected number of contacts per
unit of time
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1.2.2 Influence of demographic stochasticity
introducing a branching process on a generation
basis. The growth of the infectious population
is modelled as if newly infected individuals were
offspring of the infected. Thus the growth of the
infectious population can be described as a
normal reproduction process that has branching
like a family tree.
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Consider the finite (sub)population of infected
individuals from a generation perspective and
assume that individuals reproduce independently
from each other, the number of offspring for each
being taken from the same probability
distribution . This means that any
individual gets k offspring with probability qk
and that
The expected number of offspring R0 can be
found from qk as In words Ro equals the
sum of the number of offspring times the chance
to this number of offspring.
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Assume q0 gt 0, so there is a positive
probability that an individual will not produce
any offspring. Now start the process with one
individual then, q0 is the probability that a
population will be extinct after one
generation. Let zn denote the probability that
the population will be extinct after n steps.
Then clearly z1 q0 and it can be shown
(book) that zn can be recursively computed from
the following equation where In words g(z)
is the sum of chances to a certain number of
offspring times the number of generations before
the population goes extinct for all
branches. This function g(z) is called a
generating function, in the book some properties
of this function are stated as exercise 1.5.
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• The function g(z) increases and so does the
  sequence zn which has a limit
• probability that the population
  started by the first individual will go extinct.
• When
• , the population goes extinct with
  certainty
• , there exists a probability that the
  population will not go extinct

• We want to relate to R0, and we expect
  that
• implies and
• gives

This is going to be proved in exercise 1.7
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• Exercise 1.7
• is the smallest root in 0,1 of the
  equation (see fig and proven in ex. 1.6)
• Here we will show that for R0 lt 1.
• Look at the function , this
  function has at most two zeros, since -g(z) lt
  0.
• From exercise 1.5 we know that
• g(1) 1
• g(1) R0
• g(0) q0 gt 0
• g(z) gt 0

Since R0lt1, for z slightly larger than 0
and for z slightly less than 1. Thus the
function has two roots or no roots on the
interval 0,1). Two is impossible because of
second derivative, thus only one root at z 1
therefore z 1 is the smallest root and
for R0lt 1.

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Conclusion Even in the situation where the
infective agent has the potential of exponential
growth, i.e. R0 gt 1, it still may go extinct due
to an unlucky (for the parasite) combination of
events while numbers are low. The probability
that such an extinction happens, when we start
out with exactly one primary case, can be
computed as a specific root of the equation
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The probability that the introduction of
an infected host from outside does not lead to an
epidemic can be expressed in terms of the
parameters. Therefore function g (or,
equivalently, the probabilities qk) has to be
derived. The probabilities depends on the used
probability distrubtions, e.g. Poisson or
binomial. It is important to notice that
although the basic reproduction number R0 remains
the same, the probability depends on the
function g(z). Conclusion different kinds of
transmission models yields different values for
though the basis reproduction number Ro
remains the same.
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1.2.3 Initial growth in real time So far we have
looked at initial growth on a generation basis
and found a threshold occurring by a parameter R0
and that for R0 above this threshold, there
exists a positive probability that
introduction of one primary case does not lead to
explosive exponential growth. For simple models
the values for R0 and can be determined
explicitly in terms of the parameters of the
model. However, measuring growth on a generation
basis is not possible in real life because it
does not correspond to our observations, where
the infection generations overlap.
What is observed during the initial phase of a
real epidemic is exponential growth as
follows for some growth rate r gt 0, constant
C gt 0 were, I(t), is the prevalence, the number
of cases notified up to time t,
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The incidence, i(t), is the number of new cases
per unit of time, which is proportional to dI/dt
and can be computed as follows
Explanation (also ex 1.13) New cases at time t
result from contacts with individuals that are
infectious. The individuals that are infectious
at time t were themselves infected maximally
T2-T1 time ago. The infectious individuals make
contact at rate c with success probability p.
Draw picture of infectious!
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Trying to solve r from the previous equation is
rather difficult, therefore the substitution
below is used.

• Conclusion
• The above equation gives the exponential growth
  rate of the model
• Whereas there is an explicit formula for R0,
  there is not one for r
• r gt 0 iff R0 gt 1 and r lt 0 iff R0 lt 1
• in words there is growth in real time iff there
  is growth on a generation basis

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• A high value of R0 does not necessarily imply a
  high value of r.
• R0 pc(T2-T1) depends only on the difference
  T2-T1 (? T)
• in the equation for r the magnitude of T1 and
  T2 are important (r depends on absolute values of
  T1 and T2)
• Example (ex. 1.15)
• Choose T1, T2, T1 and T2 such that
  T2-T1T2-T1, for example, take T1T12
  and T2T22. Then R0 R0 because delta T
  remains the same.
• Now draw the graph of and then it can be
  seen that the growth rate for the second one (the
  blue line) is bigger than for the other one, so
  r lt r

Now take T2 slightly bigger, then R0gtR0 and
r lt r
R0 depends on the length of the time interval
that individuals are infectious, r depends on how
fast after infection an individual becomes
infectious to others. (demographic methaphor)
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• Now we have answered two of our questions
  initially asked
• Does this cause an epidemic?
• If so, with what rate does the number of
  infected hosts increase during the rise of the
  epidemic?
• One question still remains
• What proportion of the population will ultimately
  have experienced infection?

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• 1.3. The final size
• 1.3.1 The standard final-size equation
• In a closed population and with infection leading
  to either immunity or death, the number of
  susceptibles decreases and must therefore have a
  limit.
• Will the number of susceptible be zero at
  infinity? or
• Will some fraction of the population never get
  infected?

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For
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• Conclusion
• a certain fraction escapes from ever getting the
  disease
• this fraction is completely determined by R0
• the larger R0 is, the smaller the fraction

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1.3.2 Derivation of the final-size equation Ex.
1.21 S the size of the subpopulation of
susceptibles I the size of the subpopulation of
infectives R the size of the subpopulation of
removed Force of infection the probability per
unit of time for a susceptible to become
infected, this is proportional to I tranmission
rate (the constant of proportionality of the
force of infection)
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• Assumptions
• Infected individuals become immediately
  infectious
• Infected individuals have a constant
  probability per unit of time a to become removed
• The model used is a SIR model, also called
  compartmental model

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Since the population is closed, where N
denotes the total population size. Therefore one
of the equations of the system is redundant.
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• Determine R0
• infectious period 1/a
• transmission rate ß
• susceptibles in the beginning equals N
• ? R0 ßN/a

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Determine the (initial) growth rate r of the I
population The equation
gives the initial growth rate if one replaces S
by N. So the initial growth rate

• If r 0, then R0 1
• Threshold condition for Nthreshold a/ß.
• If
• N lt Nthreshold then R0lt1
• N gt Nthreshold then R0gt1

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It can be shown that is a conserved quantity,
by concidering dI/dS and seperating variables
(see page 182 of the book), this quantity is
indepent of time
Since this quantity is independent of time,
assuming that R0 gt1 when time runs from to
we have and
Both before and after the epidemic the number of
infected will be zero, therefore
and
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Using the previous equations we get
This identity can be re-written as
Realizing that
and
Results in the final size equation
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Drawing the phase protrait of the reduced (S, I)
system on page 16 setting dS/dt 0 gives
nullclines S0 and I0 dI/dt 0 gives
nullclines I0 and S a/ß
Given that the conserved quantity always hold we
can say
This gives the phase portrait shown on page 185.
The number of susceptibles at the peak of the
infection is at a/ß, since here the derivative of
I while
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Overshoot phenomenon The root, , of the final
size equation becomes smaller when N
increases. Why does the fráction of susceptibles
at the end of the epidemic become smaller with
increasing population size? This is due to the
overshoot-phenomenon When an epidemic starts
with a larger number of susceptibles, the number
of infected at the peak of the epidemic will be
very large. Large enough to infect a huge number
of susceptibles even though the epidemic is
decreasing.
In exercise 1.21 (viii) The SIR model of page 15
is reformulated in terms of fractions of
individuals. It is important to pay attention to
the dimensions of the parameters and to think
carefully about the new interpretation of them
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Until now we have assumed that parameters p and c
are constant. This need not be true during a real
epidemic. Lets introduce
Think of AIDS and influenza
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In the last part of section 1.3.2, also
assumptions regarding the contact rate (c) are
made more explicit and related to the validity of
the final size equation.
Until now we have assumed that the contact rate
is a constant. Does the final size equation still
hold when this is not the case, for example when
the contact rate is proportional to the
population density?
It is shown that the final size equation hold
when the disease doesnt cause any death, i.e
that it interferes in no way with the contact
process. Or if, the disease has a high mortality,
the final size equation hold when the contact
intensity is proportional to the population
density.